Which Moons Are Held Most Loosely By Their Planets?

This page is the result of my trying to confirm a claim that I read somewhere, to the effect that Earth's Moon is the most loosely held major moon of any planet in our solar system. Specifically, the Sun's gravitational effect on Earth's Moon is a larger fraction of Earth's own gravitational effect on our Moon than is the case for the tug-of-war between Sun and planet for any other major moon.

The gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (Thank you, Mr. Newton.) To compare the strength of the gravitational force of two objects on one another, say a moon and its planet, with that of the force of a different object on one of them, say of the Sun on that moon, just take the ratio of the two values; then the constant of proportionality ("Newton's gravitational constant") and the mass of the moon, which are part of the calculation of both values, cancel out and we can calculate the ratio from just the mass of the Sun, the mass of the planet, the distance of the planet from the Sun, and the distance of the moon from the planet. I obtained these values mostly from the webpages provided by the Exploring the Planets exhibition of the Smithsonian National Air and Space Museum, except for information on Pluto's moons that I found on a NASA Pluto Fact Sheet.

(Click on each figure below to view it twice as large in each dimension as is shown here.)

This figure, and the next one, organize all the moons larger than 10 km in diameter according to the planet each orbits (taking the old-skool view and including Pluto, since it has one major moon, Charon); Saturn in particular has dozens of moons smaller than this, but the plot is crowded enough as it is. The circles drawn for all moons are to the same scale, except that moons smaller than 200 km are all plotted to scale as if they had 200 km diameters. The vertical position of each moon shows the ratio of the Sun's gravitational force on it to its planet's gravitational force, and sure enough, Earth's Moon at the upper left is much higher up the chart than any of the other large moons shown. In fact, this calculation shows that the Sun pulls on our Moon twice as hard as the Earth does! Yikes - why doesn't the Sun just rip it away into interplanetary space?

The reason is that the Sun also pulls on the Earth, so that as far as the Sun is concerned the Earth and Moon pretty much move together. Imagine connecting two balls by a weak spring, and then throwing them into the air: Earth's gravity will pull on each ball much more strongly than will the spring, but it will simply make the pair arc through the air together, with their motion relative to one another still controlled by the spring. The question of how hard the Sun tries to pull a moon away from its planet has more to do with the difference between the strength of the Sun's pull as the moon moves toward and away from it in its orbit about the planet; remember that the Sun's gravitational force will be less when the moon is on the far side of the planet from the Sun than when it is between the planet and the Sun. The ratio of this difference across a moon's orbit to the strength of the planet's pull can also be calculated using the same quantities (masses and distances) as above, and is shown in the next plot.

That's more like it - all the numbers are comfortably below one (ten to the zero power), so we don't have to explain why these moons don't go wandering off. (Interactions with the gravity of other moons and planets may tweak this, so that some of the small outer moons of Jupiter in particular, the ones toward the top of the plot with the Sun's gravitational force proportionately the strongest, may indeed be asteroids that were captured in passing and that may in the future get un-captured.) Nonetheless, Earth's Moon is clearly in a class by itself, with the variation in the pull of the Sun much stronger in proportion to its planet's pull than is the case for any of the other sizable moons - the four Galilean moons of Jupiter, Saturn's Titan and several modest-sized moons, a few mid-sized moons of Uranus, Triton at Neptune, or Charon at Pluto.

While we don't have to worry about the Sun stealing our Moon, its relatively large gravitational influence shows in the fact that the Moon's orbit is subtly but constantly changing. A small body orbiting a larger body, with nothing else interfering, will trace out an ellipse whose shape maintains the same size and eccentricity (oblateness), but both of these quantities change from orbit to orbit for the Moon. This is why some "supermoons" in Earth's sky are more remarkable than others, at least if you measure carefully (the difference is hard to see by eye): a so-called supermoon occurs when the Moon is full (on the far side of Earth from the Sun, so we see it fully illuminated) at the time when it is also at perigee (closest to the Earth in its elliptical orbit). However, as noted, the shape of this orbit changes over time as the Sun pulls on the Moon with different strengths at different parts of its orbit; thus the supermoon in November, 2016 was closer than any other since 1948 or until 2034, just by happenstance of the Moon's changing orbit over time.

Back To Home Page

New 15 April 2020