Astronomy & Astrophysics A path through The Default Library ======================================== What is out there, how we know, and the physics that makes stars shine, galaxies turn and the universe expand. Step 1. Calculus I: Slopes, Totals and the Bridge Between Them -------------------------------------------------------------- Why: Calculus: orbits, light curves and expansion are rates of change. Your car's speedometer is a derivative. Its odometer is an integral. Everything a first calculus course teaches is a way of moving between the two. Look at a dashboard. The odometer says how far you have gone in total. The speedometer says how fast the total is changing right now. Press the accelerator and the speedometer moves first; the odometer follows, a little behind, like a patient accountant adding up what the speedometer reported. Nobody gave the speedometer a ruler or a clock to look at the whole trip. It reads one instant. Yet it reports a rate, which is a statement about change, from a single moment. How can a single moment say anything about change? That puzzle is what the course exists to answer, and the answer turned out to run most of modern engineering and computing. This is the second time the library has met calculus. Calculus: Derivatives, Integrals, and the Language of Motion (12/11) is the easier first read, with the tangent line and the rectangle sum. Here we go a step further: what the course trains you to see, how a computer gets it wrong, and where the idea is used afterwards. The course in one box Who takes it: first-year students in Mechanical & Aerospace Engineering and in Computer Science, and almost every other science or engineering degree. What it is for: turning "how much?" into "how fast?" and back, so that change can be calculated instead of guessed. Comes after: nothing in this series. Algebra and a little trigonometry are all it assumes. Leads to: Multivariable Calculus (34/02), Differential Equations (34/03), Probability & Statistics (34/05), Physics I: Mechanics (34/06), Thermodynamics (34/14), and from them nearly everything else on both majors. Honest line: you will not be able to differentiate a hard formula by hand after this. You will see why a flat spot on a curve is where the best answer hides, and why the area under a rate is a total. A limit is where a calculation settles The speedometer puzzle is this. Speed is distance divided by time, but in an instant no time passes and no distance is covered, so you seem to be dividing zero by zero. The way out is to stop asking what happens at zero and ask what a sequence of better and better answers is settling towards. Measure over one second, then a tenth, then a hundredth, and watch the average speed home in on a number. That number is the limit. speed now = limit of (distance change ÷ time change) as the time change shrinks to 0 A graduate does not carry the formal definition around, but carries the habit: if a quantity looks like 0 ÷ 0, ask what it is approaching. Take sin(x) ÷ x as x goes to zero, with x in radians. At x = 0.1 it is 0.99833; at 0.01, 0.99998; at 0.001, 0.9999998. It never "is" 1 at zero, since it is undefined there, but it settles on 1. That one limit is why the derivative of sine turns out to be cosine. Interactive · Shrink the gap until the computer gives up We estimate the slope of ex at x = 1 by (e1+h − e) ÷ h. The true slope is e = 2.718281828459045. Make the gap h smaller. Gap h = 10−k The error falls as the gap shrinks, as the limit promises, until about h = 10−8, where it bottoms out near 7 × 10−9. Shrink further and it gets worse: the two values being subtracted agree in almost all of their roughly 16 stored digits, so the subtraction wipes out the information. At 10−16 the answer is exactly 0. This is a computer-science lesson hiding in a calculus course: the limit is a mathematical idea, and a machine can only approach it so far. The derivative is a slope, and a slope is a rate Zoom in on any smooth curve and it straightens out. The steepness of that straight piece is the derivative. In plain words: how much the output moves per unit nudge of the input. If the curve is distance against time, the derivative is speed. If it is temperature against position, it is how sharply it warms as you walk. Rate and slope are one thing seen two ways. f′(x) = limit of [f(x+h) − f(x)] ÷ h as h → 0 The graduate's real asset is not computing derivatives but reading them. Where the curve is rising, the derivative is positive. Where it is flat, the derivative is zero. Where the curve is steepest, the derivative is at its peak. Try it: the lower graph is the derivative of the upper one, drawn point by point. Interactive · A curve and its slope, side by side x³ − 3x sin x bell: e^(−x²) Position x Check the three curves. Every time the top curve peaks or dips, the bottom one crosses zero. Every time the top is climbing, the bottom is above the axis. For x³ − 3x the slope is 3x² − 3, which is zero at x = −1 and x = 1: the hill top and the valley floor. The chain rule: rates multiply down a chain Rates often come in chains. A bicycle's pedals turn the chainring, the chain turns the rear wheel, the wheel moves the bike. If the wheel turns 3 times per pedal turn, and the bike moves 2 metres per wheel turn, the bike moves 6 metres per pedal turn. You multiply the ratios. That is the chain rule, and it is the most used rule in the course. d(outer)/d(input) = d(outer)/d(middle) × d(middle)/d(input) The classic worked case is a balloon. Its volume depends on its radius, and the radius depends on time. At radius r the surface has area 4πr², so one extra centimetre of radius adds a shell of volume about 4πr² cm³: that is dV/dr. Multiply by how fast the radius is growing and you have how fast the volume is growing. At r = 10 cm, growing 2 cm each second, volume grows at 4π × 100 × 2 ≈ 2,513 cm³ per second. Pull the sliders and confirm it. Interactive · Inflate a balloon Radius now Radius is growing at The picture shows why it works: the shaded shell is what one second adds, a skin of area 4πr² and thickness equal to the radius growth. The same rule runs inside every neural network, where the "chain" is dozens of layers long and the multiplication is called backpropagation (Machine Learning, 34/32; Multivariable Calculus, 34/02). Where the slope is zero, something is best At the top of a hill you stop going up and have not yet started going down. The slope is zero. So to find the biggest or smallest value of anything smooth (strongest, cheapest, lightest, fastest) look for where its derivative is zero. That is optimisation, and it is why engineers love the derivative. Take a 30 cm by 20 cm sheet of card. Cut a square of side x from each corner and fold up the sides to make an open box. Its volume is V = x(30 − 2x)(20 − 2x). Cut too little and the box is shallow. Cut too much and it is narrow. In between there is a best x, and the slope tells you where: at x ≈ 3.92 cm the volume peaks at about 1,056 cm³. Interactive · The best box Corner cut x Jump to the flat spot Notice what the slope does as you drag. To the left of the peak the slope is positive, so a bigger cut helps. To the right it is negative, so a smaller cut helps. At the peak it is zero and nothing nearby is better. Gradient descent, the algorithm that trains machine-learning models, is exactly this reasoning repeated: look at the slope, step downhill, repeat. The integral adds up a rate into a total The odometer is the speedometer added up over time. Draw speed against time; the area between the curve and the time axis is distance. That is the integral. Anywhere a rate is known and a total is wanted (litres per minute into a tank, watts into a battery, newtons over a distance) the answer is an area. Try it before you read on. A car accelerates steadily from 0 to 20 m/s in 10 seconds, cruises at 20 m/s for 30 seconds, then brakes steadily to a stop in 10 seconds. How far has it gone? Interactive · Predict, then reveal 600 m 800 m 1,000 m 1,200 m Choose an answer. The shape is two triangles and a rectangle: ½ × 10 × 20 + 30 × 20 + ½ × 10 × 20 = 100 + 600 + 100 = 800 m. The common wrong guess, 1,000 m, uses the top speed for all 50 seconds and forgets that the car spends time slower than that. The Fundamental Theorem: two jobs, one idea Here is the surprise that makes the course a single subject. Build a running total from a rate: the amount in the tank after time T. Then ask how fast that total is growing at T. The answer is just the rate at T. Accumulating and differentiating undo each other, the way the odometer and speedometer do. Newton and Leibniz, working separately in the late 1600s, each saw this. if A(T) = area under the rate up to T, then A′(T) = the rate at T Its practical power is that you can find a total without adding anything: find a function whose slope is your rate and subtract its values at the two ends. Below, water flows into a tank at a rate that wobbles: 4 + 3 sin(t/2) litres per minute. Drag the time and watch the right-hand graph, the tank's total, steepen exactly when the rate on the left is high. Interactive · The tank and its rate Time T The readout checks the theorem two ways: area by adding 600 thin slices, and the exact formula 4T + 6(1 − cos(T/2)). At T = 12 minutes both give 48.24 litres. The slope of the total at that moment is 4 + 3 sin 6 ≈ 3.16 litres per minute, the rate at that moment. Every smooth thing is nearly a straight line The last habit is the most useful in daily engineering. Close to any point, a smooth curve is almost its tangent line, so you can replace it with a straight line and be nearly right. The small-angle rule sin x ≈ x is why a pendulum swings at a steady rhythm. The rule √(1 + x) ≈ 1 + x/2 gives mental square roots: √1.1 ≈ 1.05, true value 1.0488. f(a + x) ≈ f(a) + f′(a) · x   (good when x is small) How small is small? The error shrinks faster than the step. Halve the step and the error falls about four times for √(1 + x) and ex, and about eight times for sine. Interactive · How good is the straight line? sin x ≈ x √(1+x) ≈ 1 + x/2 eˣ ≈ 1 + x Step x from the point What a graduate just sees A rate and a slope are the same thing, and "how fast?" always means a derivative. A total is an area under a rate, and adding up thin slices is a calculation, not a mystery. Rates multiply along a chain: gear ratios, units, layers of a network. The best value of a smooth quantity sits where its slope is zero. Differentiating and accumulating are inverse jobs, so a total can often be found without adding. Up close, everything smooth is a straight line, and engineers exploit that constantly. Subtracting two nearly equal numbers on a computer destroys precision, so a naive slope estimate gets worse when the gap gets too small. Where it shows up next Physics I (34/06) and Dynamics (34/12): velocity is the derivative of position, acceleration of velocity, and Newton's law ties force to the second one. Differential Equations (34/03): when a law is written as a rate, solving it means integrating, which is the whole next course. Thermodynamics (34/14) and Heat Transfer (34/16): heat added is an integral of power; temperature profiles are read from slopes. Probability & Statistics (34/05): the chance of a range of outcomes is the area under a density curve. Algorithms (34/24) and Machine Learning (34/32): growth rates of running time are compared with limits, and models are trained by following slopes downhill. If you want to go deeper 📘 Textbook: James Stewart, Daniel Clegg and Saleem Watson, Calculus: Early Transcendentals (9th ed., Cengage). The most-assigned text; best for worked examples and a huge problem bank. 📘 Textbook: Hass, Heil, Bogacki and Weir, Thomas' Calculus (15th ed., Pearson). Strong on applications, popular in engineering departments. ▶️ Gentler: Grant Sanderson's Essence of Calculus series on 3Blue1Brown, a visual tour of the same ideas. Also Steven Strogatz, Infinite Powers, for the story. 🌐 Free course: MIT OpenCourseWare 18.01SC Single Variable Calculus, with video lectures, notes and problems. Khan Academy also covers it. The takeaway Calculus I teaches two moves and one bridge. The derivative turns a total into a rate (a slope); the integral turns a rate into a total (an area); and the Fundamental Theorem says they undo each other. From that come the habits of mind: chain rates together, look for zero slope to find the best, and treat smooth things as lines up close. Our slope estimate for ex was best near a gap of 10−8 and useless at 10−16, a reminder that the ideas are exact and the machines are not. Try this: next time you are in a car or on a bike, watch the speedometer and ask what the area under its history means. Then on a hill, find the spot where the road is flat. That is a derivative equal to zero. Sources & Further Reading Sources. Every number in the interactives was computed in node: the forward-difference error of ex at x = 1 for h = 10−1 to 10−16 (best 6.6 × 10−9 at 10−8, exactly 0 at 10−16, in IEEE double precision); sin(x)/x at 0.1, 0.01, 0.001; the box V(x) = 4x³ − 100x² + 600x, zero slope at x = (50 − √700)/6 = 3.9237, V = 1,056.3 cm³; the balloon 4π × 10² × 2 = 2,513.27 cm³/s; the car, 800 m; the tank, A(12) = 48.239 L; small-step errors of the three approximations. Book editions checked against the publishers' pages (Cengage, Pearson) and the course numbers against MIT OpenCourseWare. 🌐 MIT OpenCourseWare 18.01SC Single Variable Calculus ▶️ 3Blue1Brown: Essence of Calculus 🌐 Khan Academy: Calculus 1 📘 Stewart, Clegg and Watson, Calculus: Early Transcendentals, 9th ed. (Cengage) 📘 Hass, Heil, Bogacki and Weir, Thomas' Calculus, 15th ed. (Pearson) 📘 Steven Strogatz, Infinite Powers, a popular history of calculus Step 2. Physics I: Mechanics — Three Ways to Keep the Books ----------------------------------------------------------- Why: Mechanics: gravity and orbits start here. Why doubling your speed quadruples your stopping distance, why a skater spins up when she pulls her arms in, and the three questions an engineer asks before touching a calculator. A car doing 50 km/h stops in about 14 metres on a dry road. Most people guess that at 100 km/h it takes about 28. It takes about 56. The extra speed doesn't just cost twice the distance, it costs four times, and the same fact is why a crash at 100 km/h is not "twice as bad" as one at 50. Nobody needs a formula to be afraid of that, but a person who has done a first mechanics course sees it: the brakes have to get rid of the car's energy, energy grows with the square of speed, and the road can only take it away at a steady rate per metre. That one chain of thought is most of the course. Physics I is less a list of topics than a way of keeping accounts. There are three ledgers, force, energy and momentum, and a good physicist picks the one in which the problem is easiest. If you want an easier first read, the library has The Four Forces (13/01), Energy: The One Currency (13/02) and Orbits (13/11). This piece goes further: it makes you use the ledgers. The course in one box Who takes it: first-year students of Mechanical & Aerospace Engineering, and in many programmes Computer Science too, as the science requirement. What it's for: turning "what will this object do?" into something you can calculate. Comes after: Calculus I (34/01). Leads to: Physics II: Electricity & Magnetism (34/07), Statics (34/10), Dynamics (34/12), Materials Science & Manufacturing (34/13), Thermodynamics (34/14). Honest line: you won't be able to design a suspension after this. You will see why a car's tyres, not its engine, decide how fast it can take a corner. Newton's laws are one accounting system Isaac Newton set out his three laws in the Principia in 1687. Read as a bookkeeper would, they say: momentum (mass times velocity) is the thing being tracked; a force is a deposit or withdrawal; and every deposit has a matching withdrawal somewhere else. The first law says that with no net force the account doesn't change: things keep their velocity. The second says that a net force changes momentum at a steady rate, which for a fixed mass is the famous rule below. The third says forces come in pairs between two bodies: if you push a wall, the wall pushes you, and the pair never acts on the same body, which is why they don't cancel. Fnet = m · athe net force on a body is its mass times its acceleration Worked number: a 1,500 kg car getting from 0 to 100 km/h (27.8 m/s) in 8 seconds accelerates at 3.47 m/s², so the road must push it forward with about 5.2 kN. Two ice skaters of 50 kg and 70 kg who push apart from rest leave with equal and opposite momentum: if the heavier moves at 1.0 m/s the lighter goes at 1.4 m/s the other way. Total momentum stays at zero. Try the classic: you on a scale in an elevator. The scale doesn't read your weight, it reads the force pushing up on you. Interactive · The elevator scale (a 70 kg rider) Speeding up, going up Cruising Speeding up, going down Cable cut Elevator acceleration (up is positive) Accelerating upward at 2 m/s² the scale reads about 827 N, like 84 kg. At constant speed it reads your true weight, whether the elevator is moving or still. In free fall it reads zero. That is exactly the "weightless" astronaut of Orbits. Free-body diagrams, and friction as a force that only pushes as hard as it must The graduate's first reflex on any problem is to draw one object, alone, and every push on it: weight, the surface pushing back, friction, tension. Nothing else. That is a free-body diagram. Add the forces as vectors, split them along the directions that matter, and F = ma does the rest. Friction is the force beginners get wrong. Static friction is not a fixed number: it pushes exactly as hard as needed to stop sliding, up to a ceiling of μs times the normal force (how hard the surface pushes back). Only once the ceiling is passed does the block slide, and then friction drops to μk × normal force. f ≤ μs · Nfriction is at most a coefficient times how hard the surfaces press together On a ramp the weight splits into a part along the slope, mg sin θ, and a part pressing into it, mg cos θ. Sliding begins when the first beats μs times the second, which gives tan θ = μs. The mass cancels. Check that in the tool: change the mass and nothing about when it slips changes. (The coefficients are round textbook values; real surfaces vary a lot.) Interactive · A block on a ramp Wood on wood Rubber on concrete Steel on ice Ramp angle Mass Arrows are drawn as fractions of the weight. With μs = 0.5 the block holds until 26.6° and then slides. At 40° it accelerates down at about 4.05 m/s² (with μk = 0.3). This is why a pile of sand has a steepest slope, and why designers of ramps, screws and wedges care so much about one number. Energy: the balance that doesn't care about the path Force problems get painful when the path is complicated. Energy is the escape. The work a force does is force times distance moved along it, and the work–energy theorem says the net work on a body equals its change in kinetic energy. When only gravity or springs are involved, the total of kinetic plus potential energy is simply conserved, whatever the route. KE = ½ · m · v²speed counts squared; this is why speed is so costly A bobsled dropping 30 m ends at √(2 × 9.81 × 30) = 24.3 m/s (87 km/h) with no friction, however many bends the track has. Now back to the opening: the brakes do work μ × m × g × d against the car's ½mv², so the stopping distance is v² ÷ (2μg). Double v, quadruple d. Interactive · Brake, or hit Dry (μ 0.7) Wet (0.5) Snow (0.2) Ice (0.1) Speed when you brake Obstacle ahead Braking only, no reaction time. μ values are round, typical figures for tyres. The sharpest result: brake from 100 km/h with an obstacle 14 m away, the distance that stops a 50 km/h car on a dry road. You still hit it at about 87 km/h, with 75% of the energy you started with: 14 metres of hard braking has removed only a quarter of it. Momentum: what survives every crash Energy has a weakness: in a collision much of it turns to heat, sound and bent metal, so the energy ledger is no use. Momentum, though, is conserved in any collision, because the two forces are equal and opposite for the same short time (the third law again). The total before equals the total after, with direction as a sign. How bouncy the collision is gets one number, the restitution e: 1 for perfectly bouncy, 0 for stuck together. Momentum doesn't care about e. Kinetic energy does. Interactive · Two carts collide Equal masses, bouncy Stick together Light hits heavy Blue mass Blue velocity Gold mass Gold velocity Bounciness e Velocity to the right is positive; arrows show velocity. With equal masses and e = 1, the moving cart stops dead and the other leaves with its speed: the toy known as Newton's cradle. Stuck together, two 1,000 kg cars (one at 10 m/s, one parked) leave at 5 m/s and half the kinetic energy, 25 kJ, becomes damage. That lost half is the reason cars have crumple zones: they spend the energy deforming metal instead of people. Rotation: the same laws with new names Everything above has a spinning twin. Force becomes torque, force times the perpendicular distance from the pivot: 80 N on a 0.25 m wrench is 20 N·m, and a longer wrench gives more for the same push. Mass becomes moment of inertia I, which counts not just how much mass there is but how far from the axis it sits. Momentum becomes angular momentum L = I × ω. τ = I · α   ·   L = I · ωwith no outside torque, L is conserved Mass far from the axis counts for much more, because I grows with the square of distance. That's why a solid disc beats a hoop of the same mass and radius down a ramp: the hoop keeps all its mass at the rim, so it spends more of its energy on spinning (the disc's acceleration is two thirds of g sin θ, the hoop's one half, whatever the size or mass). And it's why a skater spins faster when she pulls her arms in. Interactive · The skater's weights Arms out Halfway Tucked in Weights' distance from the axis A model: body I = 0.9 kg·m², plus two 2 kg weights; start at one turn per second with arms out. From arms out to tucked in, I falls from 3.15 to 0.99 kg·m², so the spin rises from 1 to about 3.2 turns per second. The energy also rises, from 62 J to 198 J, and the difference is the work her muscles do pulling the weights inward. Circular motion and gravity: a force that points at the middle Going round a bend at constant speed is still accelerating, because the velocity keeps changing direction. The acceleration points to the centre and equals v²/r, so something must push sideways with force mv²/r. On a flat road that something is friction. No more friction, no more turn: the car continues in a straight line, which feels, from inside, like being thrown outward. vmax = √(μ · g · r)fastest speed a flat bend of radius r allows Notice the engine's power isn't in the formula. On a 50 m bend with μ = 0.5 (wet) the limit is 56 km/h; dry (0.7), 67 km/h; ice (0.1), 25 km/h. Interactive · Take the bend Dry (0.7) Wet (0.5) Ice (0.1) Speed Radius of the bend Replace friction with gravity and you get orbits. Set v²/r equal to GM/r² and the period T satisfies T² ∝ r³, Kepler's third law. Check it: the Space Station's lap of 92.6 minutes at 6,779 km from Earth's centre predicts, for the Moon at 384,400 km, a period of 27.5 days. The real month is 27.3. One law, from a bend in a road to the Moon. Which ledger? Choosing force, energy or momentum This is the real skill. A hard problem in one ledger is a one-liner in another. Three rough rules: if the question asks for a force or a time, use F = ma; if it asks for a speed between two positions, and the path is messy, use energy; if there's a collision or explosion, use momentum. If something spins, use angular momentum. Try five. Interactive · Pick the first tool Force Energy Momentum Angular momentum Next question What a graduate just sees Stopping distance goes with speed squared: double the speed, four times the distance. A scale reads the push on you, not your mass: it changes in an accelerating elevator. Static friction is only as large as it has to be, up to a limit that depends on μ and the pressing force. Energy ignores the path; use it when the route is complicated. Momentum survives every collision; kinetic energy rarely does. Mass far from the axis counts for much more in rotation, because distance is squared. A turn needs a sideways force; remove the grip and you go straight. Weightless means falling together, not "no gravity". Where it shows up next Statics (34/10) is F = ma with a = 0: the free-body diagram becomes the bridge or crane that stays put. Dynamics (34/12) and Vibrations (34/17) take rotation and the second law into machines and springs, usually as differential equations (Differential Equations, 34/03). Thermodynamics (34/14) is the energy ledger with heat added: the lost half of that crash is where it starts. Physics II (34/07) redoes the same accounting for charges, with Coulomb's force and voltage as energy per charge. In Programming Fundamentals (34/09), you'll turn F = ma into a simulation loop; in Computer Science game engines and robotics code, it's the physics step. If you want to go deeper 📘 Standard textbooks. Halliday, Resnick & Walker, Fundamentals of Physics (12th ed., Wiley, 2021), or Young & Freedman, University Physics (16th ed., Pearson); the usual first-year choices, with worked examples and thousands of problems. For a sharper, more mathematical route, Kleppner & Kolenkow, An Introduction to Mechanics (2nd ed., Cambridge). 📘 Gentler, free. The Feynman Lectures on Physics, Volume I, free online at feynmanlectures.caltech.edu; chapters 4 (energy) and 10 (momentum) are a good start. 🌐 Free course. MIT OpenCourseWare 8.01SC Classical Mechanics (Fall 2016): lectures, problem sets and solutions. The takeaway Mechanics is three ledgers kept over the same world. Force tells you what changes motion and when; energy lets you skip the path; momentum survives collisions; and each has a spinning twin. Doubling speed quadruples stopping distance because energy goes with v². The graduate's skill is choosing which ledger makes this problem easy. Try this: next time you're a passenger, estimate how far the car needs to stop at the speed you're doing, using "14 m at 50 km/h, four times that at 100". Then notice how far ahead the next car really is. Sources & Further Reading Sources. Newton's laws first published in Philosophiae Naturalis Principia Mathematica (1687). Textbook physics throughout. Friction coefficients are rounded from published tables, which vary widely between sources (for example Engineering Toolbox's "Friction coefficients for common materials", which gives rubber on dry concrete 0.6–0.85, on wet concrete 0.45–0.75). Every number in the interactives was computed in node and checked by hand: 14.05 m and 56.2 m braking distances at 50 and 100 km/h (μ = 0.7); 86.6 km/h residual speed; 5.21 kN; 24.3 m/s; skater I = 3.15 and 0.99 kg·m², 3.18 turns per second, 62 J and 198 J; 56.4, 66.7 and 25.2 km/h bend limits; Kepler scaling from the Space Station (92.6 min at 6,779 km) to the Moon (27.46 days from 384,400 km). 🌐 MIT OpenCourseWare: 8.01SC Classical Mechanics 📘 The Feynman Lectures on Physics, Vol. I (free online, feynmanlectures.caltech.edu) 📘 Kleppner & Kolenkow, An Introduction to Mechanics, 2nd ed. (Cambridge University Press) 🌐 Engineering Toolbox: friction coefficients Step 3. Physics II: Electricity & Magnetism — Charge, Fields and Light ---------------------------------------------------------------------- Why: Electricity and magnetism: light is an electromagnetic wave. A rubbed balloon beats the whole Earth's gravity, electrons crawl through your wiring slower than a snail, and light turns out to be a field vibrating. The one course that explains your phone, your motor and your Wi-Fi. Rub a balloon on your hair and stick it to the wall. It hangs there. Gravity, the pull of an entire planet, is trying to drop it, and a few billion stray electrons on its surface are winning. Between a proton and an electron the electric attraction is about 2.3 × 10³⁹ times stronger than their gravitational attraction. Gravity only seems strong because the Earth is enormous and its charge is balanced out. The second surprise is quieter. When you flip a switch, the light comes on at once, yet the electrons in the copper behind the wall drift at a fraction of a millimetre per second. What travels at nearly light speed is not the electrons. It is a field. Physics II is the course where those two sentences stop being puzzling. The library has an easier first read, How Electricity Reaches Your Wall (13/04), about the grid and Ohm's law. This piece goes further: into fields, capacitors, induction, and Maxwell's discovery that light is made of the same stuff. The course in one box Who takes it: second-term first-year or second-year students of Mechanical & Aerospace Engineering and, in many programmes, Computer Science. What it's for: predicting what charges and currents do, from a battery to a radio wave. Comes after: Physics I: Mechanics (34/06) and Multivariable Calculus (34/02). Leads to: the motors, sensors and actuators of Mechanical & Aerospace Engineering, then Control Systems (34/20); for Computer Science, Computer Architecture (34/25) and Computer Networks (34/27), which are electricity at tiny and large scales. Honest line: you won't design a power supply after this. You will know why a thin wire gets hot and a thick one doesn't, and why a transformer hums. Charge and the electric field Matter carries electric charge, positive or negative. Like charges push apart and opposite charges pull together, with a force that falls with the square of the distance, the same shape as gravity. A charge doesn't need to "reach" another: it fills the space around it with a field, and any other charge placed there feels it. E = F / q = k · Q / r²the field is the force a small test charge would feel, per unit of its charge (k ≈ 8.99 × 10⁹ N·m²/C²) Worked number: 1 nC (a billionth of a coulomb) makes a field of 900 V/m at 10 cm. Drag the gold probe below. Put a positive and a negative charge side by side and the field curves from one to the other. Interactive · Drag the probe through the field A positive and a negative Two positives One alone Left charge Right charge Charges are 30 cm apart; the grid shows field direction. Equal and opposite charges: the voltage along the midline between them is exactly 0. Voltage: energy per charge Lift a book and it gains energy; the same book on a higher shelf has more. Voltage is the electrical version of height: the energy a charge gains per coulomb moved. A 1.5 V battery gives every coulomb that crosses it 1.5 joules, whatever the battery's size. V = energy ÷ charge  ·  1 volt = 1 joule per coulomban electron falling through V volts gains eV of energy That makes electrons easy to launch. Accelerate one through 100 V and it leaves at about 5.9 × 10⁶ m/s, two per cent of the speed of light. At 10,000 V its speed is 0.195 c, and the ordinary formula (½mv²) overestimates it by about 1.5%. The old cathode-ray TV worked this way, aiming such electrons at a screen. Interactive · An electron falls through a battery Voltage it falls through Computed with the relativistic energy formula, and compared with the ordinary one. Current and resistance: the slow river Current is charge flowing past a point each second (1 ampere = 1 coulomb per second). Push a voltage across a wire and a current flows; the ratio is the resistance. For most metals it's constant, which is Ohm's law, and a wire's resistance grows with its length and shrinks with its thickness. Resistance turns electrical energy into heat at a rate P = I²R. V = I · R  ·  R = ρ · L / A  ·  P = I²Rρ is the material's resistivity: for copper about 1.68 × 10⁻⁸ Ω·m Now the surprise. The electrons' average drift speed is I ÷ (n·A·e), with n about 8.5 × 10²⁸ free electrons per cubic metre of copper. Ten amps through a 2.5 mm² house wire gives a drift of 0.29 mm/s. An electron takes nearly an hour to cover one metre. The lamp lights at once because the wire is already full of electrons and the field pushes them all together, like water in a pipe that is already full. Interactive · A copper wire under load Current Wire cross-section Wire length With those defaults the wire wastes 13 W as heat. Halve the thickness and the heat doubles. That is why extension cords are rated by thickness, and why power lines run at very high voltage: a given power needs a smaller current, and heat goes as current squared. Circuits: two rules Every circuit, however tangled, obeys two bookkeeping rules from Gustav Kirchhoff. The junction rule: current flowing into a junction equals current flowing out, because charge doesn't pile up. The loop rule: going around any closed loop, the voltage rises and drops add to zero, because energy is conserved. They are conservation of charge and of energy, again. From them follow the two shapes. Resistors in series share one current, so their resistances add. Resistors in parallel share one voltage, so their currents add, and the total resistance gets smaller as you add branches. That's why plugging more appliances into one socket pulls more current from the wall, not less. Interactive · Three resistors, a 12 V battery All in series All in parallel R1, then R2 ∥ R3 Battery R1 R2 R3 With 100, 200 and 300 Ω: in series the battery supplies 20 mA; in parallel the total resistance falls to 54.5 Ω and the current rises to 220 mA, eleven times as much. Capacitors and the RC clock A capacitor is two plates close together that store charge, and with it energy in the field between them. Its capacity C is how much charge it holds per volt. Connect one through a resistor to a battery and the plates fill, but ever more slowly, because the more it holds the less voltage is left to push more in. V(t) = V₀ (1 − e−t/RC)  ·  τ = R · Cthe time constant τ sets the pace; the exponential is the solution of Differential Equations (34/03) After one τ the capacitor is 63% full, after three τ 95%, after five 99%. A 10 kΩ resistor with a 100 µF capacitor makes τ = 1 second. Camera flashes, phone touch screens, power-supply smoothing and the blink of a turn signal all use this clock. Interactive · Charge it, discharge it Charging Discharging Resistor Capacitor Time elapsed A 5 V supply. Magnetic fields: the push on moving charge A current makes a magnetic field that circles the wire. Ten amps at 5 cm makes about 40 µT, the same size as Earth's own field (about 25 to 65 µT at the surface). A magnetic field pushes only on moving charge, and sideways: perpendicular to both the motion and the field. A sideways push with constant speed is the circular motion of Physics I. F = q · v · B  ⇒  r = m · v / (q · B)the radius is bigger for faster or heavier particles, smaller for stronger fields A wire carrying current in a field gets the same push: that is a motor. And a particle's lap time, 2πm/(qB), doesn't depend on its speed, which is how cyclotron accelerators work. Interactive · A charge in a magnetic field Electron Proton Speed Field Induction: change a field, get a voltage In 1831 Michael Faraday showed that a changing magnetic field through a loop of wire drives a voltage around it. A steady field does nothing; only change counts. Faster change, more loops, and a stronger field all give a bigger voltage. ε = − N · dΦ/dtvoltage = number of turns × how fast the magnetic flux changes; the minus sign says the induced current opposes the change This one rule explains three machines. Spin a coil in a field and you have a generator: the voltage rises and falls as a sine wave, peak N·B·A·ω. Run the same machine backwards and you have a motor. Wind two coils on one iron core and an alternating current in the first induces an alternating voltage in the second, with output voltage in the ratio of the turns: a transformer. It works only with alternating current, since only change induces, and it is why the grid uses AC and why a charger hums. Interactive · Generator, then transformer Turns in the coil Magnet field Spin speed Transformer input Output coil (input has 1,000) Coil area 10 cm × 10 cm. Ideal transformer, no losses. At the defaults the generator gives 31.4 V peak at 10 Hz, and the transformer steps 240 V down to 12 V: 20 times fewer turns, 20 times the current. Maxwell: four rules and the speed of light In the 1860s James Clerk Maxwell gathered everything known into four equations. In words: (1) electric charges make electric fields that spread outward; (2) there are no isolated magnetic charges, so magnetic field lines never end; (3) a changing magnetic field makes a circulating electric field (Faraday); (4) an electric current, and, Maxwell's own addition, a changing electric field, make a circulating magnetic field. Put (3) and (4) together and something remarkable happens: a changing electric field makes a changing magnetic field, which makes a changing electric field, and the disturbance can sustain itself and travel. Maxwell calculated its speed from two measured constants of electricity and magnetism, 1/√(μ₀ε₀), and got the measured speed of light, 299,792,458 m/s. In his 1865 paper he concluded that light is such a wave. Heinrich Hertz produced and detected radio waves in the late 1880s, and the rest of the spectrum followed. c = 1 / √(μ₀ ε₀) = 299,792,458 m/sthe speed of every electromagnetic wave in empty space Radio, microwaves, infrared, visible light and X-rays differ only in frequency, and the wavelength is c ÷ f. A 100 MHz FM station has a wavelength of 3 m; Wi-Fi at 2.4 GHz, 12.5 cm; red light at 700 nm, 430 THz. This also explains antenna sizes: a quarter wavelength works well, so an FM antenna is about 75 cm. Interactive · Slide along the spectrum Mains 60 Hz AM radio FM radio Wi‑Fi Red light Frequency (log scale) What a graduate just sees Electric force is absurdly strong; things look neutral because charges balance. A voltage is energy per charge; a current is charge per second; power is their product. A thin wire heats up because P = I²R, and that's why the grid uses high voltage. Electrons drift slowly; the signal is the field, and it moves near light speed. Series adds resistance, parallel adds current. Any capacitor and resistor make a clock, and 63% in one τ is the number to remember. Magnetism acts only on moving charge, and always sideways. Only change induces: no change in the field, no voltage. So transformers need AC. Light, radio and X-rays are the same wave at different frequencies. Where it shows up next Motors, generators and sensors in Mechanical & Aerospace Engineering are the F = qvB and induction sections above; Control Systems (34/20) then closes the loop around them. Vibrations (34/17) and Differential Equations (34/03): the RC curve is the same equation as a cooling cup of coffee, and an inductor with a capacitor oscillates like a mass on a spring. Computer Architecture (34/25): a logic gate is a switch, and the delay of a wire is an RC time constant, which sets how fast a chip can be clocked. Computer Networks (34/27): Wi-Fi and cellular are Maxwell's waves with data on them; the wavelength rule sets antenna size. Thermodynamics (34/14) picks up the I²R heat; a chip's heat is why laptops have fans. If you want to go deeper 📘 Standard textbooks. Halliday, Resnick & Walker, Fundamentals of Physics (12th ed.) or Young & Freedman, University Physics (16th ed.), the usual engineering-year texts. Purcell & Morin, Electricity and Magnetism (3rd ed., Cambridge, SI units) is the elegant one for stronger students. 