Luminosity

Luminosity is a measurement of brightness.

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In astronomy

In astronomy, luminosity is the amount of electromagnetic energy a body radiates per unit of time.

The luminosity of stars is measured in two forms: apparent (counting visible light only) and bolometric (total radiant energy); a bolometer is an instrument that measures radiant energy over a wide band by absorption and measurement of heating. When not qualified, luminosity means bolometric luminosity, which is measured in the SI units watts, or in terms of solar luminosities, $L_{\odot }$; that is, how many times as much energy the object radiates than the Sun, whose luminosity is 3.846×10²⁶ W.

Luminosity is an intrinsic measurable property independent of distance, and is appraised as absolute magnitude, corresponding to the apparent luminosity in visible light of a star as seen at the interstellar distance of 10 parsecs, or bolometric magnitude corresponding to bolometric luminosity. In contrast, apparent brightness is related to the distance by an inverse square law. Onto this brightness decrease from increased distance comes an extra linear decrease of brightness for interstellar "extinction" from intervening interstellar dust. Visible brightness is usually measured by apparent magnitude. Both absolute and apparent magnitudes are on an inverse logarithmic scale, where 5 magnitudes increase counterparts a 100:th part decrease in nonlogaritmic luminosity.

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, and instead the distance and extinction can be determined without measuring it directly through the yearly parallax. Since the parallax is usually too small to be measured for many faraway stars, this is a common method of determining distances.

In measuring star brightnesses, visible luminosity (not total luminosity at all wave lengths), apparent magnitude (visible brightness), and distance are interrelated parameters. If you know two, you can determine the third. 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.

Computing between brightness and luminosity

Imagine a point source of light of luminosity $L$ 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.

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

where

$A$ is the area of the illuminated surface.

$F$ is the Flux of the illuminated surface.

For stars and other point sources of light, $A=4\pi r^{2}$ so

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

where

$r$ is the distance from the observer to the light source.

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

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

where

σ is the Stefan-Boltzmann constant 5.67×10⁻⁸ W·m^-2·K^-4

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

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

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

${\frac {L}{L_{\odot }}}\sim {\left({\frac {M}{M_{\odot }}}\right)}^{3.9}$

The magnitude of a star is a logarithmic scale of observed visible brightness. The apparent magnitude is the observed visible brightness from Earth, and the absolute magnitude is the apparent magnitude at a distance of 10 parsecs. Given a visible luminosity (not total luminosity), one can calculate the apparent magnitude of a star from a given distance:

$m_{\rm {star}}=m_{\rm {sun}}-2.5\log _{10}\left({L_{\rm {star}} \over L_{\odot }}\cdot \left({\frac {r_{\rm {sun}}}{r_{\rm {star}}}}\right)^{2}\right)$

where

m_star is the apparent magnitude of the star (a pure number)

m_sun is the apparent magnitude of the sun (also a pure number)

L_star is the visible luminosity of the star

$L_{\odot }$ is the solar visible luminosity

r_star is the distance to the star

r_sun is the distance to the sun

Or simplified, given m_sun = −26.73, dist_sun = 1.58 × 10⁻⁵ lyr:

m_star = − 2.72 − 2.5 · log(L_star/dist_star²)

Example:

How bright would a star like the sun be from 4.3 light years away? (The distance to the next closest star system Alpha Centauri)

m_sun (@4.3lyr) = −2.72 − 2.5 · log(1/4.3²) = 0.45

0.45 magnitude would be a very bright star, but not quite as bright as Alpha Centauri.

Also you can calculate the luminosity given a distance and apparent magnitude:

L_star/$L_{\odot }$ = (dist_star/dist_sun)² · 10^{[(m_sun −m_star) · 0.4]}

L_star = 0.0813 · dist_star² · 10^{(−0.4 · m_star)} · $L_{\odot }$

Example:

What is the luminosity of the star Sirius?

Sirius is 8.6 lyr distant, and magnitude −1.47.

L_Sirius = 0.0813 · 8.6² · 10^{−0.4·(−1.47)} = 23.3 × $L_{\odot }$

You can say that Sirius is 23 times brighter than the sun, or it radiates 23 suns.

A bright star with bolometric magnitude −10 has a luminosity of 10⁶ $L_{\odot }$, whereas a dim star with bolometric magnitude +17 has luminosity of 10⁻⁵ $L_{\odot }$. Note that absolute magnitude is directly related to luminosity, but apparent magnitude is also a function of distance. Since only apparent magnitude can be measured observationally, an estimate of distance is required to determine the luminosity of an object.

Computing between luminosity and magnitude

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

$M_{1}-M_{2}=-2.5\log _{10}{\frac {L_{1}}{L_{2}}}$

which makes by inversion:

${\frac {L_{1}}{L_{2}}}=10^{(M_{2}-M_{1})/2.5}.$

In scattering theory and accelerator physics

In scattering theory and accelerator physics, luminosity is the number of particles per unit area per unit time times the opacity of the target, usually expressed in either the cgs units cm^-2 s^-1 or b^{-1 s-1. The integrated luminosity is the integral of the luminosity with respect to time. The luminosity is an important value to characterize the performance of an accelerator.}

Elementary relations for luminosity

The following relations hold

$L=\rho v\,$ (if the target is perfectly opaque)

${\frac {dN}{dt}}=L\sigma$

$N=\sigma \int L\ dt$

${\frac {d\sigma }{d\Omega }}={\frac {1}{L}}{\frac {d^{2}N}{d\Omega \,dt}}$

where

$L\$ is the (instantaneous) luminosity.

$\int L\ dt$ is the integrated luminosity.

$N\$ is the number of interactions.

$\rho \$ is the number density of a particle beam.

$\sigma$ is the total cross section.

$d\Omega$ is the differential solid angle.

${\frac {d\sigma }{d\Omega }}$ is the differential cross section.

For an intersecting storage ring collider:

$L=fn{\frac {N_{1}N_{2}}{A}}$

where

$f$ is the revolution frequency

$n$ is the number of bunches in one beam in the storage ring.

$N_{i}$ is the number of particles in each bunch

$A$ is the cross section of the beam.