A different artificial gravity: Ultradense material

Question

Guests of this large space station enjoy Earth-like gravity (80-90% minimum).

How? Well the space station is so massive it has its own gravity. The disk shaped "core" is made with ultradense materials with an overall of 20-30 g/cm3 similar to Iridium or Osmium (the densest elements).

I thought that working with material with such density could provide decent gravity with relatively little amount of material so I tried crunching some numbers...

Turns out I suck at math. How massive/large does a disk with these density have to be to provide similar gravitational pull close to its surface? (Ignore edge effects and the core does not have to keep its own atmosphere, even if you can just provide some formulas it would be great since right now I don't have much time for this)

Answer 1

A disk won't really work in the sense that you want it to. Supposing that your material was massive enough to create an appreciable gravitic field, a body would be attracted to the centre of the disk. The centre of mass needs to be directly below your astronauts; they need to walk on the edge of the disk, not the flat dorsal or ventral surfaces.

Additionally, the increase in density is only a four-fold increase, while the decrease in volume will be exponential by going from a sphere to a disk. Logically, your prospective disk will have to be an order of magnitude larger in radius than the earth in order to achieve earth-like mass (and therefore gravitational attraction).

You need some fantastic substance to make this idea viable, either to take the associated interstellar punishment and associated integrity issues that would be dealt out to a disk sized like an enormous net spread out across a solar orbit, or to have some fantastically hyperdense material that will give you the gravity you want. In the latter case, this fantastic substance is entirely under your control, and you can handwave its required properties away.

EDIT: This doesn't even address the problem of how you would prevent your disk from collapsing in on itself. There's a reason celestial bodies are shaped roughly like spheres.

EDIT: @TLW points out that for a sufficiently large disk, the gravitational field changes so that gravity can no longer be approximated by a point. Very cool, but nevertheless huge, and prone to all the problems of a system-sized megastructure.

Answer 2

I believe the space station would induce nausea due to the changing distribution of the amount of gravitational force in different directions as the astronauts walk over the surface. In the center of the disk, for example, I imagine the astronaut would not feel pulled downward as much as they would feel pulled to the edges of the disk. Any effort to fix this likely results in a space station that is spheroid.

To simplify the math, let's set aside the disk shape idea and just think about a spheroid space station made entirely out of material at the upper end of the density range you provided, 30 g/cm3.

The earth has a mass of $5.972 * 10^{24} kg$. $80\%$ of this is $4.778 * 10^{24} kg$, or $4.778 * 10^{27} g$ because there are $10^3 g / kg$. To get this much mass with a $30 g / cm^3$ material, you need:

$$ \frac{4.778 * 10^{27} g}{30 g / cm^3} = \frac{4.778 * 10^{26} g}{3 g / cm^3} = \frac{1.593 * 10^{26} g}{1 g / cm^3} = 1.593 * 10^{26} cm^3$$

In a cubic meter, there are $100^3$ cubic centimeters, and in a cubic kilometer, there are $(10^3)^3$ meters or $10^9$ meters. We end up with $1.593 * 10^{11} km^3$. The earth is approximately 1 trillion $km^3$, so our space station, if spheroid, would be about $16\%$ the size of the earth.

Smashing that sphere into a disk does things to the gravitational field that I am not qualified to answer; I can only imagine it would be very disorienting to walk across.

Answer 3

Gravitation on the surface scales linearly with radius and density. The earth has a radius of 6,400 km and a density ~ 5g/cm3. Irridium and osmium have densities about 20g/cm3, thats 4 times earths density.

Therefore the radius of the sphere is 1/4 that of earth to produce normal gravity. 1,600km radius ball of osmium might be hard to find. If you want 80% gravity then 1,600*80%=1,300km radius. 4/3*Pi*r^3 tels us that the volume is 9.2 billion cubic km and as 1 billion cubic m is 1 cubic km and 1 cubic m weighs 20 tonnes. The volume is 9.2 billion billion m^3 and the density 20 million g/m^3. That gives a weight of 184 million billion billion gram's.