📘 Gentler, free. The Feynman Lectures on Physics, Volume II, free at feynmanlectures.caltech.edu. 🌐 Free course. MIT OpenCourseWare 8.02 Physics II: Electricity and Magnetism (Spring 2007), with lecture notes and problem sets. The takeaway Charges make fields; a voltage is energy per charge; a current is charge per second, resisted into heat. Two rules, conservation of charge and of energy, solve any circuit. A changing magnetic field makes a voltage, which gives generators, motors and transformers, and Maxwell showed that a changing electric field and a changing magnetic field can sustain each other as a wave travelling at the speed of light. That wave is light. Try this: look at the charger of a laptop or phone. Find the voltage and amperes printed on it, multiply them, and you have the power it can deliver. Then feel how warm the cable and the brick get, and work out which part is I²R. Sources & Further Reading Sources. Constants from the NIST CODATA 2022 listing (elementary charge 1.602 176 634 × 10⁻¹⁹ C and speed of light 299 792 458 m/s are exact; electron and proton masses; G; μ₀; ε₀). Copper's resistivity (1.68 × 10⁻⁸ Ω·m at 20 °C) from the tabulation in Wikipedia's "Electrical resistivity and conductivity"; free-electron density 8.5 × 10²⁸ m⁻³ and the drift-velocity worked example from Wikipedia's "Drift velocity". Earth's field 25–65 µT: Wikipedia's "Earth's magnetic field". Faraday's 1831 demonstration: Wikipedia's "Faraday's law of induction" (Joseph Henry independently, 1832). Maxwell's paper read to the Royal Society 8 December 1864 and published 1865: Wikipedia's "A Dynamical Theory of the Electromagnetic Field". Hertz produced and detected radio waves in 1886 and published his evidence in 1887–88: Wikipedia's "Heinrich Hertz". Computed in node and checked: electric to gravitational force ratio 2.27 × 10³⁹; drift speed 0.294 mm/s and 13.4 W; electron speeds (5.93 × 10⁶ m/s at 100 V; 0.195 c at 10 kV); 54.5 Ω and 220 mA; 63.2%, 95.0%, 99.3% RC fractions; electron orbit radius 5.7 cm and period 35.7 ns at 10⁷ m/s and 1 mT; 31.4 V peak; 1/√(μ₀ε₀) = 299,792,458 m/s to 8 digits. 🌐 MIT OpenCourseWare: 8.02 Physics II, Spring 2007 📘 Purcell & Morin, Electricity and Magnetism, 3rd ed. (Cambridge University Press, 2013) 📘 The Feynman Lectures on Physics, Vol. II (free online, feynmanlectures.caltech.edu) 🌐 NIST: fundamental physical constants (CODATA 2022) Step 4. The Sky & Observational Astronomy: Where Everything Is, and What You Can See ------------------------------------------------------------------------------------ Why: Finding your way around the sky and what the eye and a small telescope show. Before any physics of stars, there is the sky itself: a turning dome with a few thousand points of light, a backwards brightness scale, and a handful of angles that tell you almost everything. Look up on a clear night from a dark place and ask three questions. Why is that star where it is, and why will it be somewhere else in two hours? Why do some stars shine and others barely show? And how would anyone know how far away they are? People have asked these for thousands of years with no instrument at all. The Babylonians wrote down the risings of stars on clay tablets; Chinese astronomers kept star catalogues from at least the 4th century BCE; Polynesian navigators crossed the Pacific by memorised star paths. An astronomy student relearns this old, practical skill first. After it, a graduate never looks at the night sky as a ceiling again. They see a rotating sphere, tilted for their latitude, with a brightness scale and a ruler made of angles. The course in one box Who takes itAstronomy and physics students in the first year, often as an introductory "Astronomy 101" with night-sky labs, and anyone curious. Programmes such as MIT's Course 8 astrophysics track and the Institute of Physics accreditation framework expect students to know celestial coordinates, magnitudes and basic observing before they meet stellar physics. What it is forLocating and describing objects in the sky, and judging what an eye, binoculars or a small telescope can show from where you stand. Comes afterPhysics I: Mechanics (34/06), for gravity and orbits behind the sky's motions. Leads toTelescopes, Light & Spectra (36/39), Planetary Science & Exoplanets (36/40), Stars: Structure & Evolution (36/41) and Astronomical Data & Computation (36/44). What this piece gives youYou will not be able to point a telescope with setting circles from this page. You will see why the pole star sits at the height of your latitude, why stars rise four minutes earlier each night, why a magnitude 1 star is 100 times brighter than a magnitude 6 one, and why we have to go to space to measure stellar distances. Easier first read: The Big Bang and Our Expanding Universe (13/08). The sky is a sphere, and your latitude tilts it Treat the stars as painted on the inside of a huge sphere around you. The Earth spins once in about 23 hours 56 minutes, so the sphere seems to turn the other way around an axis pointing at the celestial poles. In the north, a moderately bright star, Polaris, sits within about a degree of the pole. Everything else circles it. Two coordinate systems matter. Altitude and azimuth are local: how high above the horizon, and which way round from north. They change as the sky turns and differ for every observer. Right ascension and declination are fixed to the stars, like longitude and latitude painted on the sphere. Declination is the star's "latitude" on the sphere, in degrees from −90 to +90. sin(altitude) = sin(latitude) · sin(declination) + cos(latitude) · cos(declination) · cos(H)H is the hour angle: how many hours (15° each) the star is past due south or north, its highest point Two things follow without any more algebra. A star crosses its highest point at an altitude of 90° minus the gap between your latitude and its declination. And the pole star stands at an altitude equal to your latitude: stand at 40° north and Polaris is about 40° up. A star never sets for you if it lies within your co-latitude of the pole, and never rises if it lies the same distance around the opposite pole. That is why Crux, the Southern Cross, is never seen from Europe but never sets from Australia or South Africa. Interactive · Where is that star? Your latitude Hours past its highest point Real J2000 declinations. Precession shifts positions by about 0.4° over a few decades, so Polaris's altitude wobbles within a degree of your latitude. Refraction and the star's colour are ignored. Without JavaScript: at latitude 40° north, Polaris is about 40° up all night and never sets; Sirius peaks at about 33° in the south; Acrux of the Southern Cross never rises (it would peak 13° below the horizon). The sky turns once a day and shifts once a year A solar day (noon to noon) is 24 hours. A sidereal day (a star back on the meridian) is about 23 hours 56 minutes 4 seconds. The difference is the Earth's orbit: in a day it has moved about one degree around the Sun, so it must turn about one degree extra to face the Sun again. A star therefore rises about 3 minutes 56 seconds earlier each night, roughly two hours earlier each month, and the whole sky is back in the same place after a year. That is why Orion is a winter constellation in the north and Scorpius a summer one. The same tilt gives us seasons. The Sun's declination swings between about −23.44° and +23.44° over the year (the tilt of Earth's axis). At noon the Sun is at altitude 90° minus the gap between your latitude and that declination, and day length follows. Same formula, different object. Interactive · One date, three consequences Date Your latitude Approximate formulas good to about a degree and a few minutes. Day length counts sunrise and sunset at the Sun's upper edge with standard refraction. The star hint compares right ascension only. Without JavaScript: at 40° north on 21 June the Sun is 73° high at noon and the day is about 15 hours; on 21 December it is 27° high and the day is about 9 hours 20 minutes. At 70° north in June the Sun does not set. Magnitudes: a backwards, logarithmic scale The classical scale comes from Ptolemy's star catalogue (about 150 CE), usually traced to Hipparchus: the brightest stars are "first magnitude", the faintest naked-eye ones "sixth". So a bigger number means a dimmer star. In 1856 Norman Pogson fixed the scale mathematically so that five magnitudes are exactly a factor of 100 in light. One magnitude is therefore a factor of 1001/5, about 2.512. m₁ − m₂ = −2.5 · log₁₀(F₁ / F₂)the flux ratio is 100.4 (m₂ − m₁); a difference of 5 magnitudes is a factor of 100 The scale has gone beyond its first range. Sirius, the brightest night star, is at about −1.46. Venus can reach about −4.9, the full Moon is about −12.7 and the Sun about −26.7. At the faint end, the dimmest sources of Hubble's deepest images are near magnitude 30 or beyond. Your eye follows the same kind of log scale, which is why the scale was natural. The faintest star you can see depends on two things: how dark your sky is, and how much light your instrument gathers. A telescope's aperture D collects light in proportion to D². Compared with a dark-adapted pupil of about 7 mm, that is a gain of 5 · log₁₀(D / 7 mm) magnitudes: about 4.3 for 50 mm binoculars and 7.3 for a 200 mm telescope, in the ideal case. Real gains are somewhat smaller. Interactive · What can you see tonight? Sky (Bortle class, limiting magnitude with the naked eye) Instrument ObjectMag.vs your limitSeen? Sky limits from the Bortle scale (Bortle 2001), mid-range values. Magnitudes are typical or peak values. Spread-out objects such as galaxies and nebulae are harder than their magnitude suggests, because their light is spread over an area. Without JavaScript: from a Bortle class 3 rural sky (limit about 6.8), 50 mm binoculars reach about 11.1, enough for Neptune (7.8). From a Bortle class 9 city sky (about 3.8) the naked eye sees only stars and planets brighter than about magnitude 4. Angles and parallax: measuring the sky without a ruler You cannot tell from the sky alone how big or how far anything is, only how big it looks: its angle. A degree is one of 360 in a circle; an arcminute is 1/60 of a degree; an arcsecond is 1/60 of that. Your outstretched fist covers about 10°, your little finger at arm's length about 1°, and a healthy eye can just separate two points an arcminute apart. For small angles, angle in radians = size / distance. That is the whole of the astronomer's ruler. Famously, the Sun and Moon appear almost the same size, each about half a degree. It is a coincidence of this epoch, not a law: the Sun is about 400 times wider and about 400 times farther. Both distances vary, so the match does too. When the Moon looks bigger we get a total eclipse, and when it looks smaller a ring of Sun remains (an annular eclipse). Interactive · Total or annular? Moon's distance (km) Sun's distance (million km) Moon diameter 3,474.8 km; Sun diameter 1,391,400 km (IAU 2015 nominal radius 695,700 km). The ranges are the typical extremes of the Moon's and Earth's orbits. Without JavaScript: at its nearest the Moon spans 0.558°, at its farthest 0.490°. The Sun spans 0.542° at its nearest and 0.524° at its farthest. The Moon covers the Sun only when it is the larger. The same trick gives distances to nearby stars. Look at a star in January and again in July: Earth has moved 300 million km, and a near star shifts a little against the faraway background. Half the shift is the parallax p. A star with a parallax of one arcsecond is at a distance of one parsec, 3.26 light-years. distance (parsecs) = 1 / parallax (arcseconds)1 parsec ≈ 3.26 light-years ≈ 3.09 × 10¹³ km The old argument against a moving Earth was that no parallax could be seen. It was right that none was visible; nobody could know how small it is. The nearest star, Proxima Centauri, has a parallax of 0.768 arcsecond, about 1/80 of what a good eye can resolve. Friedrich Bessel announced the first accepted stellar parallax, for 61 Cygni, in 1838: about a third of an arcsecond, a distance of around 10 light-years. The ESA satellite Gaia (launched 2013, observing until January 2025) measured parallaxes for about 1.5 billion stars, with an uncertainty near 0.02 to 0.03 milliarcseconds for stars brighter than magnitude 15. Interactive · How big is the shift? Distance (light-years) The slider is logarithmic, from 4 to 100,000 light-years. The error uses 0.03 milliarcseconds, a round figure for bright stars (Gaia DR3 gives 0.02 to 0.03 for stars brighter than magnitude 15). Without JavaScript: Proxima Centauri, 4.25 light-years, has a parallax of 768 milliarcseconds. At 1,000 light-years the parallax is 3.3 milliarcseconds, measured to about 1% by Gaia; at 10,000 light-years it is 0.33, measured to about 10%. Dark skies: what light pollution takes John Bortle's 2001 scale grades sky darkness from class 1, an excellent dark site, to class 9, an inner-city sky. A class 1 sky shows stars to about magnitude 7.6 to 8.0 and many Messier objects can be seen without a telescope; a class 9 sky hides all but magnitude 4.0 or brighter stars. In 2016 Fabio Falchi and colleagues combined satellite data with ground measurements into a world atlas and reported that more than 80% of the world's people, and more than 99% of people in the US and Europe, live under light-polluted skies. They estimated that the Milky Way cannot be seen from where about a third of humanity lives. The loss is not only for astronomers. Artificial light at night affects migrating birds, insects and human sleep, and it is wasted energy. Cheap fixes exist: shielded lamps that point down, warmer colour temperature, and lights that dim or switch off when unneeded. Protected places include the Aoraki Mackenzie Dark Sky Reserve in New Zealand and the NamibRand Nature Reserve in Namibia (both designated in 2012 by what is now DarkSky International). Big observatories sit where the sky is dry, high and dark: Chile's Atacama and Andes (ESO's Paranal and the Vera C. Rubin Observatory), La Palma and Tenerife in the Canary Islands, Hawaii's Mauna Kea, South Africa's Sutherland (SALT), and the radio-quiet Karoo and Western Australia (the SKA). Many skies: who drew which pictures A constellation is a pattern, not an object: the stars in one are usually at wildly different distances. The 88 constellations in use were fixed by the International Astronomical Union in 1922, with boundaries drawn in 1930, so that every point on the sphere belongs to exactly one. Most of the names come from Greek and Roman tradition, passed through Ptolemy's Almagest (about 150 CE) and Islamic astronomers who translated, corrected and renamed. That is why so many bright stars have Arabic names: Betelgeuse, Altair, Aldebaran, Vega. The Persian astronomer al-Sufi's Book of the Fixed Stars (964 CE) described what we now call the Andromeda Galaxy as a "little cloud". Other peoples drew other skies. Chinese astronomy grouped stars into asterisms inside a different system of the sky, with court astronomers recording comets and "guest stars" (novae and supernovae) for two thousand years. The Maya tracked Venus's 584-day cycle in the Dresden Codex. The Inca read dark lanes in the Milky Way as animals, including a llama. Many Aboriginal Australian peoples see an emu in the same dark lanes. Polynesian and Micronesian navigators memorised where stars rise and set around the horizon, so a "star compass" could hold a course across open ocean; in 1976 the canoe Hōkūle'a sailed from Hawaii to Tahiti guided by the Micronesian navigator Mau Piailug. None of these is "the real sky". They are different ways to organise the same stars, and any of them worked well enough to navigate, farm or keep time. The Western catalogue won for science mostly because it was standardised, not because the others were wrong. What a graduate just sees The sky is a sphere that turns once a day; the pole star's altitude equals your latitude. Stars rise about four minutes earlier each night, which is why the visible constellations change with the seasons. The Sun's noon height and the length of the day come from the same latitude and declination formula as any star. Magnitudes run backwards and are logarithmic: five steps is a factor of 100. Aperture sets how faint you can see, as the square of its diameter; sky darkness sets the limit you start from. Astronomers measure angles, not sizes, and turn angles into distances with parallax. Constellations are human patterns, and every culture drew its own. Where it shows up next Telescopes, Light & Spectra (36/39): the aperture, resolution and light-gathering ideas here become design numbers. Planetary Science & Exoplanets (36/40): planets are found by tiny shifts and dips, measured in the same magnitudes and angles. Stars: Structure & Evolution (36/41): the magnitude and distance of a star give its luminosity, the quantity stellar physics must explain. Astronomical Data & Computation (36/44): surveys such as Gaia and Rubin turn the sky into catalogues of billions of rows. In the world: GPS-free navigation, calendars and festivals, dark-sky lighting rules, and amateur astronomy. If you want to go deeper 📘 Textbook. Bradley Carroll and Dale Ostlie, An Introduction to Modern Astrophysics (Cambridge University Press, 2nd edition 2017), the standard upper-level text; for a first course, Andrew Fraknoi and colleagues, Astronomy 2e (OpenStax, free). 📘 Gentler. Terence Dickinson, NightWatch (Firefly Books), a practical guide to the night sky; and the free planetarium software Stellarium to see the sky from any place and date. 🌐 Free text. OpenStax Astronomy 2e at openstax.org, with chapters on the sky, scales and distances. 🌐 Free course. MIT OpenCourseWare 8.282J Introduction to Astronomy (Spring 2006). The takeaway The sky is a turning sphere whose view depends on where you stand. Brightness runs on a backwards logarithmic scale, size and distance come from angles, and the darkness of your sky decides how much of it you get. All the physics of stars begins with these plain measurements. Try this: tonight, find Polaris (or, in the south, the pole's empty patch near the Southern Cross) and hold a fist at arm's length below it. Note what the sky looks like again two hours later, and write down which stars moved and which did not. Sources & Further Reading Sources. Pogson, "Magnitudes of thirty-six of the minor planets," Monthly Notices of the RAS 17 (1856); Ptolemy, Almagest (about 150 CE); al-Sufi, Book of the Fixed Stars (964 CE); Bessel's 1838 parallax of 61 Cygni; IAU constellation boundaries (Delporte 1930); IAU 2015 Resolution B3 on nominal solar values; Bortle, "Introducing the Bortle dark-sky scale," Sky & Telescope, February 2001; Falchi et al., "The new world atlas of artificial night sky brightness," Science Advances 2 (2016); ESA, Gaia Data Release 3 (13 June 2022) documentation; DarkSky International reserve list; Meeus, Astronomical Algorithms, for solar position approximations. Star coordinates are J2000 values from standard catalogues. All outputs of the interactives were computed in node. Cultural astronomy statements follow standard surveys of the field (for example Ruggles, ed., Handbook of Archaeoastronomy and Ethnoastronomy, 2015). 📘 Carroll and Ostlie, An Introduction to Modern Astrophysics 📘 OpenStax, Astronomy 2e 🌐 MIT OCW 8.282J Step 5. Waves and Optics: When Waves Add, Light Can Make Darkness ----------------------------------------------------------------- Why: Waves and optics: how telescopes and spectra work. Why noise-cancelling headphones work, why a lens can never see everything, why your sunglasses black out a phone screen, and why one photon at a time still draws a wave pattern. Tilt your head while looking at a phone through polarised sunglasses and the screen fades to black, then returns. Nothing was switched off. The light is still coming out of the glass; your lenses are simply refusing to let that particular kind of light through. A soap bubble does something stranger: it has no pigment at all, yet it shows swirling colours that change as the film thins. Noise-cancelling headphones are stranger still. They make silence by adding sound: a microphone listens to the noise, and the speaker plays the same wave upside down. Two sounds, added together, give nothing. All three are one idea. A physics graduate looks at them and sees the same thing: waves add up, crest on crest or crest on trough, and almost everything light and sound do follows from that. It is also the doorway to quantum mechanics, where particles turn out to obey the very same arithmetic. The course in one box Who takes it: physics majors, usually in year 2 (MIT teaches it as 8.03 Vibrations and Waves; Cambridge's Natural Sciences physics route covers the same ground). Engineers meet the maths again as electromagnetics and signals; some programmes add a separate optics course for photonics or optometry. What it's for: understanding anything that oscillates and travels: sound, light, radio, water, and, later, the wavefunction itself. Comes after: Physics I: Mechanics (34/06) and Differential Equations (34/03). Leads to: Quantum Mechanics (34/83), which is wave mechanics with a new meaning for the wave; Electromagnetics & Waves (34/38); Signals & Systems (34/37); Partial Differential Equations (34/79), where the wave equation is a headline act; and, from the series, Vibrations (34/17), which is the one-spring cousin of this subject. Honest line: you won't be able to design a lens system after this. You will see why interference makes dark bands, why a telescope's size sets what it can resolve, and why a single photon is already a wave problem. A wave is a pattern that travels Throw a stone in a pond and the ripple moves outward, but the water mostly just bobs up and down. What travels is the pattern. A wave is a disturbance handed from neighbour to neighbour; each bit of the medium moves a little and returns. Three numbers describe a steady wave: the wavelength λ (crest to crest), the frequency f (crests passing a point each second) and the speed v (how fast the pattern moves). They are tied by one line of arithmetic: in one second, f crests go by, each λ long. v = f · λspeed = frequency × wavelength The medium sets the speed; the source sets the frequency; the wavelength is whatever is left. Sound in air at 20 °C runs at about 343 m/s, so middle A (440 Hz) has a wavelength of 0.78 m. An FM station at 98 MHz rides on radio waves 3.06 m long. Wi-Fi at 2.4 GHz is 12.5 cm. Green light at 550 nm is about 545 trillion cycles per second. Here is the subtlety a graduate never forgets. When a wave crosses into a new medium, its frequency stays put (the source and the boundary insist on it, crest for crest) and its speed changes, so its wavelength changes. Light in glass of refractive index n = 1.5 travels at c/1.5 and its wavelength shrinks by the same factor: 550 nm green becomes 367 nm inside the glass, with no change of colour. Colour belongs to the frequency. Interactive · Frequency, wavelength, medium Sound in airLight in glassLight in water Frequency Sound: 20 Hz to 20 kHz at 343 m/s; the strip is 2 m wide. Light: the visible range, 400 to 790 THz, with c = 299 792 458 m/s (exact) and n = 1.5 for glass, 1.333 for water. The drawing is slowed enormously. For light, the top wave is the same light in vacuum: same frequency, longer wavelength. Waves add, so they can cancel Two waves in the same place don't collide; they superpose. At every point you add the two displacements. Where a crest meets a crest the total is double; where a crest meets a trough it is zero. This is the single most useful fact in the subject, and it is a statement about sums, which is why the maths (and later, quantum mechanics) leans on linearity. Take two sources side by side, each sending out circular ripples. A point is bright if its distances to the two sources differ by a whole number of wavelengths (the waves arrive in step), and dark if they differ by a half-integer number. That is the whole rule: path difference = m · λ → bright; (m + ½) · λ → darkm = 0, ±1, ±2, … Far from the sources, the bright directions satisfy d sin θ = mλ, where d is the spacing. Thomas Young showed in 1801 that light itself forms such bands from two slits, which was the decisive evidence that light is a wave. In the lab, light of 550 nm through slits 0.5 mm apart gives bands 2.2 mm apart on a screen 2 m away (λL/d), spacing you can measure with a ruler. A useful consequence: you can only get a bright direction if mλ/d is less than 1. If the sources are closer than one wavelength apart, there is only a single bright direction, straight ahead, and the pattern is just a ball of waves. Noise cancelling is the same arithmetic in time: add a wave to its inverted copy and the sum is zero. It works best for low, steady hums, where the wavelength is long enough that the speaker and your ear are in the same phase of the wave. Interactive · Two sources, seen from above Source spacing d Wavelength λ Pause Each pixel adds the two waves using its true distances to the sources. Orange and blue are positive and negative displacement; the dark lines that don't move are where the waves always cancel. The graph below shows the average brightness far away, cos²(π d sin θ / λ). Try d smaller than λ. Light bends because it is quickest: Snell and Fermat Light slows down in glass, water and diamond. The ratio of its speed in vacuum to its speed in the material is the refractive index n: about 1.0003 for air, 1.33 for water, 1.5 for common glass, 2.42 for diamond. When a ray crosses from one to another at a slant, it bends. Snell's law gives the angles (measured from the line perpendicular to the surface): n1 sin θ1 = n2 sin θ2Snell's law Why does it bend? Pierre de Fermat's answer (1662) is the one that lasts: of all possible paths from A to B, light takes the one that needs the least time. (Strictly, a path of stationary time: usually a minimum.) Picture a lifeguard who can run fast on sand and swim slowly. The quickest route to a swimmer isn't the straight line; it bends at the shoreline to spend less time in the slow water. That bend is Snell's law. Fermat's idea is also a first glimpse of the principle that organises Classical Mechanics (34/82): nature finds a path that is stationary. Fermat explains lenses with no extra rules. A lens is shaped so that every path from a point to its image takes exactly the same time. For a thin lens with focal length f, an object at distance u makes an image at distance v where 1/f = 1/u + 1/v. Worked number: a 50 mm lens focused on something 2 m away puts the image 51.3 mm behind the lens, and the image is 0.026 times the object's size, upside down. Focusing a camera is moving the lens 1.3 mm. Going the other way matters too. From glass to air, Snell's law runs out of angle: when n1 sin θ1 would exceed n2, there is no refracted ray. This total internal reflection sets in beyond a critical angle of 41.8° for glass to air and 48.6° for water to air. It is how optical fibres trap light, and why a diamond sparkles. Interactive · Find the quickest path Air → glassAir → waterGlass → airAir → diamond Where the ray crosses the surface Target's sideways distance Snap to the quickest path Source 1 m above the surface, target 1 m below. Times are exact sums of distance ÷ speed, with c = 0.29979 m/ns. At the minimum, n₁ sin θ₁ = n₂ sin θ₂ to within the slider's step. Every opening blurs: diffraction and the limit of resolution Waves bend around corners and spread through gaps. Make the opening about as small as the wavelength and a wave leaves it as a spreading ripple. Make it large and the wave mostly carries straight on, but the edges still blur it. That spreading is diffraction, and it is the same adding-up of waves, now from every point across the opening. For a round opening (a telescope mirror, a camera lens, the pupil of your eye) a point of light does not make a point on the image. It makes a bright disc surrounded by faint rings: the Airy pattern. The first dark ring sits at an angle that depends only on the ratio of wavelength to opening: θmin ≈ 1.22 · λ / Dsmallest angle you can tell apart (Rayleigh criterion), in radians Two points closer together than that blur into one. Lord Rayleigh's rule of thumb says two equal points are "just resolved" when the centre of one falls on the first dark ring of the other; between the two peaks the combined brightness then dips to 73.5% of the peak, a 26% dip. Worked numbers for green light, computed live below: a pupil 3 mm wide cannot separate things less than 46 arcseconds apart, so two car headlights 1.5 m apart merge at about 6.7 km (real eyes do worse, because the retina and the eye's flaws add their own blur). The Hubble Space Telescope's 2.4 m mirror reaches about 0.058 arcseconds. A phone camera's few-millimetre lens is limited to about 28 arcseconds by diffraction alone. The lesson is that resolution is bought with width, not magnification. Enlarging a blurred image only enlarges the blur. That is why astronomers want 10 m mirrors, why radio telescopes are linked across continents, and why a microscope can never see an atom with visible light (atoms are far smaller than the wavelength). Interactive · Can the lens tell them apart? Eye, headlights 3.4 kmEye, headlights 11 kmHubble, Rayleigh limitPhone camera Opening width D Separation of the two points Green light, 550 nm. Brightness is the exact Airy pattern (2J₁(x)/x)² for each point, added (the two points are separate lamps, so their intensities add). The slider is in units of the Rayleigh angle; the readout converts it into arcseconds and into 1.5 m headlights at a distance. Light is a transverse wave: polarisation Sound waves push and pull along their direction of travel. Light's wave is a sideways wobble of the electric field, perpendicular to the direction it travels, and the wobble can lie along any direction in that sideways plane. Light with the wobble mostly along one line is polarised. Reflected glare off water and roads is partly polarised, which is why polarising sunglasses cut it, and why your phone (an LCD that sends polarised light) goes dark at the wrong angle. A polarising filter lets through only the component of the wobble along its axis. If the light is already polarised at angle θ to the filter's axis, the fraction of intensity that gets through is cos²θ: Malus's law, from 1809. Unpolarised light (every angle at once) loses half on its first filter and comes out polarised along that filter's axis. I = I0 cos² θMalus's law Now the surprise. Cross two filters (axes 90° apart) and no light gets through: cos²90° = 0. Slide a third filter between them, at 45°, and light appears: after the first filter the intensity is ½, after the 45° one it is ½ × cos²45° = ¼, and after the last, ¼ × cos²45° = ⅛, which is 12.5% of the original beam. Adding an obstacle let more light through, because the middle filter re-aims the wobble. This is the classical twin of a famous quantum result about spin measured in different directions, which is the reason it is in this course. Interactive · Three filters Crossed, with a 45° filterCrossed pair onlyNearly aligned First filter angle Middle filter angle Last filter angle Middle filter inMiddle filter out Ideal filters. The first filter turns unpolarised light into polarised light (losing half); each later filter multiplies by cos² of the angle between neighbouring axes. The bridge to quantum mechanics: one photon at a time Light is a wave: that was settled by interference in the early 1800s and by Maxwell's equations in the 1860s. But light is also detected in lumps. A sensitive enough camera records single dots, one photon each. So what happens if you turn the lamp down until only one photon is in the apparatus at a time? Each photon lands as a single dot, in a place that looks random. But dot by dot, the pattern builds up, and it is the interference pattern: bright bands where the wave's intensity is high, nothing where it cancels. Each photon, alone, behaves as if it went through both slits. Close one slit and the bands vanish, leaving the broad smear of a single slit. The wave doesn't say where the photon lands; it says how likely each place is. The same thing happens with electrons: Akira Tonomura's team at Hitachi published a film of the pattern building up electron by electron in 1989 (the Merli–Missiroli–Pozzi group in Bologna had done a single-electron experiment earlier, in 1974). It happens with atoms, and with large molecules. Quantum mechanics (34/83) says this is not a puzzle to be explained away but the rule: a particle comes with a wave, and the wave's squared size is the probability of finding it. What the wave is is still argued about; Quantum Mechanics and 13/06 take that up. Interactive · Fire photons one at a time Both slits openOne slit open +1+10+100+1000Reset Helium-neon laser light (633 nm), slits 0.25 mm apart and 0.06 mm wide, screen 1 m away: bands 2.5 mm apart under a broad envelope with its first zero at ±10.6 mm. Each dot is drawn at random from the exact wave intensity; the line is the prediction. Illustrative parameters, real physics. What a graduate just sees A wave is a travelling pattern; speed = frequency × wavelength, and in a new medium the frequency stays and the wavelength changes. Waves add. Equal path lengths (or whole wavelengths) mean bright; half a wavelength off means dark. Interference needs sources closer together than the pattern you want to see: d sin θ = mλ, so small spacing spreads the pattern wide. Light goes the quickest way, so it bends toward the normal when it slows down; a lens is a shape that makes all paths equally quick. Past the critical angle, light cannot leave the denser material at all. Resolution is set by wavelength divided by opening: bigger apertures, not more zoom, show finer detail. Light is a transverse wave, and a filter in the middle can create light that two filters would have blocked. One photon at a time still builds the interference pattern, which is where classical waves end and quantum mechanics begins. Where it shows up next Quantum Mechanics (34/83): the Schrödinger equation is a wave equation; standing waves in a box give discrete energies, just as a guitar string gives discrete notes. Electromagnetics & Waves (34/38) and Signals & Systems (34/37): the same wave equation for Wi-Fi, fibres and antennas; superposition is the reason Fourier analysis (see 12/12) works. Partial Differential Equations (34/79): the wave equation is one of the three classic PDEs, with the heat and Laplace equations. Technologies: lenses and cameras, fibre-optic internet, LCD and OLED screens (see 29/10), lasers, holography, anti-reflection coatings (built from thin-film interference), ultrasound scanners, radar, and gravitational-wave detectors like LIGO, which are interferometers. Astrophysics (34/88): everything we know about stars is a spectrum or a resolved image. If you want to go deeper 📘 Textbook: A. P. French, Vibrations and Waves (MIT Introductory Physics Series): short, clear, and the old standard for the second-year wave course. For optics, Eugene Hecht, Optics (Pearson, 5th edition), the most widely assigned undergraduate optics text. 