What would happen to the strength of gravity between the Earth and the moon if the moon were somehow pushed closer to Earth?

Question:  If the Moon was twice more massive and if it had the same orbital speed, would its orbit change?  Further from Earth (because of the centrifugal force), or closer to Earth (because of the gravitational force) or the same (because of the same orbital speed)?  Thank you in advance.  — Chris

Answer:  Newton’s law of universal gravitation tells us that the force between to masses is given by:

Fg = GMm/r^2

…where G is the universal gravitation constant and in our case M is the mass of the Earth, m is the mass of the Moon, and r is the distance between the Earth and Moon.  The Moon orbits the Earth without falling into the Earth (thus succumbing to Newton’s universal law of gravity) by counteracting this gravitational force with an equal an opposite centripetal force:

Fc = mv^2/r

If we set Fg = Fc we get:

GMm/r^2 = mv^2/r

Solving for v, the orbital velocity of the Moon, we get:

v^2 = GM/r

So, as you can see, the mass of the Moon (m) does not come into play.  The orbit would not change.

Jeff Mangum

The gravitational force would decrease at the point $B$ and increase at the point $A$. Actually this happens at all distances, and it's what causes the tides. Moving the Moon in until it touches the Earth is just an extreme case.

As long as the gravitational fields are weak (in this case weak means a lot less than a black hole) you can simply add the gravitational forces from the Earth and the Moon.

At point $B$ the total force is given by:

$$ F_B = F_E - F_{MB} $$

because the two forces point in different directions and oppose each other. So $F < F_E$ and the gravitational field is less than the field of the Earth alone. At point $A$ the total force is given by:

$$ F_A = F_E + F_{MA} $$

because the two forces point in the same direction and reinforce each other. So $F > F_E$ and the gravitational field is greater than the field of the Earth alone.

We can put numbers on this. I'll calculate the acceleration, which is the force per unit mass. At Earth's surface this is 1g, i.e. $9.81$ m/s$^2$, so $F_E = 9.81$ m/s$^2$. For a mass $M$ at a distance $r$ the acceleration is simply:

$$ a = \frac{GM}{r^2} \tag{1} $$

To calculate $F_{MB}$ we have to set $M$ equal to the mass of the Moon ($7.35 \times 10^{22}$ kg) and $r$ equal to the radius of the Moon ($1.74 \times 10^6$ m), and using equation (1) the acceleration is:

$$ a \approx 1.62 \text{m/s}^2 $$

To calculate $F_{MA}$ we set $r$ to the radius of the Moon plus the diameter of the Earth ($1.28 \times 10^7$ m), and using equation (1) the acceleration is:

$$ a \approx 0.023 \text{m/s}^2 $$

Combining these results and converting the accelerations to $g$ we get:

We take the moon for granted, but Earth would be a very different place had our nearest neighbor only achieved half of its present mass when it formed some 4.5 billion years ago in a titanic collision. In fact, we might not even be here to appreciate it at all.

Let's start with eclipses. In one of those bizarre cosmic coincidences, our moon today is positioned at just the right distance between Earth and the sun for its diameter to completely block out the sun during a total solar eclipse—the next of which will occur on Friday, August 1. [see ScientificAmerican.com's special report on the eclipse]. 

But what would happen if the moon had only grew to half its present mass? Assuming our half-size moon was composed of rock as dense as that of the actual moon, it would still be 80 percent as large across as the full-size version (based on the relationship between a sphere's volume and radius that you learned in grade school).

Most solar eclipses are "annular," meaning that the moon only partially blocks the sun and appears to be framed by a ring of glowing sunlight. Annular eclipses happen on average three to four times a year; total eclipses only occur about once per annum. At its current distance from Earth, if the moon were 80 percent of its current size, there could be no total eclipses—just the annular kind.

A less massive moon would also orbit closer to Earth than the real one. (This means that total eclipses could still happen, although the half-mass moon would have to be at least 20 percent closer to Earth than the actual moon is now, or closer—but that would require a coincidence on top of a coincidence) Our real moon orbits at an average distance of 238,600 miles (384,000 kilometers), but every year it drifts about 1.6 inches (four centimeters) farther away. The cause? Ocean tides.

The moon's gravity, combined with the waltz of Earth and the moon around their center of mass, forces the oceans into an oval shape, with two simultaneous high tides. One high tide is on the side of Earth facing the moon, whereas the other high tide is directly opposite, on the other side of our world. Because Earth spins so rapidly compared with the moon's orbit around us, our planet drags the high tide closest to the moon a little bit ahead of it.

