Absolute magnitude

From Wikipedia, the free encyclopedia

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 it 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. Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about −20.8.

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.

Some stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to outshine the planets and 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. 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 appearing 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. The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude M and apparent magnitude m from any distance d (in parsecs) is related by:

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

where F is the radiant flux measured at distance d (in parsecs), F₁₀ the radiant flux measured at distance 10 pc. The relation can be written in terms of logarithm:

${\displaystyle M=m-5\left(\log _{10}(d_{\text{pc}})-1\right),}$

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.

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:

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

or using apparent magnitude m and distance modulus μ:

${\displaystyle M=m-\mu }$.

Examples

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

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

Vega has a parallax p of 0.129″, and an apparent magnitude m_V of 0.03

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

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

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

The Black Eye Galaxy has a visual magnitude m_V of 9.36 and a distance modulus μ of 31.06

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

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:

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

which makes by inversion:

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

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 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. 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 L₀ = 3.0128×10²⁸ W, with the zero point luminosity L₀ set such that the Sun (with nominal luminosity 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 f₀ = 2.518021002×10⁻⁸ W/m². Using the IAU 2015 scale, the nominal total solar irradiance ("solar constant") measured at 1 astronomical unit (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:

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

where

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

L₀ 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:

${\displaystyle L_{\star }=L_{0}10^{-0.4M_{\mathrm {bol} }}}$

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. The absolute magnitude for a planet, Mv, or an asteroid, 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.

Planet	Mv
Mercury	-0.613
Venus	-4.384
Earth	-3.99
Mars	-1.601
Jupiter	-9.395
Saturn	-8.914
Uranus	-7.110
Neptune	-7.00

To convert a stellar or a 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

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.

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

where d₀ 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:

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

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

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

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.

Example

Moon:

• H_Moon = +0.25
• d_OS = d_BS = 1 AU
• d_BO = 3.845×10⁸ m = 0.00257 AU

How bright is the Moon from Earth?

• Full moon: χ = 0, p(χ) ≈ 2/3

  ${\displaystyle m_{\mathrm {Moon} }=0.25+2.5\log _{10}\left({\frac {3}{2}}\cdot 0.00257^{2}\right)=-12.26}$

  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) ≈ 2/3π (if diffuse reflector)

  ${\displaystyle m_{\mathrm {Moon} }=0.25+2.5\log _{10}\left({\frac {3\pi }{2}}\cdot 0.00257^{2}\right)=-11.02}$

  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.

Luminosity

From Wikipedia, the free encyclopedia

In astronomy, luminosity is the total amount of energy emitted per unit of time by a star, galaxy, or other astronomical object. As a term for energy emitted per unit time, luminosity is synonymous with power. 

In SI units luminosity is measured in joules per second or watts. Values for luminosity are often given in the terms of the luminosity of the Sun, L_⊙. Luminosity can also be given in terms of the astronomical magnitude system: the absolute bolometric magnitude (M_bol) of an object is a logarithmic measure of its total energy emission rate, while absolute magnitude is a logarithmic measure of the luminosity within some specific wavelength range or filter band.

In contrast, the term brightness in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both the luminosity of the object and the distance between the object and observer, and also on any absorption of light along the path from object to observer. Apparent magnitude is a logarithmic measure of apparent brightness.

Measuring luminosity

Hertzsprung–Russell diagram identifying stellar luminosity as a function of temperature for many stars in our solar neighborhood.

In astronomy, luminosity is the amount of electromagnetic energy a body radiates per unit of time. When not qualified, the term "luminosity" means bolometric luminosity, which is measured either in the SI units, watts, or in terms of solar luminosities (L_☉). A bolometer is the instrument used to measure radiant energy over a wide band by absorption and measurement of heating. A star also radiates neutrinos, which carry off some energy (about 2% in the case of our Sun), contributing to the star's total luminosity. The IAU has defined a nominal solar luminosity of 3.828×10²⁶ W to promote publication of consistent and comparable values in units of the solar luminosity.