📘 Gentler: The Feynman Lectures on Physics, Vol. I, chapters 28–33 (waves, interference, diffraction, refraction) and Vol. III, chapter 1 (quantum behaviour, the double slit), free online. Or this library's Waves and Sound (13/13) and Light and Color (13/09). ▶️ Free course: MIT OpenCourseWare 8.03SC Physics III: Vibrations and Waves (Fall 2016), with lecture videos and problem sets. 🌐 Easier first reads in this library: Waves and Sound (13/13), Light and Color (13/09), Quantum Mechanics: The Weirdness at the Bottom (13/06), and in this series Vibrations (34/17) and Electromagnetics & Waves (34/38). The takeaway Waves are patterns that travel, with v = fλ, and waves add: that one rule makes bright and dark bands, noise cancelling, and every blurry edge. Light takes the quickest path, which bends it and makes lenses; the width of an opening, measured in wavelengths, sets the finest detail it can show; and light is a transverse wave that filters can sort. Send photons one at a time and the same pattern emerges dot by dot, which is where waves hand over to quantum mechanics. Try this: hold two pencils close together and look at a bright lamp through the narrow gap between them, squeezing the gap until it nearly closes. You will see dark bands appear inside the gap: that is diffraction, and the bands spread out as the gap narrows. Then take polarised sunglasses (or a pair of lenses) and look at a phone screen while you turn them. Sources & Further Reading Sources. Wave arithmetic, Snell's law, Fermat's principle, Rayleigh's criterion (1.21967 λ/D, from the first zero of J₁ at 3.8317) and Malus's law are standard textbook material (French; Hecht; Feynman Lectures I). Constants: c = 299 792 458 m/s is exact by definition of the metre (since 1983). Young's two-slit experiment: 1801, reported to the Royal Society in 1802–1803. Single-electron build-up: A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, "Demonstration of single-electron buildup of an interference pattern", American Journal of Physics 57, 117 (1989); Merli, Missiroli and Pozzi's earlier single-electron work (1974) is credited in Physics World, "The double-slit experiment". Textbook editions: Pearson's page for Hecht's Optics, 5th edition. Computed in node: the Airy dip at Rayleigh separation (73.5% of peak), all resolution distances, the thin-lens image distance, critical angles, and the Fermat minimum (checked against Snell's law). 🌐 MIT OpenCourseWare 8.03SC Physics III: Vibrations and Waves 🌐 Physics World: The double-slit experiment 📘 French, Vibrations and Waves · Hecht, Optics · The Feynman Lectures on Physics Step 6. Telescopes, Light & Spectra: Everything We Know Arrives as Light ------------------------------------------------------------------------ Why: Everything we know arrives as light (and now gravitational waves). Nobody has visited a star. Everything astronomers know about stars, galaxies and the early universe came from what reached a mirror, and the trick of reading light is the subject of this course. In 1835 the philosopher Auguste Comte gave an example of knowledge that humans could never have: what the stars are made of. Their distance, he argued, put chemistry out of reach for ever. Within 25 years he was wrong. In 1859 Gustav Kirchhoff and Robert Bunsen showed that each chemical element prints its own pattern of lines in light, and those same lines were already known as dark gaps in sunlight. In 1868, during a total solar eclipse, astronomers saw a bright yellow line in the Sun's edge that matched no known element. It was named helium, after the Greek for Sun, and it was found on Earth only in 1895. That is the astronomer's situation in one story. You cannot touch a star, so you collect its light, spread it into a rainbow, and read it. A graduate of this course knows how big a mirror must be to see a faint thing, how sharp it can ever be, why colour tells temperature, and why the lines tell composition and motion. Telescope pictures are never just pictures. The course in one box Who takes itAstronomy and astrophysics majors in year 2, usually as "Observational Astronomy" or "Astronomical Techniques", often with a lab; also physics and engineering students interested in optics. Programmes such as MIT's Course 8 astrophysics track and Caltech's Ay courses build it on optics and quantum mechanics. What it is forKnowing how astronomical light is collected, analysed and turned into measurements of temperature, composition, motion and distance. Comes afterWaves & Optics (34/80), where diffraction and interference are taught, and The Sky & Observational Astronomy (36/38). Leads toStars: Structure & Evolution (36/41), Planetary Science & Exoplanets (36/40), Compact Objects & High-Energy Astrophysics (36/43) and Astronomical Data & Computation (36/44). What this piece gives youYou will not be able to calibrate a spectrograph. You will see why bigger mirrors see fainter, why the atmosphere limits sharpness, why a hot star is blue and a cool one red, how a line gives a speed, and why JWST looks in the infrared. Easier first reads: Light and Color (13/09) and Astrophysics and Cosmology (34/88). Aperture: a telescope is a light bucket A telescope does not mainly make things bigger. It collects more light than your pupil can. Light collected goes as the area of the mirror or lens, which is the square of its diameter D, the aperture. A dark-adapted pupil is about 7 mm across. A 70 mm lens collects 100 times as much light. A 2.4 m mirror such as Hubble's collects about 117,000 times as much. light collected ∝ D²  ·  gain in magnitudes = 5 · log₁₀(D / 7 mm)from The Sky & Observational Astronomy (36/38): five magnitudes is a factor of 100 in light This is why astronomers build ever bigger mirrors. A faint galaxy sends so few photons that a bigger bucket is the only way to collect enough in a night. The 6.5 m James Webb Space Telescope has 18 hexagonal gold-coated beryllium segments and about 25.4 m² of collecting area. The Extremely Large Telescope, being built by ESO on Cerro Armazones in Chile, will have a primary mirror of about 39 m; ESO's plan, as listed in 2025, gives technical first light in March 2029. Interactive · How big a bucket? Aperture (diameter) Ideal circular aperture, ignoring central obstructions and gaps between segments (JWST's real area is 25.4 m², less than the ideal circle). Magnitude gain compares with a 7 mm pupil. Without JavaScript: a 200 mm telescope collects about 820 times as much light as a 7 mm pupil (a gain of 7.3 magnitudes); Hubble's 2.4 m collects about 117,000 times (12.7 magnitudes); a 39 m mirror collects about 31 million times. Aperture is not the only thing that counts: detectors matter as much. Photographic plates, the astronomer's detector for a century, record only a few percent of the light that hits them. The charge-coupled device (CCD), invented at Bell Labs in 1969 by Willard Boyle and George Smith (Nobel Prize 2009), records most of it. Modern astronomical CCDs have quantum efficiencies above 90% at their best wavelengths. Switching from plates to CCDs was like building a telescope several times larger for free. Resolution: diffraction, the atmosphere and the Earth-sized dish Light collected is half the story. The other half is sharpness: how close two points can be and still look like two. Light is a wave, so a perfect lens still smears a point of light into a small disc with rings, the Airy pattern. The smallest angle you can separate is set by the wavelength λ and the aperture D. θ ≈ 1.22 · λ / Dthe Rayleigh criterion, in radians; multiply by 206,265 for arcseconds Two things follow. A bigger aperture is sharper, and a longer wavelength is blurrier. A 200 mm telescope in green light has a diffraction limit near 0.7 arcsecond. Hubble at the same wavelength: 0.058 arcsecond. A radio dish at 21 cm needs to be enormous to match: even a 40 m dish at 21 cm resolves only about 22 arcminutes, two-thirds of the width of the full Moon. On the ground there is a second limit. Air is turbulent, and turbulence blurs a star's image to about an arcsecond ("seeing") at a good site, whatever the mirror's size. Past roughly 12 cm of aperture, a ground-based optical telescope is limited by the sky, not by its mirror. Two cures exist: go above the air (Hubble, JWST), or measure the blur many times a second and flex a deformable mirror to cancel it (adaptive optics), often with a laser creating an artificial guide star. Interactive · How sharp can it ever be? Aperture Wavelength Where Seeing is taken as 1 arcsecond, a typical good site, combined with the diffraction limit in quadrature. Adaptive optics is treated as ideal, which real systems approach only partly and mostly in the infrared. Without JavaScript: in visible light a 12 cm telescope has a diffraction limit of 1.2 arcseconds, about equal to ground seeing; Hubble reaches 0.058 arcsecond and the Moon feature it could resolve is about 110 m across. The way round diffraction is to join telescopes. Combine signals from dishes thousands of kilometres apart (very long baseline interferometry) and the "aperture" becomes the separation between them. The Event Horizon Telescope used dishes across the Earth at 1.3 mm, giving a diffraction-limited resolution of roughly 25 microarcseconds: enough to image the shadow around the black hole in the galaxy M87 (announced 10 April 2019) and the one at the centre of our own galaxy (12 May 2022). The radio telescope FAST in Guizhou, China, completed in 2016, has a 500 m dish (about 300 m used at once) and a beam about 2.9 arcminutes wide at 21 cm, a good example of a very big collector that is not a sharp one. Magnification: the least important number A telescope has two lenses or mirrors: the objective that gathers light and makes an image at its focal length F, and an eyepiece of focal length f that magnifies the image. Magnification is just their ratio. Galileo's spyglass of 1609 had about 20 times; he used it to see the Moon's mountains and four moons of Jupiter, published in Sidereus Nuncius (1610). Hans Lipperhey of Middelburg had applied for a patent on a spyglass in the Netherlands in 1608, and who really invented it is argued over. magnification = F / f  ·  exit pupil = D / magnificationthe exit pupil is the width of the beam that leaves the eyepiece; it should fit your pupil (about 7 mm in the dark, less for older eyes) Refractors bend light with a lens and suffer from colour fringes, because glass bends blue more than red; the largest ever, at Yerkes (102 cm, 1897), is close to the practical limit because a big lens can only be held at its edge and sags. Isaac Newton built a reflecting telescope about 1668, using a curved mirror, which has no colour fringes and can be supported from behind. Every large research telescope since is a reflector. Higher power is not better. Beyond about twice the aperture in millimetres, the image just gets larger and dimmer, because diffraction and seeing set the detail. A very short focal-length eyepiece may need more power than the sky allows. Interactive · Choose an eyepiece Eyepiece focal length Assumes an eyepiece with a 52° apparent field, typical of a simple design. The rule of "twice the aperture in millimetres" is a common guideline, not a law; steady air can allow more. Without JavaScript: a 200 mm, 1,200 mm focal-length telescope with a 25 mm eyepiece gives 48 times, an exit pupil of 4.2 mm and a field of about 1.1°; with a 4 mm eyepiece it gives 300 times, near the practical limit of about 400 for this aperture. The spectrum: temperature written in colour Visible light is a narrow slice of the electromagnetic spectrum, which runs from gamma rays (wavelengths smaller than an atom) to radio waves (longer than a football pitch). Different parts need different telescopes, and the atmosphere lets only some through. The window is open for visible light, near-infrared patches and radio waves from about a centimetre to ten metres or so; gamma rays, X-rays and most ultraviolet and infrared are absorbed. That is why we launch telescopes, and why ALMA sits on a dry Chilean plateau at 5,000 m. Interactive · Walk the spectrum Wavelength Frequency = c / wavelength; photon energy = hc / wavelength. Band edges are conventional and differ slightly between books; atmospheric transmission depends on altitude and weather. Without JavaScript: green light at 550 nm has a frequency of about 545 THz and a photon energy of 2.3 eV; the 21.1 cm hydrogen line is 1.42 GHz; an X-ray at 1 nm has about 1,240 eV. Every hot, dense object glows with a smooth spectrum whose shape depends on one thing, its temperature. This is blackbody radiation, which Max Planck explained in 1900. Two simple results hold for stars. Wien's law (1893) says the peak wavelength is inversely proportional to temperature, so hotter means bluer. The Stefan–Boltzmann law says the power per unit area goes as T⁴, so a star's luminosity is its surface area times T⁴. λpeak = 2.898 × 10⁻³ m·K / T  ·  L = 4πR²σT⁴so L/L☉ = (R/R☉)² (T/5772 K)⁴, using the IAU's nominal solar temperature Interactive · Make a star Surface temperature Radius (in Suns) The swatch integrates the Planck curve against fitted CIE 1931 colour-matching functions (Wyman, Sloan and Shirley 2013) and converts to sRGB, brightest channel scaled to full. It is an approximation; screens differ and your eye adapts, so the Sun looks white. Presets use approximate literature values. Without JavaScript: the Sun (5772 K) peaks near 502 nm; a 3,600 K red supergiant such as Betelgeuse peaks near 805 nm in the infrared and, at about 760 solar radii, shines roughly 90,000 times brighter than the Sun; a 25,000 K white dwarf peaks in the ultraviolet near 116 nm. Spectral lines: the barcode of atoms Spread the light through a prism or grating and a star's rainbow is crossed by dark lines. Joseph von Fraunhofer mapped hundreds in sunlight in 1814. Kirchhoff and Bunsen explained them in 1859 with three rules. A hot, dense body glows with a continuous spectrum. A hot, thin gas glows only at certain wavelengths: an emission spectrum. And a cooler gas in front of a hot source takes those same wavelengths out of the continuum: an absorption spectrum. Every element has its own set. The reason came from quantum mechanics (see Quantum Mechanics, 34/83). An atom has only certain energy levels, and a photon is emitted or absorbed when an electron jumps between them. For hydrogen the jumps down to level 2 make the visible Balmer series, found by Johann Balmer in 1885 and explained by Bohr in 1913: 1/λ = RH (1/2² − 1/n²),   n = 3, 4, 5, …RH ≈ 1.0968 × 10⁷ per metre; n = 3 gives the red H-alpha line near 656 nm Interactive · Read the barcode Wavelengths in air (nm). Hydrogen lines are computed from the Rydberg formula with a standard air correction; the others are standard laboratory values. Colours are approximate. Without JavaScript: hydrogen's Balmer lines fall at about 656.3, 486.1, 434.1 and 410.2 nm; sodium's two D lines at 589.0 and 589.6 nm; calcium's H and K lines near 397 and 393 nm. Line strengths give more than composition. A star's temperature and pressure decide which ionisation states exist and which levels are occupied, so the pattern of lines orders stars into the spectral classes O, B, A, F, G, K, M (hot to cool), worked out at Harvard around 1900 and refined by Annie Jump Cannon. In 1925 Cecilia Payne showed from the lines that stars are mostly hydrogen and helium. Nebulae show strong lines of doubly ionised oxygen, a green glow first mistaken for a new element, "nebulium", until Ira Bowen explained it in 1927. Doppler and redshift: motion written in the lines A source moving toward you has its light squeezed to shorter wavelengths (a blueshift); one moving away is stretched (a redshift). For speeds well below light, the fractional shift equals the speed along the line of sight divided by the speed of light. Δλ / λ = v / c  (v ≪ c)  ·  z = (λobserved − λrest) / λrestfor large speeds use 1 + z = √((1 + v/c) / (1 − v/c)) Because the rest wavelength of every line is known from the lab, the shift gives a speed to a precision no ruler could match. Two big results rest on it. In 1995 Michel Mayor and Didier Queloz found a planet around 51 Pegasi (Nobel Prize 2019, shared) from the star's tiny back-and-forth wobble: Jupiter makes the Sun move at about 12.5 m/s, an Earth at about 9 cm/s, which at 550 nm is a shift of only about 2 × 10⁻⁵ nm for Jupiter. And in the 1920s Edwin Hubble and others found that nearly every distant galaxy is redshifted, more so the farther it is. That second kind of redshift is not quite Doppler. In an expanding universe the wavelength is stretched along with space, and the redshift z gives the factor (1 + z) by which the universe has grown since the light left. The most distant galaxy confirmed with a spectrum, as of 2025, MoM-z14, has z = 14.44, so its light has stretched by a factor of about 15.4. Interactive · Shift the line Speed along the line of sight (km/s, + is away) Cosmological redshift z Rest wavelength of H-alpha taken as 656.3 nm (air). The lower slider multiplies wavelengths by 1 + z; it is not a speed. JWST covers 0.6 to 28.5 µm. Without JavaScript: a source receding at 300 km/s moves H-alpha from 656.3 to 657.0 nm, and Earth's own 30 km/s orbit moves it by 0.07 nm. At z = 2 H-alpha is at 1.97 µm and at z = 14.44 it is at 10.1 µm, deep in the mid-infrared where JWST's MIRI instrument works. This is the practical reason for infrared telescopes. The light from the first galaxies, emitted as visible and ultraviolet, has been stretched into the infrared; to see it you need a cold telescope above the atmosphere, which is what JWST (launched 25 December 2021, first science images released 11 July 2022) is. The Hubble tension and what the lines say about the universe's expansion belong to Astrophysics and Cosmology (34/88). New messengers and better detectors Light is no longer the only messenger. On 14 September 2015 the LIGO detectors recorded gravitational waves from two merging black holes (announced 11 February 2016; Nobel Prize 2017 to Rainer Weiss, Barry Barish and Kip Thorne). On 17 August 2017 detectors saw a merger of two neutron stars, and telescopes across the spectrum saw the gamma-ray burst and the afterglow: the first event caught in both gravitational waves and light. Neutrinos from the supernova SN 1987A were seen at Kamiokande in Japan on 23 February 1987, and IceCube reported a high-energy neutrino traced to a blazar in 2018. Astronomers call the combination multi-messenger astronomy. Light itself is being collected in new ways. The Vera C. Rubin Observatory in Chile, with an 8.4 m mirror and a 3.2-gigapixel camera, released its first images on 23 June 2025, and its ten-year survey of the southern sky was scheduled to start in mid-2026. The data volumes there are why Astronomical Data & Computation (36/44) exists. What a graduate just sees A telescope is first a light bucket: gathering power goes as the square of the aperture. Sharpness is wavelength over aperture, and on the ground the air caps it near one arcsecond. Magnification is the cheapest number; past about twice the aperture in millimetres it only enlarges blur. The atmosphere blocks most of the spectrum, so astronomy happens at the windows or in space. Colour is temperature, and luminosity is surface area times T⁴. Lines are the fingerprint of elements, and a line's shift is a speed. Redshift on a cosmic scale is stretching, and it moves visible light into the infrared. Gravitational waves and neutrinos open channels light cannot. Where it shows up next Stars: Structure & Evolution (36/41): a star's colour, line pattern and luminosity place it on a diagram that records its life story. Planetary Science & Exoplanets (36/40): radial-velocity wobbles and transit dips are the same spectra and photometry, pushed to extremes. Compact Objects & High-Energy Astrophysics (36/43): X-ray and gamma-ray satellites, and gravitational-wave detectors. Astronomical Data & Computation (36/44): turning pixels, spectra and light curves into numbers. In the world: the same optics runs cameras, the same spectra identify chemicals in labs and factories, and adaptive optics now corrects eyes and lasers. If you want to go deeper 📘 Textbook. Bradley Carroll and Dale Ostlie, An Introduction to Modern Astrophysics (Cambridge University Press, 2nd edition 2017), chapters on light and telescopes and on stellar spectra; for observing techniques, Chromey, To Measure the Sky (Cambridge University Press, 2nd edition 2016). 📘 Gentler. Dava Sobel, The Glass Universe (2016), the story of the Harvard women who classified stellar spectra; and the free OpenStax Astronomy 2e. 🌐 Free text. OpenStax Astronomy 2e, chapter 6 "Astronomical Instruments", at openstax.org. 🌐 Free course. MIT OpenCourseWare 8.282J Introduction to Astronomy (Spring 2006). The takeaway Light carries almost everything we know about the sky. Aperture sets how faint, wavelength over aperture sets how sharp, the continuum sets temperature, the lines set composition, and their shifts set speed. Each bigger mirror, better detector and new window has changed what astronomers could say. Try this: look at a bright star and a planet on the same night. Notice the colours of two bright stars (Betelgeuse and Rigel in Orion, or Antares and Arcturus), and write which is cooler. Then look at a street lamp through a CD held at an angle: you are using a spectrograph. Sources & Further Reading Sources. Rayleigh criterion and Wien, Planck, Stefan–Boltzmann laws as in standard texts (Carroll and Ostlie 2017); Kirchhoff and Bunsen 1859–60; Fraunhofer 1814; Balmer 1885; Bohr 1913; Edlén's 1966 formula for the refractive index of air; Wyman, Sloan and Shirley, "Simple analytic approximations to the CIE XYZ color matching functions," Journal of Computer Graphics Techniques 2013; IAU 2015 Resolution B3 (nominal solar values); CODATA 2018 constants; Mayor and Queloz, Nature 1995; Boyle and Smith 1970 and the 2009 Nobel Prize in Physics; Abbott et al. (LIGO and Virgo), Physical Review Letters 2016 and 2017; NASA and ESA mission pages for JWST (launch 25 December 2021; first images 11 July 2022; 25.4 m²; 0.6 to 28.5 µm); ESO pages for the ELT (39 m; first light planned March 2029); the Event Horizon Telescope Collaboration 2019 and 2022; Vera C. Rubin Observatory (first look 23 June 2025); the MoM-z14 redshift (z = 14.44, 2025). All numbers in the interactives were computed in node. This piece takes no side in open questions such as the Hubble tension, which it only names. 📘 Carroll and Ostlie, An Introduction to Modern Astrophysics 📘 Sobel, The Glass Universe 🌐 MIT OCW 8.282J Step 7. Special Relativity: Everyone Gets the Same Speed of Light ----------------------------------------------------------------- Why: Special relativity: fast particles and the speed limit. Why your phone's map needs Einstein, why cosmic-ray muons reach the ground, and how a single odd fact about light stretches clocks, shrinks rulers and turns mass into energy. The satellites that tell your phone where it is carry atomic clocks, and those clocks run at the wrong rate. Seen from the ground, they tick about 38 microseconds per day too fast: roughly 7 microseconds slower because they are moving, and roughly 46 faster because they sit higher in Earth's gravity. Left uncorrected, your position would drift by around ten kilometres a day. The engineers fix it by building the correction into the clocks' design and software. Here is a quieter proof. High in the atmosphere, cosmic rays crash into air molecules and make muons, particles that fall apart in about 2.2 millionths of a second on average. Even at nearly the speed of light, that should let a muon travel only about 660 metres. Yet muons made 15 kilometres up reach sea level in huge numbers. Their short lives are stretched because they are moving. Special relativity is the theory behind both. It starts from one fact that sounds harmless, and follows it with mathematics that never once breaks. A physics graduate comes out of this course able to feel the consequences in their bones: time, length and simultaneity are not what they seemed, and the thing everybody does agree on is a different quantity altogether. The course in one box Who takes it: physics majors, usually in year 2 or 3 (MIT's 8.033 Relativity sits after Vibrations and Waves; Cambridge Natural Sciences covers it in its physics course). Every first-year Physics sequence also teaches a short version; engineers meet its results in particle accelerators, GPS and radiation. What it's for: working out what happens when speeds approach that of light, and understanding why the laws of electromagnetism are what they are. Comes after: Physics I: Mechanics (34/06) and Linear Algebra (34/04), because the Lorentz transformation is a matrix. Leads to: General Relativity (34/85), where gravity bends spacetime, using Differential Geometry (34/77); Particle & Nuclear Physics (34/86), where E = mc² is daily bread; and the relativistic side of Electromagnetics & Waves (34/38). Quantum Mechanics (34/83) meets it again in the Dirac equation. Honest line: you won't be able to solve a general-relativity problem after this, or design a collider. You will be able to say how much slower a clock runs at 0.9 c, why two events can be in a different order for two observers, and what E = mc² does and doesn't say. Two rules and a flashlight on a train Special relativity rests on two postulates. First: the laws of physics are the same for everyone moving at a steady speed. (Drop a ball on a smooth-flying aeroplane and it falls just as it does on the ground; nothing inside tells you that you are moving.) Galileo understood that in 1632. Second, the new one: light travels at the same speed, c, for every observer, no matter how the source or the observer moves. The speed is 299 792 458 m/s, exactly, because since 1983 the metre has been defined from it. The second rule collides with common sense. If you throw a ball forward at 20 m/s from a train going 30 m/s, the ground sees 50. But flash a torch forward on a train going nearly at c, and the ground still sees light at c, not 2c. The old rule of adding speeds must be only a good approximation. The exact rule, from the new postulates, is: w = (u + v) / (1 + uv/c²)speed seen from the ground = (speed on the train + speed of the train) ÷ (1 + their product ÷ c²) At everyday speeds uv/c² is vanishingly small and you get u + v back. At high speeds the denominator holds the total below c. Two spaceships each going 0.9 c towards the ground, one launching from the other, do not reach 1.8 c; the result is 0.9945 c. And if u is c, then w = c, whatever v is. The postulate was not pulled from nowhere. In 1887 in Cleveland, Albert Michelson and Edward Morley compared light travelling along two perpendicular arms of an interferometer, hoping to detect Earth's motion through the supposed ether that carried light. They found essentially nothing. Whether Einstein had that result in mind in 1905 is disputed by historians; he gave different accounts later. What is not disputed is that Hendrik Lorentz and Henri Poincaré had much of the mathematics before him, and that Einstein's contribution was to see it as a fact about space and time themselves, with no ether needed. Interactive · Adding velocities 0.5 c + 0.5 c0.9 c + 0.9 cTorch beam on a 0.9 c train Speed on the train (u) Speed of the train (v) Exact relativistic velocity addition, computed live. The line marks c. The old sum (grey) crosses it; the correct one never does. Moving clocks run slow Take the simplest clock imaginable: a pulse of light bouncing between two mirrors, one tick per round trip. Now watch it fly past you sideways. In its own frame the light goes straight up and down. In your frame, because the clock moves while the light is in flight, the light zig-zags along a longer, diagonal path. But it must still go at c. A longer path at the same speed takes more time, so each tick lasts longer for you than for the people riding with the clock. Moving clocks run slow. The size of the effect follows from Pythagoras applied to that triangle. It is captured by a single factor, which appears in nearly every formula in the subject: γ = 1 / √(1 − v²/c²)the Lorentz factor; a moving clock ticks γ times more slowly than yours Some values: at 0.1 c, γ = 1.005; at 0.5 c, 1.155; at 0.9 c, 2.29; at 0.99 c, 7.09; at 0.999 c, 22.4. It is close to 1 for anything in daily life, which is why we never see it. Aeroplanes cross oceans with their clocks off by nanoseconds. And the muons: take a typical cosmic muon at 0.995 c, so γ = 10.0, created 15 km up (an illustrative height). It needs 50 microseconds in our frame to reach the ground, which is 22.9 of its average lifetimes of 2.197 µs. The chance of surviving that without time dilation is e−22.9, about one in nine billion. With dilation, only 2.29 lifetimes pass in the muon's own time, and about 10% survive. The 1941 experiment by Bruno Rossi and David Hall on Mount Washington counted the extra muons, and it is the famous one, though the authors presented it in qualitative terms; Frisch and Smith's 1963 measurement is usually credited with the clean confirmation. It is not a trick of the muon. Atomic clocks flown around the world on airliners (Hafele and Keating, 1971) and the clocks in every GPS satellite show the same thing. Everything about the moving thing runs slow, including its biology, if it had any. Interactive · The light clock 0.5 c0.9 c0.995 c (muon) Speed of the clock Two light pulses leave together; the straight one is the clock's own tick, the slanted one is what you see. Vertical gap drawn smaller at high speed so the triangle fits; the proportions are exact. Simultaneity depends on who is looking The next consequence is stranger. Two events that happen at the same moment for you may happen one after the other for someone moving relative to you. Picture lightning striking both ends of a moving train at once, as judged from the platform. A passenger in the middle, moving towards one flash and away from the other, sees the light from the front flash first: it has less distance to cover and she is running into it. Since light has the same speed for her, she concludes that the front strike happened first. Neither is wrong. The arithmetic is the Lorentz transformation. For two events separated by a distance Δx and a time Δt, an observer moving at v finds a time separation Δt′ = γ(Δt − vΔx/c²) and a distance Δx′ = γ(Δx − vΔt). Worked: two flashes 3 light-seconds apart, simultaneous for you; for an observer moving at 0.6 c (γ = 1.25), they are 2.25 seconds apart, with the order reversed if she runs the other way. The same transformation shrinks lengths. A rod measured while it moves along its length is shorter than when at rest, by the same factor γ: a 100 m spaceship at 0.8 c measures 60 m for you, and at 0.99 c only 14 m. It is not that the ship is squashed on its own terms; it is that "its length" requires marking both ends at the same instant, and simultaneity is what moved. You cannot rescue common sense by clever tricks, but there is a limit on how bad things get. Events that could be linked by a signal at c or slower, such as one causing the other, keep their order for everyone. Events too far apart for light to connect (a "spacelike" separation) can swap order for some observers, but they can never influence each other. The speed of light is not an engineering limit; it is the thing that keeps cause ahead of effect. Interactive · A spacetime diagram Two flashes, same momentCause and effectLight signal Distance between events (Δx) Time between events (Δt) Observer's speed (v) Axes: distance in light-seconds across, time in seconds up, so light rays are the 45° lines. Orange: the moving observer's axes; dashed lines are her "same moment" lines through each event. Event A is at the origin. Proper time: the clock you carry If time and distance both change with viewpoint, what is real? There is a combination that all observers agree on. For two events, the squared interval is c²Δt² − Δx² (the time part minus the space part). It is the same number whichever frame you compute it in, which the diagram above shows in its readout. For a clock moving along a path, the sum of intervals along its route is the time it actually records: its proper time. Δτ = Δt / γproper time on a moving clock = time in your frame ÷ the Lorentz factor This is the cure for the famous twin "paradox". One twin stays; the other rockets to a star and returns. Each sees the other's clock run slow, so who is younger? The answer is the traveller, because only the traveller accelerated: she turned around, changing frames, so the situation is not symmetric. Compare paths, and the one that stays in a single frame covers the longest proper time. The numbers can be dramatic: to a star 4.25 light-years away at 0.99 c, one leg takes 4.29 years on Earth's clocks and 0.61 years on the ship's. Go fast enough (γ = 1000) and you reach the centre of the galaxy in 26 years of ship time, while about 26,000 years pass at home. Do not read this as a way to cheat death for free. The ship's journey needs energy, which is the next section's subject. Interactive · One-way trip planner Nearest star, 0.99 cGalactic centre, γ = 1000Andromeda, γ = 10 000 Distance Lorentz factor γ Constant speed, acceleration ignored. Earth time = D/v, ship time = D/(γv). Distances are rounded: nearest star about 4.25 ly, galactic centre about 26 000 ly, Andromeda about 2.5 million ly. Energy per kilogram is the kinetic energy (γ − 1)c². Energy, momentum and E = mc² Push on something and it speeds up, but never past c. Where does the extra effort go? Into energy, and the theory's accounting says energy has weight. Momentum is p = γmv and the total energy of a moving object is E = γmc². Stand still (γ = 1) and an object still has E = mc²: rest energy. The kinetic energy is the rest of it, (γ − 1)mc². The tidy relation tying them is: E² = (pc)² + (mc²)²total energy squared = (momentum × c) squared + (rest energy) squared For small speeds it reduces to Newton's ½mv² plus the constant mc². For light itself, which has no mass, E = pc. Rest energy is huge: 1 kg corresponds to c² = 9.0 × 10¹⁶ joules, about 21 megatons of TNT. That is why a nuclear reaction that converts even a tenth of a percent of mass releases so much energy. The Sun converts about 4.26 million tonnes of mass into light every second (its luminosity 3.828 × 10²⁶ W, divided by c²). The slider below pushes an electron or proton with a given amount of kinetic energy. Newton says that a 1 MeV electron should travel at nearly twice the speed of light; in fact it reaches 0.941 c. To get to 0.999 c needs about 10 MeV; 1 GeV gives 0.99999987 c. In the Large Hadron Collider, each proton carries 6.8 TeV (since 2022) and has γ = 7247. It travels only about 3 metres per second slower than light. Almost all the extra energy goes into γ, not into speed. Interactive · Pushing a particle ElectronProton TV tube, 25 keV1 MeV electronProton at the LHC, 6.8 TeV Kinetic energy Electron rest energy 0.51099895 MeV, proton 938.27209 MeV (CODATA 2018). Solid line: relativistic speed; dashed: Newton's ½mv². The orange dots mark your setting on each. What a graduate just sees Light has the same speed for every observer, so speeds don't add the old way: w = (u+v)/(1+uv/c²). A moving clock ticks γ times slower; γ is 1.005 at 0.1 c, 2.3 at 0.9 c, and 22 at 0.999 c. Two events that are simultaneous for you needn't be for someone moving; the order of unrelated events is a matter of viewpoint. Cause always precedes effect for everyone, because nothing carries a signal faster than light. What everybody agrees on is the interval c²Δt² − Δx², and a clock's own (proper) time. The twin who turns around is the younger one: acceleration breaks the symmetry. E = mc² says rest mass is a store of energy; as speed climbs, the energy goes into γ, not into going faster. At everyday speeds all of this vanishes into Newton, which is why Newton was right for 200 years. Where it shows up next General Relativity (34/85): drop the restriction to steady motion and gravity becomes the curvature of the spacetime you have just met; the muon and GPS stories both carry a gravity half. Particle & Nuclear Physics (34/86): collisions are computed with four-vectors, mass is the invariant, and every particle-decay lifetime in the tables is a proper time. Electromagnetics & Waves (34/38): magnetism is what electricity looks like from a moving frame; Maxwell's equations were already relativistic before relativity. Quantum Mechanics (34/83) and the Standard Model (13/07): Dirac combined the two in 1928 and predicted antimatter. The real world: GPS and other satellite navigation, PET scanners (matter-antimatter annihilation), synchrotrons, and electron microscopes, whose beams need relativistic corrections at 100 kV and more. If you want to go deeper 📘 Textbook: Edwin Taylor and John Archibald Wheeler, Spacetime Physics (2nd edition, W. H. Freeman, 1992): built around the invariant interval, and free to download from the authors' site. David Morin's Special Relativity for the Enthusiastic Beginner is a friendly, problem-rich alternative. 