Gravitation Part IV

1. Gravitation Part IV Tides

2. Newtonian mechanics explains why the Moon and Earth orbit each other (or their center of mass, to be precise). It also explains why the Moon shows one face to the Earth, and why there are ocean tides – tidal forces. Tidal forces are differences in the gravitational pull at different points in an object.

3. Gravitational force on ball 3 due to planet > force on ball 2 > force on ball 1.

4. What would this look like from the perspective of ball 2? You’d see ball 1 and ball 3 moving away from you in opposite directions.

5. The Moon’s gravitational force is unequal over the Earth due to the Earth’s size.

6. From the perspective of the Earth, this force stretches the Earth, producing tides.

7. As the Earth rotates underneath the oceans, any particular place gets alternating low- and high tides.

8. If you live by the sea you'd notice: • Sea level highest twice a day - high tide • Sea level lowest twice a day - low tide • Time between high tides 12h25m • Time between moonrises 24h50m

9. Just as the Moon raises tides on the Earth, the Earth causes tides on the Moon. When Moon was molten, tide due to Earth deformed it into slight "football" shape (2-3 km elongation), with long axis pointing at the Earth (like a pendulum). If spin period faster than orbit period, tidal bulge would have to move around surface (like Earth’s ocean tides), creating friction, which slows the Moon’s spin down until tidal bulge stops migrating. Thus, we always see one face of the Moon.

10. "Tidally locked". • Synchronous rotation. Most large satellites in Solar System do this!

11. What happens if the body is, unlike the Earth or the Moon, not very rigid?

12. Shoemaker-Levy 9 broke apart due to tidal forces in 1992, before plunging into Jupiter in 1994.

13. But wait – why is the Moon more important for tides than the Sun? Isn’t the gravitational pull of the Sun stronger on Earth? Important point is that the tidal force is the difference in gravitational force across the Earth, and this force has a stronger dependence on distance than does the gravitational force, and the Moon is closer to us.

14. Near rock Far rock Consider 2 small rocks near a planet. Gravitational force due to planet on the near rock will be stronger than force on the far rock.

15. If rocks are separated by distance d, and distance from near rock to planet is r, then force on the near rock is and force on the far rock is The tidal force felt by the rock combo is the difference between the gravitational force felt by the near vs. far rock.

16. Thus,

17. Because r is much larger than d, we can approximate So

18. The Point: tidal force  inverse cube of the distance. This is why influence of Moon is greater than influence of Sun for our tides. Aside: derivation is really easy with calculus. The “differential gravitational force” is given by or where dF is the differential gravitational force directed along r (minus sign: F decreases with r)

19. Quantitative example: Why is tidal force on Earth from Moon more important than from Sun? Tidal force on Earth by Sun: Similarly, tidal force on Earth by the Moon:

20. To compare, take ratio of the two forces: So the tides are mainly produced by the Moon, but the Sun is not negligible (0.45).

21. How do the tides due to the Moon and Sun work together?

22. Neap tides occur when the Moon and the Sun are at right angles (during first and last quarter Moon). Spring tides occur when the Moon and the Sun are lined up (at new and full Moon). These are more extreme than neap tides.

23. Tidal braking of the Earth • Earth rotates faster than the Moon orbits the Earth (1 day versus 27 days) • This causes friction between water and ocean floor • => will drag the "ocean bulge" in eastward direction (“forward” in rotation)

24. Causes the ocean tides to lead the Moon by about 10 degrees.

25. Tidal braking effect 1 • This friction takes away rotational energy from the Earth and acts like a brake • It therefore slows the rotation of the Earth by a small amount What does that mean in terms of the length of a day? The day is getting longer by 0.0023 seconds per century.

26. Tidal braking effect 2 • Additional mass in the ocean bulges leading the Moon causes a net forward pull. => net forward acceleration of the Moon => Moon will move to a larger orbit This is called lunar recession, and the Earth-Moon distance increases 3.8 cm per year. What effect will this have on solar eclipses?

27. Day gets longer by 2 seconds per 1000 years • The Moon recedes by 3.8 meters per 100 years After a few billion years: The Moon will be 50% farther away The lunar sidereal month will be 47 days (from which law?) The Earth's day will be 47 days long. Braking effect will stop. Earth and Moon will be locked in a 1:1 tidal resonance. (Estimate: at closest approach, month was 6.5 hours long, day was 5 hours long, Moon was only 18,000 km away and covered 11°!)

28. Tidal phenomena • Tidal locking (Pluto - Charon system) • Tidal resonances determining rotation periods (Moon, Mercury) • Tidally induced heating (Io-Jupiter, Triton-Neptune) Tides are important to understand the dynamical evolution of the Solar system!

29. Tides are important for stars and galaxies too

30. "During a full Moon,the Moon has a higher gravitational pull, creating a higher tide. Miami Beach psychiatrist Arnold Lieber says that pull affects oceans and people in a similar way, since the human body is mostly water. " Miami Herald, 1996

31. Earth's rotation speed vs. latitude • Equator rotation speed at equator: 1670 km/h • Rotation speed greatest at the equator, and decreases with increasing latitude, as the distance covered in 1 day is less. • Figure shows rotation speed decreases as cos(latitude) • In Albuquerque (35º N), the rotation speed is (1670 km/h) x (cos(35º)) =1370 km/h.

32. The Coriolis Effect • First described by the French physicist Gustave Coriolis 1835 • An effect of viewing straight-line motion from a rotating frame of reference. • On Earth, it results in an apparent deflection of a projectile • Example: fire a cannonball due north from the Equator.

33. The cannon is also moving east with the rotation of the Earth at 1670 km/h. • The cannonball retains this initial eastward speed as it travels north (Newton's first law). • But the further north it gets, the slower is the eastward motion of the Earth's surface beneath. • Result is an eastward deflection of the cannonball with respect to its initial trajectory northward.

34. The same happens if you fire your cannonball to the south. What about projectile fired south from north pole?

35. Consequences of the Coriolis effect • Affects storms and low/high pressure systems: • Hurricanes in the northern hemisphere rotate counter-clockwise • In the southern hemisphere they rotate clockwise Flow around a low pressure system in N hemisphere

37. No - an erroneous urban legend • The size of a sink or a toilet bowl is too small compared to the size of Earth • The Coriolis effect is order of magnitudes smaller than geometry, or other motions such as swirling the water with your hand. • Try: fill a sink with water, set it swirling and then pull the plug. Redo swirling in other direction.

38. Foucault's pendulum • 1851, Foucault. 67 m long pendulum with a 25kg weight. • Started swinging N-S • A few hours later it was swinging NE-SW • Even later it was swinging E-W => Caused by the rotation of the Earth! (Why the experiment was setup).

39. Pantheon, Paris

40. Imagine a Foucault pendulum at North Pole, start swinging towards a bright star. • Observer on the star would see that the direction of the swing is constant, always towards the star. Also that the Earth rotates counter-clockwise every 24 hrs.

41. Observer on NP would see the pendulum rotate clockwise every 24hr (the Earth is stationary wrt you). • At the equator, the pendulum movement is fixed wrt Earth. • Middle latitudes: pendulum does precess, but slower than at the poles (takes longer than 24 hrs to go around once).