The gravitational pull of the water on the trailing moon imparts energy to it. This makes it spiral a little farther outward with every orbit around Earth. (Each lunar revolution takes about 29.5 days). If the moon were half its mass, then the ocean tides would have been correspondingly smaller and imparted less energy to it. Given the moon's lesser mass, this means that less energy would be required to push it away from Earth; however, it turns out that the half-as-big high tide would actually contain less water than our high tide, therefore it would have less mass to influence the half-pint moon's orbit. So a less massive moon would nonetheless end up closer than the real one to Earth.

The energy given to the moon comes from Earth's rotation—and to compensate, our planet is slowing down. In other words, days are getting longer. Geologists believe that an Earth day was originally five to six hours long. If the moon had been less massive, thereby creating less drag on Earth, our planet wouldn't have slowed down as much. The day would be, perhaps, 15 hours long.

Weaker tides (of a half moon) would also have caused less erosion of Earth's landmasses over the past few billion years—and the continents' shorelines would likely look quite different for it. Less soil and minerals from land leaching into the ocean might have had profound effects on the origin of life, too. Some organic (carbon-based) compounds thought to have seeded life may not have made it into the primordial soup of the early oceans, which would also have mixed less thanks to the reduced tides.

Assuming life had still arisen, it would have had to contend with more frequent ice ages as well more extreme warm snaps. Large moons stabilize planets. Mars, which sports only two tiny moons, wobbles a lot on its axis, and as a result it has bigger climatic swings and seasonal temperature changes than Earth does. Without the full-mass moon to hold us steady, life on Earth might have experienced greater seasonal fluctuations.

The outlook for life would have been dim—literally. A smaller moon means less scattered sunlight at night—that's all moonlight is—which would mean darker nighttimes. Whatever life forms did evolve on this altered Earth would have had to develop bigger or more sensitive eyes to help them navigate, forage and spawn at night under this diminished glow.

Neil F. Comins is the author of several books, including What If the Moon Didn't Exist?: Voyages to Earths that Might Have Been; Heavenly Errors: Misconceptions About the Real Nature of the Universe; and The Hazards of Space Travel: A Tourist's Guide. He teaches astronomy at the University of Maine, Orono, and despite his lunar fascination, he swears he isn't a lunatic.

Note: In a weird coincidence, I was working on this post when I got word about the similar Saturn video I wrote about earlier. Since Saturn was at opposition last weekend, I wrote that up and posted it first. The article that follows is based on a video by the same animator, and has some of the same themes but covers different ground and dips into the math a bit more as well.

One of the fun things to do with astronomy (and science in general) is to imagine what it would be like if things were different. For example, right now the Moon orbits the Earth at an average distance of about 384,000 kilometers (238,000 miles). Even though that makes it the nearest astronomical object in the Universe, that’s still pretty far—a four-day ride in a space capsule, for example.

But what if it were closer? YouTube user “yeti dynamics” (real name: Nick) is an animator who created a fantastic video showing what it would look like from Earth if the Moon orbited us as the same distance as the International Space Station. The results are pretty amazing! Make it hi-res and fullscreen for the full effect.

Right off the bat, let me give my kudos to the animator! That was very well done, very realistic looking, and—my favorite bit—really cool, so it’s likely to grab people’s attention and get them thinking about the Moon and space.

And given its intent (just showing what it would look like, without any extrapolation on the physics), it’s pretty accurate, too. Indulge me while I dabble in some (fun) math.

When the Moon Hits Your Eye

If the Moon were that close—420 kilometers (260 miles) over the surface of the Earth—it would be more than 100° in size, literally more than half the sky! Right now it’s a mere 0.5° in size, for comparison (which is actually even smaller than you think). It’s neat how it appears to rotate, too, though that’s really just perspective; it’s the same effect that makes it look like features are sinking below the horizon as you orbit. Yeti dynamics also explains that the color of the Moon is from reflected Earthlight: The blue is from the Gulf of Mexico, and the greenish-tan from the United States. It’s dark in the middle because with the Moon blocking the Sun for so much of the Earth, there’s no light to reflect and illuminate the Moon there!

