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How can I estimate the endpoint energy for the proton in the neutron beta decay?

Following this answer, I can write the system of equation as:

$$p_e+p_p+p_\nu=0$$

$$m_n=E_p+E_e+E_\nu$$

Assuming that the neutrino has no mass and that it has $p_\nu=0$, I have: $$m_n=E_e+E_p=E_p+m_e+T_e$$ where $T_e$ is the electron kinetic energy. Now how do I proceed? I cannot impose $T_e=\dfrac{p_e^2}{2m_e}$ since the electron is relativistic

mattiav27
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  • A massless particle cannot be at rest.italways moves at light speed. – my2cts Aug 14 '19 at 06:46
  • @my2cts that is correct. So I have to assume that $p_\nu=0$ – mattiav27 Aug 14 '19 at 06:50
  • The neutrino is not actually massless, just light enough relative the dominate energy terms in the problem to ignore its mass. – dmckee --- ex-moderator kitten Aug 14 '19 at 15:18
  • The full relativistic expression for kinetic energy is (in $c=1$ units): $T = E - m = \sqrt{m^2 + p^2} - m = (\gamma - 1) m$, where $\gamma = [1 - (v/c)^2]^{-1/2}$ is the Lorentz factor. Both expressions are messy, but they can be approximated reasonably when the energy is large compared to the mass (which kinda the case here). – dmckee --- ex-moderator kitten Aug 14 '19 at 15:31

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