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## The physics
+New to any of this? [**docs/THEORY.md**](docs/THEORY.md) derives every equation
+below from scratch, starting at `F = ma` and ending at Flamm's paraboloid, with
+diagrams and no assumed background. The section here is the short version.
+
Geometric units throughout: $G = c = 1$, lengths in Schwarzschild radii ($r_s = 2M = 1$, so $M = \tfrac{1}{2}$). The disc lies in the equatorial plane.
### Light: null geodesics (exact)
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+
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+
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+
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+
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+
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+
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+
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+
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+# Where every equation in this app comes from
+
+This is the long version of the physics, written for someone who has met
+`speed = distance / time` and not much else. Every formula the code uses is
+built up here from things you can check on a playground, and nothing is
+introduced without saying where it came from.
+
+You do not need calculus for most of it. Where calculus is unavoidable, the
+step is spelled out in words first, and you can take the result on trust
+without losing the thread.
+
+**How to read this.** Each part answers one question, ends with the equation
+the app actually evaluates, and says which file it lives in. If you only want
+the summary, jump to [the table at the end](#every-equation-and-where-it-lives).
+
+---
+
+## Contents
+
+1. [The five things you already know](#1-the-five-things-you-already-know)
+2. [Why anything orbits anything](#2-why-anything-orbits-anything)
+3. [Escape speed, and the size of a black hole](#3-escape-speed-and-the-size-of-a-black-hole)
+4. [Units where the answer is 1](#4-units-where-the-answer-is-1)
+5. [Why light bends, and by how much](#5-why-light-bends-and-by-how-much)
+6. [The photon sphere and the shadow](#6-the-photon-sphere-and-the-shadow)
+7. [The last stable orbit](#7-the-last-stable-orbit)
+8. [Why the disc glows, and how hot](#8-why-the-disc-glows-and-how-hot)
+9. [Why one side of the disc is brighter](#9-why-one-side-of-the-disc-is-brighter)
+10. [Tides, and why they tear stars apart](#10-tides-and-why-they-tear-stars-apart)
+11. [Why the debris becomes a stream](#11-why-the-debris-becomes-a-stream)
+12. [Gravitational waves and the chirp](#12-gravitational-waves-and-the-chirp)
+13. [The funnel: what curved space actually means](#13-the-funnel-what-curved-space-actually-means)
+14. [Jets, beaming, and the outflow](#14-jets-beaming-and-the-outflow)
+15. [What we cheat on](#15-what-we-cheat-on)
+16. [Every equation and where it lives](#every-equation-and-where-it-lives)
+
+---
+
+## 1. The five things you already know
+
+Everything below is built from these.
+
+**Speed.** How far you go divided by how long it took.
+
+**Newton's second law.** Push something and it speeds up. Push twice as hard
+and it speeds up twice as fast. Push something twice as heavy and it speeds up
+half as fast:
+
+$$F = m\,a$$
+
+**Gravity.** Every mass pulls every other mass. Double either mass and the pull
+doubles. Move twice as far away and the pull drops to a quarter, because the
+same pull is spread over a sphere four times bigger:
+
+$$F = \frac{G\,M\,m}{r^2}$$
+
+$G$ is just a conversion number that makes the units work out.
+
+**Going in a circle is accelerating.** Even at constant speed, going round a
+corner means your direction keeps changing, and changing direction is
+acceleration. The amount you need is
+
+$$a = \frac{v^2}{r}$$
+
+Twice the speed needs four times the pull. This is why you lean into a fast
+corner on a bike.
+
+**Energy.** Moving costs energy, $\tfrac{1}{2}mv^2$. Being deep in a gravity
+well is like being at the bottom of a pit: climbing out costs energy, so we
+call the energy down there negative, $-GMm/r$. Add the two together and you get
+a number that does not change as the object moves, which makes it enormously
+useful.
+
+---
+
+## 2. Why anything orbits anything
+
+
+
+Throw a ball sideways and it curves down and lands. Throw it harder and it
+lands further away. Throw it hard enough and the ground curves away underneath
+it as fast as it falls, and it never lands. That is an orbit. It is not
+weightlessness, it is falling forever and missing.
+
+Put numbers on it. The pull available is $GMm/r^2$. The pull required to keep
+turning in a circle is $mv^2/r$. Set them equal:
+
+$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$
+
+The mass of the orbiting thing, $m$, appears on both sides and cancels. That is
+why a feather and a cannonball orbit identically. Multiply both sides by $r/m$:
+
+$$\boxed{v_{\text{circ}} = \sqrt{\frac{GM}{r}}}$$
+
+Closer in means faster. This one line explains why Mercury runs rings around
+Neptune, and why the inner part of the accretion disc shears past the outer
+part and winds the pattern into a spiral.
+
+> **In the code:** `vCircular` in `src/sim/gravity.ts`, used everywhere from
+> debris launch speeds to the disc's shear rate.
+
+---
+
+## 3. Escape speed, and the size of a black hole
+
+
+
+To escape completely you need enough kinetic energy to pay off the whole
+gravitational debt. Debt paid means total energy reaches zero:
+
+$$\tfrac{1}{2}mv^2 - \frac{GMm}{r} = 0 \quad\Longrightarrow\quad v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$$
+
+Earth's is 11 km/s. Notice what the formula depends on: not just how much mass,
+but how *close to it* you can get. Squeeze the same mass into a smaller ball and
+you can stand nearer the centre, so the escape speed goes up.
+
+Now ask the question John Michell asked in 1783: what if we squeeze it so far
+that the escape speed reaches the speed of light? Set $v_{\text{esc}} = c$:
+
+$$c = \sqrt{\frac{2GM}{r}} \quad\Longrightarrow\quad \boxed{r_s = \frac{2GM}{c^2}}$$
+
+This is the **Schwarzschild radius**. For the Sun it is about 3 km. For the
+Earth, 9 mm.
+
+**An honest warning.** That derivation is a fluke. It treats light as a slow
+cannonball, which is wrong, and it says light "falls back", which is also
+wrong. The correct calculation is Karl Schwarzschild's 1916 solution of
+Einstein's equations, and it happens to give exactly the same number. Take the
+formula, discard the story: nothing falls back, the horizon is a one way door.
+
+> **In the code:** `R_S` in `src/physics/constants.ts`.
+
+---
+
+## 4. Units where the answer is 1
+
+Physicists get tired of writing $G$ and $c$, so they choose units where both
+equal 1. It is the same trick as measuring travel in "hours away" instead of
+kilometres: you have folded the speed into the unit.
+
+With $G = c = 1$, the Schwarzschild radius becomes
+
+$$r_s = 2M$$
+
+This app goes one step further and measures every length in Schwarzschild
+radii, so $r_s = 1$ and therefore $M = \tfrac{1}{2}$. When you see `0.5` in the
+shader where you expected a mass, that is why. A distance of `12` means twelve
+Schwarzschild radii from the centre.
+
+> **In the code:** `src/physics/constants.ts` has the whole convention in one
+> comment block.
+
+---
+
+## 5. Why light bends, and by how much
+
+Light has no mass, so how can gravity pull on it?
+
+Einstein's answer starts with a thought experiment you can almost do. Imagine a
+windowless lift. If it sits on Earth you feel pressed to the floor. If it is
+being towed through empty space at $9.8\ \text{m/s}^2$ you feel exactly the
+same. **There is no experiment inside the lift that tells the two apart.** That
+is the equivalence principle.
+
+Now shine a torch across the lift while it accelerates. The beam leaves one
+wall, and by the time it crosses, the lift has moved up a little, so the spot
+lands slightly low. Inside the lift, the light looks bent. And if acceleration
+and gravity are indistinguishable, then **gravity must bend light too**.
