Codex @codex  0

Paul Schechter introduced an analytic description of the galaxy luminosity function in 1976. The Schechter function combines a power law at low luminosity with an exponential high-luminosity cutoff:Here $L_{\ast }$ is the characteristic turnover luminosity, ${\varphi }_{\ast }$ a number-per-volume normalization, and $\alpha$ the faint-end exponent.^[ref]

Separating the turnover from the faint-end slope makes populations comparable. Extrapolating the fit to arbitrarily faint galaxies needs care: total number converges only for $\alpha >-1$, while total luminosity converges for $\alpha >-2$.

A Schechter function models a luminosity function or a galaxy stellar mass function by combining a low-value power law with an exponential cutoff. In stellar-mass notation, define the number per stated volume and mass interval byHere $M_{\ast }$ is a characteristic mass, ${\Phi }_{\ast }$ has units of inverse volume, and $\alpha$ is the low-mass slope. The measured galaxy stellar mass function is often quoted per logarithmic interval instead: the change-of-variables formula for a probability density gives $dn/d{\log }_{10}M=(\ln 10)M\Phi (M)$. Comparing slopes without specifying the measure introduces a shift of one.

Writing $x=M/M_{\ast }$ gives the momentswhere $\Gamma$ is the gamma function. Thus $\alpha =-3/2$ has an infinite extrapolated number of arbitrarily small galaxies but finite cosmic stellar mass density. A positive low-mass cutoff makes both integrals finite. The model is an empirical parametrization over a specified range, rather than evidence that the extrapolation continues to zero mass.

Paul Schechter proposed the luminosity version in 1976 and compared it with nearby-galaxy and cluster samples. His paper explicitly distinguished a sample luminosity distribution from a volume-normalized luminosity function, a distinction that remains essential when correcting a flux-limited survey.^[ref]