SPACE ELEVATORS: REAL OR NOT, HERE’S WHAT YOU NEED TO KNOW ABOUT THEM

From the inception of mankind, humanity has been yearning for its place amongst the stars. Curiosity, indeed, has been one of the guiding motivations for innovation, invention, leading our species through the thousands and thousands of epiphanies that make up the lost history of the litany of science we have present today. The desire to spread out, to explore, naturally gained new steam with the inception of the Space Age. The first satellite, the Sputnik 1, was launched in 1957, followed over a decade later by humanity’s first steps on the moon. However, a much wilder, more theoretical form of orbital entry was hypothesized not in the 20th century, but much, much farther back. In 1895, Konstantin Tsiolkovsky proposed the idea of a space elevator. You may have heard of him from the famous Tsiolkovsky equation, a cornerstone of rocket science (do scientists always have to do similar things, never mind).

As of now, the only methods of orbital entry are rockets — payloads attached to thrusters. However, Tsiolkovsky presented a much, much more radical means of orbital entry — attaching an elevator car to a tether, and then shooting it straight up to an orbital station. Imagine, moving from your lowly Earth-tethered life to space, hundreds of kilometres above your miniscule house (or whatever form of shelter you might utilize), by entering an elevator, and then ding! Just like a skyscraper, you move up, wait a few minutes, and ding! You see earth and its mighty continents, its mightier blue ocean from a platform whirling about this blue planet at roughly 3000 m/s — don’t mind the Coriolis drag, you’ll learn to enjoy it.

Image credit: https://www.joalda.space/post/space-elevators

UNTETHERED

Full disclaimer — with materials available today, the creation of a space tether is not just unlikely, but impossible, inefficient and a waste of money. What it isn’t, however, is a waste of brain space and brain time. Indeed, this whole article was partly inspired by none other than exercise number 5.65 in Introduction to Classical Mechanics by David Morin. The solution to the problem, of course, required that the necessary length of cable required to keep a uniform tether/rod in geosynchronous orbit around earth such that there is no stress at the ground is 23 times the radius of the Earth, but that will not be a problem today. Instead, we will be analysing the mechanics of a space tether, the beautiful exercise it presents in analysing rotating bodies.

1) CLIMBING THE SPACE ELEVATOR

The proposed system for this particular problem is a space elevator — a rigid rod, made of some extremely strong material (carbon nanofibers, perhaps), of uniform density. The mass of the elevator shaft itself doesn’t exert any respectable amount of gravitational force, so it can be disregarded. Let the rate of earth’s rotation be ω, the mass of the earth be M, and the distance of the elevator car from the center of the earth be r. In an inertial frame of reference, the following equations of motion hold (the car is constrained on the elevator shaft, so it experiences a tangential reaction force N).

Thus, it is easy to see that, at geostationary orbit, the elevator car is in equilibrium, as

is evidently zero. Therefore, if the car is to be move at a constant velocity throughout its upward journey, this is the point at which the input power required is zero and can thus be held in rotational equilibrium. Seems like the natural docking point for the car, does it not? Well, there is more nuance to the fact that having a huge platform at geostationary orbit isn’t exactly…ideal. As the elevator car moves through the structure, it exerts a tangential contact force on the sides, due to the Coriolis effect. This is something that will have to be countered from the ground on which the base of elevator is located, as this force can exert unwanted moments on the sides of the shaft. As such, any orbital platform stationed at the other end of the elevator must have some kind of internal damping system to prevent feedback from the elevator cable.

Propulsion techniques:

One method of propulsion that has been proposed is a power transfer to the elevator car via any conventional means — solar, nuclear, electrical, etc. Further, researchers at Nihon university, Japan, who have suggested using conductive nanomaterials to supply power through the shaft. In any case, the gradual swaying of the structure is something that must be inhibited, and a secondary elevator car might do just that. From equation (2), it is clear that the direction of the contact force on the structure depends on the velocity of the cable. Thus, if the car is moving in the opposite direction while still in contact with the surface of the shaft, it exerts a tangential force in the opposite direction as a car travelling upwards. Thus, a pulley-like system of mass/population transfer in such elevator systems might actually assist in curbing Coriolis effects on the shaft.

This problem presents itself like a classical Atwood’s machine problem. Now, there are some truly terrifying pulley problems I have seen in Morin, however, this problem is considerably simpler. Let us go back to equation (1), and modify it a little bit:

T is obviously the tension in the string while the counterweight cart is moving downwards, , and r1 is simply the radial position of the counterweight with respect to the center of the earth. We can do a similar thing for the car moving up.

However, from equation (2), it is easy to see that m1 = m2 = m. Since the radial positions of both carts will differ (there is actually a constraint involved in this), the tensions required to keep both carts moving with a constant speed will thus be different. Thus, T1 does not equal T2 at all points in the journey. However, assuming a massless cable, T1 = T2. So, with just two objects in our system, there is no way for the cart to remain at equilibrium…or is there?

We could, of course, make the pulley wheel of radius Rp rotate along with the motion of the cart. This will ensure that something in the system does actually accelerate — the pulley wheel, in this case. Thus,

From equation (3), it is clear that another torque must be supplied in order to counteract the torque arising from a difference in tension. Thus,

Now, the constraint on r1, r2 are that r1 + r2 = k, where k is a constant that depends on the length of the rope connecting the two carts, as well as the length of the space elevator itself. Thus, (4) becomes

And the required input power to maintain stability is

A simple application of Newton’s law, really! Let us look at another such application.

2) OPTIMAL COUNTERWEIGHT — TETHER MASS RATIO FOR STABLE SPINNING

As mentioned earlier, in order for a tether to be supported by only its own weight in revolving about the earth in a geosynchronous orbit, it has to have a length of nearly 1.2 million km! That goes even beyond the distance between the earth and the moon. Why can’t a freestanding cable be made smaller? Simple. If it were any smaller, the gravitational force at its center of mass would be much, much more than what is required to hold it in circular orbit, and thus, it will snap. However, as mentioned before, no one will be zipping away in an elevator to nowhere — there will be a counterweight, after all! We shall now attempt to break down the physics of the tether-counterweight system.

Let the tether have a mass density μ, length l, and mass m1. The counterweight has mass m2 and exerts an outward tension T on the cable. Applying Newton’s laws of motion for the cable alone gives:

Applying the same laws to the counterweight gives:

Thus, combining (6) and (7) we get

Equation (8) can be rearranged easily to give the ratio m2/m1 of the counterweight orbital station mass to the tether function of l:

Thus, for a given length of tether required, the sufficient counterweight mass ratio needed can be easily calculate with (9). And all this with a simple application of Newton’s laws! The beauty of (9) is that it allows the cable to be as long as one wants, and you can easily find out the required mass ratio. Of course, there is only one problem — if we want geosynchronous orbiting, the ratio is undefined. This makes sense, of course, because the orbital station can move in orbit under its own power at this height, making m1 zero. However, even beyond geosynchronous orbit, there is a value of l that makes the ratio zero — this, of course, is the 22 times earth radius distance I spoke about before. However, for any reasonable distance beyond geosynchronous orbit, one can easily fix a counterweight with the give specifications!

Space elevators might not be realistic at all. However, what they do offer is an extremely fun exercise in Newton’s laws, rotation dynamics, and I feel at this point such concepts exist mainly to amuse the engineer’s mind. But who knows? Someday, some engineer somewhere might be working the counterweight ratio, because they have to design a whole space elevator!