Astroparticle Physics

140 6 Primary Cosmic Rays1 − cos ϕ iω f ≈ ω i , (6.88)1 − cos ϕ fwhere ϕ i and ϕ f are the angles between the incomingelectron and the incoming and outgoing photon. Theabove approximation holds if E i ≫ m i ≫ ω i .7. What is the temperature of a cosmic object if its maximumblackbody emission occurs at an energy of E =50 keV?(Hint: The solution of this problem leads to a transcendentalequation which needs to be solved numerically.)Problems for Sect. 6.51. A photon propagating to a celestial object of mass Mwill gain momentum and will be shifted towards theblue. Work out the relative gain of a photon approachingthe Sun’s surface from a height of H = 1km. Analogously,a photon escaping from a massive object will begravitationally redshifted.(Radius of the Sun R ⊙ = 6.9635 × 10 8 m, Mass of theSun M ⊙ = 1.993 × 10 30 kg, acceleration due to Sun’sgravity g ⊙ = 2.7398 × 10 2 m/s 2 .)2. Accelerated masses radiate gravitational waves. Theemitted energy per unit time is worked out to beP = G ...5c 2 Q 2 ,where Q is the quadrupole moment of a certain massconfiguration (e.g., the system Sun–Earth). For a rotatingsystem with periodic time dependence (∼ sin ωt)each time derivative contributes a factor ω, henceP ≈ G5c 2 ω6 Q 2 .For a system consisting of a heavy-mass object like theSun (M) and a low-mass object, like Earth (m), thequadrupole moment is on the order of mr 2 . Neglectingnumerical factors of order unity, one getsP ≈ G c 2 ω6 m 2 r 4 .Work out the power radiated from the system Sun–Earthand compare it with the gravitational power emittedfrom typical fast-rotating laboratory equipments.