Absolute Magnitude

**Dominic Ford**, Editor

## Deep sky objects

\[ m = M + 5 \log d_\textrm{pc} - 5 \] Distance modulus \(\mu\) is defined as \[ \mu = 5 \log d_\textrm{pc} - 5 \] Distance \(d_\textrm{pc}\) is measured in parsecs.

## Asteroids

\[ m = g + 5 \log \Delta + \kappa \log d_\textrm{S} - 2.5 \log p \]
where
\[ p=\frac{1 + \cos\beta}{2}, \]
\( \beta \) is the phase angle – the Sun-Body-Earth angle, and

\( p \) is the fraction of the object's visible disk which is illuminated by the Sun.
\( \kappa = 2.5 n \)

In 1985, the International Astronomical Union's Commission 20 adopted a more flexible formula, of the form
\[ m = H + 5 \log d_\textrm{E} - 2.5 \log \left( (1-G)\Psi_1 + G\Psi_2 \right) \]
where:
\[ \Psi_1 = \exp \left[ -3.33 \left( \tan \frac{\beta}{2} \right)^{0.63} \right] \]
and
\[ \Psi_2 = \exp \left[ -1.87 \left( \tan \frac{\beta}{2} \right)^{1.22} \right] \]
the brightness of an asteroid is defined by its absolute magnitude \(H\), and its slope parameter \(G\).
All distances measured in astronomical units.
See chapter 33 (page 231) of *Astronomical Algorithms* (1991) by Jean Meeus.

## Comets

\[ m = g + 5 \log d_\textrm{E} + 2.5 n \log d_\textrm{S} - 2.5 \log p \]
where, as above,
\[ p=\frac{1 + \cos\beta}{2}, \]
\( \beta \) is the phase angle – the Sun-Body-Earth angle, and

\( p \) is the fraction of the object's visible disk which is illuminated by the Sun.