📘 Gentler: The Feynman Lectures on Physics, Vol. I, chapters 15–17 (relativity, space-time, and relativistic dynamics), free online. Or this library's Why Time Is Weirder Than You Think (13/03). ▶️ Free course: MIT OpenCourseWare 8.033 Relativity (Fall 2006). 🌐 Easier first reads in this library: Why Time Is Weirder Than You Think (13/03) and The Big Bang and Our Expanding Universe (13/08); in this series Electromagnetics & Waves (34/38). The takeaway Make the speed of light the same for everyone, and everything else bends to fit: moving clocks run slow by γ, moving rods shrink by γ, simultaneity depends on the observer, and energy and mass are two faces of one thing. What stays fixed is the interval, the clock you carry, and the order of cause and effect. At daily speeds γ is so close to 1 that Newton was enough; near c, nothing works without it. Try this: next time your phone shows your blue dot on a map, remember there are clocks overhead being corrected by 38 microseconds a day, and that without the correction the dot would wander off by kilometres. Then compute γ for a jet at 250 m/s: it is 1 + 3.5 × 10⁻¹³, which is why nobody ever noticed. Sources & Further Reading Sources. Lorentz transformation, velocity addition, E² = (pc)² + (mc²)² and the twin analysis are standard textbook material (Taylor and Wheeler; Feynman Lectures I). c = 299 792 458 m/s: exact since the 1983 definition of the metre. Michelson and Morley, "On the relative motion of the Earth and the luminiferous ether", American Journal of Science (1887); historians disagree on how much it influenced Einstein, whose paper "Zur Elektrodynamik bewegter Körper" appeared in Annalen der Physik in 1905. Muon experiments: Rossi and Hall, Physical Review 59 (1941); Frisch and Smith, American Journal of Physics 31 (1963) (see Wikipedia, "Experimental testing of time dilation"). Muon mean lifetime 2.197 µs (Particle Data Group). GPS: N. Ashby, "Relativity in the Global Positioning System", Living Reviews in Relativity 6 (2003), 1; we recomputed the orbit effects in node: about −7.2 µs/day for orbital speed and +45.7 µs/day from gravity, a net of about +38 µs/day, in line with the usual figure. Proton energy at the LHC: CERN, Run 3 (6.8 TeV per beam, since 2022). Solar luminosity: IAU 2015 nominal value. CODATA 2018 for particle masses. Computed in node: every γ, the muon survival fractions, trip times, the 2.25 s simultaneity shift, and all slider values. 🌐 Taylor and Wheeler, Spacetime Physics, 2nd edition (authors' site) 🌐 MIT OpenCourseWare 8.033 Relativity 🌐 Experimental testing of time dilation (overview of the muon and clock tests) Step 8. Planetary Science & Exoplanets: Worlds by the Thousands, Read from a Dip in Light ----------------------------------------------------------------------------------------- Why: The Solar System and the thousands of worlds around other stars. Nobody has ever photographed a typical exoplanet. Yet more than six thousand are known, and we can say how hot they are, what they are made of and why we find the ones we find. Six ideas make that possible. Look west after sunset on the right evening and a steady white point stands above the horizon. It does not twinkle like the stars around it. That is Venus, a world with a surface hot enough to melt lead, shining only because it reflects sunlight. Babylonian scribes logged its comings and goings on clay tablets, and the Maya built a calendar table around its 584-day cycle. For all of human history the planets were the five or six wanderers you could see, and a puzzle about the sky. In October 1995 Michel Mayor and Didier Queloz announced a planet around a Sun-like star, 51 Pegasi. It was about half the mass of Jupiter and went round its star in 4.2 days. Nobody had expected such a thing, and they shared the 2019 Nobel Prize in Physics for it. As of 2 November 2026, the NASA Exoplanet Archive lists 6,451 confirmed planets. Almost none was seen as a picture. Most were found as a star dimming by a fraction of a per cent, or a star's speed changing by metres per second. This piece is about how to read a world from almost nothing. It starts with the clock every orbit obeys, then asks how hot a planet is, what it is made of, where planets come from, how they are found, and what the catalogue does and does not tell us. The course in one box Who takes itAstronomy and astrophysics majors in year 2 or 3, and planetary-science students in Earth and planetary science departments, which also teach it as an introductory survey. Physics majors take it as an option. What it is forTurning orbits, brightness and spectra into the size, temperature, composition and history of planets, in our Solar System and around other stars. Comes afterClassical Mechanics (34/82) and Earth Systems (36/13). Leads toStars: Structure & Evolution (36/41), because a planet's heat and orbit are set by its star, and Astronomical Data & Computation (36/44), where transit and velocity signals are dug out of noise. What this piece gives youYou will not be able to fit a planet to real data. You will see why a year lasts as long as it does, how to estimate a planet's temperature on the back of an envelope, why density gives away composition, and why the planets we find are a biased sample. Easier first reads: Orbits: Why the Moon Keeps Falling (13/11), The Physics of the Greenhouse Effect (13/15) and Astrophysics & Cosmology (34/88), which this piece goes beyond rather than repeats. Kepler's third law: every orbit has a clock Johannes Kepler published the third of his laws in 1619, after years of working through Tycho Brahe's measurements of Mars. It says that the farther a planet is from its star, the longer its year, and in a very particular way: the square of the period grows as the cube of the distance. Newton later showed why. Gravity weakens with the square of distance, and a heavier star pulls harder, so a planet around a heavy star goes round faster. P² = a³ ÷ MP in years, a in astronomical units (Earth–Sun distances), M in Sun masses Worked: Neptune is 30.07 AU out. The cube is about 27,200, and its square root is 165, so Neptune takes about 165 years. That is correct: it completed its first lap since its 1846 discovery in 2011. Exoplanets use the same law backwards. A planet that crosses its star every 4.2 days, around a star of 1.1 Suns, must orbit at 0.053 AU, eight times closer than Mercury is to our Sun. That is how a world half the mass of Jupiter ended up cooking at the temperature of a lava lake. Interactive · Choose a star and a distance MercuryEarthJupiterNeptuneTRAPPIST-1 b51 Peg b Distance Star mass Gold dots: the eight planets of our Solar System. Blue line: the law for the chosen star. Circular orbits; P² = 4π²a³ ÷ (G × total mass), planet mass neglected. TRAPPIST-1 b: 0.0898 Sun masses, 0.01154 AU (Agol et al. 2021). Without JavaScript: Earth, 1 AU, 1 year. Jupiter, 5.2 AU, 11.9 years. Neptune, 30 AU, 165 years. A planet at 0.012 AU around a star of 0.09 Suns has a year of 1.5 days. Notice the straight line on the log chart. A power law is a straight line when both axes are logarithmic, and the Solar System's planets sit on one. Change the star's mass and the whole line shifts, which is why a 1.5-day year around a dim red dwarf is not a contradiction. Equilibrium temperature: sunlight in, heat out A planet is a rock in a stream of light. It absorbs some, reflects the rest, and warms until it radiates away exactly as much as it takes in. That balance fixes a temperature, and the only inputs are the star's brightness, the distance and the reflectivity, called the albedo. T = 278 K × L¼ × (1 − A)¼ ÷ √aL star luminosity in Suns, A fraction of light reflected, a distance in AU; heat shared evenly around the planet The fourth roots are the surprise. Move a planet twice as far away and the sunlight drops to a quarter, but the temperature drops only by the square root of two, to 71%. Worked for Earth: albedo 0.30 gives 278 × 0.915 = 255 K, or −18 °C. The real global mean is about 288 K. The 33 degrees between them is the greenhouse effect (see 13/15), and it is the difference between a frozen world and a living one. Venus is the warning. With a thick cloud deck, its albedo is about 0.76, and the formula gives only 229 K, colder than Earth's. Its surface is about 737 K. The equilibrium temperature tells you what a bare, airless rock would do; the atmosphere can add anything from a few degrees to hundreds. Interactive · How warm is this world? EarthVenusMarsTRAPPIST-1 eJupiter Distance Reflectivity Star brightness Equilibrium temperature of a bare rock, T = (L(1−A) ÷ 16πσa²)¼, which gives 278.3 K at 1 AU for A = 0. Preset albedos are Bond albedos from the NASA planetary fact sheets, rounded; TRAPPIST-1 e uses A = 0 and L = 0.000553 (Agol et al. 2021), so 250 K is an upper bound for a bare rock. Without JavaScript: Earth 255 K without a greenhouse (288 K with); Venus 229 K from sunlight but 737 K at the surface; Mars about 210 K. The so-called habitable zone is the band of distances where a rocky planet with an Earth-like atmosphere could have liquid water on its surface. Climate models that include how water vapour builds up put the Sun's zone at roughly 0.95 to 1.7 AU (Kopparapu and colleagues, 2013), which is starlight from about 1.1 down to 0.35 times Earth's. Those edges depend on the model, the atmosphere and the planet's spin. A planet in the zone is a candidate for an ocean, not a world known to have one. Density: what a world is made of You cannot weigh a planet on a scale, but you can weigh it by its pull on something else. Orbits give mass (Kepler again, for a moon or a neighbour), and a transit gives radius. Divide one by the volume of the other and you have the planet's mean density, which is the closest thing to a composition reading from a distance. density = mass ÷ ( ⁴⁄₃ π × radius³ )Earth: 5.51 g/cm³. In Earth units, density = 5.51 × mass ÷ radius³ The numbers sort worlds at once. Water is 1. Rock is around 3. Iron is nearly 8 before pressure squeezes it. Earth is 5.5 because rock wraps an iron core. Mercury is 5.4 because it has a bigger core. Saturn is 0.69, less than water, and would float in a big enough bathtub. Neptune is 1.6, which tells you it is mostly water, ammonia and methane ices under a hydrogen skin. The catch is that density is a single number and compositions are lists. A rocky planet with a deep ocean can match a rocky planet with a thick light atmosphere. Gravity compresses material, so a bigger planet is denser than the same stuff in a smaller one. The best honest statement is a range of mixtures. The other catch is the surprises. Kepler-51 d has about 5.6 Earth masses spread across about 9.5 Earth radii, giving 0.04 g/cm³, lighter than cork. Planets like it are called super-puffs, and how they form is open. Interactive · Weigh it, measure it MercuryEarthMarsNeptuneSaturnTRAPPIST-1 eKepler-51 d Mass Radius Radii are volumetric means. Values: NASA planetary fact sheets for Solar System planets; Agol et al. 2021 for TRAPPIST-1 e (0.692 Earth masses, 0.920 Earth radii); Libby-Roberts et al. 2020 for Kepler-51 d (masses are uncertain for this one). Without JavaScript: Earth 5.51 g/cm³, Mars 3.93, Neptune 1.64, Saturn 0.69, TRAPPIST-1 e about 4.9 (Earth-like), Kepler-51 d about 0.04. Where planets come from: the snow line and the great migration Planets are leftovers. A cloud of gas and dust collapses into a star, and the spinning disk around it clumps into grains, rocks, and then bodies the size of mountains and worlds. The Solar System's age comes from meteorites: the oldest dated grains, from primitive meteorites, give about 4.567 billion years, and the planets finished building within some tens of millions of years of that. What a body can be made of depends on the temperature of the disk where it grows. Iron and silicates survive almost anywhere inside a few AU. Water freezes only beyond the snow line, near 2.7 AU for the young Sun. Beyond it, solid material is much more plentiful, so growing cores get big quickly, big enough to capture hydrogen and helium gas before the disk disperses. That is the classic picture of rocky planets inside and giants outside, and our Solar System fits it. Interactive · Slide out through the disk Distance in the disk Star brightness A toy: dust heated only by starlight, T = 278 K × L¼ ÷ √a. Real disks have heating from inside, thick gas and changing temperature over time, so the real snow line moved. Freezing points at disk pressures are rounded: silicates and iron near 1,300 K, water ice near 170 K, CO₂ ice near 70 K, CO ice near 25 K. Without JavaScript: for the Sun, dust is at 170 K at 2.7 AU, so rock is available everywhere and water ice beyond about 2.7 AU. Then exoplanets broke the tidy picture. 51 Pegasi b, a Jupiter-mass planet sitting 0.05 AU from its star, should not have formed there: it is far inside the snow line, where there is not enough solid material. The usual explanation is migration: planets form farther out and move inward, dragged by the disk or thrown by gravitational encounters. The Solar System may have had its own shuffle. The Nice model (Tsiganis and others, 2005) has Jupiter and Saturn crossing a resonance that rearranged the outer planets, and the Grand Tack model (Walsh and others, 2011) has Jupiter moving in and then back out. Both are active research. Neither is settled. Another surprise appears in the radii. The Kepler data show a gap: far fewer planets have radii between about 1.5 and 2 Earth radii than just below or above (Fulton and others, 2017). Planets seem to come as bare rocky worlds or as rocky worlds wrapped in thick gas. Why the line falls there, and whether it comes from heat stripping atmospheres or from formation, is debated. The transit: a shadow measured in parts per million If a planet's orbit happens to line up with our line of sight, the planet crosses its star once per orbit and blocks a little of its light. The dip is the planet's shadow area over the star's area. A Jupiter-size planet crossing a Sun-size star hides about 1%. Earth hides 84 parts per million, about 0.008%. A telescope on the ground struggles below about 0.1%; space telescopes such as Kepler (launched 2009) and TESS (2018) reached tens of parts per million. depth ≈ (planet radius ÷ star radius)²to first order; limb darkening (a star is dimmer at its edge) makes a central transit deeper Two more things come free. The time between dips is the period, and with Kepler's law, the distance. The length of the dip depends on star size and orbit, and tells you the path the planet took across the star. The price is geometry: the orbit must be tilted just right. For a randomly tilted orbit the chance is about the star's radius divided by the orbit's radius, 0.47% for Earth around the Sun. For a hot Jupiter at 10 stellar radii it is 10%. That is why transit surveys need to watch tens of thousands of stars at once. Interactive · Cross a star Red dwarfOrange dwarfSunF star EarthJupiterHot JupiterTRAPPIST-1 b Planet radius Orbital period Path across the star Computed by integrating a limb-darkened stellar disk (quadratic law, coefficients 0.4 and 0.25, illustrative) minus the planet's disk. Stars: red dwarf 0.1 Sun masses and 0.12 radii, orange 0.7 and 0.7, F 1.3 and 1.4; circular orbits. Path 0 crosses the centre; 0.9 skims the edge. TRAPPIST-1 b's radius is about 1.1 Earth radii. Without JavaScript: Earth in front of the Sun, a dip of about 100 ppm lasting 13 hours once a year, with a 0.5% chance of being seen at all. A hot Jupiter dips about 1.3 to 2% for 3 hours every 3.5 days. Notice the red-dwarf button. A planet crossing a small star hides far more light, and its orbit at a given period is tighter, so both the dip and the odds improve. Most of the nearest rocky candidates are around red dwarfs for that reason. It is a statement about our instruments, not only about nature. The wobble: a star's speed measured in metres per second A planet does not orbit a motionless star. Both circle their common centre of mass, so the star traces a small circle too, and from far away you see its speed toward and away from you change. The change is read from the Doppler shift of lines in its spectrum, in the same way an ambulance siren shifts. 51 Pegasi b was found this way. K ≈ 28.4 m/s × (m ÷ Jupiter) × (P ÷ 1 yr)−⅓ × (Mstar ÷ Sun)−⅔K is the half-swing of the star's velocity; m sin i is the planet mass times the sine of its orbit's tilt The dependence is sensible. A heavier planet tugs harder. A planet on a shorter orbit pulls faster. A heavier star is harder to move. Jupiter makes the Sun swing by 12.5 m/s, about the speed of a good sprinter, and Earth by 0.09 m/s, a slow crawl. A 12.5 m/s shift on a 500 nm line is a change of 4 parts in a hundred million, or 0.00002 nm. Detecting it means calibrating a spectrograph to a few parts in a billion. The method has a built-in blind spot. You see only the motion along your line of sight, so what you measure is the planet's mass times the sine of the orbit's tilt, a minimum mass. A transit tells you the tilt is near 90°, and then the mass is the true mass. Combining the two methods gives mass and radius, so density, which is why the best-known planets are found by both. Interactive · Shake a star JupiterEarth51 Peg b Planet mass Orbital period Orbit tilt Star mass Circular orbits, K = 28.43 m/s × (m sin i ÷ MJup) × (P ÷ yr)−⅓ × (Mstar ÷ MSun)−⅔, Jupiter = 317.8 Earth masses. Instrument remarks are rough: good spectrographs such as HARPS in Chile reach about 1 m/s, and newer ones such as ESPRESSO aim lower, but the star's own surface activity adds noise of its own. Without JavaScript: Jupiter moves the Sun by 12.5 m/s, Earth by 0.09 m/s, and 51 Pegasi b moves its star by about 55 m/s. The census is biased: what the catalogue can and cannot say Now look at what has been found. The map below plots every confirmed planet in the NASA Exoplanet Archive that has a measured mass or minimum mass (3,229 of the 6,451), by its distance from its star and its mass. Choose a method and watch how the dots move. Transits and velocities crowd the top-left corner: big planets, close to their stars. Microlensing finds planets near 1 to 10 AU. Direct imaging finds only the heaviest planets, very far out. The bottom right, the place where our Earth and the other small planets of our Solar System would sit, is nearly empty. Interactive · Who finds what AllTransitRadial velocityMicrolensingImaging Data: NASA Exoplanet Archive, composite planetary systems table, queried 2 November 2026. Only planets whose mass is measured or a minimum mass from velocity data are included (3,229 in the plotted range); planets whose mass was only estimated from their radius are left out, which under-represents small transiting planets. Tap a cell for counts by method. Without JavaScript: of the 6,451 known planets, 4,785 were found by transits, 1,204 by radial velocity, 292 by microlensing and 98 by direct imaging. The empty corner is not evidence that small, cool planets are rare. It is the shape of what our tools can do. A transit needs a lucky tilt and a tiny dip; a wobble needs a tiny speed; imaging needs a planet bright enough and far enough from its star to separate. Astronomers correct for this with careful statistics, and the corrected answer is that planets are common: most stars appear to have them, and small planets outnumber giants. How many Earth-size planets sit in habitable zones is still uncertain, with published estimates running from a few per cent to more than half of Sun-like stars, depending on the definition of "Earth-size" and "habitable". And what counts as a planet at all? In August 2006 the International Astronomical Union voted at Prague that a planet orbits the Sun, is massive enough to be nearly round, and has cleared its orbit of other bodies. Pluto fails the third test and became a dwarf planet, along with Ceres and Eris. The IAU has no equally formal rule for exoplanets; its working guidance sets an upper limit near the mass where deuterium fusion starts, about 13 Jupiter masses. Finally, the frontier. The James Webb Space Telescope reads starlight filtered through a planet's atmosphere as it crosses its star, picking out water, carbon dioxide and methane. Claims of biosignatures need more care than any other result in the field. In 2023 a team reported a possible dimethyl sulfide signal in the atmosphere of K2-18 b; later analyses by other teams did not find the signal statistically convincing, and the question remains open. A single gas rarely proves life, because chemistry that needs no life can often make it. What a graduate just sees The square of a year is the cube of the distance, divided by the star's mass. Any period gives a distance. A planet's temperature goes as the fourth root of the sunlight and the inverse square root of distance, and the atmosphere changes it, sometimes enormously (Venus). Mass and radius give density, and density is the first clue to composition, though never the whole answer. Rock inside the snow line, giants outside it, and a lot of planets that moved. Depth is radius ratio squared; probability is star radius over orbit radius. Big, close planets are easy to find. A wobble gives only a minimum mass, until a transit supplies the tilt. Every catalogue is the product of its instruments. What is missing may be missing from the sky, or only from the telescope. Where it shows up next Stars: Structure & Evolution (36/41): a planet's temperature and the width of its habitable zone move as its star brightens. In about a billion years the Sun's brightness will have raised Earth's temperature enough to end liquid oceans. Astronomical Data & Computation (36/44): transit and velocity signals are found by fitting models to noisy light curves, and TESS delivers more than anyone can look at by eye. Telescopes, Light & Spectra (36/39): the transmission spectrum is the same spectroscopy, applied to a planet's atmosphere. In the world: Mars rovers and sample return, asteroid missions such as OSIRIS-REx and Hayabusa2, and plans for telescopes that image Earth-size planets directly. If you want to go deeper 📘 Textbook. Imke de Pater and Jack Lissauer, Planetary Sciences (Cambridge University Press, 2nd edition 2010; check for a newer edition), for the Solar System. Michael Perryman, The Exoplanet Handbook (Cambridge University Press, 2nd edition 2018), for detection methods. 📘 Gentler. Lisa Kaltenegger, Alien Earths (St. Martin's Press, 2024), on how astronomers search for habitable worlds. 🌐 Free data. The NASA Exoplanet Archive, where the numbers in this piece come from and where you can plot your own. 🌐 Free course. MIT OpenCourseWare 12.425 Extrasolar Planets: Physics and Detection Techniques (Fall 2007, Sara Seager). The takeaway Orbits obey a clock, a planet's temperature obeys a balance, and its density gives away what it is made of. Planets are found as a dip in light or a wobble in speed, and both methods favour big planets close to their stars, so the thousands we know are a sample, not a portrait. The honest picture is that small planets are common, and the question of how many are like Earth is open. Try this: find Venus or Jupiter in the evening sky with a sky app. Next to it note the bright "star" nearest to it and ask which of the two is a planet: the planet does not twinkle, because it is a disc and not a point. Sources & Further Reading Sources. NASA Exoplanet Archive, composite planetary systems table (queried 2 November 2026: 6,451 planets; methods transit 4,785, radial velocity 1,204, microlensing 292, imaging 98); Kepler, Harmonices Mundi (1619); Mayor and Queloz, Nature 378, 355 (1995); Nobel Prize in Physics 2019; Wolszczan and Frail, Nature 355, 145 (1992); Kopparapu and others, Astrophysical Journal 765, 131 (2013); Fulton and others, Astronomical Journal 154, 109 (2017); Agol and others, Planetary Science Journal 2, 1 (2021); Libby-Roberts and others, Astronomical Journal 159, 57 (2020); Tsiganis and others, Nature 435, 459 (2005); Walsh and others, Nature 475, 206 (2011); Bouvier and Wadhwa, Nature Geoscience 3, 637 (2010); IAU Resolution B5 (2006); NASA planetary fact sheets; Winn, "Transits and Occultations" (in Seager, ed., Exoplanets, 2010); Madhusudhan and others, Astrophysical Journal Letters 956, L13 (2023). Numbers in the interactives are computed in the page from the formulas shown. 📘 de Pater & Lissauer, Planetary Sciences 📘 Kaltenegger, Alien Earths 🌐 NASA Exoplanet Archive Step 9. Quantum Mechanics: Waves of Probability and the Limits of Knowing ------------------------------------------------------------------------- Why: Quantum mechanics: spectral lines and nuclear fusion. Everything small is described by a wave that you can only ask questions of, one at a time. That single move explains why atoms have colours, why you cannot know everything at once, and why two particles can share a fate. Quantum Mechanics: Waves of Probability and the Limits of Knowing :root{ --bg:#faf7f2; --card:#ffffff; --ink:#1c1a17; --muted:#6b645b; --accent:#2a3f8f; --accent2:#b5842a; --line:#e7e0d6; --hi:#e9ebf7; --panel:#eef0f8; --hot:#c4452d; --cold:#2f6fb0; --good:#3f7f63; --onacc:#ffffff; --shadow:0 8px 30px rgba(0,0,0,.06); } @media (prefers-color-scheme:dark){ :root{ --bg:#0e0f16; --card:#181a24; --ink:#eef0f8; --muted:#a3a8bd; --accent:#8fa0f0; --accent2:#e6b969; --line:#282b3a; --hi:#171a2b; --panel:#10132a; --hot:#ee7b62; --cold:#6fa6e0; --good:#5fb08b; --onacc:#0e0f16; --shadow:0 8px 30px rgba(0,0,0,.45); } } *{box-sizing:border-box} body{margin:0;background:var(--bg);color:var(--ink); font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Georgia,serif; line-height:1.65;-webkit-font-smoothing:antialiased} .wrap{max-width:680px;margin:0 auto;padding:48px 16px 80px} .kicker{color:var(--accent);font-weight:700;letter-spacing:.08em;text-transform:uppercase;font-size:.72rem;font-family:-apple-system,sans-serif} h1{font-size:2.1rem;line-height:1.12;margin:.3em 0 .2em;letter-spacing:-.02em} .dek{font-size:1.15rem;color:var(--muted);margin-bottom:.6em} .meta{font-size:.8rem;color:var(--muted);font-family:-apple-system,sans-serif;border-bottom:1px solid var(--line);padding-bottom:20px;margin-bottom:26px} p{font-size:1.06rem;margin:1em 0} h2{font-size:1.45rem;line-height:1.2;margin:1.9em 0 .3em;letter-spacing:-.01em} .pull{font-size:1.4rem;line-height:1.28;font-weight:600;color:var(--accent);border-left:4px solid var(--accent);padding-left:20px;margin:1.4em 0} .card{background:var(--card);border:1px solid var(--line);border-radius:16px;padding:20px 16px;margin:26px 0;box-shadow:var(--shadow)} .lab{font-family:-apple-system,sans-serif;font-size:.72rem;font-weight:700;letter-spacing:.08em;text-transform:uppercase;color:var(--accent2);margin-bottom:.6em} .course{border-left:5px solid var(--accent2)} .course dl{margin:0;font-family:-apple-system,sans-serif;font-size:.95rem} .course dt{font-weight:700;color:var(--accent);margin-top:.7em} .course dd{margin:.1em 0 0 0} .eq{font-family:Georgia,"Times New Roman",serif;font-style:italic;font-size:1.25rem;text-align:center;background:var(--hi);border-radius:12px;padding:12px 8px;margin:14px 0;overflow-x:auto} .eq small{font-style:normal;font-family:-apple-system,sans-serif;font-size:.8rem;color:var(--muted);display:block;margin-top:2px} .chips{display:flex;flex-wrap:wrap;gap:8px;margin-bottom:6px} button{font-family:-apple-system,sans-serif;font-weight:600;cursor:pointer;border:1px solid var(--line);border-radius:999px;padding:8px 14px;font-size:.88rem;background:var(--hi);color:var(--ink);min-height:44px} button[aria-pressed=true]{background:var(--accent);color:var(--onacc);border-color:var(--accent)} .ctl{font-family:-apple-system,sans-serif;font-size:.9rem;margin-top:8px} .ctl .top{display:flex;justify-content:space-between;gap:8px} input[type=range]{width:100%;accent-color:var(--accent);min-height:44px;margin:0} svg{width:100%;height:auto;display:block;margin:10px 0;border-radius:12px;background:var(--panel)} svg text{font-family:-apple-system,sans-serif;font-size:11px;fill:var(--muted)} .read{font-family:-apple-system,sans-serif;font-size:.92rem;background:var(--hi);border-radius:10px;padding:10px 12px;font-variant-numeric:tabular-nums} .read p{font-size:.92rem;margin:.3em 0} table.t{width:100%;border-collapse:collapse;font-family:-apple-system,sans-serif;font-size:.9rem;margin:12px 0;font-variant-numeric:tabular-nums} table.t th,table.t td{border-bottom:1px solid var(--line);padding:6px 4px;text-align:left;vertical-align:top} table.t th{color:var(--muted);font-weight:600} .note{font-size:.85rem;color:var(--muted);font-family:-apple-system,sans-serif} .take{background:var(--hi);border-radius:16px;padding:22px 18px;margin-top:34px} .take .lab{color:var(--accent)} .do{font-family:-apple-system,sans-serif;font-weight:700;font-size:1.02rem;margin-top:.4em} ul.deep,ul.see{font-family:-apple-system,sans-serif;font-size:.96rem;line-height:1.6;padding-left:1.2em} ul.see li{margin:.35em 0} a{color:var(--accent2)} button:focus-visible,input:focus-visible,select:focus-visible{outline:3px solid var(--accent2);outline-offset:2px} @media (max-width:640px){table.t{display:block;overflow-x:auto;max-width:100%}} Look at a sodium street lamp and you see a colour that is not a blend of anything: a narrow, almost pure yellow-orange. Heat a pinch of table salt in a gas flame and the flame turns the same yellow. Nothing in Newton's mechanics says why a hot atom should give out only certain colours, or why every sodium atom in the universe gives out the same ones. Quantum mechanics says why, and the answer is stranger than the question. A small thing such as an electron is not a tiny ball on a path. It is described by a wave, and the wave is not a ripple in anything. It is a list of how likely each outcome would be if you looked. Confine that wave and, like a guitar string, it can only vibrate in certain patterns. Each pattern has its own energy. Those energies are the colours. A physics graduate carries a handful of pictures from this course for life. This piece builds six of them with sliders you can play with. If you want the gentlest possible start, read Quantum Mechanics: The Weirdness at the Bottom (13/06) first. This one goes further: it computes. The course in one box Who takes itPhysics majors, usually in year 2 or 3 (at MIT it is 8.04, then 8.05). Chemists, materials scientists and electrical engineers meet a shorter version. Programmes in the style of MIT Course 8 and the Cambridge Natural Sciences physics route teach it before statistical physics and particle physics. What it is forPredicting what a small system can do (its energies, its spins, its odds) from a wavefunction and the equation that moves it. Comes afterLinear Algebra (34/04), Waves & Optics (34/80) and Partial Differential Equations (34/79). Leads toStatistical Mechanics (34/84), Particle & Nuclear Physics (34/86) and Condensed Matter Physics (34/87). What this piece gives youYou will not be able