# Double Schechter luminosity function for the Milky Way

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.

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

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.