Astroparticle Physics

362 17 Solutions7. An observer in empty space measures times with an atomic clock of frequency ν 0 .Thesignals emitted by an identical clock on the surface of the pulsar reach the observer inempty space with a frequency ν = ν 0 − ν where νin this chapter) ⇒f pulsar= ν 0 − νf empty space ν 0ν 0= Uc 2= GM (see Problem 5Rc2 = 1 − GMRc 2 = 1 − ε.For the pulsar this gives ε = 0.074, i.e., the clocks in the gravitational potential on thesurface of the pulsar are slow by 7.4%. For our Sun the relative slowing-down rate, e.g.,with respect to Earth, is 2 × 10 −6 . At the surface of the Earth clocks run slow by 1.06 ×10 −8 with respect to clocks far away from any mass, from which just 7 × 10 −10 resultsfrom the Earth’s gravitation and the main contribution is caused by the gravitationalpotential of the Sun.8. Gravitational forcedF =−G M(r)dmr 2 =− GM(r)r 2 ϱ(r) dr 4πr 2 , inward forcedp =dF4πr 2 =−GM(r) dpr 2 ϱ(r) dr ,dr =−GM(r) r 2 ϱ(r) . (∗)dmrdrRM(r)On the other hand dpdr≈p(R) − p(r = 0)R=− p . Compare with (∗) and with the re-Rplacement r → R and ϱ(r) → average density ϱ one gets p R = GMR 2 ϱ.If a uniform density ϱ(r) = ϱ is assumed, (∗) can be integrated directly usingM(r) = 4π 3 ϱr3 and thus dpdr =−4π 3 Gϱ2 r,p(0) =∫ 0Rdpdr dr = 4π ∫ R3 Gϱ2 r dr = 4π 30R2Gϱ22 = 1 GM2 R ϱ.This leads to p = 1 {GM 1.3 × 10 142 R ϱ ≈ N/m 2 for the Sun1.7 × 10 11 N/m 2 for the Earth .