Physics 554/ Astronomy 510Nuclear AstrophysicsAssignment 4Due: Friday, November 231. In class we derived a formula for the survival probability of an electron neutrino under-going vacuum oscillations. Following that calculation exactly, calculate the probability thatit has become a muon neutrino at point x. (That is, take the overlap of the wave functionwith a |νµi.) Show that the sum of the two probabilities is one. Graph the electron andmuon probabilities as a f unction of x.2. Rewrite our formula for the vacuum neutrino oscillation length by using the followingnormalized dimensionless quantities: E/MeV; δm2/eV2; Lo/1 meter. If you do things right,all that should remain is some dimens ionless number relating Loto the other quantities.Stopped pions produce neutrinos of about 35 MeV average energy. If you are searching forneutrino oscillations corresponding to δm2∼ 2.5 × 10−3eV2, how far from the source shouldyou place your detector? Do you think it might be helpful to use more than one location?What if the neutrinos had energies of 2.5 GeV? This last answer tells you how far an under-ground lab should be from Fermilab to test the atmospheric neutrino oscillation results ofSuperKamiokande, assuming a new proton driver and neutrino superbeam.3. Consider a monoenergetic electron neutrino beam of energy E. It travels from a sourcethrough vacuum a distance Lo, then through matter a distance Lo, with Lodefined as inproblem 2. It exits the matter and is immediately measured. Calculate the electron neutrinosurvival probability as a function of the electron density in the matter. Note this problem isvery different from the MSW problem considered in class. It involves propagation throughconstant density, except for an abrupt change in density as the neutrino enters or exits mat-ter. The appropriate approximation to use is the sudden approximation. (See me for anexplanation if you are not familiar with this.) From the result you find, describe the effectsof the matter. In this exploration you can pick a δm2of 10−5eV2, a mixing angle of 45degrees, a neutrino energy of 1 GeV, and consider only two flavors (say electron and muon).Note that, if the matter is removed, the experimentalist would measure the full flux of elec-tron neutrinos. Is it possible for the experimentalist to measure a smaller flux of electronneutrinos, because of the matter?4. A supernova explodes in the Large Magellanic Cloud, 50 kpc from earth. The burstof neutrinos turns on suddenly at t = 0 and then declines in time as e−t/t0, where t0∼ 2seconds. The distribution in energy can be taken as a Fermi-Dirac characterized by a tem-perature T ∼ 5 MeV. The neutrinos are observed on earth in a detector, and the spreadin the arrival times of the neutrinos recorded. Estimate what neutrino mass would causethe duration of the observed events to have approximately doubled, due to spreading of the1signal during transit from the LMC to earth. Note this is about the kinematic effects of theneutrino mass, not about oscillations.5. We mentioned in class the explosive burning of28Si into iron-group nuclei. It was alsonoted that the peak temperature achieved when the shock wave passes through the Si shellis about T9∼ 6.a) Consider the process28Si +28Si →56Ni + γ. Estimate the rate for this reaction in sufficientdetail that you can answer the question: will this reaction make significant56Ni under theconditions described above? Feel free to assume resonant production, leaving the widths asunknowns but basing your conclusions on the Coulomb physics.b) Similarly, give a rough estimate of the rates of the reaction28Si (γ, α)24Mg relative tothat in a). Again, leave the widths unspecified, but otherwise handle the Coulomb physicsproperly.c) Reactions such as those in b) generate free αs, which can in turn do (α, γ) reactions.Calculate the ratioN(28Si)N(α)N(32S)where N represents a number density, in terms of the mass difference Q between28Si andα+32S. Evaluate this expression at T9=6. If you use the Saha equation, explain why this isok based on b). Could I also apply the Saha equation to the reaction in a)?d) We noted that the binding energy per nucleon increases until one reaches iron. Give asimple argument, based on physical insight or thermodynamics, about how Si burns to