The orbit that cannot be made smaller

Newtonian gravity has no innermost orbit. Give a body more angular momentum and its stable circular path sits further out; give it less and the path shrinks, without limit and without any radius at which the arrangement stops working. General relativity adds one term to the expression that says so, and the conclusion is not modified but reversed: below a definite radius there is no stable circular path at any angular momentum whatever.

Everything below is where that radius comes from, what the matter reaching it has had to give up, and where the statement stops being true.

The Newtonian well always has a bottom #

The reason a central-force problem is tractable is that it can be reduced to one dimension. Angular momentum is conserved, so the quantity $\tilde L = r^2\,\mathrm{d}\phi/\mathrm{d}t$ never changes, and the energy of the orbit can be written using it as a constant rather than as a second unknown:

$$E = \tfrac12\left(\frac{\mathrm{d}r}{\mathrm{d}t}\right)^2 + \frac{\tilde L^2}{2r^2} - \frac{GM}{r}.$$

The middle term is the kinetic energy of the angular motion, re-expressed as a function of radius. Written that way it behaves exactly like a potential: it climbs steeply as $r$ falls, and a body approaching the centre runs out of radial kinetic energy against it and turns round. That is what a centrifugal barrier is — not a force but the bookkeeping of a conserved quantity, and the same relabelling that makes the forces that are not there useful in a rotating frame.

An orbit needs a force because the velocity is turning, and at eight points around a circular path the picture is the same: velocity tangent and constant in length, acceleration inward and constant in length. That is the Newtonian starting point, and it has no smallest orbit in it — any radius will do, provided the speed is chosen to match.

The sum of the two terms is the effective potential, and its shape settles the whole question. Where it has a minimum, a body sitting there with no radial velocity stays, and a body displaced slightly oscillates about it rather than departing — which is the statement that a minimum is a place a system returns to.

The attraction on its own, with no angular momentum in it, is a potential falling monotonically towards the centre with a single turning point where the total energy meets it. Add angular momentum and a centrifugal term appears which rises faster as the radius shrinks — so the well acquires a bottom, and the bottom is the circular orbit. The Newtonian well always has one, at every angular momentum, which is why there is no innermost orbit in Newtonian gravity.

Differentiate the sum and set it to zero. The condition for a circular orbit is $GM/r^2 = \tilde L^2/r^3$, which rearranges to $r = \tilde L^2/GM$ — exactly one root for every positive $\tilde L$, and that root is always a minimum. Read it the other way and it is a recipe: name any radius at all, set $\tilde L = \sqrt{GMr}$, and a stable circular orbit exists there. There is no smallest one, and nothing in the expression hints that there could be. Near the bottom the well is quadratic — every minimum is — so the radial oscillation of a nearly circular orbit has a definite frequency, and that frequency will matter later.

The one term general relativity adds #

Outside a spherical mass the geometry differs from flat space in two places at once. The time part is stretched by $(1 - r_s/r)$, which is what makes a clock run slow lower down, and the radial space part by the reciprocal of the same factor, which has no Newtonian counterpart and is the half of the deflection of light that falling cannot supply.

Carrying the same reduction through in that geometry gives an equation of exactly the Newtonian form — a radial kinetic term plus an effective potential — with one addition:

$$\frac{V(r)}{c^2} = -\frac{GM}{rc^2} + \frac{\tilde L^2}{2r^2c^2} - \frac{GM\tilde L^2}{c^4r^3}.$$

The first two terms are Newton’s. The third is the whole of the difference between the two theories on this question, and three things about it decide everything that follows. It is negative, so it works with the attraction rather than against it. It falls as $1/r^3$, so far away it is negligible and close in it dominates. And it carries $\tilde L^2$, the same factor the centrifugal barrier carries.

That third property is the easiest to read past. The obvious defence against a term that wins at small radius is to add angular momentum until the barrier holds the body out. It does not work: raising $\tilde L$ raises the new term by the same factor, so the two scale together and the contest between $1/r^2$ and $1/r^3$ at small $r$ is settled by the powers alone.

