# Apparent magnitudes and absolute magnitudes

The brightness of a star of a given luminosity L, radiated in all directions, falls off as one over the distance to the object squared:, that is b(D) is proportional to L / D2.

Objects of the same luminosity that are located at different distances from us will have different apparent magnitudes. We therefore need to define the absolute magnitude M as the apparent magnitude an object would have if it were at a certain distance which we shall arbitrarily adopt to be 10 pc.

Remember: A parsec is the distance at which a star would have a parallax of one second of arc:

1 parsec = 1 pc = 3.26 light years = 3.09 x 1018 cm = 206265 A.U.

The basic formula relating the apparent (m) and absolute (M} magnitudes then is

M = m + 5 - 5 log D

where D is the distance to the object in pc.

Consider that we already know that the Sun has m = -26.8, and it is located at 1 A.U. ( astronomical unit) from us.

1 A.U. = 1.5 x 1013 cm = 4.85 x 10-6 pc = semimajor axis of earth's orbit.

The sun has a luminosity of 1 solar luminosity Lsun = 3.9 x 1033 erg s-1. We can calculate the absolute magnitude of the Sun Msun by considering how much fainter the Sun would appear if it were located at 10 pc from us instead of 1 A.U. For the Sun:

Thus, the absolute magnitude of the sun is Msun = +4.77. Similarly, for other stars, a star of a certain absolute magnitude M, is more or less luminous than the sun according to:

M = +4.77 - 2.5 log (L / Lsun).