# Luminosity

In physics, luminosity (more properly called luminance) is the density of luminous intensity in a given direction.

In astronomy, luminosity is the amount of energy a body radiates in unit time. It is typically expressed in the SI units watts, in the cgs units ergs per second, or in terms of solar luminosities, Ls; that is, how many times more energy the object radiates than the Sun, whose luminosity is 3.827×1026 W.

Luminosity is an intrinsic constant independent of distance, while in contrast apparent brightness observed is related to distance with an inverse square relationship. Brightness is usually measured by apparent magnitude, which is a logarithmic scale.

In measuring star brightnesses, luminosity, apparent magnitude (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 [itex]L[itex] 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 brightess.

[itex]b = \frac{L}{A}[itex] where [itex]A[itex] is the area of the illuminated surface. For stars and other point sources of light, [itex]A = 4\pi r^2[itex] so [itex]b = \frac{L}{4\pi r^2} \,[itex] where [itex]r[itex] is the distance from the observer to the light source.

It has been shown that the luminosity of a star [itex]L[itex] is also related to temperature [itex]T[itex] and radius [itex]R[itex] of the star by the equation [itex]L = 4\pi\R^2\sigma T^4[itex]

Dividing by the luminosity of the sun [itex]L_s[itex] and cancelling constants, we obtain the relationship

[itex]\frac{L}{L_s} = {\left ( \frac{R}{R_s} \right )}^2 {\left ( \frac{T}{T_s} \right )}^4[itex].

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

[itex]\frac{L}{L_s} \sim {\left ( \frac{M}{M_s} \right )}^{3.9}[itex]

It is easy to see that a star's luminosity, temperature, radius, and mass are all related.

The magnitude of a star is a logarithmic scale of observed brightness. The apparent magnitude is the observed brightness from Earth, and the absolute magnitude is the apparent magnitude at a distance of 10 parsecs. Given a luminosity, one can calculate the apparent magnitude of a star from a given distance:

[itex]m_{star}=m_{sun}-2.5\log_{10}\left({ L_{star} \over L_{sun} } \cdot \left({ {Dist}_{sun} \over Dist_{star} }\right)^2\right)[itex]

where

mstar is the apparent magnitude of the star, measured in

msun is the apparent magnitude of the reference sun, measured in

Lstar is solar luminosity of the star, measured in multiples of the Sun's luminosity

Lsun is solar luminosity of the reference sun, which can be taken as 1

Diststar is the distance to the star, measured in any units

Distsun is the distance to the reference sun, measured in the same units

Or simplified, given msun = −26.73, distsun = 1.58 × 10−5 lyr:

mstar = − 2.72 − 2.5 · log(Lstar/diststar2)

Example:

How bright would a star like the sun be from 4.3 light years away? (The distance to the next closest star Alpha Centauri)
msun (@4.3lyr) = −2.72 − 5 · log(1/4.3) = 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:

Lstar/Lsun = (diststar/distsun)2 · 10{(msun −mstar) · 0.4}
Lstar = 0.0813 · diststar2 · 10(−0.4 · mstar) · Lsun

Example:

What is the luminosity of the star Sirius?
Sirius is 8.6 lyr distant, and magnitude −1.47.
Lum(Sirius) = 0.0813 · 8.62 · 10−0.4·(−1.47) = 23.3 × Lumsun
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 Ls, whereas a dim star with bolometric magnitude +17 has luminosity of 10−5 Ls. 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.

## Hertzsprung-Russell diagram

The Hertzsprung-Russell diagram relates luminosity to color, stellar classification or surface temperature.

## SI light units

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