= Schechter function
{c}
{title2=$\Phi(M)$}
{wiki}
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 by
$$
\Phi(M)\,dM=\frac{\Phi_*}{M_*}\left(\frac M{M_*}\right)^\alpha e^{-M/M_*}\,dM.
$$
Here $M_*$ is a characteristic <mass>, $\Phi_*$ 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 $dn/d\log_{10}M=(\ln10)M\Phi(M)$. Comparing slopes without specifying the measure introduces a shift of one.
Writing $x=M/M_*$ gives the moments
$$
n_{\rm gal}=\Phi_*\Gamma(\alpha+1),\quad \alpha>-1,\qquad
\rho_*=\Phi_*M_*\Gamma(\alpha+2),\quad \alpha>-2,
$$
where $\Gamma$ 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.<An Analytic Expression for the Luminosity Function for Galaxies by Paul Schechter>[Section I, equation (1), p. 297; section II(a), equations (2)–(5), p. 298]{ref}
Back to article page
