# STELLAR MASSES

• To understand how we determine stellar masses you have to understand the Laws of Motion and Gravity from Newton and Kepler that relate orbital speeds and orbital sizes to the masses of objects in bound systems. But, everyone has a pretty good intuition for this.

• The Sun and Earth orbit about the "center of mass" of their system (which is deep within the Sun), but the Sun's motion is very small compared to the Earth's because M >> MEarth.

• You can apply some Laws of Motion and use the speed of the Earth along its orbit to calculate:

M   =   2 x 1033 grams   =   300,000 MEarth

Now, to measure the masses of the other stars out there the first thing we need to find are some Binary (two) Systems - two or more stars in orbit around one another. It turns out that binary systems are quite common and make up more than 1/3 of the stars in the Galaxy.

### Binary Systems

There are several kinds of binaries:

• Optical Doubles - these are just chance superpositions on the sky. They are not real binary systems.

• Visual Doubles - This is the case of a binary star system where the separation of the stars is great enough that we can see both members, and over a loooooong period of time, we can watch them move about in their orbits. The orbital periods of known visual binaries range from a few 10's of years to 1000 years.

• Spectroscopic Binaries - Now we are getting to the interesting types.

1. Single-lined Spectroscopic Binary: Sometimes you take a spectrum of a star over several different nights and discover that the positions of the absorption lines change with time. This is ascribed to a Doppler Shift that changes periodically with time because we are seeing one star in orbit around another.

2. Double-lined Spectroscopic Binaries: Sometimes you take a spectrum of an apparently single star and see two sets of absorption lines where each set is moving back and forth in time. This is interpreted as the result of two stars in orbit around one another. In analogy with the Sun/Earth system, the star with the larger orbital velocity is the less massive.

With a double-lined binary it is possible to measure masses of the stars to within a factor that depends on the inclination of the orbit with respect to the line of sight.

3. Double-lined Eclipsing Spectroscopic Binaries: These are a rare goldmine of binary systems.
Because the two stars eclipse each other, the orbital inclination of the system is known, and it is possible to solve for the mass of each member of the binary. The details of the light curves also give direct information about the details of the light distribution across the face of each star and on the sizes and temperature of each star.

### Mass and the HR Diagram

Measure the masses for as many stars as you can find in these systems and discover the Mass-Luminosity relation for main-sequence stars.

There is a pretty "steep" relationship between Mass and Luminosity in the sense that more massive stars are more luminous, thus

The Main-Sequence in the H-R Diagram is a Mass Sequence!

For each position along the main sequence there is a corresponding stellar mass: