A Planet Is Not A Dot

A lot of what I said in describing how we might fall into the Moon, if you and I were in the same room and suddenly the rest of the world stopped existing, was incorrect. That isn’t to say it was wrong or even bad to consider; it just means that the equations that I produced and the numbers that came out from them aren’t exactly what would happen if the sudden-failure-of-planet-Earth case were to happen. I knew the wouldn’t be exactly right going in, which leaves us the question of what I thought I was doing and why I bothered doing it.

The first reason, and the reason why it wasn’t a waste of time to consider these simple approximations of how strongly the Moon is attracting us — how fast we are falling into it, and how fast we would be falling if the Earth weren’t falling into the Moon along with us — is thanks to something which Isaac Asimov perfectly described. In an essay called “The Relativity Of Wrong”, he wrote about — well, the title says it. Ideas are not just right or wrong; they can be wrong by differing amounts, and can be wrong by such a tiny amount that it isn’t worth the complications to get it exactly right. Probably the most familiar example is the flatness of the Earth. To model the globe, or a large nation, the idea that the Earth is nearly flat is sufficiently wrong as to produce measurable, important errors where plots of land are justifiably claimed by multiple owners, maybe from multiple governments, or aren’t claimed at all and form the basis for nowhere towns in which mild fantasy or comic stories can be set. But if one wants to draw a map of the town, or of one’s own property, the curvature of the Earth is not worth considering. We can pretend the Earth is flat and get our work done a lot sooner. Other sources of error will mess up the precise result before that does.

Granted, it’s usually better form to only be wrong in ways you know about, and in ways for which you can estimate the size, so that you know you aren’t going to be too horribly wrong. We call this description of the things we know to be wrong in how we represent a system but trust aren’t too wrong to make our results meaningless a model. Henry Petroski has pointed out that a common feature in great engineering disasters — the Tacoma Narrows Bridge, and then a bunch of other ones familiar to fans of great engineering disasters — is a new design in which some minor factor, previously ignorable even when it was known to happen in the existing models, is allowed to suddenly be so big that its features govern the behavior of the whole system. Then we need to build a new model and bridge. It may sound odd, then, that I can say I know some of the things which would happen, were the Earth to suddenly cease to exist, and know the kinds of corrections I have to make for them. Here’s one.

We started out with the force of gravity between two objects, which is just a constant — the Gravitational Constant — times the mass of the first object times the mass of the second, divided by the distance between them. This is a nice familiar force law, called the inverse-square law, and I remember knowing it in elementary school. I grant I was very strange in elementary school. But the equation, while not being any E = mc2 as far as popular recognition goes, is not far behind anyway, certainly not compared to such physics-culture favorites as the Biot-Savart Law or Wien’s Displacement Law.

The thing is, it doesn’t actually describe the attraction between the Earth and the Moon, or between people standing on the former Earth and the Moon. At least not right away. Let your inner obnoxious ten-year-old self stare at the equation, and the terms. What does she or he say?

If your inner obnoxious ten-year-old is arguing something about whether the distance is between the nearest point on the Earth and the nearest point on the Moon, or the points farthest away, or the middle points, or what, the ten-year-old is right and deserves a reward, such as not being strangled right away. The equation describing how strong the pull of gravity between two objects, as written above and in the previous column, is talking about point masses: what would happen if you took all the mass of an object and squeezed it down to a dot which had no length, no width, no height, no cross-sectional area, no volume. It works perfectly well for that case.

But there aren’t any point masses. Even black holes aren’t that much of a dot; they have a minimum volume and there’s a clear difference between things within that volume and outside it. We have a perfect description of the behavior of things which don’t exist. But they are much better-behaved than the things which do.

Besides, it isn’t meaningless, on several grounds, to treat a whole planet (or the Moon) as if it were a dot. The first and simplest reason to do that is also one of the great common features of many mathematical models: from far enough away, everything looks like a dot.

I certainly discovered that as a child, looking over globes and maps of particularly places we were going to visit, and finding that it took so much longer to get there and that places were so much bigger than the globe made them look. From the distance of the imaginary space station far above North America represented by staring down on a globe, the sixty-something square miles of Washington, DC, vanish into a blot of ink.

And what makes a point, a dot, such a perfect point? It’s a point because it isn’t any farther to the front of the point than it is to the back, or to the top, or to the left side, or to anywhere within it. If it’s very nearly the same distance to every spot on the Moon, then it’s reasonable to think that the gravity from the Moon is going to act very much like the gravity from a point.

The average distance to the center of the Moon is around 384,400 kilometers. It actually varies a good bit, sometimes drawing twenty thousand kilometers closer or receding another twenty thousand kilometers, which might be hard to visualize even if you recognize that as about half the distance around the Earth, or about the longest plane flight one can take before the plane just goes the other way. It might be more convincing that the distance is a “good bit” to know the diameter of the Moon is only about 3500 kilometers, so there are times it’s twelve times its entire size closer to the Earth than at other times. But since it’s never less than a hundred times its diameter away from the Earth it has a dozen or so radiuses to spare.

But suppose the Moon sits at 384,400 kilometers away. Its diameter is about 1730 kilometers, which means the nearest point to us that’s on the Moon is about 382,660 kilometers away, or about 99.5 percent of the Moon’s average distance away. The farthest point is about 386,140 kilometers our about 100.4 percent of the Moon’s average distance away. Distances left to right, or top to bottom, are … a little subtler to figure out; we have to be careful what we mean by the left or right edge. Think about it. But it comes to pretty much the same answer anyway.

A total of one percentage point doesn’t seem like very much of an error, and we could analyze it more carefully and prove to ourselves that it is a really tiny error. Treating the Moon as a point for us is doing pretty well.

Of course, a person makes an even better point, from the Moon’s point of view. A person we can say is … well, a person is all long in one direction and short in the other two. But in that long direction they’re not more than two meters, head to toe, with a few exceptions. If it’s 384,400 kilometers from the Moon to the center of a person standing just underneath it, it’s 384,400.001 kilometers to their feet and 384,399.999 kilometers to their head. The distance from the Moon to the person’s head is 99.999 999 7 percent the distance from the Moon to the center of the person; the distance from the Moon to the feet is 100.000 000 3 percent that center distance. The distances side to side are even closer matches.

The wrong-ness of treating the Moon as a dot is fairly small. The wrong-ness of treating a person as a dot would have to be a thousand times bigger to be negligible. All that work is in good shape, except for the obvious factor, and some subtle ones.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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