Absolute magnitude

Absolute magnitude is a measure of the luminosity of a celestial object, on a logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if were viewed from a distance of exactly 10 parsecs (32.6 light years), with no extinction (or dimming) of its light due to absorption by interstellar dust particles. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared on a magnitude scale. As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength ranges corresponding to specified filter bands or passbands; for stars a commonly quoted absolute magnitude is the absolute visual magnitude, which uses the visual (V) band of the spectrum (in the UBV photometric system). Absolute magnitudes are denoted by a capital M, with a subscript representing the filter band used for measurement, such as M_V for absolute magnitude in the V band.

The more luminous an object, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities, and a difference of n magnitudes in absolute magnitude corresponds to a luminosity ratio of 100^(n/5). For example, a star of absolute magnitude M_V=3 would be 100 times more luminous than a star of absolute magnitude M_V=8 as measured in the V filter band. The Sun has absolute magnitude M_V=+4.83.^[1] Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about −20.8.^[2]

An object's absolute bolometric magnitude represents its total luminosity over all wavelengths, rather than in a single filter band, as expressed on a logarithmic magnitude scale. To convert from an absolute magnitude in a specific filter band to absolute bolometric magnitude, a bolometric correction is applied.

For Solar System bodies that shine in reflected light, a different definition of absolute magnitude (H) is used, based on a standard reference distance of one astronomical unit.

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 milliarcseconds). 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 outshine the planets and be able to cast shadows if they were at 10 parsecs from the Earth. Examples include 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.^[3]^[4] 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).

Apparent magnitude

The Greek astronomer Hipparchus established a numerical scale to describe the brightness of each star appeared in the sky. The brightest stars in the sky were assigned an apparent magnitude $m = 1$ , and the dimmest stars visible to the naked eye are assigned $m = 6$ .^[5] The difference between them corresponds to a factor of 100 in brightness. For objects within the Milky Way, the absolute magnitude $M$ and apparent magnitude $m$ from any distance $d$ (in parsecs) is related by:

100^{\frac {m-M}{5}}={\frac {F_{10}}{F}}=\left({\frac {d}{10\;\mathrm {pc} }}\right)^{2},

where $F$ is the radiant flux measured at distance $d$ (in parsecs), $F 10$ the radiant flux measured at distance $d = 10 pc$ . The relation can be written in terms of logarithm:

M=m-5\left(\log _{10}d-1\right),

where the insignificance of extinction by gas and dust is assumed. Typical extinction rates within the galaxy are 1 to 2 magnitudes per kiloparsec, when dark clouds are taken into account.^[6]

For objects at very large distances (outside the Milky Way) the luminosity distance $d L$ must be used instead of $d$ (in parsecs), because the Euclidean approximation is invalid for distant objects and general relativity must be taken into account. Moreover, 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.

The absolute magnitude $M$ can also be approximated using apparent magnitude $m$ and stellar parallax $p$ :

M=m+5\left(\log _{10}p+1\right),

or using apparent magnitude $m$ and distance modulus $μ$ :

M=m-\mu

Examples

Rigel has a visual magnitude $m V$ of 0.12 and distance about 860 light-years

M_{\mathrm {V} }=0.12-5\left(\log _{10}{\frac {860}{3.2616}}-1\right)=-7.0.

Vega has a parallax $p$ of 0.129″, and an apparent magnitude $m V$ of 0.03

M_{\mathrm {V} }=0.03+5\left(\log _{10}{0.129}+1\right)=+0.6.

Alpha Centauri A has a parallax $p$ of 0.742″ and an apparent magnitude $m V$ of −0.01

M_{\mathrm {V} }=-0.01+5\left(\log _{10}{0.742}+1\right)=+4.3.

The Black Eye Galaxy has a visual magnitude $m V$ of 9.36 and a distance modulus $μ$ of 31.06

M_{\mathrm {V} }=9.36-31.06=-21.7.

Bolometric magnitude

The bolometric magnitude $M bol$ , takes into account electromagnetic radiation at all wavelengths. It includes those unobserved due to instrumental pass-band, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the luminosity of the stars. In the case of stars with few observations, it must be computed assuming an effective temperature.

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

M_{\mathrm {bol,\star } }-M_{\mathrm {bol,\odot } }=-2.5\log _{10}\left({\frac {L_{\star }}{L_{\odot }}}\right)

which makes by inversion:

{\frac {L_{\star }}{L_{\odot }}}=10^{0.4\left(M_{\mathrm {bol,\odot } }-M_{\mathrm {bol,\star } }\right)}

where

L ⊙

is the Sun's luminosity (bolometric luminosity)

L ★

is the star's luminosity (bolometric luminosity)

M bol, ⊙

is the bolometric magnitude of the Sun

M bol,★

is the bolometric magnitude of the star.