+
+The next step is the one that made Einstein famous. He found that light bends
+by *twice* what you get from pretending a photon is a cannonball. The full
+weak-field result, for a ray that passes at closest distance $b$:
+
+$$\alpha \simeq \frac{4GM}{bc^2} = \frac{2 r_s}{b}$$
+
+Eddington measured it during the 1919 eclipse, got the factor of two rather
+than one, and Einstein was on the front page of every newspaper.
+
+Close to the hole, "bends a bit" is no longer enough and we need the exact
+path. Writing $u = 1/r$ (so big $u$ means close in) and tracking the ray by the
+angle $\phi$ it has swept around the hole, general relativity gives
+
+$$\frac{d^2u}{d\phi^2} + u = \tfrac{3}{2} r_s u^2$$
+
+Read it as two pieces. **Without** the right-hand side, this is the equation for
+a straight line written in polar coordinates, which you can check by putting in
+$u = \sin\phi / b$. **With** it, the line curves, and the correction grows as
+$u^2$, so it is negligible far away and dominant close in. All of relativistic
+lensing is in that one extra term.
+
+The app does not integrate in $\phi$, because a shader ray needs to move in
+$x, y, z$. The same equation in vector form is
+
+$$\ddot{\mathbf{x}} = -\tfrac{3}{2}\, r_s\, h^2\, \frac{\mathbf{x}}{r^5}, \qquad h = |\mathbf{x} \times \dot{\mathbf{x}}|$$
+
+where $h$ is the ray's angular momentum about the hole, which stays constant
+along the path. Every pixel on your screen runs this equation with a
+fourth-order Runge-Kutta integrator, a few hundred steps, every frame.
+
+> **In the code:** `centerAccel` in `src/render/shaders/geodesic.frag`, and the
+> same equation on the CPU in `src/physics/geodesic.ts` so the drawn photon
+> paths land exactly on the features the shader renders.
+
+---
+
+## 6. The photon sphere and the shadow
+
+
+
+Aim a ray at the hole and the only thing that matters is $b$, how far off
+centre you aimed. Three things can happen, and there is a sharp line between
+them.
+
+To find the line, use energy bookkeeping. For light, the "how close can I get"
+question reduces to comparing $1/b^2$ against the function
+
+$$V(r) = \frac{1}{r^2}\left(1 - \frac{r_s}{r}\right)$$
+
+A ray can reach radius $r$ only where $1/b^2 > V(r)$. So the deciding feature is
+the **highest point** of $V$, the hill the ray has to get over.
+
+Find it by differentiating and setting to zero:
+
+$$\frac{dV}{dr} = -\frac{2}{r^3} + \frac{3r_s}{r^4} = 0 \quad\Longrightarrow\quad \boxed{r_{\text{ph}} = \tfrac{3}{2} r_s = 3M}$$
+
+That is the **photon sphere**: the radius where light can orbit in a circle.
+It is a knife-edge, not a place to park. Nudge inward and you spiral in; nudge
+outward and you spiral away.
+
+The height of the hill sets the dividing aim. Put $r = 3M$ back into $V$:
+
+$$V_{\max} = \frac{1}{(3M)^2}\left(1 - \frac{2M}{3M}\right) = \frac{1}{9M^2}\cdot\frac{1}{3} = \frac{1}{27M^2}$$
+
+and a ray just clears the hill when $1/b^2 = V_{\max}$, giving
+
+$$\boxed{b_{\text{crit}} = \sqrt{27}\,M = 3\sqrt{3}\,M = \frac{3\sqrt{3}}{2} r_s \approx 2.598\, r_s}$$
+
+This is the single most important number in the picture, because **it is the
+size of the black shadow you see**. Note that it is 2.6 times the horizon, not
+1. The dark disc in the image is considerably bigger than the hole itself,
+because rays that would have missed a merely dark ball are still bent in.
+
+Just outside that edge sits the **photon ring**, the bright thin circle in
+every image, made of light that looped one or more times before escaping to
+your eye.
+
+> **In the code:** `B_CRIT` and `R_PHOTON` in `src/physics/constants.ts`.
+
+---
+
+## 7. The last stable orbit
+
+Around a star, every circular orbit is stable: nudge a satellite and it
+wobbles a bit, then carries on. Around a black hole this stops being true
+below a certain radius, and the reason is the same hill from Part 6, now for
+matter instead of light. Below
+
+$$r_{\text{ISCO}} = 6M = 3\, r_s$$
+
+the hill has no dip left to sit in, so the smallest nudge means falling in.
+This is the **innermost stable circular orbit**, and it is why the accretion
+disc has an inner edge with nothing inside it.
+
+Solving the full relativistic orbit for every debris particle would be
+expensive, so the app uses a shortcut that gets the important behaviour exactly
+right: the **Paczyński-Wiita potential**
+
+$$\Phi(r) = -\frac{GM}{r - r_s}$$
+
+The only change from Newton is $r \to r - r_s$, which makes gravity blow up at
+the horizon instead of at the centre. That small edit reproduces the true ISCO
+at $3\,r_s$, so matter destabilises and plunges in exactly the right place.
+
+Redo Part 2's balance with this potential. The inward pull is now
+$GM/(r-r_s)^2$, so
+
+$$\frac{v^2}{r} = \frac{GM}{(r - r_s)^2} \quad\Longrightarrow\quad \boxed{v_{\text{circ}} = \frac{\sqrt{GM\,r}}{r - r_s}}$$
+
+and the energy bookkeeping that decides whether debris is bound or escaping is
+
+$$\varepsilon = \tfrac{1}{2}v^2 - \frac{GM}{r - r_s}$$
+
+Negative means bound and it will come back. Positive means gone. The app
+computes exactly this for every particle at the moment it is born, and that
+single sign is what splits a shredded star into the half that returns and the
+half that escapes.
+
+> **In the code:** `R_ISCO` in `src/physics/constants.ts`, the potential in
+> `src/sim/gravity.ts`, the bound test in `boundFlag` in `src/sim/debris.ts`.
+
+---
+
+## 8. Why the disc glows, and how hot
+
+Gas cannot fall straight in. It arrives with some sideways motion, so it ends
+up circling, and collisions between gas on slightly different orbits flatten
+the whole mess into a disc, the same way a spinning ball of pizza dough
+flattens.
+
+Now the important part. Neighbouring rings orbit at *different* speeds (Part 2),
+so they rub. Rubbing converts orbital energy into heat, the gas radiates that
+heat away, loses energy, and sinks a little closer in, where it rubs even
+harder. The disc is a machine for converting gravitational energy into light,
+and it is a spectacularly efficient one: several percent of the rest mass of
+everything it swallows, against 0.7% for hydrogen fusion.
+
+The temperature profile follows from three statements:
+
+1. Energy released by matter sinking from $r + dr$ to $r$ is set by gravity.
+2. In a steady state the heat generated in a ring must leave it as light, so
+ heat in equals $\sigma T^4$ out (Stefan-Boltzmann).
+3. At the inner edge there is nothing further in to grip, so no torque acts
+ there. This "no-torque inner boundary" forces the emission to zero at
+ $r_{\text{in}}$.
+
+Together they give the **Shakura-Sunyaev** profile:
+
+$$T(r) \propto r^{-3/4}\left(1 - \sqrt{\frac{r_{\text{in}}}{r}}\right)^{1/4}$$
+
+The $r^{-3/4}$ is the "closer is hotter" part. The bracket is the no-torque
+boundary, and it is why the disc fades right at its inner rim instead of being
+brightest there.
