---
title: Binary Systems and Mass Transfer
draft: false
module: Binaries and Gravitational Waves
moduleNumber: 9
lessonNumber: 1
order: 901
summary: >
  Most stars are born in pairs, and a binary is the only setting where a stellar
  mass can be measured directly. Visual, spectroscopic, and eclipsing binaries each
  expose a different combination of the orbital elements, and together they
  calibrate the mass-luminosity relation. When one star swells to fill its Roche
  lobe, gas streams through the inner Lagrange point onto its companion. Conservative
  transfer widens or shrinks the orbit depending on the mass ratio, and the sign of
  that response explains the Algol paradox.
topics: [Binaries and Gravitational Waves]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 7 — Binary Systems and Stellar Parameters; Ch. 18 §18.1 Roche Lobes"
  - book: Maoz
    ref: "Ch. 6 — Binary Stars and Accretion"
---

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

$$
m_1 a_1 = m_2 a_2,
\qquad a = a_1 + a_2.
$$

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,

$$
P^2 = \frac{4\pi^2 a^3}{G(m_1 + m_2)}.
$$

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.

> **Definition (Binary classes).** Binaries are named by what makes the orbit
> observable:
> - **Visual binary**: both stars are resolved and the two elliptical paths are
>   traced on the sky over years. The angular sizes give $a_1$ and $a_2$ once the
>   distance is known.
> - **Spectroscopic binary**: the components are unresolved, but periodic Doppler
>   shifts of the spectral lines reveal the radial-velocity variation of one
>   (single-lined) or both (double-lined) stars.
> - **Eclipsing binary**: the orbital plane lies near the line of sight, so each star
>   periodically blocks the other and the combined light dips.

$$
% caption: The three binary classes and the signal each produces: a resolved
% astrometric orbit on the sky, periodic radial-velocity curves in the spectrum, and
% a light curve with primary and secondary eclipse minima.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% --- panel 1: visual/astrometric orbit ---
\draw[very thick] (1.5,3.4) ellipse (1.1 and 0.65);
\fill[black] (1.5,3.4) circle (1.2pt);
\node[black, anchor=south] at (1.5,4.15) {astrometric orbit};
\fill[black] (2.55,3.55) circle (1.6pt);
\node[black, anchor=west] at (2.65,3.55) {star};
\node[black, anchor=north] at (1.5,2.65) {Visual};
% --- panel 2: radial-velocity curves ---
\draw[->, black] (4.2,2.8) -- (7.3,2.8) node[right, black!70] {phase};
\draw[->, black] (4.2,2.4) -- (4.2,4.2) node[above, black!70] {Vr};
\draw[very thick] (4.4,3.5) .. controls (5.0,4.1) and (5.6,4.1) .. (6.2,3.5)
  .. controls (6.6,3.1) and (7.0,3.1) .. (7.2,3.5);
\draw[black, very thick, densely dashed] (4.4,3.5) .. controls (5.0,2.9) and (5.6,2.9) .. (6.2,3.5)
  .. controls (6.6,3.9) and (7.0,3.9) .. (7.2,3.5);
\node[black, anchor=north] at (5.7,2.35) {Spectroscopic};
% --- panel 3: eclipse light curve ---
\draw[->, black] (0.4,0.6) -- (3.6,0.6) node[right, black!70] {phase};
\draw[->, black] (0.4,0.5) -- (0.4,2.0) node[above, black!70] {brightness};
\draw[very thick] (0.6,1.7) -- (1.1,1.7) -- (1.25,0.95) -- (1.4,1.7)
  -- (2.3,1.7) -- (2.45,1.3) -- (2.6,1.7) -- (3.4,1.7);
\node[black, anchor=north] at (1.25,0.9) {primary};
\node[black, anchor=south] at (2.45,1.35) {secondary};
\node[black, anchor=north] at (2.0,0.3) {Eclipsing};
\end{tikzpicture}
$$

### 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

$$
K_1 = \frac{2\pi a_1 \sin i}{P\,\sqrt{1 - e^2}},
$$

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**.

