I would like to calculate the density of MW-like galaxies using the double Schechter luminosity function:

$ \Phi(m) d m=\left[\phi_1\left(\frac{m}{m_*}\right)^{\alpha_1}+\phi_2\left(\frac{m}{m_*}\right)^{\alpha_2}\right] e^{-m / m_*} \frac{d m}{m_*}$

However, I'm not sure what values to assume for the various parameters for the MW.

$ log(m_*/M_\odot)$ = ? $ \alpha_1, \alpha_2$ = ? $ \phi_1, \phi_2$ = ?

Any suggestions about how to go about this?


1 Answer 1


The Schechter luminosity function (LF) — double or single — is a phenomenological description of observed galaxy populations, without much physical foundation1, and therefore its parameters must be determined observationally. That means that the answer to your question will depend on which observations you prefer/trust. Using a double rather than a single introduces more parameters, but can be used to model an observed, slight excess around (slightly on the faint side) of the characteristic turn-over magnitude (which you call $m_*$).

A luminosity function describes the galaxy population in a given wavelength band, e.g. in the UV. So if you want to use the LF to get the density of Milky Way (MW)-like galaxies, you should compare to the MW's luminosity in the same band. Since galaxies evolve, so does the LF, and hence you also need to decide which epoch you're interested in. I assume "today", i.e. at redshift $z\sim0$. Here, most LFs are determined using the SDSS survey.

Since luminosity is related to stellar mass (modulo dust, AGN, …), one instead sometimes consider the stellar mass function (SMF), which can also be described by a Schechter function.

With these things in mind, you could for example use the parameters of Blanton et al. (2005) who fit SDSS galaxies with a double Schechter law. You will find their fitting parameters in Table 3 for all ugriz bands. Other options include Weigel et al. (2016), who do a similar analysis for a later data release, Ting-Wen et al. (2016) who look specifically at the r band, and Hanwen et al. (2024) who look at higher-redshift galaxies.

1 The function does mimic the dark matter halo mass function (HMF) which is a similar power law with an exponential cutoff, and which can be derived almost from first principles. However, the LF is not just a mirror image of the HMF, because it is modified by many physical processes, e.g. stellar and AGN feedback, ionization suppression, and merging.


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