Luminosity

Encyclopedia : L : LU : LUM : Luminosity

Luminosity has different meanings in several different fields of science.

Contents

1 In photometry
2 In astronomy

2.1 Computing between brightness and luminosity

3 In scattering theory and accelerator physics

3.1 Elementary relations for luminosity

In photometry

In photometry, "luminosity" is sometimes incorrectly used for luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.

In astronomy

In astronomy, luminosity is the amount of energy a body radiates per unit time. It is typically expressed in the SI units watts, in the cgs units ergs per second, or in terms of solar luminosities, [L_]; 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 [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 brightess.

[b = \frac] where [A] is the area of the illuminated surface. For stars and other point sources of light, [A = 4\pi r^2] so [b = \frac \,] 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 an 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_] and cancelling constants, we obtain the relationship

[\frac} = } \right )}^2 } \right )}^4].

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

[\frac} \sim } \right )}^]

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:

[m_=m_-2.5\log_\left( \over L_ } \cdot \left(\frac } }\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 luminosity of the star

[L_] is the solar 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 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_] = (dist_star/dist_sun)² · 10

L_star = 0.0813 · dist_star² · 10^{(−0.4 · m_star)} · [L_]

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 × [L_]

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_], whereas a dim star with bolometric magnitude +17 has luminosity of 10⁻⁵ [L_]. 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.

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
[L] is the Luminosity.
[N] is the number of interactions.
[\rho] is the number density of a particle beam, e.g. within a bunch.
[\sigma] is the total cross section.
[d\Omega] is the differential solid angle.
[\left(\frac\right)] is the differential cross section.
Then the following relation holds:

[L = \rho v] (if the target is perfectly opaque)
[\frac = L \sigma]
[\left(\frac\right) = \frac \fracN}]

For an intersecting storage ring collider:

[f] is the revolution frequency
[n] is the number of bunches in one beam in the storage ring.
[N_] is the number of particles in each beam
[A] is the cross section of the beam.
[L = f n \frac N_}]

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