Astroparticle Physics

286 13 Dark Matter3. Derive the Fermi energy of a classical and relativisticgas of massive neutrinos. What is the Fermi energy ofthe cosmological neutrino gas?4. If axions exist, they are believed to act as cold dark matter.To accomplish the formation of galaxies they mustbe gravitationally bound; i.e., their velocities should notbe too high.Work out the escape velocity of a 1 µeV-mass axion froma protogalaxis with a nucleus of 10 10 solar masses ofradius 3 kpc!5. Consider a (not highly singular) spherically symmetricmass density ϱ(r), the total mass of which is assumed tobe finite.a) Determine the potential energy of a test mass min the gravitational field originating from ϱ(r). Theforce is given by Newton’s formula in which onlythe mass M(r) inside that sphere enters, whose radiusis given by the position of m.b) With the previous result calculate the potential energyof the mass density, where the test mass is replacedby dM = M ′ (r) dr and the r integration isperformed to cover the whole space. Show with anintegration by parts in the outer integral that the totalpotential energy can be written as∫ ∞M 2 (r)E pot =−Gr 2 dr.0c) Determine the potential energy of a massive sphericalshell of radius R and mass M, i.e., M(r) = 0forrR.d) Work out the potential energy of a sphere of radiusR and mass M with homogeneous mass density.(This problem is a little difficult and tricky.)6. Motivate the lower mass limit of neutrinos based on theirmaximum escape velocity from a galaxy (13.17), if it isassumed that they are responsible for the dynamics ofthe galaxy.7. In the discussion on the search for MACHOs there is astatement that the radius of the Einstein ring varies asthe square root of the mass of the deflector. Work outthis dependence and determine the ring radius for• distance star–Earth = 55 kpc (LMC),• distance deflector–Earth = 10 kpc (halo),• Schwarzschild radius of the deflector = 3 km (correspondingto the Schwarzschild radius of the Sun).