Astronomy 161:

An Introduction to Solar System Astronomy

Prof. Richard Pogge, MTWThF 2:30

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

How big is the Tidal Bulge of the Earth?

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.

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.

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 length of the day is getting gradually longer by about 2.3 milliseconds per century at the present time.

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.

[20.2]

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 length of the day has gotten longer at a rate of about 1.7 milliseconds per century [see Note 20.3] averaged over the past 2700 years.

The Moon recedes by about 3.84 meters/century on average.

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!).

[20.4]

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: