Absolute Magnitude
Clearly, a star that is very bright in our sky could be so primarily because
it is very close to us (the Sun, for example), or because it is rather distant
but is intrinsically very bright (Betelgeuse, for example). It is the "true"
brightness, with the distance dependence factored out, that is of most interest to
us as astronomers.
Therefore, it is useful to establish a convention that allows us to
compare two stars on the same footing, without variations in brightness due to
differing distances complicating the issue. Astronomers do this by defining the
absolute magnitude of a star:
Absolute Magnitude: the apparent magnitude
that a star would have if it were, in our imagination, placed at a distance of
10 parsecs or 32.6 light years from the Earth.

The Distance Modulus
From the definitions for absolute magnitude M
and apparent magnitude m, and some algebra,
m and M are related by the logarithmic equation
M = m  5 log [d(pc) / 10]
which permits us to calculate the absolute magnitude from the apparent
magnitude and the distance.
This equation can
be rewritten as
d(pc) = 10^{(m  M + 5) / 5}
The quantity m  M is called the distance modulus,
since it is a measure of how
distant the star is.


Determining Absolute Magnitudes
We can determine the absolute magnitude if we know
the distance to the star because the
inverse square intensity law can be used to
determine how its apparent brightness would change if we moved it from its true
position to a standard distance of 10 parsecs. Thus, the absolute magnitude, like the
luminosity, is a measure of the true brightness of the star.
There is nothing magic about the
standard distance of 10 parsecs. We could as well use any other distance as a
standard, but 10 parsecs is the distance astronomers have chosen.
A common convention, and one that we will mostly follow, is to use a lowercase
m to denote an apparent magnitude and an
uppercase M to denote an absolute
magnitude.
Absolute Magnitudes for Familiar Stars


Name 
Spectral Class 
Absolute Magnitude

Sun 
G2V 
+4.8

Proxima Centauri 
M5.5V 
+15.53

Betelgeuse 
M2Iab 
5.14

Sirius 
A1V 
+1.47

61 Cygni A 
K5V 
+7.5 



Some Absolute Magnitudes
The absolute magnitudes for some familiar stars are listed in the adjacent table
(spectral class will be defined later in this chapter).
At the standard distance of 10 pc our Sun would be a rather faint magnitude 4.8. (Recall that
magnitude 6 or 7 is the faintest the naked
eye can see under ideal conditions.)
Luminosity
Since both the absolute magnitude and the luminosity are measures of the true energy output of
a star, they should be related. In fact, they obey a logarithmic equation completely
analogous to the general equation that we wrote down earlier for magnitudes.
M_{2}  M_{1} =
2.5 log (L_{1} / L_{2})
which can also be written in the equivalent exponential form
L_{1} / L_{2} =
10^{0.4(M2 
M1)}
where "1" and "2" denote two different stars, M is an absolute magnitude,
and L is a luminosity. We often wish to express luminosities in units of the Sun's
luminosity. If we choose star 2 to be the Sun and use the Sun's absolute magnitude of
4.85, the preceding equation gives
L_{} / L_{sun} =
10^{0.4(4.85 
M)}
where M is the absolute magnitude and L is the luminosity of the star in
question. Given the absolute magnitude, we can use this equation to calculate the
luminosity of a star relative to that of the Sun. Therefore, to determine a luminosity we
may
first calculate the absolute magnitude from the apparent magnitude and distance, and then use
the absolute magnitude to calculate the luminosity.
The Most Luminous Stars
The following table lists the most luminous
stars known with some certainty.
Stars with the Highest Luminosities


Bayer Name 
Common Name 
m_{v} 
D (pc) 
Distance Error 
M_{v} 
Luminosity (Solar Units)

beta Ori 
Rigel 
0.18 
237 
19% 
 6.69 
41,000

gamma Cyg 
 
2.23 
467 
24% 
 6.12 
24,000

zeta Pup 
 
2.21 
429 
22% 
 5.95 
21,000

upsilon Car 
 
2.92 
498 
20% 
 5.56 
15,000

alpha Car 
Canopus 
0.62 
95.9 
5% 
 5.53 
14,000

beta Cen 
Agena 
0.61 
161 
9% 
 5.42 
13,000

alpha Lep 
Arneb 
2.58 
393 
28% 
 5.40 
13,000

phi Vel 
 
3.52 
592 
30% 
 5.34 
12,000

gamma Vel 
 
1.75 
258 
14% 
 5.31 
12,000

zeta Ori 
 
1.74 
251 
20% 
 5.26 
11,000

alpha Ori 
Betelgeuse 
0.45 
131 
22% 
 5.14 
10,000 


Based on parallax from Hipparcos/Tycho Catalog. Distance error is the percentage
uncertainty
in the parallax measurement. This translates into a larger error for the
luminosity. For example,
the quoted uncertainty in the distance to Rigel implies
an uncertainty of around 40 percent
for the inferred luminosity.

As noted in the footnote of this table, absolute magnitudes and luminosities reflect any
uncertainties in determining the distance to a star. Because of the dependence of brightness
on the inverse square of the distance,
a given percentage uncertainty in distance
translates into a much larger uncertainty in the luminosity. Therefore, the luminosities of
more distant stars are known only approximately from direct measurements.