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Flux density

  1. Feb 21, 2007 #1
    1. The problem statement, all variables and given/known data
    A population of sources in a flat matter-dominated universe has a number-density [itex]n_0[/itex] at the present epoch and a monochromatic luminosity [itex]P(\nu) \propto \nu^{-\alpha}[/itex] at frequency [itex]\nu[/itex]. Show that the flux density [itex]S(\nu_0)[/itex] observed at the present epoch from a source at redshift z satisfies

    [tex]S(\nu_0) = P(\nu_0) (1+z)^{1-\alpha}D_L^{-2}[/tex],

    where [itex]D_L[/itex] is the luminosity distance.

    2. Relevant equations
    Luminosity distance is defined by
    [tex]D_L = \left( \frac{L}{4 \pi l} \right)^{1/2} = a_0^2 \frac{r}{a} =a_0 r (1+z)[/tex]
    where [itex]L[/itex] is the power emitted by a source at coordinate distance [itex]r[/itex] at time [itex]t[/itex], [itex]l[/itex] is the power recieved per unit area (flux) at present time and [itex]a[/itex] is the scale factor.

    Redshift is defined as
    [tex]1+z = \frac{a_0}{a} = \frac{\nu_e}{\nu_0} = \frac{\lambda_0}{\lambda_e}[/itex]

    3. The attempt at a solution
    The flux density has the units of power per unit area per unit frequency so is
    [tex]l = \int S d\nu[/tex]?
    We should also have
    [tex]l = \frac{L}{4 \pi D_L^2}[/tex]
    and I guess that [itex]L = P[/itex] but here I'm stuck. Has someone got any ideas?
  2. jcsd
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