I have the power spectrum for a galaxy of radius $R$, which we can approximate as a sphere at some distance $D$. Now, on earth we can measure the power spectrum which is given as a list of tuples, ( $Energy\;$(eV) , $S_\nu \; $(Jy) ), where the Jansky (Jy) is defined as ($\frac{10^{-26}\;W}{m^{2}\cdot Hz}$).

I would like to calculate an estimate for the average number density of photons $n$ ($ \frac{photons}{m^{3}\cdot eV}$) inside the spherical galaxy.

my first attempt was to first calculate the luminosity spectrum of the galaxy $L_{Galaxy}\; (\frac{W}{eV})$.

$L_{Galaxy} = \frac{S_{\nu}\times 4\pi D^2 \times 6.242\times 10^{-8}}{h}$

where $h\; (s\cdot eV)$ is Planck's constant, and the numerical multiplier converts to $Joules$ to $eV$.

Assuming, there is no mechanism to absorb the photons at any time, the luminosity is independent of distance to the source.

Further, I assume that there is a point like source of the radiation inside the galaxy at the center, and thus the time for a photon to reach the "surface" of the galaxy is:

$\tau = \frac{R}{c}$

I then argue that the total energy per unit photon energy in the volume of the galaxy at any time, is:

$G_{E} = L_{Galaxy}\times \tau$

I then convert this into the number of photons per photon energy:

$N_{e} = \frac{G_{E}}{E_{\gamma}}$

where $E_{\gamma}$ is the energy of the photon bin, (i.e. $Energy$ in the tuple)

Finally, I divide by the volume of the galaxy to obtain a number density per unit photon energy:

$n = \frac{N_{E}}{\frac{4}{3} \pi R^3} = \frac{S_{\nu} 4 \pi D^2 R( 6.242\times 10^{-8})}{\frac{4}{3} \pi R^3 E_{\gamma} hc} = \frac{3 S_{\nu} D^2 ( 6.242\times 10^{-8})}{R^2 E_{\gamma} hc} $

The units check out, however, I get values that seem too low to be physically true, perhaps as a result of my naive model. Im hoping someone can help me see where my model fails or maybe even point me in the right direction to doing this calculation in the "accepted' way, if there is such a thing.


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