+
+> **In the code:** `discEmission` in `src/render/shaders/geodesic.frag`, with
+> $r_{\text{in}} = 3 r_s$.
+
+---
+
+## 9. Why one side of the disc is brighter
+
+
+
+Look at any picture of an accretion disc and one side is glaringly brighter.
+That is not artistic licence. Three effects stack, and all three are in this
+app.
+
+**Doppler shift.** A source coming toward you has its light waves squashed
+together, so it looks bluer, and going away it looks redder. You hear this with
+sirens. At the ISCO the gas is orbiting at half the speed of light, so the
+effect is enormous.
+
+**Beaming.** Fast-moving sources do not radiate evenly. Their light is swept
+into a forward-pointing cone, like rain on a windscreen appearing to come from
+straight ahead when you drive fast. The approaching side of the disc aims its
+cone at you.
+
+**Gravitational redshift.** Climbing out of the gravity well costs energy, and
+a photon pays by getting redder. Deeper in means redder.
+
+The app folds all three into one number, the ratio of received to emitted
+frequency:
+
+$$g = \frac{\sqrt{1 - \dfrac{3M}{r}}}{\gamma\left(1 - \beta \cos\alpha\right)}, \qquad \gamma = \frac{1}{\sqrt{1 - \beta^2}}$$
+
+The top is the gravitational part for gas on a circular orbit. The bottom is
+the Doppler part, with $\beta = v/c$ and $\alpha$ the angle between the gas's
+motion and the direction the light leaves in. Brightness then scales as a power
+of $g$, and colour shifts by moving the blackbody temperature by the same
+factor.
+
+Two details this app gets right that are easy to get wrong:
+
+- $\beta$ is taken from the **Paczyński-Wiita** circular speed, so it reaches
+ 0.5 at the ISCO as it should.
+- $\alpha$ uses the **bent** photon direction at the point where the ray crosses
+ the disc, not the straight line back to the camera. That is why the lensed
+ image of the disc's far side, the part arcing over the top of the hole,
+ beams correctly too.
+
+> **Honest note.** The exact bolometric scaling for a moving blackbody surface
+> is $g^4$. The app uses $g^3$, which softens the contrast slightly and is a
+> deliberate art choice; it is the `BEAM_EXP` define in
+> `src/render/blackHolePass.ts` if you want the honest exponent.
+
+---
+
+## 10. Tides, and why they tear stars apart
+
+
+
+The Moon does not pull the Earth apart, but it does pull the near ocean harder
+than it pulls the planet's centre, and the far ocean less. That difference is
+the tide, and it stretches.
+
+Put numbers on the difference. Gravity at distance $r$ goes as $1/r^2$. For a
+body of radius $R$ whose centre is at $r$, the near side sits at $r - R$:
+
+$$\Delta a = \frac{GM}{(r-R)^2} - \frac{GM}{r^2} \approx \frac{2GMR}{r^3}$$
+
+(the approximation is just "$R$ is small compared to $r$", which holds until the
+very end). Notice the $1/r^3$: **tides grow much faster than gravity itself as
+you approach**. This is the entire story of spaghettification.
+
+A star fights back with its own gravity, which holds its surface down with
+about $Gm_\star/R^2$. Set the tide equal to the self-gravity and solve for
+where the star loses:
+
+$$\frac{2GMR}{r^3} = \frac{Gm_\star}{R^2} \quad\Longrightarrow\quad \boxed{r_T \approx R_\star \left(\frac{M}{m_\star}\right)^{1/3}}$$
+
+This is the **tidal radius**. Outside it the star survives, inside it comes
+apart.
+
+**The surprising bit.** Compare $r_T$ to the horizon. Since $r_s \propto M$ and
+$r_T \propto M^{1/3}$:
+
+$$\frac{r_T}{r_s} \propto M^{-2/3}$$
+
+So the *bigger* the black hole, the *closer in* the tidal radius sits relative
+to the horizon. Around a stellar-mass hole a star is shredded far outside, in a
+violent flash. Around a hole of ten billion suns, $r_T$ has moved inside the
+horizon: the star crosses intact and nothing is seen from outside at all. The
+spectacular disruptions come from the middle of the range, around a million
+solar masses.
+
+This is also why the app's tidal radii sit outside the disc: it is drawing the
+supermassive case.
+
+> **In the code:** `TDE_TUNING` in `src/config.ts`, phase thresholds in
+> `src/sim/tidal.ts`.
+
+---
+
+## 11. Why the debris becomes a stream
+
+
+
+Here is the part that took the longest to get right in this project, and the
+part most illustrations quietly skip.
+
+When the star comes apart it is already stretched, so its near tip sits deeper
+in the gravity well than its far tip. Deeper means more tightly bound. The
+spread in orbital energy across the star is roughly the tidal potential
+difference across its diameter:
+
+$$\Delta \varepsilon \approx \frac{GM R_\star}{r_p^2}$$
+
+where $r_p$ is how close the star got. Compare that to the energy the star had
+on the way in, which for a star wandering in from far away is about zero, and
+you get a startling result: **the spread is much larger than the original
+energy**. So the debris straddles zero. Half of it ends up bound, half of it
+ends up unbound, and the split is close to fifty-fifty.
+
+The bound half swings back out on long stretched orbits, returns, and wraps
+around the hole. That returning ribbon is the stream you see. The rate at
+which it comes back follows a famous power law, $\dot M \propto t^{-5/3}$
+(Rees, 1988), which is how real tidal disruption flares are identified.
+
+**The trap.** Energy is not the only thing that matters: angular momentum
+decides how close the debris passes on its return. Give a particle a kick along
+its direction of travel and you change its energy *and* strip its angular
+momentum, so its return pass dives into the hole and the stream never forms.
+Give it a **radial** kick and the angular momentum is untouched, so it keeps the
+star's pericenter and swings back out.
+
+This app applies the spread radially for exactly that reason, and launches each
+particle onto the star's own orbit at its own radius rather than copying the
+star's velocity vector. Both decisions were arrived at by watching the stream
+fail without them.
+
+> **In the code:** `spawnFromBody` in `src/sim/debris.ts`, the orbit
+> reconstruction in `src/sim/orbit.ts`, and the tests in
+> `src/sim/__tests__/stream.test.ts` that hold the resulting shape in place.
+
+---
+
+## 12. Gravitational waves and the chirp
+
+
+
+If mass curves space, then *moving* mass makes ripples in it. Wiggle a mass and
+the curvature nearby wiggles, and that wiggle travels outward at the speed of
+light. Those are gravitational waves.
+
+**Why two crests per orbit.** A spinning dumbbell looks identical after half a
+turn. The wave it makes therefore repeats twice per orbit, so the gravitational
+wave frequency is **twice** the orbital frequency. This is not a detail: it is
+how you know a detected chirp came from an orbit.
+
+Waves carry energy, and that energy is stolen from the orbit, so the pair must
+spiral together. Peters worked out the rate in 1964:
+
+$$\frac{da}{dt} = -\frac{64}{5}\,\frac{G^3\, m_1 m_2 (m_1 + m_2)}{c^5\, a^3}$$
+
+Look at the $1/a^3$. When the holes are far apart the decay is glacial. As they
+close, the shrinking accelerates violently, which drives the frequency up, which
+radiates harder still, which shrinks it faster. The runaway ends in a merger,
+and the rising tone it produces is the **chirp** that LIGO heard in 2015.
+
+At the end, the merged hole is *lighter* than the sum of its parts, because the
+missing mass left as waves:
+
+$$E_{\text{rad}} \approx 0.048\, M_{\text{tot}} \cdot \frac{\eta}{0.25}, \qquad \eta = \frac{m_1 m_2}{(m_1+m_2)^2}$$
+
+normalised to GW150914, which radiated about 4.6% of itself, roughly three
+solar masses converted to ripples in space, briefly outshining every star in
+the observable universe put together.
