Mass and Surface Gravity of a Dyson Sphere

[Drakkith]

TL;DR

Mass and Surface Gravity of a Dyson Sphere

I'm re-watching Star Trek TNG and I just started the episode where they encounter Scotty aboard a ship that's crashed into a Dyson sphere.
That got me thinking. What would the mass and external surface gravity of a Dyson Sphere be? I've done the math myself, but I'd appreciate someone double checking my numbers.

Assumptions:
Internal Radius of 0.5 AU
Thickness, based on a very, very crude estimate by comparing the Enterprise-D to the sphere: 2 Km
Density of approximately 7 g/cm³ (I don't know what the density of carbon-neutronium is, but I'm using a value a little less than iron's because reasons).

If my calculation is correct, the mass is roughly 5x10²⁹ kg, about 0.25 solar masses, and the surface gravity is about 0.0015 m/s².

 

[phinds]

Dude, you have way too much spare time on your hands

 

[Drakkith]

Dude, you have way too much spare time on your hands

You have no idea.

 

[Ibix]

Hm. I make the shell volume $4\pi×(75×10^9)^2×2×10^3\mathrm{m}^3$, or $(9\pi/2)×10^{25}\mathrm{m}^3$. With a density of $7000\mathrm{kgm^{-3}}$ that makes a mass of $31.5\pi×10^28\mathrm{kg}=10^{30}\mathrm{kg}$, which is 0.5 solar masses.

 

[Drakkith]

Well, glad to know we're within an order of magnitude of each other. Really goes to show the insane amount of material you'd need to build a Dyson Sphere like this.

Of course, I assume the shell of a *real* Dyson Sphere wouldn't be a solid mass, but have significant empty space. Either that or it would be significantly thinner than 2 km. Still, even if it's merely a 10th the amount of material that we've calculated here that still puts it in the realm of literally stellar amounts of material.

 

[DrStupid]

Of course, I assume the shell of a *real* Dyson Sphere wouldn't be a solid mass, but have significant empty space. Either that or it would be significantly thinner than 2 km. Still, even if it's merely a 10th the amount of material that we've calculated here that still puts it in the realm of literally stellar amounts of material.

That's not the only problem with this kind of DS. It requires a lot of magical technology (handwavium with sufficient mechanical strength, harvesting matter from the star or import from other star system, artificial gravity for the habitable inner surface and so on). A real Dyson Sphere would need to be build completely different (especially no rigid shell and nobody living inside) and even a 10th of the area density would be way too much.

 

[stefan r]

...
Thickness, based on a very, very crude estimate by comparing the Enterprise-D to the sphere: 2 Km
Density of approximately 7 g/cm³ (I don't know what the density of carbon-neutronium is, but I'm using a value a little less than iron's because reasons).

If my calculation is correct, the mass is roughly 5x10²⁹ kg, about 0.25 solar masses, and the surface gravity is about 0.0015 m/s².

Metal coated graphene or metalized plastic can be inflated with helium or hydrogen. That can lower your estimate by more than 4 orders of magnitude. Could be made to look like the sets in scifi. I find it unlikely that Hollywood hires the welders who make battleship armor. I suspect we are seeing pictures of painted plastic or CGI based on earlier sets made out of plastic.

...Really goes to show the insane amount of material you'd need to build a Dyson Sphere like this...

...Still, even if it's merely a 10th the amount of material that we've calculated here that still puts it in the realm of literally stellar amounts of material.

That limits us to building at most a few hundred billion of them in the Milky Way.

 

[DrStupid]

That limits us to building at most a few hundred billion of them in the Milky Way.

The problem is not the available material in the Milky Way but in a single planetary system. In case of the solar system all plantes and moons together have only 0.13 % of the mass of Sun and only about 5 % of that mass is solid. That means without transmutation the maximum mass of a Dyson Sphere would be 0.007 % of the mass of Sun (assuming that all metals can be used). Even with transmutation of H and He from the gas giants to heavier elements, the mass limit would still be two orders of magnitide below a 10th of the mass of the Sun.

Building havier Dyson Spheres would require large-scale gas mining and transmutation of matter from the star or (even worse) hauling matter between stars. Maybe there is a possibility to get the requires materials during the accretion phase of a protostar. But that would imply a lot of other problems (e.g. the strong variations of the luminosity during the transition of the protostar to a main sequence star).

 

[stefan r]

The problem is not the available material in the Milky Way but in a single planetary system. In case of the solar system all plantes and moons together have only 0.13 % of the mass of Sun and only about 5 % of that mass is solid. That means without transmutation the maximum mass of a Dyson Sphere would be 0.007 % of the mass of Sun (assuming that all metals can be used). Even with transmutation of H and He from the gas giants to heavier elements, the mass limit would still be two orders of magnitide below a 10th of the mass of the Sun.

Building havier Dyson Spheres would require large-scale gas mining and transmutation of matter from the star or (even worse) hauling matter between stars. Maybe there is a possibility to get the requires materials during the accretion phase of a protostar. But that would imply a lot of other problems (e.g. the strong variations of the luminosity during the transition of the protostar to a main sequence star).

2.8 x 10²³ m² for a sphere with Earth orbital distance.

Graphene has 7.6 x 10^-7 kg/m² specific density. So we need something like 2.1 x 10¹⁷ kg of carbon per atomic layer. The asteroid belt has estimated 2.3 x 10²¹ kg of material. Meteors recovered on Earth have 1.5% carbon content. Asteroids are likely to average much higher carbon content. We have enough material in the belt to do a hundred layers of graphene foam. Add few atoms thick layer of metallic coating. We would need to weigh this down so that it did not get blow away by solar light pressure. That could be done by inflating the foam with gas giving it that bulky Star Trek look.

Neptune has 10^26 kg. Several hundred kg/m² would be more like the concrete decks we build in cities. There may not be enough calcium in Neptune for a thick concrete deck. Concrete could not support itself and needs some sort of "active support structure". There is enough material to do wood floorboard, tortoise shell/bone, a variety of thick polymers, or sheet metal steel. All of those would require active support too.

Building with active support has not been demonstrated yet. Paul Birch wrote some articles for Journal of the British Interplanetary Society. He gets into making spheres around gravity wells in section 3.4 of the third article published in 1983. The shell mass can be made arbitrarily thick so long as the rotor mass or rotor velocity is able to be increased by the same amount. Not "impossible" just "prone to catastrophic failure".

 

[DrStupid]

2.8 x 10²³ m² for a sphere with Earth orbital distance.

Graphene has 7.6 x 10^-7 kg/m² specific density. So we need something like 2.1 x 10¹⁷ kg of carbon per atomic layer. [...]

Yes, bulding a light-weight Dyson Sphere (down to a minimum of 1.3 g/m² for a closed Dyson Bubble with 100% absorption) is not limited by the available material. But there are not enough resources for a rigid Dyson Shell (not to speak of the required tensile strength).

We would need to weigh this down so that it did not get blow away by solar light pressure. That could be done by inflating the foam with gas giving it that bulky Star Trek look.

Looking bulky is not sufficient. We are talking about a DS with a habitable inner surface: