The size forgets how the mass was lost, and the shape does not

A star like the Sun ends its life by shedding about half its mass. Most of the loss happens at the very end, on the asymptotic giant branch, where the star is a swollen, cool, pulsating envelope round a small carbon–oxygen core, and the outer layers are driven away by the pressure of starlight on dust that condenses in them. What is left is a white dwarf of about 0.6 solar masses, and whatever planets, asteroids and comets survived the giant phase are left orbiting it.

What happens to their orbits has two clean answers. If the mass is lost slowly compared with an orbital period, each orbit swells in proportion to the mass lost and keeps its shape exactly: the product of the star’s gravitational parameter and the orbit’s semi-major axis is an adiabatic invariant, and so is the eccentricity. If the mass is lost suddenly — in much less than an orbit — the planet’s position and velocity are unchanged at the instant and its new orbit is whatever those values define in the shallower well, which can be much more eccentric, much less eccentric, or not bound at all. A supernova is the sudden case, and a quiet giant’s wind is usually treated as the slow one.

The giant’s wind is not steady, and that is the whole difficulty. On the asymptotic giant branch the helium-burning shell ignites in flashes — thermal pulses — every ten to a hundred thousand years, and after each one the star’s luminosity and radius swell and its mass-loss rate rises by orders of magnitude for a few hundred to a few thousand years before subsiding. The loss comes in episodes. Whether an episode is slow or sudden depends on the planet: a thousand-year burst is slow for a planet with a ten-year orbit and sudden for a comet with a hundred-thousand-year one. Most planets are in between for at least part of the history, and every planet sees a sequence of episodes rather than one. Neither limit describes that, and there is no closed form for what does. The calculations below are integrations of the orbit through prescribed mass-loss histories, each one checked against the piece of theory it tests.

One history, three answers #

The figure takes one history of mass loss — six equal pulses, together removing 45 per cent of the star’s mass, at intervals of about a hundred pulse-lengths — and three planets that differ only in their orbital periods. For the innermost planet each pulse lasts three orbits. Its eccentricity oscillates a little while the mass is going, as an osculating eccentricity does when the well is changing under it, and returns to 0.050 each time the mass loss stops: the slow limit, exactly. For the outermost each pulse lasts three hundredths of an orbit, and each one changes the eccentricity abruptly, sometimes up and sometimes down, ending at 0.298. The middle planet, for which a pulse lasts about a third of an orbit, ends in between.

The x-axis is the mass lost, not the time, and it is the natural one: it makes all three planets see the same history, and on it the sudden planet’s changes appear as ramps rather than jumps, because its osculating eccentricity is a function of the instantaneous mass while the mass is falling faster than the planet can move. The thing to take from the figure is the spread. A star whose total loss is known — and the initial–final mass relation gives it well — leaves planets whose eccentricities are not determined by that total, even in principle, unless the timing of the loss is known too.

A kick whose sign is the orbital phase #

The building block is a single sudden loss, and its effect on the eccentricity depends on where the planet is.

For a circular orbit the answer has no phase dependence at all. The planet was moving at exactly the circular speed for the old mass, which is faster than circular for the new one, so it is now at the pericentre of an ellipse with eccentricity $\delta/(1-\delta)$, where δ is the fractional loss — 0.111 for a ten per cent loss, whichever way it happens to be facing. An eccentric orbit is different. At pericentre the planet is moving fast and deep in the well; weakening the well there throws it onto a wider, more eccentric orbit. At apocentre it is moving slowly, below even the new circular speed, and weakening the well moves it towards circular. For an orbit of eccentricity 0.2 the same ten per cent loss changes the eccentricity by anything from +0.133 to −0.089, and the sign is set by the phase.

That is the fact the rest of the argument is built on. In the vis-viva relation — one equation for the speed anywhere in an orbit — a sudden change of the mass is a change of the potential at fixed kinetic energy, and whether that energy now over- or under-fills the new well is a question about where in the old one the body was. A star cannot choose the phase of a planet it does not know about, so from the planet’s point of view a sequence of sudden losses is a sequence of kicks with random signs.

Where sudden turns into slow #

A real pulse is neither instantaneous nor infinitely slow; it has a width, and the question is how the kick depends on it.

The answer is sharper than the phrase “slow compared with a period” suggests. For a nearly circular orbit, linearising the equations of motion about the circle turns the eccentricity into a driven oscillator: the radial displacement oscillates at the orbital frequency, and the mass-loss rate is the force driving it. A driven oscillator responds to the Fourier component of its forcing at its own frequency, and to nothing else. A Gaussian pulse of width σ has a Fourier amplitude at frequency $n = 2\pi/P$ that falls as $\exp[-(n\sigma)^2/2]$, so the kick is

$$e \approx \frac{\delta}{1-\delta}\,\exp\!\left[-\tfrac{1}{2}\left(\frac{2\pi\sigma}{P}\right)^{2}\right].$$

The integrations agree with that curve to a few per cent at every width drawn. The crossover is not at one period but at about a sixth of one, where the exponent reaches a half, and above it the kick falls off faster than any power. By a third of a period it is down to twelve per cent; by a whole period, to a few parts in a billion. A pulse that lasts one orbit is already, for practical purposes, perfectly slow.