+
+The wave pattern drawn on the curvature grid has amplitude falling as $1/r$
+(not $1/r^2$: waves carry energy, and energy spreads over an area, so the
+*amplitude* falls as the square root of that), two crests per turn, and the
+crests wound into a spiral because the source keeps rotating while the ripples
+travel outward.
+
+> **In the code:** `src/sim/binary.ts` for the inspiral, and
+> `src/sim/gravitationalWave.ts` for the pattern on the grid, with tests for
+> both.
+
+---
+
+## 13. The funnel: what curved space actually means
+
+
+
+You have seen the rubber sheet with a bowling ball on it. It is a bad analogy
+(it explains gravity using gravity), but there is a real, precise version of
+that picture, and this app draws the real one.
+
+Here is the idea. Around a black hole, if you measure the circumference of a
+circle and divide by $2\pi$, you get a number. But if you then physically walk
+outward and measure the distance to the next circle, **you get more distance
+than the difference in those numbers says you should**. Space is stretched in
+the radial direction.
+
+We can draw exactly that. Take a flat 2D sheet, and bend it in a third
+direction so that walking outward along the *bent* sheet covers the extra
+distance. The extra vertical rise stores the stretch.
+
+The Schwarzschild solution says radial distance and circumference are related by
+
+$$ds^2 = \frac{dr^2}{1 - r_s/r}$$
+
+so the true radial distance for a step $dr$ is longer than $dr$ by that factor.
+Our bent sheet must satisfy Pythagoras: rise squared plus run squared equals
+true distance squared,
+
+$$dz^2 + dr^2 = \frac{dr^2}{1 - r_s/r}$$
+
+Solve for the slope:
+
+$$\left(\frac{dz}{dr}\right)^2 = \frac{1}{1 - r_s/r} - 1 = \frac{r_s}{r - r_s}$$
+
+Take the square root and integrate ($\int \sqrt{r_s/(r-r_s)}\,dr$, substituting
+$w = r - r_s$ makes it a plain $\int \sqrt{r_s/w}\,dw$):
+
+$$\boxed{z(r) = 2\sqrt{r_s\,(r - r_s)}}$$
+
+This is **Flamm's paraboloid**, and it is the funnel the app draws when you
+press `G`. Far away the slope goes to zero, so the sheet flattens into ordinary
+space. At $r = r_s$ the slope goes vertical, which is the throat.
+
+**What it is not.** It is a picture of *one slice of space at one moment*, not
+of spacetime, and the vertical direction is not a direction you can move in.
+Nothing "rolls downhill" on this surface. Gravity in relativity comes from the
+bending of *time* at least as much as space, and none of that is in the
+picture. It is an honest visualisation of one true thing, not of everything.
+
+> **In the code:** `embeddingDepth` in `src/render/spacetimeGrid.ts`, with the
+> wave from Part 12 added on top.
+
+---
+
+## 14. Jets, beaming, and the outflow
+
+Two very different things come out of the poles, and this app draws both.
+
+**The jet** is a narrow beam of plasma moving at nearly the speed of light.
+Because it is so fast, the beaming from Part 9 applies in an extreme form. For
+a steady jet the observed brightness scales as
+
+$$\delta^3, \qquad \delta = \frac{1}{\gamma\left(1 - \beta\cos\theta\right)}$$
+
+where $\theta$ is the angle between the jet and your line of sight. Point it at
+you and $\delta$ is large; point it away and $\delta$ is small. Cubed, this
+turns a symmetric pair of jets into one blazing beam and one you can barely
+find. Real radio galaxies show exactly this asymmetry, and the app reproduces
+it: the two cones are physically identical and only the geometry differs.
+
+**The outflow** is a different animal: slow, wide, and ragged. It appears when
+the hole is being fed faster than it can comfortably swallow. There is a limit,
+the **Eddington luminosity**, where the outward push of radiation on infalling
+gas balances gravity:
+
+$$L_{\text{Edd}} = \frac{4\pi G M m_p c}{\sigma_T}$$
+
+Above that, radiation pressure wins and blows material away. A tidal disruption
+dumps half a star onto a disc in a few weeks, which is far above the limit, so
+a broad wind is launched. In this app the outflow's strength is tied to the
+disc's feed rate for exactly that reason: it swells during fallback and dies
+away with it.
+
+> **In the code:** `jetEmission` and `windEmission` in
+> `src/render/shaders/geodesic.frag`.
+
+---
+
+## 15. What we cheat on
+
+A physics document that only lists what it gets right is advertising. Here is
+the other column.
+
+| Cheat | Why | What would fix it |
+|---|---|---|
+| **No spin.** Schwarzschild, not Kerr. | Kerr geodesics are a rewrite of the core integrator. | Real holes spin, which drags space around with them, skews the shadow, and moves the ISCO in to $1.24\,r_s$. |
+| **Two holes are superposed, not solved.** | The real two-body problem in GR needs supercomputers. | Numerical relativity. Our double shadows and eyebrow images are qualitatively right, quantitatively not. |
+| **Matter is pseudo-Newtonian.** | Full geodesic motion for 48,000 particles is too slow. | Paczyński-Wiita gets the ISCO exactly right, which is the part that matters visually. |
+| **The disc is infinitely thin.** | A volumetric disc multiplies the per-pixel cost. | Raymarched volume with real optical depth. |
+| **The jet is drawn, not launched.** | Blandford-Znajek needs spin, which we do not have, plus magnetohydrodynamics. | GRMHD simulation data. The beaming, at least, is computed and not painted. |
+| **Two clocks run fast.** | A disruption takes days, an inspiral from $8\,r_s$ takes about 1600 time units. | Nothing: the trajectories are exact, only the clock is compressed, and the app says so in the interface. |
+| **No light travel-time delay.** | You see the whole disc at one instant rather than each part as it was when its light left. | Track photon arrival times through the march. |
+| **The sky is invented.** | It follows real structure (luminosity function, dust extinction, clustering) but it is not a star catalogue. | A real survey texture, at the cost of every image looking identical. |
+
+---
+
+## Every equation and where it lives
+
+| Quantity | Equation | Code |
+|---|---|---|
+| Circular orbit speed | $v = \sqrt{GM/r}$ | `sim/gravity.ts` |
+| Escape speed | $v = \sqrt{2GM/r}$ | Part 3, sets $r_s$ |
+| Schwarzschild radius | $r_s = 2GM/c^2$ | `physics/constants.ts` |
+| Light deflection, weak field | $\alpha = 2r_s/b$ | Part 5 |
+| Photon path | $\ddot{\mathbf{x}} = -\tfrac{3}{2} r_s h^2 \mathbf{x}/r^5$ | `shaders/geodesic.frag`, `physics/geodesic.ts` |
+| Photon sphere | $r = 1.5\,r_s$ | `physics/constants.ts` |
+| Shadow radius | $b_{\text{crit}} = 3\sqrt{3}M \approx 2.6\,r_s$ | `physics/constants.ts` |
+| ISCO | $r = 3\,r_s$ | `physics/constants.ts` |
+| Paczyński-Wiita potential | $\Phi = -GM/(r - r_s)$ | `sim/gravity.ts` |
+| PW circular speed | $v = \sqrt{GMr}/(r - r_s)$ | `sim/gravity.ts` |
+| Specific energy | $\varepsilon = v^2/2 - GM/(r-r_s)$ | `sim/orbit.ts`, `sim/debris.ts` |
+| Disc temperature | $T \propto r^{-3/4}(1 - \sqrt{r_{\text{in}}/r})^{1/4}$ | `shaders/geodesic.frag` |
+| Doppler + gravity shift | $g = \sqrt{1 - 3M/r}\,/\,\gamma(1 - \beta\cos\alpha)$ | `shaders/geodesic.frag` |
+| Tidal radius | $r_T \approx R_\star (M/m_\star)^{1/3}$ | `config.ts` |
+| Tidal energy spread | $\Delta\varepsilon \approx GMR_\star/r_p^2$ | `sim/debris.ts` |
+| Peters inspiral | $da/dt = -\tfrac{64}{5} G^3 m_1m_2(m_1{+}m_2)/c^5a^3$ | `sim/binary.ts` |
+| Radiated mass | $E_{\text{rad}} \approx 0.048 M_{\text{tot}}\,\eta/0.25$ | `sim/binary.ts` |
+| Flamm's paraboloid | $z = 2\sqrt{r_s(r - r_s)}$ | `render/spacetimeGrid.ts` |
+| Relativistic beaming | $\delta = 1/\gamma(1 - \beta\cos\theta)$, $I \propto \delta^3$ | `shaders/geodesic.frag` |
+| Eddington luminosity | $L = 4\pi GMm_pc/\sigma_T$ | Part 14, motivates the outflow |
+
+---
+
+## Where to go next
+
+- **Luminet (1979)**, *Image of a spherical black hole with thin accretion
+ disk*. The first computed picture of what this app draws, made with a
+ pencil, a mainframe, and no graphics hardware at all.
