Question

2. Fundamental Planetary Science problem 2.5 The most easily observable effect of general relativity is the precession of orbits because it is nil in the Newtonian two-body approximation. The first-order (weak-field) general relativistic corrections to Newtonian grav- ity imply a precession of the periapse of orbit of a small body (m2 << mı) about a primary of mass mı at the rate as/(1- where c is the speed of the light. Calculate the general relativistic precession of the periapse of the following object. a) Mercury b) Earth c) Io's orbit around Jupiter, Quote your answer to arcseconds per year.

Answer 1

General relativity gives rise to small correction to the Force per unit mass exerted by the delta in (mass M_o) as a planet in a circular orbit of radius r, angular man. Per unit mass h. In fact, the modified formula for F is F Tilde equal - G M_o/r^2 - 3 M_o h^2/c^2 r^4 It follows that rF' = -2 (1 + 3 h^2/c^2 r^2 + ...) The angle is psi = pi (1 + 3 h^2/c^2 r^2) Thus, the Peri helian advances by delta psi = 6 pi M_o/c^2 r The rate of Peri helian precession due to general relativity correction is delta psi = 0.383/RT where, R = mean orbital Radius in mean earth orbital radii, T = orbital Period in years. Now, we have delta psi_mercury = 0.41 arc seconds/year delta psi_earth = 0.0383 arc s/year delta psi_Jupiter = 0.000621 arc sec/year delta psi_Io's around Jupiter = 27.9 arc sec/year