ASTR 498Problem Set 5Due Thursday, April 171. [4 points] In accretion onto nonrelativistic and weakly magnetic nonrotating stars, half of theaccretion energy is released in the inspiral to just above the surface, and half is released in theboundary layer formed when the matter slows down to a stop on the surface. What about for aneutron star? Assuming that matter (a) spirals in nearly-circular orbits to the ISCO, (b) drops tothe surface without releasing further energy, then (c) releases any additional energy in a boundarylayer so that it comes to rest on the surface, calculate in full general relativity the fraction ofthe total energy released in the boundary layer for neutron star radii of R = 6M , R = 5M , andR = 4M . Assume the star is nonrotating. These fractions suggest that high-amplitude timingphenomena in the accretion-powered X-rays (such as quasiperiodic oscillations) may have theirclock set in the disk, but the energy release must primarily be on the stellar surface.2. [4 points] Planetary Solutions, Inc. has hired you as a consultant. An alien race has decidedthat it would be chic to put their planet into a circular orbit around the Crab Pulsar. Their planetis very similar to Earth, so they want to be placed where the flux from the total spindown energyof the pulsar is equal to the flux from the Sun received at Earth. Using the standard dipolarspindown model, calculate the distance that is needed (and show all your work). You’ll need tolook up the spin rate of the Crab as well as its magnetic field (please give me the URL of anyreferences); note that the pulse shape from the Crab tells us that it is an orthogonal rotator, withsin α = 1.This isn’t part of your grade on this problem, but can you think of any special difficultiesabout this environment? What warning might you give?3. [4 points] The most rapidly rotating known pulsar spins at 716 Hz. Suppose that the magneticdipole moment of this pulsar is 3 × 1026G cm3and that the current spin rate was the equilibriumspin rate (i.e., the orbital frequency at the Alfv´en radius) during accretion. As a function ofthe unknown mass of the pulsar (measured in solar masses), compute the average accretion rateduring this phase.4. [4 points] Dr. Sane has realized that there is another possible energy source for supernovae,which has been overlooked by lesser scientists. His calculations suggest that during the collapse ofthe core, various processes result in about 1% of the neutrons escaping from nuclei as free particles(i.e., basically in zero density environments). His proposal is that these neutrons then decay intoprotons, electrons, and electron antineutrinos, and that this energy release powers the supernova.J. Craig Wheeler, president of the American Astronomical Society, is investigating this modelfor a possible award of a major prize to Dr. Sane. He asks you to do a calculation to see if thismodel works. The input: the core itself starts at 1.4M¯of mass, of which initially about half isneutrons. Supernovae produce about 1051ergs of energy in photons and kinetic energy. You’llneed to look up the masses of neutrons, protons, and electrons. Give Dr. Sane the maximumbenefit of the doubt by assuming that all of the energy in his neutrinos couples to matter andhelps drive the