Phases of the Moon

Spin and orbital frequencies

Libration of the moon: observed by Galileo

Orbital acceleration

Acceleration = V2 / r

Measure the magnitude of the force

Force= m V2 / r

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).

Basic mathematics Two bodies in an inertial frame,

Fi,j=-GMiMj(ri– rj)/dij3

In a frame centered on the Sun, equation of motion:

d2 r /dt2 = -m r/r3

m= - G(Mi+ Mj) includes an indirect term.

Nonlinear force: but has analytic solutions

Elliptical orbits: semi major axis a,

Eccentricity e & longitude of peri apse v r=a(1-e2)/(1+e cos q)

Conservation of angular momentum:

L = r x dr/dt = rVq =[ma(1-e2)]1/2,

energy E= (Vq2+Vr

2)/2 – m /r = - m/2a,

and the longitude of periapse v for a point-mass potential

Period P= 2p(a3/ m)1/2

Mean motion n=2p/P

Solutions to the Kepler’s laws 5/24

Lunar precession: Laplace-Lagrange secular perturbation theory

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

Force in different directions

Torque and angular momentum

Orbital planes and lunar eclipse

Nodal precession

Restricted 3 body problem Newton’s nightmare: precession of moon’s orbit

3-body problem => more complex dynamics

Restricted 3 bodies: M3<< M1,2 , 1 & 2 circular

orbit, 2 sources of gravity m1,2= M1,2/(M1+M2)

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(M1+M2)=1 & a12= 1, eq of motion:

d2x/dt2 - 2n dy/dt –n2x = -[m1(x+m2)/r13+m2(x-m1)/r23]

d2y/dt2+ 2n dx/dt –n2y = -[m1/r13+m2/r23] y

d2z/dt2 = -[m1/r13+m2/r23] z

where r12= (x+m2)2+y2+z2 and r2

2= (x+m1)2+y2+z2

Note: 1)additional Corioli’s and centrifugal forces.

2)coordinate can be centered on the Sun6/24

Equi-potential surface and Roche lope Energy & angular momentum are not conserved.

Conserved quantity: Jacobi ``energy’’ Integral

CJ = n2(x2 + y2) + 2(m1/r1+m2/r2)-(x2+y2+z2)

Roche radius: distance between the planet and L1

rR = (m2/3m1)1/3 a12 (to first order in m2/m1)

Hill’s equation is an

approximation m1 =1

7/24

. . .

Lagrangian points and tadpole orbits

Bound & horseshoe orbits, capture zone

= +

Guiding center epicycle

v and W precession e and a variations

Match CJ at L2 and Keplerian orbit at superior conjunction => Da = (12)1/2 rR

8/24

Kepler’s trigon

Jupiter-Saturn relative position in the skySunSaturnIn a frame moving with Jupiter

Jupiter and Saturn’s perturbation