to solve the hydrogen atom. You will see why it has the colours it has, why uncertainty is a property of waves, and what a Bell test actually tested. Easier first reads: Quantum Mechanics (13/06) and The Standard Model (13/07). The state is a wave, and its square is a probability In classical mechanics the state of a ball is two lists: where it is and how fast it moves. In quantum mechanics the state of an electron is a function ψ(x), the wavefunction, a number (in general a complex one) for every place. You cannot see ψ. What you can get from it is the odds of finding the electron near x. Max Born proposed this reading in 1926, and it is called the Born rule. P(x) = |ψ(x)|²the chance of finding the particle near x is the square of the wave's size there The wave is fixed; each look gives one random place. Collect many looks and the pile follows the square of the wave. Here is the simplest wave worth knowing: a particle trapped on a line of length L, with walls it cannot pass. The allowed waves are standing waves, ψn = √(2/L) sin(nπx/L), with n = 1, 2, 3 … Interactive · Look at the particle Wave pattern n Look onceLook 50 timesLook 1,000 timesClear The curve is |ψn|² computed from the formula. The bars are positions drawn at random from that curve, in 20 bins. The middle third of the box is where n = 1 is most likely to be found and where n = 2 is least likely to be. With n = 1 you find the particle in the middle third of the box 61% of the time. With n = 2 only 20%. A classical bead rattling between walls would be found there 33% of the time, wherever you look. The wave has nodes, places where it is exactly zero, and the particle is never found at a node, even though it passes from one side to the other. Squeeze a wave and its energy comes in steps A standing wave that fits the box has a whole number of half-wavelengths: L = nλ/2. A short wavelength means a large momentum (de Broglie, 1924: p = h/λ), and momentum means energy. So the allowed energies are not a continuum. They are a ladder. En = n² h² ÷ (8 m L²)energy grows as the square of n, and grows as the box shrinks or the particle gets lighter Worked number: an electron in a box 1 nm wide (a few atoms across) has E1 = 0.376 eV. The step from n = 2 to n = 1 is 3 × 0.376 = 1.13 eV, which is a photon of wavelength 1,099 nm, in the near infrared. Shrink the box to 0.1 nm, the size of an atom, and the same step is 113 eV, in the ultraviolet. Squeezing a wave raises its energy, which is why atoms do not collapse: the electron's lowest wave is never at zero energy. Interactive · The energy ladder ElectronProton1-milligram grain Box width L En = n²h²/(8mL²) with CODATA h and masses. The slider runs from 0.01 nm to 1 cm. For the grain the ladder is drawn, but its real steps are far below anything measurable. The grain row is the point. Quantization is not absent for large things; it is unmeasurably fine. The same formula that gives an electron a 1 eV step gives a milligram grain in a 1 cm box a step of about 10−38 eV. Hydrogen is the real-world version. Its levels are En = −13.6 eV ÷ n², and a jump from n = 3 to n = 2 releases a photon at 656.5 nm in vacuum (656.3 nm in air), the red line astronomers call H-alpha. Sodium's yellow doublet near 589 nm comes from the same logic with a more complicated atom. Superposition: a wave can be a mix, and the mix sloshes Waves add. If ψ1 and ψ2 are both allowed, so is any blend a·ψ1 + b·ψ2. That is a superposition, and it is the same fact as adding two guitar notes, not a mystical one. What is odd is the reading: the particle is not "secretly in one of the two states". The blend is the state. The square of a sum contains a cross term, and the cross term is where interference lives. Each pattern's energy sets how fast its phase turns, so the two parts drift in and out of step and the probability sloshes back and forth at the beat frequency (E2 − E1) ÷ h. For the 1 nm electron box that is one slosh every 3.67 femtoseconds. This is how an excited atom radiates: its charge cloud oscillates, and an oscillating charge makes light. Interactive · Mix two states and let time pass Share of n = 2 in the mix Time (fraction of one slosh) |ψ|² = (1−a)ψ₁² + aψ₂² + 2√(a(1−a)) ψ₁ψ₂ cos(2πt). The average position is integrated numerically. At a = 0 or 1 nothing sloshes: a single energy has a frozen density. At 50:50 the average position swings by about 18% of the box width either way, and then back. At 0% or 100% nothing moves at all: a state with one energy is a stationary state. That is why atoms are stable in their lowest level. Uncertainty: not clumsiness, but what waves are A short wave packet, a pulse, is a sum of many wavelengths. A pure single wavelength is spread over all space. You cannot have a wave that is both narrow in place and sharp in wavelength, and this is true of sound, of water and of a radio signal long before quantum mechanics. Quantum mechanics adds one thing: for a particle, wavelength is momentum (p = h/λ). So the same trade-off becomes Heisenberg's relation (1927). Δx · Δp ≥ ħ ÷ 2ħ = h/2π = 1.0546 × 10⁻³⁴ J·s · the spread in position times the spread in momentum has a floor It is not about the disturbance of measuring. It says the quantity "exact position and exact momentum at once" does not exist for the wave. Worked number: pin an electron into an atom-sized region, Δx = 0.1 nm, and the smallest possible momentum spread gives a velocity spread of 580 km/s. That is a quarter of the electron's speed in hydrogen's lowest orbit (2.19 × 10⁶ m/s). This is why an atom has its size: squeeze the electron closer and its motion grows faster than the squeeze can pay back. Interactive · Pin a particle down ElectronProtonBaseball Position spread Δx Δp = ħ/(2Δx), the minimum for a Gaussian packet; Δv = Δp/m. The two curves are schematic: their widths are drawn so that the product stays fixed. Now slide to a baseball and 1 mm. The velocity spread is about 10−30 m/s, which is why nobody ever noticed uncertainty in daily life. Spin and measurement: asking changes the answer An electron has an inner two-valued property called spin. Send a beam through a magnet whose field changes with position and each electron comes out on one of exactly two paths, "up" or "down" relative to the magnet's axis. That is the Stern–Gerlach experiment (Frankfurt, 1922, with silver atoms). Rotate the magnet and it is still two paths, never a smear. The state of a spin is a two-component arrow, and measuring along another axis gives "up" with probability cos²(θ/2), where θ is the angle between the two axes. Measuring also changes the state: after you filter for up along z, the electron really is z-up, and an x measurement then gives 50:50 and erases the z information. Put three filters in a row and the odd result follows: z-up, then x-up, then z again lets some z-down electrons through, even though the first filter removed all of them. A pair of crossed polarising sunglasses with a third lens slid between them does exactly this to light. Interactive · Three filters in a row Angle of the middle filter from z Send 1,000 electrons Start with a beam that is already z-up. Middle filter passes "up along angle θ" with probability cos²(θ/2). The last filter asks z again. Final z-down fraction = cos²(θ/2) · sin²(θ/2) = sin²(θ)/4. Runs use the browser's random numbers, so they differ each time. At 90 degrees, a quarter of the beam ends up z-down. With no middle filter, none do. At 0 or 180 degrees the middle filter either does nothing or flips the beam cleanly, and the final count is 0 or 100%. Entanglement: correlations no pre-agreed script can fake Prepare two spins in the combined state with total spin zero (the "singlet"), and fly them apart. Each one alone looks random. But when both are measured along the same axis they always disagree, and for axes at angle Δ the correlation is E = −cos Δ (+1 means always agree, −1 always disagree). Is this like two sealed envelopes, one with a red card and one with a blue? Einstein, Podolsky and Rosen argued in 1935 that quantum mechanics must then be incomplete: hidden "instruction sets" travelling with the particles. In 1964 John Bell showed that any such local scheme must obey a bound. Using the CHSH form (Clauser, Horne, Shimony and Holt, 1969), pick two settings a, a′ on one side and b, b′ on the other: S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)any local hidden-variable script: |S| ≤ 2 · quantum mechanics can reach 2√2 ≈ 2.83 Interactive · Beat the bound Best quantum settingsAll the samePerpendicular Alice, first setting a Alice, second setting a′ Bob, first setting b Bob, second setting b′ Run 2,000 pairs E(α, β) = −cos(α − β) for the singlet. The run draws each pair's two ±1 results with the matching probabilities; it estimates S with a statistical error of about ±0.1. Experiments sided with quantum mechanics: Clauser and Freedman (1972), then Aspect (1982), then "loophole-free" tests that closed the remaining gaps in 2015 (Delft, NIST and Vienna). The 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser and Anton Zeilinger for this line of work. Bell's result does not say you can send a message faster than light: each side's results alone are random. It says the world cannot be explained by local pre-agreed instructions alone. Four stories about what it means Everyone who uses the maths gets the same numbers. What the maths is about is still argued. The main readings, in their holders' own terms: Copenhagen (Bohr, Heisenberg and descendants): the wavefunction is a tool for predicting results of measurements; asking what the electron "really did" between measurements has no answer. Many-worlds (Everett, 1957; later DeWitt, Deutsch, Wallace): the wave never collapses. Every outcome happens, in branches that no longer interact. Pilot-wave (de Broglie 1927; Bohm 1952): particles always have positions, guided by the wave. It reproduces the same predictions, at the price of non-local guidance, which Bell's theorem says any such theory needs. QBism (Fuchs, Mermin and others): the wavefunction is an agent's personal degrees of belief about what they will experience. No experiment so far separates these as they are usually stated, and physicists are split. Treat any claim that physics has settled the question with caution. What a graduate just sees The state is a wave; the square of the wave is the odds. Nodes are places the particle is never found. Confine a wave and its energy comes in steps; the smaller the box, the larger the steps and the higher the lowest energy. Spectra of atoms are the fingerprints of their energy ladders. Superposition is addition of waves, and interference lives in the cross term. Uncertainty is the narrow-versus-sharp trade-off of any wave, with ħ setting the exchange rate. Measuring along one axis resets the state; a third filter can restore what a second one hid. Entangled outcomes are correlated more strongly than any pre-agreed script allows (|S| up to 2.83 against a bound of 2), yet no message travels. The maths is agreed; the meaning is not. Where it shows up next Statistical Mechanics (34/84): counting quantum levels is how temperature and entropy are defined for real matter. Particle & Nuclear Physics (34/86): fields are quantum waves; spin and superposition run the Standard Model. Condensed Matter Physics (34/87): billions of boxes coupled together make the energy bands of a chip. In the world: LEDs and lasers (levels and photons), MRI (spin), atomic clocks (the hydrogen-like levels again), flash memory (tunnelling), quantum cryptography (Bell tests), and the first small quantum computers. If you want to go deeper 📘 Textbook. David Griffiths and Darrell Schroeter, Introduction to Quantum Mechanics (Cambridge, 3rd edition, 2018). The most widely assigned first course, with worked problems and a clear style. 📘 Textbook. J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics (Cambridge, 3rd edition). Starts from spin and states, closer to the linear-algebra view; a standard for the next course. 📘 Gentler. Leonard Susskind and Art Friedman, Quantum Mechanics: The Theoretical Minimum (Basic Books, 2014). Real maths, written for self-study, with no university needed. 🌐 Free course. MIT OpenCourseWare 8.04 Quantum Physics I (Spring 2016), followed by 8.05 Quantum Physics II. The takeaway A quantum system is a wave whose square gives the odds. Confine it and its energies become a ladder, which is why atoms have fixed colours and sizes. Mix waves and they interfere; ask a question and the answer rewrites the state; entangle two systems and their answers correlate beyond any script. The equations are agreed; what they mean is still open. Try this: this week, take three polarising lenses (old polarised sunglasses can be cut or popped out, or use a pair of passive 3D cinema glasses) and cross two of them until the light goes dark, then slip the third between them at 45 degrees. Light comes back. That is the three-filter result, in photons instead of electrons. Sources & Further Reading Sources. The box, hydrogen, uncertainty and spin results are standard textbook quantum mechanics (Griffiths & Schroeter; Sakurai). Constants are CODATA 2018 values: h = 6.62607015 × 10⁻³⁴ J·s, electron mass 9.1093837015 × 10⁻³¹ kg, proton mass 1.67262192369 × 10⁻²⁷ kg, 1 eV = 1.602176634 × 10⁻¹⁹ J. Hydrogen H-alpha from the Rydberg formula with the reduced-mass correction (656.47 nm vacuum). Born 1926, Heisenberg 1927, Stern–Gerlach 1922, Bell 1964, CHSH 1969, Clauser–Freedman 1972, Aspect 1982, and the 2022 Nobel Prize in Physics (Aspect, Clauser, Zeilinger): Nobel Prize press release. The loophole-free tests of 2015 (Hensen et al., Delft; Shalm et al., NIST; Giustina et al., Vienna) are cited from the literature, not re-checked for this piece. Interpretation summaries follow the Stanford Encyclopedia of Philosophy entries on Copenhagen, many-worlds and Bohmian mechanics. Every number an interactive shows was computed in node: for example E1 = 0.376 eV at 1 nm, beat period 3.67 fs, 61% and 20% in the middle third for n = 1 and 2, Δv = 579 km/s at 0.1 nm, final z-down fraction 25% at 90 degrees, and S = 2.828 at the best settings. 🌐 Nobel Prize in Physics 2022: press release 🌐 Stanford Encyclopedia of Philosophy: Copenhagen Interpretation 🌐 MIT OCW 8.04 Quantum Physics I 📘 Griffiths & Schroeter, Introduction to Quantum Mechanics Step 10. Statistical Mechanics: Temperature Is What Counting Looks Like ----------------------------------------------------------------------- Why: Statistical mechanics: gases, radiation and the inside of a star. Nothing in a single atom is hot. Temperature, pressure and the arrow of time appear when you count the ways trillions of atoms can arrange themselves, and the biggest count wins by a margin so large it looks like a law. Statistical Mechanics: Temperature Is What Counting Looks Like :root{ --bg:#faf7f2; --card:#ffffff; --ink:#1c1a17; --muted:#6b645b; --accent:#2a3f8f; --accent2:#b5842a; --line:#e7e0d6; --hi:#e9ebf7; --panel:#eef0f8; --hot:#c4452d; --cold:#2f6fb0; --good:#3f7f63; --onacc:#ffffff; --shadow:0 8px 30px rgba(0,0,0,.06); } @media (prefers-color-scheme:dark){ :root{ --bg:#0e0f16; --card:#181a24; --ink:#eef0f8; --muted:#a3a8bd; --accent:#8fa0f0; --accent2:#e6b969; --line:#282b3a; --hi:#171a2b; --panel:#10132a; --hot:#ee7b62; --cold:#6fa6e0; --good:#5fb08b; --onacc:#0e0f16; --shadow:0 8px 30px rgba(0,0,0,.45); } } *{box-sizing:border-box} body{margin:0;background:var(--bg);color:var(--ink); font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Georgia,serif; line-height:1.65;-webkit-font-smoothing:antialiased} .wrap{max-width:680px;margin:0 auto;padding:48px 16px 80px} .kicker{color:var(--accent);font-weight:700;letter-spacing:.08em;text-transform:uppercase;font-size:.72rem;font-family:-apple-system,sans-serif} h1{font-size:2.1rem;line-height:1.12;margin:.3em 0 .2em;letter-spacing:-.02em} .dek{font-size:1.15rem;color:var(--muted);margin-bottom:.6em} .meta{font-size:.8rem;color:var(--muted);font-family:-apple-system,sans-serif;border-bottom:1px solid var(--line);padding-bottom:20px;margin-bottom:26px} p{font-size:1.06rem;margin:1em 0} h2{font-size:1.45rem;line-height:1.2;margin:1.9em 0 .3em;letter-spacing:-.01em} .pull{font-size:1.4rem;line-height:1.28;font-weight:600;color:var(--accent);border-left:4px solid var(--accent);padding-left:20px;margin:1.4em 0} .card{background:var(--card);border:1px solid var(--line);border-radius:16px;padding:20px 16px;margin:26px 0;box-shadow:var(--shadow)} .lab{font-family:-apple-system,sans-serif;font-size:.72rem;font-weight:700;letter-spacing:.08em;text-transform:uppercase;color:var(--accent2);margin-bottom:.6em} .course{border-left:5px solid var(--accent2)} .course dl{margin:0;font-family:-apple-system,sans-serif;font-size:.95rem} .course dt{font-weight:700;color:var(--accent);margin-top:.7em} .course dd{margin:.1em 0 0 0} .eq{font-family:Georgia,"Times New Roman",serif;font-style:italic;font-size:1.25rem;text-align:center;background:var(--hi);border-radius:12px;padding:12px 8px;margin:14px 0;overflow-x:auto} .eq small{font-style:normal;font-family:-apple-system,sans-serif;font-size:.8rem;color:var(--muted);display:block;margin-top:2px} .chips{display:flex;flex-wrap:wrap;gap:8px;margin-bottom:6px} button{font-family:-apple-system,sans-serif;font-weight:600;cursor:pointer;border:1px solid var(--line);border-radius:999px;padding:8px 14px;font-size:.88rem;background:var(--hi);color:var(--ink);min-height:44px} button[aria-pressed=true]{background:var(--accent);color:var(--onacc);border-color:var(--accent)} .ctl{font-family:-apple-system,sans-serif;font-size:.9rem;margin-top:8px} .ctl .top{display:flex;justify-content:space-between;gap:8px} input[type=range]{width:100%;accent-color:var(--accent);min-height:44px;margin:0} svg{width:100%;height:auto;display:block;margin:10px 0;border-radius:12px;background:var(--panel)} svg text{font-family:-apple-system,sans-serif;font-size:11px;fill:var(--muted)} .read{font-family:-apple-system,sans-serif;font-size:.92rem;background:var(--hi);border-radius:10px;padding:10px 12px;font-variant-numeric:tabular-nums} .read p{font-size:.92rem;margin:.3em 0} table.t{width:100%;border-collapse:collapse;font-family:-apple-system,sans-serif;font-size:.9rem;margin:12px 0;font-variant-numeric:tabular-nums} table.t th,table.t td{border-bottom:1px solid var(--line);padding:6px 4px;text-align:left;vertical-align:top} table.t th{color:var(--muted);font-weight:600} .note{font-size:.85rem;color:var(--muted);font-family:-apple-system,sans-serif} .take{background:var(--hi);border-radius:16px;padding:22px 18px;margin-top:34px} .take .lab{color:var(--accent)} .do{font-family:-apple-system,sans-serif;font-weight:700;font-size:1.02rem;margin-top:.4em} ul.deep,ul.see{font-family:-apple-system,sans-serif;font-size:.96rem;line-height:1.6;padding-left:1.2em} ul.see li{margin:.35em 0} a{color:var(--accent2)} button:focus-visible,input:focus-visible,select:focus-visible{outline:3px solid var(--accent2);outline-offset:2px} @media (max-width:640px){table.t{display:block;overflow-x:auto;max-width:100%}} Shake a box of coins and they land about half heads. Shake a box of a hundred coins and you will rarely get exactly fifty, but you will almost never get ninety. Shake a mole of coins and the fraction of heads will be one half to twelve decimal places, every single time, and nobody calls that a coincidence. They call it a law. The air in your room works the same way. Its 1026 or so molecules each wander at random, yet the pressure on your eardrum is steady and the temperature is a single number. Nothing in one molecule is "warm". Statistical mechanics is the course that shows temperature, pressure, heat flow and the second law all emerging from one move: count the ways. If you read Entropy: Why You Can't Unstir Your Coffee (13/12) and Thermodynamics (34/14), you have met the results. This piece shows where they come from, and lets you watch the numbers do it. The course in one box Who takes itPhysics majors, usually in year 3 (MIT's 8.044 Statistical Physics I), and chemists and materials scientists in their own versions. Programmes in the style of MIT Course 8 and the Cambridge Natural Sciences physics route teach it after a first quantum course. What it is forDeriving the laws of heat, and the properties of gases, solids and radiation, from the rules governing the particles inside them. Comes afterProbability & Statistics (34/05), Thermodynamics (34/14) and Quantum Mechanics (34/83). Leads toCondensed Matter Physics (34/87) and Astrophysics & Cosmology (34/88). What this piece gives youYou will not be able to compute a partition function for a real material. You will see why temperature is a slope, why large numbers make the second law near-certain, and what the Boltzmann factor is doing in everything from chemistry to stars. Count the ways: macrostates and microstates Split the job in two. A microstate is the full detail: which coin shows heads, which molecule is on the left. A macrostate is what you can measure: how many heads, how many molecules on the left. Many microstates make the same macrostate, and the number of them, W, is the multiplicity. For N coins, the number of ways to get exactly r heads is "N choose r". There is one way to get all heads and an astronomical number of ways to get half. If every microstate is equally likely (the founding assumption), the macrostate with the most microstates wins. As N grows the peak does not just rise: it narrows. The relative width of the heads fraction is 1/(2√N). Interactive · How sharp does the peak get? Number of coins (or gas molecules) N Exact sums of "N choose r" (log-gamma function), shaded band 45% to 55%. The curve is the bell shape of width 1/(2√N), drawn at equal height to show its width. For a mole of molecules the same width is 6.4 × 10−13. With 100 coins the chance of landing between 45 and 55 heads is about 73%, and exactly 50 only 8%. With 1,000 coins the band holds 99.86%. With 10,000 it holds all but about 2 in 10 billion. Now take N = 6 × 1023: the width is 6.4 × 10−13. This is why "average" and "actual" are the same thing for a gas. Entropy is a count, and the second law is a bet that wins Boltzmann's definition (formalised by Planck) is the one formula a graduate remembers, and it is carved on Boltzmann's grave in Vienna: S = k ln Wentropy = Boltzmann's constant × the logarithm of the number of microstates · k = 1.380649 × 10⁻²³ J/K, exact by definition since 2019 The logarithm is there for a reason: two separate systems multiply their counts, and logs turn products into sums, so entropy adds. Notice what the second law has become. It is not a force. A system drifts to the macrostate with the most microstates because there are overwhelmingly more of them, the way a shuffled deck is not found sorted. Try it with a small model: N molecules in a box with two halves, all starting on the left. At each step one molecule picked at random hops to the other side (the Ehrenfest urn model of 1907). Nothing in the rule prefers equal halves. Interactive · Release the gas Number of molecules Step10 steps100 stepsReset Each step picks one molecule uniformly at random (browser random numbers) and moves it across. S/k = ln(N choose left count). The mean time to return to "all on the left" is 2N steps. With 10 molecules the count wanders and does return to all-left now and then: on average once in 1,024 steps. With 100 molecules, 2100 is about 1.3 × 1030 steps, which at a million steps a second is about 4 × 1016 years, millions of times the age of the universe. The "law" that gas never gathers on one side is not a prohibition. It is a bet at odds that no one will ever see paid out. In the opposite direction, a 100-molecule system really can fluctuate, which is why small machines (molecular motors, nanoscale devices) behave differently from large ones. Temperature is how fast entropy grows with energy Temperature has just been a reading on a thermometer. Here it gets a definition. Put energy into a system and its count of microstates grows. The steepness of that growth is the thing that tells you whether it will give up or accept energy: 1/T = dS/dEtemperature is the inverse of the slope of entropy against energy The classic model is an Einstein solid: N oscillators sharing q quanta of energy. The number of ways to distribute them is Ω = (q + N − 1)! ÷ (q! (N − 1)!). Check: 3 oscillators sharing 3 quanta can do it 10 ways. Now put two such solids in contact and let quanta move. The combined count is ΩA × ΩB, and the split with the largest product is where the system sits. At that point the two slopes are equal, so the two temperatures are equal. Heat flows from hot to cold because that is the direction that raises the total count. Interactive · Two blocks trade energy Oscillators in A Oscillators in B Total quanta Quanta currently in A ln Ω from the log-gamma function. Temperature in units of ε/k: kT/ε = 1/ln((q+N)/(q+1)), the energy needed per unit entropy gain, found from adding one quantum. Move the quanta slider and watch heat want to flow toward the peak. Setting A to 100 oscillators and B to 200 with 300 quanta, the peak sits at about 99 quanta in A, one third of the total, matching A's one third of the oscillators. The two temperatures read about 1.45 and 1.46 there (in units of ε/k). Off the peak the hotter block has the lower slope and is the one that loses quanta. The Boltzmann factor: why cold things are boring Now stop looking at the whole box and look at one small system (an atom, a molecule) touching a big reservoir at temperature T. The reservoir can lend the small system energy, but lending a lot costs the reservoir many microstates, and each extra bit of energy taken multiplies its count by a fixed factor. The result is the most reused formula in physics. P(E) ∝ e−E / kTthe odds of finding a state fall off exponentially with its energy, measured in units of kT kT at 300 K (room temperature) is 25.85 meV (0.0259 eV): the natural thermal energy of a molecule. A state one kT above the ground state is about 37% as likely as the ground state; a state 10 kT up is 0.005% as likely. A molecular vibration (nitrogen's is about 0.29 eV) sits more than 11 kT up, so at room temperature only about 1 molecule in 75,000 is vibrating. That is why room-temperature air heat capacity barely feels vibrations. Interactive · Warm up a ladder Spacing = kT at 300 KN₂ vibration at 300 KN₂ vibration at 2000 K Temperature Level spacing Six equally spaced, non-degenerate levels; population ∝ exp(−nΔE/kT), normalised by the partition function Z. k = 8.617 × 10⁻⁵ eV/K. The same factor drives chemistry: reaction rates grow like e−Ea/kT (the Arrhenius law), so a reaction with a barrier of 50 kJ/mol goes about 1.9 times faster when 25 °C becomes 35 °C, which is where the rule of thumb "rates double for each 10 degrees" comes from. It drives stars too. Cecilia Payne's 1925 thesis used the Boltzmann factor, together with the matching rule for ionisation, to show that stars are mostly hydrogen: the 10.2 eV needed to lift hydrogen's electron to the level that gives the visible Balmer lines means that at a sun-like 5,772 K only about 5 in a billion neutral hydrogen atoms are in that level at any instant, yet it is enough to produce the lines. A gas is a distribution, not a speed Apply the Boltzmann factor to kinetic energy ½mv² and you get Maxwell's 1860 distribution of molecular speeds: a hump that starts at zero, peaks, then has an exponential tail. Two results are worth a lifetime. First, equipartition: every independent way a molecule can store energy as a square (each direction of motion, each rotation) holds on average ½kT. So the average translational energy is 3/2 kT = 38.8 meV at 300 K, for every gas, and heavier molecules simply move slower. Second, the tail: the fraction beyond some speed is controlled by e−mv²/2kT, so it is violently sensitive to mass and temperature. Worked number: nitrogen at 300 K has a root-mean-square speed of 517 m/s and the most likely speed is 422 m/s. Hydrogen molecules at the same temperature average 1,927 m/s. Interactive · Tune a gas H₂HeN₂O₂CO₂ Temperature f(v) = 4π (m/2πkT)3/2 v² e−mv²/2kT. Curve drawn at equal height to show its shape; axis 0 to 12,000 m/s with Earth's escape speed 11,186 m/s marked. The tail fraction is an upper-end measure only: real atmospheres lose gas from the thin top layer, so treat it as a trend, not a loss rate. At 1,000 K (roughly the temperature of the thin outer atmosphere, which swings with the sun), the fraction of hydrogen molecules above Earth's escape speed is about 10−6, helium about 10−12, and nitrogen about 10−90. That gap is why Earth keeps its nitrogen and oxygen and has lost almost all its free hydrogen and helium. Information has a heat price: demons and bits In 1867 James Clerk Maxwell imagined a tiny doorkeeper who watches molecules and opens a gate only for fast ones going one way and slow ones going the other. Without doing any work, it would sort a gas into hot and cold and defeat the second law. The paradox was famous for a century. The resolution came from information: the demon must remember what it saw, and memory is finite, so eventually it must erase. Rolf Landauer (1961) showed that erasing one bit must release at least kT ln 2 of heat, and Charles Bennett (1982) used that to close the case. Eerase ≥ k T ln 2= 2.87 × 10⁻²¹ J (0.018 eV) per bit at 300 K That is far below what any chip dissipates, but it is real. In 2012 a team in Lyon trapped a single glass bead in a double well to act as a one-bit memory and measured the heat from erasing it, approaching the limit as they erased slowly (Bérut et al., Nature 483, 187). The idea that information is physical is now a working principle, and it ties statistical mechanics to Computer Architecture (34/25) and Theory of Computation (34/29). What a graduate just sees Macrostates are piles of microstates; the biggest pile wins, and it wins by more the bigger N gets. Fluctuations scale like 1/√N, so a mole of anything obeys sharp laws. Entropy is ln of a count, and the second law is probability at astronomical odds. Temperature is the slope of entropy against energy; equal slopes mean equilibrium. Every thermally accessible state has weight e−E/kT; compare any energy to kT (about 26 meV at room temperature). Each quadratic way of storing energy holds ½kT, and mass sets speed, not energy. Tails of distributions decide rare events: reactions, escape, failure. Information is physical: erasing a bit costs at least kT ln 2. Where it shows up next Condensed Matter Physics (34/87): electrons in solids obey quantum versions of these counts (Fermi–Dirac), which decide conductors, insulators and semiconductors. Astrophysics & Cosmology (34/88): stellar spectra, the cosmic microwave background and the entropy of black holes are all statistical mechanics. Thermodynamics (34/14) and Heat Transfer (34/16): their laws are the averaged form of what you just counted. In the world: why batteries and chips are temperature-sensitive, how fridges and heat pumps are limited, the Arrhenius ageing tests used on electronics, and the sampling methods in machine learning that borrow the Boltzmann distribution. If you want to go deeper 📘 Textbook. Daniel Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 1999). The standard first course, built on exactly the counting approach used here. 📘 Textbook. Charles Kittel and Herbert Kroemer, Thermal Physics (2nd edition, W. H. Freeman, 1980). Compact, and a common second book. 📘 Gentler. The Feynman Lectures on Physics, Volume I, chapters 39 to 46 (kinetic theory and statistical mechanics), free at feynmanlectures.caltech.edu. 🌐 Free course. MIT OpenCourseWare 8.044 Statistical Physics I (Spring 2013). The takeaway Count microstates. Entropy is the logarithm of the count; temperature is how steeply it climbs with energy; the Boltzmann factor e−E/kT weights every state; and for 1023 particles the most likely macrostate is the only one you will ever see. Heat flowing downhill is not a decree but an overwhelming majority. Try this: this week, ask of anything you meet, "how big is this energy compared with kT?" A 1.5 V battery, a chemical bond, a photon of red light are all hundreds of kT. A hot cup or a warm room is a few. The answer tells you whether heat can wreck it. Sources & Further Reading Sources. The counting, Einstein solid, Boltzmann factor, equipartition and Maxwell–Boltzmann results are standard (Schroeder; Kittel & Kroemer). k = 1.380649 × 10⁻²³ J/K exact since the 2019 SI redefinition; other constants CODATA 2018. Planck evaluated k in 1900–1901 and S = k ln W is on Boltzmann's grave in Vienna (Zentralfriedhof). The Ehrenfest urn is from 1907, Maxwell's speed distribution from 1860, and Cecilia Payne's thesis from 1925. Landauer 1961 and Bennett 1982 are cited from the literature. Experiment: Bérut et al., "Experimental verification of Landauer's principle linking information and thermodynamics," Nature 483, 187–190 (8 March 2012). Course contents: MIT OCW 8.044. Every number an interactive shows was computed in node and checked: P(45–55%) = 72.9% for N = 100, 99.86% for N = 1,000; relative width 6.4 × 10⁻¹³ for a mole; 2¹⁰⁰ = 1.27 × 10³⁰; Ω(3,3) = 10; kT = 25.85 meV at 300 K; N₂ at 300 K vrms = 517 m/s; tail fractions 10−5.9 (H₂), 10−12.3 (He), 10−90 (N₂) at 1,000 K; Landauer 2.87 × 10⁻²¹ J at 300 K; Arrhenius ratio 1.92 for 50 kJ/mol between 25 and 35 °C. 