Astroparticle Physics

14.1 Problems 2913. What determines whether a planet has an atmosphere?Hint: Gas atoms or molecules will be lost from an atmosphereif their average velocity exceeds the escapevelocity from the planet.4. Rotationally invariant long-range radial forces: (nonrelativistic)two-body problem in n dimensions. 2a) How does such a force F between two bodies dependon the distance between two point-like bodies?Recall from three dimensions how the force dependson the surface or the solid angle of a sphere, respectively(‘force flow’). The (hyper)surface s n (r)of the n-dimensional (hyper)sphere can be writtenas s n (r) = s n r g(n) ,whereg(n) needs to be fixed.b) What is the corresponding potential energy and howdoes the effective potential (which includes the centrifugalbarier) look like? Find conditions for stableorbits with attractive forces.c) Consider the introduction of a field-energy densityw ∼ F 2 (mediated by a field strength ∼ F )andintegrate w within the range of the two radii λ andΛ. Then-dimensional volume element is given bydV n = s n (r) dr, see also part (a). At what n doesthis expression diverge for λ → 0orΛ → ∞?If quantum corrections are supposed to compensatethe divergences, at what limit should they becomerelevant?2 This problem discusses unconventional dimensions. It is quitetricky and mathematically demanding.