Estimating Neutrino Flux Density

[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".

Homework Equations

n_v = U_v(T) / <E_v>

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. m_v < 1 eV), since the question does not specify. Although I believe the T_v = 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 traveling in all directions with the same velocity.

 

[mfb]

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

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

 

[Joeseye]

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

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

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 . u_v(T)) / (3 . <E_v>) = (c . u_v(T)) / (3 . k_B . 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 u_v(T)) divided by the average energy of a neutrino <E_v>.

Do you think this is correct?

 

[mfb]

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.

 

[Joeseye]

Flux should be something "per time*area".

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 traveling 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?

 

[mfb]

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