The well moves, and the body keeps its share of it

Every picture of the well angular momentum builds treats the curve as furniture. The body slides along a level and the curve holds still, which is right as long as the mass at the centre is a constant and the angular momentum is a constant, and both are — in the two-body problem, exactly.

Neither is a constant in the situations the two-body problem is used to describe. A star loses about half its mass before it finishes; a satellite in a low orbit loses angular momentum to the air; a moon gains it from a tide. The well those bodies move in is therefore a curve that changes shape underneath them, and the question this essay is about is what, if anything, survives the change.

The obvious candidates both fail. The energy is not conserved, because the potential is explicitly a function of time and there is work being done on the body by a field that is changing. The semi-major axis is not conserved, because it is fixed by the energy. What does survive is a quantity nobody would think to look for, it survives only under a condition, and the condition is not about how much the well changes.

What a slowly changing well conserves #

The quantity is an area. Draw the radial motion in the plane of position against radial momentum — the radial phase plane — and a bound orbit traces a closed loop there once per radial period, in from $r_+$ to $r_-$ along the negative half and back along the positive. The area enclosed by that loop is the radial action,

$$J_r = \oint p_r \, dr = \oint \sqrt{2\left(E - U_{\rm eff}(r)\right)}\,dr .$$

The theorem, which goes back to Ehrenfest and to the old quantum theory that needed it, is that if the parameters of the system are changed slowly enough, $J_r$ stays put while $E$ and everything else moves. It is not approximately conserved in the way a small perturbation is small; the error is exponentially small in the slowness, which is a far stronger statement and the reason the result is useful rather than decorative.

The strength of that statement is worth unpacking, because “approximately conserved” is usually a warning that the approximation degrades as the change accumulates, and here it does not. If the parameters change over a time $T$ and the orbital period is $P$, an ordinary perturbative estimate would put the change in $J_r$ at some power of $P/T$. What the analysis gives instead is a change going as $e^{-cT/P}$ for a smoothly varying parameter, so doubling the slowness does not halve the error — it squares it. A loss spread over a hundred orbits and the same loss spread over four hundred are not two points on a gentle trend, and that is why the flat end of the figure above is flat rather than merely shallow.

The condition on the smoothness is not decorative either. The exponential holds for a parameter with no sharp corners in it; a rate that switches on and off abruptly excites the radial oscillation at its own period and leaves a residue that no amount of overall slowness removes. Real stellar mass loss has exactly that character, which is the reason the last section of this essay is about what has not been settled.

For the Kepler problem the integral can be done, and it comes out as

$$J_r = 2\pi\left(\sqrt{GMa} - L\right).$$

That is the whole of the argument in one line. $L$ is exactly conserved whatever the mass does. If $J_r$ is also conserved then $\sqrt{GMa}$ is conserved, so

$$GM \cdot a = \text{constant},$$

and the semi-major axis grows in exact inverse proportion to the mass that is left. Put that into $L^2 = GMa(1-e^2)$ with $L$ and $GMa$ both fixed and the eccentricity cannot move either.

So a slowly emptying well leaves an orbit that is a scaled copy of itself. Not a similar orbit, not an approximately similar one: the same eccentricity, the same orientation, the same shape, at a larger size. The body’s share of its own well — how far up the sides its level sits — is what it keeps.

Doing it rather than deriving it #

The derivation above is a manipulation of closed forms, and a closed form can hide an assumption. The figures here do not use it. They integrate the body’s motion in the plane with the central mass changing during the integration, read the osculating elements out of the state vector at every step, and report what comes out.

The integration is checked before it is used, on the one case whose answer is known in advance: with the mass held fixed, twenty orbits of a fourth-order integrator have to return the semi-major axis and the eccentricity they started with. A figure about what a slow change preserves would be worthless if the integrator were quietly changing things on its own, and an integrator that drifted would produce exactly the graph the adiabatic claim predicts — flat at the slow end, wrong at the fast one — for reasons that had nothing to do with the physics.

The second check is the claim itself. At the slowest rate the product $GM\cdot a$ has to come back to what it began at, and at the fastest it has to fail to. A test that only ever confirmed the first would be a test with no way of failing, which is why the figure is drawn at both ends of the range rather than at the end that behaves.

Why the criterion is not about size #

The condition on adiabatic invariance is often stated as “the change must be small”, and it is not. Forty-five per cent of the central mass is not a small change by any reading, and the slow run in the figure above preserves the eccentricity to four decimal places anyway.

