Planets or Orbiting Habitats?

In the only original Star Trek Nearly all the action takes place on a planet or on the Enterprise itself. While there are starbases mentioned, and action takes place on a starbase in at least one episode, The Trouble With Tribbles the impression is given that the overwhelming majority of humans and humanoids live on a planet. There are no ringworlds or Dyson spheres.

And the starbases shown are not the giant cylinders or toruses with open skies and forests. They are cramped affairs, less spacious than a modern cruise ship or an Embassy Suites hotel. This is despite having artificial gravity and plentiful quantities of unobtanium.

Was this a lack of vision on the part of Star Trek's writers?

Or was it accidental prescience?

Sorry, No Ringworlds

Without some stupendously strong unobtanium, there is no way to spin a ring around a star fast enough to get Earthlike gravity. You cannot even put a ring around our moon.

Let's do the math. Let's look at a section of a ringworld looking down the polar axis.

And if we slice our ring it has a cross section like this:

In these diagrams:

$R$	Radius of the ring. We need tens of millions of kilometers to do a ring around a star. Something like 150 million kilometers for a star as bright as our own.
$T$	Tension on the ring.
$w$	Width of the ring.
$h$	Height, aka thickness, of the ring. Note that as the thickness grows, the ring gets stronger, but it also gets heavier.
$A$	The area of the cross section of the ring.

Before we go deep into the math, let's do some analogies. A ringworld is like a rope bridge, like in those old jungle adventure movies. The rope bridge is held up purely by the tension on the two ends. There are no supports in the middle. For a spinning ring, the tension is provided by the ring as a whole, but we can look at a small section of the ring as analogous to one of those scary rope bridges.

The rope bridge is trying to hold the people crossing it up but the bridge is going across a river/gorge. To have an up component, the bridge must sag. The less it sags, the stronger the ropes need to be.

Well, for a ringworld, the bigger the radius, the less the sag and the greater tension required to hold up the ring itself -- not counting the people and materials on it.

Now let's do the math. In the first figure, we have a ring, with a vertical line and a symmetric segment spanning $\pm \theta$ . Where $\theta$ is a very small angle. The total weight of this range of the ring is

Where

$\rho$	is the density of the material which makes up the ring.
$g$	is the desired pseudo gravitational acceleration. For full Earth gravity this would be $9.8\text{m}/{\text{s}}^2$

Holding up this weight is the force:

The factor of two comes from the fact that both ends of the segment are holding it "up."

For a very small angular range $\text{tan}\theta \approx \theta$ . Putting this all together we get

Cancelling we get:

where $S_{\text{max}}$ is the maximum stress we want to place on the material we use to build the ring. This should be below the yield strength of a ductile material and well below the tensile strength of a more brittle material.

Let's turn to Wikipedia's handy dandy article on tensile strength. We see that a really good, but not too brittle, steel can had a yield strength of around 2000 MPa and a density of around 8000 kg / m³. Plugging these numbers we get:

When it comes to ringworlds, we are off by over a factor of a million.

Let's go for an exotic material. It appears that graphene maxes out at 130,000 MPa with a density of 1000 kg / m³. Note that this is tensile strength, not yield strength, so we double plus want a safety margin!

With graphene we could make a ringworld circling Earth, but not a star.

But I would double plus want some safety margin! When it comes to brittle materials like carbon fiber, imperfections lower strength bigly. And even for more forgiving/self-healing materials like steel, you need to figure in both the extreme temperatures of outer space and cosmic rays. (Steel beats extended molecules in this regard. Steel has some self-healing properties.)

Note that all of this analysis is just for the ring itself. If people are to live on the ringworld, you need soil, buildings, lakes and atmosphere!

O'Neill Colonies

Let's consider something less ambitious: a cylindrical O'Neill colony in the Earth-Moon L5 or L4 region. There was talk back in the day of making these things a kilometer in diameter. Is this feasible?

Let's treat a slice of the cylinder as a ring, and load it down with additional weight. Let $D$ be the mass per square meter of dirt, trees, buildings, etc. sitting on "top" of the cylinder wall. Our mass becomes:

Now, let's equate this with the y component of the tension as before:

Cancelling:

Let's solve for the required thickness $h$ .

Note that this equation is only meaningful if the denominator is positive, which is the case if

For good steel, $R_{\text{max}}$ was 25 km, which is well above the half kilometer radius for the proposed O'Neill cylinder.

For the cylinder in the picture, there is enough dirt to contain a lake. Let's go with four meters thick with a density 1.5 times that of water. That would be:

Let's play it a little safe and dial our maximum stress down to 1000 MPa. When building something this huge, there's bound to be defects in the steel. That's still four times the yield strength of structural steel. Required thickness becomes:

Good news! That second term in the denominator is insignificant. So let's ignore it and round the numerator upward to get

A bit over an inch thick! Wheee!

But I neglected something very important: atmosphere! The Earth's atmosphere at sea level weighs about the same a 10 meters of water, or 33 feet. I could merge this equivalent mass per square meter into the equation above, but in case we want to play with $g$ let's just make it a second term. We now have:

We could dial the air pressure back significantly. A quick search indicates that Denver averages 83,400 Pascals of pressure and is a reasonably pleasant place to live. On the other hand, if you want human powered flight, it's better to keep the air pressure up.

Finally, let's fret about radiation. Here on Earth we are shielded by the equivalent of ten meters of water. Maybe we should increase the amount of dirt to the equivalent mass, or even more. On Earth, we are also shielded by a big magnetic field. Whether we could generate enough magnetic field to shield an L5 colony is an exercise for another day.

Anyway, let's up the dirt to 10,000 kilograms per square meter. (And note that the windows need to be thick too, or cover them in water to add shielding.) I'm going round $g$ up to 10 just to make things cleaner.

OK, I'm regaining my excitement about building giant space colonies. This is looking feasible!

But I fear we are still looking at close to a century in the future. It's going to take a LOT of infrastructure to build at this scale in space.

By the way, I am not an engineer, so don't design your space habitat based on this article. This is just a bit of back of the envelope thinking to double check if O'Neill and company were being realistic. Apparently, they were.