In supernovae, neutrinos are produced in huge amounts. They...

In supernovae, neutrinos are produced in huge amounts. They were detected from the 1987 A supernova in the Magellanic Cloud, which is about 120,000 lightyears away from Earth (relatively close to our Milky Way Galaxy). If neutrinos have a mass, they cannot travel at the speed of light, but if their mass is small, their velocity would be almost that of light. (a) Suppose a neutrino with a $7-\mathrm{eV} / c^{2}$ mass has a kinetic energy of $700 \mathrm{keV}$. Find the relativistic quantity $\gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}}$ for it. (b) If the neutrino leaves the 1987 A supernova at the same time as a photon and both travel to Earth, how much sooner does the photon arrive? This is not a large time difference, given that it is impossible to know which neutrino left with which photon and the poor efficiency of the neutrino detectors. Thus, the fact that neutrinos were observed within hours of the brightening of the supernova only places an upper limit on the neutrino's mass. (Hint: You may need to use a series expansion to find $v$ for the neutrino, since its $\gamma$ is so large.)

Key Concepts

Time-of-flight Measurements in Astrophysics

Time-of-flight measurements involve comparing the travel times of particles moving nearly at the speed of light over vast distances. Small differences in velocity can result in measurable time delays when propagation occurs over astronomical scales. This technique is crucial for setting upper bounds on the masses of particles and for testing the predictions of special relativity, as even a minuscule mass can cause a particle to travel at a speed slightly less than c, leading to observable delays in arrival times compared to light.

Series Expansion Approximations in Special Relativity

When dealing with particles that move at speeds very close to the speed of light, the Lorentz factor becomes very large and direct computation of v from ? can be challenging. Series expansion techniques provide approximate solutions under the assumption that the deviation from the speed of light is very small. This method simplifies the calculations, making it possible to estimate slight differences in velocity and the corresponding arrival times over astronomical distances.

Lorentz Factor (?)

The Lorentz factor is a critical concept in special relativity that quantifies how time, length, and relativistic mass change for an object moving at a significant fraction of the speed of light. It is defined as ? = 1/?(1 - v²/c²), where v is the velocity of the particle and c is the speed of light. This factor is central in relating the kinetic energy of a particle to its rest mass energy and is essential for understanding the behavior of particles traveling at relativistic speeds.

Relativistic Energy and Its Relation to Mass

Relativistic energy encompasses both the kinetic energy and the rest energy of a particle, following the relation E = ?mc². This equation highlights how, as a particle’s velocity approaches the speed of light, its energy increases dramatically due to the Lorentz factor. The interplay between rest mass and kinetic energy is especially important when dealing with particles that have very small masses but extremely high energies, as even small mass values can lead to large ? factors.

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