Astronomy & Celestial Mechanics
the sky, computed
Concept

Mass transfer — where it appears

The flow of material from a star that has expanded to fill its own Roche lobe onto its companion, through the inner Lagrange point. It runs away when the donor is the heavier star, because losing mass moves the donor's lobe inwards rather than outwards, and that is the mechanism behind the Algol paradox.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

Zero-velocity curves, mass fraction 0.15. Level sets of the Jacobi constant in the frame that rotates with the two bodies. A particle with a given value of C is confined to the side of its own curve where the kinetic energy would be positive; as C falls the curves open, first at the inner point between the bodies, then behind the smaller one, and finally around the whole system.

The curve that says where a body cannot go

The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time.

gravitation · The three-body problem The surface a star stops at. The equipotential through L₁ — the Roche lobe — at mass fractions 0.50, 0.20, 0.05, in the frame that rotates with the pair. Each is a level set of exactly the same function the zero-velocity curves are level sets of, at exactly the critical value, so nothing here is a new construction: the Roche lobe is the last closed equipotential, and it is closed only because the two lobes touch at a single point. Material that reaches that point is no longer bound to the star it came from, and it leaves through an opening of zero area. The lobes are drawn in the orbital plane; in three dimensions each is a teardrop, and its volume-equivalent radius is what "the size of a Roche lobe" means. As the mass ratio becomes extreme the smaller star's lobe shrinks towards it, which is why a white dwarf accreting from a companion has a lobe smaller than the Sun.

The surface a star stops at

Around each star of a close pair there is a last closed equipotential, and the two touch at a single point. A star that swells to reach it hands its outer layers to its companion through an opening of zero area — and the transfer, once started, makes itself worse.

gravitation · Lagrange points Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses.

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

stars · Binary stars Four kinds of explosion, told apart by which line is missing. Spectra of 4 supernova types — Ia, II, Ib, Ic — near maximum light, stacked, over the same wavelength range and drawn with the same photospheric expansion speed of 10,000 kilometres per second. Every feature is a P Cygni profile: a trough blueshifted by the material expanding towards the observer and a peak at rest wavelength from the material moving across the line of sight, and the width of both is the expansion speed. Reading the silicon trough of the type Ia off the drawing puts it at 614.0 nanometres, which recovers 10,131 km/s from a rest wavelength of 635.5. The classification is a decision tree on absences. Hydrogen present makes it a type II; no hydrogen but silicon makes it a Ia; no hydrogen and no silicon but helium makes it a Ib; none of the three makes it a Ic. That tree was written down before anybody knew what these objects were, and it separates a detonating white dwarf from a collapsing massive core almost perfectly — because what the letters are actually reading is how much of the star's outer envelope was still there when it exploded. The type II kept its hydrogen, the Ib had lost it and shows the helium beneath, the Ic had lost that too.

Two explosions told apart by a missing line

The classification of supernovae is a decision tree on absences — is hydrogen there, is silicon there, is helium there — and it was drawn up decades before anybody knew what any of these events were. It nevertheless separates a detonating white dwarf from a collapsing stellar core almost perfectly, and the reason it does is worth the essay.

stars · Supernovae 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.

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.

stars · Mass transfer A line nothing spun up by accretion can lie above, and the millisecond pulsars beneath it. The period–period-derivative diagram with the spin-up line drawn on it. An accreting neutron star is torqued by the disc until its magnetosphere turns at the same rate as the material arriving there, which fixes an equilibrium period as a function of the magnetic field and the accretion rate. Eliminating the field between that relation and the dipole formula that every point in this diagram is already read through leaves a straight line of slope 1.33, drawn here for accretion at the Eddington rate — the fastest a star can be pushed. The 7 recycled pulsars all sit below it, which is what the figure is for: none of them was spun up faster than the limit allows, and their positions are a record of how much mass each one received rather than of how old it is. The young pulsars are in the opposite corner, above the line and to the right, spinning down from birth. The two populations are not two stages of one life. A star that reaches the bottom left has been fed by a companion for a hundred million years, which is why almost every millisecond pulsar has one and almost no young pulsar does.

