Binary Systems and Mass Transfer

More than half of all stars above a solar mass belong to binary or higher-multiple systems. The binary is not an observational nuisance but the single most productive laboratory in stellar astrophysics: it is the only configuration in which a star's mass can be weighed directly, through the gravitational pull it exerts on a visible companion. Every entry in the empirical mass-luminosity relation traces back to a binary orbit. This lesson develops the three classes of binary and the masses they yield, then turns to the geometry of the shared gravitational potential, the onset of mass transfer when a star overflows its Roche lobe, and the orbital evolution that follows.

The two-body orbit and the observables

Two stars of masses $m_{1}$ and $m_{2}$ orbit their common center of mass on similar ellipses of semimajor axes $a_{1}$ and $a_{2}$ , related by

m_{1} a_{1} = m_{2} a_{2}, a = a_{1} + a_{2} .

The relative orbit, the ellipse of one star as seen from the other, has semimajor axis $a$ and obeys Kepler's third law in its exact Newtonian form,

P^{2} = \frac{4 π ^{2} a ^{3}}{G ( m _{1} + m _{2} )} .

A single measured period and separation fix only the total mass. The individual masses require the mass ratio $m_{1} / m_{2} = a_{2} / a_{1}$ , and that ratio is accessible only when the orbit is resolved or the two spectra are separately measured. Three observational classes deliver different subsets of the elements.

The three binary classes and the signal each produces: a resolved astrometric orbit on the sky, periodic radial-velocity curves in the spectrum, and a light curve with primary and secondary eclipse minima.

Visual binaries

When both ellipses are resolved and the distance $d$ is measured by parallax, the angular semimajor axes convert to physical sizes and the ratio $a_{2} / a_{1}$ gives the mass ratio directly. Combining with Kepler's law yields both masses. The catch is geometric: the true orbit is projected onto the sky at an unknown inclination $i$ , so the observed ellipse is a foreshortened image of the real one. The position of the primary star at the focus of the apparent ellipse, however, is preserved under projection, and fitting the full apparent orbit recovers $i$ along with the elements. Visual binaries have periods of decades to centuries and populate the solar neighborhood; they anchor the mass-luminosity relation for main-sequence stars.

Spectroscopic binaries and the mass function

In an unresolved system the orbital motion appears only as a periodic Doppler shift. The radial velocity of star 1 varies with a semi-amplitude

K_{1} = \frac{2 π a _{1} sin i}{P 1 - e ^{2}},

and similarly for star 2. For a double-lined system both $K_{1}$ and $K_{2}$ are measured, and their ratio gives the mass ratio without any distance,

\frac{K _{1}}{K _{2}} = \frac{a _{1}}{a _{2}} = \frac{m _{2}}{m _{1}} .

Only the products $a_{1} sin i$ and $a_{2} sin i$ are observable, because the Doppler effect senses only the line-of-sight velocity component. Eliminating $a_{1}$ between $K_{1}$ and Kepler's law collects everything measurable into the mass function.

With only one spectrum the inclination and the ratio remain degenerate, and $f$ sets a floor on the companion mass. This floor is decisive in searches for stellar-mass black holes: an X-ray binary whose optical star shows a mass function exceeding the Chandrasekhar mass, roughly $1.4 M_{⊙}$ , cannot have a white-dwarf or neutron-star companion, forcing a black hole.

Eclipsing binaries

When $i \approx 9 0^{\circ}$ the stars eclipse, and $sin i \approx 1$ removes the inclination ambiguity. A double-lined eclipsing binary is the gold standard: $K_{1}$ , $K_{2}$ , and $sin i \approx 1$ together yield both masses with no distance assumed. The eclipse geometry adds the stellar radii. The duration of ingress fixes the smaller radius through the projected orbital speed, and the flat bottom of a total eclipse fixes the larger, so the light curve alone constrains $R_{1}$ and $R_{2}$ in units of $a$ . The relative depths of the primary and secondary minima give the ratio of surface brightnesses, hence of effective temperatures. Detached double-lined eclipsing binaries are the primary source of accurate stellar masses and radii across the H-R diagram.

The Roche potential

When the stars are close enough to interact, the relevant description is the shape of the gravitational-plus-centrifugal potential in the frame that co-rotates with the orbit. Assume a circular orbit and treat the stars as point masses; work in the rotating frame with angular velocity

Ω = \frac{G ( m _{1} + m _{2} )}{a ^{3}}

about the center of mass. A test particle at position $r$ feels the two gravitational potentials plus the centrifugal potential, giving the Roche potential

Φ (r) = - \frac{G m _{1}}{∣ r - r _{1} ∣} - \frac{G m _{2}}{∣ r - r _{2} ∣} - \frac{1}{2} Ω^{2} s^{2},

where $s$ is the distance from the rotation axis through the center of mass. The Coriolis force does no work and drops out of the potential; it deflects moving gas but not the equipotential surfaces themselves. The surfaces $Φ = const$ organize the entire flow.

Equipotentials of the Roche potential in the co-rotating frame: the two Roche lobes meet at the inner Lagrange point L1, with the outer collinear points L2 and L3 and the triangular points L4 and L5 marked.

