Orbital
Feb21-07, 07:58 AM
1. The problem statement, all variables and given/known data
A population of sources in a flat matter-dominated universe has a number-density n_0 at the present epoch and a monochromatic luminosity P(\nu) \propto \nu^{-\alpha} at frequency \nu. Show that the flux density S(\nu_0) observed at the present epoch from a source at redshift z satisfies
S(\nu_0) = P(\nu_0) (1+z)^{1-\alpha}D_L^{-2},
where D_L is the luminosity distance.
2. Relevant equations
Luminosity distance is defined by
D_L = \left( \frac{L}{4 \pi l} \right)^{1/2} = a_0^2 \frac{r}{a} =a_0 r (1+z)
where L is the power emitted by a source at coordinate distance r at time t, l is the power recieved per unit area (flux) at present time and a is the scale factor.
Redshift is defined as
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?
We should also have
l = \frac{L}{4 \pi D_L^2}
and I guess that L = P but here I'm stuck. Has someone got any ideas?
A population of sources in a flat matter-dominated universe has a number-density n_0 at the present epoch and a monochromatic luminosity P(\nu) \propto \nu^{-\alpha} at frequency \nu. Show that the flux density S(\nu_0) observed at the present epoch from a source at redshift z satisfies
S(\nu_0) = P(\nu_0) (1+z)^{1-\alpha}D_L^{-2},
where D_L is the luminosity distance.
2. Relevant equations
Luminosity distance is defined by
D_L = \left( \frac{L}{4 \pi l} \right)^{1/2} = a_0^2 \frac{r}{a} =a_0 r (1+z)
where L is the power emitted by a source at coordinate distance r at time t, l is the power recieved per unit area (flux) at present time and a is the scale factor.
Redshift is defined as
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?
We should also have
l = \frac{L}{4 \pi D_L^2}
and I guess that L = P but here I'm stuck. Has someone got any ideas?