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.

• 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/r2 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/r2.” A sensitive test of Newton’s Law of Gravitation!

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°

• 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.

• 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):

(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) = -(3GMm2)/(c2 r4)

= -(3GMm2)u4/(c2)

• Put Fg(r) + FGR(r) into (1):

(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 an approximate soln 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!)

• 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)

• 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!