Are Space Elevators Possible?

This episode of real engineering is brought to you by a brilliant problem-solving website that teaches you how to think like an engineer. Space

elevators

are one of those technologies that sci-fi nerds like me obsess over, straddling the line between outlandish impossibility and genuine engineering. Potential is a technology that could cross the divide between science fiction and scientific reality if we somehow improved on existing technologies. It's the kind of thought experiment and grand engineering challenge that could drive the development of future technologies, after all. Necessity is the mother of invention before jumping In to the technologies that must emerge to facilitate

space

elevators

, let's first explore what a

space

elevator actually is.

A space elevator is exactly that, a giant elevator shaft that we can climb up to reach space, eliminating our dependence on rocket fuel to reach orbit and, hopefully, in the process reducing the cost of space travel. This is not your typical construction that relies on the compressive strength of a material to remain standing. Our buildings are very restricted in height as a result of the compressive strength of our construction materials. The higher we build, the more weight accumulates on the foundation of the building. We can counteract this by widening the base of the building to distribute the weight over a larger area and then narrowing the building as it rises to reduce the weight that is added as we add more floors, most obviously.

Examples of this are the pyramids, but even the Burj Khalifa uses the same principle being widest at its base and gradually narrowing to its seemingly im

possible

height. We can build taller with current materials if we widen the base, but this becomes uneconomical quite quickly as the base takes up an excessive amount of space, so how would a space elevator solve this problem by counteracting the weight of the structure by pulling upwards? We can do this thanks to centrifugal force. Let's imagine a gripping ball swinging around a pole at a certain angular velocity. It is held straight and taut against the post because centrifugal force, an apparent force that appears in a rotating frame of reference, pulls outward.

Now the problem is that the goal of the feather ball is to wrap the rope around the pole if the rope cannot rotate around the center. of twist will simply wrap around the pole. Basically we are trying to recreate this dynamic but on an astronomical scale and to do so we have to work with the natural rotation of the Earth, so our structure will have to be located at the equator. Let's imagine a base located in the middle of the Atlantic Ocean from here we are going to draw a straight line into space for now this is just a line there is no structure but any structure that is built will need to exist along this line if not In sync with the Earth's rotation, the structure will curve and break or, in some sort of cartoon world, wrap around the Earth as in the Tetherball example, our orbit will have to be circular rather than elliptical, as an elliptical orbit would require a tether capable of constantly changing length without breaking we can find an orbit that will achieve this with some simple algebra to stay in a stable circular orbit we need our centrifugal force to be equal to the gravitational force the centrifugal force is defined by this equation where ms is the mass of the satellite omega is the angular velocity and r is the distance to the center of the earth, while the force due to gravity is defined by this equation where g is the gravitational constant and mp is the mass of the satellite planet, the mass of the satellite cancels out as we manipulate the equations to get a value for r our orbital radius now we have an equation with all the known values ​​that we can solve by entering the Earth values ​​and we find a value of 42,168 kilometers, this is the distance from the center of the planet, so it will be about 36,000 kilometers above the planet's surface at the equator.

Well, this gives us a starting point for our construction. We are going to put some kind of massive satellite with a cable into this orbit and start the building process from the planet is not an option that we should build now, this is where things get complicated. If we extend our tether directly toward Earth, we will shift our center of mass and disrupt our orbit to counteract this. We will need to extend our tether in both directions, this keeps our center of mass in geostationary orbit and therefore maintains our circular orbit. If we put a counterweight on the other end, we don't have to have equal lengths of tether on both sides to balance our load and this counterweight could be a useful platform for operations, so let's do it now, something interesting happens when we start to extend our tethers , since this is our neutral point where the gravitational force and the centripetal force are equal.

Any material that extends towards the Earth will experience more gravitational force, while any material that extends outward. from the Earth will experience more centrifugal force, this creates tension in our tether, which will reach its maximum at our neutral point in the geostationary orbit, as everything below it pulls towards the Earth and everything above it pulls towards space, We can calculate the maximum tension in the cable with a uniform cross section with this equation where g is the gravitational constant m is the mass of the earth rho is the density of our material of choice r is the radius of the earth and rg is the radius from the geostationary orbit there is an explanation of how this was derived in this document, which you can find by matching the reference number now on the screen with the list of references in the description.

All of these numbers are fixed, bar 1, the density of the material we choose if we choose to build this cable. of steel with a density of 7900 kilograms per meter cubed, our maximum tensile stress would be 382 gigapascals, that is, 240 times the maximum tensile strength of steel; In other words, steel can't do the job, so can we solve this problem? Steel is one of the strongest. Materials that we have we certainly do not have a 240 times stronger material, but we do have less dense materials that will reduce the tensile stress that we have to withstand, in addition to this, we do not have to have a uniform cross section to attach our tensile strap.

