Astronomy 161 An Introduction to Solar System Astronomy Prof. Scott Gaudi

# Lecture 20: Tides

## Key Ideas

Tides are caused by differences in the gravitational pulls of the Moon and Sun between near and far sides of the Earth.
• Earth's Tidal Bulge
• Spring & Neap Tides

Tidal Effects:

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

## Seashore Astronomy

Ocean Tides are a familiar phenomenon to those who make their homes near the sea:
• 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
This folk intuition is correct: Tides are in fact caused primarily by the gravitational pull of the Moon.

## Differential Gravity

The gravitational force exerted by the Moon on the near and far sides of the Earth is different:
• 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
This causes a net front-to-back differential gravitational force felt by the Earth:
• 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

The most extreme ocean tides are experienced in Canada's Bay of Fundy between Nova Scotia and New Brunswick, where the shape of the bay leads to average high tides of 12 meters compared to low tide, and maximum high tides of up to 17 to 18 meters!

## Sun Tides

Gravity is a universal force, so tides are raised between any two bodies.

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.
The Sun and Moon work together to give different kinds of tides and different times of a Lunar Month.

The highest High Tides are the Spring Tides:

• Occur when the Moon and Sun are lined up at New and Full Moon

The lowest High Tides are the Neap Tides:

• Occur when the Moon and Sun are at right angles during First and Last Quarter Moon.

## Tidal Locking of the Moon

Similarly, the Earth raises tides on the Moon
• The Earth is much bigger, so the Earth tides on the Moon are larger
If the Moon were rotating faster than its orbit, the constant stretching and squeezing of the Moon by Earth tides would have generated heat.
• This heat would take energy away from the Moon's rotation.
• The Moon would slow down until its rotation and orbital periods were the same, stopping the squeezing.
The end result is that the Moon is Tidally Locked to the Earth.

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

The Earth rotates faster than the Moon orbits the Earth (24 hours compared to 27 days).

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

Another effect of the Tidal Braking is that the extra mass in the ocean bulges leading the Moon causes a small net forward tug.
• 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 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 2.3 milliseconds every 100 years
• The Moon recedes by about 3.84 meters/century.
After a few Billion Years, this adds up until:
• 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
The Earth & Moon will be locked together in a 1:1 Tidal Resonance, and always keep the same face towards each other.

## Dynamical Evolution

Tidal phenomena are extremely important throughout the Solar System.

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)
Tides are essential to understanding the dynamical evolution of the Solar System.
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