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Space elevators are one of those technologies that sci-fi nerds, like me, obsess over. They
straddle the line between outlandish impossibility and genuine engineering potential. It’s
a technology which could cross the divide of science fiction to science reality, if
we somehow improved on existing technologies.
It’s the kind thought experiment and lofty engineering challenge that could drive the
development of future technologies. Necessity is the mother of invention after all.
Before jumping into the technologies that need to 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 to reach
space. Eliminating our dependence on rocket fuel to reach orbit and hopefully, in the
process, lower the cost of space travel.
This isn’t your typical construction that relies on the compressive strength of a material
to remain standing.
Our buildings are largely restricted in height as a result of the compressive strength of
our building materials. The higher we build the more weight is piled onto the foundations
of the building. We can counteract this by widening the base of the construction, to
spread the weight over a larger area and then taper the building as it rises to reduce the
weight being added as we add more floors. The most obvious examples of this are the
pyramids, but even the burj khalifa uses the same principle, being widest at its base and
gradually narrowing to it’s seemingly impossible height.
We can build higher with current materials if we widen the base, but that becomes uneconomic
pretty quickly as the base would take up an unreasonable amount of space.
So how would a space elevator solve this problem? By counter-balancing the weight of the structure
by pulling upwards. We can do this thanks to centrifugal force.
Imagine a tether ball swinging around a poll. At a certain angular velocity the string will
be held straight and taut against the poll, because centrifugal force, an apparent force
that appears in a rotating reference frame, pulls outwards.
Now, the problem is, the whole point of tetherball is to wrap the string around the poll, if
the string can’t rotate around the centre of spin it will simply wrap itself around
the poll.
We are essentially trying to recreate this dynamic, but on an astronomical scale and
to do that we have to work with earth’s natural rotation.
So, our structure will need to be located on 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 out into space. For now, this is just
a line, no structure exists. But, any structure that is constructed will need to exist along
this line. If it is not insync with earth’s rotation the tether will curve and break,
or in some sort of cartoon world wrap around the earth like our tetherball example.
Our orbit will also need 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 remain in a steady
circular orbit we need our centrifugal force to equal the gravitational force. [1]
Centrifugal force is defined by this equation. Where ms is the mass of the satellite, w (omega)
is the angular velocity and r is the distance to the centre 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 planet. The mass of the satellite cancels out while
we manipulate the equations to get a value for r, our orbital radius. [Reference Image
1]
Now we have an equation with all known values, which we can solve for by inputting the values
for earth, and we find a value of 42, 168 kilometres. This is the distance from the
centre of the planet, so this will be about 36 thousand kilometres above the surface of
the planet at the equator.
Okay, so this gives us a starting point for our construction. We are going to put some
form of massive satellite into this orbit and begin the construction process. Building
up from the planet is not an option, we need to build down.
Now this is where things get tricky. If we extend our tether directly down to earth,
we will shift our centre of mass and disrupt our orbit. To counter this we are going to
extend our tether in both directions. This keeps our centre of mass in geostationary
orbit and so maintains our circular orbit. If we place a counter weight on our far end,
we won’t have to have equal lengths of tether on either side, to balance our load, and this
counterweight could be a useful platform for operations, so let’s do that.
Now, something interesting happens when we start to extend our tethers out. Since this
is our neutral point, where gravitational force and centripetal force equal, any material
extended toward earth will experience more gravitational force, while any material extending
away from earth will experience more centrifugal force. This creates tension in our tether,
which will reach its maximum at our neutral point at geostationary orbit, as everything
below it is pulling towards the earth and everything above it is pulling outwards towards
space.
We can calculate the max tension in a 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 earth’s radius and Rg is the radius of geostationary
orbit. There is an explanation of how this was derived in this paper, which you can find
by matching the reference number appearing on screen now, to the reference list in the
description.
All of these numbers are fixed, bar one. The density of the material we chose. If we chose
to build this cable out of steel, with a density of 7,900 kg/m^3. Our maximum tensile stress
would be 382 gigapascals. That’s 240 times the ultimate 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 we have. We certainly
don’t have a material 240 times stronger. But we do have less dense materials, which
will reduce the tensile stress we have to endure. On top of this, we don’t have to
have a uniform cross section tether.
Our tensile stress approaches 0 at it’s endpoints, but material at these points have
the highest effect on our stress as gravitational force and centrifugal forces increase as we
move further from our geostationary orbit neutral point. So it makes sense to minimise
materials at the end points and maximise it where it’s needed most.
This will result in an improved design called the tapered tower.
