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Galaxy Luminosity Function

  1. Mar 7, 2009 #1

    TFM

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

    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.

    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


    [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?
     
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