# Measuring the Stars Continued

### Spectroscopic Parallax

Spectroscopic parallax isn't exactly a parallax (angle) measurement, but it
does present a means of getting a distance estimate for a star that is too
far away for actual parallax measurement. You do it by examining the spectrum
of a star and determining its spectral type.
Looking at the Main Sequence band at that type
allows you to look to the left of the diagram and get an
estimate of the absolute magnitude. Since you always can find m (apparent
magnitude) and you have an estimate of M (absolute magnitude), the formula
will yield a distance. The Main Sequence does have some width (is not a thin
line), so the estimate of M, and therefore the distance, is somewhat uncertain
(has an error bar). That is, however, a LOT better than knowing nothing about
the distance.

### Finding Stellar Masses

There's only one way to determine the mass of a star directly - measure its
gravitational influence on something else. For the Sun it's easy; use
the orbits of its planets to find its gravity. For other stars that do
not have planets we can see it is a problem. The only thing we are likely
to be able to use in finding the star's gravitational influence is another
star, and that requires having two stars orbiting each other in a binary
star system. Fortunately, over half the stars in the Galaxy are in binary
systems, so there are a lot of opportunities.

The masses are determined by studying the orbit of the binary system. Remember
that the two stars orbit around the common center of mass (CM). Astronomers will
determine the following.

- Orbital period - by observation over time.
- Distance from Earth - by parallax or spectroscopic parallax.
- Size of orbit - by apparent size and measured distance or doppler measurement
of orbital velocity.
- Ratio of distances from CM - by measuring angle.

Kepler's 3rd law will give the sum of the two masses. The ratio of distances
from the CM will give the mass ratio. This gives two simple equations in
two unknowns, which can be quickly solved for the masses.
Once masses have been determined for a significant number of stars, we find
that mass and luminosity are related. See Figure 10.21 for an indication
of the Mass-luminosity relationship
(or use this graphic);
type O stars are the most massive and
type M stars (red dwarfs) are the least massive.

Also note that stellar radius and spectral type are related. The more massive
stars are the largest, while the least massive stars are smallest.

Be sure you understand this chapter. Use the questions at the end of the
chapter for practice.