By Don Lincoln, Ph.D., University of Notre Dame

The orbits of planets aren’t simple ellipses. They move by a very small degree after every completed loop. This happens due to the oblateness of planets and stars. The oblateness of the Sun doesn’t have much of an impact on the solar system, but there is greater impact between planets and moons. Is the impact of orbital motion significant in these cases?

Newton’s universal law of gravity, which he used to show that the orbit of planets should be elliptical, was based on the idea that he could ignore the shape and sizes of the stars and planets, and that the strength of gravity weakened as the square of the distance between the two objects, and those ideas are true to a large degree. But they’re not completely true.

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Saturn Doesn’t Fit into Newton’s Law

Saturn is clearly not a point-like mass, nor is it spherical. Saturn is a big ball of gas that spins very quickly, its day is just 10 hours long, and it’s distorted. It is what we call an oblate spheroid, which means that the distance from North to South Pole differs from its distance across the equator.

We can see how that works by imagining how a flexible hoop distorts as it is spun faster and faster around an axis across its diameter. For Saturn, that means that the polar diameter is about 109 million meters, while the equatorial diameter is larger, about 120 million meters. It’s about a 10% difference.

This effect is biggest for Saturn, but all of the planets experience this effect except for Mercury and Venus, which rotate too slowly to distort. The Sun also experiences such an effect, although it is extremely tiny. However, since the Sun is the dominant thing in the solar system, we have to be careful about assuming its small distortion doesn’t matter.

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Orbits in Motion

Why does this matter? It matters because it changes the shape of the gravitational field by a little bit. It turns out that the changes in the shape of the Sun and planets affect the shape of the gravitational field as well. It no longer falls off as one over the distance squared. In reality, it falls off as a mix of the distance squared, and the distance to the fourth power, and even the distance to the sixth power. Mathematically, we can write it with a more complicated equation for gravity with more terms.

In the equation, the one over distance squared behavior is still the most important. The others are just small corrections. But they are corrections that can’t be ignored, if we want to do precise measurements. And those corrections do a surprising thing. They have the net effect of spinning the long axis of the ellipse of a planet or a moon’s orbit. For those of you who played with a Spirograph as a kid or with your own kids, that’s kind of like what this effect does.

The orbit isn’t the simple ellipse, rather it is one that moves a little, loop after loop. This motion is called precession, and you may have heard about it when astronomers talk about the precession of the orbit of Mercury.

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Earth, Moon, and the Spirograph Effect

Just how big are the changes in orbit for real planets and moons? Well, as you’d probably expect, it depends. That’s because the size of the effect depends on the distance raised to a power. If the distance is just one, well one over one square is just one, but so is one over one to the fourth power, and one over one to the sixth power. So, for that distance, the three terms all have about the same effect.

However, if we do the same exercise for a distance, let’s just call it r less than 1, and let’s use r equals 0.1 as an example, we see that 1 over r-squared is 100, 1 over r to the fourth is 10,000, and 1 over r to the sixth is a million. In the close realm, the new terms are a super big deal. In contrast, for r greater than 1, let’s use r equals 10 this time. We see that 1 over r-squared is 1 over 100; for r to the fourth, we get 1 over 10,000; and for r to the sixth, we get one over a million. So, for long distances, the terms from an oblate object are just super small and, well, don’t matter very much.

In the real case of our solar system, it turns out that the oblateness of the Sun is just not much of an effect. The planets are too far away to make a difference. But there are objects orbiting other objects with a much smaller distance. For example, the Moon orbits the Earth much more closely than the Earth orbits the Sun.

In the case of the orbit of the Moon around the Earth, the Spirograph effect is much bigger. The perigee of the Moon’s orbit is about 360 million meters from the center of the Earth, and its apogee is about 400 million meters. Perigee and apogee are like perihelion and aphelion, except for orbits that aren’t around the Sun because helios means Sun. Perigee and apogee are more generic terms that mean closest and furthest distance from the central body.

The pretty cool thing is that the ellipse that the Moon orbit walks completely around the Earth every 8.9 years. That’s simply an enormous effect. In that amount of time, the Moon orbits the Earth a little over 100 times. Since there are 360° in a circle, that means that in each orbit, the location of the perihelion of the Moon moves about 3.5°.

Other moons also experience this Spirographic action, but there are caveats. While the Martian moons Deimos and Phobos experience this sort of motion, the two moons’ orbits are nearly circular around the planet. Accordingly, it is difficult to see this effect in the system of the Martian moons.

This is a transcript from the video series Understanding the Misconceptions of Science. Watch it now, on The Great Courses Plus.

The Orbit of Mercury Moves 15 Degrees Per Century

While the oblateness of the Sun doesn’t affect the orbit of the planets, there is another effect that causes deviations of their orbit from Kepler’s ellipses. The planet affected most is Mercury. Here the thing causing deviations is not the Sun, but rather the effect of all of the other planets.

Mercury is closest to the Sun, and the other planets are at greater distances. That means, on average, the other planets pull Mercury outwards. There are a lot of ways to approach this, but one way is to use the fact that the changes to the elliptical orbit of Mercury are very slow.

However, because that’s true, we don’t have to think of the outer planets as little objects sitting out far away from the Sun. Instead, we can think of them as objects that zoom around the Sun many times before Mercury’s orbit changes very much. So, mathematically at least, we can think of them as lines of mass the size of each planet’s orbit.

To help you understand what that means, let’s take the Earth as an example. The average radius of the Earth’s orbit is 93 million miles or about 150 million kilometers. The Earth’s mass is 6 times 10 to the 24 kilograms. So, from that, we can figure out the circumference of the Earth’s orbit. It’s 584 million miles or 940 million kilometers.

For purposes of this calculation we can then replace the Earth, which is a small single object with a mass of 6 times 10 to the 24 kilograms, with a loop with a radius of 150 million kilometers with the same mass. That’s a loop with a mass of about 7 trillion kilograms for every meter.

So, you do that with all of the planets, replacing them with loops where their mass is spread out all around their orbit. And when you do that, you can calculate their gravitational tug on Mercury compared to that of the Sun. It’s not much. The Sun pulls on Mercury just shy of two million times harder than the other planets do. But it’s enough to have a real effect on Mercury’s orbit.

The effect of the other planets on the orbit of Mercury causes the perihelion of Mercury’s ellipse to move by 531 arc seconds per century. You may not have an intuitive grip on what an arc second is. It turns out that it is a measure of angle. Of course, you know that a circle has 360°. Well each degree can be cut into 60 arc minutes and each arc minute can be split into 60 arc seconds. So each degree can be split into 3600 arc seconds, which means that 531 arc seconds is about 0.15°.

So, this means that the gravitational effect of the other planets makes the ellipse of the orbit of Mercury move 0.15° per century or about a degree every 700 years. That means it will take 250,000 years for the perihelion of Mercury to move all around the Sun and come back to where it started. So, this is a very small effect.

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