+- **James, von Tunzelmann, Franklin and Thorne (2015)**, the *Interstellar*
+ paper. The full version of Parts 5 to 9, written by the people who did it
+ for the film.
+- **Rees (1988)** on tidal disruption, for Parts 10 and 11.
+- **Peters (1964)** for Part 12, four pages that predicted a sound nobody heard
+ for fifty-one years.
+
+Throw a ball sideways and it curves down and lands. Throw it harder and it
+lands further away. Throw it hard enough and the ground curves away underneath
+it as fast as it falls, and it never lands. That is an orbit. It is not
+weightlessness, it is falling forever and missing.
+
+Put numbers on it. The pull available is $GMm/r^2$. The pull required to keep
+turning in a circle is $mv^2/r$. Set them equal:
+
+$$\frac{GMm}{r^2} = \frac{mv^2}{r}$$
+
+The mass of the orbiting thing, $m$, appears on both sides and cancels. That is
+why a feather and a cannonball orbit identically. Multiply both sides by $r/m$:
+
+$$\boxed{v_{\text{circ}} = \sqrt{\frac{GM}{r}}}$$
+
+Closer in means faster. This one line explains why Mercury runs rings around
+Neptune, and why the inner part of the accretion disc shears past the outer
+part and winds the pattern into a spiral.
+
+> **In the code:** `vCircular` in `src/sim/gravity.ts`, used everywhere from
+> debris launch speeds to the disc's shear rate.
+
+---
+
+## 3. Escape speed, and the size of a black hole
+
+
+
+To escape completely you need enough kinetic energy to pay off the whole
+gravitational debt. Debt paid means total energy reaches zero:
+
+$$\tfrac{1}{2}mv^2 - \frac{GMm}{r} = 0 \quad\Longrightarrow\quad v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$$
+
+Earth's is 11 km/s. Notice what the formula depends on: not just how much mass,
+but how *close to it* you can get. Squeeze the same mass into a smaller ball and
+you can stand nearer the centre, so the escape speed goes up.
+
+Now ask the question John Michell asked in 1783: what if we squeeze it so far
+that the escape speed reaches the speed of light? Set $v_{\text{esc}} = c$:
+
+$$c = \sqrt{\frac{2GM}{r}} \quad\Longrightarrow\quad \boxed{r_s = \frac{2GM}{c^2}}$$
+
+This is the **Schwarzschild radius**. For the Sun it is about 3 km. For the
+Earth, 9 mm.
+
+**An honest warning.** That derivation is a fluke. It treats light as a slow
+cannonball, which is wrong, and it says light "falls back", which is also
+wrong. The correct calculation is Karl Schwarzschild's 1916 solution of
+Einstein's equations, and it happens to give exactly the same number. Take the
+formula, discard the story: nothing falls back, the horizon is a one way door.
+
+> **In the code:** `R_S` in `src/physics/constants.ts`.
+
+---
+
+## 4. Units where the answer is 1
+
+Physicists get tired of writing $G$ and $c$, so they choose units where both
+equal 1. It is the same trick as measuring travel in "hours away" instead of
+kilometres: you have folded the speed into the unit.
+
+With $G = c = 1$, the Schwarzschild radius becomes
+
+$$r_s = 2M$$
+
+This app goes one step further and measures every length in Schwarzschild
+radii, so $r_s = 1$ and therefore $M = \tfrac{1}{2}$. When you see `0.5` in the
+shader where you expected a mass, that is why. A distance of `12` means twelve
+Schwarzschild radii from the centre.
+
+> **In the code:** `src/physics/constants.ts` has the whole convention in one
+> comment block.
+
+---
+
+## 5. Why light bends, and by how much
+
+Light has no mass, so how can gravity pull on it?
+
+Einstein's answer starts with a thought experiment you can almost do. Imagine a
+windowless lift. If it sits on Earth you feel pressed to the floor. If it is
+being towed through empty space at $9.8\ \text{m/s}^2$ you feel exactly the
+same. **There is no experiment inside the lift that tells the two apart.** That
+is the equivalence principle.
+
+Now shine a torch across the lift while it accelerates. The beam leaves one
+wall, and by the time it crosses, the lift has moved up a little, so the spot
+lands slightly low. Inside the lift, the light looks bent. And if acceleration
+and gravity are indistinguishable, then **gravity must bend light too**.
+
+The next step is the one that made Einstein famous. He found that light bends
+by *twice* what you get from pretending a photon is a cannonball. The full
+weak-field result, for a ray that passes at closest distance $b$:
+
+$$\alpha \simeq \frac{4GM}{bc^2} = \frac{2 r_s}{b}$$
+
+Eddington measured it during the 1919 eclipse, got the factor of two rather
+than one, and Einstein was on the front page of every newspaper.
+
+Close to the hole, "bends a bit" is no longer enough and we need the exact
+path. Writing $u = 1/r$ (so big $u$ means close in) and tracking the ray by the
+angle $\phi$ it has swept around the hole, general relativity gives
+
+$$\frac{d^2u}{d\phi^2} + u = \tfrac{3}{2} r_s u^2$$
+
+Read it as two pieces. **Without** the right-hand side, this is the equation for
+a straight line written in polar coordinates, which you can check by putting in
+$u = \sin\phi / b$. **With** it, the line curves, and the correction grows as
+$u^2$, so it is negligible far away and dominant close in. All of relativistic
+lensing is in that one extra term.
+
+The app does not integrate in $\phi$, because a shader ray needs to move in
+$x, y, z$. The same equation in vector form is
+
+$$\ddot{\mathbf{x}} = -\tfrac{3}{2}\, r_s\, h^2\, \frac{\mathbf{x}}{r^5}, \qquad h = |\mathbf{x} \times \dot{\mathbf{x}}|$$
+
+where $h$ is the ray's angular momentum about the hole, which stays constant
+along the path. Every pixel on your screen runs this equation with a
+fourth-order Runge-Kutta integrator, a few hundred steps, every frame.
+
+> **In the code:** `centerAccel` in `src/render/shaders/geodesic.frag`, and the
+> same equation on the CPU in `src/physics/geodesic.ts` so the drawn photon
+> paths land exactly on the features the shader renders.