🌐 MIT OCW 8.044 Statistical Physics I 📘 Schroeder, An Introduction to Thermal Physics 📘 Feynman, Lectures on Physics, vol. I (free online) Step 11. Particle & Nuclear Physics: What Stuff Is Made Of, and What Can Happen to It ------------------------------------------------------------------------------------- Why: Nuclear and particle physics: what makes stars shine. Everything you can touch is built from a short list of particles, and a short list of rules says which collisions and decays may happen. Almost nothing else is allowed, and that is why the world is stable. A banana is slightly radioactive. It holds potassium, and about one atom in 8,500 of that potassium is the isotope potassium-40, which decays on its own, now and then, with a half-life of about 1.25 billion years. Every day, somewhere in your body, a few thousand of those atoms do it. Nothing pushed them. Nothing about the atom changed. One of its neutrons simply turned into a proton, an electron and an almost undetectable neutrino, and flew apart. Now ask a stranger question. Why does your banana not do other things? Why does a proton in it sit still for longer than any experiment has been able to measure, when the proton is heavy and could, as far as raw energy goes, fall apart into lighter pieces? A graduate in particle and nuclear physics sees the banana, the proton and the sun as one subject: a short list of ingredients, and a short list of bookkeeping rules about what can turn into what. The course in one box Who takes itPhysics majors, usually in year 3 or 4, often as a choice after the quantum core. Programmes in the style of MIT's Course 8 (the course 8.701 Introduction to Nuclear and Particle Physics) and the Cambridge physics route teach it late, because it needs quantum mechanics and relativity. Nuclear engineers meet the nuclear half in a more applied form. What it is forKnowing what the smallest things are, how they interact, and how to tell from a collision or a decay what is going on. Comes afterQuantum Mechanics (34/83) and Special Relativity (34/81). Leads toAstrophysics & Cosmology (34/88): stars burn on nuclear reactions, and the early universe was a particle collider. What this piece gives youYou will not be able to compute a cross-section or a Feynman diagram. You will see what the Standard Model's table contains, how to check whether a decay is allowed with three lines of arithmetic, and why the nuclear energy per kilogram is millions of times that of any chemical fuel. Easier first reads in the library: Quantum Mechanics: The Weirdness at the Bottom (13/06), The Standard Model: What Everything Is Made Of (13/07) and How a Nuclear Reactor Works (13/14). This one goes further and lets you test the rules. Energy is mass, so collisions can make new things Einstein's best-known equation is usually read as a statement about bombs. Particle physicists read it as a price list. Mass and energy are the same currency, and the exchange rate is the speed of light squared. If you hit two things together hard enough, some of their motion can be spent on a brand-new particle, which then exists until it decays. E = m c²energy = mass × (speed of light)². So masses are quoted in energy units: an electron is 0.511 MeV, a proton 938.3 MeV, about 0.94 GeV The catch is that not all your energy is spendable. Momentum must still balance after the collision, so a particle fired at a stationary target keeps most of its energy as leftover motion. Fire two protons at each other head-on and the whole lot is spendable, because the total momentum is zero. The usable energy is the centre-of-mass energy, written √s. For two beams of energy E head-on it is just 2E. For one beam of energy E on a stationary proton it is only √(2m² + 2mE), in units where c = 1. A worked number. To make a proton–antiproton pair in a proton–proton collision you need √s of at least four proton masses, 3.75 GeV. With a stationary target that takes a beam kinetic energy of 5.63 GeV, which is why the Bevatron in Berkeley, built to about 6 GeV, found the antiproton in 1955. At a collider you need only about 1.9 GeV in each beam. Try heavier targets below. Producing a Higgs boson of mass about 125 GeV needs roughly 63.5 GeV per beam in a collider, but about 8,600 GeV (8.6 TeV) in a fixed-target beam. That gap is why the biggest accelerators are colliders. Interactive · Collider or fixed target? Antiproton pair W boson Higgs Top pair Mass of the new stuff to make Two protons in, the new stuff must have mass M, so √s ≥ 2mp + M (mp = 0.938272 GeV). Collider: each beam needs √s/2. Fixed target: E = (s − 2mp²)/(2mp). This is a threshold: real production at the LHC also depends on how the proton's quarks and gluons share the energy, so real machines run well above it. For scale, the Large Hadron Collider has run at 13.6 TeV collision energy since July 2022. Reaching that with a stationary target would need a beam of roughly 98,600 TeV. No ring could hold it. Quarks and leptons: the short list of what stuff is made of The Standard Model's list is shorter than the periodic table. Matter particles come in two families. Quarks (up, down, charm, strange, top, bottom) feel the strong force and are never seen alone. Leptons (the electron, muon and tau, each with its own neutrino) do not feel the strong force. Each family comes in three "generations" with the same charges and growing mass; ordinary matter uses only the first: up and down quarks, the electron and its neutrino. Every particle has an antiparticle with opposite charges. Quark charges are odd: up-type quarks carry +2/3 and down-type −1/3 of a proton's charge. But you never see a third of a charge, because quarks are confined. They only appear glued into bundles with total strong "colour charge" zero: three quarks (baryons), three antiquarks, or a quark with an antiquark (mesons). A proton is two up and one down: 2/3 + 2/3 − 1/3 = +1. A neutron is up, down, down: 0. Build some yourself. Interactive · Build a hadron udscb ūd̄s̄c̄b̄ Clear Charges: u, c +2/3; d, s, b −1/3; antiquarks opposite. Baryon number ±1/3 per quark. Strangeness −1 per s quark. The top quark is left out because it decays before it can bind. The list of named hadrons is a small sample. The quarks' bound pairs and triples are by far the biggest source of everyday mass. The up and down quarks together are only about 1% of a proton's mass; the rest is the energy of the strong force binding them, via E = mc². Most of you is binding energy. Conservation laws decide what can happen A reaction is allowed if three things hold. Add up the electric charge before and after: it must match. Add up baryon number (quarks count +1/3, antiquarks −1/3) and lepton number, counting each lepton family separately: they must match. And the total energy in the rest masses must be at least that of the products. If a rule is not forbidden by one of these, it happens, at some rate. If it is forbidden, it has never been seen. Those rules are why the proton, the lightest baryon, is stable. It has nothing lighter with the same baryon number to decay into. Experiments such as Super-Kamiokande in Japan have searched for it to fall apart and found nothing, setting a lifetime limit above 1034 years, many orders of magnitude longer than the universe has existed. In the same spirit, the muon should be able to decay into an electron and a photon on energy grounds alone, but flavour-specific lepton number forbids it, and searches find none at the level of about 1 in a trillion. Predict first, then check. Pick a reaction, say whether you think it happens, and the page runs the arithmetic. Interactive · Allowed or forbidden? I think it happens I think it never does Rest masses in MeV from the Particle Data Group (neutrinos treated as massless; their real masses are tiny). Charge, baryon number and electron/muon lepton numbers are summed on each side. Heavy messengers have short reach In quantum field theory, forces are carried by particles. Two electrons repel by swapping photons. The strong force binds quarks by swapping gluons. The weak force, behind the neutron's decay, is carried by the W and Z bosons. And here is the part a graduate remembers: the heavier the carrier, the shorter the force's reach. Borrowing a heavy particle from the vacuum is expensive in energy, so the loan must be repaid quickly, and it cannot travel far. The range is about ħc divided by the carrier's rest energy. range ≈ ħc / (m c²)ħc = 197.327 MeV·fm. One femtometre (fm) is 10⁻¹⁵ m, about the width of a proton (radius ≈ 0.84 fm) The photon is massless, so electric forces reach across the galaxy. The pion, the lightest meson at 139.6 MeV, gives 1.41 fm, which is why the nuclear force binds neighbours in a nucleus and fades beyond. The W boson (80.4 GeV) gives 0.0025 fm, about 300 times smaller than a proton. That is why the weak force is "weak" at ordinary distances: it hardly reaches. Interactive · How far does a force reach? Pion Kaon W boson Photon Carrier rest energy Yukawa estimate, range = ħc/mc². The strong force between quarks (gluons) is a different case: gluons are massless but the force is still confined, so this is the classic picture for pions between nucleons and for the weak force, not for gluons. Binding energy: why the sun shines and reactors run A nucleus weighs a little less than its separate protons and neutrons. The missing mass, multiplied by c², is the binding energy, the work needed to pull it apart. Divide by the number of nucleons A and you get a curve with a shape every nuclear scientist knows: it rises steeply for light nuclei, peaks around iron and nickel (about 8.8 MeV per nucleon), then falls slowly to uranium. Anything that moves a nucleus up the curve releases energy. Light nuclei going up by fusing; heavy ones going up by splitting. Worked numbers from measured masses: helium-4 has 7.07 MeV per nucleon, iron-56 8.79, uranium-235 7.59. Deuterium plus tritium fusing into helium-4 and a neutron frees 17.6 MeV, about 3.4 × 10¹⁴ joules per kilogram of fuel. Fission of uranium-235 frees roughly 200 MeV per atom, about 8 × 10¹³ J/kg. TNT is 4.2 × 10⁶ J/kg: fusion fuel carries about 80 million times as much. Interactive · The curve of binding energy Mass number A Line: the semi-empirical mass formula (volume, surface, Coulomb and asymmetry terms; pairing left out for a smooth curve), with stable Z. Dots: measured values computed from atomic masses (He-4, C-12, O-16, Fe-56, Ni-62, U-235, U-238). The formula is poor below A ≈ 20 (it gives 5.7 for helium-4 where the real value is 7.07), a reminder that it is a model. The same curve explains the sun, which fuses hydrogen into helium, and why stars much heavier than the sun make iron and stop: past iron, fusion costs energy instead of paying it. That is the hinge to Astrophysics & Cosmology (34/88). Decay is random, yet the curve is exact An unstable nucleus has no clock. A carbon-14 atom that has survived 5,730 years is exactly as likely to decay in the next minute as one made a second ago. So nothing in any single atom tells you when it will go. Yet a sample of trillions behaves with high precision: after one half-life, half remain; after two, a quarter. N(t) = N₀ × (½)t / t½survivors = starting number × one half, raised to the number of half-lives elapsed Carbon dating uses it: living things hold carbon-14 at the atmosphere's level and stop taking more when they die. A bone with a quarter of the level has lived two half-lives, about 11,460 years. After 20,000 years only about 8.9% remain, which is why the method runs out near 50,000 years. The half-lives span an enormous range: below, from microseconds to billions of years. Watch the randomness shrink as the sample grows. Interactive · Atoms that cannot be predicted, in a crowd that can Po-214I-131Cs-137C-14U-238 10 atoms1001,000Roll again Time passed (half-lives) Each atom draws its own random lifetime (exponential), so no atom's fate is known in advance. Half-lives: Po-214 164 µs, I-131 8.02 days, Cs-137 30.04 years, C-14 5,730 years, U-238 4.468 billion years (standard tables). With 10 atoms, the survivors after one half-life can easily be 3 or 7. With 1,000, they are close to 500 every time. That is the whole content of "the law of large numbers", and the reason a particle detector needs many events before it says anything. Where the Standard Model runs out The Standard Model is the most precisely tested theory in science, and every physicist expects it to be incomplete. The decisive experiments: neutrinos oscillating between flavours, which means they have mass (Super-Kamiokande and SNO, Nobel Prize 2015); the Higgs boson, announced on 4 July 2012 by ATLAS and CMS at about 125 GeV; and the muon's magnetic moment, where Fermilab's final 2025 result is 127 parts per billion precise and where theory predictions were revised, shrinking a famous disagreement. What is left? Neutrinos have mass but tiny: the KATRIN experiment's 2025 limit is under 0.45 eV, at 90% confidence, at least a million times lighter than the electron, and the Standard Model does not say why. Dark matter makes up about five times as much as ordinary matter, and no Standard Model particle fits. The universe has far more matter than antimatter, and the model's known effects are too small to explain it. Gravity is not in the model at all. These are open, and physicists disagree on which gap will give first. What a graduate just sees Mass is a form of energy, and a collision can turn motion into new particles, if the centre-of-mass energy is enough. Colliders beat fixed targets because head-on collisions waste no energy on leftover motion. Quarks only come in colourless bundles. A proton's mass is mostly binding energy, not quark mass. Before asking if a reaction happens, check charge, baryon number, lepton number and energy. The heavier the force carrier, the shorter the force's range: ħc over mass. Energy per nucleon peaks at iron: fusion pays below it, fission pays above it. Each atom's decay is random with no memory; a crowd's decay is exponential and exact. A very large number of events is the only way to see rare processes. Where it shows up next Astrophysics & Cosmology (34/88): the binding-energy curve decides which stars make which elements, and the first minutes of the universe were particles interacting at colliders' energies and above. Condensed Matter Physics (34/87): the same quantum field ideas return as "quasi-particles" inside solids. Medicine: PET scanners detect the photons from positron annihilation, proton therapy aims beams at tumours, and technetium-99m (half-life 6 hours) is the commonest imaging isotope. Energy: fission reactors, fusion projects such as ITER, and radiocarbon dating all lean on the same curve and the same decay law. Computing: the web was invented at CERN, and detectors produce so much data that the trigger systems discard most events in microseconds. If you want to go deeper 📘 Textbook. David Griffiths, Introduction to Elementary Particles (Wiley-VCH, 2nd revised edition, 2008). Clear and short; the usual first particle-physics book after quantum mechanics. 📘 Textbook. Mark Thomson, Modern Particle Physics (Cambridge, 2013). More modern and more demanding, and the course text at Cambridge. 📘 Nuclear. Kenneth Krane, Introductory Nuclear Physics (Wiley, 1988). The standard undergraduate nuclear text. 📘 Gentler. Frank Close, Particle Physics: A Very Short Introduction (Oxford, 2004). No equations, one afternoon. 🌐 Free course. MIT OpenCourseWare 8.701 Introduction to Nuclear and Particle Physics (Fall 2020): notes, videos, problem sets and solutions. 🌐 Reference. The Particle Data Group's Review of Particle Physics: every measured mass and lifetime, free. The takeaway Matter is a short list of quarks and leptons, held together by forces whose reach is set by the mass of their carriers. A small set of conservation laws says what may happen; energy conversion by E = mc² says what it costs. The binding-energy curve explains why both the sun and a reactor work, and random decay with an exact crowd behaviour explains why radioactivity is unpredictable and measurable at once. Try this: next time you see a smoke detector, a banana or a bag of potassium-based salt substitute, say aloud that some atoms in it are decaying right now, each one at a random moment, and that the count per second is nevertheless steady. That is the exponential law at work. Sources & Further Reading Sources. Constants are CODATA values (ħc = 197.3269804 MeV·fm; atomic mass unit 931.49410242 MeV/c²). Masses and half-lives are standard tables (Particle Data Group; NNDC). Higgs: ATLAS and CMS announced a boson of about 125 GeV on 4 July 2012 at CERN. KATRIN: Science 388, 180 (2025), upper limit 0.45 eV (90% CL). Muon g-2: Fermilab final result, 3 June 2025, precision 127 ppb; the Standard Model prediction was revised by newer lattice calculations. Super-Kamiokande proton-decay searches give lifetimes above 10³⁴ years for the channels searched. The LHC began Run 3 at 13.6 TeV in July 2022. Antiproton found at the Bevatron in 1955 (Chamberlain, Segrè, Wiegand, Ypsilantis), threshold kinetic energy 6 mp = 5.63 GeV. Every number an interactive shows was computed in node: threshold beams for the Higgs 63.5 GeV (collider) and 8,604 GeV (fixed target); 13.6 TeV collider equals 9.86 × 10⁷ GeV fixed target; Yukawa ranges 1.41 fm (pion), 0.399 fm (kaon), 0.00245 fm (W); binding energies per nucleon He-4 7.074, C-12 7.680, O-16 7.976, Fe-56 8.790, Ni-62 8.795, U-235 7.591, U-238 7.570 MeV; D-T release 17.59 MeV, 3.37 × 10¹⁴ J/kg; C-14 after 20,000 years = 8.9%. 🌐 MIT OCW 8.701 🌐 Particle Data Group 📘 Griffiths, Introduction to Elementary Particles 📘 Close, Particle Physics: A Very Short Introduction Step 12. Stars: Structure & Evolution: Balance Sheets of Weight, Heat and Time ------------------------------------------------------------------------------ Why: How stars are built, live and die. A star is a ball of gas that has been losing a war against its own weight for billions of years and winning each battle by getting hotter. Five ideas explain why it shines, how long it lasts, how it ends and why you are made of its ashes. In the 1850s and 1860s Hermann von Helmholtz and Lord Kelvin asked where the Sun's energy comes from. Burning coal would last a few thousand years. Their best answer was gravity: let the Sun shrink slowly, and the weight of its outer layers falls inward and turns into heat. It could shine that way for tens of millions of years. Geologists and Darwin wanted hundreds of millions at least, and the argument ran for decades, until nuclear physics in the 1920s and 30s showed a far bigger reservoir. The lesson is lasting: a star is a bookkeeping problem. Weight pushes in, heat pushes out, and the sums have to balance at every layer, in every moment, for billions of years. Once you see how the books are kept, a surprising amount of stellar life follows from a few lines of arithmetic. You can see the results with a naked eye. Orion's Betelgeuse is red because it is cool and swollen; Rigel is blue-white because it is hot. The Pleiades are young and blue; the old star clusters are yellow and red. Astronomers read those colours as ages. The course in one box Who takes itAstronomy and astrophysics majors and physics majors who choose astrophysics, usually in year 3. Programmes such as MIT's Course 8 offer it as a stellar-structure course; the textbook route is Carroll and Ostlie or Kippenhahn and Weigert. What it is forPredicting a star's size, temperature, brightness, lifetime and fate from its mass and composition, using the laws of gases, radiation and nuclear reactions. Comes afterStatistical Mechanics (34/84), Particle & Nuclear Physics (34/86) and Telescopes, Light & Spectra (36/39). Leads toGalaxies & the Milky Way (36/42) and Compact Objects & High-Energy Astrophysics (36/43), which start where stars end. What this piece gives youYou will not be able to integrate the equations of stellar structure. You will see why contraction heats a star, why fusion works at all, why a thermostat holds the Sun steady, how long light takes to leave, how clusters give ages, and where the elements come from. Easier first reads: Astrophysics & Cosmology (34/88), whose main-sequence and mass–lifetime toys this piece builds on rather than repeats, and How a Nuclear Reactor Works (13/14). Hydrostatic equilibrium: a star that loses heat gets hotter Cut a slice through a star. The gas below the slice must hold up all the gas above it, so pressure rises toward the centre. The rule is hydrostatic equilibrium: at each layer, the pressure difference across the layer balances the weight of the layer. If it did not, the layer would fall or rise at once. If the balance failed, the Sun would collapse or expand within about an hour, its free-fall time, so any star we see is in this balance. A first consequence comes from comparing sizes. Gravity's grip on a gas particle grows with the star's mass and shrinks with its radius. For the gas to hold, each particle must be moving fast enough to resist, which means a temperature that scales as mass divided by radius. This is the virial theorem. kT ≈ G M m ÷ ( 5 R )for a uniform ball: the typical thermal energy of a gas particle of mass m is a fraction of its gravitational energy. A real star is centrally concentrated, so its core is several times hotter Worked for the Sun: with an average particle mass of 0.6 proton masses, this gives 2.8 million K for a uniform Sun. The real core is 15.7 million K, about 5.7 times hotter, because mass piles up toward the centre. Now the strange part. If a star radiates energy and shrinks, R goes down, so T goes up. The star gets hotter as it loses energy. Physicists describe this as negative heat capacity, and it is the reason stars can start fusion at all: a cloud collapsing under its own weight heats until the centre is hot enough to ignite. Interactive · Squeeze a young star Sun, settledSun-mass cloud, 30× biggerStill contractingToo light to ignite Mass Size Mean temperature T = GMμmp ÷ 5kR for a uniform ball with μ = 0.6, a rough scaling. The core estimate multiplies by 5.7, the ratio that matches the Sun's 15.7 million K, so it is a calibration to the Sun, not a stellar model. Settled sizes and brightness use the main-sequence scalings of 34/88. Time = GM² ÷ (RL), the Kelvin–Helmholtz time; for the Sun it is about 31 million years. Without JavaScript: for the Sun, contraction alone could power it for about 31 million years. The nuclear fuel lasts about 10 billion. That 31 million years is Kelvin's number. It was close to right for what gravity alone can supply, and it was wrong only about what the Sun uses. In 1920 Arthur Eddington argued that hydrogen turning to helium is the real source, and in 1939 Hans Bethe worked out the reaction chains. Fusion at fifteen million degrees: tunnelling and a thermostat Two protons repel each other. To fuse they must approach within about a femtometre, where the electric repulsion is about 1.4 million electron-volts of energy. At the Sun's core the typical thermal energy is 1.35 thousand electron-volts. Classically, the fraction of protons energetic enough is about e−1000, around 10−460. In a Sun of 1057 protons, none would ever fuse. The way out is quantum mechanics (see 34/83). A particle's wave leaks through the barrier. George Gamow worked out the tunnelling in 1928, and Robert Atkinson and Fritz Houtermans applied it to stars in 1929. The reaction rate is the product of two competing things: fewer and fewer particles at high energy, and a tunnelling chance that rises with energy. Their product peaks at a middle energy, the Gamow peak, here about 6 thousand electron-volts. That is about four and a half times the thermal energy, and a few hundred times below the barrier top. rate ∝ exp( −3 (EG ÷ 4kT)⅓ )EG = 2μc²(π α Z₁Z₂)²: it grows with the product of the two charges, so heavier nuclei tunnel much worse Out of this formula comes a striking result: how fast fusion speeds up as the core gets hotter. For two protons, the rate rises as about the fourth power of temperature. For a proton hitting nitrogen-14 (charge 7), the barrier is 49 times higher in this measure, and the rate rises as about the twentieth power. The Sun mostly uses the proton–proton chain, and the carbon–nitrogen–oxygen (CNO) cycle supplies about 1%, a share the Borexino experiment confirmed by detecting CNO neutrinos in 2020. In stars of about 1.3 Suns and heavier the extreme temperature sensitivity lets CNO take over. Interactive · Turn up the core Core temperature Computed from the Gamow factor, rate ∝ τ² e−τ with τ = 3 (EG ÷ 4kT)⅓, with CODATA constants; EG is 493 keV for p+p and 45.1 MeV for p+¹⁴N. Both curves are normalised to the Sun's core at 15.7 million K, with CNO set to 1% there. The slow weak-interaction step in the pp chain and other details are left out, so the crossover near 21 million K is indicative; textbooks put it at roughly 17 to 20 million K. Without JavaScript: at 15.7 million K the pp rate goes as T3.8 and CNO as T19.6. A 10% hotter core makes 44% more pp energy and 6.5 times more CNO energy. That steep dependence works as a thermostat. Suppose the core makes a bit too much energy. The gas heats and expands, which lowers the temperature, which cuts the fusion rate sharply back. Too little energy, and the core shrinks and heats, and the rate rises. The Sun has held steady for 4.6 billion years this way. The same steepness is why massive stars burn out so fast: a bit more mass means a hotter core and a rate that grows like a high power of temperature. Getting the energy out: a hundred-thousand-year random walk The energy made at the Sun's centre must reach the surface. Most of the way it travels as light, but the interior is so dense that a photon cannot go far. It is absorbed and re-emitted within a fraction of a centimetre, in a random direction. A random walk gets out slowly: after N steps of length l you are only about √N × l from the start, so to cover the Sun's radius R takes about (R ÷ l)² steps. time ≈ R² ÷ ( l c )a diffusion time: R the star's radius, l the photon's mean free path, c the speed of light With l about 0.03 cm, a typical value averaged over the Sun, this gives about 170,000 years. Estimates found in the literature run from tens of thousands to millions of years, depending on how the opacity is averaged; 170,000 years is the figure of Mitalas and Sills (1992). Without scattering the light would take 2.3 seconds. The sunshine on your face began its trip long before the first farms. The photons that finally reach you are new ones, made in the last scatterings at the surface. Interactive · A photon's drunken walk Distance to the surface Mean free path in the Sun New random walk The walk is a real three-dimensional simulation with isotropic steps of equal length, averaged over 150 trials; the Sun's number applies the diffusion rule R² ÷ (lc) with R = 6.957 × 10⁸ m. A uniform mean free path is a simplification: the real one varies by orders of magnitude between core and surface. Without JavaScript: a walker must take about (R ÷ l)² steps, so doubling the distance takes four times as many. For the Sun, that is about 170,000 years. Deep in the star, in some layers, radiation is not the best way to carry heat; the gas itself rises and sinks, boiling like a pot. This is convection. It takes over where the temperature falls steeply with radius. In the Sun it carries the heat through the outer 30% by radius, which is why the surface is a seething grain of cells. Stars lighter than about a third of the Sun are convective all the way through, which mixes their fuel and lets them burn for trillions of years. In stars heavier than about 1.3 Suns it is the core that boils. Leaving the main sequence: a cluster is a clock A star's hydrogen lasts, in round figures, 10 billion years divided by its mass to the power 2.5 (see 34/88). When the core runs out, the core contracts and heats, hydrogen burns in a shell around it, and the outer layers swell and cool. The star leaves the main sequence for the giant branch and gets brighter and redder. Astronomers use this on clusters: groups of stars born together from one cloud. All the stars have the same age but different masses, so at any time the heavy ones have left the main sequence and the light ones have not. The mass at the point where the main sequence bends off, the turn-off, gives the age of the cluster, because the star at the turn-off is just now running out of fuel. A young cluster like the Pleiades, about 100 million years old, has a main sequence reaching up into hot blue stars. An old globular cluster such as 47 Tucanae, about 12 billion years old, has a main sequence cut off at a star lighter than the Sun. Interactive · Age a cluster PleiadesHyadesM6747 Tucanae Cluster age Illustrative: 700 stars drawn with a Salpeter-like mass function (number ∝ M−2.35, 0.1 to 20 Suns), main-sequence positions from the 34/88 scalings, and schematic giants and white dwarfs. Real turn-offs also have a sub-giant branch and depend on composition. Cluster ages are rounded from the literature: Pleiades about 100 to 125 million years, Hyades about 650 million, M67 about 4 billion, 47 Tucanae about 12 billion. Without JavaScript: stars of 1 Sun mass leave the main sequence after about 10 billion years in this model, stars of 5 Suns after about 100 million. The oldest globular clusters come out at roughly 12 to 13 billion years by this method, a little under the 13.8 billion years that cosmology gives for the universe. Two independent clocks that do not contradict each other are a strong check on both. Endings: what the mass decides What happens next depends almost entirely on mass. The Sun will swell into a red giant in about 5 billion years, losing a good fraction of its mass in winds. Mercury and Venus will be swallowed; whether Earth survives is borderline in published models. By then the oceans will long since be lost, because the Sun brightens by roughly 10% every billion years. The Sun will shed its outer layers as a glowing shell, a planetary nebula, and its core will remain as a white dwarf of about half a Sun mass, supported by electron degeneracy pressure. A star above about 8 Suns goes further. It fuses carbon, neon, oxygen and silicon in turn, each stage faster than the last, until an iron core forms. Iron cannot release energy by fusing (see 34/86), so the core collapses in a fraction of a second into a neutron star, and the rebound launches a supernova. The energy is huge: the collapse releases about 3 × 1046 joules, of which only about 1% goes into the explosion, 99% leaving as neutrinos. When Supernova 1987A exploded in the Large Magellanic Cloud, about 168,000 light-years away, detectors on Earth caught 25 neutrinos on 23 February 1987 hours before the light arrived: 12 at Kamiokande II, 8 at IMB and 5 at Baksan. The first look inside a collapsing core came from a couple of dozen particles. Interactive · Choose a mass Initial mass Boundaries are rough and shift with composition, rotation and binary companions: about 8 Suns separates white dwarfs from supernovae, and the neutron star–black hole border (shown near 25) is uncertain by several Suns. White-dwarf masses use the initial–final relation Mf ≈ 0.109 Mi + 0.394 (Kalirai and others, 2008), good for roughly 1 to 8 Suns. Lifetimes from the 34/88 scaling. Without JavaScript: below about 0.075 Suns, brown dwarf; to about 8 Suns, white dwarf; to roughly 20–25 Suns, usually a neutron star; above that, often a black hole. White dwarfs cannot exceed about 1.4 Suns, the Chandrasekhar limit. A white dwarf in a binary that gains matter can approach it and explode as a type Ia supernova, with a fairly standard brightness. That is why these explosions work as distance markers across the universe (see 34/88). What they leave, and how neutron stars and black holes behave, is the work of Compact Objects & High-Energy Astrophysics (36/43). Stars make the elements The Big Bang made hydrogen, helium and a trace of lithium. Everything heavier was made in stars or in the events that end them. The oxygen you breathe and the calcium in your bones were made in massive stars, and carried out by supernovae and winds into the gas clouds that later formed the Sun. In 1957 Margaret and Geoffrey Burbidge, William Fowler and Fred Hoyle published the foundational paper on how it all works (usually called B²FH), after Hoyle had predicted in 1953 the special energy level in carbon that makes the triple-alpha reaction possible. Iron is the end of the line for fusion, and heavier elements need neutrons. Slow neutron capture in giant stars builds barium and lead. Rapid capture needs a flood of neutrons for a second or less, and for gold that source was long a mystery. In August 2017 the LIGO and Virgo detectors recorded gravitational waves from two neutron stars merging (GW170817), and telescopes then watched the afterglow, a kilonova, whose light matched freshly made heavy elements. So mergers make r-process elements. Whether they make most of them is not settled. Interactive · Where did this atom come from? HHeLiCOFeBaAu Tap an element to see where it came from. Summarised from the standard picture (for example Kobayashi, Karakas and Lugaro, 2020). Fractions are uncertain for most elements. Without JavaScript: hydrogen, helium and some lithium from the Big Bang; carbon in giant stars; oxygen and much iron in supernovae; barium in giant stars; gold in neutron-star mergers and possibly other events. What a graduate just sees A star is a balance of weight and pressure at every layer. Lose energy and it contracts and heats. Temperature scales as mass over radius, so a collapsing cloud always finds the temperature to start fusion if it has enough mass. Fusion works by tunnelling, and its rate climbs as a steep power of temperature, which makes the Sun a thermostat. Light takes about a hundred thousand years to leave the Sun, but eight minutes to cross space afterwards. A cluster is one age and many masses, so the turn-off reads its age. Mass is destiny: brown dwarf, white dwarf, neutron star or black hole, with a thin fuzzy border between them. Most of the elements heavier than helium were forged in stars and scattered by their deaths. Where it shows up next Compact Objects & High-Energy Astrophysics (36/43): white dwarfs, neutron stars and black holes are the remnants this piece stops at. Galaxies & the Milky Way (36/42): a galaxy's colour and history are the sum of its stars' lifetimes, and the elements stars make enrich each generation. Planetary Science & Exoplanets (36/40): a star's brightening over billions of years moves its habitable zone outward. In the world: solar neutrino detectors, fusion energy research (which copies the Sun's tricks at lower density and higher temperature), and stellar models behind the age of the oldest known stars. If you want to go deeper 📘 Textbook. Bradley Carroll and Dale Ostlie, An Introduction to Modern Astrophysics (Cambridge University Press, 2017 reissue of the 2nd edition), for the standard undergraduate treatment. Rudolf Kippenhahn, Alfred Weigert and Achim Weiss, Stellar Structure and Evolution (Springer, 2nd edition 2012), for the full structure equations. 📘 Gentler. James Kaler, Stars and Their Spectra (Cambridge University Press, 2nd edition 2011), a readable tour of the stars and what their light says. 🌐 Free tool. MESA, the open-source stellar evolution code used in research, with its documentation and tutorials. 🌐 Free course. MIT OpenCourseWare 8.901 Astrophysics I (Spring 2006), covering stellar structure and evolution. The takeaway A star is gravity held off by heat, with fusion paying the bill. Contraction heats a star until tunnelling lights its core, a steep temperature dependence keeps it steady, and the light takes a hundred thousand years to escape. Its mass decides its lifetime and its end, clusters turn that into a clock, and the dying stars scatter the elements that planets and people are made of. Try this: find Betelgeuse (red) and Rigel (blue-white) in Orion on a winter evening and compare their colours. One is a cool, swollen star near the end of its fuel, and the other a hot, young star. Both are heavier than the Sun and will end in supernovae. Sources & Further Reading Sources. Eddington, "The Internal Constitution of the Stars," The Observatory 43 (1920); Bethe, "Energy Production in Stars," Physical Review 55, 434 (1939); Gamow, Zeitschrift für Physik 51 (1928); Atkinson and Houtermans, Zeitschrift für Physik 54 (1929); Borexino Collaboration, "Experimental evidence of neutrinos produced in the CNO fusion cycle in the Sun," Nature 587, 577 (2020); Mitalas and Sills, Astrophysical Journal 401, 759 (1992); Kalirai and others, Astrophysical Journal 676, 594 (2008); Burbidge, Burbidge, Fowler and Hoyle, Reviews of Modern Physics 29, 547 (1957); Abbott and others (LIGO–Virgo), GW170817, Physical Review Letters 119, 161101 (2017); SN 1987A neutrino detections (Hirata and others 1987; Bionta and others 1987; Alekseev and others 1987); Kobayashi, Karakas and Lugaro, Astrophysical Journal 900, 179 (2020). Solar values: G, k, mp, c from CODATA 2018; M☉, R☉, L☉ from IAU 2015 nominal values. Numbers in the interactives are computed in the page from the formulas shown. 