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Q. 16.7

Astrophysicists were able to deduce a limit on the mass of electron neutrinos in the 1990s based on the spread of arrival times of neutrinos from supernova 1987A using detectors in both Japan and the United States. Find the difference in travel times between a neutrino with very low mass and a neutrino having a mass of 16 $eV / c^{2}$. Assume the energy of the neutrino is 20 MeV.

Strategy Let t be the travel time for the low-mass neutrino, t’ be the travel time for the neutrino having mass, and Δt be the time difference. We use relativistic relations and relate the mass-energy E with the speed v, which allows us to calculate Δt. The distance of SN 1987A from Earth is d = 160,000 ly.

Verified Solution

We assume the speed of the neutrino with small mass is c. If β is the value of v/c for the neutrino having greater mass, the time relations are

$t=\frac{d}{c} \quad t^{\prime}=\frac{d}{\beta c}$

$\Delta t=t^{\prime}-t=\frac{d}{c}\left(\frac{1}{\beta}-1\right)=\frac{d}{c}\left(\frac{1-\beta}{\beta}\right)$

From our earlier study of relativity, we have the total energy $E=\gamma m c^{2}, \gamma^{2}=1 /\left(1-\beta^{2}\right), \text { and } 1-\beta^{2}=1 / \gamma^{2}$. The last equation can be written

$(1-\beta)(1+\beta)=\frac{1}{\gamma^{2}}$

or

$(1-\beta)=\frac{1}{1+\beta} \frac{1}{\gamma^{2}} \approx \frac{1}{2 \gamma^{2}}$

because $\beta \approx 1$. The equation for Δt becomes, with $\beta \approx 1$,

$\Delta t=\frac{d}{c} \frac{1}{2 \gamma^{2}}=\frac{d}{2 c}\left(\frac{m c^{2}}{E}\right)^{2}$

We put in the appropriate numbers to obtain

$\Delta t=\frac{1.6 \times 10^{5} ly }{2 c}\left(\frac{16 eV }{20 MeV }\right)^{2}=5.1 \times 10^{-8} y =1.6 s$

This result is consistent with the actual spread in arrival times of a few seconds. Some assumptions had to be made in the actual calculation. For example, what if the slower neutrinos were emitted first and the faster neutrinos emitted last due to some effect within the supernova? Then the neutrinos might tend to arrive bunched together. A Monte Carlo simulation considering a wide class of possible neutrino emission models was used to assign an upper mass limit of 16 $eV / c^{2}$ for the electron neutrinos. The electron neutrino mass is now believed to be less than 2.2 $eV / c^{2}$.