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.

$ \underbrace{m - M}_{\text{Distance Modulus}} = 5\log_{10} (\frac{r}{10}) $ 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:

Star mv Mv d (pc)
Sun -26.8 4.83
Alpha Centauri -0.3 4.1 1.3
Canopus -0.72 -3.1 30.1
Rigel 0.14 -7.1 276.1
Deneb 1.26 -7.1 490.8

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

$ \frac{F_1}{F_2} = 100^{(m_2 - m_1)/5} $ where $ F_{1, 2} = $ flux from stars 1 and 2
$ m_{1, 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:

$ \frac{F_{10}}{F} = 100^{(m - M)/5} $ 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:

$ F = \frac{L}{4\pi r^2} $

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:

$ (\frac{r}{10})^2 = 100^{(m - M)/5} $, with $ r $ in parsecs

Rearranging gives the distance modulus:

$ m - M = 5 \log_{10} (\frac {r}{10}) $, with $ r $ in parsecs


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