A Walk on the Beach, a Walk on the Moon, and Newton’s Third Law

Anyone who has spent some time at an ocean beach knows about the tides. At most beaches, the water level rises and falls rhythmically twice a day. And tides are synchronized with the moon. Nearly everyone knows that it’s the earth’s gravity that causes the moon to orbit the earth each month.

Tides by Vi Nguyen via NASA

Fewer people are aware that each month the earth also orbits the moon. That is due to Newton’s third law of motion: For every action, there is an equal and opposite action. The third law means that as the earth’s gravity pulls on the moon, the moon’s gravity pulls back with an equal and opposite force. However, since the earth has about 80 times more mass than the moon, the moon does most of the moving (that is Newton’s second law of motion). The moon is approximately 240,000 miles away, so each month the earth moves in roughly a circle having a radius of 240,000/80 = 3,000 miles. Since the earth’s radius is around 4,000 miles, the center of the earth-moon system (its barycenter, meaning the common center of mass around which both the earth and moon orbit) is about 1,000 miles below the earth’s surface. Consequently, while the moon clearly orbits the earth each month, the earth sort of wobbles a little in response.

How Tides Work

But there’s more. Since the earth and moon are so close, not all parts of either body are affected equally by the other body’s gravity. Consider the moon. The moon has a diameter of about 2,000 miles, so the side of the moon that is closest to the earth is about 2,000/240,000 = 1/120 closer to the earth than the side of the moon that is farthest from the earth. This causes the near side of the moon to be pulled more toward the earth than the far side of the moon is pulled toward the earth. Physicists call this a differential force, meaning a difference in force. The differential force stretches the moon along the direction between the earth and moon. Therefore, the diameter of the moon is greatest along a line connecting the earth and moon. The moon has synchronous rotation, meaning that it rotates on its axis at the same rate that it orbits the earth, so the moon’s longest diameter is always oriented toward the earth. This tidal lock means that there is a tidal bulge in the rocks of the moon that is always aligned in the same direction. And note that there is a tidal bulge on either side of the moon: the side facing the earth and the side facing away from the earth.

Since the earth is about four times larger than the moon, the side of the earth facing the moon is 1/30 closer to the moon than the opposite side of the earth is. Hence, the moon also exerts a differential force on the earth. Like the moon, this differential gravitational force stretches the earth in the earth-moon direction, producing a tidal bulge on either side of the earth. But unlike the moon, the earth does not rotate synchronously—the earth rotates nearly 30 times faster than it orbits the moon. The earth’s rapid rotation causes the tidal bulges to spin ahead of the earth-moon line. Unlike the rocks on the moon that are tidally locked, the earth’s rocks must heave up and down twice each day as the earth rapidly spins. This takes time, and earth’s rocks are not flexible enough to allow them to fully respond to the demands of the tide. As a result, materials that are more fluid respond more easily to the tidal forces. Since the earth’s oceans and atmosphere are more fluid, they rise and fall in rhythm to the tides each day. Yes, there are atmospheric tides, but they are far less noticeable than ocean tides.

The Long-Term Effect of Tides

Tides by Vi Nguyen via NASA

This is the short-term result of the tides, but there is a long-term effect too. The earth’s rapid rotation causes its tidal bulge to run ahead of the direction of the moon. This misaligned tidal bulge is a sort of “handle” that the moon’s gravity pulls on. But the tidal bulge on the side of the earth facing the moon is closer to the moon than the tidal bulge on the side opposite the moon. Consequently, the moon’s gravity pulls more on the tidal bulge on the earth’s side closer to the moon than it does on the tidal bulge on the earth’s side farther from the moon. Thus, this introduces a second differential force on the earth. When speaking of forces like this acting on a spinning body, it is more proper to describe these effects in terms of torques (a twisting force that changes rotational speed). The force on the tidal bulge on the side of the earth opposite the moon experiences a weaker pull and produces a positive torque that slightly accelerates the earth’s rotation. But the force on the tidal bulge on the side of the earth facing the moon experiences a stronger pull and produces a negative torque that slows the earth’s rotation. Since the negative torque is greater than the positive torque, the net effect is a gradual slowing of the earth’s rotation, causing the length of the day to increase—a change that we can actually measure.