Where the minimum stops existing #

Setting the derivative of the full potential to zero and clearing denominators gives a quadratic rather than the Newtonian linear condition:

$$r^2 - \frac{\tilde L^2}{GM}\,r + \frac{3\tilde L^2}{c^2} = 0.$$

Its roots are $\tfrac12(\tilde L^2/GM)\left(1 \pm \sqrt{1 - 12G^2M^2/\tilde L^2c^2}\right)$. The outer root is a minimum and the inner one a maximum, so a body with enough angular momentum has two circular orbits available: a stable one in the well, and an unstable one balanced on the top of the barrier. As the angular momentum falls the two move toward each other. And then the discriminant goes negative and both are gone.

The condition for the roots to be real is $\tilde L \geq \sqrt{12}\,GM/c$, and at equality they coincide at $r_{\text{ISCO}} = 6GM/c^2 = 3r_s$, the innermost stable circular orbit.

Two features of that number matter, and the second is the surprising one. The first is that it exists at all: no amount of angular momentum, energy or engineering produces a smaller stable orbit. The second is that it has no Newtonian analogue whatsoever — it is not a correction to a known radius. Newton’s theory says the stable circular orbits are every radius above zero; relativity says every radius above $6GM/c^2$; and one term separates the two.

Between $4GM/c^2$ and $6GM/c^2$ circular orbits still exist, on the maxima rather than in the wells, and they are unstable in the strict sense: displace one inward by any amount and it does not oscillate, it plunges. Below $4GM/c^2 = 2r_s$ even those are unbound, requiring more energy than a body brought in from rest far away possesses. The region between the innermost stable orbit and the horizon is not empty of solutions; it is full of solutions nothing can stay on.

Released from rest at ten Schwarzschild radii, a body reaches the horizon after 49.0 $r_s/c$ by its own clock — 4.83 milliseconds for a mass of ten suns. That is what happens once the well is gone: there is no turning point to reach and no orbit to settle into, so the fall is direct and brief. The distant observer’s account is quite different and the falling clock’s is the one that matters to the faller.

The energy of the innermost orbit #

The same expression that produced the radius produces the energy, and this is where the geometry starts making claims about quantities an engineer would recognise. The energy per unit mass of a circular orbit at radius $r$, in units of $c^2$ and against unity for a body at rest far away, is

$$\tilde E(r) = \frac{1 - 2GM/c^2r}{\sqrt{1 - 3GM/c^2r}}.$$

Newtonian intuition expects that to fall monotonically: a tighter orbit is a more bound one. It does not. It falls, reaches a minimum and climbs again without limit, and the minimum is at the radius already found.

Substituting $r = 6GM/c^2$ by hand confirms it. The numerator is $1 - \tfrac13 = \tfrac23$ and the denominator is $\sqrt{1 - \tfrac12} = 1/\sqrt2$, so the energy is $2\sqrt2/3 = \sqrt{8/9} = 0.942809$. The shortfall from unity is 5.719% of $mc^2$, and since mass is a form of energy there is nowhere else for it to have gone.

The comparison worth carrying away is with the alternative. Hydrogen fusion converts 0.7% of the mass of its fuel — the mass defect of four protons against a helium nucleus, and the whole budget of the process. Geometry, with no reaction in it, converts 5.72%. The ratio is 8.2, and every step of the calculation that produced it involved a metric, a conserved angular momentum and a minimisation. No cross-section appeared, no fuel, no reaction rate, nothing about what the orbiting matter is made of. Gravity outperforms the best nuclear process available to ordinary matter by a factor of eight, as a consequence of the shape of a curve.

Two more numbers at that orbit are worth having. A static observer at 3 rs measures the body passing at exactly $c/2$: the algebra gives $(v/c)^2 = (GM/c^2)/(r - r_s)$, which is $\tfrac14$ there. And the orbital period in distant coordinate time, $2\pi r^{3/2}/\sqrt{GM} = 92.3\,GM/c^3$, is 4.5 milliseconds per ten solar masses — the same order as the crossing time in the figure above.

The photon sphere, and the orbit light alone can have #

The denominator of the energy expression has been sitting there unremarked. It vanishes at $r = 3GM/c^2 = 1.5r_s$, and the energy of a circular orbit there is infinite — the standard signature of a quantity that is trying to describe something moving at $c$.

