A test particle is held (by some external agency

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Schwarzschild spacetime and the tests of general relativity

The Geometry of Gravity, Ch Orbits and light rays in Schwarzschild spacetime

Problem

A test particle is held (by some external agency, for the moment) to move on a circle of fixed areal radius r=1000r = 1000 km around a compact central body. Use GM⊙/c2=1.477GM_\odot/c^2 = 1.477 km. ((a)) For M=1 M⊙M = 1\,M_\odot, 5 M⊙5\,M_\odot and 20 M⊙20\,M_\odot, compute the ratio rs/rr_s/r, the relative size of the new term in VeffV_{\rm eff} compared with the centrifugal term at this radius. ((b)) Find the critical mass McritM_{\rm crit}, in solar masses, at which the innermost stable circular orbit 6GM/c26GM/c^2 coincides with r=1000r = 1000 km. ((c)) State whether a genuine (unforced) stable circular orbit can exist at r=1000r=1000 km for MM above or below McritM_{\rm crit}.

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