Absolute magnitude

Absolute magnitude is the measure of intrinsic brightness of a celestial object. It is the hypothetical apparent magnitude of an object at a standard distance of exactly 10 parsecs (32.6 light years) from the observer, assuming no astronomical extinction of starlight. This places the objects on a common basis and allows the true energy output of astronomical objects to be compared without the distortion introduced by distance. As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength intervals; for stars the most commonly quoted absolute magnitude is the absolute visual magnitude, which uses only the visual (V) band of the spectrum (UBV system). Also commonly used is the absolute bolometric magnitude, which is the total luminosity expressed in magnitude units that takes into account energy radiated at all wavelengths, whether visible or not.

The brighter the celestial object the smaller its absolute magnitude. The magnitude scale extends downward through the positive numbers and into the negative numbers as brightness increases. A difference of 1.0 in absolute magnitude corresponds to a difference of 2.512 ≈ 10^0.4 of absolute brightness. Therefore a star of magnitude -2 is 100 (or 2.512⁵) times brighter than a star of magnitude 3. The Milky Way, for example, has an absolute magnitude of about −20.5, so a quasar with an absolute magnitude of −25.5 is 100 times brighter than the Milky Way. If this particular quasar and the Milky Way could be seen side by side at the same distance of one parsec and the Milky Way's stars reduced to a single point, the quasar would be 5 magnitudes (or 100 times) brighter than the Milky Way. Similarly, Canopus has an absolute visual magnitude of about −5.5, whereas Ross 248 has an absolute visual magnitude of +14.8, for a difference of about 20 magnitudes, i.e., Canopus would be seen as about 20 magnitudes brighter; stated another way, Canopus emits more than 100 million (10⁸) times more visual power than Ross 248.

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). Galaxies (and other extended objects) are much larger than 10 parsecs, their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, treating that integrated brightness as the brightness of a single point-like or star-like source, and computing the magnitude of that point-like source as it would appear if observed at the standard 10 parsecs distance. Consequently, the absolute magnitude of any object equals the apparent magnitude it would have if it were 10 parsecs away.

The measurement of absolute magnitude is made with an instrument called a bolometer. When using an 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 usually is 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, whereas very cool stars radiate mostly infrared radiation (see Planck's law).

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 at 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 brighter than the Sun, whose absolute visual magnitude is 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

If extinction by gas and dust is not significant, 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 actual distance in parsecs (1 parsec is 206,265 astronomical units, approximately 3.2616 light-years). For very large distances, the cosmological redshift complicates the relation between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a k correction might have to be applied to the magnitudes of the distant objects.

For nearby astronomical objects (such as stars in the Milky Way) luminosity distance D_L is almost identical to the real distance to the object, because spacetime within the Milky Way 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 $m$ and the star's parallax $p$ in arcseconds:

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

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

$M_V = 0.12 - 5 \cdot (\log_{10} \frac{860}{3.2616} - 1) = -7.0.$

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 m_V=+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 the Milky Way 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 the Milky Way) the luminosity distance D_L 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, and extinction by interstellar dust. In the case of stars with few observations, it usually must be computed assuming an effective temperature.

Classically, 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:

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

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.

In August 2015, the International Astronomical Union passed Resolution B2^[3] defining the zero points of the absolute and apparent bolometric magnitude scales in SI units for power (watts) and irradiance ($W/m^2$), respectively. Although bolometric magnitudes had been used by astronomers for many decades, there had been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in bolometric corrections scales, which when combined with incorrect assumed absolute bolometric magnitudes for the Sun could lead to systematic errors in estimated stellar luminosities (and stellar properties calculated which rely on stellar luminosity, like radii, ages, etc.).

IAU 2015 Resolution B2 defines an absolute bolometric magnitude scale where $M_{Bol} = 0$ corresponds to luminosity 3.0128×10²⁸ watts, with the zero point luminosity $L_{o}$ set such that the Sun (with nominal luminosity 3.828×10²⁶ watts) corresponds to absolute bolometric magnitude $M_{Bol_{\rm Sun}} = 4.74$. Placing a radiation source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale $m_{Bol} = 0$ corresponds to irradiance $f_{o}$ = 2.518021002×10⁻⁸ $W/m^2$. Using the IAU 2015 scale, the nominal total solar irradiance ("Solar constant") measured at 1 astronomical unit ($1361 W/m^{2}$) corresponds to an apparent bolometric magnitude of the Sun of $m_{Bol_{\rm Sun}} = -26.832$.

Following IAU 2015 Resolution B2 system, the relation between a star's absolute bolometric magnitude and its luminosity is no longer directly tied to the Sun's (variable) luminosity:

$M_{Bol} = -2.5 log \frac{L_{\rm star}}{L_{o}} = -2.5 log L_{\rm star} + 71.197425...$

where

$L_{\rm star}$ is the star's luminosity (bolometric luminosity) in watts

$L_{o}$ is the zero point luminosity 3.0128×10²⁸ watts

$M_{Bol}$ is the bolometric magnitude of the star

The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal solar luminosity corresponds closely to $M_{Bol}$ = 4.74, a value that was commonly adopted by astronomers before the 2015 IAU resolution.

The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude $M_{Bol}$ as:

$L_{\rm star} = L_{o} 10^{-0.4 M_{Bol}}$

using the variables as defined previously.

Solar System bodies (H)

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

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 2/3 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.^[4]

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)
  • $m_{Moon} = 0.25 + 2.5 \log_{10}{\left(\frac{3\pi}{2} 0.00257^2\right)} = -11.02\!\,$
  • (Actually 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.^[5][6]

See also

• Photographic magnitude
• Hertzsprung–Russell diagram – relates absolute magnitude or luminosity versus spectral color or surface temperature.
• Jansky radio astronomer's preferred unit – linear in power/unit area
• Surface brightness – the magnitude for extended objects
• List of most luminous stars

References

1. Cayrel de Strobel, G. (1996), "Stars resembling the Sun", Astronomy and Astrophysics Review 7 (3): 243–288, Bibcode:1996A&ARv...7..243C, doi:10.1007/s001590050006
2. Casagrande, L.; Portinari, L.; Flynn, C. (November 2006), "Accurate fundamental parameters for lower main-sequence stars", MNRAS (Abstract) 373 (1): 13–44, Bibcode:2006astro.ph..8504C, doi:10.1111/j.1365-2966.2006.10999.x
3. IAU XXIX General Assembly Draft Resolutions Announced, retrieved 2015-07-08
4. Planetary magnitudes Calculator – update, retrieved 2013-05-16
5. "Absolute magnitude of meteors", Glossary, International Meteor Organization, retrieved 2013-05-16
6. "Absolute magnitude of Solar System bodies", Solar System Dynamics Glossary, NASA Jet Propulsion Laboratory, retrieved 2013-05-16

External links

• Reference zero-magnitude fluxes
• International Astronomical Union
• Absolute Magnitude of a Star calculator
• The Magnitude system
• About stellar magnitudes
• Obtain the magnitude of any star – SIMBAD
• Converting magnitude of minor planets to diameter
• Another table for converting asteroid magnitude to estimated diameter