Jupiter has mass kg and radius m

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

Jupiter has mass

M_J = 1.898\times10^{27}

kg and radius

R_J = 7.15\times10^7

m, so

GM_J/c^2 = 1.41

m. Use order-of-magnitude reasoning (you may use the formulas already derived for Mercury, the Shapiro delay and the Einstein ring, without rederiving them) to estimate three effects associated with Jupiter. ((a)) Io orbits Jupiter with semi-major axis

a = 4.22\times10^8

m, eccentricity

e = 0.0041

and period

T_{\rm orb} = 1.77

days. Estimate the general-relativistic perihelion (here perijove) advance of Io's orbit, in arcseconds per century. ((b)) A spacecraft signal is sent from Earth, grazing Jupiter's limb (

b = R_J

), to a probe a further

1.50\times10^{11}

m beyond Jupiter, with the Earth–Jupiter distance at the time

r_1 = 6.30\times10^{11}

m. Estimate the Shapiro time delay of the round trip. ((c)) Treat Jupiter as a point lens at distance

D_{\rm L} = 6.30\times10^{11}

m from Earth, with a background star far beyond it (

D_{\rm LS}\approx D_{\rm S}

). Estimate the angular radius of the Einstein ring, and compare it with Jupiter's own angular radius as seen from Earth at that distance, commenting on whether the ring could actually be observed.

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