📘 Carroll & Ostlie, An Introduction to Modern Astrophysics 📘 Kippenhahn, Weigert & Weiss, Stellar Structure and Evolution 🌐 MIT OCW 8.901 Astrophysics I Step 13. Galaxies & the Milky Way: Island Cities of Stars, Weighed by Their Spin -------------------------------------------------------------------------------- Why: Our galaxy and the others, and the dark matter in them. Our galaxy is a spinning disk of hundreds of billions of stars that we live inside. Six ideas show how astronomers map it from within, weigh it, sort its cousins, explain its arms and see its past and its future. On a dark night away from city lights, a pale band crosses the sky. Peoples everywhere have read it. In Chinese tradition it is the Silver River. The Inca read animal shapes in its dark dust clouds, and Aboriginal Australians saw an emu stretched along its dark lanes. All were looking at the same thing: the Milky Way's disk, seen edge-on from a point inside it. It took until the 1920s to learn that the band is only one galaxy among a great many. In 1923 and 1924 Edwin Hubble used Cepheid stars to show that the Andromeda "nebula" was far outside the Milky Way, and the universe suddenly held islands of stars. Today the study of galaxies asks how these systems were built, how fast they turn, what holds them together and how they change. This piece starts from inside the Milky Way, where we stand, and moves outward to its neighbours and to its centre. It goes beyond the dark-matter rotation curve of Astrophysics & Cosmology (34/88) rather than repeating it: here the question is the structure of galaxies, why they have their shapes, and what happens when they meet. The course in one box Who takes itAstronomy and astrophysics majors in year 4, and physics majors in an astrophysics track. MIT's Astrophysics II (8.902) is an example of a course that covers galaxies, and Caltech's Ay courses and Cambridge's astrophysics options cover similar ground. What it is forUnderstanding how stars, gas and dark matter are arranged in galaxies, how galaxies move, grow and merge, and what lives at their centres. Comes afterStars: Structure & Evolution (36/41). Leads toAstrophysics & Cosmology (34/88), where galaxies become the units of the universe, and Compact Objects & High-Energy Astrophysics (36/43). What this piece gives youYou will not be able to model a galaxy's potential or run a merger simulation. You will see how a galactic year is counted, why spin gives mass, how galaxies are sorted, why spiral arms survive, how a collision plays out and how one star weighed a black hole. Easier first read: The Big Bang and Our Expanding Universe (13/08). The Milky Way from inside: a galactic year The Milky Way is a flat disk with a bulge and a bar at the middle, wrapped in a thin, nearly invisible halo of old stars. The stellar disk is about 100,000 light-years across, and the Sun sits at roughly 8.2 kiloparsecs, about 27,000 light-years, from the centre (a kiloparsec is 3,262 light-years). The distance was pinned down by watching the star S2 orbit the black hole at the centre (GRAVITY collaboration, about 8.28 kpc in 2021). The Sun moves around the centre at about 230 km/s, with estimates from the Gaia satellite in the range of roughly 229 to 236. period = 2π R ÷ vthe length of one lap divided by the speed; R in kiloparsecs, v in km/s, 1 kpc = 3.0857 × 10¹⁶ km Worked: 2π × 8.2 kpc is 1.59 × 10¹⁸ km. Divide by 230 km/s: 6.9 × 10¹⁵ seconds, or about 219 million years. A "galactic year" is often quoted as 225 to 250 million years, depending on the speed and distance chosen. Since the Solar System formed 4.57 billion years ago, it has gone round about 21 times. The last time the Sun was where it is now, dinosaurs were new. Interactive · How long is a lap? The SunInner disk, 4 kpcOuter disk, 15 kpc Distance from centre Orbital speed Period = 2πR ÷ v for a circular orbit. The picture is a schematic top view with the disk edge drawn at 15 kpc and a bar tilted for illustration; it is not a map of the real spiral arms. The Sun's orbit is slightly elliptical in practice. Without JavaScript: a star at 8.2 kpc moving at 230 km/s takes 219 million years per lap and has made about 21 laps since the Solar System formed. Notice what the slider shows. Hold the speed fixed and a star twice as far out takes twice as long, because it has twice as far to go. Real disks are close to this: speeds stay about the same over a wide range of radii. That flatness is the whole reason galaxies need extra mass, taken up in the next section. Mapping the disk from inside is hard, because dust hides the centre in visible light. Infrared and radio see through it. The European Space Agency's Gaia satellite measured the positions and motions of about 1.8 billion stars in its third data release (13 June 2022), the most detailed map of our galaxy so far. Gaia stopped observing on 15 January 2025. Its data show that the disk is warped and rippling, and that the halo is full of streams of stars torn from smaller galaxies. One of them, the Gaia–Enceladus (or "Sausage") merger, brought in a galaxy about 8 to 11 billion years ago; the Sagittarius dwarf galaxy, found in 1994, is being pulled apart now. Weighing a galaxy by how fast it turns Orbits give mass. The faster a star circles at a given radius, the more gravity must be pulling it inward from the mass inside. Worked: with the Sun's numbers, the speed of 230 km/s at 8.2 kpc gives a mass inside the Sun's orbit of 1.0 × 10¹¹ Suns, from just the speed and radius. M(inside R) = v² R ÷ Gwith G = 4.30 × 10⁻⁶ kpc (km/s)² per Sun mass, v in km/s and R in kpc, the mass comes out in Suns Apply it far out, where stars and gas are thin on the ground, and the mass still grows. That is the dark-matter evidence of 34/88. Here is a second and stronger test. Across thousands of spiral galaxies, the speed at which the outer parts rotate predicts the total mass of the stars and gas, with little scatter. The baryonic Tully–Fisher relation says the mass of stars plus gas grows as the fourth power of the flat rotation speed: double the speed and the visible mass is 16 times bigger. Brent Tully and Richard Fisher found the original relation for luminosity in 1977; Stacy McGaugh and colleagues later showed that it is tightest when gas is counted too. The fit gives about 47 Suns for each (km/s)⁴. Interactive · How much of the mass shines? Dwarf galaxyMid-size spiralMilky Way-like Rotation speed Outermost radius measured Illustrative: Mbaryon = 47 v⁴ (McGaugh 2012; the fitted slope in later work is about 3.8 to 4), scatter about 0.1 to 0.15 in the logarithm. Presets pair typical speeds with typical outermost radii; real galaxies vary. The Milky Way's own stars and gas are lower, around 6 × 10¹⁰ Suns by recent estimates, so a single galaxy can sit below the line. Without JavaScript: at 230 km/s, the mass inside 30 kpc is about 3.7 × 10¹¹ Suns and the Tully–Fisher relation gives about 1.3 × 10¹¹ Suns of stars and gas; at 40 km/s inside 6 kpc, the mass is about 2.2 × 10⁹ Suns and the relation gives 1.2 × 10⁸. The slider tells a clear story: the smaller the galaxy, the larger the share of invisible mass inside its measured radius. The relation is tight and has a slope near four. In the standard dark-matter picture, that tightness has to arise from how galaxies form in their halos, and its precision is an open challenge for simulations. A competing idea, Mordehai Milgrom's MOND (1983), predicts a slope of exactly four as a law. Each side points to the data it explains best, and clusters and the cosmic microwave background, which MOND struggles with, are the main evidence against it. The Milky Way's total mass, dark halo included, is roughly 1 to 2 trillion Suns, uncertain by a factor of about two. The galaxy zoo: shapes that record how stars were made In 1926 Edwin Hubble sorted galaxy photographs into a scheme that later became the tuning fork. On one prong: ellipticals, from round (E0) to flat (E7). On the other: spirals, from tight arms and a big bulge (Sa) to loose arms and a small bulge (Sc), with a parallel prong of barred spirals. Between them: lenticulars (S0), disks without arms. Hubble did not mean it as an evolutionary sequence, and it is not one, but the shapes do carry history. The shapes go with colour. Ellipticals are red because their stars are old; they have little cold gas, and make few new stars. Spirals are bluer because their arms are lit by young, hot stars from gas that is still collapsing. Plot colour against brightness for many galaxies and they fall into two groups, a red sequence and a blue cloud, with a sparse "green valley" between, where galaxies are shutting off star formation. What switches it off (a black hole's heating, the loss of gas in a cluster, mergers) is under study. The Milky Way is a barred spiral near the green valley, with a bar that Spitzer Space Telescope maps confirmed in 2005. Interactive · A tuning fork of galaxies E0E5S0SaSbScSBb Schematic drawings made from random stars on rules (bulge size, spiral pitch angle from 10° for Sa to 26° for Sc, a bar for SBb); they show how the types differ, not real galaxies. For real ones, the Galaxy Zoo project has volunteers classify survey images. Without JavaScript: E0 round and old; E5 flattened; S0 disk without arms; Sa tight arms and a big bulge; Sc loose arms and many young stars; SBb has a central bar. Spiral arms: a traffic jam, not a river Spiral arms pose a puzzle. Stars near the centre go round faster than stars far out. If the arms were fixed chains of stars, they would wind up like a coil and vanish into tight rings within a few hundred million years. The Milky Way has gone round some 50 times in its life, and the arms are still open. This is the winding problem. The classic answer, by C. C. Lin and Frank Shu in 1964, is that arms are not made of fixed stars but are a wave of extra density that moves through the disk at its own speed, like a traffic jam on a motorway. Cars enter, slow, leave, and the jam stays in place. Stars and gas pile up as they pass through, and gas compressed in the wave forms new stars, which is why arms are lit blue. Estimates of the Milky Way's pattern speed run roughly from 20 to 30 km/s per kiloparsec, and not everyone agrees. Computer simulations also show arms that form, break and re-form over a few hundred million years, and grand-design spirals such as M51 are tied to the tidal pull of a nearby companion. Several mechanisms probably operate in different galaxies. Interactive · Do the arms wind up? Stars carry the armPattern wave Time elapsed A toy: 520 stars on circular orbits in a flat rotation curve of 220 km/s, with a two-arm logarithmic spiral of 20° pitch. In "pattern" mode the arm turns rigidly at 25 km/s/kpc and brightens whichever stars are inside it at that moment; real density waves also change the stars' orbits slightly, which this leaves out. Without JavaScript: in 1 billion years a star at 4 kpc goes round about 9 times and one at 12 kpc about 3 times, so an arm made of fixed stars would be wound round about six extra times. Galaxies collide: the Milky Way and Andromeda The Andromeda galaxy (M31) is about 2.5 million light-years away and the nearest large galaxy. Its light is shifted toward blue, so it is approaching us at about 110 km/s. Gravity speeds up the approach. Whether the galaxies actually collide depends on the sideways motion, which is hard to measure because Andromeda's apparent drift across the sky is almost impossibly small. Hubble Space Telescope measurements published in 2012 by Sangmo Tony Sohn, Jay Anderson and Roeland van der Marel found sideways motion small enough to be consistent with zero, and predicted a merger in about 5 billion years. A later analysis in 2025 by Till Sawala and colleagues, which includes the pull of the galaxy M33 and the Large Magellanic Cloud and allows for measurement uncertainty, concluded that there is only about a 50% chance of a merger within the next 10 billion years. Interactive · Aim the galaxies Combined mass Sideways speed Two point masses integrated in time: start at 780 kpc, approaching at 110 km/s. Real galaxies have extended halos, lose orbital energy to dynamical friction and are pulled by neighbours, none of which a point-mass orbit includes, so treat the times as a feel for the physics, not a forecast. The slider range for mass is the commonly quoted 2 to 8 trillion Suns. Without JavaScript: at the default mass, with no sideways motion, the galaxies fall straight together in about 3.3 billion years; with 100 km/s sideways, they miss by about 180 kpc on the first pass. If they do merge, nothing like a crash of stars occurs: the gaps between stars are so wide that stars almost never collide. The two disks are scrambled, gas is compressed and a burst of star formation follows, and in the end the pair settle into one big, round galaxy, an elliptical. This is how many ellipticals are thought to have formed. Mergers are a main theme of modern galaxy formation, and the Milky Way's halo already holds the remains of galaxies it absorbed. The monster in the middle: weighing Sagittarius A* with one star At the centre of the Milky Way a dense cluster of stars orbits something dark. For nearly three decades two teams, led by Reinhard Genzel and Andrea Ghez, tracked individual stars through the dust with infrared telescopes in Chile (the European Southern Observatory) and Hawaii (Keck). The star S2 goes round in 16.05 years on an elongated orbit, coming within about 120 AU of the object, which is roughly four times Neptune's distance from the Sun, at about 2.5% of the speed of light. Kepler's third law, from 36/40, does the rest: mass equals the cube of the orbit's size in AU over the square of the period in years, in Suns. M = a³ ÷ P²a the semi-major axis in AU, P the period in years, M in Sun masses (the planet's or star's own mass neglected) Worked: S2's orbit has a of about 1,035 AU and P = 16.05 years. The cube is 1.11 × 10⁹, the square of the period is 257.6, so M is 4.3 million Suns, in agreement with the published 4.3 million. That mass packed inside an orbit smaller than our Solar System is too compact to be anything but a black hole. Genzel and Ghez shared the 2020 Nobel Prize in Physics for this, with Roger Penrose. In 2022 the Event Horizon Telescope published the first image of its glowing ring of gas. Interactive · Weigh the centre S2Earth around the Sun Orbit size Orbit period M = a³ ÷ P² in solar units, then Schwarzschild radius 2GM ÷ c². S2 values: period 16.05 years, semi-major axis about 1,035 AU, eccentricity about 0.88 (GRAVITY Collaboration 2022, with M = 4.30 million Suns). The orbit drawing is a schematic of that shape. Without JavaScript: a = 1,035 AU and P = 16.05 years give about 4.3 million Suns. The Earth around the Sun gives 1, which checks the formula. Almost every large galaxy seems to have such a black hole, and the more massive its bulge's stars, the heavier it is: the black hole's mass tracks the stars' random speeds in the bulge, a relation found around 2000. That is odd, because the hole's pull reaches only a tiny part of the bulge. It hints that galaxies and their central holes grew together, perhaps through the energy the hole releases when it feeds. Astronomers are still working out how. What a graduate just sees A galaxy's spin is a scale: orbital speed gives the mass inside, and a lap for the Sun takes about 220 million years. The visible stars are a minority of the mass inside a galaxy's disk, and smaller galaxies are more dark-matter dominated. Galaxy shape records history: red and round means old stars and few new ones, blue and spiral means gas and star formation. Spiral arms are waves of density, not rivers of stars, but how they start is still debated. Big galaxies are built by merging smaller ones, and collisions mix gas and stars rather than smashing stars. Kepler's law at the very centre of a galaxy weighs its black hole. Most of what we call "the Milky Way" is a view from inside, so mapping it needs infrared, radio and careful statistics. Where it shows up next Astrophysics & Cosmology (34/88): galaxies are the tracers by which the expansion, dark matter and the cosmic web are measured. Compact Objects & High-Energy Astrophysics (36/43): central black holes switch on as quasars when they feed, and galaxy clusters shine in X-rays. Astronomical Data & Computation (36/44): Gaia's billions of stars and the Vera C. Rubin Observatory's survey in Chile are galaxy science run as data science. In the world: radio arrays such as MeerKAT in South Africa, which imaged the Galactic Centre in 2022 and is a precursor to the Square Kilometre Array, and the stars in your sky. If you want to go deeper 📘 Textbook. Linda Sparke and John Gallagher, Galaxies in the Universe: An Introduction (Cambridge University Press, 2nd edition 2007), a standard undergraduate text. James Binney and Michael Merrifield, Galactic Astronomy (Princeton University Press, 1998), for the Milky Way in detail. James Binney and Scott Tremaine, Galactic Dynamics (Princeton, 2nd edition 2008), for graduate study. 📘 Gentler. Ken Croswell, The Alchemy of the Heavens (Anchor Books, 1995), a readable account of how we mapped the Milky Way. 🌐 Free tools. Zooniverse, where Galaxy Zoo volunteers classify real galaxy images, and the free planetarium program Stellarium for finding the Milky Way's band. 🌐 Free course. MIT OpenCourseWare 8.902 Astrophysics II (Fall 2004), covering galaxy dynamics, active galaxies and cosmology. The takeaway Galaxies are weighed by how fast they turn, and the answer is that most of their mass is dark. Their shapes and colours tell of old and new stars, their spiral arms are waves rather than fixed chains, they grow by merging, and each has a heavy black hole at its centre that Kepler's law can weigh. We see our own from the inside, one lap every 220 million years. Try this: on a dark, moonless night, find the Milky Way's band and its dark lanes. Then find the faint smudge of the Andromeda galaxy (in autumn in the northern hemisphere, near the W of Cassiopeia). The light you see left it about 2.5 million years ago. Sources & Further Reading Sources. Hubble, "A spiral nebula as a stellar system: Messier 31," Astrophysical Journal 69 (1929) and the 1924 Cepheid announcement; Hubble, The Realm of the Nebulae (1936); GRAVITY Collaboration, Astronomy & Astrophysics 657, L12 (2022): M = 4.30 million Suns, distance 8.28 kpc, S2 period 16.05 years; Genzel and Ghez, Nobel Prize in Physics 2020; Event Horizon Telescope, Sgr A* images (12 May 2022); Gaia Data Release 3 (ESA, 13 June 2022; observations ended 15 January 2025); Helmi and others, Nature 563, 85 (2018); Ibata, Gilmore and Irwin, Nature 370, 194 (1994); Tully and Fisher, Astronomy & Astrophysics 54, 661 (1977); McGaugh, Astronomical Journal 143, 40 (2012); Lin and Shu, Astrophysical Journal 140, 646 (1964); Sohn, Anderson and van der Marel, Astrophysical Journal 753, 7 (2012); Sawala and others, Nature Astronomy (2025); Benjamin and others (GLIMPSE), Astrophysical Journal Letters 630, L149 (2005); Ferrarese and Merritt, Astrophysical Journal Letters 539, L9 (2000), and Gebhardt and others, Astrophysical Journal Letters 539, L13 (2000); Milgrom, Astrophysical Journal 270, 365 (1983). Numbers in the interactives are computed in the page from the formulas shown. 📘 Sparke & Gallagher, Galaxies in the Universe 📘 Binney & Merrifield, Galactic Astronomy 🌐 MIT OCW 8.902 Astrophysics II Step 14. General Relativity: Gravity Is the Shape of Spacetime -------------------------------------------------------------- Why: General relativity: gravity as geometry. Newton said masses pull on each other. Einstein said masses bend the stage, and everything simply goes straight on a bent stage. The difference shows up in the clocks in your phone, in the light of distant galaxies, and in ripples that crossed a billion years of space. General Relativity: Gravity Is the Shape of Spacetime :root{ --bg:#faf7f2; --card:#ffffff; --ink:#1c1a17; --muted:#6b645b; --accent:#2a3f8f; --accent2:#b5842a; --line:#e7e0d6; --hi:#e9ebf7; --panel:#eef0f8; --hot:#c4452d; --cold:#2f6fb0; --good:#3f7f63; --onacc:#ffffff; --shadow:0 8px 30px rgba(0,0,0,.06); } @media (prefers-color-scheme:dark){ :root{ --bg:#0e0f16; --card:#181a24; --ink:#eef0f8; --muted:#a3a8bd; --accent:#8fa0f0; --accent2:#e6b969; --line:#282b3a; --hi:#171a2b; --panel:#10132a; --hot:#ee7b62; --cold:#6fa6e0; --good:#5fb08b; --onacc:#0e0f16; --shadow:0 8px 30px rgba(0,0,0,.45); } } *{box-sizing:border-box} body{margin:0;background:var(--bg);color:var(--ink); font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Georgia,serif; line-height:1.65;-webkit-font-smoothing:antialiased} .wrap{max-width:680px;margin:0 auto;padding:48px 16px 80px} .kicker{color:var(--accent);font-weight:700;letter-spacing:.08em;text-transform:uppercase;font-size:.72rem;font-family:-apple-system,sans-serif} h1{font-size:2.1rem;line-height:1.12;margin:.3em 0 .2em;letter-spacing:-.02em} .dek{font-size:1.15rem;color:var(--muted);margin-bottom:.6em} .meta{font-size:.8rem;color:var(--muted);font-family:-apple-system,sans-serif;border-bottom:1px solid var(--line);padding-bottom:20px;margin-bottom:26px} p{font-size:1.06rem;margin:1em 0} h2{font-size:1.45rem;line-height:1.2;margin:1.9em 0 .3em;letter-spacing:-.01em} .pull{font-size:1.4rem;line-height:1.28;font-weight:600;color:var(--accent);border-left:4px solid var(--accent);padding-left:20px;margin:1.4em 0} .card{background:var(--card);border:1px solid var(--line);border-radius:16px;padding:20px 16px;margin:26px 0;box-shadow:var(--shadow)} .lab{font-family:-apple-system,sans-serif;font-size:.72rem;font-weight:700;letter-spacing:.08em;text-transform:uppercase;color:var(--accent2);margin-bottom:.6em} .course{border-left:5px solid var(--accent2)} .course dl{margin:0;font-family:-apple-system,sans-serif;font-size:.95rem} .course dt{font-weight:700;color:var(--accent);margin-top:.7em} .course dd{margin:.1em 0 0 0} .eq{font-family:Georgia,"Times New Roman",serif;font-style:italic;font-size:1.25rem;text-align:center;background:var(--hi);border-radius:12px;padding:12px 8px;margin:14px 0;overflow-x:auto} .eq small{font-style:normal;font-family:-apple-system,sans-serif;font-size:.8rem;color:var(--muted);display:block;margin-top:2px} .chips{display:flex;flex-wrap:wrap;gap:8px;margin-bottom:6px} button{font-family:-apple-system,sans-serif;font-weight:600;cursor:pointer;border:1px solid var(--line);border-radius:999px;padding:8px 14px;font-size:.88rem;background:var(--hi);color:var(--ink);min-height:44px} button[aria-pressed=true]{background:var(--accent);color:var(--onacc);border-color:var(--accent)} .ctl{font-family:-apple-system,sans-serif;font-size:.9rem;margin-top:8px} .ctl .top{display:flex;justify-content:space-between;gap:8px} input[type=range]{width:100%;accent-color:var(--accent);min-height:44px;margin:0} svg{width:100%;height:auto;display:block;margin:10px 0;border-radius:12px;background:var(--panel)} svg text{font-family:-apple-system,sans-serif;font-size:11px;fill:var(--muted)} .read{font-family:-apple-system,sans-serif;font-size:.92rem;background:var(--hi);border-radius:10px;padding:10px 12px;font-variant-numeric:tabular-nums} .read p{font-size:.92rem;margin:.3em 0} table.t{width:100%;border-collapse:collapse;font-family:-apple-system,sans-serif;font-size:.9rem;margin:12px 0;font-variant-numeric:tabular-nums} table.t th,table.t td{border-bottom:1px solid var(--line);padding:6px 4px;text-align:left;vertical-align:top} table.t th{color:var(--muted);font-weight:600} .note{font-size:.85rem;color:var(--muted);font-family:-apple-system,sans-serif} .take{background:var(--hi);border-radius:16px;padding:22px 18px;margin-top:34px} .take .lab{color:var(--accent)} .do{font-family:-apple-system,sans-serif;font-weight:700;font-size:1.02rem;margin-top:.4em} ul.deep,ul.see{font-family:-apple-system,sans-serif;font-size:.96rem;line-height:1.6;padding-left:1.2em} ul.see li{margin:.35em 0} a{color:var(--accent2)} button:focus-visible,input:focus-visible,select:focus-visible{outline:3px solid var(--accent2);outline-offset:2px} @media (max-width:640px){table.t{display:block;overflow-x:auto;max-width:100%}} The satellites that tell your phone where you are carry atomic clocks. Left alone, those clocks would run about 38 microseconds a day ahead of clocks on the ground. That is 0.000038 of a second, and it would be fatal: a navigation system measures distance as light-travel time, and light covers 11 kilometres in 38 microseconds. Within a day, the blue dot on your map would be a long way from the road. The clocks are not broken. Gravity is a little weaker 20,000 km up, and a clock in weaker gravity ticks faster. Engineers build that correction into the satellites before launch. This is general relativity, working quietly as engineering. If you want the friendlier overview first, read Why Time Is Weirder Than You Think (13/03), Orbits: Why the Moon Keeps Falling (13/11) and The Big Bang and Our Expanding Universe (13/08). This piece goes further: it computes. The course in one box Who takes itPhysics majors, usually in year 4 as an option, or as a first graduate course (MIT's 8.962 General Relativity is a typical example). Programmes in the style of MIT Course 8 and the Cambridge Natural Sciences physics route often teach a short introduction in the final year. What it is forWorking out how matter and energy shape spacetime, and how clocks, light and bodies move through it: planets, black holes, gravitational waves and the whole universe. Comes afterSpecial Relativity (34/81) and Differential Geometry (34/77). Leads toAstrophysics & Cosmology (34/88). What this piece gives youYou will not be able to solve Einstein's equations. You will see why free fall is the natural state, how to estimate clock shifts, light bending, black-hole sizes and gravitational-wave chirps, and what the equation says in words. Free fall is the natural state Stand in a lift whose cable has snapped (in thought only). You and a dropped coin fall together; the coin floats beside you, and no experiment inside the lift can tell you whether you are falling near a planet or drifting in empty space. Einstein called this the happiest thought of his life, in 1907. It is the equivalence principle: a small enough freely falling room feels no gravity at all. Why is this so surprising? Because every object falls at the same rate, whatever its mass or material. In Newton's theory that is a coincidence: the mass that resists a push and the mass that gets pulled by gravity just happen to be equal. Einstein turned the coincidence into the foundation. If everything falls the same way, then falling is not a property of things. It is a property of the place. The reverse picture helps too. In a rocket accelerating at 9.8 m/s² in empty space you feel your normal weight, and a ray of light crossing the cabin bends downward, because the floor rises to meet it. Equivalence says gravity must bend light as well. Clocks run at different rates Light climbing out of a gravity well loses energy, so its frequency drops (a redshift). Read the other way: clocks lower in a gravitational field run slow. To first order the fractional rate difference is the difference in gravitational potential divided by c². Near Earth it is a tiny number, but atomic clocks read it. Pound and Rebka measured the shift between the bottom and top of a 22.5 m tower at Harvard in 1959, and optical clocks today can see the difference for a height change of a few centimetres. Δτ/τ ≈ Δ(GM/r) ÷ c² − v² ÷ 2c²the first term: weaker gravity speeds a clock up · the second: motion slows it down (special relativity) A GPS satellite gets both. Its altitude (about 20,200 km) speeds the clock up by about 45.8 µs a day; its speed of 3.9 km/s slows it by about 7.2 µs. The net is about +38.6 µs a day. A satellite in a low orbit like the International Space Station is dominated by the second term, so its clocks run slow. Somewhere in between the two effects cancel. Interactive · Clock in orbit Ground (circular orbit at the surface)ISSGPSGeostationary Orbit altitude Circular orbit; Earth's geoid potential 6.2636856 × 10⁷ m²/s², GM = 3.986004418 × 10¹⁴ m³/s², c = 299,792,458 m/s. Weak-field formula; the satellite is compared with a clock at rest on the geoid. The "ground" preset is a hypothetical satellite just above the surface. Slide the altitude: the clock is slowest overall at the surface, breaks even near 3,170 km, and runs steadily faster beyond. For GPS the uncorrected error would build up at 11.6 km a day. Notice what this means for everyday gravity: you age slightly faster at the top of a skyscraper than at its foot, by about one part in 1014 for every 100 m of height. Nobody notices, but precision clock networks have to account for it. Straight lines on a curved stage So what is gravity? Einstein's answer: spacetime has a shape, and freely falling things follow the straightest possible paths in it, called geodesics. The Moon is not pulled off a straight line; it is going straight, on a stage bent by the Earth. Think of two airliners flying due north from the equator. They start parallel and end up meeting at the pole. Neither turned, and no force pulled them together. The curvature of the surface did it. That picture makes two predictions that differ from Newton's, and both have been measured. 1. Orbits do not close. In Newton's theory a single planet traces the same ellipse forever. In general relativity the ellipse slowly turns, so the closest point creeps forward each orbit. The weak-field result is a turn of Δφ = 6πGM ÷ (c²a(1 − e²)) per orbit. For Mercury it is 5.0 × 10−7 radians per orbit, which adds up to 43 arcseconds a century. Astronomers had measured an extra turn of that size since the 1850s; Newtonian tugs from other planets explained the rest, and general relativity explained the leftover without any adjustment, which Einstein found in 1915. Interactive · An orbit that will not close EinsteinNewton Size of orbit (a, in units of GM/c²) Eccentricity (how stretched) Computed live by integrating the Schwarzschild orbit equation u″ + u = 1/L² + 3u² (units G = M = c = 1) with a fourth-order Runge–Kutta step, over four orbits. The Newtonian case drops the 3u² term. Mercury's own orbit is about 650,000 times wider than the smallest shown here, so its turn is far too small to see. Compare the simulated turn with the weak-field formula. For wide orbits they agree to a few per cent; for tight ones the formula underestimates, because the curvature is no longer weak. 2. Light bends twice as much as Newton would say. Light follows geodesics too, and a ray passing a mass of M at distance b is deflected by an angle α = 4GM ÷ (c²b). That is twice the value you would get by treating light as fast Newtonian particles. For rays grazing the Sun the angle is 1.75 arcseconds. Arthur Eddington's expeditions photographed stars near the Sun during the total eclipse of 29 May 1919, at Príncipe (off West Africa) and Sobral (Brazil), and reported shifts close to Einstein's number (roughly 1.6 and 2.0 arcseconds, with uncertainties of a few tenths; historians still debate how the plates were chosen and reduced). Modern radio measurements agree with Einstein to a small fraction of a percent. Interactive · Bend the light SunJupiterEarthGalaxy cluster Passing distance α = 4GM/(c²b), valid when b is much larger than 2GM/c². The cluster is 1014 solar masses with distance in kiloparsecs. The bend in the drawing is exaggerated and not to scale. The cluster preset is why the effect matters to astronomers. A cluster of galaxies bends background light by tens of arcseconds, enough to smear distant galaxies into arcs and to weigh the cluster, including the mass we cannot see. Black holes: where curvature wins Squeeze a mass into a small enough space and no path leads outward. The critical radius is the Schwarzschild radius, found in 1916 within months of Einstein's equations: rs = 2GM ÷ c²about 2.95 km for each solar mass · the boundary (horizon) of a non-spinning black hole The Sun would have to be squeezed to 3 km and the Earth to under 9 mm. The strange part is that the horizon of a heavy hole is not a place of extreme conditions. The average density inside rs falls as 1/M², so the hole in M87, at about 6.5 billion Suns, is less dense than air. And the tidal force, the difference in pull between your head and feet, at the horizon falls as 1/M². A stellar black hole stretches you violently long before you reach it; a supermassive one lets you cross in comfort. Interactive · Weigh a black hole 1 Sun10 SunsSgr A* (about 4 million)M87* (about 6.5 billion) Mass rs = 2GM/c²; density = M/(4/3 π rs³); tidal stretch across 2 m at rs = 2GM·(2 m)/rs³ (Newtonian estimate, indicative only). The bar marks rs on a log scale from 1 km to 100 AU. The Event Horizon Telescope imaged the shadow of the black hole in the galaxy M87 in 2019, with a mass of about 6.5 billion Suns, and of the one at the centre of our own galaxy, Sagittarius A*, in 2022, near 4 million Suns. Both ring sizes matched general relativity's prediction within the uncertainties. Ripples: gravitational waves If mass shapes spacetime, moving mass makes the shape change, and the change spreads at the speed of light. Two black holes in orbit stir spacetime like two spoons in a pond. The waves carry energy away, so the orbit shrinks, the pair spins faster and the wave frequency climbs: a chirp. At leading order the climb depends only on one combination of the masses, the chirp mass Mc = (m₁m₂)3/5 ÷ (m₁+m₂)1/5, and the time left before merger is τ = (5/256) · (GMc/c³)−5/3 · (πf)−8/3time to merger from the moment the wave frequency is f On 14 September 2015 the two LIGO detectors recorded the first such signal, GW150914 (announced 11 February 2016): black holes of about 36 and 29 Suns merging into one of about 62, with roughly 3 solar masses of energy radiated in gravitational waves. Chirp mass: about 28 Suns. From 35 Hz the formula gives 0.19 seconds to merger, matching the roughly 0.2 s the signal lasted in the band. The distortion of space at Earth was about 10−21: a 4 km arm changes length by far less than a proton's width. The 2017 Nobel Prize in Physics went to Rainer Weiss, Barry Barish and Kip Thorne for the detector. Interactive · Make a chirp GW150914Two neutron stars10 + 10 Suns First mass Second mass Leading-order (Newtonian-quadrupole) inspiral, valid until the innermost stable orbit at f = 4,400 Hz ÷ (total mass in Suns). The plot shows the last 0.1 s of that stage, amplitude ∝ f2/3. The real merger and ringdown, up to about 250 Hz for GW150914, need numerical relativity. Gravitational waves were predicted by Einstein in 1916, and indirectly confirmed earlier by the Hulse–Taylor binary pulsar, discovered in 1974, whose orbit shrinks as general relativity says (Nobel Prize 1993). LIGO and its partners turned that into an astronomy, and in 2017 they heard two neutron stars merge at the same time as light was seen from the same event. The equation, and where it stops Everything above comes from one tensor equation, in words: matter and energy tell spacetime how to curve; curved spacetime tells matter how to move (John Wheeler's summary). Gμν + Λgμν = (8πG ÷ c⁴) Tμνcurvature (left) = energy, momentum and pressure (right) · Λ is the cosmological constant It is ten coupled equations, hard to solve in general, which is why exact solutions (Schwarzschild 1916, Kerr 1963, the expanding universe of Friedmann and Lemaître in the 1920s) are famous. Its record is remarkable: Mercury, starlight, the tower, GPS, binary pulsars, black hole images and gravitational waves. TestWhenResult Mercury's perihelion turn191543″/century explained Starlight bending1919about 1.75″ at the limb Clock rates (Pound–Rebka)1959shift seen, as predicted Binary pulsar1974 onorbit shrinks as predicted Gravitational waves2015GW150914 seen Black hole shadows2019, 2022M87*, Sgr A* match Where it stops: general relativity is a classical theory, and at a black hole's centre or at the Big Bang it predicts infinite curvature, which is a signal that something else takes over. Nobody yet has a tested theory of quantum gravity. On the largest scales it works only if most of the universe is something we have not identified (dark matter and dark energy), and its models disagree at present about how fast the universe expands (the Hubble tension). Both are the business of Astrophysics & Cosmology (34/88). What a graduate just sees Free fall is the natural state; weight is the floor pushing back. Gravity is geometry: bodies follow straightest paths on a curved stage. A clock higher in a gravity well runs faster, and a moving clock runs slow; satellites feel both. Orbits are not closed ellipses; they precess, and light bends by 4GM/(c²b), twice the Newtonian guess. rs = 2GM/c² is 3 km per solar mass, and a horizon is gentle if the hole is big. Accelerating masses radiate ripples that carry energy away and make binaries chirp. The equation says "matter curves space, space steers matter"; it fails at singularities and awaits quantum gravity. Where it shows up next Astrophysics & Cosmology (34/88): neutron stars, black holes, lensing and the expansion of the universe are all solved with these tools. Differential Geometry (34/77) and Special Relativity (34/81): the language and the local rules it was built from. Particle & Nuclear Physics (34/86): the open problem is making this theory and quantum fields agree. In the world: satellite navigation and time distribution, gravitational-lens surveys, the LIGO–Virgo–KAGRA network and the space-based LISA mission being planned. If you want to go deeper 📘 Textbook. James Hartle, Gravity: An Introduction to Einstein's General Relativity (Addison-Wesley, 2003). Physics first, with the maths introduced as needed; the usual advanced-undergraduate choice. 