Galaxy Luminosity Function

1. Mar 7, 2009

TFM

1. The problem statement, all variables and given/known data

The galaxy luminosity function $$\phi(L)$$ tells us the number density of galaxies as a function of luminosity L. The luminosity density of the universe l is thus given by the expression
$$l = \int^{\infty}_{0}L\phi(L)dl$$

a)
For a Schechter luminosity function,

$$\Phi(L)dL = \phi^*(\frac{L}{L^*})^aexp(-\frac{L}{L^*}d(\frac{L}{L^*}))$$

Show that

$$l = \phi^*L^*\Gamma(a + 2)$$

where the gamma function $$\Gamma(x) = \int^{\infty}_{0}t^{x - 1}e^{-t}dt$$

b)

The Sloan Digital Sky Survey (SDSS) has recently measured the following Schechter
parameters in the r passband: a = -1.16 $$\pm$$ 0.03, $$M^*$$ = -20.80 $$\pm$$ 0.03, $$\phi^*$$ = (1.50 $$\pm 0.13) * 10^2 h^3 Mpc^{-3}$$ 0.13). Given that the sun has absolute magnitude M = 4.62 in the SDSS r band, calculate the luminosity density in this band in solar units. Estimate the error on this quantity.
$$\Gamma$$(0.84) = 1.122, $$\Gamma$$(0.81) = 1.153, $$\Gamma$$(0.87) = 1.094.

2. Relevant equations

Given in Question

3. The attempt at a solution

Okay, I am on the first part, but I am slightly unsure what to do. They give us

$$\Phi(L)dL = \phi^*(\frac{L}{L^*})^aexp(-\frac{L}{L^*}d(\frac{L}{L^*}))$$

and

$$l = \int^{\infty}_{0}L\phi(L)dl$$

do we have to put these two equations together? How do we get Gamma out?