# Absolute magnitude

Absolute magnitude (also known as absolute visual magnitude when measured in the standard V photometric band) is the measure of a celestial object's intrinsic brightness. It is the apparent magnitude an object would have if it were at a standard luminosity distance (10 parsecs, or 32.6 light years) away from the observer, in the absence of astronomical extinction. It allows the true brightnesses of objects to be compared without regard to distance. Bolometric magnitude is luminosity expressed in magnitude units; it takes into account energy radiated at all wavelengths, whether observed or not.

The absolute magnitude uses the same convention as the visual magnitude: a factor of 100.4 (≈2.512) ratio of brightness corresponds to a difference of 1.0 in magnitude. The Milky Way, for example, has an absolute magnitude of about −20.5. So a quasar at an absolute magnitude of −25.5 is 100 times brighter than our galaxy (because (100.4)(−20.5-(−25.5)) = (100.4)5 = 100). If this particular quasar and our galaxy could be seen side by side at the same distance, the quasar would be 5 magnitudes (or 100 times) brighter than our galaxy.

## Stars and galaxies (M)

In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light years, 308.57 Petameters or 308.57 trillion kilometres). A star at 10 parsecs has a parallax of 0.1" (100 milli arc seconds). For galaxies (which are of course themselves much larger than 10 parsecs, and whose overall brightness cannot be directly observed from relatively short distances) the absolute magnitude is defined by reference to the apparent brightness of a point-like or star-like source of the same total luminosity as the galaxy, as it would appear if observed at the standard 10 parsecs distance. In other words, the absolute magnitude of any object equals the apparent magnitude it would have if it was 10 parsecs away.

In defining absolute magnitude one must specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The bolometric magnitude can be computed from the visual magnitude plus a bolometric correction, $M_{bol}=M_V+BC$. This correction is needed because very hot stars radiate mostly ultraviolet radiation, while very cool stars radiate mostly infrared radiation (see Planck's law). The dimmer an object (at a distance of 10 parsecs) would appear, the higher (more positive) its absolute magnitude becomes. The lower (more negative) an object's absolute magnitude, the higher its luminosity.

Many stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to cast shadows if they were only 10 parsecs from the Earth: Rigel (−7.0), Deneb (−7.2), Naos (−6.0), and Betelgeuse (−5.6). For comparison, Sirius has an absolute magnitude of 1.4 which is greater than the Sun's absolute visual magnitude of 4.83 (it actually serves as a reference point). The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.[1] [2] Absolute magnitudes of stars generally range from −10 to +17. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant elliptical galaxy M87 has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10).

### Computation

One can compute the absolute magnitude $M\!\,$ of an object given its apparent magnitude $m\!\,$ and luminosity distance $D_L\!\,$:

$M = m - 5 ((\log_{10}{D_L}) - 1)\!\,$

where $D_L\!\,$ is the star's luminosity distance in parsecs, wherein 1 parsec is approximately 3.2616 light-years. For very large distances, cosmological redshift complicates the relation between absolute and apparent magnitude, and an additional k correction might be required.

For nearby astronomical objects (such as stars in our galaxy) luminosity distance DL is almost identical to the real distance to the object, because spacetime within our galaxy is almost Euclidean. For much more distant objects the Euclidean approximation is not valid, and General Relativity must be taken into account when calculating the luminosity distance of an object.

In the Euclidean approximation for nearby objects, the absolute magnitude $M\!\,$ of a star can be calculated from its apparent magnitude and parallax:

$M = m + 5 (1 + \log_{10}{p})\!\,$

where p is the star's parallax in arcseconds.

You can also compute the absolute magnitude $M\!\,$ of an object given its apparent magnitude $m\!\,$ and distance modulus $\mu\!\,$:

$M = m - {\mu}.\!\,$

#### Examples

Rigel has a visual magnitude of $m_V = 0.12$ and distance about 860 light-years

$M_V = 0.12 - 5 \cdot (\log_{10} \frac{860}{3.2616} - 1) = -7.02.$

Vega has a parallax of 0.129", and an apparent magnitude of +0.03

$M_V = 0.03 + 5 \cdot (1 +\log_{10}{0.129}) = +0.6.$

Alpha Centauri A has a parallax of 0.742" and an apparent magnitude of −0.01

$M_V = -0.01 + 5 \cdot (1 +\log_{10}{0.742}) = +4.3.$

The Black Eye Galaxy has a visual magnitude of mV=+9.36 and a distance modulus of 31.06.

$M_V = 9.36 - 31.06 = -21.7.$

### Apparent magnitude

Given the absolute magnitude $M\!\,$, for objects within our galaxy you can also calculate the apparent magnitude $m\!\,$ from any distance $d\!\,$ (in parsecs):

$m = M - 5 (1-\log_{10}{d}).\!\,$

For objects at very great distances (outside our galaxy) the luminosity distance DL must be used instead of d (in parsecs).

