Astronomy 162:
Introduction to Stellar, Galactic, & Extragalactic Astronomy

Lecture 8: Binary Stars: Stellar Masses & Radii

Key Ideas:

Types of Binary Stars:
- Visual
- Spectroscopic
- Eclipsing
Can only measure masses for Binary Stars
Radii are measured for very few stars

Binary Stars

Apparent Binaries:

Chance projection of two distinct stars along the line of sight.
Often at very different distances.

True Binary Stars:

A pair of stars bound together by gravity.
Orbit each other about their center of mass.
Between 20% and 80% of all stars are binaries

Types of Binaries

Visual Binary:

Can see both stars and follow their orbits over time.

Spectroscopic Binary:

Cannot separate the two stars, but see their orbit motions as Doppler shifts in their spectral lines.

Eclipsing Binary:

Can separate the stars, but see the total brightness drop when they periodically eclipse each other.

Visual Binaries

Two stars orbiting about their center-of-mass.

Visual Binary Star

Center of Mass

Two stars orbit about their center of mass.

Center of Mass

Measure semi-major axis, a, from projected orbit & the distance.
Relative positions about the center give: M₁/M₂ = a₂/a₁

Here are two movies showing visual binary stars, one with a circular orbit, the other with an elliptical orbit, emphasizing the point about their orbits around their common center of mass in all cases.

Measuring Masses

Newton's Form of Kepler's Third Law:

Newton's Form of Kepler's 3rd Law

Procedure:

Measure the period, P, by following the orbit.
Measure semi-major axis, a, and the Mass Ratio, M₁/M₂, from the projected orbit on the sky.
Solve the equation above and separate Masses.

Problems
We need to follow the orbits long enough to trace them out in detail:

This can take decades
Need to work out the projection on the sky

Everything depends on knowing the distance:

semi-major axis depends on d
derived mass depends on d³

Small errors add up quickly.

Spectroscopic Binaries

Most binaries are too far away to be able to see both stars separately.

But, you can detect their orbital motions by the periodic Doppler shifts of the spectral lines:

Determine the orbit period & size from the pattern of orbital velocities

Spectroscopic Binary Star

Here is a movie showing a spectroscopic binary star.

Problems:
Cannot see the two stars separately:

Semi-major axis must be guessed from the orbit motions.
Can't tell how the orbit is tilted on the sky

Everything depends critically on knowing the distance.

Eclipsing Binaries

Two stars orbiting nearly edge-on to our line-of-sight.

See a periodic drop in brightness as one star eclipses the other.
Combine with spectra which measure orbital speeds

With the best data, one can find the masses of the stars without having to know the distance!!!

Eclipsing Binary Star

Here is a movie showing an eclipsing binary star and its light curve.

Problems
Eclipsing Binary stars are very rare.

Measurement of the light curves is complicated by details:

Partial eclipses yield less accurate numbers.
The atmospheres of the stars soften the edges.
Close binaries can be tidally distorted.

Stellar Masses

From a combination of visual and eclipsing binaries, masses are known for about 150 stars.

Range: ~0.1 to 50 Solar Masses

Stellar Radii

Radii are very difficult to measure because stars are so far away.

Methods:

Eclipsing binaries (need distance)
Interferometry (single stars)
Lunar Occultation (single stars)

Radii have only been measured for about 25 stars.

Advances in optical interferometry should greatly increase that number in the next decade.