While bolometers do exist, they cannot be used to measure even the apparent brightness of a star because they are insufficiently sensitive across the electromagnetic spectrum and because most wavelengths do not reach the surface of the Earth. In practice bolometric magnitudes are measured by taking measurements at certain wavelengths and constructing a model of the total spectrum that is most likely to match those measurements. In some cases, the process of estimation is extreme, with luminosities being calculated when less than 1% of the energy output is observed, for example with a hot Wolf-Rayet star observed only in the infra-red. Bolometric luminosities can also be calculated using a bolometric correction to a luminosity in a particular passband.

The term luminosity is also used in relation to particular passbands such as a visual luminosity of K-band luminosity. These are not generally luminosities in the strict sense of an absolute measure of radiated power, but absolute magnitudes defined for a given filter in a photometric system. Several different photometric systems exist. Some such as the UBV or Johnson system are defined against photometric standard stars, while others such as the AB system are defined in terms of a spectral flux density.

Stellar luminosity

A star's luminosity can be determined from two stellar characteristics: size and effective temperature. The former is typically represented in terms of solar radii, R_⊙, while the latter is represented in kelvins, but in most cases neither can be measured directly. To determine a star's radius, two other metrics are needed: the star's angular diameter and its distance from Earth, often calculated using parallax. Both can be measured with great accuracy in certain cases, with cool supergiants often having large angular diameters, and some cool evolved stars having masers in their atmospheres that can be used to measure the parallax using VLBI. However, for most stars the angular diameter or parallax, or both, are far below our ability to measure with any certainty. Since the effective temperature is merely a number that represents the temperature of a black body that would reproduce the luminosity, it obviously cannot be measured directly, but it can be estimated from the spectrum.

An alternative way to measure stellar luminosity is to measure the star's apparent brightness and distance. A third component needed to derive the luminosity is the degree of interstellar extinction that is present, a condition that usually arises because of gas and dust present in the interstellar medium (ISM), the Earth's atmosphere, and circumstellar matter. Consequently, one of astronomy's central challenges in determining a star's luminosity is to derive accurate measurements for each of these components, without which an accurate luminosity figure remains elusive. Extinction can only be measured directly if the actual and observed luminosities are both known, but it can be estimated from the observed colour of a star, using models of the expected level of reddening from the interstellar medium.

In the current system of stellar classification, stars are grouped according to temperature, with the massive, very young and energetic Class O stars boasting temperatures in excess of 30,000 K while the less massive, typically older Class M stars exhibit temperatures less than 3,500 K. Because luminosity is proportional to temperature to the fourth power, the large variation in stellar temperatures produces an even vaster variation in stellar luminosity. Because the luminosity depends on a high power of the stellar mass, high mass luminous stars have much shorter lifetimes. The most luminous stars are always young stars, no more than a few million years for the most extreme. In the Hertzsprung–Russell diagram, the x-axis represents temperature or spectral type while the y-axis represents luminosity or magnitude. The vast majority of stars are found along the main sequence with blue Class O stars found at the top left of the chart while red Class M stars fall to the bottom right. Certain stars like Deneb and Betelgeuse are found above and to the right of the main sequence, more luminous or cooler than their equivalents on the main sequence. Increased luminosity at the same temperature, or alternatively cooler temperature at the same luminosity, indicates that these stars are larger than those on the main sequence and they are called giants or supergiants.

Blue and white supergiants are high luminosity stars somewhat cooler than the most luminous main sequence stars. A star like Deneb, for example, has a luminosity around 200,000 L_⊙, a spectral type of A2, and an effective temperature around 8,500 K, meaning it has a radius around 203 R_⊙. For comparison, the red supergiant Betelgeuse has a luminosity around 100,000 L_⊙, a spectral type of M2, and a temperature around 3,500 K, meaning its radius is about 1,000 R_⊙. Red supergiants are the largest type of star, but the most luminous are much smaller and hotter, with temperatures up to 50,000 K and more and luminosities of several million L_⊙, meaning their radii are just a few tens of R_⊙. An example is R136a1, over 50,000 K and shining at over 8,000,000 L_⊙ (mostly in the UV), it is only 35 R_⊙.

