# Connecting UV Luminosity to Stellar Mass

## UV Luminosity Function

If we have a UV luminosity function, a connection between UV luminosity and star formation rate, and a relation between star formation rate and stellar mass, we can estimate a stellar mass function.

Consider a UV luminosity function $\Phi(M_{UV}) \equiv \frac{dn}{d M_{UV}}$ with units Mpc$^{-3}$ mag$^{-1}$, such that the galaxy number density is
$% $

## Star formation rate function

To convert the UV luminosity function into a star formation rate function, we need to have some connection between $M_{UV}$ and SFR. Here, we will take $L_{UV}$ to be the monochromatic UV luminosity with units ergs s$^{-1}$ Hz$^{-1}$. The connection between star formation rate and $L_{UV}$ is often written in the form
$\log_{10}\left[\frac{SFR}{M_{\odot}~\mathrm{yr}^{-1}}\right] = \log_{10} \left(\nu L_{UV}\right) + A_{SFR,UV} \equiv \log_{10} L_{UV} + B_{SFR,UV}$
assuming that $\nu$ corresponds to the monochromatic frequency that defines what we mean by “UV”. $A_{SFR,UV}$ and $B_{SFR,UV}$ are constants. Throughout, I’ll implicitly divide by $M_{\odot}~\mathrm{yr}^{-1}$ or $M_{\odot}$ such that the arguments of logarithms are unit free.

We can work in magnitudes rather than luminosity by noting that
$L_{UV} = D_{UV} 10^{-0.4 M_{UV}}$
where $D_{UV}$ is some constant. If $SFR\propto L_{UV}$ as above, then we just have
$\log_{10} SFR = -0.4 M_{UV} + C_{SFR}$
or
$d\log_{10} SFR = -0.4 dM_{UV}$
or
$\frac{dM_{UV}}{d\log_{10} SFR} = -2.5 [\frac{\mathrm{mag}}{\mathrm{dex}}].$

We can then write the SFR function as
$\frac{dn}{d\log_{10} SFR} = \frac{dn}{dM_{UV}} \frac{dM_{UV}}{d\log_{10}SFR}.$

## Stellar mass function

If we assume a one-to-one correlation between star formation rate and stellar mass, then it’s straightforward to convert between the SFR rate to the stellar mass function.

Let’s assume a connection between star formation rate and stellar mass of
$\log_{10} SFR = \gamma \log_{10} M_{\star} + K_{SFR,\star}$
or
$\frac{d\log_{10} SFR}{d\log_{10} M_{\star}} = \gamma.$
The units are $[\mathrm{dex}/\mathrm{dex}]$, or no units!

We then have the stellar mass function as
$\frac{dn}{d\log_{10} M_{\star}} = \frac{dn}{dM_{UV}} \frac{dM_{UV}}{d\log_{10}SFR} \frac{d\log_{10}SFR}{d\log_{10} M_{\star}}.$
This function has units of Mpc$^{-3}$ dex$^{-1}$.

## Scatter in stellar mass vs. SFR

Imagine we have a population of galaxies that have a distribution of star formation rate and stellar mass. We could then write that distribution as $\frac{d^2 n}{d\log_{10}SFR~d\log_{10}M_{\star}}$
such that
$\frac{dn}{d\log_{10}M_{\star}} = \int_{SFR} \frac{d^2 n}{d\log_{10}SFR~d\log_{10}M_{\star}} d\log_{10} SFR$
But we can relate the joint distribution of SFR and stellar mass $\frac{d^2 n}{d\log_{10}SFR~d\log_{10}M_{\star}}$ to the conditional distribution of stellar mass given star formation rate $p(\log_{10}M_{\star}|\log_{10}SFR)$ as
$\frac{dn}{d\log_{10} M_{\star}} = \int_{-\infty}^{\infty} \frac{dn}{d\log_{10}SFR} p(\log_{10}M_{\star}|\log_{10}SFR) d\log_{10}SFR$
Note that $p(d\log_{10}M_{\star}|d\log_{10}SFR)$ could be, e.g., a gaussian in $d\log_{10}M_{\star}$ normalized such that
$\int_{\log_{10}M_{\star}}p(\log_{10}M_{\star}|\log_{10}SFR) d\log_{10}M_{\star} = 1.$