> **Definition (Spectroscopic mass function).** For a single-lined spectroscopic
> binary the measured period, eccentricity, and velocity amplitude combine into
> $$
> f(m_1, m_2, i) = \frac{m_2^{3}\sin^{3} i}{(m_1 + m_2)^{2}}
>   = \frac{P\,K_1^{3}}{2\pi G}\,(1 - e^{2})^{3/2}.
> $$
> The right side is entirely observational; the left contains the unknowns. Because
> $\sin i \le 1$, the mass function is a strict lower bound on $m_2$ when $m_1$ is
> known, which is how the mass of an unseen compact companion is bounded from below.

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 $1.4\,M_\odot$, cannot have a white-dwarf or
neutron-star companion, forcing a
[black hole](/astrophysics-cosmology/stellar-death-and-compact-remnants/black-holes-schwarzschild-and-kerr).

### Eclipsing binaries

When $i \approx 90^\circ$ 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

$$
\Omega = \sqrt{\frac{G(m_1 + m_2)}{a^3}}
$$

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

$$
\Phi(\vec r) = -\frac{G m_1}{|\vec r - \vec r_1|}
             - \frac{G m_2}{|\vec r - \vec r_2|}
             - \tfrac{1}{2}\,\Omega^2\, s^2,
$$

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 $\Phi = \text{const}$
organize the entire flow.

> **Definition (Roche lobe and Lagrange points).** The critical equipotential that
> passes through the inner saddle point encloses two teardrop volumes, one around
> each star, joined at that point. Each teardrop is a **Roche lobe**; the largest
> volume a star can occupy while remaining bound to itself. The five stationary
> points of $\Phi$ in the rotating frame are the **Lagrange points** $L_1$ through
> $L_5$: $L_1$ is the inner saddle between the stars, $L_2$ and $L_3$ lie outside
> each star along the axis, and $L_4$, $L_5$ form equilateral triangles with the two
> masses.

$$
% caption: Equipotentials of the Roche potential in the co-rotating frame: the two
% Roche lobes meet at the inner Lagrange point L1, with the outer collinear points
% L2 and L3 and the triangular points L4 and L5 marked.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% centers
\coordinate (m1) at (2.8,2.4);
\coordinate (m2) at (5.6,2.4);
% inner critical equipotential: two lobes meeting at L1
\draw[very thick] (2.8,2.4) ellipse (1.05 and 0.85);
\draw[very thick] (5.6,2.4) ellipse (0.8 and 0.62);
% outer equipotential (dashed, encloses both)
\draw[black, densely dashed] (4.2,2.4) ellipse (3.2 and 1.9);
\fill[black!70] (m1) circle (2pt) node[above=2pt, black!70] {primary};
\fill[black!70] (m2) circle (1.6pt) node[above=2pt, black!70] {secondary};
% Lagrange points
\fill[acc] (3.85,2.4) circle (1.6pt) node[below=2pt, acc] {L1};
\fill[black] (7.1,2.4) circle (1.4pt) node[below=2pt, black] {L2};
\fill[black] (0.9,2.4) circle (1.4pt) node[below=2pt, black] {L3};
\fill[black] (4.2,4.0) circle (1.4pt) node[above=2pt, black] {L4};
\fill[black] (4.2,0.8) circle (1.4pt) node[below=2pt, black] {L5};
\draw[black, densely dotted] (0.6,2.4) -- (7.4,2.4);
\end{tikzpicture}
$$

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,

$$
\frac{R_{L,1}}{a} = \frac{0.49\, q^{2/3}}{0.6\, q^{2/3} + \ln\!\bigl(1 + q^{1/3}\bigr)},
$$

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.

$$
% caption: Roche-lobe overflow: the donor has expanded to fill its lobe and gas
% escapes through the inner Lagrange point L1, forming a stream that falls toward the
% accretor and, deflected by the Coriolis force, misses it and circularizes.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% donor (fills its lobe)
\draw[very thick] (2.4,2.2) ellipse (1.1 and 0.9);
\node[black, anchor=center] at (2.4,2.2) {donor};
% accretor lobe
\draw[black, very thick] (5.6,2.2) ellipse (0.75 and 0.6);
\fill[black!70] (5.6,2.2) circle (2pt);
\node[black, anchor=south] at (5.6,2.85) {accretor};
% L1
\fill[black!70] (3.6,2.2) circle (1.6pt) node[below=2pt, black] {L1};
% stream through L1, curving (Coriolis) into a ring
\draw[acc, very thick, ->] (3.6,2.2) .. controls (4.3,2.35) and (4.9,2.15) .. (5.35,1.75)
  .. controls (5.9,1.5) and (6.2,2.0) .. (5.95,2.45)
  .. controls (5.75,2.75) and (5.3,2.7) .. (5.1,2.4);
\node[acc, anchor=south west] at (4.1,2.35) {stream};
\end{tikzpicture}
$$

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](/astrophysics-cosmology/binaries-and-gravitational-waves/accreting-compact-objects).

