Lecture 20: Tides

Key Ideas

• Earth's Tidal Bulge
• Spring & Neap Tides

Tidal Effects in the Earth-Moon System:

• Tidal Locking of the Moon
• Tidal Braking slowing the Earth's Rotation
• Lunar Recession (increasing size of the Moon's orbit)

Seashore Astronomy

• Sea level is highest twice a day at "high tide"
• Sea level is lowest twice a day at "low tide"

People near the sea quickly notice that the timing of the tides was governed by the motions of the Moon:

• The time between successive high tides is 12h 25m
• The time between successive moonrises is 24h 50m, or twice the time between high tides

Differential Gravity

• The Moon is 12740 km closer to the near side of the Earth than the far side
• This results in a 7% stronger gravitational force on the near side compared to the far side

• Stretches the Earth along the Moon-Earth line
• Squeezes the Earth at right angles to this line

The net result is 2 tidal bulges on opposite sides of the Earth, and so 2 tides per day as the Earth rotates through the Earth-Moon line.

Land and Sea Tides

The main body of the Earth is made of rock, which is stiff and resists deformation by tides.

• "Body Tides" on Earth are only about 30 centimeters high.

The oceans are made of water which is fluid and flows easily in response to the tidal forces:

• Ocean Tides on Earth are about 1 meter high in the open sea
• Near the shore, tidal flows and the seafloor shape can work together to produce much larger local tide

Some of the most extreme ocean tides on Earth are observed in Canada's Bay of Fundy between Nova Scotia and New Brunswick. Here the shape of the bay leads to average high tides of 12 meters compared to low tide, with maximum high tides of up to 17 to 18 meters!

Sun Tides

The Sun also raises tides on the Earth:

• The difference between the gravity force on the day and night sides of the Earth are about half that due to the Moon.

Spring Tides:

• The highest High Tides
• Occur during New Moon and Full Moon, when the Moon and Sun are lined up with the Earth.

(Graphic by R. Pogge)

Neap Tides:

• The lowest High Tides
• Occur at First Quarter and Last Quarter phase, when the Moon and Sun are at right angles seen from Earth.

(Graphic by R. Pogge)

Tidal Locking of the Moon

• The Earth is more massive, and the Moon's radius is smaller.
• Earth tides are ~20x stronger than Moon tides.

• This means it was rotating through its tidal bulge.
• This generated tremendous internal friction, slowing the Moon's rotation.
• Eventually, the Moon's rotation slowed until it matched its orbital period, and the friction stopped.

This is why the Moon always keeps the same face towards the Earth, as we saw back in Lecture 8.

Because the rotation and orbit periods are the same, we say that the Moon is locked in a 1:1 Tidal Resonance with the Earth.

Tidal Braking of the Earth

There is therefore friction between the ocean and the seabed as the Earth turns out from underneath the ocean tidal bulges.

• This drags the ocean bulge in the eastward direction of the Earth's rotation.
• Result is that ocean tides lead the Moon by about 10-degrees

The friction from the ocean tides robs the Earth of rotational energy, acting like brake pads.

This effect is known as Tidal Braking

• Slows the Earth's rotation a tiny amount.
• The Day is getting gradually longer by 0.0023 seconds per century.

Lunar Recession

• Results in a net forward acceleration of the Moon
• Moves the Moon into a slightly larger orbit

This effect is known as Lunar Recession

• Steady increase in the average Earth-Moon distance by about 3.8 cm per year.

The Lunar Recession rate is measurable using Laser Ranging experiments that use retroreflector arrays left on the Moon by the Apollo missions (Apollo 11, 14, and 15), and two Soviet landers (Lunakhod 1 and 2). Telescopes on Earth bounce laser beams off the reflector arrays and measure the distance to the Moon to millimeter precision.

The rate of lunar recession quoted is the present-day measured rate. However, please use caution if using this number to extrapolate back into the past or forward into the future, as the rate depends on a number of factor that make the actual recession rate over long periods of time non-linear.

The Once and Future Moon

Lunar Recession and Tidal Braking of the Earth's rotation are coupled: the rotational energy being taken from the Earth in braking is effectively being transferred, via tides, to the Moon. This extra energy lifts it into a higher orbit.

As a result:

• The Day gets longer by about 1.7 milliseconds every 100 years. [NOTE: The length of the day has changed during Earth's history. The best data come from the study of tidal rhythmites - layered sediments laid down in tidal estuaries that record the cycle of tides, and hence a combination of the lunar month and the rotation rate of the Earth ("length of the day"). The study of paleorotation and paleotides helps us see these effects over scales of billions of years. At present, data goes back reliably to 620Myr ago, when the day was 21.9+/-0.4 hr long, 18.2+/-0.2 hours about 900Myr ago, 18.9+/-0.7 hours 2.5Gyr ago (see Williams, G.E. 1997, Geophysical Research Letters, 24, 421).]
• The Moon recedes by about 3.84 meters/century.

• The Moon will be ~50% farther away from the Earth
• The Lunar Sidereal Month will be about 47 days long
• The Earth's rotation period (the day) will be 47 days long

Once the Earth and Moon are tidally locked, further tidal evolution should stop. However, remember that the Earth and Moon orbit the Sun, and so tidal effects from the Sun will come into play, and continue to evolve the Earth-Moon system dynamically. The details are quite complicated, and beyond the scope of this course (and, to be fair, even the experts argue among themselves about the details - it is a difficult problem!).

The time required for the Moon and Earth to get into perfect tidal synchronization is actually longer than the time remaining in the Sun's life. Thus other changes will occur in the Earth-Moon system that somewhat obviate these effects, so they are shown for illustration only. The main point to carry away is that, all other things being equal, it takes a very long time for a well-separated system like the Earth and Moon to tidally lock.

Dynamical Evolution

In the remainder of the class, we will often encounter examples of tides playing a role in the dynamics of planets and their moons.

Some examples:

• Tidal Resonances determining rotation periods (Moon & Mercury)
• Tidal Locking (Pluto & Charon system)
• Tidally-induced Heating (Io around Jupiter, and Triton around Neptune)