Galaxy stellar mass ($M_{\ast }$)

Galaxy stellar mass is the total mass in a galaxy's stars and stellar remnants. It is usually inferred from luminosity and colour or a spectral-energy-distribution fit, so it depends on stellar-population models, dust treatment, and the adopted initial mass function.

The mass-to-light ratio compares the mass assigned to an astronomical system with its luminosity in a specified band, often in solar units. A stellar value uses only stellar mass and depends on the stellar population, age, composition and band. A dynamical value can include gas and dark matter, so it need not equal the stellar value. In a halo-to-galaxy mapping, $L=M_{\ast }/{{\rm Y}}_{\ast }$ requires the stellar ratio and not the total dark matter halo ratio.

A dynamical mass-to-light ratio must specify its aperture and whether the mass includes dark matter. A growing dark halo can increase ${\rm Y}(<R)$ outside the bright stellar body even when the stellar ratio is constant. Comparing a tidal-radius satellite mass with a large-aperture host mass is different from comparing both within their half-light radii. From $M\propto {\sigma }^{2}R$ and $L\propto {\sigma }^4$, the inferred ratio scales as $R/{\sigma }^2$, so equal ratios additionally require homologous radius scaling.

Write $x=m/M_{\odot }$, the initial mass function as $dN=A{x}^{-\alpha }dx$, and a main-sequence mass-luminosity relation as $L/L_{\odot }=x^{\beta }$. On $a<x<b$, the mass-to-light ratio in solar units isFor $\alpha =\beta =3$ this is exactly $1/(ab)$. The result concerns the specified unevolved stellar population; stellar remnants, evolved luminous stars and cooling substellar objects require different accounting.

The galaxy stellar mass function counts galaxies per stellar-mass interval and volume, often using comoving volume. Its stellar-mass moment gives the cosmic stellar mass density, while its number moment counts galaxies only if the low-mass population is integrable or truncated. It differs from the initial mass function of individual stars.

The cosmic stellar mass density is the mass moment of a galaxy stellar mass function in a stated volume convention. With a comoving mass function it is mass per comoving volume, while proper density is larger by $(1+z{)}^3$. Comparing it to a critical or matter density requires keeping the epoch and volume convention explicit.

A generalized mass function with low-mass slope $-7/4$ and cutoff $e^{-\sqrt{M_{\ast }/M_{\ast ,\mathrm{ch}}}}$ differs from the ordinary exponential Schechter cutoff. Its total stellar-mass density is finite, even though its untruncated total galaxy count diverges. This form arises from a scale-free Press-Schechter mass function with a constant stellar-to-halo mass mapping.

Integrating galactic stellar mass against a $x^{-7/4}{e}^{-\sqrt{x}}$ mass function requires ${\int }_0^{\infty }{x}^{-3/4}{e}^{-\sqrt{x}}dx$. Substitution $u=\sqrt{x}$ turns it into $2\Gamma (1/2)=2\sqrt{\pi }$. Thus the stellar density is $2\sqrt{\pi }$ times characteristic number density times characteristic galactic stellar mass, in the same volume convention as the mass function.

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]