The stress approaches zero at its end points, but the material at these points has the greatest effect on our stress as the gravitational force and centrifugal force increase as we move away from the neutral point of our geostationary orbit, so It makes sense to minimize materials at the end points and maximize them where it is most needed, this will result in an improved design called a conical tower, so this brings us to a new question: how can we calculate the area needed at any point at how long the strap? Our previous article has an answer once again, this is the equation they derived here.

Just like the area of ​​the belt that we choose on the surface of the Earth, this initial value will depend largely on design considerations that we can't know at this time, but we're going to want to minimize it because this right here is an exponential function that means our width. is going to increase exponentially as we go up, it is imperative that we minimize this value inside this parenthesis and we only have two values ​​that we can control in this equation, the density that we want to minimize and the stress value that we are designing for which here it is donated For t that we want to maximize normally we wouldn't use the maximum stress that the material can withstand as a design stress that leaves zero margin for error, we should design with a factor of safety, but for now I'm just going to stick with that. and I'm saying this is not going to be safe and I'm designing it right on the edge of braking so yeah keep that in mind remember the strength and density material selection diagram from our last video let's refer to that again to pick a pair .

Material selection for structural analysis with steel is cheap and well understood, so let's start with a high-quality, high-strength alloy like 350 margin steel. This steel has a maximum tensile strength that can range from 1 .1 gigapascals and 2.4 with a density of 8200. kilograms per cubic meter this document cites a steel with a maximum tensile strength of 5 gigapascals and a density of 7900 kilograms per cubic meter. I don't know which aliens they got their data from, but this is beyond the realm of reality, we'll use steel. but with more realistic material properties, then we will choose some better existing materials. They wisely chose kevlar, which is a widely available high-strength fiber that we could easily form into a strap.

We're going to add two existing materials to the mix, also titanium, which we discovered in our last video has excellent specific strength qualities, and carbon fiber composites which have even better specific strength qualities and would be used today if the sr-71 were redesigned. using these material properties. We can calculate the taper ratio, which will be the ratio of the area of ​​the belt at the bottom of our elevator to the area of ​​the belt at its widest point in geostationary orbit. I'm going to assume a circular area five millimeters in diameter at the base by multiplying the cross-sectional area at the bottom by the taper ratio.

We find the widest point of the cable for the steel. This taper ratio is so large that our cable at its widest point will be this number. Whatever it is for reference, the width of the known universe is 8.8 times 10 to the 26 meters wide, even dividing the diameter of this cable by the width of the known universe produces this number that I still can't understand. Titanium is slightly better. Kevlar and carbon fiber look much better now. They will have a circular diameter of 80 meters and 170 meters respectively. Not yet. It's quite feasible that the amount of material needed to build something like this will exceed any cost savings we can offer, and that's only assuming the fibers can even take this shape without losing a significant portion of their ultimate tensile strength, which It's a big assumption, so I think it's safe to say that at this point space elevators are

possible

in the sense that the physics of how they work is based on reality, we just have to make a material capable of making it feasible, esp. if we consider that we are analyzing this at maximum tension. resistance and we should actually use a value below our yield strength since above that value our material will start to narrow where the cross sectional area actually decreases as the material elongates.

We're not even considering the tension here, a future technology that many of What people are promoting for future use in space elevators are carbon nanotubes, whose strength is off the charts; some studies cite maximum tensile stress values ​​of up to 130 gigapascals and a low density of 1,300 kilograms per cubic meter; at that value the taper ratio is only 1.6 if this material could be manufactured on a large scale it would revolutionize life on Earth but we would still have to solve a lot of engineering challenges. Eliminating the vibrations and waves propagating through the tether is a major challenge in propelling the climber and dealing with the adverse weather of the lower atmosphere and dodging space debris in orbit are enormous challenges, even before beginning the most fundamental problem of manufacturing carbon nanotubes.

We will explore these problems and their possible solutions in future videos, one on how carbon nanotubes are made, why they are made. so strong and what needs to happen to take them from the laboratory to normal life and then we will return to this topic with a design investigation for a real space elevator using this new theoretical material during the research for this video, I noticed several errors in the article . I referenced small mistakes that anyone could easily make and overlook. I only noticed them because I applied their methods myself and noticed inconsistencies. Your rounding was so aggressive that the results were off by an inconceivably large number thanks to that exponential function in your equation and I noticed. that his material properties for steel were wrong because I recreated his calculations for titanium and noticed that it was worse even though the equation was completely determined by specific resistance.

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