# Precession of relativistic orbit in pure inverse-square force

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1. Oct 26, 2014

### RylonMcknz

PROBLEM:
Show that Special Relativity predicts a precession of π(GMm/cl)2 radians per orbit for any elliptic orbit under a pure inverse-square force.

where G is gravitational constant, M is mass of larger body, m is mass of smaller orbiting body, c is speed of light and l is angular momentum.​

HINTS:
Differential equation for Newtonian inverse-square force orbit: d2u/dθ2 + u = mk/l2 (Eq.1), where u = 1/r, k = GMm.

To make this relativistic: mk→ϒmk, where ϒ = 1/√(1 - (v2/c2)).

Also E = ϒmc2 + U.

Then right side of (Eq. 1) becomes ⇒ϒmk/l2 = (E - U)k/c2l2 = (E + ku)k/c2l2 (Eq. 2).

Precession causes the following change of phase angle (in radians/orbit):

Δθ - Δθ' = 2π/ω - 2π = 2π(1/ω - 1) (Eq. 3).​

ATTEMPT:
(Eq. 1) & (Eq. 2) d2u/dθ2 + u = (E + ku)k/c2l2

d2u/dθ2 + u - k2u/c2l2 =
Ek/c2l2

⇒ d2u/dθ2 + (1 - k2/c2l2)u = Ek/c2l2
,

where ω2 = (1 - k2/c2l2) (Eq. 4),
and let A = Ek/c2l2.

Then substitute y = u - A/ω2.

Then we have d2y/dθ2 + ω2[y + A/ω2] - A = 0

d2y/dθ2 + ω2y = 0

Solution is then y = y0cos(ωθ - θ0).

I then try substituting (Eq. 4) into (Eq. 3) to find the resulting precession in radians per orbit, but this doesn't not give the desired result. This is where I'm stuck, not sure what I'm doing wrong. Thank you for your considerations. ​

2. Oct 31, 2014