Astronomy 162: Introduction to Stars, Galaxies, & the Universe Prof. Richard Pogge, MTWThF 9:30

Lecture 8: Stellar Masses & Radii

Key Ideas

Measure stellar masses from binary stars

Only way to measure stellar masses

Only measured for ~150 stars

Types of Binary Stars:

Visual

Spectroscopic

Eclipsing

Stellar radii have been measured for ~600 stars

Measuring Masses

Examples:

• Your mass from how much the Earth's gravity pulls upon you (your "weight").

• Earth's mass from the orbital motionss of the Moon or artificial satellites.

• Sun's mass from the orbital motions of the planets.

Binary Stars

• 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 common center of mass.
• About 60% all systems have 2 or more stars

Types of True Binaries

Can see both stars and follow their orbits over time.

Spectroscopic Binary:

Too close to see as separate stars, but you can detect their orbit motions by the Doppler shifts of their spectral lines.

Eclipsing Binary:

Too close see as separate stars, but we see the total brightness of the system decrease when the stars periodically eclipse each other.

Visual Binaries

Center of Mass

Two stars orbit about their common center of mass.

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

Measuring Masses

1. Measure the period, P, by following the orbit.
2. Measure semi-major axis, a, from the observed angular separation between the two stars and the distance to the system.
3. Solve the formula above to get the Total Mass (M₁+M₂)
4. Estimate the Mass Ratio, M₁/M₂, from the proje cted orbit on the sky and the center-of-mass
5. Solve for the individual masses.

Problems:
We need to follow an orbit long enough to trace it out in detail:

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

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

Spectroscopic Binaries

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

Here is a movie showing a simulation of a spectroscopic binary star.

Problems:
Often cannot see the two stars separately:

• Semi-major axis must be estimated from the orbit parameters.
• Can't tell how the orbit is tilted on the sky

Again...

Eclipsing Binaries

• See a periodic drop in the total brightness of the system as one star eclipses the other.
• Combine with spectra to measure their orbital speeds with time.

Here is a movie showing a simulation of 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

Range: ~0.07 to 60 Solar Masses

We will explore the masses of stars more in the next unit on stellar structure.

Stellar Radii

For example, from a distance of 1pc, the Sun is only 0.0093 arcseconds in diameter.

Methods:

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

Stellar Radii have been measured for about 600 stars so far. The large number compared to a few years ago is due to advances in techniques like optical interferometry.