Redoing the reduction for light gives an effective potential proportional to $(1 - r_s/r)/r^2$, and its structure differs in one decisive way: it has a single maximum and no minimum at all, at $r = 1.5r_s$ for every value of the angular momentum. So light has exactly one circular orbit, at a radius fixed by the mass alone, and it is unstable — a photon placed on it circles indefinitely while nothing disturbs it, and any perturbation sends it out to infinity or in through the horizon. The photon sphere is the limit of the unstable orbits found on the barrier tops above: as a massive particle’s angular momentum rises its unstable orbit moves inward, and $1.5r_s$ is where it would arrive at the speed of light.

Light passing a solar mass is deflected by 1.75″ at one body radius and 0.58″ at three, and the relation is a straight line of slope $-1$ on log axes. That is the weak-field regime, far outside anything this essay is about — and it is worth the comparison because it shows how far the innermost stable orbit is from the situations where the familiar formulae apply.

The same term, read from far away #

The $1/r^3$ term does not switch on at small radius. It is there at every radius, and far from the mass it produces something that looks nothing like a vanished orbit: a nearly circular orbit that fails to close.

The mechanism is the radial oscillation mentioned earlier. A body displaced slightly from a circular orbit oscillates radially at frequency $\kappa$ while going round at $\Omega$, and if the two are equal the orbit closes into a repeating ellipse. In the Schwarzschild geometry they are not equal, and the ratio is exactly $\kappa^2/\Omega^2 = 1 - 6GM/c^2r$. That one expression contains both halves of this essay. Far away, $\kappa$ is slightly less than $\Omega$, so the radial cycle takes slightly longer than the orbital one and the point of closest approach arrives a little late each time — it advances, by $2\pi(1 - \kappa/\Omega)$ per orbit, which for large $r$ is $6\pi GM/c^2r = 3\pi r_s/r$. At a million Schwarzschild radii that is 1.94 arcseconds per orbit. And close in, $\kappa$ falls to zero, at exactly $r = 6GM/c^2$ — because a stable orbit whose radial oscillation has no frequency is an orbit whose well has flattened out, which is the definition of the merge already computed.

So the perihelion advance and the innermost stable orbit are one phenomenon seen from two distances. At 6 rs the ratio is $\sqrt{1/2} = 0.7071$ and the advance is 105 degrees per orbit; at 10 rs it is 0.8367 and 59 degrees. Nothing changes on the way in except the size of a number that was always there.

What it costs, and where the model stops #

The energy has to be radiated, and something has to take the angular momentum. A body cannot spiral in on its own: circular orbits are exact solutions, so a body left alone stays on the one it is on for ever. Reaching 3 rs from far away needs a mechanism that removes both energy and angular momentum, and the 5.72% is what must have been emitted somewhere for the descent to have happened. The figure gives the end state of a process it does not contain.

The result assumes no rotation and no charge. Schwarzschild’s geometry has one parameter, and real collapsed objects have at least two. Spin moves the innermost stable orbit substantially and in opposite directions depending on which way the orbit goes: around a maximally rotating hole a co-rotating orbit’s limit falls to $GM/c^2$, a sixth of the non-rotating value, and the efficiency rises to 42.3%; a counter-rotating orbit’s limit moves out to $9GM/c^2$ and the efficiency falls to 3.77%. That factor of eleven is why the efficiency of gravity is quoted as a range, with 5.72% one point inside it.

Everything here is a test particle. The orbiting body has a mass small enough to ignore at every step: it does not contribute to the geometry, its self-gravity is absent, and so is the back-reaction of the radiation it emits on its own path. The first approximation fails when the two masses are comparable and the last fails whenever the emission is strong — which is the regime in which the descent to the innermost orbit is fast enough to matter. A body massive enough to hold itself together against tides is massive enough to be part of the problem, in the way self-gravity becomes a shaping force above a certain size, and the radiation carries momentum as well as energy, as a stretching and squeezing wave must.

Real matter is not a test particle but a fluid. Anything spread out has pressure, viscosity and magnetic fields, and in that problem the innermost stable orbit is a boundary condition rather than an answer — the radius at which the fluid equations must be handed something else. Pressure supports material in a way a point mass cannot be supported, and magnetic stress carries angular momentum outward across radii no orbit connects. None of that appears in a curve computed from a metric, and the curve is not wrong about what it does describe.