What matters is the comparison between two timescales: how long the well takes to change appreciably, against how long the body takes to go round. The body has to complete many orbits while the parameters are still nearly what they were, so that each orbit is a closed loop in the phase plane and the loop’s area is well defined. Change the well over hundreds of orbits and every individual orbit is a Kepler orbit to high accuracy, and the sequence of them is a slow migration through a family. Change it over one and there is no family to migrate through.

That is a genuinely different criterion from smallness, and the difference is what makes the result applicable. A star may throw away most of itself and still leave its planets on the same-shaped orbits, because the throwing away takes ten thousand years and the planets go round in one.

The Sun’s own case #

The arithmetic is worth doing on the system it will happen to.

The Sun will leave the main sequence in about five thousand million years, spend a few hundred million as a giant, and end as a white dwarf of a little over half a solar mass. The loss happens almost entirely at the end, during the asymptotic giant phase and the ejection of the envelope, over something like $10^5$ years.

An orbit at one astronomical unit has a period of one year, so the loss is spread over of order $10^5$ orbits. The ratio is enormous, the loss is adiabatic by a wide margin, and the outcome follows: every surviving planet’s orbit expands by the inverse of the mass ratio — a factor of about 1.85 for a final mass of 0.54 — with its eccentricity untouched — and an eccentricity is a statement about how far an orbit’s energy sits above the floor of its own well.

The Earth’s orbit, if the Earth survives, moves out to 1.85 astronomical units. Jupiter goes to 9.6. The spacing is preserved because everything scales together, so the resonances between the planets are preserved too, and the outer solar system arrives at the white-dwarf era looking like a photographic enlargement of itself.

The factor by which a planetary system swells is therefore read off a white dwarf’s mass, and the relation above says a 3-solar-mass star’s planets expand by 4.0 while a 1-solar-mass star’s expand by 1.85. That is a prediction about where planets should be found around white dwarfs, and it is testable: debris and major planets have now been detected around several, and the orbital radii are the observable the argument delivers.

The survival question is separate and less settled. Expansion does not protect a planet that is engulfed on the way, and the Sun’s giant radius will exceed the Earth’s current orbit, which is the swelling a shrinking core forces. Whether the Earth escapes depends on a race between the orbit moving out and the star’s surface moving out, complicated by tidal drag from the giant’s envelope, which removes angular momentum and pulls the planet back in. The adiabatic argument gives one side of that race exactly and says nothing about the other.

What has actually been seen #

Everything above is arithmetic about a future, and it would be a weak essay if that were all. The expansion has observable consequences now, around stars that have already been through it, and they are among the more surprising results in the study of planetary systems.

A quarter to a half of all white dwarfs have metal lines in their spectra. That should not happen. A white dwarf’s surface gravity is four orders of magnitude above the Sun’s, so anything heavier than helium sinks out of the visible atmosphere in days to a few million years — a time so short compared with the star’s cooling age that the metals cannot be left over from before. They are being delivered now, and the delivery rate implied by the line strengths is of order $10^{8}$ grams a second, continuously.

What is being delivered is rock. The relative abundances measured in the best-studied cases match the bulk composition of the terrestrial planets rather than that of the star or of the interstellar medium: iron, magnesium, silicon and oxygen in chondritic proportions, with the volatiles depleted. Several white dwarfs show an infrared excess from a disc of dust inside the tidal disruption radius, and a handful show the disc in absorption as gas. One has a planet transiting it every thirty-four hours.

The adiabatic argument is what makes that population intelligible. A surviving asteroid belt around a white dwarf sits 1.8 times further out than it did, still with its original eccentricities, and so does every surviving planet — so the resonances that pumped eccentricities in the main-sequence system are still there, still doing the same work, on a system that now has a target 100 times smaller in radius at its centre. A body scattered onto a star-grazing orbit passes inside the tidal disruption radius, comes apart, and its debris circularises and rains in. The metal lines are evidence that the planetary system survived, and they are evidence of it precisely because the expansion left the scattering mechanism intact.

The negative case is as informative. The Sun’s loss is slow for the planets and not for everything: an object in the outer Oort cloud has an orbital period of millions of years, which is far longer than the $10^5$ years over which the mass will go, so for that population the loss is impulsive. The outer cloud will be unbound, and the same statement applies to the widest binaries, whose separations of tens of thousands of astronomical units give periods long enough to put them on the wrong side of the boundary. One mass loss, one star, and two populations with opposite fates decided by nothing except their own periods.

What a sudden loss does instead #

The opposite limit is as clean and much more violent, and it is the one that has turned up already without being named.

If the mass vanishes in a time short compared with an orbital period, the body’s position and velocity are unchanged at the instant — nothing has had time to act on it — and what changes is the well it finds itself in. Its new orbit follows immediately from the vis-viva relation with the new $GM$ and the old speed, and the result depends entirely on where in its orbit the body happened to be.