📘 Textbook. Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley, 2004; Cambridge University Press, 2019). A fuller, more mathematical course. 📘 Gentler. Kip Thorne, Black Holes and Time Warps (W. W. Norton, 1994). History and ideas, with almost no maths. 🌐 Free course. MIT OpenCourseWare 8.962 General Relativity (Spring 2020). The takeaway Gravity is not a pull but the shape of spacetime. Free fall is the straight path; clocks, light and orbits all follow from the geometry, and it is precise enough to correct a satellite's clock by 38 microseconds a day. The same geometry gives black holes with a size of 3 km per Sun and ripples that shake detectors by a thousandth of a proton. Try this: this week, when you check a maps app, remember that its blue dot needs two relativity corrections, one for speed and one for gravity, and that without them it would drift by about 11 km a day. Sources & Further Reading Sources. Equivalence principle, geodesics, the perihelion formula, light deflection, Schwarzschild radius and the leading-order chirp are standard (Hartle; Carroll). Constants from CODATA 2018 and IAU nominal values (GMsun = 1.32712440018 × 10²⁰ m³/s²). GPS: about 45 µs a day from gravity, about 7 µs a day from speed, net about 38 µs a day (Ashby, Living Reviews in Relativity, as reported in GPS World). GW150914: LIGO–Virgo, 36 and 29 solar masses to about 62, about 3 solar masses radiated (Phys. Rev. Lett. 116, 061102, 2016, and the follow-up properties paper). Eddington's 1919 results: Dyson, Eddington and Davidson, 1920; values of about 1.98″ (Sobral) and 1.61″ (Príncipe), with uncertainties of a few tenths of an arcsecond reported differently in secondary sources. EHT: M87* mass (6.5 ± 0.7) × 10⁹ solar masses (2019), Sgr A* about 4 × 10⁶ solar masses (2022). Pound–Rebka 1959, Hulse–Taylor 1974 and Nobel 1993, Mercury 1915, Schwarzschild 1916 and the 2017 Nobel Prize are cited from the literature and not re-checked for this piece. Every number an interactive shows was computed in node: for example net clock shift +38.59 µs/day at 20,200 km, break-even at 3,167 km, 43.0″ a century for Mercury, 1.751″ at the solar limb, 0.016″ for Jupiter, 39″ for a 1014 solar mass cluster at 100 kpc, rs = 2.95 km for the Sun, and 0.19 s from 35 Hz for GW150914. 🌐 MIT OCW 8.962 General Relativity 🌐 Nobel Prize in Physics 2017: press release 📘 Hartle, Gravity 📘 Thorne, Black Holes and Time Warps Step 15. Compact Objects & High-Energy Astrophysics: The Densest Things and the Hardest Light --------------------------------------------------------------------------------------------- Why: White dwarfs, neutron stars, black holes and the violent universe. A star's corpse can weigh a mountain per thimble, spin 700 times a second, and feed on gas until it outshines a galaxy. The graduate's skill is reading all of that from a flicker, a period and a ratio. On 17 August 2017 two detectors in the United States and one in Italy shook, very slightly, for about a minute and a half. Less than two seconds later a gamma-ray satellite flagged a short flash from the same patch of sky. Within hours telescopes in Chile had found a new point of light in a galaxy 130 million light-years away, and over the next weeks that light faded from blue to red. Two neutron stars, each the leftover of an exploded star and each about as wide as a city, had spiralled together and merged. That one event showed ripples in spacetime, a burst of light, a glowing cloud of freshly made heavy elements, and a new way to measure how fast the universe expands. It is the whole subject in miniature: compact objects are the dead stars (white dwarfs, neutron stars, black holes), and high-energy astrophysics is what they do to the gas, the magnetic fields and the particles around them. Nobody has been near any of this. A graduate does it by scaling arguments: if the flicker lasts a millisecond, the thing is smaller than a country; if the period is 33 milliseconds and slowing, here is its magnetic field. This piece lets you do those sums yourself. The course in one box Who takes itAstronomy, astrophysics and physics majors, usually as a fourth-year option or an early graduate course. Programmes in the style of MIT's Course 8 astrophysics track and the UK Institute of Physics framework offer it as high-energy astrophysics, or split it into compact objects, accretion and radiation processes. What it is forWorking out what holds a dead star up, how matter falling onto it turns into light, and how the universe makes particles far more energetic than any laboratory can. Comes afterGeneral Relativity (34/85) and Stars: Structure & Evolution (36/41). Leads toNothing later on this list; it meets Galaxies & the Milky Way (36/42) at the black holes in galactic centres, Astrophysics & Cosmology (34/88) in the early universe, and Astronomical Data & Computation (36/44) in how the signals are processed. What this piece gives youYou will not be able to model an accretion disk or a pulsar magnetosphere. You will see why a white dwarf shrinks as it gains mass, how to get a magnetic field and an age from a spin period, why accretion is the best engine nature has and why light caps its speed, how timing alone gives size and speed, what limits a cosmic accelerator, and how a merger measures distance. The Schwarzschild radius, the Chandrasekhar limit as a headline and the gravitational-wave chirp are already in General Relativity (34/85) and Astrophysics & Cosmology (34/88); read those first and this goes on from there. Degeneracy: what holds up a dead star A star avoids collapse because hot gas pushes back. When the fuel is gone and the heat leaks away, something else must push, and it is not heat. It is a quantum rule: no two electrons may occupy the same state. Squeeze electrons into a smaller space and each one's momentum must rise, because confining a particle makes its momentum uncertain. That momentum is a pressure, and it does not need any temperature. This is degeneracy pressure. It has a strange consequence. For ordinary (slow) electrons, the pressure rises as density to the power 5/3. Work through the balance with gravity and a heavier white dwarf is a smaller one: R ∝ M−1/3a white dwarf shrinks as it gains mass · but only until the electrons approach light speed Near the top of the range the electrons move at nearly the speed of light, the pressure law softens to density to the power 4/3, and that exactly matches gravity's pull: no radius is stable at any mass beyond a critical value, the Chandrasekhar mass of about 1.4 Suns for a carbon–oxygen star. Subrahmanyan Chandrasekhar worked it out in 1930–31 on a ship to England. The practical form is a curve you can plot. A good fit to his full calculation (Nauenberg 1972) is R = 0.0112 R☉ √(x−2/3 − x2/3), with x = M/1.44 M☉. Interactive · A white dwarf that shrinks as it gains weight Mass Nauenberg's approximation to Chandrasekhar's carbon–oxygen white dwarf (two nucleons per electron), cold. Real stars differ by several per cent. Sirius B: measured mass about 1.02 M☉ and radius about 0.0081 R☉ (Bond et al. 2017); the curve gives about 6% less. Without JavaScript: at 0.6 Suns the curve gives a radius of about 8,700 km (a little more than Earth); at 1.0 Sun, 5,500 km; at 1.4 Suns, about 1,500 km, and the radius falls toward zero as the mass approaches 1.44. Why does this matter beyond textbooks? A white dwarf in a binary that gathers gas toward the limit can explode as a Type Ia supernova. Because the trigger mass is nearly the same each time, the explosions are similar enough to be used as distance markers (the rung in Astrophysics & Cosmology, 34/88), though they need a calibration based on how fast each one fades. Note the honest caveat: whether the usual parent is a white dwarf growing by accretion or two white dwarfs merging is still argued, and likely both happen. Pulsars: a lighthouse that tells you its field and its age Past the limit, electrons and protons are crushed into neutrons, and neutron degeneracy plus the strong force holds the star up: a neutron star of roughly 1.2 to 2.3 Suns (perhaps a little more) and a radius near 12 km. The density is about 4 × 1017 kg/m³, a mountain's mass in a thimble. A neutron star's gravity gives a compactness GM/Rc² of about 0.17: the escape speed is 59% of the speed of light. Among the heaviest measured is PSR J0952−0607, about 2.35 solar masses with an uncertainty near 0.17 (Romani et al. 2022). Radii from NASA's NICER X-ray telescope fall near 12–13 km. The upper limit for neutron stars is not known exactly because the physics of matter at that density is not. Squeeze a spinning star from the size of the Sun to 12 km and it spins up enormously (angular momentum is conserved), and its magnetic field is squeezed too. The beam of radio waves from the magnetic poles sweeps the Earth like a lighthouse: a pulsar, discovered in 1967 by Jocelyn Bell Burnell and Antony Hewish. The pulses are extremely regular, but the period slowly grows because the rotating magnet radiates away spin energy. Just two measured numbers, the period P and how fast it grows (Ṗ), give three things: B ≈ 3.2 × 1019 √(P Ṗ) gauss · age τ = P / 2Ṗ · power Ė = 4π² I Ṗ / P³field at the surface, characteristic age, spin-down luminosity · for a star with moment of inertia I = 1038 kg m² (the standard value) and P in seconds Worked for the Crab pulsar, the core left by the supernova that Chinese and other astronomers recorded in 1054: P = 33.4 ms, Ṗ = 4.2 × 10−13. That gives B near 4 × 1012 gauss, a characteristic age of 1,260 years (the true age is about 970; the formula assumes the star started spinning infinitely fast and braking in a particular way, so treat it as a rough guide), and a spin-down power of 4.5 × 1031 watts, about 100,000 Suns, which is what lights the whole Crab Nebula. Interactive · The P–Ṗ diagram Period P Period growth Ṗ Dashed lines: constant field. Dotted lines: constant characteristic age. Preset values are rounded catalogue values (ATNF Pulsar Catalogue; Israel et al. 2016 for SGR 1935+2154); the plot axes are logarithmic. Equator speed uses R = 12 km. Without JavaScript: the Crab (33 ms) has B near 4 × 1012 G; the millisecond pulsar B1937+21 (1.56 ms) has about 4 × 108 G and is hundreds of millions of years old; the magnetar SGR 1935+2154 (3.2 s) has about 2 × 1014 G. Slide it. Young, strong-field pulsars sit at the top; they slow, drift to the right and down, and fade out of the radio as the period reaches several seconds. Some are rejuvenated: a neutron star in a binary can be spun back up by gas from its companion into a millisecond pulsar, which is why the lower left fills with old stars with weak fields. The fastest known spins 716 times a second; its equator moves at about 18% of light speed. At the top right, with fields of 1014–1015 G, are the magnetars. In April 2020 one of them, SGR 1935+2154, emitted a radio burst resembling the mysterious fast radio bursts, the first such burst traced to a known object in our own galaxy. Accretion: the best engine, and the speed limit on it Fusion turns 0.7% of hydrogen's mass into energy. Dropping gas onto a compact object does far better. A kilogram that falls onto a neutron star's surface releases about GM/R in energy per kilogram, which is 0.17 of its rest energy. Onto a black hole there is no surface, so the gas swirls in a disk and radiates as it spirals in until it reaches the innermost stable circular orbit (ISCO), after which it plunges in without radiating much. How much it radiates depends on how close the ISCO is, and that depends on the black hole's spin: η = 1 − √(1 − 2 / 3risco)fraction of rest-mass energy radiated, with risco in units of GM/c² · 6 for no spin, down to 1 for the fastest prograde spin, up to 9 for the opposite spin (Bardeen 1972) With no spin, η = 5.7%. At maximum spin it reaches 42%, and an astrophysical limit near 32% (Thorne 1974) is more realistic, because the hole's own radiation tends to cap its spin. That is eight to sixty times better than fusion, and it is why a quasar can outshine all the stars of its host galaxy from a region not much bigger than our solar system. There is a ceiling. Light carries momentum, and the infalling gas is ionised. If the luminosity is too large, the light pushes outward on the electrons harder than gravity pulls the matching protons inward, and the inflow stalls. Setting the two forces equal gives the Eddington luminosity, which depends only on mass: LEdd = 4π G M mp c / σT ≈ 1.26 × 1031 W × (M / M☉)mp proton mass, σT the Thomson cross-section of the electron · the ceiling for steady spherical inflow, which real discs can exceed briefly The limit gives a speed limit on growth. A hole shining at the Eddington rate grows by a factor of e (2.7) in a time t = 0.45 Gyr × η / (1 − η), about 50 million years for η = 0.1 (the Salpeter time). That clashes with a surprise: quasars with a billion Suns already existed when the universe was less than 700 million years old. ULAS J1342+0928 (redshift 7.54, about 800 million Suns; Bañados et al. 2018) and J0313−1806 (redshift 7.64, about 1.6 billion Suns; Wang et al. 2021) are examples. Start from a 100-Sun remnant of an early star and you need ln(107) = 16 e-folds. Try it. Interactive · How efficient is the engine, and can it grow in time? Spin a/M (negative: disc runs against the spin) Brightness as a fraction of LEdd η from Bardeen's ISCO formula (prograde and retrograde); fusion 0.7%; neutron star = GM/Rc² for 1.4 M☉ and 12 km. Time available: from redshift 30 (0.099 Gyr, a rough guess for the first stars) to redshift 7.54 (0.688 Gyr), flat ΛCDM with Planck 2018 values. Radiation is assumed to leave freely, and a continuous run at the chosen rate is an upper bound on real growth. Without JavaScript: η is 5.7% for no spin, about 15.6% at spin 0.9, 32% at 0.998. A 100-Sun seed shining at Eddington needs about 0.44 Gyr to reach a billion Suns if η = 5.7% and 0.81 Gyr if η = 10%, against 0.59 Gyr available. Notice the twist: a faster spin makes the engine more efficient, so more of the infalling mass is turned into light and less is kept, and the black hole grows slower at a given brightness. Astronomers therefore consider massive seeds (direct collapse of a gas cloud into a black hole of 10,000 or more Suns), super-Eddington bursts and mergers. Which wins is open, and the James Webb Space Telescope keeps finding black holes that look heavy for their age. Timing and beaming: size and speed from light alone Take a source that doubles its brightness in a time Δt. Its parts can only act in step if light has time to cross it, so it cannot be much larger than cΔt. The reasoning needs nothing else. A flicker of 1 millisecond means a region under 300 km, which is smaller than a country, so it can only be a compact object. Quasars brightening in a few days are smaller than the solar system. This is how the central engines were known to be tiny long before anyone could see them. Now suppose the source is a blob moving close to the speed of light almost toward you. It nearly catches up with its own light, so the time between two flashes it emits is squeezed when you receive them, and it seems to cross the sky faster than light: βapp = β sin θ / (1 − β cos θ) · δ = 1 / (Γ (1 − β cos θ))β = v/c · θ the angle to the line of sight · Γ the Lorentz factor · δ the Doppler factor: a moving blob is brightened roughly as δ³ Nothing travels faster than light. The illusion is geometry, and it is a measurement tool: the apparent speed and the brightening together give Γ and θ. Mirabel and Rodríguez saw blobs from the X-ray binary GRS 1915+105 in 1994 move at an apparent 1.25 c; with β = 0.92 and θ = 70° the formula gives 1.26. The effect is strong in jets of galaxies (apparent speeds up to several times c in M87 and in many quasars), and in 2018 radio images of the GW170817 afterglow showed motion of about four times c, which constrained the jet's angle. Interactive · Size from a flicker, speed from a jet Flicker time Δt Lorentz factor Γ Angle to the line of sight θ Size bound R ≲ cΔt (order of magnitude; relativistic motion can loosen it by a factor of δ). The horizon radius uses 2.95 km per solar mass. The plot shows βapp against θ for the chosen Γ; the dot marks your angle. Without JavaScript: a 1 ms flicker bounds the size at about 300 km, a day at about 173 AU (one light-day), a year at one light-year. A jet with Γ = 10 seen at 6° looks about 9.9 times faster than light. The other half of this is the counter-jet. The blob heading away is dimmed by the same factor, so a jet that shows up on one side only tells you it is moving fast. For Γ = 10 at 6° the approaching side is brighter by a factor of several million. Cosmic accelerators: how big, how strong Cosmic rays are charged particles, mostly protons, that arrive from space with energies up to tens of millions of times the energy of a Large Hadron Collider beam. Because magnetic fields bend them, they do not point back to their source, and for a century their origin was a mystery. A graduate has one neat sum for what an accelerator would need. A particle of charge Ze and energy E in a field B moves in a circle of radius E / (ZecB). If that is larger than the accelerator, the particle leaves. So the biggest energy that a region of size R and field B can hold is the Hillas limit (A. M. Hillas, 1984): Emax ≈ Z e β B R cin electron-volts, Emax = Z · β · B[tesla] · R[metres] · c[m/s] · β is the speed of the moving shock or magnet in units of c Check it on the LHC. Its bending magnets are 8.33 tesla over a bending radius of 2,804 m. The formula gives 7.0 TeV, the design energy per beam. Now plug in the sky: a supernova-remnant shock, with a field of about 10 microgauss (1 nanotesla) over a few parsecs, is one of the presets below. The result is a ceiling, not a prediction: real accelerators lose energy to radiation and rarely reach it. Interactive · The Hillas ceiling Field B (tesla) Size R (metres) Speed β of the moving magnet or shock Presets are order-of-magnitude values (SNR: 1 nT = 10 μG over 5 pc; lobe: 10 nT over 10 kpc; cluster: 0.1 nT over 1 Mpc). The ticks are real milestones: LHC design 7 TeV; the cosmic-ray "knee" at about 3 PeV; the "ankle" near 5 EeV; the GZK region near 50 EeV; the magnetar surface is a ceiling no real accelerator reaches; the record events, 320 EeV (Fly's Eye, 1991) and 244 EeV (Telescope Array, 2021). Without JavaScript: the LHC sum gives 7.0 TeV; a supernova shock (1 nT, 5 pc, β = 0.03) gives about 1.4 PeV, the right region for the knee; a radio-galaxy lobe (10 nT, 10 kpc, β = 0.1) gives about 90 EeV. The results are the story. Supernova remnants can reach the knee at about 3 × 1015 eV, and they are widely thought to supply most Galactic cosmic rays. The highest-energy particles, near 1020 eV, need something far bigger or stronger, such as the lobes and jets of active galaxies, or something exotic; the Pierre Auger Observatory in Argentina reported in 2017 that cosmic rays above 8 EeV arrive preferentially from a direction pointing outside our galaxy. Their sources are not yet identified. Neutrinos have no charge, so they point straight back: IceCube traced one 2017 event to the blazar TXS 0506+056 (2018) and found a Seyfert galaxy, NGC 1068, in 2022. Mergers: gravitational waves plus light Chirps from black holes show that the waves are there. A neutron-star merger adds light, and the pair together does what neither can alone. The chirp gives the masses and, because the strain h (the stretch of space) drops as 1/d, the distance as well: the waveform tells you its own loudness at the source, so the observed loudness gives how far it is. That is a standard siren, with no distance ladder. The light gives the galaxy, and the galaxy gives the redshift. h ≈ (4 / d) (G𝓜c / c²)5/3 (πf / c)2/3 and H₀ = v / dstrain from the leading-order inspiral · 𝓜c is the chirp mass · v the recession velocity of the host galaxy, corrected for its own motion For GW170817 the numbers were: chirp mass 1.188 solar masses, a distance of about 40 Mpc (43.8 Mpc is the central value of the best-fit analysis, with a large lopsided error because the tilt of the orbit and the distance are partly degenerate), and a host galaxy, NGC 4993, receding at about 3,000 km/s. That gives H₀ near 70 km/s/Mpc, with a range of +12 and −8 from a single event (later narrowed to roughly ±5 with radio images of the jet). Slide the numbers. Interactive · A siren for the Hubble tension Distance d Host recession velocity v v = 3017 km/s and d = 43.8 Mpc are the values used by the LIGO–Virgo analysis (Abbott et al. 2017), and h is for a face-on orbit at 100 Hz. Planck 2018 gives 67.4 ± 0.5 and the SH0ES distance ladder 73.04 ± 1.04 (Riess et al. 2022); a single event cannot separate them. Without JavaScript: 3,017 km/s over 43.8 Mpc is 68.9 km/s/Mpc. A distance 15% smaller gives 81; 15% larger, 60. One merger cannot decide between 67 and 73; dozens could. The rest of the event was a gift. The gamma-ray burst came 1.74 s after the merger, from a journey of 130 million years. That fixes the speed of gravity equal to the speed of light to about one part in 1015, which killed a family of alternative gravity theories. The glow that followed, a kilonova, was powered by the radioactive decay of newly made heavy elements, evidence that neutron-star mergers are one of the places where gold, platinum and many other heavy nuclei are made; how much of the universe's supply they account for is still being worked out. And the masses are changing the picture of the stellar graveyard: in 2024 the LIGO–Virgo–KAGRA team announced a merger seen in May 2023, GW230529, with a partner of 2.5 to 4.5 Suns, in the gap between the heaviest neutron stars and the lightest known black holes. What a graduate just sees Degenerate stars shrink as they gain mass, and there is a mass beyond which no radius is stable. A pulsar's period and its slowdown give its field, its rough age and its power; a plot of the two is a family tree of neutron stars. Falling onto a compact object releases about 6 to 40 per cent of the rest mass as light, eight to sixty times more than fusion. Light pushes back: Eddington luminosity is about 1031 W per solar mass, and growth time is tens of millions of years at best. A flicker of Δt means a size under cΔt, and an apparent speed above c is a clue to a nearly head-on jet. An accelerator cannot hold a particle whose gyro-radius exceeds the accelerator: size times field is the budget. A merger with a chirp and a flash gives a distance and a redshift together. Where it shows up next Galaxies & the Milky Way (36/42): the supermassive black hole at a galaxy's centre, its jets and the Eddington-limited growth shape the galaxy around it. Astrophysics & Cosmology (34/88): Type Ia supernovae, standard sirens and the Hubble tension all rest on compact-object physics. Astronomical Data & Computation (36/44): pulsar searches, transient alerts and the Event Horizon Telescope are signal-processing problems. Particle & Nuclear Physics (34/86): nuclear matter at neutron-star density, and the origin of heavy elements. In the world: pulsar timing arrays, which clock dozens of millisecond pulsars and in 2023 reported evidence of a background hum probably from pairs of supermassive black holes, X-ray space telescopes, the Pierre Auger and IceCube observatories, and the LIGO–Virgo–KAGRA network. If you want to go deeper 📘 Textbook. Malcolm Longair, High Energy Astrophysics (Cambridge, 3rd edition 2011). The standard single-volume course, from radiation processes to cosmic rays and black holes. 📘 Textbook. Juhan Frank, Andrew King and Derek Raine, Accretion Power in Astrophysics (Cambridge, 3rd edition 2002), the classic on discs; and Stuart Shapiro and Saul Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (Wiley, 1983), the standard on the stars themselves. 📘 Gentler. Janna Levin, Black Hole Blues and Other Songs from Outer Space (Knopf, 2016), the human story of LIGO; Kip Thorne's Black Holes and Time Warps (1994) covers the ideas. 🌐 Free course. MIT OpenCourseWare 8.901 Astrophysics I (Spring 2006, Deepto Chakrabarty), a graduate introduction to stellar astrophysics and galactic dynamics. The takeaway Dead stars are held up by quantum rules rather than heat, which makes them smaller as they gain weight and caps their mass. Matter falling onto them is the most efficient engine in nature but is limited by its own light, and the signals we get, a period, a flicker, a chirp, are enough to weigh, size and date objects nobody will ever visit. Try this: pick any fast-changing thing you know of, a flickering candle or a blinking light, and ask how big it can be if the flicker is real. Light travels 300,000 km in a second: what time would have to pass for the same trick to work on something the size of the Moon? Sources & Further Reading Sources. Chandrasekhar 1931 (Astrophysical Journal 74); Nauenberg 1972 (ApJ 175); Bond et al. 2017 (Sirius B, ApJ 840); Romani et al. 2022 (PSR J0952−0607, ApJ Letters 934); NICER radius results (Miller et al. and Riley et al. 2019, 2021); ATNF Pulsar Catalogue; Hewish, Bell et al. 1968 (Nature); Israel et al. 2016 (SGR 1935+2154); CHIME/FRB and STARE2 2020 (FRB 200428); Bardeen 1972 and Thorne 1974 (spin and efficiency); Eddington luminosity and Salpeter time (standard; Frank, King and Raine); Bañados et al. 2018 (Nature, J1342+0928); Wang et al. 2021 (ApJ Letters, J0313−1806); Mirabel and Rodríguez 1994 (Nature); Mooley et al. 2018 (Nature, GW170817 jet); Hillas 1984 (Annual Review of Astronomy and Astrophysics 22); Pierre Auger Collaboration 2017 (Science); Telescope Array 2023 (Science); IceCube 2018 and 2022 (Science); Abbott et al. 2017 (GW170817: PRL 119; the Hubble constant: Nature 551); LVK 2024 (GW230529). Planck Collaboration 2020 and Riess et al. 2022 for H₀. Constants: CODATA 2018; M☉ = 1.98847 × 1030 kg. 📘 Longair, High Energy Astrophysics 📘 Frank, King and Raine, Accretion Power in Astrophysics 🌐 MIT OCW 8.901 Astrophysics I Step 16. Astrophysics & Cosmology: Reading the Universe from Light Alone ------------------------------------------------------------------------ Why: Cosmology: the universe's history and fate. Nobody has visited a star, weighed a galaxy on a scale or watched the universe begin. Everything we know comes from light and a few ideas that hold from the kitchen to the Big Bang. Look up on a clear night and every point of light is a sun. The faint smudge you can sometimes spot in Andromeda is a galaxy of a trillion suns about 2.5 million light-years away. The light reaching your eye left it before our species existed. There is no way to travel there, take a sample, or send a probe. And yet astronomers can tell you the star's mass, its temperature, how long it has left to live and how far away it is, from the colour and the brightness of that light. That is the odd thing about this subject: it is the one place in physics where you may not touch, and the whole discipline is built on inference. A graduate learns to see a point of light as an equation: brightness falls with the square of distance, colour tells temperature, and the stretching of its wavelengths tells how the space between us is growing. Out of three such ideas comes the age of the universe. The course in one box Who takes itPhysics majors in year 4, and astronomy majors earlier. MIT's Course 8 offers it in courses such as 8.282 (Introduction to Astronomy) and 8.286 (The Early Universe); the Cambridge Natural Sciences physics route teaches it in the later options. What it is forTurning light, and a few laws of physics, into the mass, size, history and fate of stars, galaxies and the universe. Comes afterStatistical Mechanics (34/84), General Relativity (34/85) and Particle & Nuclear Physics (34/86). Leads toNothing later on this list: it is the end of the physics major, and the door to research, where the open questions in the last section live. What this piece gives youYou will not be able to model a stellar interior or solve the Friedmann equations. You will see how astronomers get distance from brightness, why a star's mass fixes its life, what the evidence for dark matter really is, and how to turn two numbers into the age of the universe. Easier first reads in the library: The Big Bang and Our Expanding Universe (13/08), Orbits: Why the Moon Keeps Falling (13/11) and The Far Future: How the Universe Ends (13/10). This one goes further and lets you move the numbers. Light is all we get: the cosmic distance ladder A lamp twice as far away looks four times fainter, because its light spreads over four times the area. This inverse-square law is the first tool of astronomy: if you know how bright something really is, its apparent brightness tells you its distance. Magnitudes are astronomers' logarithmic scale for brightness, and the working form is the distance modulus: m − M = 5 log₁₀( d / 10 pc )apparent magnitude minus absolute magnitude, for a distance d in parsecs (1 pc = 3.26 light-years). Smaller magnitudes are brighter; 5 magnitudes is a factor of 100 The hard part is knowing the true brightness, M. For nearby stars, geometry gives it directly: as the Earth goes round the sun, a nearby star seems to shift against the background. Its parallax angle p, in arcseconds, gives distance 1/p parsecs. The Gaia satellite has measured Proxima Centauri's parallax as about 768 milliarcseconds, so it is 1.30 pc, 4.25 light-years away. Past a few thousand parsecs parallax fails, and astronomers climb a ladder instead: Cepheid variable stars, whose pulsation period fixes their luminosity (Henrietta Leavitt, 1912), and then Type Ia supernovae, exploding white dwarfs that are all about equally bright, at around absolute magnitude −19.3, visible across billions of light-years. Interactive · Two rungs of the ladder Parallax shift of a star Sun-like star (M ≈ +4.8) Cepheid, 10-day period (M ≈ −4) Type Ia supernova (M ≈ −19.3) How bright it looks (apparent magnitude) Parallax: d (pc) = 1000 ÷ p (mas). Candle: d = 10(m − M + 5)/5 pc. Ignores dust dimming, which real work must correct. Absolute magnitudes are rounded textbook values. Each rung is calibrated on the one before. That chain is why a 5% mistake near the bottom spreads to everything above, which matters later in this piece. A star is a balance, and its mass sets its life A star is a ball of gas in which two things fight. Gravity pulls everything inward. Pressure from the hot gas pushes outward. The heat is replenished by fusion in the core (the binding-energy story from Particle & Nuclear Physics, 34/86). If the core cools, gravity squeezes it, it heats up and burns faster: the balance is self-correcting. Because of that, mass is almost the only thing that decides a main-sequence star's luminosity, size and temperature. The fuel runs out faster than you would think. A star's luminosity grows steeply with mass, roughly as M3.5 for stars a few times the sun. Its fuel grows only as M. So the lifetime, fuel divided by burn rate, drops as about M−2.5. Worked for the sun: it fuses about 10% of its hydrogen, converting 0.7% of the mass to energy, so its lifetime is 0.1 × 0.007 × Mc² ÷ L ≈ 10 billion years. A star of ten solar masses burns 4,400 times brighter and lasts of order tens of millions of years. Move the mass. Interactive · Pick a star's mass Red dwarf 0.1Sun 1Sirius-like 2Blue giant 10 Mass (solar masses) Rough main-sequence scalings: L/L☉ = 0.23 M2.3 (M < 0.43), M4 (to 2), 1.4 M3.5 (above); R ∝ M0.9 below 1 and M0.8 above; T = 5772 K × (L/R²)1/4; lifetime = 10 Gyr × M/L. Good to within a factor of a few. Real red dwarfs mix their fuel and last far longer than this gives. The plot is a Hertzsprung–Russell diagram: temperature (hot on the left) against luminosity. Stars burning hydrogen fall on a band across it, the main sequence, where the sun sits in the middle. Every star ever catalogued is a point on that diagram, and its position, together with its mass, tells the stage of its life. When the balance fails: white dwarfs, neutron stars, black holes When the core's fuel is spent, nothing replaces the heat, and gravity wins. What stops the collapse depends on the mass left behind. For stars like the sun, electrons are squeezed until the Pauli principle resists: a white dwarf the size of the Earth. In 1930–31 Subrahmanyan Chandrasekhar showed that this support fails above about 1.4 solar masses (the Chandrasekhar limit; Nobel Prize 1983). Beyond that, the core collapses until protons and electrons merge into neutrons, a neutron star about 12 km across, with a teaspoon weighing billions of tonnes. And above a limit near two to three solar masses, nothing known can hold it up: a black hole. A black hole has one size, the radius inside which not even light escapes. In general relativity (34/85) it is: rs = 2GM / c²the Schwarzschild radius: about 2.95 km per solar mass The sun would be a black hole 3 km wide, and the Earth one under 9 mm. Black holes are not cosmic vacuum cleaners; from afar they pull as any other mass does. In 2015 the LIGO detectors recorded the gravitational waves of two black holes of about 36 and 29 solar masses merging (GW150914), the first direct detection, and in 2019 and 2022 the Event Horizon Telescope imaged the shadows of the black holes at the centres of the galaxy M87 and of our own. Interactive · How big is a black hole? EarthSunWhite dwarf (Sirius B)Neutron starGW150914 resultSgr A*M87* Mass rs = 2GM/c² with G = 6.6743 × 10⁻¹¹, c = 299,792,458 m/s, M☉ = 1.98847 × 10³⁰ kg. Masses: Sirius B about 1.02 M☉; neutron star 1.4 M☉ with a radius near 12 km; GW150914's remnant about 62 M☉; Sgr A* about 4.3 million and M87* about 6.5 billion M☉ (EHT, GRAVITY). The density line is the surprise. A black hole's average density, mass over the volume inside its horizon, falls as 1/M². The 6.5-billion-sun hole in M87 is, on average, lighter than air. Galaxies spin too fast: the dark matter evidence Kepler's law for planets says the farther out you go, the slower you orbit. Stars in a galaxy should do the same: outside the bright disk, where little mass is visible, their orbital speed should fall. In the 1970s Vera Rubin and Kent Ford measured the Andromeda galaxy, and later Rubin, Ford and Norbert Thonnard measured many more. The speeds did not fall. They stayed flat to the edge of the visible disk. The simplest reading is that there is unseen mass, more than the visible stuff, spread in a huge halo. The same hint, from the Coma cluster, had been noticed by Fritz Zwicky in 1933. v² = G M(