Total luminosity of total population of galaxies!

Question

I'm trying to get the total luminosity of a total population of galaxies. I have the Schechter Function like

$$\Phi(L) = \frac{\Phi^{*}}{L^{*}} \left(\frac{L}{L^{*}}\right)^{\alpha} \exp \left(\frac{L}{L^{*}}\right)$$

and I need to the total luminosity:

$$L_{tot} = \int_0^\infty L\Phi(L)dL = \Phi_{0}L^{*}\Gamma(2+\alpha)$$

but when I substitute the Schecter functions I don't get the $\Gamma$ function form. Could you have any tips for that?

And I need to show that the galaxies that more contributed to the luminosity are those have $L^{*}$

Thank you in advance.

Answer 1

I think there's a minus sign missing in the exponent of the expression for $\Phi(L)$ (see Schechter luminosity function). I will be including it here: $$\Phi(L) = \frac{\Phi^*}{L^*}\left(\frac{L}{L^*}\right)^\alpha \exp\left( -\frac{L}{L^*} \right)$$

The integration in question is: $$L_{tot} = \int_{0}^{\infty} L\Phi(L) \,dL$$

$$L_{tot} = \int_{0}^{\infty} L\frac{\Phi^*}{L^*}\left(\frac{L}{L^*}\right)^\alpha \exp\left( -\frac{L}{L^*} \right) \,dL$$

$$L_{tot} = \frac{\Phi^*}{(L^*)^{\alpha+1}} \int_{0}^{\infty} L^{(\alpha+1)} \exp\left( -\frac{L}{L^*} \right) \,dL$$

$$L_{tot} = \frac{\Phi^*}{(L^*)^{\alpha+1}} \left[ (L^*)^{\alpha+2} \Gamma(\alpha+2) \right]$$

$$L_{tot} = \Phi^* L^* \Gamma(\alpha+2)$$

This seems to be what you have, assuming $\Phi_0=\Phi^*$.