The extremes are worth having. Lose the mass while the body is at apoapsis, where it is moving slowest, and the old apoapsis becomes the new periapsis — the eccentricity goes up. Lose it at periapsis and the eccentricity goes down. Lose more than half the total mass while the body is anywhere near apoapsis and the orbit is not bound at all.

The astronomical case is a supernova. The exploding star sheds several solar masses in seconds while its companion goes round in days or years, which is about as impulsive as anything gets, and the consequence is the one the impulsive arithmetic gives: the surviving companion is left on a wildly eccentric orbit, or on no orbit at all. A companion that survives and is moving is the observable, and its speed is the orbital speed it had at the instant the binding disappeared.

The contrast with the white-dwarf case is the cleanest illustration of a rate mattering more than an amount. A star that loses half its mass over $10^5$ years leaves its planets on the same orbits, larger. A star that loses half its mass in ten seconds unbinds them. The masses are identical and the two outcomes have nothing in common.

What the invariant is, physically #

There is a reading of the radial action that makes its conservation less surprising, and it is worth having because the formal proof is an exercise in averaging.

$J_r$ counts how far up the well the body sits, in a scale-free way. A circular orbit has $J_r = 0$: no radial motion, no loop, no area. A nearly parabolic orbit has $J_r$ approaching $2\pi L$ from below. So $J_r/2\pi L$ runs from 0 to 1 across the whole range of bound orbits at a given angular momentum, and it is a pure number that says how eccentric the orbit is — indeed $J_r/2\pi L = 1/\sqrt{1-e^2} - 1$ exactly.

Conserving it is therefore the statement that a slow change moves the well without moving the body’s place in it. The floor drops or rises, the walls move in or out, and the level the body occupies moves with them, keeping the same fractional height. A body at the bottom stays at the bottom; a body two-thirds of the way up stays two-thirds of the way up.

That last observation is doing real work in the subject. It is why a planet on a nearly circular orbit stays on one while its star empties, why the outer planets will arrive at the white-dwarf era still nearly circular, and why the eccentricities that are observed around white dwarfs need a cause — scattering between planets, or a companion — rather than being what the mass loss produced.

Where the argument gives out #

Slow is measured against the orbital period, so the same star is adiabatic for one planet and impulsive for another. A star losing mass over $10^4$ years is slow for anything inside a few hundred astronomical units and fast for a companion in the outer Oort cloud with a period of millions of years. There is no rate that is slow for a whole system, and the boundary between the two behaviours falls at a particular distance from the star, which sorts the system into two populations with different fates.

The invariant is a statement about one degree of freedom, and a real system has several. With two planets both expanding, the mutual perturbations are not what they were and a pair that was outside a resonance can be swept into one. The single-body result is exact and the system-level result is not a consequence of it.

And the well has been assumed to stay spherical. Mass leaving a star is only a change in $GM$ if it leaves isotropically and escapes the system entirely. Asymmetric ejection gives the star a recoil, which is a different problem — an acceleration of the centre rather than a change of its strength — and the observed kicks to newly formed neutron stars are hundreds of kilometres a second, which no adiabatic treatment contains.

The same invariant somewhere else entirely #

The radial action is one member of a family, and the family is worth naming because the same argument is made about very different things.

A pendulum whose string is slowly shortened keeps the ratio of its energy to its frequency — the same theorem, applied to an oscillation rather than an orbit, and the case Einstein and Lorentz argued about at the 1911 Solvay conference. A charged particle spiralling in a slowly strengthening magnetic field keeps its magnetic moment, which is the action of its gyration, and that single fact is the whole of why particles are trapped in a planet’s radiation belts: the invariant turns a converging field into a mirror. A star in a galaxy whose potential deepens slowly as the galaxy assembles keeps the actions of its orbit, which is the basis of every method that reconstructs a galaxy’s history from the present distribution of its stars.

What the three share is the structure rather than the subject. In each case a system oscillates, something about the system is changed over many oscillations, and the area of the loop in phase space is what carries through. The conserved quantity in a changing system is not usually one of the quantities the unchanging system conserved, and looking for it among them is the reliable way to conclude that nothing is conserved at all.

Still open: how fast is fast enough #

The two limits are exact and the crossover between them is not. Everything above treats the loss as either negligible in one orbit or complete within one, and real mass loss is neither: a star’s rate varies by orders of magnitude through thermal pulses, so a planet can find itself adiabatic for a century and impulsive for a decade within the same episode. What survives an intermittent change is not the average of the two answers, and there is no closed form for it — the honest route is the integration the figures here already run, done at the rate history a stellar model supplies rather than at a constant one.