Fun Fact - Tidally Locked and Receding

Before we get started, make sure you have read the previous article, which describes the earth's orbit around the sun. This article focuses on the moon's orbit around the earth, and as you might imagine, there are similarities. For instance, the moon orbits the earth in an ellipse, just as the earth orbits the sun in an ellipse - and the moon's orbit, like the earth's orbit, is nearly a circle. It's not a perfect circle, 362,000 KM at perigee (closest approach) and 405,000 km at apogee (farthest distance), but the difference is only 10%, so if you step back it pretty much looks like a circle having a radius of 385 km, or 240,000 miles. Now a quarter million miles seems far away, especially if you're trying to get there in a rocket ship, but it's close compared to the planets orbiting the sun. Mercury, the closest in, is 33 million miles from the sun, and Saturn is 1.4 billion miles away. Those are big numbers compared to a quarter million miles, so let's just say that our moon is pretty close to us. This proximity introduces some new gravitational effects that you don't see in the planets as they orbit the sun. Things happen in a close orbit that don't happen in a far orbit.

Back up about 7 thousand years, when Egyptians bravely set sail upon the open sea. They learned that the water rose and fell, relative to the coastline, in predictable patterns. This was particularly true of the ocean, but the effect could also be seen in large lakes and seas. These undulations were called tides. The high water mark was called high tide, and the low water mark was called low tide. The difference between high and low tides is usually about a meter, but varies widely, from almost no tidal activity in some locations, to 16 meter tides recorded at the Bay of Fundy. If the beach presents a gradual slope, then a rise in sea level of a meter could bring water inland by 20 or 30 meters. Say goodbye to your sand castle! An understanding of tides was important for anyone living by the sea.

People quickly noticed a connection between the moon and the tides. When the moon was overhead, the tide was high, as though the moon was pulling up on the water. Apparently some mysterious lunar force caused the tides. However, the tide was also high 12 hours later, when the moon was nowhere to be seen. In fact, the moon was on the far side of the earth. If the earth was made of glass, and you could see through it, then look down, and the moon would be below your feet. The tide was high when the moon was overhead or below your feet, zenith and nadir. This could not be explained by a mysterious lunar force that pulls on the water, for that would produce a low tide when the moon was situated beneath. The tides would cycle once a day as the earth turned beneath the moon. Instead, there are two high tides per day, one when the moon is above and one when the moon is below - and low tides occur 6 hours after high tides, when the moon is on the horizon, rising in the east or setting in the west. The tidal cycle is 12 hours long, not 24. This was known for millenia, but it defied explanation until Isaac Newton formulated his universal theory of gravity in 1670.

As Newton explained, the earth pulls on the moon, and the moon pulls on the earth. Everything pulls on everything else, proportional to mass and inversely proportional to distance squared. Two bodies, such as the earth and the moon, orbit each other about their common center of mass. The earth is heavier than the moon, 81 times as massive, so the center of mass is actually below the surface of the earth. The earth wabbles just a little, while the moon orbits around it in a wide circle.

Return to the props of the previous article, but this time place the earth at the center of your diningroom table, and find a smaller spherical object to act as the moon. Move the sun to a far corner of the room, or just don't worry about it at all for now. We're interested in the moon and the earth. The moon orbits the earth approximately once a month, hence the connection between the words moon and month. If you were acting out this celestial drama in real time you would walk around the table once a month with the moon in your hand, while the earth turned at the center of the table once a day. We don't have that kind of time, so we're going to speed things up.

An exercise in integral calculus, which I will mercifully skip, shows that the orbit is the same if you shrink a sphere down to a point. Crunch all the mass of the moon down to a marble at its center, a very dense marble to be sure, and crunch all the mass of the earth down to a marble at its center, and the orbits are the same. In other words, the moon's orbit is determined by its center. A tiny rock at the center of the moon is perfectly balanced in its orbit. The gravitational pull from the earth matches circular acceleration precisely, as the center of the moon travels along in its path. The near side of the moon however, the side we see as we look up into the night sky, is 1,000 miles closer to the earth than its orbit might suggest. The pull of the earth on those particular rocks is greater, being closer to us, and at the same time, the radius of revolution is smaller, thus the circular acceleration is less. The net effect is an upward tug on the moon rocks from the earth. If a moon rock, on the near side of the moon, was suddenly isolated from the moon, i.e. if the rest of the moon disappeared, then this rock would have to travel a little faster around the earth if it wanted to stay in that particular orbit. But the moon is a solid body, it moves as one, and the rocks on the near side travel at the angular speed dictated by the center of the moon, thus they experience a slight pull from the earth.

Now move to the far side of the moon, the side you don't see as you look up into the night sky. These rocks are farther from the earth than the center of the moon, and feel less gravitational pull from the earth. (Remember that gravity falls off as distance squared.) At the same time, the orbital radius is increased, and with it the circular acceleration. The rocks are orbiting a little faster than they should, for the track they are on, and they feel an outward tug towards space. If there was water on the moon, which there is not, but if there was, it would be pulled up by the earth's gravity on the near side, and pulled up by net centrifugal force on the far side. An astronaut on the moon would observe high tide when the earth was overhead, and when the earth was underfoot.

