# Flux density

1. Feb 21, 2007

### Orbital

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?