Astroparticle Physics

190 8 CosmologyGM 2R > 3 2 kT M µ ,where 3 2 kT is the thermal energy of a molecule, M µis thenumber of molecules, and µ is the molecule mass. Derivefrom this condition the stability limit of a gas cloud(Jeans criterion).3. Let us assume that a large astrophysical object of constantdensity not stabilized by internal pressure is aboutto contract due to gravitation. Estimate the minimum rotationalvelocity so that this object is stabilized againstgravitational collapse! How does the rotational velocitydepend on the distance from the galactic center?4. Derive the acceleration equation (8.29) from the Friedmannequation (8.15) with the help of the fluid equation(8.28).5. Show that the gravitational redshift of light emitted froma massive star (mass M)ofradiusR isνν= GMc 2 R .6. Estimate the classical value for the deflection of starlightpassing near the Sun.7. Clocks in a gravitational potential run slow relative toclocks in empty space. Estimate the slowing-down ratefor a clock on a pulsar (R = 10 km, M = 10 30 kg)!8. Estimate the gravitational pressure at the center of theSun (average density ϱ = 1.4g/cm 3 ) and the Earth (ϱ =5.5g/cm 3 ).9. Estimate the average density ofa) a large black hole residing at the center of a galaxyof 10 11 solar masses,b) a solar-mass black hole,c) a mini black hole (m = 10 15 kg).10. The orbital velocity v of stars in our galaxy varies upto distances of 20 000 light-years as if the density werehomogeneous and constant (ϱ = 6 × 10 −21 kg/m 3 ). Forlarger distances the velocities of stars follow the expectationfrom Keplerian motion.a) Work out the dependence of v(R) for R