This is a relatively small effect, but its effect on the earth accumulates over time. As I’ve pointed out before, we can precisely calculate when and where total solar eclipses occur. These predictions closely match where eclipses are visible now. But if we extrapolate the earth’s current rate of rotation into the past, we find that they don’t match historical records of where eclipses were observed. There is a growing shift in longitude between where eclipses were predicted to occur and where they were actually observed, especially for events that happened centuries ago. This shift provides a direct measurement of the rate at which the earth’s rotation is slowing. That rate is about one ten-thousandths of a second per century. As I said, the rate is small, but with the baseline of many centuries, it is measurable. By the way, eclipse predictions far into the past or future take the earth’s slowing rotation into account.

This shift provides a direct measurement of the rate at which the earth’s rotation is slowing.

But then there is Newton’s third law again, this time modified to apply to torques. Since the moon exerts a torque on the earth to slow its rotation, the earth must pull back on the moon with an opposite and equal torque, accelerating the moon forward in its orbit. To see this, the tidal bulge on the earth on its side facing the moon exerts a positive torque on the moon, while the tidal bulge on the earth opposite the moon exerts a negative torque on the moon. Since the positive torque is greater than the negative torque, the moon accelerates. In orbital mechanics, when an object accelerates, it moves into a higher orbit, so as the moon accelerates, its orbital distance increases.

A laser ranging retroreflector. Image via NASA

Again, this is a very small effect. Measuring it requires measuring the distance to the moon with high precision, which is not easy to do. To aid in this, astronauts on three of the Apollo missions (Apollo 11, 14, and 15) left retroreflectors, highly reflective mirrors, on the moon. In addition, four unmanned missions to the moon have left retroreflectors on the lunar surface. Periodically, astronomers use powerful lasers attached to telescopes to send brief laser pulses to one of the retroreflectors on the moon and collect the light of the reflected laser beams. The returned beams are very faint, only a few photons. The delay in time between the sent and returned signals reveals the distance between the observatory and the retroreflector being used. This precise distance must be corrected for the center-to-center distance between the earth and moon, as well as where the moon is in its orbit at the time of the experiment. The tests have been repeated often over the past half century, resulting in a precise measurement of the moon’s rate of recession—about four centimeters per year. That is less than two inches per year.

Two inches per year is not much, but the change in distance to the moon accumulates year in and year out. Over 6,000 years or so since creation, that amounts to only 1,000 feet. Even over 4.5 billion years, the age of the earth and moon supposed by most scientists, that would result in a change of only 140,000 miles. Thus, one might have expected that 4.5 billion years ago the moon was only 100,000 miles from the earth. That is close, but not catastrophically close. However, these simple calculations assumed that the two inches per year rate of lunar recession is linear. It isn’t. The rate of lunar recession is highly dependent upon the distance of the moon from the earth. The tidal recession rate varies inversely with the sixth power of that distance, meaning that even small changes can dramatically affect how fast the moon moves away. That is a very steep function of distance. The upshot is that the rate of lunar recession decreases as the distance between the earth and moon increases, so in the past, the rate of lunar recession was greater than the measured rate today.

The earth-moon system could not have been in existence for more than 1.5 billion years.

Properly solving this problem of the nonlinear rate of lunar recession requires use of differential equations, something that math students study after a few semesters of calculus. Over a few thousand years, the correct solution doesn’t differ significantly from assuming a constant rate of lunar recession. However, over many millions of years, the solution greatly differs from the linear assumption. The correct solution shows that nearly 1.5 billion years ago, the moon would have been in contact with the earth. No one (secularist or creationist) believes that to have happened, so that should tell us that the earth-moon system could not have been in existence for more than 1.5 billion years. That doesn’t automatically mean that the earth-moon system is only thousands of years old as indicated by Scripture, but it does eliminate a 4.5-billion-year lifetime.

Conclusion

As you might expect, those committed to billions of years have an answer to this problem. They claim that for most of the first four billion years, the moon was much closer to the earth and that it was in a tidal lock, maintaining the same distance from the earth. But then nearly a billion years ago, the tidal lock stopped, and the moon has been spiraling away ever since. There is no evidence for this claim, but those committed to billions of years are convinced that it must have happened, or else we wouldn’t be here contemplating this problem. They offer computer simulations of how that might have happened, but simulations are not evidence. Rather, simulations merely are scenarios that might have happened. These scenarios require certain conditions. Almost all simulations like this prove is that someone can simulate a preconceived outcome. This commits the informal fallacy of special pleading. Critics of recent creationists often accuse us of special pleading, but they rarely recognize when they do it.

If you go to the seashore on vacation this summer, pay attention to the tides. And reflect on the fact that the tidal interaction provides evidence that the creation is only thousands of years old, not billions of years old.