Astroparticle Physics

6.6 Problems 139the star and its temperature. What would be the luminosityof a star ten times larger than our Sun (R = 10 R ⊙ )but at the same temperature? What would be the luminosityof a star of the size of our Sun but with ten timeshigher temperature?3. The power radiated by a relativistic electron of energy Ein a transverse magnetic field B through synchrotron radiationcan be worked out from classical electrodynamicsto beP = e2 c 32π C γ E 2 B 2 ,wherer eC γ = 4 3 π (m e c 2 ) 3 ≈ 8.85 × 10−5 mGeV −3 .Work out the energy loss due to synchrotron emissionper turn of a 1 TeV electron in a circular orbit around apulsar at a distance of 1000 km. What kind of magneticfield had the pulsar at this distance?4. Consider the energy loss by radiation of a particle movingin a transverse homogeneous time-independent magneticfield, see also Problem 3 for this section. The radiatedpower is given bye 2P = 2 3 m 2 γ 2 |ṗ| 2 .0 c3Calculate the time dependence of the particle energy andof the bending radius of the trajectory for thea) ultrarelativistic case,b) general case. 165. An X-ray detector on board of a satellite with collectionarea of 1 m 2 counts 10 keV photons from asource in the Large Magellanic Cloud with a rate of1/hour. How many 10 keV photons are emitted fromthe source if isotropic emission is assumed? The detectoris a Xe-filled proportional counter of 1 cm thickness(the attenuation coefficient for 10 keV photons isµ = 125 (g/cm 2 ) −1 , density ϱ Xe = 5.8 × 10 −3 g/cm 3 ).6. X rays can be produced by inverse Compton scatteringof energetic electrons (E i ) off blackbody photons (energyω i ). Show that the energy of the scattered photonω f is related to the scattering angles ϕ i and ϕ f by16 The solution to this problem is mathematically demanding.