Now come back down to earth, where there is indeed water. The physics is the same. The near side of the earth, closest to the moon, is pulled on by the moon more than the center of the earth, thus water on the near side of the earth is pulled up and the tide is high. On the far side of the earth, away from the moon, the pull of the moon is less than average, as though the earth as a whole was being pulled away from the water, and the water rises in high tide. Intermediate regions on the earth, coplanar with the center, experience no tidal forces. In other words, the water is low at longitudes perpendicular to the earth moon line.

Set the moon at the right edge of your diningroom table, and gently spin the earth counterclockwise. You are a seaside observer, stationed at a maritime country close to the equator, such as Spain. The tide rises and falls, rises and falls, as the moon, relative to your vantage point, climbs high in the sky, sets in the west, moves to the back of the earth, then rises in the east. The moon pulls on the water, leaves it alone, pulls the earth away from the water, and leaves it alone. You smile, because you understand the connection between the moon and the tides. Newton's mathematics explains yet another mystery of nature.

It is clear that the tides would be higher if the moon were closer, but how much higher? Gravity is inversely related to distance squared, but tidal forces are inversely related to distance cubed. Verify this with some approximate mathematics. The distance from the center of the earth to its surface is s. The distance from the center of the earth to the center of the moon is r, the radius of the orbit. The pull of gravity from the moon, at the center of the earth, is b over r2, for some value b. At the far side, the pull is only b over (r+s)2. Subtract these two fractions and get:

b × (2rs + s2) / (r2 × (r+s)2)

Assuming s is small relative to r, this simplifies to 2bs/r3. The tidal forces caused by the moon, on the near and far sides of the earth, are inversely related to the cube of the distance between the moon and the earth. Cut the distance in half, so that the moon is only 200,000 km from the earth, and the tides are 8 times as high. when combined with storms and other factors, this would put coastal cities at risk. It is fortunate that the moon keeps its distance.

There is a factor that I neatly omitted in this back of the envelope calculation. I only considered the change in the moon's gravity from one side of the earth to the other. Yet the earth also orbits the center of mass of the earth moon system, just as the moon does. This circular motion also adds to the tides. Since this motion is barely a wabble, it is not significant. However, it becomes a factor in the tides of the moon, or in the smaller partner of any two body system. Let t be the distance from the center of the moon to its surface. As before, the decreasing gravity of the earth contributes 2bt/r3 to the tides of the moon, for some value b. In addition to this, circular acceleration varies from one side of the moon to the other. Twirl about, in the privacy of your own home, and see for yourself. As you spin about at a fixed rate, so many turns per second, the more you stretch out your arms, the greater the pull on your hands. With this in mind, there is some value c, such that circular acceleration is c times distance. In particular, circular acceleration at the center of the moon is c times r. Since the center of the moon is in balance, cr = b/r2. Thus c = b/r3. The circular acceleration at the far side of the moon is c×(r+t). Subtract c×(r+t) - cr to get a tidal difference of ct. Then replace c with b/r3. Put this all together, and the tidal force induced by gravity, and by the change in circular motion, is 2bt/r3 + bt/r3, or 3bt/r3. With gravity and circular motion both taken into account, tidal force is still inversely related to r3.

So far we have set the sun to the side, but it also induces tides on the earth. The sun is massive, but far away. It's tides are about half those produced by the moon. When the sun and the moon are in line, at new moon or full moon, their tides add together, thus increasing high tide. This is called a spring tide. (This is poorly named, having nothing to do with spring.) Conversely, when the sun and the moon are 90 degrees apart, a half moon waning or waxing, the tides are not as high. This is called a neap tide. The tides vary with the phase of the moon.

Water, like all matter, has inertia. When it rises, it does not instantly retreat as the earth turns away from the moon. The bulge in the ocean persists ahead of the moon, for perhaps two hours, and the higher water attracts the moon by gravity, and drags the moon forward in its orbit. On the other side of the earth, the high tide persists as well, and pulls the moon back in its orbit, but the near side has the greater effect. This is because gravity falls off as distance squared. It is similar to the uneven teeter totter described in the previous article. Thus the turning earth pulls the moon forward, causing it to move faster than its orbit would prescribe. This in turn puts the moon in a higher orbit. The tides cause the moon to move away from us at a rate of 3.8 centimeters per year. This has been measured precisely, thanks to laser reflectors placed on the moon during the 1970's. We can actually see the moon receding.

Don't forget Newton's third law of motion, for every action there is an equal and opposite reaction. If the earth drags the moon forward in its orbit, then the moon pulls the earth back in its rotation. In other words, the moon pulls on the bulge in the ocean, as the elevated water rotates out from under the moon. This acts like friction on a spinning top, slowing down the day. When the earth and moon were newly formed, the terrestrial day was about ten hours long, and the moon was just a few thousand miles from the earth. Tidal forces were huge, enough to stretch the solid earth itself. The bulge in the earth rotated away from the moon, and pulled the moon forward in its orbit, which in turn pulled back on the earth's rotation. The moon put the brakes on the spinning earth. four billion years later, the day is 24 hours long, and the moon is a quarter million miles away. At this distance the recession of the moon and the slowing of the earth are small, yet still measurable. The moon recedes at 3.8 centimeters per year, and our day slows down by 1.7 milliseconds per century. These changes will decrease further, as the moon moves farther away from us.

In any closed system, angular momentum is conserved. The moon gains angular momentum as it rises to a higher orbit, and in exchange, the earth loses angular momentum as its day slows down. However, as these two bodies perform their celestial dance, energy is lost. The spinning earth and the revolving moon have less energy today than they did yesterday, and they will have less energy tomorrow. Where does the energy go? Heat of course, in the turbulence of the water as it sloshes about. The "friction" of the earth moon system is literal, to the tune of 3.75 terawatts. Plans to capture this energy have been on the drawingboard for 40 years, but technological challenges remain. See the Scientific American article below.

What if the moon's orbit was retrograde? What if the earth, like almost everything else in the solar system, rotated counterclockwise (when viewed from the north pole), yet the moon went around the earth clockwise? This is rare, but retrograde orbits do exist sporadically throughout the solar system. Most of these are small, distant satellites, formed elsewhere and captured on the "wrong" side of the planet. A notable exception is Triton, an unusually large retrograde moon that is close to its host planet Neptune. The Neptunian high tide drags Triton backwards, not forward, in its orbit. This slows Triton down, and lowers its orbit. In 3.6 billion years or so, Triton will crash into Neptune, or fragment into a ring system that encircles the planet.

Let's return to the earth, and look at another effect of a close orbit. The moon puts the brakes on the earth's rotation, but at the same time, the earth slows down the rotation of the moon. This was especially true in the early solar system, when the moon was quite close to the earth. Also, the earth is 81 times as massive as the moon, and generates much stronger tides. The moon stretched and flexed, and the rocks themselves slid past each other and melted with the heat. The near side of the moon rose toward us in response to tidal forces, then rotated away under the moon's early spin, whereupon the earth pulled back on the slightly elongated moon, putting the brakes on its rotation. Within a billion years or so, the moon's rotation had stopped completely, relative to the earth. One side of the moon faces the earth at all times. This is called "tidally locked". As long as there have been eyes to see, one side of the moon faces us, and the other side remains hidden from view. Nobody had ever seen the far side of the moon until 1959, when the Soviet probe, Luna 3, sent pictures back to earth. The far side of the moon is sometimes called the dark side of the moon, though that appellation is rather misleading. The moon rotates on its axis as it revolves around the earth, thus presenting the same hemisphere to us at all times. Since the moon rotates, it has a day and a night, like any other body that rotates. The far side of the moon is sunlit when the moon is new, and the near side is sunlit when the moon is full. There is no dark side of the moon, But there is a far side that we never see, because the moon is tidally locked, thanks to its close orbit about a much heavier body.

When I was young we thought that Mercury was tidally locked to the sun. Yes, Mercury is 33 million miles from the sun, but the sun is massive enough, we thought, to stop Mercury's spin. My textbook summarized, "One side of Mercury faces the sun at all times, and is hot enough to melt lead, while the dark side is ice cold." In this case there really would have been a dark side of the planet, forever facing away from the sun. We now know that Mercury is not tidally locked, but its rotation is quite slow, as one might expect. The solar tides have definitely put the brakes on a Mercurial day. Its stellar day, one rotation relative to the fixed stars, is 58 earth days, while its year, one trip around the sun, is 88 earth days. Mercury spins on its axis 3 times while it orbits the sun twice. Put this all together and one solar day on Mercury is two Mercury years.

Astronomers have asked whether other stars, much cooler than the sun, might harbor life? An earth-like planet would have to huddle close to the star to keep warm, and would be tidally locked. At that point one side might broil under the heat, while the other side froze solid. This could prevent the evolution of life, or snuff it out of existence if locking occurred after life began. However, some speculate that a substantial atmosphere could disperse heat around the planet, making life possible across large portions of its surface. If this is the case, then the little creatures living on this planet are lucky indeed, for a low mass star, 8% the mass of our sun, can shine on for ten trillion years. That's plenty of time to evolve, and develop, and build a thriving civilization, and unravel the mysteries of the universe. We are not so lucky. In just a billion years we will have to leave our solar system, as the sun heats up and enters its red giant phase. Who knows where we will go at that time. Perhaps our technology will enable a small colony of humans to venture out to another star. If we can find a low mass star with a nearby habitable planet, we could safely call that home for the next 10 trillion years. Let's aim for that, shall we - because I don't really want to move again, at least not for a while.

Further Reading

Ocean exploration
Tides and tidally locked
Giant impact hypothesis
Earth's rotation
Horsepower from the Tides Scientific American
Prograde and retrograde
Far side of the moon
Different kinds of stars