Radio luminosity

The luminosity of a radio source is measured in W Hz⁻¹, to avoid having to specify a bandwidth over which it is measured. The observed strength, or flux density, of a radio source is measured in Jansky where 1 Jy = 10⁻²⁶ W m⁻² Hz⁻¹.

For example, consider a 10W transmitter at a distance of 1 million metres, radiating over a bandwidth of 1 MHz. By the time that power has reached the observer, the power is spread over the surface of a sphere with area 4πr² or about 1.26×10¹³ m², so its flux density is 10 / 10⁶ / 1.26×10¹³ W m⁻² Hz⁻¹ = 10⁸ Jy.

More generally, for sources at cosmological distances, a k-correction must be made for the spectral index α of the source, and a relativistic correction must be made for the fact that the frequency scale in the emitted rest frame is different from that in the observer's rest frame. So the full expression for radio luminosity, assuming isotropic emission, is

${\displaystyle L_{\nu }={\frac {S_{\mathrm {obs} }4\pi {D_{L}}^{2}}{(1+z)^{1+\alpha }}}}$

where L_ν is the luminosity in W Hz⁻¹, S_obs is the observed flux density in W m⁻² Hz⁻¹, D_L is the luminosity distance in metres, z is the redshift, α is the spectral index (in the sense ${\displaystyle I\propto {\nu }^{\alpha }}$, and in radio astronomy, assuming thermal emission the spectral index is typically equal to 2.)

For example, consider a 1 Jy signal from a radio source at a redshift of 1, at a frequency of 1.4 GHz. Ned Wright's cosmology calculator calculates a luminosity distance for a redshift of 1 to be 6701 Mpc = 2×10²⁶ m giving a radio luminosity of 10⁻²⁶ × 4π(2×10²⁶)² / (1+1)⁽¹⁺²⁾ = 6×10²⁶ W Hz⁻¹.
To calculate the total radio power, this luminosity must be integrated over the bandwidth of the emission. A common assumption is to set the bandwidth to the observing frequency, which effectively assumes the power radiated has uniform intensity from zero frequency up to the observing frequency. In the case above, the total power is 4×10²⁷ × 1.4×10⁹ = 5.7×10³⁶ W. This is sometimes expressed in terms of the total (i.e. integrated over all wavelengths) luminosity of the Sun which is 3.86×10²⁶ W, giving a radio power of 1.5×10¹⁰ L_⊙.

Magnitude

Grouping	Range	Example	VMag
First magnitude	< 1.5	Vega	0.03
Second magnitude	1.5 to 2.5	Denebola	2.14
Third magnitude	2.5 to 3.5	Rastaban	2.79
Fourth magnitude	3.5 to 4.5	Sadalpheretz	3.96
Fifth magnitude	4.5 to 5.5	Pleione	5.05
Sixth magnitude	5.5 to 6.5	54 Piscium	5.88
Seventh magnitude	6.5 to 7.5	HD 40307	7.17
Eighth magnitude	7.5 to 8.5	HD 113766	7.56
Ninth magnitude	8.5 to 9.5	HD 149382	8.94
Tenth magnitude	9.5 to 10.5	HIP 13044	9.98

Luminosity is an intrinsic measurable property of a star independent of distance. The concept of magnitude, on the other hand, incorporates distance. First conceived by the Greek astronomer Hipparchus in the second century BC, the original concept of magnitude grouped stars into six discrete categories depending on how bright they appeared. The brightest first magnitude stars were twice as bright as the next brightest stars, which were second magnitude; second was twice as bright as third, third twice as bright as fourth and so on down to the faintest stars, which Hipparchus categorized as sixth magnitude. The system was but a simple delineation of stellar brightness into six distinct groups and made no allowance for the variations in brightness within a group. With the invention of the telescope at the beginning of the seventeenth century, researchers soon realized that there were subtle variations among stars and millions fainter than the sixth magnitude—hence the need for a more sophisticated system to describe a continuous range of values beyond what the naked eye could see.

