1. Phases of the Moon

2. Spin and orbital frequencies

3. Libration of the moon: observed by Galileo

4. Orbital acceleration Acceleration = V 2 / r Measure the magnitude of the force Force= m V 2 / r

5. Elliptical orbits * A – Minor, orbiting body * B – Major body being orbited by A * C – Reference plane, e.g. the * D – Orbital plane of A * E – Ascending node * F – Periapsis * ω – Argument of the periapsis The red line is the line of apsides; going through the periapsis (F) and apoapsis (H); this line coincides wíth the major axix in the elliptical shape of the orbit The green line is the node line; going through the ascending (G) and descending node (E); this is where the reference plane (C) intersects the orbital plane (D).

6. Basic mathematics Two bodies in an inertial frame, F i,j =-GM i M j (r i – r j )/d ij 3 In a frame centered on the Sun, equation of motion: d 2 r /dt 2 = -  r/r 3  = - G(M i + M j ) includes an indirect term. Nonlinear force: but has analytic solutions Elliptical orbits: semi major axis a, Eccentricity e & longitude of peri apse   r=a(1-e 2 )/(1+e cos  ) Conservation of angular momentum: L = r x dr/dt = rV  =[  a(1-e 2 )] 1/2, energy E= (V   +V r 2 )/2 –  /r = -  /2a, and the longitude of periapse  for a point-mass potential Period P= 2  (a 3 /  ) 1/2 Mean motion n=2  /P Solutions to the Kepler’s laws 5/24

7. Lunar precession: Laplace-Lagrange secular perturbation theory

8. Perturbation of the Moon by the Sun By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as in the motion of the Moon, and each orbit depends on the combined motions of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind. Newton 1684 By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as in the motion of the Moon, and each orbit depends on the combined motions of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind. Newton 1684

9. Force in different directions

10. Torque and angular momentum

11. Orbital planes and lunar eclipse

12. Nodal precession

13. Restricted 3 body problem Newton’s nightmare: precession of moon’s orbit 3-body problem => more complex dynamics Restricted 3 bodies: M 3 << M 1,2, 1 & 2 circular orbit, 2 sources of gravity  1,2 = M 1,2 /(M 1 +M 2 ) Symmetry & conservation: Time invariance is required for ``energy’’ conservation => rotating frame In a co-rotating frame centered on the Center of Mass, normalized with G(M 1 +M 2 )=1 & a 12 = 1, eq of motion: d 2 x/dt 2 - 2n dy/dt –n 2 x = -[    x+    r      x-    r 2 3 ] d 2 y/dt 2 + 2n dx/dt –n 2 y = -[    r      r 2 3 ] y d 2 z/dt 2 = -[    r      r 2 3 ] z where r 1 2 = (x+     +y 2 +z 2 and r 2 2 = (x+     +y 2 +z 2 Note: 1)additional Corioli’s and centrifugal forces. 2)coordinate can be centered on the Sun 6/24

14. Equi-potential surface and Roche lope Energy & angular momentum are not conserved. Conserved quantity: Jacobi ``energy’’ Integral C J = n 2 (x 2 + y 2 ) + 2(    r     r 2 )-(x 2 +y 2 +z 2 ) Roche radius: distance between the planet and L 1 r R = (       a 12 (to first order in     ) Hill’s equation is an approximation    7/24...

15. Lagrangian points and tadpole orbits

16. Bound & horseshoe orbits, capture zone = + Guiding center epicycle  and  precession e and a variations Match C J at L 2 and Keplerian orbit at superior conjunction =>  a = (12) 1/2 r R 8/24

17. Kepler’s trigon

18. Jupiter-Saturn relative position in the sky Sun Saturn In a frame moving with Jupiter

19. Jupiter and Saturn’s perturbation