Iridium costs £10 per gram and osmium £5 per gram. (ballpark) This makes the project cost in the region of £1 billion billion billion or £10^27.

With a world gdp of £50 thousand billion this project could be paid for in 20 thousand billion years(assuming all funds are devoted to it and osmium prices and gdp remain constant) About when the last stars are dying out.

Answer 4

Given a disk of radius $R$ and thickness $T$ made out of a material of density $\rho$, the gravitational acceleration a distance $h$ directly above (or below) the center of the dist is given by the integral $$\int_0^T\int_0^R\frac{2\pi\rho G}{\sqrt{x^2+(t+h)^2}}\,dx\,dt.$$ The solution is lengthy, but what really interests us is the limit as $h\to 0$, which is $$a=2\pi\rho G\left(R \log \left(\frac{\sqrt{R^2+T^2}+T}{R}\right)+T \log \left(\sqrt{R^2+T^2}+R\right)-T \log (T)\right)$$ If we assume that the thickness of the disk is very small compared to the radius of the disk, We can make some simplifications which reduce this to $$a=2\pi\rho G\left(R(1-\log 2)+R\log(R/T)\right).$$ Anyway, here are the equations you asked for.

Answer 5

If you can stabilize neutronium, or something equally dense, this will work.

Big if, however.

Otherwise, it won't work - the density simply isn't high enough to build it on a sane scale.

Build the flooring of your station out of a sheet of neutronium ~58nm in thickness and ~100m in diameter.

A cautionary note: this will mass somewhere on the order of three-quarters of a quadrillion metric tons (~7.5 * 10¹⁴ kg) - somewhere in the range of a decent asteroid (several km in radius, depending on the density of said asteroid). You had best ensure that it is safely bound to the station, lest it tear the station apart. Remember, it's 60nm thick.

Also, one had best hope that said stabilization is stable, lest it explode. That much evaporating neutronium would release on the order of 5.6*10²⁸ J of energy. (That's somewhere around the amount of energy required to stop the Moon in its orbit around the Earth, just to give you a comparison.) That's enough to melt a cm of aluminum at 45 light-seconds away (!), if I did my math correctly.

This will give you a difference in acceleration of something like 20cm/s² between head and feet, which may be annoying. Though you can mitigate this by making it larger.

In actuality, you'll want to vary the thickness depending on the radius, as well as spin it slightly (which reduces the total material required). Otherwise you'll get weird effects anywhere but the center of the disk. But with the spinning and thickness variation you can get something like 90 % of the disk's radius to be locally flat, which is good enough for most purposes.

Note that you could accomplish much the same thing with an array of small black holes - although they would have to be rather small in order to not have a noticeably "bumpy" field, and the required stabilization would be even more absurd for them than for neutronium (among other things, the Hawking radiation would mean that they would decay almost instantaneously unless stopped)

Answer 6

Robert L Forward wrote about using ultra dense materials for controlling and manipulating gravity, but when he was speaking of ultra dense materials the subject was the sort of degenerate material from White Dwarf stars or neutronium as you would find in a Neutron star. One example in his book "Future Magic" suggests that to nullify a 1g field you could take a 4 million ton mass and compress it to a sphere 32cm in diameter, or a disc 45cm in diameter and 10 cm thick.

Forward was always a bit playful, so some of his suggestions included supporting a disc of degenerate matter with massive diamond pillars to counteract the gravity of Earth under the disk. This would make for an amazing amusement park ride (although the ticket price would be a bit steep). Small matters like how to keep degenerate matter or neutronium from expanding at close to the speed of light outside of their environments were conveniently overlooked.

At any rate, many other posters provided the mathematical tools, so simply plug in the desired "g" field to find out what unnatural material the gravity disc will have to be made from.