Luminosity function (astronomy)

White dwarf luminosity function

The white dwarf luminosity function (WDLF) gives the number of white dwarf stars with a given luminosity. As this is determined by the rates at which these stars form and cool, it is of interest for the information it gives about the physics of white dwarf cooling and the age and history of the Galaxy.

Schechter luminosity function

The Schechter luminosity function ${\displaystyle \varphi }$ provides an approximation of the abundance of galaxies in a luminosity interval ${\displaystyle [L+dL]}$. The luminosity function has units of a number density ${\displaystyle n}$ per unit luminosity and is given by a power law with an exponential cut-off at high luminosity

${\displaystyle dn(L)=\varphi dL={\varphi }^{\ast }{\left(\frac{L}{L^{\ast }}\right)}^{\alpha }{\mathrm{e}}^{-L/L^{\ast }}d\left(\frac{L}{L^{\ast }}\right),}$

where ${\displaystyle L^{\ast }}$ is a characteristic galaxy luminosity controlling the cut-off, and the normalization ${\displaystyle {\varphi }^{\ast }}$ has units of number density.

Equivalently, this equation can be expressed in terms of log-quantities with

${\displaystyle dn(L)=\ln (10){\varphi }^{\ast }{\left(\frac{L}{L^{\ast }}\right)}^{\alpha +1}{\mathrm{e}}^{-L/L^{\ast }}d\left({\log }_{10}L\right).}$

The galaxy luminosity function may have different parameters for different populations and environments; it is not a universal function. One measurement from field galaxies is ${\displaystyle \alpha =-1.25,{\varphi }^{\ast }=1.2\times {10}^{-2}{h}^3{\mathrm{M}\mathrm{p}\mathrm{c}}^{-3}}$.

It is often more convenient to rewrite the Schechter function in terms of magnitudes, rather than luminosities. In this case, the Schechter function becomes:

${\displaystyle n(M)dM=(0.4\ln 10){\varphi }^{\ast }[{10}^{0.4(M^{\ast }-M)}{]}^{\alpha +1}\exp [-{10}^{0.4(M^{\ast }-M)}]dM.}$

Note that because the magnitude system is logarithmic, the power law has logarithmic slope ${\displaystyle \alpha +1}$. This is why a Schechter function with ${\displaystyle \alpha =-1}$ is said to be flat.

Integrals of the Schechter function can be expressed via the incomplete gamma function

${\displaystyle {\int }_a^b{\left(\frac{L}{L^{\ast }}\right)}^{\alpha }{e}^{-\left(\frac{L}{L^{\ast }}\right)}d\left(\frac{L}{L^{\ast }}\right)=\mathrm{\Gamma }(\alpha +1,a)-\mathrm{\Gamma }(\alpha +1,b)}$

Historically, the Schechter luminosity function was inspired by the Press–Schechter model. However, the connection between the two is not straight forward. If one assumes that every dark matter halo hosts one galaxy, then the Press-Schechter model yields a slope ${\displaystyle \alpha \sim -3.5}$ for galaxies instead of the value given above which is closer to -1. The reason for this failure is that large halos tend to have a large host galaxy and many smaller satellites, and small halos may not host any galaxies with stars. See, e.g., halo occupation distribution, for a more-detailed description of the halo-galaxy connection.