The Sun pulls the Moon twice as hard as the Earth does
Two numbers out of the same formula, and the second one is bigger. The Moon has been in the wrong hands for four and a half billion years, and the reason it stays is not that the Earth is winning.
The gravity calculator opens on the Earth and the Moon: 5.97 × 10²⁴ kg, 7.34 × 10²² kg, 384,400 km apart. It returns F = 1.98 × 10²⁰ N. That is the grip the Earth has on the Moon.
Leave the Moon's mass alone and change the other two. The Sun is 1.989 × 10³⁰ kg; one astronomical unit is 1.496 × 10¹¹ m. The same panel returns 4.35 × 10²⁰ N.
The Sun pulls our Moon more than twice as hard as we do.
Why mass wins
Nothing subtle is happening. Newton's law puts mass on top and distance squared underneath, so the contest is one fraction. The Sun is 333,000 times heavier than the Earth. It is also 389 times further from the Moon. Squaring the distance handicap gives 151,000, and 333,000 divided by 151,000 is 2.20.
The Sun starts at an enormous disadvantage and wins comfortably, because the mass ratio is 333,000 and the distance penalty is only quadratic. The calculator's own distance table makes the trade legible: it reruns the force at half, double, triple and ten times the separation, and ten times further is a hundredth of the force. A hundredth looks devastating right up until you notice what the Sun is holding.
Nor does the result wobble much over a month. At new moon the Moon sits one lunar orbit nearer the Sun and the ratio is 2.21; at full moon, 2.19. The Sun never stops winning — not for a minute, not anywhere on the orbit.
So why is the Moon still ours?
The reflex answer is that the Earth is closer, and the reflex answer has just been refuted. The real answer is that the question was the wrong one.
The Sun is not pulling the Moon away from the Earth, because it is pulling the Earth as well. It gives the Earth an acceleration of 5.931 × 10⁻³ m/s². It gives the Moon between 5.901 and 5.962 × 10⁻³ m/s², depending where the Moon is in its month. The two agree to half a percent. Both bodies are falling around the Sun together, side by side, at the same rate, and a passenger cannot feel a lift accelerate if the whole lift accelerates with them.
What the Earth competes against is therefore not the Sun's pull. It is the difference between the Sun's pull on the Moon and on the Earth, which is what survives after the shared motion cancels. That tidal term goes as the separation over the cube of the distance to the Sun, and it comes to 2.24 × 10¹⁸ N.
Set that beside the Earth's 1.98 × 10²⁰ N and it is 1.1%. The Sun's ability to prise the Moon off us is a one-percent effect, and the two-to-one figure never enters the calculation. The same competition — a body's own grip against the differential pull of something larger — is what the Roche calculator solves when it decides whether a moon survives at all.
The tidy summary is the Hill sphere, the radius inside which the Earth can hold something against the Sun. For the Earth it is about 1,500,000 km. The Moon orbits at 384,400 km, a quarter of the way out. It is not clinging on.
The part that is hard to picture
Here is what the two-to-one ratio does buy.
At any instant the Moon feels 4.35 × 10²⁰ N toward the Sun and 1.98 × 10²⁰ N toward the Earth. At full moon the Earth lies between the Sun and the Moon, so both pulls point the same way. At new moon the Moon is on the sunward side and the Earth pulls back against the Sun, with 1.98 against 4.38, leaving 2.4 × 10²⁰ N still directed sunward. Everywhere else in the month the Earth's share is partly sideways and counts for even less.
The net force on the Moon points toward the Sun. Always. In no month of any year does the Moon accelerate away from it.
A path whose acceleration never reverses cannot double back on itself. The Moon's route around the Sun has no loops and no cusps: it is convex everywhere, a very slightly scalloped circle. The spirograph in the back of the textbook, with the Moon curling round and round the Earth as the pair drift along, is not a rough drawing. It is the wrong shape.
Scale is what does it. The Moon swings 384,400 km to either side of the Earth's track; in the same month the Earth carries both of them about 76 million km along that track, a figure you can rebuild from the orbital period and a circumference. The sideways swing is half a percent of the forward motion. In speeds it is the same story: 1.02 km/s around us against 29.8 km/s around the Sun, so the lunar orbit is a 3.4% correction on a solar one.
So the Moon does not orbit the Earth while the Earth orbits the Sun. The Moon orbits the Sun on a track almost identical to ours, and the Earth's contribution is to nudge it half a percent to either side, a bit over twelve times a year. We are not its centre. We are its escort.
References (3)
- the proof that the Moon’s path around the Sun is convex everywhere Brannen (2001). The Sun, the Moon, and Convexity. The College Mathematics Journal 32(4), 268–272.
- the same path drawn to scale, and why the classroom picture is the wrong shape Rovšek (2024). Trajectory of the Moon around the Sun. Physics Education 59(4), 045016.
- the masses and distances every number here is built from Luzum et al. (2011). The IAU 2009 system of astronomical constants. Celestial Mechanics and Dynamical Astronomy 110(4), 293–304.