Connecting UV Luminosity to Stellar Mass

Brant Robertson bio photo By Brant Robertson

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 with units Mpc mag, 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 to be the monochromatic UV luminosity with units ergs s Hz. The connection between star formation rate and is often written in the form

assuming that corresponds to the monochromatic frequency that defines what we mean by “UV”. and are constants. Throughout, I’ll implicitly divide by or such that the arguments of logarithms are unit free.

We can work in magnitudes rather than luminosity by noting that

where is some constant. If as above, then we just have

or

or

We can then write the SFR function as

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

or

The units are , or no units!

We then have the stellar mass function as

This function has units of Mpc dex.

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
such that

But we can relate the joint distribution of SFR and stellar mass to the conditional distribution of stellar mass given star formation rate as

Note that could be, e.g., a gaussian in normalized such that