6.3 Eclipsing binaries and spectroscopic binaries

Binary Star Systems

Binary star systems give astronomers a direct way to measure stellar masses, radii, and temperatures. Because two stars orbit a shared center of mass, their gravitational interaction encodes physical properties into observable signals: periodic brightness changes and Doppler-shifted spectral lines. This section covers how eclipsing and spectroscopic binaries work, and how to extract stellar parameters from their data.

Geometry of eclipsing binaries

An eclipsing binary occurs when two stars orbit their common center of mass with the orbital plane nearly aligned to our line of sight. From Earth, one star periodically passes in front of the other, blocking some of its light and producing measurable dips in the system's total brightness.

The primary eclipse is the deeper dip. It happens when the hotter, brighter star is obscured by its companion. The secondary eclipse is shallower, occurring when the fainter star is blocked. A common point of confusion: the primary eclipse isn't necessarily caused by the larger star passing in front. It's defined by which eclipse produces the bigger brightness drop, which depends on surface brightness (and therefore temperature).

Light curve shapes tell you about the system's geometry:

• Detached binaries have flat regions between eclipses, because the stars are well-separated and roughly spherical. Brightness stays constant outside of eclipse events.
• Contact binaries show continuous brightness variation with no flat portions. The stars are so close that tidal forces distort them into elongated shapes, and the projected area facing Earth changes throughout the orbit.

Eclipse duration depends on the stellar radii relative to the orbital separation and on the orbital eccentricity. Eclipse depth depends on the temperature contrast and size ratio between the two stars. A large, cool companion transiting a small, hot star produces a very different light curve than two similar Sun-like stars.

Properties from binary light curves

Light curves contain a surprising amount of information if you analyze them carefully. Here's what you can extract and how:

Orbital period is the most straightforward measurement. You identify the time between successive primary eclipses (or secondary eclipses) using photometric time-series data.

Stellar radii come from combining eclipse duration with orbital velocity. If you know how fast the stars move (from spectroscopy) and how long an eclipse lasts, you can calculate the physical size of each star. For a circular orbit with known velocity $v$ and eclipse duration $\Delta t$, the stellar radius scales with $v \cdot \Delta t$.

Temperature ratio is encoded in the relative depths of primary and secondary eclipses. For total eclipses, the ratio of eclipse depths is related to the ratio of surface brightnesses, which follows from the stars' effective temperatures. Color information (like the B-V index) measured during and outside eclipses helps pin down individual temperatures.

Orbital inclination $i$ can be constrained from eclipse shape. A system viewed exactly edge-on ($i = 90°$) produces flat-bottomed eclipses (total eclipses), while a slightly inclined system produces V-shaped or rounded eclipses (partial/grazing eclipses).

Several secondary effects also shape the light curve and need to be accounted for in detailed modeling:

• Limb darkening: Stars appear dimmer near their edges than at their centers. This rounds off the ingress and egress portions of each eclipse.
• Reflection effect: In close binaries, each star irradiates the facing hemisphere of its companion, making that hemisphere brighter. This creates a smooth brightness modulation between eclipses.
• Ellipsoidal variations: Tidal distortion in close systems (like the Algol system) changes the projected area of each star as it orbits, producing a double-humped brightness variation even outside eclipses.

Principles of spectroscopic detection

Not all binaries eclipse. If the orbital plane is tilted away from our line of sight, you won't see brightness dips. But you can still detect the system through the Doppler effect on spectral lines.

As each star orbits the center of mass, it alternately moves toward and away from Earth. This shifts its spectral lines to shorter wavelengths (blueshift) and longer wavelengths (redshift). By tracking these shifts over time, you build a radial velocity curve: a plot of line-of-sight velocity versus time.

Spectroscopic binaries fall into two categories:

• Single-lined (SB1): Only one star's spectral lines are visible (the companion is too faint or its lines are blended). You measure one radial velocity curve.
• Double-lined (SB2): Both stars contribute visible spectral lines. You see two sets of lines shifting in antiphase. SB2 systems are more valuable because they give you velocity information for both components.

For a circular orbit, the radial velocity curve is sinusoidal. The orbital period $P$ equals the time between successive velocity maxima (or minima). The semi-amplitude $K$ measures the peak velocity shift from the systemic velocity.

For an eccentric orbit, the radial velocity curve deviates from a pure sine wave, becoming asymmetric. The degree of asymmetry encodes the eccentricity $e$ and the argument of periastron $\omega$.

Detecting small velocity shifts requires high-resolution spectrographs. Modern instruments achieve precision on the order of m/s, which is essential for low-mass companions.

Mass determination in spectroscopic binaries

Mass is the most fundamental stellar property, and spectroscopic binaries are one of the few ways to measure it directly. The approach differs depending on whether you have an SB1 or SB2 system.

For SB1 systems, you can measure the semi-amplitude $K_1$ and the period $P$ from the radial velocity curve. These combine into the mass function:

$f(M) = \frac{(M_2 \sin i)^3}{(M_1 + M_2)^2} = \frac{P K_1^3}{2\pi G}$

The right-hand side is entirely observable. The left-hand side contains three unknowns: $M_1$, $M_2$, and the inclination $i$. With only one radial velocity curve, you cannot solve for individual masses. The mass function gives a minimum mass for the unseen companion, $M_2 \sin i$, since $\sin i \leq 1$.

For SB2 systems, you measure both $K_1$ and $K_2$. The mass ratio follows directly:

$q = \frac{M_2}{M_1} = \frac{K_1}{K_2}$

This works because the less massive star orbits faster (larger velocity amplitude) around the center of mass. With the mass ratio and the mass function, you can solve for $M_1 \sin^3 i$ and $M_2 \sin^3 i$ individually.

Resolving the inclination ambiguity is the key remaining challenge. If the system also happens to be an eclipsing binary, you can determine $i$ from the light curve geometry (as described above). Combining eclipsing and spectroscopic data gives you true masses, not just minimum masses. This is why eclipsing spectroscopic binaries are so valuable to stellar astrophysics.

Additional parameters and practical considerations:

• The semi-major axis (projected) for a circular orbit is $a_1 \sin i = \frac{K_1 P}{2\pi}$, with the analogous expression for $a_2$.
• Eccentricity is estimated by fitting the radial velocity curve shape. A circular orbit gives a pure sinusoid; deviations indicate $e > 0$.
• Measurement uncertainties propagate through these equations. Because the mass function involves $K_1^3$, even small errors in velocity amplitude translate into significant mass uncertainties. Rigorous error analysis (including techniques like Monte Carlo simulations) is standard practice.