Show that if a neutrino has a mass mv that is very small, a...

Show that if a neutrino has a mass $m_v$ that is very small, a spread in energy $\Delta E_v$ for a set of detected neutrinos having an average flight time $t$ from source to detector will lead to a spread in arrival times given by $$ \frac{\Delta t}{t}=\frac{\Delta v}{v} \simeq \frac{m_v^2}{E_v^2} \frac{\Delta E_v}{E_v}, $$ where $E_v$ is the average energy and $\Delta E_v$ is the spread in energy. Some $19 \bar{v}_{\mathrm{e}}$ events were detected on Earth from Supernova 1987A. These neutrinos arrived over about a 10 -second time span with an average energy of about $25 \mathrm{MeV}$ and a spread in energy of about 10 $\mathrm{MeV}$. Use these results to place an upper limit on the $v_{\mathrm{e}}$ (and thus $v_{\mathrm{e}}$ ) mass, assuming the supernova to have occurred about 170,000 lightyears away.

Show that if a neutrino has a mass $m_v$ that is very small, a spread in energy $\Delta E_v$ for a set of detected neutrinos having an average flight time $t$ from source to detector will lead to a spread in arrival times given by
$$
\frac{\Delta t}{t}=\frac{\Delta v}{v} \simeq \frac{m_v^2}{E_v^2} \frac{\Delta E_v}{E_v},
$$
where $E_v$ is the average energy and $\Delta E_v$ is the spread in energy. Some $19 \bar{v}_{\mathrm{e}}$ events were detected on Earth from Supernova 1987A. These neutrinos arrived over about a 10 -second time span with an average energy of about $25 \mathrm{MeV}$ and a spread in energy of about 10 $\mathrm{MeV}$. Use these results to place an upper limit on the $v_{\mathrm{e}}$ (and thus $v_{\mathrm{e}}$ ) mass, assuming the supernova to have occurred about 170,000 lightyears away.

Key Concepts

Time-of-Flight Measurements in Astrophysics

Astrophysical observations, such as detecting neutrinos from distant supernovae, employ time-of-flight measurements to gather constraints on particle properties. By accurately measuring the time span over which neutrinos arrive and knowing the distance from the source, one can deduce limits on the neutrino mass. This method has been particularly useful in analyzing data from events like Supernova 1987A, where the flight time over astronomical distances amplifies the impact of tiny differences in speed.

Astrophysical Constraints on Neutrino Mass

Observations of neutrinos from astrophysical sources provide a natural laboratory for constraining neutrino mass. When neutrinos from events like Supernova 1987A are observed to arrive within a narrow time window despite having a spread in energy, the upper limit on the neutrino mass can be inferred. This approach utilizes the relationship between energy, velocity, and mass derived from relativistic principles, thus helping to set stringent upper bounds on the neutrino mass.

Neutrino Kinematics and Relativistic Approximations

When considering particles with extremely small mass, such as neutrinos, one must use the framework of relativistic kinematics. In this context, the energy of a particle is given by E² = p²c² + m²c?. For neutrinos whose rest mass is negligibly small compared to their kinetic energy, approximations can be made (such as neglecting higher order terms in m/E) that allow for simple expressions relating mass, energy, and velocity. These approximations underlie the derivation of the relationship between the fractional spread in arrival time and the fractional energy uncertainty.

Energy Uncertainty and Its Impact on Propagation Time

A spread in the energy of neutrinos translates into a spread in their velocities because of the dependence of speed on energy for a massive particle. Even a small mass leads to slight differences in velocity for neutrinos of different energies, which over very long distances results in measurable differences in arrival times. This concept is crucial for deriving the relation ?t/t ? (m_v²/E_v²) (?E_v/E_v), linking energy spread to time spread via the small but nonzero mass.

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