29:50 Modern Astronomy
Fall 1999
Lecture 10 ...September 15, 1999
Stellar Brightness, Other Stellar Properties
Watch the skies! Watch the skies!
Planets in the morning sky. You can see Jupiter, Saturn, and Venus.
Mars is now in the constellation of Scorpius, not far from Antares.
Vega, Altair, and 
Ophiuchi form a triangle.
Absolute Magnitude and Stellar Distances
Last time defined the Absolute Magnitude as the Apparent Magnitude a star
would have if it were at a distance of 10 parsecs. It is a measure of the
intrinsic brightness of a star. The absolute magnitude of the Sun is +4.85.
Let see how the Sun stacks up against other stars.
Look at Appendix 12 of your textbook, a list of the brightest
stars. Stars with Absolute Magnitudes of 
are not uncommon. These are
about 10 magnitudes more luminous than the Sun. This equals a factor of 
in
luminosity!
This indicates clearly the range in stellar properties. Stars
span a huge range in their properties.
Why is the absolute magnitude of a star (or other types of objects)
important? Because it provides a way of determining the distance to an object, and
thus permits us to map out the universe.
The case of Alpha Ophiuchi. Alpha Ophiuchi is close to Vega on the night
sky, but is markedly fainter. Vega has a visual magnitude of 0.03, while
Ophiuchi has m=2.08, so it is almost exactly 2 magnitudes fainter, or a factor of
6.3 in brightness.
Nonetheless, if we take a spectrum of each of these two stars and compare them,
we find they are almost identical. They are so called ``A stars''.
Spectrum of A star.
The conclusion is that Alpha Ophiuchi is a clone of Vega, but is
further away. Let's get quantitative and determine how much further.
The inverse square law for light
As a source of light gets further away, it becomes fainter.
Illustration with demonstration
Physics tells us exactly how this happens. It happens via the inverse square law.
Let's let L be the luminosity of a source of light, which is just like the power output
of a lightbulb, and is expressed in Watts. The source of light is a distance r
away. b is the brightness that we measure (to be exact, the flux of radiation
in Watts/m 
). The brightness is then given by
If we consider the brightness of the same object at two distances 
and 
, we
then have 
Let's apply these ideas to the stars. Ophiuchi is a factor of
6.3 fainter than Vega, although of the same intrinsic brightness. This means that the
ratio of the distance to Ophiuchi to that of Vega is
So Ophiuchi is 2.51 times further away than Vega. Since the distance to Vega is
8 parsecs (See Appendix 12), this means that Ophiuchi is 20 parsecs away.
The difference between the apparent and absolute magnitude is similarly
a measure of the distance to an object. This difference m - M tells you how many
magnitudes fainter (or brighter) a star is relative to what it would be at a distance
of 10 parsecs. Using arguments similar to those above, we can find the ratio of the
true distance of a star to a distance of 10 parsecs.
In view of this, the difference between the apparent and absolute magnitude is called the distance modulus in astronomy. The formula for the relation between distance and distance modulus is
where d is the distance in parsecs.
This equation can be used to generate the following table.
Distance Modulus - Distance Relation
These equations are of extreme importance in contemporary astronomy. In the last year,
there have been articles on the front page of the New York Times describing the results
of the Hubble Space Telescope ``Key Project''. They have observed, for the first time,
a class of star called Cepheid variables, with absolute magnitudes of 
, in
galaxies of the ``Virgo Cluster'' at apparent magnitudes 
. You can figure
out for yourself what the significance of this observation is.
The Diversity of Stellar Properties
By this time we have gotten the idea that there is a good deal of variety in
the characteristics of stars. They vary greatly in temperature and luminosity. They
also vary greatly in mass. Can we see any patterns in this?
The observational approach to answering this question is to make a graph, in which we plot the absolute magnitude of a star versus its surface temperature. We could also plot it as luminosity (relative to the Sun) versus Spectral Type. The spectral classes of stars are discussed in your book and are O,B,A,F,G,K,M.
The result of this plot is one of the most important experimental finding of astronomy, and is referred to as the Hertzsprung-Russell diagram. It is shown in Figure 16-20 of your textbook.