#### 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 lower-case m to denote an apparent magnitude and an upper-case 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.

M2 - M1 = 2.5 log (L1 / L2)

which can also be written in the equivalent exponential form

L1 / L2 = 100.4(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 / Lsun = 100.4(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
mv D (pc) Distance
Error
Mv 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.