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Tides, the Earth, the Moon, and why our days are getting longer

Every few months, one of a series of questions comes up on the USENET group sci.astro involving tides, or the rotation of the Moon, or the recession of the Moon from the Earth. In an effort to make a generic answer to this, I have compiled a description of the tidal evolution of the Earth Moon system. This description is long, and I am not taking as much time as I should to edit it, but maybe after a while I'll take another look at it and streamline it. For now though, I'll let it be.

This missive explains the following:

  • Why the Moon always shows the same face to the Earth.
  • Why the Moon's rotation period is the same as the length of time it takes to orbit the Earth (same as number 1, but phrased differently).
  • Why there are two tides a day.
  • Why the Earth's rotation rate is slowing.

  • [Note added September 28, 1999: as if to prove how difficult this concept can sometimes be, I made a mistake in the original version of this page when I talked about leap seconds. I then corrected it, but made an error in the correction. Then I found out my whole correction was wrong! Arg! Usually I keep such things in the pages so you can see just where I made my mistakes, but this has gotten so confusing I have simply changed the page and left it at that. Hopefully this will clear everything up! My thanks to Bad Reader Charlie Kluepfel for pointing this all out to me!]

    The strength of gravity depends on the distance from the source. The closer you are, the stronger the "pull" you feel. The Moon's gravity acts on the Earth; but the diameter of the Earth is large enough in relation to the distance of the Moon that the side of the Earth nearer the Moon feels the Moon's gravity significantly more strongly than the side of the Earth away from the Moon. If you could stand at the center of the Earth you would feel the Moon's gravity somewhere between the two.

    This part is tricky, and is the hardest part of this explanation to understand. A drawing of these forces looks like this:

       -->      ---->      ------->
       far      center     near
       side     of Earth   side

    where the arrows represent the force (and direction) of the Moon's gravity on these three points of the Earth. Now, we measure the gravity of the Earth relative to the center of the Earth; everywhere on the Earth, the center is "down". In a sense, we see the center of the Earth as "at rest". It is mathematically correct to then subtract the force of the Moon on the center of the Earth from the force felt on the near and far sides. This is called vector addition. If we do that, our diagram will look like this:

       <-         X          ->
       far	      center     near
       side       of Earth   side

    (Note that this drawing is not meant to be exact, but just to give a feel for what's happening).

    Now we see that in this sense, the Earth is stretched by the difference in the Moon's gravity across the Earth. We call this effect "tides". Tides are a differential force, that is, they result by the difference in the force of gravity between two points.

    That is why there are two tidal bulges on the Earth, one on the near side, and one on the far side. Since water is more flexible than rock, we see the tidal effect strongly in the oceans of the Earth, but barely at all in the ground. However, the rock does bend, by as much as 30 centimeters (about a foot) up and down twice a day!

    As it it turns out, the tidal bulges do not line up exactly between the center of the Earth and the Moon. Since the Earth rotates, the bulges are swept forward a bit along the Earth. So picture this: the bulge nearest the Moon is actually a bit ahead of the Earth-Moon line. That bulge has mass; not a lot, but some. Since it has mass, it has gravity, and that pulls on the Moon. It pulls the Moon forward in its orbit a bit. This gives the Moon more orbital energy. An orbit with higher energy has a larger radius, and so as the bulge pulls the Moon forward, the Moon gets farther away from the Earth. This has been measured and is something like a few centimeters a year.

    Of course, the Moon is pulling on the bulge as well. Since the Moon is "behind" the bulge (relative to the rotation of the Earth), it is pulling the bulge backwards, slowing it down. Because of friction with the rest of the Earth, this slowing of the bulge is actually slowing the rotation of the Earth! This is making the day get longer. The effect is small, but measurable.

    This is also why every few years people that measure such things (chronologists?) need to add a leap second to the year. We use atomic clocks to measure time now, and to do this scientists needed to set these clocks to a standard time. The time chosen was 1900. However, the Earth's rotation is decelerating at a rate of about 0.002 seconds per day per century. It's been about a century since the atomic clocks' standard time, so the Earth is slowing relative to an atomic clock by about 0.002 seconds per day, or about 0.7 seconds per year. Note that this does not mean the Earth is actually slowing its rotation by that amount; it means that a clock set by the rotating Earth loses time at that rate relative to an atomic clock. We add leap seconds to our calendar to get the two clocks aligned. Still confused? The United States Naval Observatory has an excellent web page talking all about leap seconds. Take a look there and see what they say. To save time, I'll quote their example here:

    As an example, the situation is similar to what would happen if a person owned a watch that lost two seconds per day. If it were set to a perfect clock today, the watch would be found to be slow by two seconds tomorrow. At the end of a month, the watch will be roughly a minute in error (thirty days of the two second error accumulated each day). The person would then find it convenient to reset the watch by one minute to have the correct time again.

    Despite all this confusion, the Earth's rotation is in fact slowing down. Eventually, the Earth's rotation will slow down so much that the bulge will line up exactly between the centers of the Earth and the Moon. When this happens, the Moon will no longer be pulling the bulge back, and the Earth's spin will stop slowing. But when this happens, the time it takes for the Earth to rotate once will be slowed to exactly the same time it takes for the Moon to go around the Earth once! If you were to stand on the Moon and look at the Earth, you would always see the same face of the Earth.

    Does this sound familiar?

    It should. Since Earth's gravity is much stronger than the Moon's, the tides from the Earth on the Moon are much stronger than the Moon's tides on the Earth. The Moon has tidal bulges just like the Earth, and so it too was slowed by the Earth's pull on its nearer bulge. Eventually, the Moon's rotation was locked so that it took the same time to spin once on its axis as it takes to go around the Earth. This is why we always see the same face of the Moon! And this happened to the Moon before the Earth because the Earth's tides are so much stronger.

    If this is hard to picture, grab two oranges; one for the Earth and one for the Moon. Let one go around the other, first without any rotation and then letting the "Moon" rotate just once on its axis for every time it goes around the "Earth". See how if the Moon does NOT rotate, then eventually we see all sides? Therefore the Moon does rotate, but it does very slowly: once a month! [Note: by month, I mean the time it takes the moon to go around the Earth once, about 27 days or so. Don't confuse this with what we call a calendar month of 30 or 31 days!]

    Once the Earth is rotationally locked with the Moon, there will be no more evolution of the system from either the Earth or the Moon. However, there are tides from the Sun, which are actually about half as strong as those of the Moon (the Sun is much farther away than the Moon, but a LOT bigger). This will continue to affect the system, but at this point you're on your own!


    Another (excellent) page about tides and gravity is at Mikolaj "Mik" Sawicki's website (that link takes you directly to a PDF file).

    ©2008 Phil Plait. All Rights Reserved.

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