Tidal Forces: Why the Moon Raises Two Bulges

The tide that shouldn't be there#

Stand on any ocean coast and the sea rises and falls roughly twice a day. One high tide makes obvious sense: the Moon is overhead, its gravity tugs the water upward, the ocean bulges toward it. But there is a second high tide about twelve hours later — when the Moon is on the far side of the Earth, pulling in the opposite direction. If the Moon simply lifts the near ocean, where does the far bulge come from?

The usual mental picture — the Moon pulls the near ocean up, so there is one bulge — is wrong, and fixing it reveals what a tide actually is. There are two bulges, on opposite sides of the Earth, and they exist for the same reason at the same time. The key is that gravity is not uniform across the Earth: the Moon pulls harder on whatever is closer.

It's the difference that matters#

The Moon's gravitational acceleration on a chunk of matter at distance $r$ is

$$g(r) = \frac{GM_{\text{Moon}}}{r^2}.$$

The Earth is not a point. Its near side sits about one Earth radius closer to the Moon than its centre, and its far side one Earth radius farther. So the Moon pulls the near ocean a little harder than it pulls the Earth's centre, and pulls the far ocean a little weaker than the centre.

Now stand in the Earth's own reference frame, which falls freely toward the Moon at exactly the rate the centre is pulled. What you feel is not the raw pull but the difference between the local pull and the pull on the centre — the residual, or tidal, force:

• Near side: pulled toward the Moon more than the centre → stretched toward the Moon.
• Far side: pulled toward the Moon less than the centre → the centre accelerates away from it, so it is left behind, stretched away from the Moon.
• The flanks (top and bottom) are pulled slightly inward toward the Earth–Moon line → squeezed.

The result is a stretch along the Earth–Moon axis and a squeeze across it: an egg-shaped envelope with a bulge on both ends. The Earth rotates through this fixed pair of bulges once a day, so most coastlines see two high tides every ~25 hours (a little over 24 because the Moon has moved along its orbit).

Why it falls off as 1/r³#

The size of that difference is what makes tides special. Take the gravitational acceleration $g(r) = GM/r^2$ and ask how much it changes across a small span $\Delta r$ (like the Earth's radius). That is a derivative:

$$\frac{dg}{dr} = -\frac{2GM}{r^3},$$

so the tidal (differential) acceleration across a body of size $\Delta r$ is approximately

$$\Delta g \approx \frac{2GM}{r^3}\,\Delta r.$$

Raw gravity weakens as $1/r^2$; the tidal force weakens far faster, as $\mathbf{1/r^3}$. Distance matters much more for tides than for ordinary gravity — which sets up the most famous surprise in the subject.

The Sun is bigger, so why does the Moon win?#

The Sun is about 27 million times more massive than the Moon. Naively it should own our tides. But tides scale as $M/r^3$, and the Sun is about 390 times farther away. Compare the two:

$$\frac{\Delta g_{\text{Moon}}}{\Delta g_{\text{Sun}}} = \frac{M_{\text{Moon}}}{M_{\text{Sun}}}\left(\frac{r_{\text{Sun}}}{r_{\text{Moon}}}\right)^3 \approx \frac{1}{2.7\times10^7}\times(390)^3 \approx 2.2.$$

That cube is decisive: $390^3 \approx 5.9\times10^7$, more than twice the Sun's mass advantage. So the Moon's tidal effect is about 2.2 times the Sun's, and the nearby, feeble Moon beats the distant, mighty Sun. The belief that the bigger Sun must dominate the tides fails precisely because tides live on $1/r^3$, not gravity's $1/r^2$.

The Sun is not irrelevant, though. When Sun, Earth, and Moon line up (new and full Moon), the two tidal stretches add, giving extra-high spring tides. When the Sun and Moon pull at right angles (first and last quarter), they partly cancel, giving weaker neap tides. The same geometry that governs orbits in Kepler's laws sets this monthly rhythm.

Tidal locking: one face forever#

Tides do not just move water — they act on the solid body too, and over time they reshape orbits. The Earth raises tidal bulges on the Moon, and because the Moon once spun faster than it orbited, those bulges were dragged slightly ahead of the Earth–Moon line. The Earth's gravity tugging back on the offset bulges applied a torque that braked the Moon's spin, until its rotation period exactly matched its orbital period.

That is tidal locking: the Moon now turns once per orbit, so it keeps the same face toward Earth permanently — the reason there is a "far side" we never see from the ground. The same process is slowly lengthening Earth's day (about 1.7 ms per century) and pushing the Moon outward at ~3.8 cm per year, as angular momentum shifts from Earth's spin into the Moon's orbit. Most large moons in the solar system are tidally locked to their planets for the same reason.

The Roche limit: when tides win#

Push the differential force far enough and it stops merely stretching a body — it tears it apart. A moon is held together by its own self-gravity. But self-gravity is a fixed internal budget, while the tidal stretch grows as $1/r^3$ as the moon spirals inward. At some critical distance the stretch across the moon exceeds what its gravity can hold, and the body disintegrates.

For a fluid or loosely bound "rubble pile" moon, that Roche limit is

$$d \approx 2.44\,R_{\text{planet}}\left(\frac{\rho_{\text{planet}}}{\rho_{\text{moon}}}\right)^{1/3},$$

where $R_{\text{planet}}$ is the planet's radius and $\rho$ the densities. Inside this radius, a fluid moon shears into a ring of debris.

This is very likely the origin of Saturn's rings: a moon (or a comet) that wandered inside the Roche limit and was shredded into the billions of orbiting ice chunks we see today. It is also why the giant planets have their rings inside the limit and their intact moons outside it. Bodies held by material strength rather than gravity — small, rigid, rocky ones — can survive closer in, which is why a strong spacecraft is not ripped apart by Earth. The extreme version of this stretch, "spaghettification," is what awaits anything falling toward a black hole, where the $1/r^3$ tidal gradient becomes lethal. And it is these same tidal interactions, playing out over billions of years, that sculpted the moons and rings we map across the solar system.