Given the absolute magnitude $M\!\,$, you can also compute apparent magnitude $m\!\,$ from its parallax $p\!\,$:

$m = M - 5 ( 1+ \log_{10}p).\!\,$

Also calculating absolute magnitude $M\!\,$ from distance modulus $\mu\!\,$:

$m = M + {\mu}.\!\,$

### Bolometric magnitude

Bolometric magnitude corresponds to luminosity, expressed in magnitude units; that is, after taking into account all electromagnetic wavelengths, including those unobserved due to instrumental pass-band, the Earth's atmospheric absorption, or extinction by interstellar dust. For stars, in the absence of extensive observations at many wavelengths, it usually must be computed assuming an effective temperature.

The difference in bolometric magnitude is related to the luminosity ratio according to:

$M_{bol_{\rm star}} - M_{bol_{\rm Sun}} = -2.5 \log_{10} {\frac{L_{\rm star}}{L_{\odot}}}$

which makes by inversion:

$\frac{L_{\rm star}}{L_{\odot}} = 10^{((Mbol_{\rm Sun} - Mbol_{\rm star})/2.5)}$

where

$L_{\odot}$ is the Sun's (sol) luminosity (bolometric luminosity)
$L_{\rm star}$ is the star's luminosity (bolometric luminosity)
$M_{bol_{\rm Sun}}$ is the bolometric magnitude of the Sun
$M_{bol_{\rm star}}$ is the bolometric magnitude of the star.

## Solar System bodies (H)

For planets and asteroids a different definition of absolute magnitude is used which is more meaningful for nonstellar objects.

In this case, the absolute magnitude (H) is defined as the apparent magnitude that the object would have if it were one astronomical unit (AU) from both the Sun and the observer. Because the object is illuminated by the Sun, absolute magnitude is a function of phase angle and this relationship is referred to as the phase curve.

To convert a stellar or galactic absolute magnitude into a planetary one, subtract 31.57. A comet's nuclear magnitude (M2) is a different scale and can not be used for a size comparison with an asteroid's (H) magnitude.

### Apparent magnitude

The absolute magnitude can be used to help calculate the apparent magnitude of a body under different conditions.

$m = H + 2.5 \log_{10}{\left(\frac{d_{BS}^2 d_{BO}^2}{p(\chi) d_0^4}\right)}\!\,$

where $d_0\!\,$ is 1 au, $\chi\!\,$ is the phase angle, the angle between the Sun–Body and Body–Observer lines. By the law of cosines, we have:

$\cos{\chi} = \frac{ d_{BO}^2 + d_{BS}^2 - d_{OS}^2 } {2 d_{BO} d_{BS}}.\!\,$

$p(\chi)\!\,$ is the phase integral (integration of reflected light; a number in the 0 to 1 range).

Example: Ideal diffuse reflecting sphere. A reasonable first approximation for planetary bodies

$p(\chi) = \frac{2}{3} \left( \left(1 - \frac{\chi}{\pi}\right) \cos{\chi} + \frac{1}{\pi} \sin{\chi} \right).\!\,$

A full-phase diffuse sphere reflects ⅔ as much light as a diffuse disc of the same diameter.

Distances:

• $d_{BO}\!\,$ is the distance between the observer and the body
• $d_{BS}\!\,$ is the distance between the Sun and the body
• $d_{OS}\!\,$ is the distance between the observer and the Sun

Note: because Solar System bodies are never perfect diffuse reflectors, astronomers use empirically derived relationships to predict apparent magnitudes when accuracy is required.[3]

#### Example

Moon:

• $H_{Moon}\!\,$ = +0.25
• $d_{OS}\!\,$ = $d_{BS}\!\,$ = 1 au
• $d_{BO}\!\,$ = 384.5 Mm = 2.57 mau

How bright is the Moon from Earth?

• Full Moon: $\chi\!\,$ = 0, ($p(\chi)\!\,$ ≈ 2/3)
• $m_{Moon} = 0.25 + 2.5 \log_{10}{\left(\frac{3}{2} 0.00257^2\right)} = -12.26\!\,$
• (Actual −12.7) A full Moon reflects 30% more light at full phase than a perfect diffuse reflector predicts.
• Quarter Moon: $\chi\!\,$ = 90°, $p(\chi) \approx \frac{2}{3\pi}\!\,$ (if diffuse reflector)
• $m_{Moon} = 0.25 + 2.5 \log_{10}{\left(\frac{3\pi}{2} 0.00257^2\right)} = -11.02\!\,$
• (Actual approximately −11.0) The diffuse reflector formula does for smaller phases.

## Meteors

For a meteor, the standard distance for measurement of magnitudes is at an altitude of 100 km (62 mi) at the observer's zenith.[4][5]