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Tides

Effects of Tidal Forces

The tidal force is a universal consequence of Newton’s law of gravity, and we can see its effects throughout the universe. The force that causes the Earth’s

oceans to rise and fall also operates elsewhere in the solar system and beyond. Large objects in close proximity exert the strongest tidal forces, but any timetwo large objects orbit each other closely, tides are important. This is true of planets, stars, and even entire galaxies!

The same tidal force that stretches a satellite also tends to slow its rotation until the longest axis of the satellite lines up with the planet. Just as the Earth’s

rotation is slowing due to the Moon’s tidal force on it, the Moon’s rotation has slowed until it is locked into this position. This is why most satellites, like the

Moon, face toward their planet – they are “tidally locked” in that orientation. We always see the same face of the Moon from the Earth because the

Moon’s rotation period is the same as the time it takes to complete one orbit around the Earth. Another way of saying this is that the Moon is in a 1:1 spin

orbit resonance – the ratio of its rotational (spin) period to its orbital period is 1 to 1. Examples of this are common in our solar system. Pluto and Charon

are tidally locked to each other. Mercury’s eccentric orbit prevents it from being in a 1:1 spin orbit resonance. Instead, it’s in a 3:2 resonance – in other

words, Mercury’s day is two-thirds as long as its year.

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If a satellite (or a passing body) comes very close to a planet, the tidal forces can be destructive. The closer two objects are in space, the stronger thegravity between them, and the stronger the tidal force. So the closer an object comes to a planet, the more it’s stretched. Within a certain distance called

the Roche limit, stretching forces can break it apart. That’s why we don’t see satellites orbiting too close to planets – instead, we see ring systems within a

certain distance of the planet. These rings are the remnants of bodies that were broken up by tidal forces.

Even when there’s no water to respond to tidal forces, the solid mass of a planet feels the stress caused by these forces. In fact, tidal forces can heat the

interior of a satellite in an elliptical orbit. In that case, the satellite comes closer to the planet during one part of its orbit, and there it’s subjected to strong

stretching forces. As it moves away from the planet, the stress is partly released, and the body relaxes back toward a more spherical shape. This continual

flexing of the satellite creates heat through internal friction, in the same way that if you flex a tennis ball enough times, it becomes warm. This effect is called

tidal heating. The more elliptical the orbit, the stronger the tidal heating. Jupiter’s large Galilean satellites experience this kind of heating – enough to produceextensive volcanism on Io, and possibly create liquid-water oceans beneath the surface of Europa.

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Calculating Tidal Forces

If the Sun keeps the Earth in its orbit, why is it the Moon that causes tides? To understand this, we need to compare the strength of the gravitational force

of the Sun and the Moon on the Earth. The force of gravity is proportional to the mass of two bodies and inversely proportional to the square of the

distance between them. In this equation, there is also a numerical constant, G. The subscripts S, E, and M represent the Sun, Earth, and Moon,

respectively. The force of gravity caused by the Sun on the Earth is:

FSE = G MS ME / (RSE)2

The force of gravity caused by the Moon on the Earth is:

FME = G MM ME / (RME)2

Some quantities will cancel out if we take the ratio of the Sun’s force on the Earth to the Moon’s force, FSE/FME. (In general, when you are doing algebra

problems, you should wait until you have simplified the relations as much as you can before plugging in numbers and solving the equation.) The ratio of these

forces is:

FSE / FME = (MS / MM)(RME / RSE)2

Now we can insert the values to get the answer. The masses of the Sun and Moon are MS = 2.0 × 1030 kilograms and MM = 7.4 × 1022 kilograms. The

distances from the Earth are RSE = 1.5 × 108 kilometers (1 Astronomical Unit or A.U., by definition) and RME = 3.8 × 105 kilometers. We get the result

FSE / FME = 173. So the Sun’s attractive force on the Earth is over a hundred times the size of the Moon’s attractive force. There is no question that the

Sun controls the orbit of the Earth. So how can the Moon cause the tides on the Earth? The gravitational force depends on the inverse square of distance.So the gravitational force on the near side of a large object is larger than the gravitational force on the far side – the result is a stretching force called a tidalforce. Tides are caused by the difference between the gravitational force on one side of an object and the other side.

We can approximate the strength of the tidal force by taking the gravitational force we just calculated, and multiplying it by the ratio of the front-to-back

distance of the Earth divided by its distance from the Sun or Moon. (You’d need calculus to derive a precise result.) Let’s call the Earth’s diameter DE. For

the stretching of the Sun on the Earth, we get:

DE / RSE = 12,700 / 1.5 × 108 = 8.5 × 10-5

For the stretching of the Moon on the Earth, we get:

DE / RME = 12,700 / 384,000 = 0.033

The ratio of these two numbers is 390. The size of the Earth is a much larger fraction of the Earth-Moon distance than it is of the Earth-Sun distance. In

other words, the difference between the Moon’s gravitational force at the near side of the Earth and the far side is a much larger fraction of the total force

exerted on the Earth. This difference causes tides.

While the Sun has a larger force on the entire Earth than the Moon, the Moon has a larger stretching force. Larger by what factor? It is larger by the ratio of

390 to 173, or roughly a factor of two. Even though the Moon controls the Earth’s tides, the Sun is a significant contributor – the Sun raises tides that are

about half as large as the Moon’s tides. This is the reason that tides are more extreme near a full Moon or a new Moon (“spring tides”), when the stretching

forces due to the Moon and Sun line up in the same direction. “Neap” tides occur when these stretching forces are perpendicular to each other, so they

partially cancel out.

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