## Absolute Magnitudes

Thus far we have discussed only apparent magnitudes. They do not tell us anything about the true brightness of stars, since the distances differ. A quantity measuring the intrinsic brightness of a star is the absolute magnitude . It is defined as the apparent magnitude at a distance of 10 parsecs from the star (Fig. 4.7).

We shall now derive an equation which relates the apparent magnitude m, the absolute magnitude M and the distance r. Because the flux emanating from a star into a solid angle w has, at a distance r, spread over an area wr2, the flux density is inversely proportional to the distance squared. Therefore the ratio of the flux density at a distance r, F(r), to the flux density at a distance of 10 parsecs, F(10), is

Thus the difference of magnitudes at r and 10 pc, or the distance modulus m — M, is m — M = —2.5lg

5lg(

10 pc

Fig. 4.7. The flux density at a distance of 10 parsecs from the star defines its absolute magnitude

For historical reasons, this equation is almost always written as m — M = 5lgr — 5 ,

which is valid only if the distance is expressed in parsecs. (The logarithm of a dimensional quantity is, in fact, physically absurd.) Sometimes the distance is given in kiloparsecs or megaparsecs, which require different constant terms in (4.12). To avoid confusion, we highly recommend the form (4.11).

Absolute magnitudes are usually denoted by capital letters. Note, however, that the U, B and V magnitudes are apparent magnitudes. The corresponding absolute magnitudes are Mu, MB and MV.

The absolute bolometric magnitude can be expressed in terms of the luminosity. Let the total flux density at a distance r = 10 pc be F and let Fe be the equivalent quantity for the Sun. Since the luminosity is L = 4nr2 F, we get

The absolute bolometric magnitude Mbol = 0 corresponds to a luminosity L0 = 3.0 x 1028 W.

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