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

# Lecture 8: Stellar Masses & Radii

Readings: Chapter 19, section 19-9,10, & 11

## 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

Masses are measured by using the effects of gravity on objects:

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

Apparent 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

Visual Binary:
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

Two stars orbiting about their center-of-mass.

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

## Measuring Masses

Recall Newton's Version of Kepler's Third Law:

Procedure:
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:

• Need to work out the projection on the sky
Measurements depend on knowing the distance:
• semi-major axis depends on d
• derived mass depends on d3
Small errors add up quickly (10% error in distance translates into a 30% error in the mass!).

## 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

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
Everything depends critically on knowing the distance.

Again...

## Eclipsing Binaries

Two stars orbiting nearly edge-on to our line-of-sight.
• 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.
With the best data, one can find the masses of the stars without having to know the distance!!!

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.
Despite this, most of the best stellar masses are from eclipsing binary systems.

## Stellar Masses

From a combination of visual and eclipsing binaries, masses are known for about 150 stars.
Range: ~0.07 to 60 Solar Masses
A few very massive stars are known, with masses of 80-120 Solar Masses, but these are very unusual and rare.

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

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

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)
The details of these techniques are beyond the scope of this course to discuss.

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.

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Updated: 2006 January 8