Is it possible to build a bridge between planets?

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I'm very tempted to say that no, that's not possible in practice. At least not without severely stretching the laws of physics. But like Nobody proposed, you could perhaps make it work if you are willing to introduce a super-strong material, plus some other paraphernalia.

Let me introduce you to a few things that belong to the field of orbital mechanics. For simplicity, I will consider only two-body systems, not n-body systems (which are far more complicated to model).

An orbit is an ellipse
The ellipse can be more or less circular, as described by its eccentricity
One of the foci of the ellipse is at the system's center of mass

One of the problems that was solved by the work of Johannes Kepler was that before him, we generally thought of orbits as circular. That isn't the case, and forcing orbits to be perfectly circular leads to all sorts of problems in the long term. By realizing that orbits are elliptical, not circular, Kepler was able to derive a model that described the behavior of orbiting bodies much more accurately.

There is a simple reason why orbits are ellipses. When one body is moving away from another, the gravitational attraction between them is reduced, but so is their relative velocity. Eventually they turn and start falling toward each other instead of away from each other. We call the point farthest from the system's center of mass apoapsis and the point nearest to the center of mass periapsis. When discussing orbits around the Earth specifically, the terms are apogee and perigee, respectively.

Notice how I said "the system's center of mass" above? That was not by accident. We generally think of for example the Earth as orbiting the Sun, or the Moon as orbiting the Earth, but that is a simplification. What really happens is that the Earth orbits the center of mass of the Earth-Sun system (again, simplifying by ignoring all the other bodies in the solar system), and that the Moon orbits the center of mass in the Earth-Moon system. In these cases, the center of mass happens to lie within the more massive body. In other cases, such as the Pluto-Charon system, the center of mass lies outside of either body. Even a small man-made satellite in orbit of the Earth perturbs (tugs at) the Earth ever so slightly. This is Newton's famous apple pulling the Earth toward it in action.

To have a bridge-like construction between two celestial bodies, you need the two bodies to be tidally locked with each other; in other words, they have to always present the same side to each other. Tidal locking happens when one of the bodies in the system is significantly larger (specifically, more massive) than the other, and the distance between them is comparatively small; for example, Earth-Moon, or Sun-Mercury. Tidal locking is a gradual process, but eventually, the smaller body stops rotating relative to the larger body.

The problem with that is that the larger body is still rotating relative to the smaller one. So there is no good anchor point on the larger body!

The only possible solution I can see to this is to use either rotational pole as the anchor point, on both bodies. Then you need to figure out a way to keep the anchor points of the "bridge" from being torn off, but that's an engineering problem, not a physics one. It only becomes a physics problem when you are trying to find a material to build the bridge with.

Now, you've got a bridge. But there's another problem: The elliptical shape of the orbit between the two bodies! Take the Moon's orbit around the Earth, for example; perigee is at 356.4 Mm and apogee is at 406.7 Mm, with a nominal semi-major axis (orbital radius) of 384.4 Mm. The distance to the Moon changes from -8.3% to +5.8% compared to its "normal" value! And the Moon is pretty massive; at about $7.34 \times 10^{22}$ kg, it masses about a percent of the Earth's $5.97 \times 10^{24}$ kg. Unless you can perfectly circularize the Moon's orbit around the Earth first, whatever material you build the bridge out of is going to be subjected to extreme stresses.

You can solve that by introducing that super-strong material Nobody mentioned, and that I alluded to in the beginning of my answer. Now the bridge itself will hold, and may even theoretically be able to hold the Moon at the distance of the bridge's length.

But how are you going to anchor the bridge? If you anchor it to the surface (of the Earth in our example), that surface isn't going to be equally super-strong. And even if it was, the Moon is already massive enough to tug the Earth back and forth a bit in the dance between the two.

Much of that could be solved if you put the system's center of mass exactly in the middle, right there in space between the two. The easiest way to do that is to give both celestial bodies the exact same mass. Now you are looking at a true double-planet system, rather than a planet and its satellite, but I suspect that would be okay, since you specifically said "planets" in your question. This would take an incredibly unlikely coincidence during planetary formation, somewhere along the lines of the infinite improbability drive, but I suppose if you handwave sufficiently, it could in principle happen. It won't be a stable situation, though. A handful of large asteroid impacts could potentially upset the balance, and those happen, in terms of time relevant for planetary formation, all the time.

Barring that, you need the material that you build the bridge from to be sufficiently strong to withstand such forces, and you need the anchor points at both ends to be sufficiently strong to withstand those same forces, and you need the two bodies to be in a perfectly circular orbit about each other, and you need them to be tidally locked with each other (not just one of them tidally locked with the other).

That's sufficiently unlikely to be possible that I would say that while what you propose may be possible in theory it won't be possible in practice.

And walking that bridge would be taxing, but perhaps possible.

posted over 7 years ago