The motion in the video is sped up; at that distance the Moon would orbit the Earth in about 90 minutes or so. It would cross the sky in very roughly five minutes. Note: Nick sent me a note saying that, accidentally, he rendered the Moon upside-down! I’m generally pretty good at spotting these things, but I’m having a hard time picturing that even with a map in front of me. Maybe you can do better: At 35 seconds, Mare Crisium dominates the view near the bottom (and rotates back into view at 45 seconds), then at 1:00, right after the Moon blocks the Sun, the dark crater Plato can be seen to the upper right. Does that help orient you?

All in all, that would be a very impressive sight! Unfortunately, if you saw it, you’d be very, very dead.

Why?

The answer is gravity, specifically, tides.

The Pull of the Moon

The force of gravity you feel from an object depends on how massive and how far away it is (measured from its center). The Earth has about 80 times the mass of the Moon, so if you could situate yourself exactly halfway between them, the Earth would pull on you 80 times harder than the Moon. But it’s worse than that; gravity drops as the square of the distance, and the Moon is pretty far away. Right now, the center of the Earth is roughly 6,400 kilometers below you, and the Moon’s center is about 380,000 kilometers above you (actually that can vary depending on where the Moon is in its orbit, but let’s ignore that). Take the ratio and square it, and you see that the Earth pulls on you 3,500 times harder just because it’s closer. Add in the fact that the Earth is more massive, and you’ll find it pulls on you about 300,000 times harder than the Moon!

That’s why you don’t notice the gravity of the Moon. It’s only 0.0003 percent as strong as what you feel from the Earth.

But what if the distance were closer? In the video, the Moon is 420 kilometers (260 miles) from the Earth—in this case, that’s measured from the surface of the Earth to the surface of the Moon. The center of the Moon would then be an extra 1,738 kilometers away (the distance of the Moon’s surface to its center, in other words its radius). So now the center of the Moon is 2,158 kilometers (1,340 miles) away, which is close.

If you redo the gravity calculation, you’d find the force of gravity from the Moon on you is 1/10th that of Earth! When the Moon passed overhead, you’d weigh 10 percent less. I’ll note this depends on your latitude and other factors, but again I’m trying to keep it simple here. You’ll see in a moment why worrying over details isn’t important.

The Tide Waits for No Moon

Weighing a little less every time the Moon goes over you might sound like fun, but then you have to remember about tides. The force of gravity gets weaker with distance. For example, right now the far side of the Earth feels less gravity from the Moon than the side facing it. The difference isn’t much, but it’s enough to stretch the Earth a little bit (it’s actually more complicated than that, of course, but for now that’s close enough). That’s what we call the tidal force. Right now, the Moon’s tides on Earth pull water up and down by roughly a meter or two between high and low tide as the Earth rotates under the Moon every day.

But if we bring the Moon in really close, suddenly one side of the Earth is a lot closer to the Moon than the other: The Earth’s near side is 2,158 kilometers from the Moon’s center, and the far side is nearly 15,000 kilometers away. That’s a huge difference, and the tides felt by the Earth would be amplified enormously—nearly 100,000 times what we experience now! There would be global floods as a tidal wave kilometers high sweeps around the world every 90 minutes (due to the Moon’s closer, faster orbit), scouring clean everything in its path. The Earth itself would also be stretched up and down, so there would be apocalyptic earthquakes, not to mention huge internal heating of the Earth and subsequent volcanism. I’d think that the oceans might even boil away due to the enormous heat released from the Earth’s interior, so at least that spares you the flood … but replaces water with lava. Yay?

In the video, you’re standing in a pastoral park enjoying the view as the Moon passes silently overhead. In reality you’d be drowned, vaporized, and shaken to bits. So, yeah.

And it would be even worse for the Moon. The Earth is more than 80 times more massive than the Moon, and so the tides the Moon feels would be even bigger. In fact, at that distance the Moon would be well inside the Earth’s Roche limit, the distance from the Earth where its tides could break another object apart. In other words, the tides from the Earth would literally rip the Moon to pieces! So we wouldn’t even have a Moon; we’d have a thick debris ring composed of ex-Moon. That would be cool to see, too, except for the whole everyone being dead thing.

I can understand why Nick left that part out. It might distract from what he was trying to show. Animating it might have been something of a chore, too.

As far as even more pedantic scientific nitpickery goes, there are tons of details to consider here—the Earth’s rotation, dependence on latitude, whether the Moon suddenly appeared much closer, or if it spiraled in over a few years—but I’ll leave that as exercise for the ambitious reader. As for me … I think I’ll go listen to some Debussy.