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

Transcription

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

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

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

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

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

6 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

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8 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?

9 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?

10 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

11 Precession of Moon s Orbit

12 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 Moon s orbit perigee precesses with period of about 8.85 years

13 Precession of Moon s Orbit

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

15 Bay of Fundy

16 Mont St. Michel, Normandy, France

17 Mont St. Michel, Normandy, France

18 Mont St. Michel, Normandy, France

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

20 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

21 Formation of Tides

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

24 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

25 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

26 Tidal Flow of Oceans

27 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)

28 Tides in the Bay of Fundy

29 Bay of Fundy

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

31 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

32 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 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 R d cosϑ + R2 d 2 Drop terms O R2 d 2

33 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 R d cosϑ GM d R d cosϑ GM d 2 GM d 2 M GM d R cosϑ + GM d d GMR d 3 cosϑ GM d 2

34 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

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

36 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

37 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

38 Both the Moon and the Sun contribute to tides

39 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

40 Spring Tide

41 Neap Tide

42 The Moon s Orbit and its Phases

43 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)

44 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

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

46 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.

47 Near Side of the Moon

48 Far Side of the Moon

49 no rotation rotation

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

51 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!

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53 Far Side of the Moon

54 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

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

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

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

58 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

59 Tidal Friction on Earth

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61 Tidal Friction on Earth Moon s gravity on bulges slows down Earth s rotation Day lengthens by 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

62 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

63 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

64 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

65 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 > 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 more exact fluid result

66 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