Apsidal Angles & Precession

2. Apsidal Angles & Precession • Brief Discussion! • Particle undergoing bounded, non-circular motion in a central force field  Always have r1< r < r2 • V(r) vs r curve Only 2 apsidal distances exist for bounded, noncircular motion.

3. φ -  • Possible motion: Particle makes one complete revolution in θ but doesn’t return to original position (r & θ). That is, the orbit is not closed! • Angle between any 2 consecutive apsides φ  Apsidal Angle • Closed orbit = Symmetric about any apsis  All apsidal angles must be equal for a closed orbit. • Ellipse: Apsidal angle = π

4. If the orbit is not closed  The mass gets to apsidal distances at different θ in each revolution  Apsidal angle is not a rational fraction of 2π. • If the orbit is almost closed Apsides Precess Rotate Slowly in the plane of motion. • 1/r2 force  All elliptic orbits must beEXACTLYclosed!  The apsides must stay fixed in space for all time. • If the apsides move with time, no matter how slowly  Force law is not exactly the inverse square law! Newton: “Advance or regression of a planet’s perihelion would require deviation of the force from 1/r2.” A sensitive test of Newton’s Law of Gravitation!

5. Precession • FACT:Planetary motion: Total force experienced by a planet deviates from 1/r2 (r measured from sun), because of perturbations due to gravitational attractions to other planets, etc. For most planets, this effect is very small. Celestial (classical) dynamics calculations: Account for this (very accurately!) by perturbation theory. • Largest effect is for Mercury: Observed perihelion advances 574 of arc length PER CENTURY! • Accurate classical dynamics calculations using perturbation theory, as mentioned, predict 531 of arc length per century! 1° = 60´ = 3600´´  574´´ 0.159° 531´´ 0.1475°

6. Discrepancy between observation & classical dynamics calculations: 574 - 531 = 43 ( 0.01194°) of arc length per century! Neither calculational nor observational uncertainties can account for this difference! Until early 1900’s this was the major outstanding difficulty with Newtonian Theory! • Einstein in early 1900’s: Showed that General Relativity (GR) accounts (VERY WELL) for this difference! A major triumph of GR! • Instead of doing GR, we can approximately account for this by (in ad-hoc manner) by inserting GR (plus Special Relativity) effects into Newtonian equations.

7. Back a few lectures: General central force f(r): Differential eqtn which for orbit u(θ)  1/r(θ) (replacing mass μwith the planetary mass m): (d2u/dθ2) + u = - (m/2)u-2 f(1/u) (1) Alternatively: (d2[1/r]/dθ2) + (1/r) = - (m/2)r2 F(r) Put the universal gravitation law into this: fg(r) = -(GMm)/(r2) & get: (d2u/dθ2) + u = (Gm2M)/(2) (2) • GR (+ SR) modification of (2) (lowest order in 1/c2, c = speed of light): Add additional force to Fg(r) varying as (1/r4) = u4 : FGR(r) = -(3GMm2)/(c2 r4) = -(3GMm2)u4/(c2) • Put Fg(r) + FGR(r) into (1):

8.  (2) is replaced by: (d2u/dθ2) + u = (Gm2M)/(2) + (3GM)u2/(c2) (3) • Let: (1/α)  (Gm2M)/(2), δ (3GM)/(c2)  (3) becomes: (d2u/dθ2) + u = (1/α) + δu2 (4) • Now some math to get anapproximatesoln to (4): (Like a nonlinear harmonic oscillator eqtn!) • Note: The nonlinear term δu2 is very small, since δ (1/c2)  Use perturbation theory as outlined on pages 319-320 of Marion (handed out in class!)

9. After a lot of work (!), find the apsidal angle (perihelion precession) of θ = (2π)/[1-(δ/α)]  (2π)[1+(δ/α)]  Effect of the GR term is to displace the perihelion in each revolution by Δ 2π(δ/α) = 6π[(GmM)/(c)]2 (5) • Results for elliptic orbit (μ = m): e = eccentricity, 2 = mka(1- e2);k = GMm, a = semimajor axis  (5) becomes: Δ  [6πGM]/[ac2(1- e2)] (6)

10. Prediction for an elliptic orbit, e = eccentricity, a = semimajor axis Δ [6πGM]/[ac2(1- e2)] (6) • (6)  An enhanced Δ for small semimajor axis & large eccentricity.  Earlier table: Mercury (e = 0.2056, a = 0.3871 AU) should have the largest effect! Get: Mercury: Calc.: Δ  43.03/century. Obs.: Δ  43.11 /century!