## Distance Modulus

• The 'distance modulus' is the difference between the apparent magnitude and absolute magnitude of a celestial object (m - M), and provides a measure of the distance to the object, r.

 where m = apparent magnitude of the star M = absolute magnitude of the star, and r = distance to the star in parsecs

This table shows the apparent and absolute visual magnitudes of some stars and their distances:

StarmvMvd (pc)
Sun-26.84.83
Alpha Centauri-0.34.11.3
Canopus-0.72-3.130.1
Rigel0.14-7.1276.1
Deneb1.26-7.1490.8

We can derive the expression for distance modulus by using the relation between the flux ratio of two stars and their apparent magnitudes:

 where flux from stars 1 and 2 apparent magnitude of stars 1 and 2

Consider a star of luminosity L and apparent magnitude m, at a distance r. Now we apply the relation for the ratio of the flux we receive from the star, F, and the flux we would receive if the star was at a distance of 10 parsec, F10. Identifying m1 as the apparent magnitude of the star and m2 as the absolute magnitude, the last equation becomes:

 where m = apparent magnitude of the star M = absolute magnitude of the star F = flux we receive from the star, and F10 = flux we would receive if the star was at 10 parsecs

The flux and luminosity of a star are related by:

Substituting for F and F10, L cancels out (luminosity is an intrinsic property of the star and does not depend on the distance to the observer), and we have:

, with in parsecs

Rearranging gives the distance modulus:

, with in parsecs