In August 2015, the International Astronomical Union passed Resolution B2^[7] defining the zero points of the absolute and apparent bolometric magnitude scales in SI units for power (watts) and irradiance (W/m²), 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, such as radii, ages, and so on).

Resolution B2 defines an absolute bolometric magnitude scale where $M bol = 0$ corresponds to luminosity $L 0 =$ 7028301280000000000♠3.0128×10²⁸ W, with the zero point luminosity $L 0$ set such that the Sun (with nominal luminosity 7026382800000000000♠3.828×10²⁶ W) corresponds to absolute bolometric magnitude $M bol, ⊙ =$ 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 $f 0 =$ 6992251802100199999♠2.518021002×10⁻⁸ W/m². Using the IAU 2015 scale, the nominal total solar irradiance ("solar constant") measured at 1 astronomical unit (7003136100000000000♠1361 W/m²) corresponds to an apparent bolometric magnitude of the Sun of $m bol, ⊙ =$ −26.832.

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

M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{\star }}{L_{0}}}=-2.5\log _{10}L_{\star }+71.197425...

where

L ★

is the star's luminosity (bolometric luminosity) in watts

L 0

is the zero point luminosity 3.0128×10²⁸ W

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_{\star }=L_{0}10^{-0.4M_{\mathrm {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 non-stellar 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, and in conditions of ideal solar opposition. In fact, one has to take into account that Solar System bodies are illuminated by the Sun, therefore the magnitude varies as a function of illumination conditions, described by the phase angle. This relationship is referred to as the phase curve. The absolute magnitude is defined for the ideal case of phase angle equal to zero.

To convert a stellar or galactic absolute magnitude into a planetary one, subtract 31.57. A comet's nuclear magnitude ( $M 2$ ) is a different scale and can not be used for a size comparison with an asteroid's ( $H$ ) magnitude.

Apparent magnitude

Diffuse reflection on sphere and flat disk

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

m=H+2.5\log _{10}{\left({\frac {d_{\mathrm {BS} }^{2}d_{\mathrm {BO} }^{2}}{p(\chi )d_{0}^{4}}}\right)}

where $d 0$ is 1 AU, $χ$ is the phase angle, the angle between the body-Sun and body–observer lines. By the law of cosines, we have:

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

$p (χ)$ 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.^[8]

Example

Moon:

$H Moon$ = +0.25
$d OS$ = $d BS$ = 1 AU
$d BO$ = 7008384500000000000♠3.845×10⁸ m = 7008384466527698999♠0.00257 AU

How bright is the Moon from Earth?

Full moon: $χ = 0$ , $p (χ) \approx 2 / 3$
$m_{\mathrm {Moon} }=0.25+2.5\log _{10}\left({\frac {3}{2}}\cdot 0.00257^{2}\right)=-12.26$

Actual value: −12.7. A full Moon reflects 30% more light than a perfect diffuse reflector predicts.
Quarter moon: $χ = 90° = π / 2$ , $p (χ) \approx 2 / 3π$ (if diffuse reflector)
$m_{\mathrm {Moon} }=0.25+2.5\log _{10}\left({\frac {3\pi }{2}}\cdot 0.00257^{2}\right)=-11.02$

Actual value: approximately −11. The diffuse reflector formula does well 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.^[9]^[10]

References

↑ "Sun Fact Sheet". NASA Goddard Space Flight Center. Retrieved 25 February 2017.
↑ Karachentsev, I. D.; et al. (2004). "A Catalog of Neighboring Galaxies". The Astronomical Journal. 127: 2031–2068. doi:10.1086/382905. Retrieved 27 February 2017.
↑ 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.
↑ 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. arXiv:astro-ph/0608504 . doi:10.1111/j.1365-2966.2006.10999.x.
↑ Carroll, Bradley W.; Ostlie, Dale A. An Introduction to Modern Astrophysics (2nd ed.). Pearson. p. 60. ISBN 0-321-44284-9.
↑ Unsöld, Albrecht; Baschek, Bodo (2013), The New Cosmos: An Introduction to Astronomy and Astrophysics (5 ed.), Springer Science & Business Media, p. 331, ISBN 3662043564
↑ "IAU XXIX General Assembly Draft Resolutions Announced". Retrieved 2015-07-08.
↑ "Planetary magnitudes Calculator – update". Archived from the original on April 19, 2012. Retrieved 2013-05-16. CS1 maint: Unfit url (link)
↑ "Glossary – Absolute magnitude of meteors". International Meteor Organization. Retrieved 2013-05-16.
↑ "Solar System Dynamics Glossary – Absolute magnitude of Solar System bodies". NASA Jet Propulsion Laboratory. Retrieved 2013-05-16.

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Absolute magnitude

Stars and galaxies (M)

Apparent magnitude

Examples

Bolometric magnitude

Solar System bodies (H)

Apparent magnitude

Example

Meteors

See also

References

External links

Stars and galaxies ( $M$ )

Solar System bodies ( $H$ )