In 1856 Norman Pogson, noticing that photometric measurements had established first magnitude stars as being about 100 times brighter than sixth magnitude stars, formalized the Hipparchus system by creating a logarithmic scale, with every interval of one magnitude equating to a variation in brightness of 100^1/5 or roughly 2.512 times. Consequently, a first magnitude star is about 2.5 times brighter than a second magnitude star, 2.5² brighter than a third magnitude star, 2.5³ brighter than a fourth magnitude star, et cetera. Based on this continuous scale, any star with a magnitude between 5.5 and 6.5 is now considered to be sixth magnitude, a star with a magnitude between 4.5 and 5.5 is fifth magnitude and so on. With this new mathematical rigor, a first magnitude star should then have a magnitude in the range 0.5 to 1.5, thus excluding the nine brightest stars with magnitudes lower than 0.5, as well as the four brightest with negative values. It is customary therefore to extend the definition of a first magnitude star to any star with a magnitude less than 0.5, as can be seen in accompanying table.

Artist impression of a transiting planet temporarily diminishing the star's brightness, leading to its discovery.

The Pogson logarithmic scale is used to measure both apparent and absolute magnitudes, the latter corresponding to the brightness of a star or other celestial body as seen if it would be located at an interstellar distance of 10 parsecs. The apparent magnitude is a measure of the diminishing flux of light as a result of distance according to the inverse-square law. In addition to this brightness decrease from increased distance, there is an extra decrease of brightness due to extinction from intervening interstellar dust.

By measuring the width of certain absorption lines in the stellar spectrum, it is often possible to assign a certain luminosity class to a star without knowing its distance. Thus a fair measure of its absolute magnitude can be determined without knowing its distance nor the interstellar extinction, allowing astronomers to estimate a star's distance and extinction without parallax calculations. Since the stellar parallax is usually too small to be measured for many distant stars, this is a common method of determining such distances.

To conceptualize the range of magnitudes in our own galaxy, the smallest star to be identified has about 8% of the Sun’s mass and glows feebly at absolute magnitude +19. Compared to the Sun, which has an absolute of +4.8, this faint star is 14 magnitudes or 400,000 times dimmer than our Sun. Our galaxy's most massive stars begin their lives with masses of roughly 100 times solar, radiating at upwards of absolute magnitude –8, over 160,000 times the solar luminosity. The total range of stellar luminosities, then, occupies a range of 27 magnitudes, or a factor of 60 billion.

In measuring star brightnesses, absolute magnitude, apparent magnitude, and distance are interrelated parameters—if two are known, the third can be determined. Since the Sun's luminosity is the standard, comparing these parameters with the Sun's apparent magnitude and distance is the easiest way to remember how to convert between them.

Luminosity formulae

Point source S is radiating light equally in all directions. The amount passing through an area A varies with the distance of the surface from the light.

The Stefan–Boltzmann equation applied to a black body gives the value for luminosity for a black body, an idealized object which is perfectly opaque and non-reflecting:

${\displaystyle L=\sigma AT^{4}}$ ,

where A is the area, T is the temperature (in Kelvins) and σ is the Stefan–Boltzmann constant, with a value of 5.670367(13)×10⁻⁸ W⋅m⁻²⋅K⁻⁴.

Imagine a point source of light of luminosity ${\displaystyle L}$ that radiates equally in all directions. A hollow sphere centered on the point would have its entire interior surface illuminated. As the radius increases, the surface area will also increase, and the constant luminosity has more surface area to illuminate, leading to a decrease in observed brightness.

