How Does Earth Move Through Space?

As it turns out, by some phenomenon of carelessness, ambiguity, or groupthink, science popularizers have disseminated information based on a flawed assumption about the average distance between planets. Using a mathematical method that we devised, we determine that when averaged over time, Earth’s nearest neighbor is in fact Mercury.

That correction is relevant to more than just Earth’s neighbors. The solution can be generalized to include any two bodies in roughly circular, concentric, and coplanar orbits. By using a more accurate method for estimating the average distance between two orbiting bodies, we find that this distance is proportional to the relative radius of the inner orbit. In other words, Mercury is closer to Earth, on average, than Venus is because it orbits the Sun more closely. Further, Mercury is the closest neighbor, on average, to each of the other seven planets in the solar system.
Simple but wrong

To calculate the average distance between two planets, The Planets and other websites assume the orbits are coplanar and subtract the average radius of the inner orbit, r1, from the average radius of the outer orbit, r2. The distance between Earth (1 astronomical unit from the Sun) and Venus (0.72 AU) comes out to 0.28 AU. The table at the bottom of the article shows the calculated distance between each pair of planets using that method.

Although it feels intuitive that the average distance between every point on two concentric ellipses would be the difference in their radii, in reality that difference determines only the average distance of the ellipses’ closest points. Indeed, when Earth and Venus are at their closest approach, their separation is roughly 0.28 AU—no other planet gets nearer to Earth. But just as often, the two planets are at their most distant, when Venus is on the side of the Sun opposite Earth, 1.72 AU away. We can improve the flawed calculation by averaging the distances of closest and farthest approach (resulting in an average distance of 1 AU between Earth and Venus), but finding the true solution requires a bit more effort.
A better approach

To more accurately capture the average distance between planets, we devised the point-circle method. The PCM treats the orbits of two objects as circular, concentric, and coplanar. For our solar system, that’s a pretty reasonable assumption: The eight planets have an average orbital inclination of 2.6° ± 2.2°, and the average eccentricity is 0.06 ± 0.06. An object in a circular orbit maintains constant velocity, which means that over a sufficiently long period, it is equally likely to be in any position in that orbit. We consider a planet’s position at any given time as a uniform probabilistic distribution around a circle defined by the average orbital radius, as shown in figure 1a. The average distance between two planets can therefore be described as the average distance of every point on the circle c2, defined by r2, to every point on the circle c1, defined by r1.

Venus is on average 1.14 AU from Earth, but Mercury is a much closer 1.04 AU.

First, the bad news: the inner solar system is unstable. Given enough time, Jupiter's gravity could yank Mercury out of its present orbit.

Two new computer simulations of long-term planetary motion — one by Jacques Laskar (Paris Observatory), the other by Konstantin Batygin and Gregory Laughlin (University of California, Santa Cruz) — have both reached the same disturbing conclusion.

Says Laughlin, "The solar system isn't as stable as we'd thought." Both teams have found that Jupiter's gravity can increase Mercury's orbital eccentricity over time. Mercury's path around the Sun is already nearly as elliptical as Pluto's. But Jupiter can make Mercury's orbit so out of round that it overlaps the path of Venus. A close encounter between them could send the innermost planet careening off wildly.

"Once Mercury crosses Venus's orbit," Laughlin says, "Mercury is in serious trouble."

So is Earth.

At that point, the simulations predict Mercury will suffer generally one of four fates: it crashes into the Sun, gets ejected from the solar system, it crashes into Venus, or — worst of all — crashes into Earth.

To call this catastrophic is a gross understatement. Such an impact would kill all life on our planet. Nothing would survive. By contrast, the asteroid that doomed the dinosaurs 65 million years ago was likely just 6 miles in diameter; Mercury is 3,032 miles across. The last time an object about that size hit the Earth, the resulting debris formed our Moon.

Think we'll escape the chaos by fleeing to Mars? Think again. Even Mars might not be safe. In one of the computer simulations, the Red Planet was tossed into the cold of interstellar space.

Now, the good news: there's only about a 1% chance that Mercury will go crazy before the Sun bloats into a red giant billions of years from now. "If you're an optimist," says Laughlin, "then you say the glass is 99 percent full."

Laskar, who discovered that Mercury could go wild back in 1994, will publish his paper in Icarus; Batygin (who's still an undergraduate) and Laughlin will publish theirs in The Astrophysical Journal.

Ken Croswell is the author of a new book about the solar system, Ten Worlds: Everything That Orbits the Sun (Boyds Mills Press, 2007).