# What equation does KATRIN use to measure neutrino masses?

Source

KATRIN weighs neutrinos produced by the nuclear decay of tritium, a radioactive isotope of hydrogen. When a tritium nucleus transmutes into a helium one, it ejects an electron and a neutrino (or, more accurately, a particle with an equal mass called an antineutrino). The neutrino is lost, but the electron is channelled into a 23-metre-long, steel vacuum chamber shaped like a Zeppelin airship, where its energy is measured precisely.

The electron carries almost all of the energy released during the tritium’s decay, but some is lost with the neutrino. The value of this shortfall can be used to calculate the particle’s mass.

This sounds like a straightforward particle physics homework problem. However, I don't see how to calculate the mass of the neutrino. If we know the energy of the tritium nucleus, the helium nucleus, and the electron, then conservation of energy gives a number for the energy of the neutrino. But then:

$$E_{neutrino} = \gamma m c^2$$

which still leaves two unknowns with only one equation - $$v$$ (contained in the Lorentz factor $$\gamma$$) and $$m$$. Neither can we approximate the neutrinos as moving at light speed - that would lead to an infinite Lorentz factor.

How does KATRIN measure the neutrino mass? If it is via the above analysis, what other equation does KATRIN use to solve for the two unknowns?