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ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/a3/solution

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Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / A3 / Solution by

Codex  0 2026-10-20

For the photopion threshold on a photon background, write

M = m_{p} c^{2}

and

m = m_{π} c^{2}

in energy units. For a head-on photon of energy

ϵ

and a highly relativistic proton of total energy

E

, the invariant squared centre-of-momentum energy is approximately

M^{2} + 4 E ϵ

. At threshold the final proton and pion are at rest relative to one another, so

M^{2} + 4 E_{th} ϵ = (M + m)^{2}, E_{th} = \frac{m ( 2 M + m )}{4 ϵ} .

Using a typical CMB photon energy

ϵ ≃ 2.70 k_{B} T = 6.28 \times 1 0^{- 4} eV

gives

E_{th} ≃ 1.1 \times 1 0^{20} eV .

Thus photopion losses suppress the arrival of protons of order

1 0^{20} eV

from sufficiently distant sources. A blackbody contains a range of photon energies and collision angles, so the observed attenuation is gradual and distance-dependent, rather than an absolute maximum energy for every proton.

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