Astronomy & Celestial Mechanics
the sky, computed
Generator

The roche-lobe generator

The critical surface of a binary at mass ratio 0.6, pinched at L1
The critical surface of a binary at mass ratio 0.6, pinched at L1. Equipotentials of a synchronously rotating binary of mass ratio 0.6, drawn in the plane of the orbit and in the frame that turns with it, so the centrifugal term is part of the surface rather than a force acting on it. The heavy contour is the critical one, the level of the inner Lagrange point, and its shape is the whole argument: two closed lobes that touch at a single point on the line of centres and enclose nothing else. Inside a lobe the surfaces close around one star and matter belongs to that star; on the critical surface itself the two are joined at L1, and a star that grows until its photosphere reaches that level has nowhere for the next layer to go but across. The volume-equivalent radius of the lobe around the more massive star is 0.4255 separations and of the other 0.3358, both obtained by integrating the drawn surface rather than from a formula. The outer contours are levels above L1, where a single surface encloses both stars, and the two further saddles beyond the stars are L2 and L3, at which that common envelope opens to infinity in turn.

Equipotentials of a synchronously rotating binary of mass ratio 0.6, drawn in the plane of the orbit and in the frame that turns with it, so the centrifugal term is part of the surface rather than a force acting on it. The heavy contour is the critical one, the level of the inner Lagrange point, and its shape is the whole argument: two closed lobes that touch at a single point on the line of centres and enclose nothing else. Inside a lobe the surfaces close around one star and matter belongs to that star; on the critical surface itself the two are joined at L1, and a star that grows until its photosphere reaches that level has nowhere for the next layer to go but across. The volume-equivalent radius of the lobe around the more massive star is 0.4255 separations and of the other 0.3358, both obtained by integrating the drawn surface rather than from a formula. The outer contours are levels above L1, where a single surface encloses both stars, and the two further saddles beyond the stars are L2 and L3, at which that common envelope opens to infinity in turn.

2 essays call roche-lobe. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through. Stars

The flow that narrows its own channel

Two stars close enough share a surface, and the point where that surface pinches is a valve. What comes through it changes the orbit, and the orbit changes the valve — with a sign that reverses at equal masses, which is why some binaries transfer quietly for a hundred million years and others tear themselves apart in a thousand.

 The same equation, run away at q = 0.8 and settled at q = 0.5. The donor's overfill of its own Roche lobe against time, integrated for 3 mass ratios at a donor adiabatic response of −0.33 — a star with a deep convective envelope, which expands as it loses mass. The overfill sets the transfer rate and the transfer rate changes the overfill, and the sign of that feedback is the stability criterion: where the lobe shrinks faster than the star does, ζ_L > ζ_ad, the overfill grows and the growth is exponential. At q = 0.8 the lobe responds at 0.03 and the overfill runs away; At q = 1.4 the lobe responds at 1.32 and the overfill runs away. At q = 0.5 it responds at -0.62 and the transfer throttles itself back. The critical ratio for this donor is 0.63 — below equal masses, which is the result the whole subject turns on: a giant transferring to a lighter companion is already unstable before the mass ratio has reversed, and what follows is not accretion but a common envelope. Nothing here is a fitted rate. The vertical scale is logarithmic and the runaway is a straight line on it, which is what an exponential is. Stars

Three clocks and a runaway

Whether mass transfer between two stars is stable is a comparison of two logarithmic derivatives. How fast it runs is a separate question with three possible answers fourteen orders of magnitude apart — and the answer decides whether the companion accretes, is buried, or is swallowed.

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