Binary Stars

We discussed 3 kinds of binaries. A binary system is one in which 2 stars are in orbit about each other (bound gravitationally). The 2 stars orbit around their common center-of-mass. The first kind is "visual binaries", in which both stars are imaged (either just in a telescope, or perhaps in an interferometer). We can watch the stars go around their common center of mass and see the orbital period, orbital tilt, and angular separation. If we know the distance to the system, we can convert the angular separation to AU, and thus get the mass of the 2 stars. The mass ratio could be found using the distance of each star from the center of mass. To get the mass, you just apply Kepler's Third Law in its more general form:
M₁ + M₂ = a³ / P²  (here "a" is the semimajor axis of one star about the other, or half their average separation)
Another way to get the mass ratio, even if you can't see the 2 stars separately is to observe the stars' spectra, and watch the Doppler shift change as they orbit (a "spectroscopic binary"). The ratio of the 2 velocities will be in the ratio of the masses. The orbital tilt affects the value of the velocities, since Doppler shifts only occur for line-of-sight motions. Thus, if we look down on the orbit from above, there is no Doppler shift at all! This means that we cannot convert velocities to a mass unless we know the orbital tilt (visual binary). If we have only a spectroscopic binary, but see both stars, we can get the orbital period, mass ratio, and a lower limit on the masses. With only one star visible, we can still get the period and a limit on the mass.
Eclipsing binaries are the most useful. We know that the orbit is not tilted, since the stars eclipse, so there is no ambiguity about the Doppler shifts. Furthermore, the timing of how long the eclipses last gives the size of the stars as well. And with timing and Doppler shifts, the scale of the system and thus the masses can be found without even knowing the distance to the system. Here are some cool Websites which have much more on binary stars (including simulations):

 Beginners Guide to Binaries
 Simulation Software
 Eclipsing Binaries and other links
 Web Simulations

Stellar Parameters
It is hard to figure out the basic stellar parameters below. Some of them depend on knowing others, so we try to locate stars or systems in which several can be measured. The idea is that the spectrum of a stars contains a lot of information about it, and is a fairly unique signature of all the below parameters except distance. So we try to measure the parameters for stars of each spectral type, and then in almost all cases use the "known" characteristics of a given spectral class (eg. G2V) to assign properties to a new star.

Distance
  Parallax: the angle induced by the shift of viewpoint caused by the Earth's orbit.    d (parsecs) = 1/p (arcsec)
 Appearance: if intrinsic luminosity L is known from the spectral type, the distance can be found from the apparent brightness B using   B = L / 4pd²

Radius
  Can be found from eclipsing binaries. Or for stars of a known temperature and angular diameter, can use brightness to infer radius using L  a R²  T⁴. Calibrate for all spectral types (then can use just spectrum).

Temperature
  Can be found from color or spectrum of star. Calibrate for all spectral types (then can use just spectrum).  T(K) = 3x106 / lpeak (nm)

Luminosity
 Must know the distance to convert B (which is measured) to L using  B = L / 4pd². Or if you know the radius and temperature from a binary and the spectrum, can use  L  a R²  T⁴

Mass
  Can be found from eclipsing or visual binaries. Mass ratios can be found for spectroscopic binaries. Use the general form of Kepler's Third Law:
   M₁+M₂ (suns)  = a(AU)³ / P(yr)²
 
 

The HR Diagram (see Figs. 12.17-12.21)

Named for Hertzsprung and Russell, this diagram shows luminosity on the vertical axis, and temperature (or often spectral type) on the horizontal axis. Luminosity increases upward, and temperature decreases to the right. We find that most stars lie in a band going from upper left to lower right: the main sequence. This band represents the location of stars in the diagram when they are fusing hydrogen in their cores. Since that is by far the lengthiest stage of a star's life, you are likely to see most stars during that phase. The main sequence is also a mass sequence, from high mass at the upper left to low mass at the lower right. Only on the main sequence can you infer a star's mass from its position in the HR diagram. Also represented on the HR diagram is stellar size, running from smaller in the lower left to larger in the upper right. We find a sequence of very small objects below and to the left of the main sequence; these white dwarfs are the burned out remnants of low and intermediate mass stars. In the upper right are the red giants and supergiants. These are stars which have exhausted hydrogen in their cores, and are now consuming heavier elements (starting with helium). Thus, the HR diagram is also an "evolutionary" diagram: you can tell what phase of a star's life it is in by its position in the diagram.