Absolute Magnitude

**Dominic Ford**, Editor

An astronomical object's brightness depends on both its intrinsic light output and its distance from the Earth. This means that an object's apparent brightness is not a good measure of its intrinsic luminosity.

For example, two objects which appear with similar brightness might have very different intrinsic luminosities if they lie at very different distances. Distant galaxies, meanwhile, appear much fainter than nearby stars, despite shining with the light of hundreds of billions of stars.

The *absolute magnitude* of an astronomical object is a measure of its
intrinsic light output, independent of its distance. The definition of
absolute magnitude differs for asteroids, comets, and deep sky objects which
lie beyond the solar system.

## Deep sky objects

For deep sky objects – i.e. stars, galaxies, star clusters, and any other nebulae that lie outside the solar system – absolute magnitude is defined as the brightness the object would have if it lay at a distance of exactly 10 parsecs from the Earth. The absolute magnitude \( M \) of an object with apparent magnitude \( m \), lying at a distance of \( d_\textrm{pc} \) parsecs from the Earth, can be calculated as

\[ M = m - 5 \log d_\textrm{pc} + 5. \]
The difference between the object's absolute magnitude
\( M \)
and its apparent magnitude
\( m \)
is called the object's *distance modulus*, which is
represented by the Greek letter mu
(\( \mu \))

## Asteroids

The apparent brightness of solar system objects varies in a more complicated way because they reflect the Sun's light rather than producing their own.

This means that such objects change in brightness over time as a result of both their changing distances from the Earth and their changing distances from the Sun.

To a crude approximation, the brightness of an asteroid decreases with the inverse square of its distance from the Earth, and also with the inverse square of its distance from the Sun:

\[ \textrm{Light flux} \propto \left( \frac{1}{d_\textrm{E}^2} \right) \left( \frac{1}{d_\textrm{S}^2} \right) \]where

\( d_\textrm{E} \)
is the distance of the object from the Earth, and

\( d_\textrm{S} \)
is the distance of the object from the Sun,

both measured in
astronomical units.

However, some objects brighten more rapidly than this when they approach the Sun. For example, comets contain ice which vapourises on heating, producing a spherical halo and gas and dust around the nucleus, greatly increasing the object's brightness.

To account for this, the absolute magnitude of an asteroid can be defined as shown below. Here, the absolute magnitude is denoted by the letter \( g \), to distinguish it from the very different definition of absolute magnitude \( M \) used for deep sky objects:

\[ m = g + 5 \log d_\textrm{E} + \kappa \log d_\textrm{S} - 2.5 \log p. \]\( p \) is the fraction of the object's visible disk which is illuminated by the Sun, which can be calculated as

\[ p=\frac{1 + \cos\beta}{2}, \]where \( \beta \) is the phase angle – i.e. the Sun-Body-Earth angle.

The quantity
\( \kappa \)
is called the *curvature parameter*. This
defines how rapidly the object brightens as it approaches the Sun. For a solid
lump of rock which does not change when heated by the Sun,
\( \kappa=5 \).

Sometimes – most usually for comets – the term \( \kappa \) is replaced by \( 2.5 n \). In this version of the formula, \( n=2 \) for a lump of rock with unchanging size and albedo.

In 1985, the International Astronomical Union's Commission 20 adopted a more complicated, but also 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

\[ \Psi_2 = \exp \left[ -1.87 \left( \tan \frac{\beta}{2} \right)^{1.22} \right]. \]In this parameterisation, the absolute magnitude of the asteroid is denoted \( H \) and its slope parameter is denoted \( G \). As before, all distances are measured in astronomical units.

The advantage of this definition is that it accounts for the way in which rocky bodies do not reflect light uniformly in all directions, but tend to scatter it preferentially backwards in the direction that it came.

For more information, see chapter 33 (page 231) of
*Astronomical Algorithms* (1991) by Jean Meeus.

## Comets

The definition of absolute magnitude used for comets is similar to the first definition given for asteroids above, in which the absolute magnitude is denoted \( g \). The curvature parameter is denoted \( n \):

\[ m = g + 5 \log d_\textrm{E} + 2.5 n \log d_\textrm{S} - 2.5 \log p \]As above, \( p \) is the fraction of the object's visible disk which is illuminated by the Sun, which can be calculated as

\[ p=\frac{1 + \cos\beta}{2}, \]where \( \beta \) is the phase angle – i.e. the Sun-Body-Earth angle.