+
+---
+
+## 6. The photon sphere and the shadow
+
+
+
+Aim a ray at the hole and the only thing that matters is $b$, how far off
+centre you aimed. Three things can happen, and there is a sharp line between
+them.
+
+To find the line, use energy bookkeeping. For light, the "how close can I get"
+question reduces to comparing $1/b^2$ against the function
+
+$$V(r) = \frac{1}{r^2}\left(1 - \frac{r_s}{r}\right)$$
+
+A ray can reach radius $r$ only where $1/b^2 > V(r)$. So the deciding feature is
+the **highest point** of $V$, the hill the ray has to get over.
+
+Find it by differentiating and setting to zero:
+
+$$\frac{dV}{dr} = -\frac{2}{r^3} + \frac{3r_s}{r^4} = 0 \quad\Longrightarrow\quad \boxed{r_{\text{ph}} = \tfrac{3}{2} r_s = 3M}$$
+
+That is the **photon sphere**: the radius where light can orbit in a circle.
+It is a knife-edge, not a place to park. Nudge inward and you spiral in; nudge
+outward and you spiral away.
+
+The height of the hill sets the dividing aim. Put $r = 3M$ back into $V$:
+
+$$V_{\max} = \frac{1}{(3M)^2}\left(1 - \frac{2M}{3M}\right) = \frac{1}{9M^2}\cdot\frac{1}{3} = \frac{1}{27M^2}$$
+
+and a ray just clears the hill when $1/b^2 = V_{\max}$, giving
+
+$$\boxed{b_{\text{crit}} = \sqrt{27}\,M = 3\sqrt{3}\,M = \frac{3\sqrt{3}}{2} r_s \approx 2.598\, r_s}$$
+
+This is the single most important number in the picture, because **it is the
+size of the black shadow you see**. Note that it is 2.6 times the horizon, not
+1. The dark disc in the image is considerably bigger than the hole itself,
+because rays that would have missed a merely dark ball are still bent in.
+
+Just outside that edge sits the **photon ring**, the bright thin circle in
+every image, made of light that looped one or more times before escaping to
+your eye.
+
+> **In the code:** `B_CRIT` and `R_PHOTON` in `src/physics/constants.ts`.
+
+---
+
+## 7. The last stable orbit
+
+Around a star, every circular orbit is stable: nudge a satellite and it
+wobbles a bit, then carries on. Around a black hole this stops being true
+below a certain radius, and the reason is the same hill from Part 6, now for
+matter instead of light. Below
+
+$$r_{\text{ISCO}} = 6M = 3\, r_s$$
+
+the hill has no dip left to sit in, so the smallest nudge means falling in.
+This is the **innermost stable circular orbit**, and it is why the accretion
+disc has an inner edge with nothing inside it.
+
+Solving the full relativistic orbit for every debris particle would be
+expensive, so the app uses a shortcut that gets the important behaviour exactly
+right: the **Paczyński-Wiita potential**
+
+$$\Phi(r) = -\frac{GM}{r - r_s}$$
+
+The only change from Newton is $r \to r - r_s$, which makes gravity blow up at
+the horizon instead of at the centre. That small edit reproduces the true ISCO
+at $3\,r_s$, so matter destabilises and plunges in exactly the right place.
+
+Redo Part 2's balance with this potential. The inward pull is now
+$GM/(r-r_s)^2$, so
+
+$$\frac{v^2}{r} = \frac{GM}{(r - r_s)^2} \quad\Longrightarrow\quad \boxed{v_{\text{circ}} = \frac{\sqrt{GM\,r}}{r - r_s}}$$
+
+and the energy bookkeeping that decides whether debris is bound or escaping is
+
+$$\varepsilon = \tfrac{1}{2}v^2 - \frac{GM}{r - r_s}$$
+
+Negative means bound and it will come back. Positive means gone. The app
+computes exactly this for every particle at the moment it is born, and that
+single sign is what splits a shredded star into the half that returns and the
+half that escapes.
+
+> **In the code:** `R_ISCO` in `src/physics/constants.ts`, the potential in
+> `src/sim/gravity.ts`, the bound test in `boundFlag` in `src/sim/debris.ts`.
+
+---
+
+## 8. Why the disc glows, and how hot
+
+Gas cannot fall straight in. It arrives with some sideways motion, so it ends
+up circling, and collisions between gas on slightly different orbits flatten
+the whole mess into a disc, the same way a spinning ball of pizza dough
+flattens.
+
+Now the important part. Neighbouring rings orbit at *different* speeds (Part 2),
+so they rub. Rubbing converts orbital energy into heat, the gas radiates that
+heat away, loses energy, and sinks a little closer in, where it rubs even
+harder. The disc is a machine for converting gravitational energy into light,
+and it is a spectacularly efficient one: several percent of the rest mass of
+everything it swallows, against 0.7% for hydrogen fusion.
+
+The temperature profile follows from three statements:
+
+1. Energy released by matter sinking from $r + dr$ to $r$ is set by gravity.
+2. In a steady state the heat generated in a ring must leave it as light, so
+ heat in equals $\sigma T^4$ out (Stefan-Boltzmann).
+3. At the inner edge there is nothing further in to grip, so no torque acts
+ there. This "no-torque inner boundary" forces the emission to zero at
+ $r_{\text{in}}$.
+
+Together they give the **Shakura-Sunyaev** profile:
+
+$$T(r) \propto r^{-3/4}\left(1 - \sqrt{\frac{r_{\text{in}}}{r}}\right)^{1/4}$$
+
+The $r^{-3/4}$ is the "closer is hotter" part. The bracket is the no-torque
+boundary, and it is why the disc fades right at its inner rim instead of being
+brightest there.
+
+> **In the code:** `discEmission` in `src/render/shaders/geodesic.frag`, with
+> $r_{\text{in}} = 3 r_s$.
+
+---
+
+## 9. Why one side of the disc is brighter
+
+
+
+Look at any picture of an accretion disc and one side is glaringly brighter.
+That is not artistic licence. Three effects stack, and all three are in this
+app.
+
+**Doppler shift.** A source coming toward you has its light waves squashed
+together, so it looks bluer, and going away it looks redder. You hear this with
+sirens. At the ISCO the gas is orbiting at half the speed of light, so the
+effect is enormous.
+
+**Beaming.** Fast-moving sources do not radiate evenly. Their light is swept
+into a forward-pointing cone, like rain on a windscreen appearing to come from
+straight ahead when you drive fast. The approaching side of the disc aims its
+cone at you.
+
+**Gravitational redshift.** Climbing out of the gravity well costs energy, and
+a photon pays by getting redder. Deeper in means redder.
+
+The app folds all three into one number, the ratio of received to emitted
+frequency:
+
+$$g = \frac{\sqrt{1 - \dfrac{3M}{r}}}{\gamma\left(1 - \beta \cos\alpha\right)}, \qquad \gamma = \frac{1}{\sqrt{1 - \beta^2}}$$
+
+The top is the gravitational part for gas on a circular orbit. The bottom is
+the Doppler part, with $\beta = v/c$ and $\alpha$ the angle between the gas's
+motion and the direction the light leaves in. Brightness then scales as a power
+of $g$, and colour shifts by moving the blackbody temperature by the same
+factor.
+
+Two details this app gets right that are easy to get wrong:
+
+- $\beta$ is taken from the **Paczyński-Wiita** circular speed, so it reaches
+ 0.5 at the ISCO as it should.
+- $\alpha$ uses the **bent** photon direction at the point where the ray crosses
+ the disc, not the straight line back to the camera. That is why the lensed
+ image of the disc's far side, the part arcing over the top of the hole,
+ beams correctly too.