${\displaystyle F={\frac {L}{A}}}$ ,

where

${\displaystyle A}$ is the area of the illuminated surface.

${\displaystyle F}$ is the flux density of the illuminated surface.

The surface area of a sphere with radius r is ${\displaystyle A=4\pi r^{2}}$, so for stars and other point sources of light:

${\displaystyle F={\frac {L}{4\pi r^{2}}}\,}$ ,

where ${\displaystyle r}$ is the distance from the observer to the light source.

It has been shown that the luminosity of a star ${\displaystyle L}$ (assuming the star is a black body, which is a good approximation) is also related to temperature ${\displaystyle T}$ and radius ${\displaystyle R}$ of the star by the equation:

${\displaystyle L=4\pi R^{2}\sigma T^{4}\,}$ ,

where σ is the Stefan–Boltzmann constant 5.67×10⁻⁸ W·m⁻²·K⁻⁴.

Dividing by the luminosity of the Sun ${\displaystyle L_{\odot }}$ and cancelling constants, we obtain the relationship:

${\displaystyle {\frac {L}{L_{\odot }}}={\left({\frac {R}{R_{\odot }}}\right)}^{2}{\left({\frac {T}{T_{\odot }}}\right)}^{4}}$ ,

where ${\displaystyle R_{\odot }}$ and ${\displaystyle T_{\odot }}$ are the radius and temperature of the Sun, respectively.

For stars on the main sequence, luminosity is also related to mass:

${\displaystyle {\frac {L}{L_{\odot }}}\approx {\left({\frac {M}{M_{\odot }}}\right)}^{3.5}}$ .

Magnitude formulae

The magnitude of a star, a unitless measure, is a logarithmic scale of observed visible brightness. The apparent magnitude is the observed visible brightness from Earth which depends on the distance of the object. The absolute magnitude is the apparent magnitude at a distance of 10 parsecs, therefore the bolometric absolute magnitude is a logarithmic measure of the bolometric luminosity.
The difference in bolometric magnitude between two objects is related to their luminosity ratio according to:

${\displaystyle M_{\text{bol1}}-M_{\text{bol2}}=-2.5\log _{10}{\frac {L_{\text{1}}}{L_{\text{2}}}}}$

where:

${\displaystyle M_{\text{bol1}}}$ is the bolometric magnitude of the first object

${\displaystyle M_{\text{bol2}}}$ is the bolometric magnitude of the second object.

${\displaystyle L_{\text{1}}}$ is the first object's bolometric luminosity

${\displaystyle L_{\text{2}}}$ is the second object's bolometric luminosity

This can be used to derive a luminosity in solar units:

${\displaystyle M_{\text{*}}-M_{\odot }=-2.5\log _{10}{\frac {L_{*}}{L_{\odot }}}}$

which makes by inversion:

${\displaystyle {\frac {L_{*}}{L_{\odot }}}=10^{(M_{\odot }-M_{*})/2.5}}$

where

${\displaystyle L_{*}}$ is the star's bolometric luminosity

${\displaystyle L_{\odot }}$ is the Sun's bolometric luminosity

${\displaystyle M_{*}}$ is the bolometric magnitude of the star.

${\displaystyle M_{\odot }}$ is the bolometric magnitude of the Sun (approximately 4.7554).

Although the absolute bolometric magnitude of the sun is approximately 4.7554, the zero point of the absolute magnitude scale is actually defined as a fixed luminosity of 3.0128×10²⁸ W. Therefore, the absolute magnitude can be calculated from a luminosity in watts:

${\displaystyle M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{*}}{L_{0}}}\approx -2.5\log _{10}L_{*}+71.1974}$

where L₀ is the zero point luminosity 3.0128×10²⁸ W and the luminosity in watts can be calculated from an absolute magnitude (although absolute magnitudes are often not measured relative to an absolute flux):

${\displaystyle L_{*}=L_{0}\times 10^{-0.4M_{\mathrm {bol} }}}$