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
- Measure stellar masses from binary stars
- Only way to measure stellar masses
- Only measured for ~150 stars
- Types of Binary Stars:
- Stellar radii have been measured for ~600 stars
Masses are measured by using the effects of gravity on objects:
- Your mass from how much the Earth's gravity pulls upon you
- Earth's mass from the orbital motionss of the Moon or
- Sun's mass from the orbital motions of the planets.
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
- Can see both stars and follow their orbits over time.
- Too close to see as separate stars, but you can detect
their orbit motions by the Doppler shifts of their spectral lines.
- Too close see as separate stars, but we see the total brightness
of the system decrease when the stars periodically eclipse each other.
Two stars orbiting about their center-of-mass.
Center of Mass
Two stars orbit about their common center of mass.
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.
- Measure semi-major axis, a, from projected orbit & the
- Relative positions about the center give:
M1/M2 = a2/a1
Recall Newton's Version of
Kepler's Third Law:
- Measure the period, P, by following the orbit.
- Measure semi-major axis, a, from the observed angular separation
between the two stars and the distance to the system.
- Solve the formula above to get the Total Mass (M1+M2)
- Estimate the Mass Ratio, M1/M2, from the proje cted orbit on the sky and the center-of-mass
- Solve for the individual masses.
We need to follow an orbit long enough to trace it out in detail:
Measurements depend on knowing the distance:
- This can take decades
- Need to work out the projection on the sky
Small errors add up quickly (10% error in distance translates into
a 30% error in the mass!).
- semi-major axis depends on d
- derived mass depends on d3
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
Here is a movie showing a simulation of
a spectroscopic binary star.
Often cannot see the two stars separately:
Everything depends critically on knowing the distance.
- Semi-major axis must be estimated from the orbit parameters.
- Can't tell how the orbit is tilted on the sky
Two stars orbiting nearly edge-on to our line-of-sight.
With the best data, one can find the masses of the stars
without having to know the distance!!!
- 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.
Eclipsing Binary stars are very rare.
Measurement of the light curves is complicated by details:
Despite this, most of the best stellar masses are from eclipsing binary
- Partial eclipses yield less accurate numbers.
- The atmospheres of the stars soften the edges.
- Close binaries can be tidally distorted.
From a combination of visual and eclipsing binaries, masses are
known for about 150 stars.
A few very massive stars are known, with masses of 80-120 Solar Masses,
but these are very unusual and rare.
- Range: ~0.07 to 60 Solar Masses
We will explore the masses of stars more in the next unit on stellar
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
The details of these techniques are beyond the scope of this course to
- 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
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Updated: 2006 January 8
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