It is possible that you might not have some of the information
required to solve parts of these problems.
Please please please ask me for help if that is the case.
Perhaps we will discuss them in class before they are due.
- Oh, no! All the protons and electrons in the Sun
spontaneously and completely
turn into neutrons, generating a flash of neutrinos!
- How many neutrino-neutron interactions will be recorded
in the SAGE detector?
- Some of these neutrinos will interact with the
nuclei within human bodies.
For a typical adult human, estimate
the increase in body temperature due to the
interactions.
(Don't worry too much about this one,
everybody will soon die anyway -- the Sun has stopped shining)
Just for fun, you might read
What If's take on lethal neutrino radiation.
Note that the approach we are taking (raising body temperature)
is different than the one taken by Randall Munro (which involves
the ionizing effects of the radiation).
- Detecting supernova explosions by the neutrino pulses they emit
sounds like a great idea. But just how far could we push
the technique? The nearest big cluster of galaxies,
the Virgo Cluster, is at a distance of about 20 Mpc.
Could we detect a core-collapse SN in that cluster using
current neutrino telescopes?
The actual number of neutrinos emitted during a core-collapse
event is a complicated quantity to estimate, and can vary
considerably depending on the model one uses.
Let's adopt the models of
Sukhbold et al., ApJ 821, 38 (2016).
Look at their Figure 8 to get a rough idea for the
typical energy emitted by a supernova in neutrinos.
- What is a typical value for this energy, in ergs?
- Assume that all neutrinos have an energy of
10 MeV. How many neutrinos are emitted?
- The Ice Cube detector has a volume of roughly
1 cubic kilometer. Assume the ice has a density
of 970 kg per cubic meter.
How many electrons are contained within Ice Cube?
- Assume the scattering cross section between electrons
and neutrinos of energy 10 MeV is 9 x 10-48
square meters.
What is the total cross section of Ice Cube?
- Suppose that scientists require a total of
N = 10 interactions over the course of a few seconds
to declare that an event has been detected.
How far away, in meters, could Ice Cube
detect a supernova explosion?
- Could it detect an event in the Virgo Cluster?
- Gamma rays have a lot of energy -- much more than optical photons.
Is that a good thing or a bad thing from an observational point of view?
Let's see ...
The Crab Pulsar is the remnant of a supernova explosion that was witnessed
humans in the year 1054. The tiny neutron star spins very rapidly,
with a period of about P = 33 milliseconds.
It emits both optical and gamma-ray photons.
How easy is it to detect the flashes of this spinning neutron star
in each section of the electromagnetic regime?
According to
Sollerman et al. (2019),
if one were to observe the Crab Pulsar in the optical R-band,
at about 7000 Angstroms, one would collect a flux of
- Suppose that one uses an optical telescope with an aperture of
radius R = 50 cm to observe in the R-band.
Treat all the photons as having a wavelength of exactly
7000 Angstroms.
How many photons would the telescope collect each second?
- Let's simplify and pretend that all the energy emitted by the pulsar
during each period
comes out in a brief flash when the magnetic pole of the pulsar
points toward the Earth.
How many photons would we collect from each of these individual
flashes?
(If this is a large number, then it ought to be easy to notice
the periodic changes in the brightness of the pulsar over time)
- Now pretend that the pulsar emits the same amount of energy
in a gamma-ray passband, centered on an energy of
10 MeV. Suppose one uses a gamma-ray telescope of exactly
the same size: an aperture of radius R = 50 cm.
How many gamma-ray photons will this telescope collect
each second?
- How many gamma-ray photons will be collected from a single
flash of the pulsar? Will it be easy to detect the
rotation of the neutron star using this instrument?