A corner of the diagram that has to be earned

A pulsar spinning a thousand times a second cannot have been born that way and stayed that way, because its own radiation would have slowed it in a few million years. It got there by being fed, and the line it cannot lie above is where the accretion torque balances the magnetic one.

stars · Pulsars Three causes, three shapes, one residual. An eclipse-timing residual: the observed minus the computed time of each eclipse, against cycle number, in days. Three effects are superposed and each has its own functional form. A slow change in the orbital period — from mass transfer or from magnetic braking — integrates to a parabola. A third body in a wide orbit moves the whole binary towards and away from the observer, so its light-travel time adds a sinusoid at the third body's period. And an eccentric orbit whose apsides are precessing moves the two eclipses in opposite directions, which is a sinusoid that changes sign between primary and secondary minima. The last is why both eclipses have to be timed: a third body moves them together and apsidal motion moves them apart, and a series of primary minima alone cannot tell the two apart at all.

Three causes with three shapes in one curve

The times of eclipse in a binary star are a clock, and the clock runs late and early. Three completely different things make it do so — a third body, a changing period, and a slowly turning orbit — and they are separable only because each imposes a different shape on the residual.

stars · Binary stars Two critical mass ratios that move with where the mass goes. The logarithmic response of the donor's Roche lobe to its own loss of mass, ζ, against the mass ratio donor over accretor, for conservative transfer (dashed) and for all of the transferred mass lost from the accretor, from the donor and to a ring. The horizontal lines are two donors' own responses: −1/3 for a deep convective envelope, which swells as it is stripped, and +2 for a radiative one, which shrinks. Transfer is stable where the donor's line lies above the lobe's curve, so each crossing is a critical mass ratio — conservatively 0.63 and 1.72. Re-emission from the accretor moves them to 0.96 and 2.25, a wind from the donor to 1.24 and beyond the range drawn; and with everything fed to a ring the convective donor is unstable at every mass ratio drawn and the radiative one only below 0.19. Nothing about the donor has changed between the curves; only the question of where a gram that is not accreted leaves from.

Mass that leaves takes the orbit's spin with it

Most of the gas a star pours onto its companion never stays there. Where it leaves from decides how much angular momentum it carries, and that one bookkeeping entry moves the largest donor that can transfer stably by a factor of two — without anything about either star having changed.

stars · Mass transfer A period set by the donor's density, running in two directions. The orbital period at which a donor exactly fills its Roche lobe, against the donor's mass, onto a 0.7 M☉ white dwarf. A lobe-filling star's period is fixed by its mean density alone, so the period follows the donor's mass–radius relation. A main-sequence donor shrinks as it is stripped and grows denser, and the period falls: from 6.0 hours at 0.6 M☉ to 1.5 hours at the hydrogen-burning limit. A white-dwarf donor swells and grows less dense, and the period rises — 3.5 minutes at 0.2 M☉, 13 at 0.05, 47 at 0.01. The binaries with helium-rich spectra and periods between five minutes and an hour lie along the second curve and nowhere near the first, and their period is a mass gauge for a star that cannot be seen.

A donor that grows as it is eaten

A white dwarf that loses mass gets bigger, so a binary in which one white dwarf feeds another runs backwards — its period lengthens as the donor shrinks. Whether the pair survives first contact at all depends less on the donor than on whether the stream finds room to form a disc, and for most pairs it does not.

stars · Mass transfer Orbital energy spent against the envelope it has to lift. The orbital energy a 0.6 M☉ companion releases by spiralling in from the onset of a common envelope down to a final separation, against that separation (curve), and the binding energy of the 0.85 M☉ envelope of a 1.2 M☉ giant with a 0.35 M☉ core, divided by the efficiency, for αλ of 0.25, 0.5, 1 (horizontal lines). Energies are in units of GM☉²/R☉. The giant fills its lobe at a separation of 118 R☉, and the companion releases nothing net until the orbit has shrunk below 34 R☉, because the envelope's mass leaves with the envelope; from there it falls until the energy released pays for the envelope, so each crossing is a final orbit — 1.29, 2.48, 4.63 R☉. The released energy grows only as one over the separation, so halving the efficiency halves the final orbit: a factor of four in αλ is a factor of four in the outcome, and there is no measurement of αλ that does not run through outcomes like these.

The least known number in binary evolution

When a giant swallows its companion, the companion spirals inward until the orbital energy it has released is enough to throw the envelope off. The final orbit is set by one product of two unknowns, it moves in proportion to that product, and the merger time of the white dwarfs it leaves moves as its fourth power.

stars · Mass transfer

Named alongside it

The objects these essays reach for when they reach for this one.

Roche lobe overflowAccretionAccretion discLagrange pointsMass ratioRoche lobeWhite dwarfAngular momentumBinary pulsarCompact binaryEddington limitEquipotential

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