Absolute Magnitude

by 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.

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