None of the coordinates here are measurements. The radius $r$ is a Schwarzschild radial coordinate, defined as circumference divided by $2\pi$, and no ruler can be laid along it — the radial stretch factor plotted earlier is the reason. “The innermost stable orbit is at 88.6 km” is a statement about the circumference of that orbit and not about a distance from anything, and the horizontal axis of every figure here is a label.

And the whole treatment is classical. Quantum effects have been switched off throughout, which is legitimate at these radii and is not the last word.

A solar-mass horizon has a temperature of $6.17\times10^{-8}$ K and an evaporation time of $2.1\times10^{67}$ years, so nothing quantum enters anything above. That is the size of what has been left out: the geometry here is entirely classical, and the one quantum effect attaching to a horizon is smaller than every other term by a margin no experiment will ever close.

One trick, and the man who wrote the geometry down #

The effective potential is not a relativistic device. It is the standard reduction of any central-force problem: use the conserved angular momentum to eliminate the angular coordinate, and what is left is a one-dimensional motion in a modified potential whose stationary points are the circular orbits. The same construction handles a pendulum, whose small lie is the parabola at the bottom of its well, and the general question of what a landscape of potential energy permits, and the radial equation of the hydrogen atom, where the barrier is $\ell(\ell+1)\hbar^2/2mr^2$ and keeps states of high angular momentum away from the nucleus.

What relativity contributes is one term, and the lesson is how much a term can do. Every previous use of the construction produced a well whose position moved as the parameters changed; this one produces a well that can be abolished, and the abolition is not gradual. At $\tilde L$ just above $\sqrt{12}\,GM/c$ there is a stable orbit; just below, there is none anywhere. One added power of $1/r$ turns a family of solutions that exists for all parameters into one that exists only above a threshold.

Karl Schwarzschild found the geometry within weeks of the field equations being published in late 1915, while serving on the Russian front; he sent it to Einstein in December and died of an illness contracted there in May 1916. Reading the consequences out took much longer. The complete analysis of the geodesics of his metric — every orbit it permits, stable, unstable and plunging — was published by Yusuke Hagihara in 1931, which is when the innermost stable circular orbit became a known property of a solved equation. For decades afterwards it was a fact about a differential equation and nothing else, since nothing was known to exist that could have anything orbiting near one. The radius was derived first, and the objects were found later.

What the picture cannot show #

Every figure here is a radial cut. The horizontal axis is one coordinate of a four-dimensional geometry, and an orbit is a path in that geometry rather than a point sitting in a well — so the well’s shape describes how the radius changes and the path itself, going round, is nowhere in the drawing. The merge at 3 rs is two roots of a quadratic coinciding, not anything an observer could watch happen.

The curve is also not a potential energy in the sense the Newtonian one is. It appears in an equation for $(\mathrm{d}r/\mathrm{d}\tau)^2$, where $\tau$ is the proper time of the orbiting body and not the time of anybody who could plot the graph.

Nothing in these figures shows a body losing angular momentum. The descent to the innermost orbit is stated in the captions and absent from the drawings, because the drawings contain only exact solutions and the descent is a sequence of departures from them. The energy curve gives the energy circular orbits have, not the trajectory of anything that visited them.

And the geometry drawn is the non-rotating one throughout, so the factor of eleven in efficiency is invisible in all ten figures. Spin would change every curve on the page, and the page has no way of saying so.

The ladder from here #

Later rungs on this anchor: the rotating geometry, where the innermost stable orbit splits into a co-rotating and a counter-rotating one and the efficiency spans a factor of eleven; the marginally bound orbit at $4GM/c^2$ and the plunge trajectories below it; the epicyclic frequencies as a pair, radial and vertical, and how a slightly eccentric or slightly tilted orbit precesses; the effective potential for light in full, including the capture cross-section it fixes; and the failure of the test-particle approximation, where the orbiting mass is large enough to deform the geometry it is orbiting in.

The neighbouring ladders are the horizon, the other radius this geometry defines and the one that is not an orbit; the deflection of light, whose weak-field expansion breaks at the photon sphere reached here from the inside; and the equivalence principle, which supplies the falling half of every gravitational effect and cannot see the term this essay is about.