+
+> **Honest note.** The exact bolometric scaling for a moving blackbody surface
+> is $g^4$. The app uses $g^3$, which softens the contrast slightly and is a
+> deliberate art choice; it is the `BEAM_EXP` define in
+> `src/render/blackHolePass.ts` if you want the honest exponent.
+
+---
+
+## 10. Tides, and why they tear stars apart
+
+
+
+The Moon does not pull the Earth apart, but it does pull the near ocean harder
+than it pulls the planet's centre, and the far ocean less. That difference is
+the tide, and it stretches.
+
+Put numbers on the difference. Gravity at distance $r$ goes as $1/r^2$. For a
+body of radius $R$ whose centre is at $r$, the near side sits at $r - R$:
+
+$$\Delta a = \frac{GM}{(r-R)^2} - \frac{GM}{r^2} \approx \frac{2GMR}{r^3}$$
+
+(the approximation is just "$R$ is small compared to $r$", which holds until the
+very end). Notice the $1/r^3$: **tides grow much faster than gravity itself as
+you approach**. This is the entire story of spaghettification.
+
+A star fights back with its own gravity, which holds its surface down with
+about $Gm_\star/R^2$. Set the tide equal to the self-gravity and solve for
+where the star loses:
+
+$$\frac{2GMR}{r^3} = \frac{Gm_\star}{R^2} \quad\Longrightarrow\quad \boxed{r_T \approx R_\star \left(\frac{M}{m_\star}\right)^{1/3}}$$
+
+This is the **tidal radius**. Outside it the star survives, inside it comes
+apart.
+
+**The surprising bit.** Compare $r_T$ to the horizon. Since $r_s \propto M$ and
+$r_T \propto M^{1/3}$:
+
+$$\frac{r_T}{r_s} \propto M^{-2/3}$$
+
+So the *bigger* the black hole, the *closer in* the tidal radius sits relative
+to the horizon. Around a stellar-mass hole a star is shredded far outside, in a
+violent flash. Around a hole of ten billion suns, $r_T$ has moved inside the
+horizon: the star crosses intact and nothing is seen from outside at all. The
+spectacular disruptions come from the middle of the range, around a million
+solar masses.
+
+This is also why the app's tidal radii sit outside the disc: it is drawing the
+supermassive case.
+
+> **In the code:** `TDE_TUNING` in `src/config.ts`, phase thresholds in
+> `src/sim/tidal.ts`.
+
+---
+
+## 11. Why the debris becomes a stream
+
+
+
+Here is the part that took the longest to get right in this project, and the
+part most illustrations quietly skip.
+
+When the star comes apart it is already stretched, so its near tip sits deeper
+in the gravity well than its far tip. Deeper means more tightly bound. The
+spread in orbital energy across the star is roughly the tidal potential
+difference across its diameter:
+
+$$\Delta \varepsilon \approx \frac{GM R_\star}{r_p^2}$$
+
+where $r_p$ is how close the star got. Compare that to the energy the star had
+on the way in, which for a star wandering in from far away is about zero, and
+you get a startling result: **the spread is much larger than the original
+energy**. So the debris straddles zero. Half of it ends up bound, half of it
+ends up unbound, and the split is close to fifty-fifty.
+
+The bound half swings back out on long stretched orbits, returns, and wraps
+around the hole. That returning ribbon is the stream you see. The rate at
+which it comes back follows a famous power law, $\dot M \propto t^{-5/3}$
+(Rees, 1988), which is how real tidal disruption flares are identified.
+
+**The trap.** Energy is not the only thing that matters: angular momentum
+decides how close the debris passes on its return. Give a particle a kick along
+its direction of travel and you change its energy *and* strip its angular
+momentum, so its return pass dives into the hole and the stream never forms.
+Give it a **radial** kick and the angular momentum is untouched, so it keeps the
+star's pericenter and swings back out.
+
+This app applies the spread radially for exactly that reason, and launches each
+particle onto the star's own orbit at its own radius rather than copying the
+star's velocity vector. Both decisions were arrived at by watching the stream
+fail without them.
+
+> **In the code:** `spawnFromBody` in `src/sim/debris.ts`, the orbit
+> reconstruction in `src/sim/orbit.ts`, and the tests in
+> `src/sim/__tests__/stream.test.ts` that hold the resulting shape in place.
+
+---
+
+## 12. Gravitational waves and the chirp
+
+
+
+If mass curves space, then *moving* mass makes ripples in it. Wiggle a mass and
+the curvature nearby wiggles, and that wiggle travels outward at the speed of
+light. Those are gravitational waves.
+
+**Why two crests per orbit.** A spinning dumbbell looks identical after half a
+turn. The wave it makes therefore repeats twice per orbit, so the gravitational
+wave frequency is **twice** the orbital frequency. This is not a detail: it is
+how you know a detected chirp came from an orbit.
+
+Waves carry energy, and that energy is stolen from the orbit, so the pair must
+spiral together. Peters worked out the rate in 1964:
+
+$$\frac{da}{dt} = -\frac{64}{5}\,\frac{G^3\, m_1 m_2 (m_1 + m_2)}{c^5\, a^3}$$
+
+Look at the $1/a^3$. When the holes are far apart the decay is glacial. As they
+close, the shrinking accelerates violently, which drives the frequency up, which
+radiates harder still, which shrinks it faster. The runaway ends in a merger,
+and the rising tone it produces is the **chirp** that LIGO heard in 2015.
+
+At the end, the merged hole is *lighter* than the sum of its parts, because the
+missing mass left as waves:
+
+$$E_{\text{rad}} \approx 0.048\, M_{\text{tot}} \cdot \frac{\eta}{0.25}, \qquad \eta = \frac{m_1 m_2}{(m_1+m_2)^2}$$
+
+normalised to GW150914, which radiated about 4.6% of itself, roughly three
+solar masses converted to ripples in space, briefly outshining every star in
+the observable universe put together.
+
+The wave pattern drawn on the curvature grid has amplitude falling as $1/r$
+(not $1/r^2$: waves carry energy, and energy spreads over an area, so the
+*amplitude* falls as the square root of that), two crests per turn, and the
+crests wound into a spiral because the source keeps rotating while the ripples
+travel outward.
+
+> **In the code:** `src/sim/binary.ts` for the inspiral, and
+> `src/sim/gravitationalWave.ts` for the pattern on the grid, with tests for
+> both.
+
+---
+
+## 13. The funnel: what curved space actually means
+
+
+
+You have seen the rubber sheet with a bowling ball on it. It is a bad analogy
+(it explains gravity using gravity), but there is a real, precise version of
+that picture, and this app draws the real one.
+
+Here is the idea. Around a black hole, if you measure the circumference of a
+circle and divide by $2\pi$, you get a number. But if you then physically walk
+outward and measure the distance to the next circle, **you get more distance
+than the difference in those numbers says you should**. Space is stretched in
+the radial direction.
+
+We can draw exactly that. Take a flat 2D sheet, and bend it in a third
+direction so that walking outward along the *bent* sheet covers the extra
+distance. The extra vertical rise stores the stretch.
+
+The Schwarzschild solution says radial distance and circumference are related by
+
+$$ds^2 = \frac{dr^2}{1 - r_s/r}$$
+
+so the true radial distance for a step $dr$ is longer than $dr$ by that factor.
