Schechter luminosity function (integration)

[iloveannaw]

Suppose the luminosity function of galaxies is given by the approximation:

dn = \varphi L dL = \varphi* (L / L*)^-\alpha dL/L* when L < L*
and
dn = \varphi L dL = 0 when L > L*

where \alpha = 1.25, L* = 2.0E10 solar luminosities and \varphi* = (3Mpc)^-3

Integrate the above eqn to find the total number density of galaxies, integrated over all luminosities. Ypu should find that the integral diverges (e.g. the number of galaxies is infinite)

Well the integration part seemed fairly straight forward, taking x = (L/L*) and dx = dL/L* I got:

I = \int \varphi* (L / L*)^-\alpha dL/L* = \varphi*\int x^-\alpha dx

I = \varphi* (x^(1-\alpha)) / (1-\alpha)
I = ((3 Mpc)^-3 . (2.0E10 SLs)^-0.25) / -0.25 = (very small negative value)(1 SL)

Im pretty sure the number of galaxies is not less than zero just can't figure this question out. As limits I've taken:
upper limit = L*
lower limit = 0

thanks in advance

 

[clamtrox]

It really hurts my eyes when you write the latex code like that :D Just write the entire equation inside the brackets next time. Also, don't forget to plug in your integration limits since they are usually important for getting the right answer. You get an answer which is "\infty - something small", where the infinity comes from the lower limit (as you might expect, since there are many galaxies with near-zero luminosity as seen from earth.)