Astroparticle Physics

17.15 Chapter 15 3790 =−F eff (r) = dV eff(r)dr= (n − 2)C nr n−1( ) 1− L2mn−4˜C nmr 3 ⇒ r orb =L 2with ˜C n = (n − 2)C n , n ̸= 4, also applicable for n = 2. Stable orbits are given forpotential minima:∣ ()∣0 < d2 V eff (r) ∣∣∣∣rorbdr 2 = − (n − 1) ( ) 4˜C nr n + 3L2 ∣∣∣∣rorbmr 4 = (4 − n) L2 L2 n−4m m ˜C nThus, stable motion is only possible for n2 ,which is similar to the expression of the potential energy. For n > 2 the limitΛ →∞leads to a finite result, whereas W diverges for λ → 0. In the case n = 2both limits are divergent. Quantum corrections, here for the so-called self-energycorrections, are supposed to become significant for small distances. Therefore, thelimit λ → 0 may diverge, whereas the limit Λ →∞should exist. Thus, n>2dimensions are considered valid from this aspect. The degree of divergence is thensmallest for n = 3.λ.17.15 Chapter 151. There is not yet a solution. If you have solved the problem successfully, you shouldbook a flight to Stockholm because you will be the next laureate for the Nobel Prize inphysics.