Apsidal Angles & Precession Brief Discussion! Particle undergoing bounded, non-circular motion in a central force field  Always have r 1 < r < r 2 V(r) vs r curve  Only 2 apsidal distances exist for bounded, noncircular motion.

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 φ -  

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/r 2 force  All elliptic orbits must be EXACTLY closed!  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/r 2.” A sensitive test of Newton’s Law of Gravitation!

Precession FACT: Planetary motion: Total force experienced by a planet deviates from 1/r 2 (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´´  °

Discrepancy between observation & classical dynamics calculations: 574   = 43  (  °) 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.

Back a few lectures: General central force f(r): Differential eqtn which for orbit u(θ)  1/r(θ) (replacing mass μ with the planetary mass m): (d 2 u/dθ 2 ) + u = - (m/ 2 )u -2 f(1/u) (1) Alternatively: (d 2 [1/r]/dθ 2 ) + (1/r) = - (m/ 2 )r 2 F(r) Put the universal gravitation law into this: f g (r) = -(GMm)/(r 2 ) & get: (d 2 u/dθ 2 ) + u = (Gm 2 M)/( 2 ) (2) GR (+ SR) modification of (2) (lowest order in 1/c 2, c = speed of light): Add additional force to F g (r) varying as (1/r 4 ) = u 4 : F GR (r) = -(3GMm 2 )/(c 2 r 4 ) = -(3GMm 2 )u 4 /(c 2 ) Put F g (r) + F GR (r) into (1):

 (2) is replaced by: (d 2 u/dθ 2 ) + u = (Gm 2 M)/( 2 ) + (3GM)u 2 /(c 2 ) (3) Let: (1/α)  (Gm 2 M)/( 2 ), δ  (3GM)/(c 2 )  (3) becomes: (d 2 u/dθ 2 ) + u = (1/α) + δu 2 (4) Now some math to get an approximate soln to (4): (Like a nonlinear harmonic oscillator eqtn!) Note: The nonlinear term δu 2 is very small, since δ  (1/c 2 )  Use perturbation theory as outlined on pages of Marion (handed out in class!)

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- e 2 ); k = GMm, a = semimajor axis  (5) becomes: Δ  [6πGM]/[ac 2 (1- e 2 )] (6)

Prediction for an elliptic orbit, e = eccentricity, a = semimajor axis  Δ  [6πGM]/[ac 2 (1- e 2 )] (6) (6)  An enhanced Δ for small semimajor axis & large eccentricity.  Earlier table: Mercury (e = , a = AU) should have the largest effect! Get: Mercury: Calc.: Δ   /century. Obs.: Δ   /century!