Earth-Moon System Fun with Gravity Sarazin. Sizes of Earth and Moon

Earth-Moon System Fun with Gravity Sarazin Sizes of Earth and Moon

Precession: Gravity not parallel to rotation axis rotation axis gravity gravity

Precession of Earth Both Moon and Sun matter, but precession due to Moon is ~ twice that due to Sun. Precession period ~25,800 years

Precession and Climate Seasons on Earth due to tilt of Earth s axis and orbit around Sun

Precession and Climate Seasons on Earth due to tilt of Earth s axis and orbit around Sun Precession changes when season s occur relative to perihelion of orbit Can affect long term climate Milankovitch Cycles

Precession and Climate Seasons on Earth due to tilt of Earth s axis and orbit around Sun Changes when season s occur relative to perihelion of orbit Can affect long term climate Milankovitch Cycles Cause of recent ice ages?

Precession and Climate Mars: Orbit more eccentric, inclination similar to Earth è larger climate changes? Titan (Saturn s moon): Climate change causes methane oceans to change locations?

Precession of Moon s Orbit Moon s orbit in inclined about 5 o to ecliptic plane (orbit of Earth-Moon around Sun) Moon s orbit ~ spinning top Sun is perturber Moon s orbital inclination precesses with period of about 18.6 years

Precession of Moon s Orbit

Tides ASTR 2120 Sarazin Mont St. Michel, Normandy, France

Bay of Fundy

Mont St. Michel, Normandy, France

Mont St. Michel, Normandy, France Causeway at low tide Causeway at high tide

Twice Daily Lunar Tides Due to gravity of Moon Gravitation attraction decreases as distance squared Different parts of the Earth are attracted to the Moon by different amounts

Formation of Tides

Two Tidal Bulges Side closest to Moon, side furthest from Moon Low tide on circle half way between bulges Prolate shape, like a football

Two Tides Per Day Earth s rotation move into and out of bulges 2 high tides per day, 2 low tides per day low high high low

Daily Lunar Tides Due to Moon s gravity Two tides per day Tides in solid (rock) Earth about 20 cm Ocean tides involve flow of water

Tidal Flow of Oceans

Daily Lunar Tides (Cont.) Ocean tides depend on local conditions Ocean tides range from 30 cm to 16 meters Really big tides due to channeling of water, like a shock wave (tidal bore)

Tides in the Bay of Fundy

Bay of Fundy

Tidal Bores Upstream on Rivers Tidal bore rafting, Nova Scotia Surfing a tidal bore, Amazon River

Tidal Gravitational Forces f a x y x R q a d r a y f M Assume static sphere R << d Separate! a =! a cm +! a tidal! a cm = + GM d 2! a = GM r 2 ˆ e r ˆ e x a y tidal = asinφ Sine law : sinφ R a tidal y = GM r 2 = sinϑ r R r sinϑ a tidal y = GMR sinϑ r 3

Tidal Gravitational Forces f a x y x R q a d r a y f M a x tidal = a cosφ GM d 2 = GM r 2 = GM r 2 GM 1 1 r 2 2 1 sin 2 φ GM d 2 1 R2 r 2 sin2 ϑ GM d 2 R 2 r 2 sin2 ϑ + GM d 2 Cosine Law r 2 = d 2 + R 2 2rd cosϑ = d 2 1 2 R d cosϑ + R2 d 2 Drop terms O R2 d 2

Tidal Gravitational Forces f a x y a x tidal = acosφ GM d 2 = GM r 2 = GM r 2 GM 1 1 r 2 2 x 1 sin 2 φ GM d 2 1 R2 r 2 sin2 ϑ GM d 2 R q R 2 r 2 sin2 ϑ + GM d 2 a a x tidal d r a y f r 2 d 2 1 2 R d cosϑ GM d 2 1 2 R d cosϑ GM d 2 GM d 2 M GM d 2 1+ 2 R cosϑ + GM d d 2 + 2 GMR d 3 cosϑ GM d 2

Tidal Gravitational Forces on Earth a x tidal = +2 GMR d 3 a y tidal = GMR d 3 General pattern cosϑ sinϑ y q x a tidal ~ R d a cm

Solar Tides Sun also raises tides on Earth Sun much more massive than Moon Sun much more distant than Moon