This is the classical adiabatic theorem in its sharpest form: for a change that is smooth — whose rate has no sharp features — the adiabatic invariant is violated by an amount exponentially small in the ratio of the change’s timescale to the oscillation’s. What matters is not the pulse’s length but how much of its rate varies on the orbit’s own timescale. A pulse with a sharp onset — a shock, a sudden ignition — has more high-frequency content than a Gaussian of the same width and kicks harder; a pulse with a gently rounded profile kicks less. For the planets it is the shape of the pulse’s leading and trailing edges, not its duration, that decides whether it counts as sudden.

A random walk in the eccentricity #

A star that loses its mass in N pulses, each sudden for a given planet, subjects that planet to N kicks of random sign, each about $1/N$ as large as a single loss of the same total would give.

Kicks with random signs add like steps in a random walk: the expected square of the sum is the sum of the squares. N kicks each of size about $\Delta/N$ therefore sum to about $\Delta/\sqrt{N}$ rms. The simulations follow that law from four pulses on: forty per cent of the star’s mass lost in one sudden event changes the eccentricity by 0.57 rms — often unbinding the planet’s orbit entirely or nearly so — while the same loss split into 128 sudden pulses changes it by 0.031. A single pulse is dominated by nonlinear effects, which is why the first point sits above the line.

The random walk explains the spread in the first figure. The outer planet’s six pulses were each sudden, each kick was about seven per cent in δ, and six random kicks of that size produce an eccentricity change of order 0.1 to 0.3, depending on the phases. A different sequence of phases — the same star, a planet started a quarter-orbit further round — would have given a different number. The eccentricity after the mass loss is a random variable whose distribution is set by the number and sharpness of the pulses, and whose particular value is set by an accident of timing.

This is the same arithmetic as the one that governs round-off error in a long integration: errors of random sign accumulate as the square root of their number, and splitting one large error into many small ones reduces the total even though the sum of their sizes is unchanged. Here the errors are physical, and the thing they accumulate in is the shape of an orbit.

The size is settled much sooner #

The semi-major axis behaves differently, and the difference is the essay’s title.

In the slow limit $GM \cdot a$ is conserved exactly, so after losing forty per cent of its mass the star’s planets sit at $1/0.6 = 1.67$ times their old distances. A single sudden loss leaves $GM\cdot a$ nearly twice its old value — the orbit is ninety-five per cent larger than the slow answer, because a sudden loss always raises the energy relative to the new well, whatever the phase. That is the key difference from the eccentricity. The change in size from one pulse is a sum of a term that averages to zero over phase and a term second order in δ that is always positive, and it is the second-order term that survives. N pulses of size $\Delta/N$ give a total excess of $N\cdot(\Delta/N)^2 = \Delta^2/N$, which falls as $1/N$. After eight pulses the orbit is within four per cent of the slow answer; after 128, within a third of a per cent.

So a planet round a white dwarf whose progenitor lost its mass in many sudden pulses sits almost exactly where the slow theory puts it, with an eccentricity that remembers how the loss was delivered. The two elements carry different information. The size records the total mass lost, and the initial–final mass relation supplies that. The shape records the history of the loss as the planet saw it: the number of pulses, their sharpness, and the particular phases they caught it at.

Which bodies are on which side #

The crossover at a sixth of a period turns the question of which limit applies into arithmetic about the Solar System’s own future. The Sun will lose about 45 per cent of its mass, most of it in the last few hundred thousand years of the asymptotic giant branch, in a series of thermal pulses whose enhanced mass loss lasts of order a thousand years each. A pulse of that length is effectively slow for any orbit shorter than a few thousand years — every planet, and the Kuiper belt out to a few hundred astronomical units, where periods are at most a few thousand years. For those bodies the pulses merge into one smooth loss and the slow answer holds: the orbits swell by a factor of about 1.8 and keep their shapes, and the only eccentricity change is the part of the random walk that survives the exponential cut-off, which is negligible.

The Oort cloud is on the other side. Its comets orbit at ten to a hundred thousand AU, with periods of a million years or more, and to them each thousand-year pulse is instantaneous. They receive the full random walk: a few tens of kicks, each a couple of per cent in the fractional loss, adding to eccentricity changes of ten to twenty per cent and, for comets that happen to be near the far ends of already elongated orbits when several pulses arrive, to escape. Between the two populations — the scattered disc and the inner cloud, at a few hundred to a few thousand AU — sits the crossover, where a pulse’s edges decide whether it counts, and where the outcome for a given body depends on the detailed shape of each episode as much as on its length.

