EXAM 1 EQUATION REVIEW
The last few chapters contained more than the usual number of equations,
so here is a review of the most important mathematical relations. You should
know how to use these equations, but they will all be printed
on the exam, so you don't need to memorize them.
Parsecs
1 parsec is defined as the distance at which 1 A.U. subtends an angle of
1 arc second, meaning that the parallax angle for this distance is 1 arc
second. In general, if p is the parallax angle in seconds for a star, the
distance to the star is given in parsecs by
d = 1/p .
1 parsec = 3.26 light-years.
Distance - Magnitude Relations
A number of equations appear in the relation between distances and magnitudes
which are worth remembering. Three of these are most important, and are
summarized here. It is a good idea to remember what they mean, because in
a multiple choice problem, often only one answer will be reasonable, so that
no calculation is actually needed. You just have to recognize which answer
is most reasonable.
There are just three equations:
First, there is the magnitude comparison for two stars of known
luminosity (intensity): for two stars of luminosity
L1 and L2, the magnitude difference will be
m1 - m2 = 2.5 log (L2 / L1) .
If star 2 is brigher, the log will be positive, so that star 1 has
the greater magnitude. Remember that small magnitudes mean bright stars.
The apparent magnitude (m) is how bright a star appears from Earth. The absolute
magnitude (M) is how bright the star would appear if it were 10 pc away.
If you know the distance d in parsecs, then the difference between the apparent and absolute magnitudes is
m - M = 5 log(d/10).
If you know the magnitudes and need the distance, the inverse
of this equation is
d = 10(1 + (m - M)/5)
These relations come from the inverse square law, which says that the
intensity of light from a star at distance d is proportional to 1/d2.
In multiple choice questions, the equation is often unnecessary
if you remember a few things:
-
m is bigger than M if the star is further than 10 pc.
- This is because the star appears dimmer at greater distances.
- A change in 5 magnitudes corresponds to a brightness change of a
factor of 100.
- Logarithms don't vary very rapidly: log(1) = 0, log(10) = 1, log(100) = 2,
etc.
Finally, there are some equations relating luminosity to properties of stars.
Luminosity-Size-Temperature Relation
If L, R, and T are the temperature of a star in units of the solar luminosity,
solar radius, and solar surface temperature, then Stefan-Boltzmann's Law gives
L = R2 T4.
Mass-Luminosity Relationship
If L and M are the luminosity and mass of a star in units of the solar
luminosity and mass, then
L = M3.5