Luminosity

Luminosity is a measurement of brightness.

In photometry and color imaging

Main article: Luminance

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

The luminosity function a.k.a. luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. There are two luminosity functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies.

In graphics and video

Main articles: Luma (video) and HSL color space

In Adobe Photoshop's imaging operations, luminosity is the term used incorrectly to refer to the luma component of a color image signal; that is, a weighted sum of the nonlinear red, green, and blue signals. It seems to be calculated with the Rec. 601 luma co-efficients (Rec. 601: Luma (Y’) = 0.299 R’ + 0.587 G’ + 0.114 B’).

The "L" in HSL color space is sometimes said incorrectly to stand for luminosity. "L" in this case is calculated as 1/2 (MAX + MIN), where MAX and MIN refer to the highest and lowest of the R'G'B' components to be converted into HSL color space.

In astronomy

In astronomy, luminosity is the amount of electromagnetic energy a body radiates per unit of time. The word "luminosity" may also refer to spectral luminosity, measured either in W/Hz or W/nm.

The luminosity of stars is measured in two forms: apparent (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 either in the SI units, watts; or in terms of solar luminosities, $L_{\odot}$, that is, how many times as much energy the object radiates as 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. In addition to this brightness decrease from increased distance there is 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 100th part decrease in nonlogarithmic 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 stellar parallax is usually too small to be measured for many far away 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

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.

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 density of the illuminated surface.

The surface area of a sphere with radius r is A = 4πr², so for stars and other point sources of light

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

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:

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

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:

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

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

d_star is the distance to the star

d_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)

Computing between luminosity and bolometric magnitude

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

$Mbol_{\rm star} - Mbol_{\rm sun} = -2.5 \log_{10} {\frac{L_{\rm star}}{L_{\odot}}}$

which makes by inversion:

$\frac{L_{\rm star}}{L_{\odot}} = 10^{((Mbol_{\rm sun} - Mbol_{\rm star})/2.5)}$

where

$L_{\odot}$ is the Sun's (sol) luminosity (bolometric luminosity)

L_star is the star's luminosity (bolometric luminosity)

Mbol_sun is the bolometric magnitude of the Sun

Mbol_star is the bolometric magnitude of the star

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:

$L = f n \frac{N_{1} N_{2}}{A}$

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.