ASTR 498Problem Set 1Due Thursday, February 141. 4 points Problem 1.8 from book: In a spaceship shuttle service from Earth to Mars, eachspaceship is equipped with two identical lights, one at the front and one at the back. Thespaceships usually travel at a speed v relative to Earth, such that the headlight of a spaceshipapproaching the Earth appears green (λ = 500 nm) and the taillight of a departing spaceshipappears red (λ = 600 nm).(a) 2 points Show that v/c=1/11.(b) 2 points One spaceship accelerates to overtake the spaceship ahead of it. Show that theovertaking spaceship has to travel with a speed of 0.18c relative to Earth so that the taillight ofthe Mars-bound spaceship ahead of it looks like a headlight (i.e., green).2. 4 points Problem 1.9 from book: A particle as observed in a certain reference frame has atotal energy of 5 GeV and a momentum of 3 GeV/c.(a) 1 point What is its mass in GeV/c2?(b) 1 point What is its energy in a frame in which its momentum is equal to 4 GeV/c?(c) 2 points Use the velocity addition formulae, or the energy-momentum transformation, to findthe relative speed of the two frames of reference, if the particles are moving in the same direction.3. 4 points Problem 3.11 from book:The energy loss rate (i.e., energy per time) for a single, ultrarelavistic electron to synchrotronradiation isdEdt≈ −23r20cγ2B2(1)where r0is the classical radius of the electron r0= e2/(mec2), γ = 1/p1 − v2/c2is the Lorentzfactor, and B is the strength of the magnetic field. For an ultrarelativistic electron, E ≈ γmec2and γ À 1.Show that the energy changes with time asE(t) =E01 + t/τ(2)where τ is the synchrotron loss time and E0is the energy at t = 0. Also derive τ; for each variablethat τ depends on (examples would be the magnetic field strength B, or the charge e of theelectron), discuss whether the dependence should be direct (increasing as the variable increases)or inverse (decreasing as the variable increases).4. 4 points The well-known theoretical gadfly Dr. I. M. N. Sane has realized that a fundamentalerror is being made in studies of core-collapse supernovae. When collapse to a neutron starhappens, a characteristic temperature is T = 1012K. Ordinary theorists think that neutrinosproduced in this environment have an energy of 3–5 MeV (where 1 eV=1.6 × 10−12ergs, andthe mass-energy of an electron is 511 keV), but Dr. Sane asserts that the real energy is of orderkT . There are about 1057electrons uniformly distributed in the inner 100 km of the star, andneutrinos scatter off of them, delivering momentum and energy. Everyone, including Dr. Sane,agrees that about 1% of neutrinos interact in this way; more than that, and supernovae would betoo energetic.The Washington Post wants to do a story on this, but has consulted you first. Do aquantitative calculation to determine if Dr. Sane’s idea hangs