physics, Space, thought experiment

The Moon Cable

It was my cousin’s birthday. In his honor, we were having lunch at a slightly seedy Mexican restaurant. Half of the people were having a weird discussion about religion. The other half were busy getting drunk on fluorescent mango margaritas. As usual, me and one of my other cousins (let’s call him Neil) were talking absolute nonsense to entertain ourselves.

“So I’ve got a question,” Neil said, knowing my penchant for ridiculous thought experiments, “Would it be physically possible to tie the Earth and Moon together with a cable?” I was distracted by the fact that the ventilation duct was starting to drip in my camarones con arroz, so I didn’t give the matter as much thought as I should have, and I babbled some stuff I read about space elevators until Neil changed the subject. But, because I am an obsessive lunatic, the question has stuck with me.

The first question is how much cable we’re going to need. Since the Earth and Moon are separated, on average, by 384,399 kilometers, the answer is likely to be “a lot.”

It turns out that this isn’t very hard to calculate. Since cable (or wire rope, as the more formal people call it) is such a common and important commodity,  companies like Wirerope Works, Inc. provide their customers (and idiots like me) with pretty detailed specifications for their products. Let’s use two-inch-diameter cable, since we’re dealing with a pretty heavy load here. Every foot of this two-inch cable weighs 6.85 pounds (3.107 kilograms; I’ve noticed that traditional industries like cabling and car-making are stubborn about going metric). That does not bode well for the feasibility of our cable, but let’s give it a shot anyway.

Much to my surprise, we wouldn’t have to dig up all of North America to get the iron for our mega-cable. It would have a mass of 3,919,000,000 kilograms. I mean, 3.918 billion is hardly nothing. I mean, I wouldn’t want to eat 3.919 billion grains of rice. But when you consider that we’re tying two celestial bodies together with a cable, it seems weird that that cable would weigh less than the Great Pyramid of Giza. But it would.

So we could make the cable. And we could probably devise a horrifying bucket-brigade rocket system to haul it into space. But once we got it tied to the Moon, would it hold?

No. No it would not. Not even close.

The first of our (many) problems is that 384,399 kilometers is the Moon’s semimajor axis. Its orbit, however, is elliptical. It gets as close as 362,600 kilometers (its perigee, which is when supermoons happen) and as far away as 405,400 kilometers. If we were silly enough to anchor the cable when the Moon was at perigee (and since we’re tying planets together, there’s pretty much no limit to the silliness), then it would have to stretch by 10%. For many elastic fibers, there’s a specific yield strength: if you try to stretch it further than its limit, it’ll keep stretching without springing back, like a piece of taffy. Steel is a little better-behaved, and doesn’t have a true yield strength. However, as a reference point, engineers say that the tension that causes a piece of steel to increase in length by 0.2% is its yield strength. To put it more clearly: the cable’s gonna snap.

Of course, we could easily get around this problem by just making the cable 405,400 kilometers long instead of 384,399. But we’re very quickly going to run into another problem. The Moon orbits the Earth once every 27.3 days. The Earth, however, revolves on its axis in just under 24 hours. Long before the cable stretches to its maximum length, it’s going to start winding around the Earth’s equator like a yo-yo string until one of two things happens: 1) So much cable is wound around the Earth that, when the moon his apogee, it snaps the cable; or 2) The pull of all that wrapped-up cable slows the Earth’s rotation so that it’s synchronous with the Moon’s orbit.

In the second scenario, the Moon has to brake the Earth’s rotation within less than 24 hours, because after just over 24 hours, the cable will have wound around the Earth’s circumference once, which just so happens to correspond to the difference in distance between the Moon’s apogee and perigee. Any more than one full revolution, and the cable’s gonna snap no matter what. But hell, physics can be weird. Maybe a steel cable can stop a spinning planet.

Turns out there’s a handy formula. Torque is equal to angular acceleration times moment of inertia. (Moment of inertia tells you how hard an object is to set spinning around a particular axis.) To slow the earth’s spin period from one day to 27.3 days over the course of 24 hours requires a torque of 7.906e28 Newton-meters. For perspective: to apply that much torque with ordinary passenger-car engines would require more engines than there are stars in the Milky Way. Not looking good for our cable, but let’s at least finish the math. Since that torque’s being applied to a lever-arm (the Earth’s radius) with a length of 6,371 kilometers, the force on the cable will be 1.241e22 Newtons. That much force, applied over the piddling cross-sectional area of a two-inch cable, results in a stress of 153 quadrillion megapascals. That’s 42 trillion times the yield strength of Kevlar, which is among the strongest tensile materials we have. And don’t even think about telling me “what about nanotubes?” A high-strength aramid like Kevlar is 42 trillion times too weak. I don’t think even high-grade nanotubes are thirteen orders of magnitude stronger than Kevlar.

So, to very belatedly answer Neil’s question: no. You cannot connect the Earth and Moon with a cable. And now I have to go and return all this wire rope and get him a new birthday present.

Standard

4 thoughts on “The Moon Cable

    • That’s true. I’ll have to put that in an addendum. It actually turns out that there’s a fundamental limit to the strength-to-wait ratio of any material or force-field, which imposes a breaking length of about nine trillion kilometers. But no physical substance can supply anything close to that.

  1. I think Larry Niven ran into the tensile strength problem with ‘Ringworld’, which works out to be the same engineering as a suspension bridge with no ends, only on a slightly bigger scale. Somebody did the math and it turned out the required tensile strength was about equal to the weak nuclear force, except it had to apply over greater distances. So this meant his ‘scrith’ was one of those ‘indestructible metals’ so beloved of sci-fi, which is no problem (flapping the hands in the direction of the problem makes it go away) except that Niven already had another ultimate metal in his universe in the form of the indestructible Puppeteer spaceship hulls… and that’s without the ‘shadow squares’ which were held in place by multi-million kilometre long wires. Kind of awkward.

    • That’s one of the reasons I never got into Larry Niven: from the little bit I’ve read and heard through the grapevine, he’s got the big ideas that make science fiction good, but he also commits most of the major science fiction sins while he’s doing it. I think I read somewhere that, even if you allow the handwavium Ring, it wouldn’t be stable: move it a little off-center, and it falls into the star.

      For my money, a much more forgivable instance of handwavium, I recommend Alastair Reynolds. Without spoiling anything, the Conjoiner engines are pretty handwavey, but he’s a decent enough writer to make that a plot point, and to draw attention to it. Which I think is the graceful solution.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s