Bernard – Looking out my window at the setting moon just now, and wondering: if the moon is tidally locked to the earth hence we always see only one face of the moon, why is the earth not similarly tidally locked to the sun? -Wayne

Wayne,

A good question, but I should say that we don’t see “exactly” the same face of the Moon. Because of details, the Moon shows us somewhat more than 60% of its face as it moves in its orbit. It wobbles a bit.

The Earth is not yet tidally locked to the Moon, which I know is not your question, yet. But it eventually will be, some billions of years from now. The Sun causes tides on the Earth, but they are about half the size of those of the Moon, so whatever the effects of tides are that lead to locking, the Sun’s will operate more slowly than the Moon’s on the Earth.

Here’s the big picture. I’m going to try to show you the Earth-Moon system to scale. The Earth is about 8,000 miles in diameter. The Moon about 2,000. The distance from the Earth to the Moon is about 240,000 miles, or 30 times the Earth’s diameter.

If the Earth is a circle a quarter inch in diameter, the Moon will be a circle a sixteenth of an inch about 7 ½ inches away.

As the volume of a sphere is proportional to the cube of the radius, the Earth’s volume is about 4^{3} or 64 times larger than the Moon. Shall we say that the Earth’s mass is similarly about 60 or 70 times the Moon’s mass? I could look up more exact numbers, but this will do.

The force holding the two in orbit around each other is proportional to product of the masses and inversely proportional to the square of the distance: Newton’s famous law of Universal Gravitation. Consider for the moment, tides in the solid rock of the Moon. They arise because the nearer side of the Moon is closer by, say, 1000 miles than is the far side. The force on the nearer side is greater than the force on the farther side. While the Moon’s center of mass still follows its orbit, this tidal force pulls the Moon into a slightly elongated shape along the line between the Earth and the Moon, and into a slightly squashed shape perpendicular to that line. This tidal force is proportional to the Earth’s mass, and to the difference between the two sides of the Moon, (which is a derivative), so it turns out to be inversely proportional to the cube of distance between the Earth and Moon, 1/r^{3}.

Suppose the Moon is not yet tidally locked to the Earth. It rotates with a different period than it revolves around the Earth. Depending upon whether the rotation period is shorter or longer than the revolution, the pulled and squashed shape of the Moon will not lie exactly along the line between the Earth and Moon. Perhaps the near side will lead a bit and the far side will lag. The Earth will exert a force on that leading bulge closer to it that tends to pull it back to the Earth-Moon line, but the force on the lagging bulge farther away will tend to pull the lagging bulge away from that line. But, the first force is a tiny amount greater than the second force, so pulling back to the Earth-Moon line wins.

Pulling on the Moon’s rocks, stretching them a bit, and then moving that stretch around as the Moon rotates relative to the Earth, involves a little friction. Mostly the rocks are elastic, which means they return to their shapes when you remove the force, but not 100% elastic. That means some of the rotational energy of the Moon gradually transfers from rotation energy into heat energy in the Moon. If you grab a metal strip and flex it rapidly, you will warm it. Thus, the Moon will gradually slow its rotation.

This will keep up, slowing, until the rotation rate and the revolution rate are about the same. Once the tidal bulge is no longer being pulled around the Moon, the frictional transfer of energy from rotation to heating will slow, and things will stay about the same. We see the same face of the Moon, mostly, and never see most of the far side. All this has to do with the Earth-Moon distance, and to the mass of the Earth.

Consider the reverse case. The Earth-Moon distance, of course, is the same as the Moon-Earth distance, but we guesstimated that the Moon is much less massive than the Earth. I looked it up, and a more correct figure is that the Moon is about 1.2% of the Earth’s mass. Or the Earth is about 80 times more massive than the Moon.

The tidal force pulling the solid Earth into an elongation along the Earth-Moon line and into a squashed shape perpendicular to that line is about 80 times smaller than the tidal force of the Earth on the Moon. The vertical displacement of the solid Earth’s tide is about a foot. Hardly noticeable, but scientists with sensitive instruments detect them.

The Earth has oceans and the tidal accelerations cause much larger displacements in the oceans than they do in the solid Earth. To bulge in one place and sink in another, water must flow from what will be lower to what will be higher. These currents experience frictional forces along the ocean floor, and again there is transfer of energy from rotation to heating.

Because, however, of that factor of 80 in the masses, these tidal forces and frictions are about 1% of the forces and frictions on the Moon. The Earth has more rotational energy too. The upshot is that it will take a long time before the Earth locks to the Moon.

What about your actual question, which was why isn’t the Earth tidally locked to the Sun? Now we know the factors to look at to estimate about the Sun-Earth system, as compared to the Earth-Moon system. The Sun is 93 million miles from the Earth, or about 400 times farther from the Earth than the Moon is from the Earth. We must cube this to get the relative tidal force: (4 X 10^{2})^{3} = 4^{3} X 10^{6} = 64 million. Oh yes. Don’t forget, inverse too. The Sun’s mass is about 3.3 X 10^{5} times the mass of the Earth. So, the Sun is much more massive than either the Earth or the Moon, but it is much farther away.

That difference between r^{3} and r^{2} matters. The gravitational force of the Sun on the Earth is much greater than the force of the Moon on the Earth. After all, the Earth, we say, orbits the Sun. But the tidal force of the Sun on the Earth is about half or a third of the tidal force of the Moon on the Earth. We see these solar tides in the seas, and when the solar and lunar tides happen to line up, the tide range is highest, and when they are crosswise, the tides range is lowest.

The upshot of this, is that it will take much longer for the Earth’s day to become the same as an Earth year, and, indeed, much longer than it will take for an Earth’s day to become the same as a lunar month. No need to buy tickets for the event just yet.

Bernard