The three collinear points $L_{1}$ , $L_{2}$ , $L_{3}$ are saddle points, unstable in every direction. The triangular points $L_{4}$ and $L_{5}$ are maxima of the effective potential yet are dynamically stable when the mass ratio is extreme enough ( $m_{1} / m_{2} > 24.96$ ), because the Coriolis force curves any drifting particle into a closed loop around them; the Trojan asteroids occupy the Sun-Jupiter $L_{4}$ and $L_{5}$ .

The size of the Roche lobe depends only on the mass ratio $q = m_{1} / m_{2}$ and the separation $a$ . A widely used fit to the volume-equivalent lobe radius of star 1 is Eggleton's formula,

\frac{R _{L, 1}}{a} = \frac{0.49 q ^{2/3}}{0.6 q ^{2/3} + ln ( 1 + q ^{1/3} )},

accurate to better than one percent over all $q$ . For a comparable-mass system the lobe radius is roughly $0.38 a$ ; the more massive star claims the larger lobe.

Roche-lobe overflow and mass transfer

A star evolves at fixed mass by expanding: a main-sequence star swells as it ascends toward the giant branch, while its Roche lobe is fixed by the orbit. When the stellar radius reaches $R_{L}$ , the outermost layers find themselves beyond the critical surface, where the companion's gravity and the centrifugal term overwhelm the star's own binding. Gas flows through the $L_{1}$ saddle, the one low point in the potential wall separating the two lobes, and falls toward the companion. This is Roche-lobe overflow, the dominant channel for mass exchange in close binaries.

Roche-lobe overflow: the donor has expanded to fill its lobe and gas escapes through the inner Lagrange point L1, forming a stream that falls toward the accretor and, deflected by the Coriolis force, misses it and circularizes.

Because the transferred gas carries the orbital angular momentum of the $L_{1}$ point, it does not fall straight onto the companion. The Coriolis deflection turns the stream, and it settles into a rotating ring at the radius where its specific angular momentum matches a circular orbit. Viscosity then spreads the ring into an accretion disk, the subject of the next lesson.

Orbital evolution under conservative transfer

Mass transfer changes the orbit, and the direction of change follows from angular momentum. Take conservative transfer: the total mass $M = m_{1} + m_{2}$ is conserved (no mass leaves the system) and so is the orbital angular momentum

J = m_{1} m_{2} \frac{G a}{M} .

Differentiate $ln J$ with $M$ and $J$ held fixed, and let star 1 be the donor, so $\overset{m}{˙}_{1} < 0$ and $\overset{m}{˙}_{2} = - \overset{m}{˙}_{1} > 0$ :

\frac{a ˙}{a} = - 2 \overset{m}{˙}_{1} (\frac{1}{m _{1}} - \frac{1}{m _{2}}) = - 2 \overset{m}{˙}_{1} \frac{m _{2} - m _{1}}{m _{1} m _{2}} .

The sign hinges on the mass ratio.

The turning point is $m_{1} = m_{2}$ , where the separation is momentarily stationary. Transfer from the initially more massive star therefore begins as a rapid, runaway phase on the donor's thermal timescale, reverses the mass ratio, and then continues as a slow phase driven by the donor's continued expansion or by angular momentum losses.

The response of orbital separation to conservative mass transfer: the orbit contracts while the donor is the more massive star and expands once transfer has reversed the mass ratio, with the separation stationary at equal masses.

The Algol paradox

Algol is a close eclipsing binary in which the less massive star, a $0.8 M_{⊙}$ subgiant, is the more evolved of the pair, while its $3.7 M_{⊙}$ companion still sits on the main sequence. Single-star evolution insists that the more massive star evolves faster and should be the more advanced. The observed pairing inverts that expectation.

The resolution is mass transfer. The subgiant was originally the more massive star. It exhausted its core hydrogen first, expanded to fill its Roche lobe, and began transferring mass to its companion. Because the donor was then the more massive member, the early transfer was rapid and the orbit contracted, feeding the runaway until the mass ratio reversed. What remains is the stripped, evolved core of the former primary, now the lighter star, orbiting a rejuvenated companion that has gained most of the transferred mass. The paradox is the fossil record of a completed episode of Roche-lobe overflow.

Summary

A binary is the direct route to stellar masses. Kepler's third law fixes the total mass; the mass ratio, and thus the individual masses, follows from resolving the two orbits (visual), measuring both radial-velocity amplitudes (double-lined spectroscopic), or observing eclipses that pin $sin i \approx 1$ . The spectroscopic mass function $f = m_{2}^{3} sin^{3} i / (m_{1} + m_{2})^{2}$ places a hard lower bound on an unseen companion, the tool that identifies black holes in X-ray binaries. In the co-rotating frame the shared potential is the Roche potential, whose inner critical surface defines the Roche lobes joined at $L_{1}$ . A star that expands to fill its lobe spills gas through $L_{1}$ onto its companion. Conservative transfer contracts the orbit while the donor is the more massive star and expands it afterward, and this sign reversal resolves the Algol paradox: the presently lighter, more evolved star was once the massive primary that drove the transfer.