Earth-Moon Systen

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Motion of the moon

• 27.3 sidereal orbit
• 29.5 synodic orbit
• rises in east and sets in west diurnally, but moves eastwards by about 12 deg per day rel to stars
• rises hour later per night

Precession

• earth is an oblate spheroid with equatorial bulge of .3% cause by separation
• sun, moon, and planets exert a torque $\tau$ on earth
• $\vec{\tau }=\vec{r}\times \vec{F}$
• results in precession of spin axis of earth around ecliptic pole
• NCP moves. Polaris will not always be at NCP
• moves through stars with $P\approx 28500yr$
• opening angle
• $47°(=2\times 23.5°)$

Tidal Forces

• Moon exerts diff tidal forces on matter on earth
• esp noticeable on earths ocean surface as tides
• when sun and moon align (sun-earth-moon at $0°and180°$) high-amp tides result, called spring tides
• when sun and moon are at $90°$ they sum destructively, producing neap tides

Diff gravitational tidal forces

• arise from the $r^{-2}$ dependence of grav force
• Taylor expansion about center of earth $r_0$
• $\delta F=\frac{2G{M}_{moon}m}{r^3}(r-r_0)$
• Sun exerts about half as strong as moon tidal forces

Rotation of tides

• tidal bulges produced on earths by the moon rotate at the same angular rate as the moons orbit around earth
• but the earth is rotating faster at once per sidereal day by 4 minutes. Drags the tides forward from where they would otherwise be by about $10°$ by friction \begin{list}{-}{}
• therefore high tides occur shortly after upper transit of moon
• the misalignment drives angular momentum transfer between earth and Moon \begin{list}{-}{}
• moon pulls strongly on nearer tidal bulge than farther tidal bulge
• net torque to slow earth rotation
• but conversely the tidal bugle pulls more strongly on the moon, pulling it forward, increasing its angular momentum

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Earth Shape

• moon stretches earth in a prolate deformation
• spin of the earth causes an oblate deformation
• oblate is much greater the prolate def

Roche Limit

• object get too close, forces on one side much greater then other, rip object apart
• approx a planet as two spheres 2m
• $\Delta F=\frac{dF}{dr}\Delta r=\frac{2GMm}{r^3}\Delta r$
• Is there a force holding 2m together? yes, self grav
• $F=-\frac{Gmm}{(\Delta r{)}^2}$

% Preamble needs:

Hill radius

• Tidal forces of sun on earth-moon systems means that there is a maximum orbital distance for the moon, if it is to remain bound to the earth

Plane of lunar orbit

• Inclined by 5.1$°$
• the moon is near the cele equator so the moon is above the horizon about 50% of the time for most observers on earth
• moves north and south in the sky in addition to its motion around the earth. greatest dec is $23.5+5.1=28.6$ and min is $-23.5-5.1=-28.6$
• causes eclipses to be retrograde

Tidal forces: earth vs moon

• earth exerts greater tidal forces on moon than the moon does on the earth.
• $\Delta g_{moon\to earth}=\frac{\Delta F}{m}\propto \frac{M_{Moon}{R}_{Earth}}{r^3}$
• $\Delta g_{earth\to moon}=\propto \frac{M_{Earth}{R}_{Moon}}{r^3}$
• $\frac{\Delta g_{moon\to earth}}{\Delta g_{earth\to moon}}\frac{M_{moon}{R}_{earth}}{M_{earth}{r}_{moon}}\approx \frac{1}{20}$

lunar librations

• E-w and n-s nodding motions of the moon seen from earth, caused by parallax
• tidal locking is not perfect, so the libration happens in longitude
• because the rotation axis is inclined there is libration in lat