+Our bent sheet must satisfy Pythagoras: rise squared plus run squared equals
+true distance squared,
+
+$$dz^2 + dr^2 = \frac{dr^2}{1 - r_s/r}$$
+
+Solve for the slope:
+
+$$\left(\frac{dz}{dr}\right)^2 = \frac{1}{1 - r_s/r} - 1 = \frac{r_s}{r - r_s}$$
+
+Take the square root and integrate ($\int \sqrt{r_s/(r-r_s)}\,dr$, substituting
+$w = r - r_s$ makes it a plain $\int \sqrt{r_s/w}\,dw$):
+
+$$\boxed{z(r) = 2\sqrt{r_s\,(r - r_s)}}$$
+
+This is **Flamm's paraboloid**, and it is the funnel the app draws when you
+press `G`. Far away the slope goes to zero, so the sheet flattens into ordinary
+space. At $r = r_s$ the slope goes vertical, which is the throat.
+
+**What it is not.** It is a picture of *one slice of space at one moment*, not
+of spacetime, and the vertical direction is not a direction you can move in.
+Nothing "rolls downhill" on this surface. Gravity in relativity comes from the
+bending of *time* at least as much as space, and none of that is in the
+picture. It is an honest visualisation of one true thing, not of everything.
+
+> **In the code:** `embeddingDepth` in `src/render/spacetimeGrid.ts`, with the
+> wave from Part 12 added on top.
+
+---
+
+## 14. Jets, beaming, and the outflow
+
+Two very different things come out of the poles, and this app draws both.
+
+**The jet** is a narrow beam of plasma moving at nearly the speed of light.
+Because it is so fast, the beaming from Part 9 applies in an extreme form. For
+a steady jet the observed brightness scales as
+
+$$\delta^3, \qquad \delta = \frac{1}{\gamma\left(1 - \beta\cos\theta\right)}$$
+
+where $\theta$ is the angle between the jet and your line of sight. Point it at
+you and $\delta$ is large; point it away and $\delta$ is small. Cubed, this
+turns a symmetric pair of jets into one blazing beam and one you can barely
+find. Real radio galaxies show exactly this asymmetry, and the app reproduces
+it: the two cones are physically identical and only the geometry differs.
+
+**The outflow** is a different animal: slow, wide, and ragged. It appears when
+the hole is being fed faster than it can comfortably swallow. There is a limit,
+the **Eddington luminosity**, where the outward push of radiation on infalling
+gas balances gravity:
+
+$$L_{\text{Edd}} = \frac{4\pi G M m_p c}{\sigma_T}$$
+
+Above that, radiation pressure wins and blows material away. A tidal disruption
+dumps half a star onto a disc in a few weeks, which is far above the limit, so
+a broad wind is launched. In this app the outflow's strength is tied to the
+disc's feed rate for exactly that reason: it swells during fallback and dies
+away with it.
+
+> **In the code:** `jetEmission` and `windEmission` in
+> `src/render/shaders/geodesic.frag`.
+
+---
+
+## 15. What we cheat on
+
+A physics document that only lists what it gets right is advertising. Here is
+the other column.
+
+| Cheat | Why | What would fix it |
+|---|---|---|
+| **No spin.** Schwarzschild, not Kerr. | Kerr geodesics are a rewrite of the core integrator. | Real holes spin, which drags space around with them, skews the shadow, and moves the ISCO in to $1.24\,r_s$. |
+| **Two holes are superposed, not solved.** | The real two-body problem in GR needs supercomputers. | Numerical relativity. Our double shadows and eyebrow images are qualitatively right, quantitatively not. |
+| **Matter is pseudo-Newtonian.** | Full geodesic motion for 48,000 particles is too slow. | Paczyński-Wiita gets the ISCO exactly right, which is the part that matters visually. |
+| **The disc is infinitely thin.** | A volumetric disc multiplies the per-pixel cost. | Raymarched volume with real optical depth. |
+| **The jet is drawn, not launched.** | Blandford-Znajek needs spin, which we do not have, plus magnetohydrodynamics. | GRMHD simulation data. The beaming, at least, is computed and not painted. |
+| **Two clocks run fast.** | A disruption takes days, an inspiral from $8\,r_s$ takes about 1600 time units. | Nothing: the trajectories are exact, only the clock is compressed, and the app says so in the interface. |
+| **No light travel-time delay.** | You see the whole disc at one instant rather than each part as it was when its light left. | Track photon arrival times through the march. |
+| **The sky is invented.** | It follows real structure (luminosity function, dust extinction, clustering) but it is not a star catalogue. | A real survey texture, at the cost of every image looking identical. |
+
+---
+
+## Every equation and where it lives
+
+| Quantity | Equation | Code |
+|---|---|---|
+| Circular orbit speed | $v = \sqrt{GM/r}$ | `sim/gravity.ts` |
+| Escape speed | $v = \sqrt{2GM/r}$ | Part 3, sets $r_s$ |
+| Schwarzschild radius | $r_s = 2GM/c^2$ | `physics/constants.ts` |
+| Light deflection, weak field | $\alpha = 2r_s/b$ | Part 5 |
+| Photon path | $\ddot{\mathbf{x}} = -\tfrac{3}{2} r_s h^2 \mathbf{x}/r^5$ | `shaders/geodesic.frag`, `physics/geodesic.ts` |
+| Photon sphere | $r = 1.5\,r_s$ | `physics/constants.ts` |
+| Shadow radius | $b_{\text{crit}} = 3\sqrt{3}M \approx 2.6\,r_s$ | `physics/constants.ts` |
+| ISCO | $r = 3\,r_s$ | `physics/constants.ts` |
+| Paczyński-Wiita potential | $\Phi = -GM/(r - r_s)$ | `sim/gravity.ts` |
+| PW circular speed | $v = \sqrt{GMr}/(r - r_s)$ | `sim/gravity.ts` |
+| Specific energy | $\varepsilon = v^2/2 - GM/(r-r_s)$ | `sim/orbit.ts`, `sim/debris.ts` |
+| Disc temperature | $T \propto r^{-3/4}(1 - \sqrt{r_{\text{in}}/r})^{1/4}$ | `shaders/geodesic.frag` |
+| Doppler + gravity shift | $g = \sqrt{1 - 3M/r}\,/\,\gamma(1 - \beta\cos\alpha)$ | `shaders/geodesic.frag` |
+| Tidal radius | $r_T \approx R_\star (M/m_\star)^{1/3}$ | `config.ts` |
+| Tidal energy spread | $\Delta\varepsilon \approx GMR_\star/r_p^2$ | `sim/debris.ts` |
+| Peters inspiral | $da/dt = -\tfrac{64}{5} G^3 m_1m_2(m_1{+}m_2)/c^5a^3$ | `sim/binary.ts` |
+| Radiated mass | $E_{\text{rad}} \approx 0.048 M_{\text{tot}}\,\eta/0.25$ | `sim/binary.ts` |
+| Flamm's paraboloid | $z = 2\sqrt{r_s(r - r_s)}$ | `render/spacetimeGrid.ts` |
+| Relativistic beaming | $\delta = 1/\gamma(1 - \beta\cos\theta)$, $I \propto \delta^3$ | `shaders/geodesic.frag` |
+| Eddington luminosity | $L = 4\pi GMm_pc/\sigma_T$ | Part 14, motivates the outflow |
+
+---
+
+## Where to go next
+
+- **Luminet (1979)**, *Image of a spherical black hole with thin accretion
+ disk*. The first computed picture of what this app draws, made with a
+ pencil, a mainframe, and no graphics hardware at all.
+- **James, von Tunzelmann, Franklin and Thorne (2015)**, the *Interstellar*
+ paper. The full version of Parts 5 to 9, written by the people who did it
+ for the film.
+- **Rees (1988)** on tidal disruption, for Parts 10 and 11.
+- **Peters (1964)** for Part 12, four pages that predicted a sound nobody heard
+ for fifty-one years.