Luminosity distance

Luminosity distance D_L is defined in terms of the relationship between the absolute magnitude M and apparent magnitude m of an astronomical object.

${\displaystyle M=m-5\log _{10}{\frac {D_{L}}{10\,{\text{pc}}}}\!\,}$

which gives:

${\displaystyle D_{L}=10^{{\frac {(m-M)}{5}}+1}}$

where D_L is measured in parsecs. For nearby objects (say, in the Milky Way) the luminosity distance gives a good approximation to the natural notion of distance in Euclidean space.

The relation is less clear for distant objects like quasars far beyond the Milky Way since the apparent magnitude is affected by spacetime curvature, redshift, and time dilation. Calculating the relation between the apparent and actual luminosity of an object requires taking all of these factors into account. The object's actual luminosity is determined using the inverse-square law and the proportions of the object's apparent distance and luminosity distance.

Another way to express the luminosity distance is through the flux-luminosity relationship,

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

where F is flux (W·m⁻²), and L is luminosity (W). From this the luminosity distance (in meters) can be expressed as:

${\displaystyle D_{L}={\sqrt {\frac {L}{4\pi F}}}}$

The luminosity distance is related to the "comoving transverse distance" ${\displaystyle D_{M}}$ by

${\displaystyle D_{L}=(1+z)D_{M}}$

and with the angular diameter distance ${\displaystyle D_{A}}$ by the Etherington's reciprocity theorem:

${\displaystyle D_{L}=(1+z)^{2}D_{A}}$

where z is the redshift. ${\displaystyle D_{M}}$ is a factor that allows calculation of the comoving distance between two objects with the same redshift but at different positions of the sky; if the two objects are separated by an angle ${\displaystyle \delta \theta }$, the comoving distance between them would be ${\displaystyle D_{M}\delta \theta }$. In a spatially flat universe, the comoving transverse distance ${\displaystyle D_{M}}$ is exactly equal to the radial comoving distance ${\displaystyle D_{C}}$, i.e. the comoving distance from ourselves to the object.[1]