The same reasoning applies to any star that loses mass episodically, with the numbers scaled. A massive star’s winds are stronger and more erratic, and its eruptions can shed a solar mass in a decade, sudden for anything beyond a few AU. The white dwarf left by a Sun-like star inherits the whole spectrum, from a smoothly expanded inner system to an outer cloud whose orbits were shaken by every convulsion of the giant that preceded it.

Why the well picture still helps #

It is worth saying what the effective potential contributes to all this, since the calculation is done with an integrator rather than with a picture. The radial motion of an orbit is motion in the effective potential, the gravitational well plus the wall that angular momentum builds, and eccentricity is the amplitude of the radial oscillation in that well. A slow loss makes the well shallower and wider without disturbing the oscillation: its action, the area enclosed by one radial cycle in phase space, is conserved, and a conserved action in a scaled well means a conserved shape. A sudden loss changes the well while the body is at one particular point in its oscillation, leaving it with more or less amplitude depending on whether it was at the top or the bottom of the swing — which is the phase dependence of the second figure, seen as a pendulum whose pivot is jerked.

Seen that way, the random walk is not surprising: jerk a swinging pendulum’s pivot at random moments and its amplitude wanders, sometimes up and sometimes down, growing on average as the square root of the number of jerks. The size of the orbit is the well’s width, which the mass loss sets directly, and it does not wander because it does not depend on the phase at first order. The picture predicts which element remembers the history before any integration is run.

What the white dwarfs show #

White dwarfs are the one place where planetary systems that went through this can be examined. Between a quarter and a half of them have atmospheres polluted with heavy elements — magnesium, silicon, iron, calcium — that sink out of sight in days to millions of years, so something is delivering rocky material onto them continuously. The material has to arrive on orbits that reach within about a solar radius of the star, where tides pull a body apart, and it has to come from asteroid belts or planetary systems that started at several astronomical units.

Getting rocky bodies from several AU to a solar radius needs eccentricities near one. Mass loss by itself produces them only in the sudden limit, which for bodies at a few AU is rare. What mass loss does do, in the pulsed picture, is give every orbit an eccentricity kick of a few per cent, randomly oriented, and double every orbit’s size. That changes the stability of whole systems: planets that were stably spaced around the main-sequence star are, around the white dwarf, spaced by a different number of their own Hill radii, because the Hill radius scales with the cube root of the planet-to-star mass ratio and that ratio has nearly doubled while every orbit grew in the same proportion. Systems that were marginally stable become unstable, and planets begin scattering each other and the asteroids near them onto star-grazing orbits — a process that continues for billions of years and explains why even old white dwarfs are still polluted.

The eccentricity kicks from pulsed mass loss are a small ingredient in that, but not negligible: they seed the instability, and for the outermost bodies — the comets of an Oort-like cloud, for which every pulse is sudden and whose first passages through the inner system are how such a cloud is known at all — they are the dominant effect, and a white dwarf’s cloud is left more eccentric, more loosely bound and partly stripped.

What the calculation leaves out #

The integrations treat the star as a point whose mass falls, and the planets as test particles in a plane. They leave out everything else the giant phase does: tides raised on the swollen star, which drag in and destroy planets within a few AU; drag from the wind itself on small bodies; the star’s radius passing through the orbits of the innermost planets; and the recoil if the wind is lost asymmetrically, which gives the white dwarf a velocity kick of its own. They also prescribe a history of Gaussian pulses at irregular intervals, which is a caricature of a real thermal-pulse cycle, whose mass-loss profile rises steeply after each flash and decays slowly, with more high-frequency content than a Gaussian and correspondingly harder kicks for planets near the crossover.

The random-phase assumption is the one most worth stating. It holds if the pulse intervals are long and irregular compared with the orbital period, so that each pulse catches the planet at an unrelated point. If the intervals were regular and commensurate with the orbit, the kicks would add coherently, growing as $N$ rather than $\sqrt{N}$. Thermal pulses are regular to within tens of per cent, so a planet whose period is a simple fraction of the pulse interval could be in a weak resonance with its star’s convulsions for the few cycles that matter.

Still open: whether a white dwarf’s planets remember their star’s pulses #

The argument predicts that the eccentricities of planets and belts round white dwarfs carry a signature of how their progenitor lost its mass, and that the signature should depend on orbital distance: negligible for bodies whose orbits are much shorter than a pulse, a random walk of known amplitude for bodies whose orbits are longer. The few planets now known round white dwarfs are too few, and too likely to have been moved since, for their eccentricities to test it. The debris discs and the pollution rates are indirect. The most direct test would be the white dwarfs’ wide binary companions — stars at hundreds to thousands of AU, for which every pulse is sudden — whose orbital eccentricity distribution, measured statistically from their projected separations and velocities, should be broadened relative to that of main-sequence binaries of the same separation by an amount set by the number of pulses. Whether the distributions now being measured are broadened by that much, or by more because of the white dwarf’s own recoil, or by less because the losses were smoother than the pulses drawn here, is what the next samples will say.