Gravitation Part IV

Gravitation Part IV

Tides

1

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.

2

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

3

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.

4

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

5

From the perspective of the Earth, this force stretches the Earth, producing tides (and slight squeezing at poles).

6

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

7

(“top” views)

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 12h 25m (not 12h because time between moonrises is 24h 50m

8

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.

9

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

10

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

11

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

12

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.

2SE

SESunEarth

r

MGMF

2

ME

MEMoonEarth

r

MGMF

13

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.

Near rockFar rock

14

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 is the difference between the gravitational force felt by the near vs. far rock.

2r

mGMF rockplanet

near

2)( dr

mGMF rockplanet

far

15

Thus,

22 )( dr

GMm

r

GMmFFF farnear

22

22

2

22

222

22

22

22

)(

)2(

)(

2

)(

2

)(

)(

)(

11

drr

drdGMm

drr

drdGMm

drr

rdrdrGMm

drr

rdrGMm

drrGMm

16

Because r is much larger than d, we can approximate

So

rdr

rdr

22

3

22

2

)2(

r

dGMmF

rr

rdGMmF

17

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 (dF is in opposite direction from dr, hence minus sign)

drr

MmGdF

r

MmG

dr

dF

3

3

2

2

18

Quantitative example: How much more important is the tidal force on Earth from Moon than from Sun?

Tidal force on Earth by Sun:

Similarly, tidal force on Earth by the Moon:

32SE

ESESunEarth

r

DMGMF

32ME

EMEMoonEarth

r

DMGMF

19

To compare, take ratio of the two forces:

FE S

FE M

2GME MSDE

rE S3

2GME MM DE

rE M3

MS

rE S3

MM

rE M3

MS

MM

rE M

rE S

3

1.99 1030 kg

7.35 1022 kg

3.84 105 km

1.50 108 km

3

0.45

So the tides are mainly produced by the Moon, but the Sun is not negligible.

20

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

21

Spring tides occur when the Moon and the Sun are lined up (at new and full Moon). These are more extreme than neap tides.Spring tides occur when the Moon and the Sun are lined up (at new and full Moon). These are more extreme than neap tides.

Neap tides occur when the Moon and the Sun are at right angles (during first and last quarter Moon).Neap tides occur when the Moon and the Sun are at right angles (during first and last quarter Moon).

22

Tidal braking of the Earth

• Earth rotates faster than the Moon orbits the Earth (1 day versus 27.3 days)

• This causes friction between water and ocean floor

• => will drag the "ocean bulge" in eastward direction (“forward” in rotation)

23

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

24

top view

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?What does that mean in terms of the length of a day?

The day is getting longer by 0.0023 seconds per century.The day is getting longer by 0.0023 seconds per century.

25

Tidal braking effect 2

• Additional mass in the ocean bulges – the one leading the Moon is closer to it, causing 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?What effect will this have on solar eclipses?

26

• 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% further 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°!) 27

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!

28

29

Tides are important for stars and galaxies too

"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

30

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.

31

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.

32

• 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.

33

What about projectile fired south from north pole? 34

The same happens if you fire your cannonball to the south.

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 lowpressure system inN hemisphere

35

36

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.

37

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).

38

Pantheon, Paris

39

• 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.

40

• 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).

41