How Does the Schechter Luminosity Function Determine Galaxy Luminosity Density?

[TFM]

Homework Statement

The galaxy luminosity function [tex] \phi(L) [/tex] 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
[tex] l = \int^{\infty}_{0}L\phi(L)dl [/tex]

a)
For a Schechter luminosity function,

[tex] \Phi(L)dL = \phi^*(\frac{L}{L^*})^aexp(-\frac{L}{L^*}d(\frac{L}{L^*})) [/tex]

Show that

[tex] l = \phi^*L^*\Gamma(a + 2) [/tex]

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

b)

The Sloan Digital Sky Survey (SDSS) has recently measured the following Schechter
parameters in the r passband: a = -1.16 [tex]\pm[/tex] 0.03, [tex]M^*[/tex] = -20.80 [tex]\pm[/tex] 0.03, [tex]\phi^*[/tex] = (1.50 [tex]\pm 0.13) * 10^2 h^3 Mpc^{-3}[/tex] 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.
[tex]\Gamma[/tex](0.84) = 1.122, [tex]\Gamma[/tex](0.81) = 1.153, [tex]\Gamma[/tex](0.87) = 1.094.

Homework Equations

Given in Question

The Attempt at a Solution

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


[tex] \Phi(L)dL = \phi^*(\frac{L}{L^*})^aexp(-\frac{L}{L^*}d(\frac{L}{L^*})) [/tex]

and

[tex] l = \int^{\infty}_{0}L\phi(L)dl [/tex]

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

 

[BigJDubs]

EDIT: I am thinking we should put them together, then we can integrate from 0 to infinity and the Gamma function will appear when we solve it.

 

[nlightner]

Yes, in order to solve for l, we need to combine these two equations. We can start by substituting the Schechter luminosity function into the second equation:

l = \int^{\infty}_{0}L\phi^*(\frac{L}{L^*})^aexp(-\frac{L}{L^*}d(\frac{L}{L^*}))dl

Then, we can rearrange the terms and pull out the constants to get:

l = \phi^*L^*\int^{\infty}_{0}L^{a+1}e^{-\frac{L}{L^*}}dl

Next, we can use the gamma function to solve the integral, giving us:

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

So, to answer your question, we get the gamma function out by solving the integral using its definition. I hope this helps! Let me know if you have any further questions.