Tides on Earth Compare tides due to Moon & Sun a Sun = GM Sun R Å / (AU) 3 a Moon = GM Moon R Å / d Moon 3 a Sun / a Moon = (M Sun / M moon ) (d Moon / AU) 3 ~ 1/2 Tides due to Sun 1/2 of those due to Moon a Moon / g = (GM Moon R Å / d Moon3 ) / (GM Å / R Å2 ) = (M Moon / M Å ) (R Å / d Moon ) 3 ~ 6 x 10-8 h tides ~ (a Moon / g) R Å ~ 1 meter

Solar Tides Sun also raises tides on Earth Sun much more massive than Moon Sun much more distant than Moon Tides from Sun ~ ½ those from Moon Solar tides either add or subtract from lunar tides

Both the Moon and the Sun contribute to tides

Spring and Neap Tides Spring Tide: new and full moons. Particularly high high tides and low low tides Neap Tide: first and third quarter moons. Weaker high and low tides

Spring Tide

Neap Tide

The Moon s Orbit and its Phases

Motion of the Moon Phase Possible Eclipse Tides Moonrise Transit Moonset New Solar Spring (high) Sunrise Noon Sunset First Quarter (none) Neap Noon 6 p.m. Midnight Waxing Half Moon (low) Full Moon Lunar Spring (high) 6 p.m. Midnight 6 a.m. Third Quarter (None) Neap Midnight 6 a.m. Noon Waning Half Moon (low)

Tides on Moon due to Earth Equal distances, but Earth higher mass Tides on Moon due to Earth much bigger than tides on Earth due to Moon Solid (rock) tides on Moon 20 meters vs. 20 cm on Earth

Rotation of the Moon Everyone is a Moon, and has a dark side which he never shows to anybody. Mark Twain

Rotation of the Moon The Moon always presents the same face towards Earth. To keep the same face toward Earth, the Moon must rotate once every orbit. The far side of the Moon is not dark, it gets just as much sunlight as the near side.

Near Side of the Moon

Far Side of the Moon

no rotation rotation

Rotation of the Moon Rotation period = orbital period 29.5 days synodic (seen from Earth) 27.3 days (sidereal period, or actual rotation)

Misconception: Dark Side of the Moon The Dark Side of the Moon should really be called the Far Side. Far Side gets just as much light as the near side. Example: new moon, the near side dark & far side of fully illuminated!

Far Side of the Moon

Tidal Friction Synchronous rotation of Moon Why? Tidal friction If rotation revolution, material moves in and out of tidal bulges friction slows down rotation Moon

Synchronous Rotation Permanent tides No friction Tidal friction leads to synchronous rotation Moon

Synchronous Rotation Examples: Earth s Moon Jupiter s larger moons Pluto and its moon Charon both rotate synchronously

Tidal Friction on Earth Complex, hard to calculate. Most friction occurs at shore lines in narrow regions

Tidal Friction on Earth Friction drags tidal bulge of Earth beyond direction of Moon Tidal bulge doesn t point directly at Moon tidal bulges without friction

Tidal Friction on Earth

Tidal Friction on Earth Moon s gravity on bulges slows down Earth s rotation Day lengthens by 0.0023 seconds per century Reason for Leap Seconds Earth bulges speed up Moon in orbit Moon moves out 4 cm per year Month is getting longer

Tidal Friction: Past Several billion years ago Day was about 10 hours Month was about 15 days Moon was much closer Moon rotated in about 10 hours

Tidal Friction: Far Future In ~50 billion years Day = Month 60 days Earth and Moon completely synchronized Moon fixed over one side of Earth But, don t worry, Sun will swallow Earth in 5 billion years anyway

Roche Limit Tidal forces increase as distance decreases If a moon gets too close to a planet, it is destroyed Closest distance = Roche limit or tidal disruption limit

Roche Limit R 2 Object in danger d R 1 M 2 M 1 a max tidal = 2GM 1R 2 d 3 g 2 = GM 2 R 2 2 g 2 > a max tidal GM 2 > 2GM 1R 2 2 R 2 d 3 M 1 = 4π 3 ρ 1R 1 3, M 2 = 4π 3 ρ 2R 2 3 average densities G R 2 2 4π 3 ρ R 3 2 2 > 2GR 2 d 3 2ρ d > R 1 1 ρ 2 1/3 ρ d > 2.44R 1 1 ρ 2 1/3 4π 3 ρ R 3 1 1 more exact fluid result

Tidal Friction: Far, Far Future Tidal friction due to Solar tides on Earth Day year Earth rotation slower than Moon s orbit Moon moves in towards Earth Moon reaches Roche limit, is torn apart Still not to worry, Sun will swallow Earth in 5 billion years anyway