ASTR 680Problem Set 5Due Thursday, April 161. Fermi energy.Suppose that you have a star that is supported by nonrelativistic degeneracy (of electrons orneutrons, it doesn’t matter). Derive the depe ndence of the radius R of the star in equilibrium onits mass M given in the notes (for example, if radius depended on mass to the seventh power,which it doesn’t, you would write R ∝ M7).To do this, first make the simplifying assumption that the density is constant, and that themass is carried by protons and neutrons and the degenerate particles form a fixed fraction ofthe total particles. Determine the total energy per particle (including both Fermi energy andgravitational potential energy). Then, at a fixed stellar mass M, minimize with respect to theradius R of the star.2. What about magnetic fields?Dr. Sane has once again stunned the astrophysical community with his unmatched breadth andvision. Ordinary researchers have noted the “ankle” in the cosmic ray spectrum above ∼ 1019eVwithout realizing its significance. Dr. Sane has rushed to fill this well-needed gap in the literature.More specifically, he notes that a neutron star with a magnetic field B, radius R, and rotationfrequency Ω (in radians per second) can produce a potential drop of BΩR2/c in statvolts (recallthat one statvolt equals 300 volts). He asserts that neutron stars can therefore accelerate protonsto > 1019eV, and in his model it is necessary that a given star be able to do this for a thousandyears to explain the ankle.The cosmic ray community is contemplating awarding Dr. Sane the Yodh Prize for careercontributions to cosmic ray research. They have, howe ver wisely consulted you first. To makeyour evaluation, assume R = 106cm. However, you should be open-minded about B (allowing itto be between 108G and 1016G) and Ω (allowing it to be anywhere between 0 and 104rad s−1).With these parameters, what can you say about Dr. Sane’s latest idea? Assume an orthogonalrotator, meaning that the magnetic axis is perpendicular to the rotation axis.3. Spinup of neutron star. Suppose you have a neutron star of mass M and moment of inertiaI that is initially nonrotating. It has a dipolar moment µ, and matter accretes on the star at arate˙M. Assuming that the matter couples to the field exactly at the Alfv´en radius, derive thecharacteristic time for the star to reach spin equilibrium. By “characteristic time” we mean theequilibrium spin frequency divided by the initial rate at which the spin frequency changes (inreality the spin frequency would asymptote to equilibrium, but we want just the characteristictime). Assume that the Alfv´en radius is large enough that a Newtonian expression for the specificangular momentum there is okay, and that you can neglect the change of mass in the neutron starduring the accretion. What is the characteristic time for a star with M = 1.4 M⊙, I = 1045g cm2,µ = 1030G cm3, and˙M = 10−8M⊙yr−1, which might be typical for a young neutron star in ahigh-mass X-ray binary?4. Helium core flash.Let’s get some insight into thermonuclear flashes in a slightly different context: the evolution of amassive star. Given the equations below (which indicate the energy generation rate per time intriple-alpha fusion of helium, plus how the temperature adjusts to that energy), write a code tofollow the temperature with time.I’ll need from you (a) a plot of the log temperature as a function of time (in units of days),and (b) an e-mail copy of your code, which has to be able to compile and run on the astromachines. Note that the temperature rises very suddenly during the flash, so you need to resolvethe rise; you don’t have to have a tiny time step the whole way, though.The details:Fix the density at ρ = 2 × 105g cm−3and assume the composition is always pure helium. Theinitial temperature is T = 1.5 × 108K. The energy generation rate for the triple-alpha reaction isǫ3α=5.1 × 108ρ2Y3T39e−4.4027/T9erg g−1s−1. (1)where T9≡ T/109K and Y = 1 is the helium mass fraction. For a given time step, in which somenumber of ergs per gram is produced, the change in temperature is determined by the sum of thespecific heats of the helium and the electrons. That is:dTdt=1cV(He) + cV(e)ǫ3α. (2)Here cV(He) = 3.1 × 107erg g−1K−1is the specific heat for helium andcV(e) = 0.346(T/1 K) erg g−1K−1(3)is the specific heat for the electrons (these are not general formulae; the coefficients are specific tothis