## 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

$$
J = m_1 m_2 \sqrt{\frac{G a}{M}}.
$$

Differentiate $\ln J$ with $M$ and $J$ held fixed, and let star 1 be the donor, so
$\dot m_1 < 0$ and $\dot m_2 = -\dot m_1 > 0$:

$$
\frac{\dot a}{a} = -2\,\dot m_1\!\left(\frac{1}{m_1} - \frac{1}{m_2}\right)
                = -2\,\dot m_1\,\frac{m_2 - m_1}{m_1 m_2}.
$$

The sign hinges on the mass ratio.

> **Result.** Under conservative mass transfer the orbital separation responds to
> the mass ratio:
> - **Donor more massive** ($m_1 > m_2$): the factor $m_2 - m_1 < 0$ and, with
>   $\dot m_1 < 0$, the orbit **shrinks** ($\dot a < 0$). The tightening drives the
>   donor deeper into its lobe, accelerating transfer; the process runs away.
> - **Donor less massive** ($m_1 < m_2$): the orbit **widens** ($\dot a > 0$), which
>   relaxes the overflow and makes transfer self-limiting and slow.

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.

$$
% caption: The response of orbital separation to conservative mass transfer: the
% orbit contracts while the donor is the more massive star and expands once transfer
% has reversed the mass ratio, with the separation stationary at equal masses.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0.4,0.6) -- (8.4,0.6) node[right, black!70] {mass transferred};
\draw[->, black] (0.4,0.6) -- (0.4,4.2) node[above, black!70] {separation a};
% U-shaped separation curve, minimum at equal-mass point
\draw[acc, very thick] (0.8,3.7) .. controls (2.4,2.3) and (3.4,1.3) .. (4.4,1.25)
  .. controls (5.4,1.2) and (6.6,2.2) .. (8.0,3.9);
\fill[acc] (4.4,1.25) circle (2pt);
\draw[black, densely dotted] (4.4,0.6) -- (4.4,1.25);
\node[black, anchor=north] at (4.4,0.55) {equal masses};
\node[black, anchor=east] at (2.2,2.0) {contracts};
\node[black, anchor=west] at (6.4,2.4) {expands};
\end{tikzpicture}
$$

## The Algol paradox

Algol is a close eclipsing binary in which the less massive star, a
$0.8\,M_\odot$ subgiant, is the more evolved of the pair, while its
$3.7\,M_\odot$ 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.

> **Worked example.** A binary starts with $m_1 = 3\,M_\odot$ (donor) and
> $m_2 = 2\,M_\odot$ at separation $a_i$. Conservative transfer proceeds until the
> masses reverse to $m_1 = 2\,M_\odot$, $m_2 = 3\,M_\odot$. Using constancy of
> $J = m_1 m_2 \sqrt{Ga/M}$ with $M = 5\,M_\odot$ throughout, the product $m_1 m_2$
> returns to its starting value ($6\,M_\odot^2$ at both endpoints), so $a$ returns to
> $a_i$. Between the endpoints $m_1 m_2$ is larger (it peaks at $6.25\,M_\odot^2$ when
> $m_1 = m_2 = 2.5\,M_\odot$), and since $a \propto (m_1 m_2)^{-2}$ at fixed $J, M$,
> the separation dips to a minimum of $a_i (6/6.25)^2 \approx 0.92\,a_i$ at the
> equal-mass point before recovering. The orbit first tightens, then re-widens to its
> original size.

## 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.

[^co-binary]: Carroll & Ostlie, Ch. 7 — the classification of binaries, the mass function, and the determination of stellar masses and radii from visual, spectroscopic, and eclipsing systems.
[^co-roche]: Carroll & Ostlie, Ch. 18 §18.1 — the Roche potential, Lagrange points, Roche-lobe overflow, and the orbital response to mass transfer.
[^maoz-binary]: Maoz, Ch. 6 — close binaries, Roche geometry, Eggleton's lobe-radius approximation, and the Algol paradox.