# Number Density of Photons (Cosmology)

1. Oct 22, 2014

### roam

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

Given the maximum possible number density of stars in the present universe, assume that they have been radiating light for 10 billion years at a solar rate of $3.9 \times 10^{26} \ W$. Photons emitted all have a restframe wavelength of 500 nm. Find a crude upper bound for the number density of photons emitted by stars over the history of universe.

First find the maximum number density of stars in the present universe if all baryons were assembled into solar mass stars.

2. Relevant equations

3. The attempt at a solution

I have previously found a number density of stars to be:

$n= \frac{\rho}{m}= \frac{\rho_{crit} \Omega_b}{m}=2.225 \times 10^{-58} \ m^-3$

$6.54 \times 10^{-9} \ pc^{-3}$

The calculation was made given the sun weighs $2 \times 10^{30}$, and a baryon fraction $\Omega_b=0.05$ today. ($\rho_{crit}$ was worked out by finding $H_0$)

So, now what equation do I need to use to find the number density of photons emitted? Do I need to multiply the number density by solar rate? How do I take into account the wavelength of the radiation?

Any help is appreciated.

2. Oct 22, 2014

### Simon Bridge

You are given the rate each star radiates energy, not photons. What do you have to do to get the rate each star radiates photons?

3. Oct 23, 2014

### roam

Frankly, I'm not sure what equation to use here for finding the photon rate. Do we need to somehow use the Planck relation $E=hc/\lambda$? (that's the only way I can think of that wavelength comes into the picture)

4. Oct 23, 2014

### Simon Bridge

It looks like you are trying to solve physics problems by looking up equations instead of by understanding the physics involved.
This is not a good strategy.

Lets turn it around:
Consider a single photon with wavelength 440nm ... how much energy does it carry?
If a source were to radiate 1W or power, then how many of those photons per second need to be radiated?

5. Oct 23, 2014

### roam

I need to divide the required power (1 W) by the energy of a single photon ($hc/440 \ nm = 4.5 \times 10^{-19} W$). So ~2.2x1018 photons are required per second.

So, in my problem I simply need to divide the given solar rate by the energy of a single 550 nm photon? Is that right?

i.e.: (3.9 x 1026)/(hc/500 nm)= 9.8039 x 1044

6. Oct 23, 2014

### Simon Bridge

Well that's what the physics is telling you. See how talking about it as the energy of one photon is better as a guide to your thinking than thinking of it as this equation that seems to have the right bits in it somehow?

Don't forget units, don't forget correct rounding.

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