(a) An object is observed to have a total synchrotron luminosity L. Show that the total energy of the system is minimized when UeUB, i.e. when the electrons and magnetic field are close to equipartition. (b) The radio lobes of the radio galaxy Cyg A are roughly 50kpc across and have a total luminosity 1045ergs1 observed at radio frequencies of several GHz. Assuming a typical magnetic field B104G what else can you deduce about the physical conditions of the plasma (e.g. mean electron density, average energy of the electrons, etc.)? What is the cooling time of the electrons? Crab Nebula Spectral Energy Distribution (SED) Figure 1: Spectrum energy distribution (SED) from Crab Nebula c) Consider a source of synchrotron radiation which is self-absorbed. Write the luminosity as roughly L=pL, where p is the frequency at which the spectrum peaks. Also take the brightness temperature in the self-absorbed part of the spectrum to be given by kBTbmcc2. Calculate the value of Tb at which the energy density in the synchrotron photons is equal to UB (you should find Tb1012K for p=1GHz ). What would happen if the brightness temperature exceeded this value (this is known as the inverse Compton catastrophe; see e.g. Readhead 1994, ApJ, 426.51)