So this brings us to a new question. How can we calculate the area needed at any point
along the tether. Our previous paper has the answer once again. This is the equation they
derived. Here As is the area of the tether we chose at earth’s surface. This starting
value will largely depend on design considerations that we can’t possibly know right now, but
we are going to want to minimise it, because this right here is an exponential function.
Meaning, our width is going to increase exponentially as we rise. It is imperative that we minimise
this value inside this bracket, and we only have two values we can control in this equation.
The density, which we want to minimise and stress value we are designing for, which he
is donated by T, which we want to maximise.
Normally, we wouldn’t use the maximum stress a material can hold as the design stress.
That leaves zero margin for error. We should be designing in a safety factor. But for now,
I’m just gonna go with it and say this thing isn’t gonna be safe and I’m designing
it riiiiight on the edge of breaking. So, yeah….bear that in mind.
Remember that strength and density material selection diagram from our last video? Let’s
refer to that again to pick a couple of materials to analyse this structure with.
Steel is cheap and well understood, so let’s start there with a high grade high strength
alloy like 350 maraging steel.
This steel has an ultimate tensile strength that can range from 1.1 GPa to 2.4 GPa with
a density of 8,200 kilogram per meter cubed.
This paper quotes a steel with a UTS of 5 GPa and a density of 7,900 kilogram per meter
cubed. I don’t know what aliens they got their data from, but this is beyond the realm
of reality. We will use steel, but with more realistic material properties.
Then we will pick some better existing materials. They wisely picked Kevlar, which is a widely
available high strength fibre we could easily form into a tether.
We are going to throw two existing materials into the mix too. Titanium, which as we discovered
in our last video has excellent specific strength qualities, and carbon fibre composites, which
have even better specific strength qualities and would be used today if the SR-71 was redesigned.
Using these material properties. We can calculate the taper ratio, which will be the ratio of
the area of the tether at the bottom of our elevator to the area of the tether at its
widest at geostationary orbit.
I’m going to assume a circular area 5 millimeters in diameter at the base. By multiplying the
cross sectional area at the bottom by the taper ratio we find the cables widest point.
For steel, this taper ratio is so huge that our cable at it’s widest point will be this
number, whatever that is. For reference the width of the known universe is 8.8 by ten
to the power of 26 meters wide. Even dividing the diameter of this cable by the width of
the known universe yields this number, which I still can’t comprehend. Titanium is marginally
better.
Now Kevlar and Carbon Fibre are looking a lot better. They will have a circular diameter
of 80 meters and 170 metres respectively still not quite feasible.
The amount of material required to build something like this would outstrip any cost savings
it could possibly supply. And that’s just assuming the fibres could even be formed into
this shape without losing a significant portion of their maximum tensile strength, which is
a big assumption. Footage: I don’t know...
So I think it’s safe to say that right now space elevators are possible in the sense
the physics of how they work is based in reality, we just don’t have a material capable of
making it feasible.
Especially when you consider we are analysing this at ultimate tensile strength in reality
we should be using a value below our yield strength, as above that value our material
will begin necking where the cross sectional area actually decreases as the material elongates.
We aren’t even considering strain here.
One future tech that a lot of people are hyping up for future use in space elevators is carbon
nanotubes. Whose strength is off the charts with some studies quoting ultimate tensile
stress values as high as 130 Gigapascals and a low density of 1300 kg per metre cubed.
At that value the taper ratio is just 1.6.
If this material could be manufactured on a massive scale, it would revolutionise life
on earth, but we would still have to solve a huge number of engineering challenges.
Eliminating vibrations and waves propagating through the tether is a huge challenge. Powering
a climber and dealing with the adverse weather of the lower atmosphere and dodging space
debris in orbit are all massive challenges, before even starting on the most fundamental
problem of all. Manufacturing carbon nanotubes.
We will explore these problems and potential solutions in future videos. One on how carbon
nanotubes are made, why they are so strong, and what needs to happen to take them from
the laboratory to regular life. And then, we will revisit this subject with a design
investigation for an actual space elevator using this new theoretical material.
During the research of this video I noticed several mistakes in the paper I referenced,
small mistakes that anyone could make and easily overlooked. I only noticed them because
I applied their methods myself and noticed inconsistencies. Their rounding was so aggressive
that their results were off an inconceivably large number thanks to the exponential function
in their equation and I noticed their material properties for steel was incorrect because
I recreated their calculations for titanium and noticed it was worse, despite the equation
being entirely determined by specific strength.
This is the power of applying knowledge yourself, rather than being a passive observer. You
begin to understand things on a fundamental deeper level and this is why I love Brilliant
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I found myself struggling to remember core parts of calculus while following the derivations
in the paper, as it’s been years since I had to integrate anything. So, I have committed
myself to completing Brilliants course on calculus and advanced mathematics to brush
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