# Cosmic microwave background fits the blackbody radiation spectrum

## Homework Statement

(a)The cosmic microwave background fits the blackbody radiation spectrum well with a temperature of 2.7 K and a corresponding peak wavelength at 1.9nm. Applying the relationship between the radiant emittance, i.e. the total power emitted per unit area, and the photon energy density for the blackbody radiation to estimate the number of CMB photons per $\mathrm{cm}^3$ in outer space.

(b) The high energy cosmic rays are believed to be mostly protons and the max energies observed on earth seem to have a bound. This upper bound is explained by collisions between the high energy protons traveling from extra-galactic space and the CMB photons. If the proton energy is high enough and exceed a threshold value, the following process could occur $p+\gamma\rightarrow\Delta^+, \Delta^+\rightarrow p\pi^0\text{ (or }n\pi^0\text{)}$. Therefore, the proton energy observed on earth is decreased and there exists a so called GZK cut-off. Please determine this energy threshold in eV. The masses of the proton and $\Delta$ baryons are about 938Mev/c^2 and 1232 MeV/c^2.

## The Attempt at a Solution

Is the answer of (a) be $\sigma T^4/hc^2/\lambda$?
I have no idea about (b).
Any help would be appreciated.

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vela
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Homework Helper

## Homework Statement

(a)The cosmic microwave background fits the blackbody radiation spectrum well with a temperature of 2.7 K and a corresponding peak wavelength at 1.9nm. Applying the relationship between the radiant emittance, i.e. the total power emitted per unit area, and the photon energy density for the blackbody radiation to estimate the number of CMB photons per $\mathrm{cm}^3$ in outer space.

(b) The high energy cosmic rays are believed to be mostly protons and the max energies observed on earth seem to have a bound. This upper bound is explained by collisions between the high energy protons traveling from extra-galactic space and the CMB photons. If the proton energy is high enough and exceed a threshold value, the following process could occur $p+\gamma\rightarrow\Delta^+, \Delta^+\rightarrow p\pi^0\text{ (or }n\pi^0\text{)}$. Therefore, the proton energy observed on earth is decreased and there exists a so called GZK cut-off. Please determine this energy threshold in eV. The masses of the proton and $\Delta$ baryons are about 938Mev/c^2 and 1232 MeV/c^2.

## The Attempt at a Solution

Is the answer of (a) be $\sigma T^4/hc^2/\lambda$?
I don't know if your expression is anywhere near correct, but I just wanted to ask if you meant
##\sigma T^4/(hc^2/\lambda)## or ##(\sigma T^4/hc^2)/\lambda##.

Explain how you came up with your answer.

I have no idea about (b).
Any help would be appreciated.
You're asked to find the energy threshold. Why would ##p+\gamma\rightarrow\Delta^+## happen at some energies and not others?

By the way, ##\Delta^+ \to n\pi^0## violates conservation of charge. It can't happen. Perhaps you meant ##\Delta^+ \to n\pi^+##.