# Estimating Neutrino Flux Density

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1. Apr 28, 2013

### Joeseye

1. Problem

"Estimate the flux of neutrinos passing through your body per second if the present energy density of neutrinos from the Big Bang is 0.2 MeV/m3. Assume that you are a standard size covering 0.01 m2".

2. Relevant equations

nv = Uv(T) / <Ev>

3. The attempt at a solution

I've assumed that the neutrinos have a temperature of 1.95 K. Now I'm not sure whether to presume that the neutrinos are relativistic (hence, zero mass and velocity of c) or non-relativistic (i.e. mv < 1 eV), since the question does not specify. Although I believe the Tv = 1.95 K value comes from assuming neutrinos are massless (I think).

I've attempted both and have different answers (although I doubt whether they are correct). Regardless, I've not had much success converting the neutrino density to a flux density. I assume that the neutrinos are travelling in all directions with the same velocity.

Last edited: Apr 28, 2013
2. Apr 28, 2013

### Staff: Mentor

You can assume that the neutrinos are ultra-relativistic, I think.

A human body with 0.01m^2 surface area is... strange.

3. Apr 29, 2013

### Joeseye

I thought that 0.01 m2 was quite low, too. Perhaps he meant 0.1 m2.

Assuming the neutrinos are ultra-relativistic I got a flux of 1.19 x 1017 m-2 s-1... which I'm pretty sure is higher than the solar neutrino flux.

I used:

Flux ϕ = (c . uv(T)) / (3 . <Ev>) = (c . uv(T)) / (3 . kB . T)

The factor of 1/3 comes from assuming the neutrinos are isotropic. Essentially, this is the power density (which is c/3 times the radiation pressure uv(T)) divided by the average energy of a neutrino <Ev>.

Do you think this is correct?

Last edited: Apr 29, 2013
4. Apr 29, 2013

### Staff: Mentor

Flux should be something "per time*area".

2K correspond to about .5meV, therefore we have ρ=0.4*10^9 primordial neutrinos per m^2, moving at nearly the speed of light. Using only one direction, the flux is 1/2 ρ c or about 10^17/(m^2*s).
Looks good.

Those neutrinos are hard (or even impossible) to detect as they have a very low energy.

5. Apr 29, 2013

### Joeseye

Ah yeah. Sorry I meant neutrinos per meter squared per second - I'll edit my post.

Thanks for your reply. Is it appropriate to assume the neutrinos are travelling at a velocity of c and are massless? I thought that when neutrinos decoupled (2s after the Big Bang) they had a velocity close to c, but have since slowed to approximately 105 - 106 m s-1?

6. Apr 29, 2013

### Staff: Mentor

If they are slow, they are not relativistic - with 2K, they would need some significant mass to be so slow.

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