How big is the 'Crucible' from Doctor Who?

Question

The Crucible is a large, spherical object used by the Daleks as a ship/superweapon. I would like to know how big it is.

Answer 1

The minimum radius to have the spheroidal shape characteristic of a planet depends in whether it is made of weak ice or strong rock and thus might vary from 200 to 400 kilometers and even more. Jupiter, the largest planet in our solar system, has a radius of 69,911 kilometers and thus a diameter of 139,822 kilometers. Jupiter has nearly the largest possible radius for a planet since more massive ones will get denser and not much larger.

It was once expected that any icy body larger than approximately 200 km in radius was likely to be in hydrostatic equilibrium (HE).1 However, Ceres (r = 470 km) is the smallest body for which detailed measurements are consistent with hydrostatic equilibrium,2 whereas Iapetus (r = 735 km) is the largest icy body that has been found to not be in hydrostatic equilibrium.9 Earth's moon (r = 1,737 km) is also not in hydrostatic equilibrium, but—unlike icy Ceres and Iapetus—it is composed primarily of silicate rock, which has a much higher tensile strength than ice.

https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size#Larger_than_400_km[3]

All imaged icy moons with radii greater than 200 km except Proteus are clearly round, although those under 400 km that have had their shapes carefully measured are not in hydrostatic equilibrium.[24] The known densities of TNOs in this size range are remarkably low (1–1.2 g/cm3), implying that the objects retain significant internal porosity from their formation and were never gravitationally compressed into fully solid bodies.[25]

https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size#From_200_to_399_km[4]

As said above, Jupiter, the largest planet in our solar system, has a radius of 69,911 kilometers and thus a diameter of 139,822 kilometers. Jupiter has nearly the largest possible radius for a planet since more massive ones will get denser and not much larger.

Planets more massive than Jupiter, up to about 13 times themass of Jupiter, and brown dwarfs more massive than planets up to about 75 or 80 times the mass of Jupiter, and even the lowest mass stars above that limit, will not be much larger or smaller than Jupiter. Their diameters will only vary by about 15 percent over that whole great range of mass.

https://www.discovermagazine.com/the-sciences/how-big-is-the-biggest-possible-planet[5]

The only exception to that rule is when gas giant planets orbit very close to their stars and get very hot. Their dense atmospheres expand and increase the diameters of the planets greatly.

Gas giants with a large radius and very low density are sometimes called "puffy planets"[42] or "hot Saturns", due to their density being similar to Saturn's. Puffy planets orbit close to their stars so that the intense heat from the star combined with internal heating within the planet will help inflate the atmosphere. Six large-radius low-density planets have been detected by the transit method. In order of discovery they are: HAT-P-1b,[43][44] COROT-1b, TrES-4, WASP-12b, WASP-17b, and Kepler-7b. Some hot Jupiters detected by the radial-velocity method may be puffy planets. Most of these planets are below two Jupiter masses as more massive planets have stronger gravity keeping them at roughly Jupiter's size.

https://en.wikipedia.org/wiki/Hot_Jupiter#Puffy_planets[6]

The largest known puffy planets have up to 3 times the radius of Jupiter, while two borderline candidates that might be brown dwarfs instead of planets have radii up to 10 times the radius of Jupiter.

https://en.wikipedia.org/wiki/List_of_largest_exoplanets[7]

So if you count "puffy planets" a planet could have a radius of up to roughly 209,733 kilometers and a diameter of up to 419,466 kilometers, or even a radius of up to roughly 699,110 kilometers and a diameter of up to 1,398,220 kilometers.

I note that the largest planets, ice giants or gas giants, have very dense and massive atmospheres which surround their rocky cores and extend hundreds and thousands of miles above their more or less "solid" "surfaces", to use those terms loosely. Thus such giant planets can become "puffy planets" when orbiting very close to their stars.

I note that a planet shaped object that is a machine made out of metal can not be infinitely large. Eventually the object will become massive enough and have a strong enough gravity to have intense internal pressure in its innermost layers and the metal there will melt and flow into a sold spherical shape with no voids. Building more layers on top of the outer layers will compress more internal layers into the dense core.

Of course, since a machine or spaceship would be mostly hollow chambers, it would have a much lower density per unit of volume than a planet made out of solid rock with no voids. Thus such a machine could be built to a much greater size than a planet built out of solid rock before its own gravity became strong enough to compress its inner regions.

The Tardis Wiki describes the Crucible as: "It was a mostly spherical space station the size of a planet".

Suppose that the Crucible's spherical part was so massive that it had a surface gravity as strong as Earth's. In that case the tallest metal tower or projection that could be built above its metal spherical surface would be no taller than the tallest metal tower or projection that could be built above the surface of Earth.

The tallest mountains on Earth are almost as tall as the tallest possible mountains on Earth despite being solid rock. Their weight is so great that they would sink into the underlying rocks. But tall building are only about 15 percent structure and 85 percent air by volume, and could get a lot taller than mountains before starting to sink into the earth under their own weight. So theoretically it might be possible to build buildings a few tens of kilometers or miles high.

https://www.citylab.com/design/2012/08/there-limit-how-tall-buildings-can-get/2963/[8]

Since Earth has a radius of 6,371 kilometers and a diameter of 12,742 kilometers, even a 100 kilometer tall structure would be only 1/63.71 or 0.0156 of the radius and 1/127.42 or 0.0078 of the diameter of Earth. If one looked at the whole Earth from a distance such a structure would be barely visible as a bump on the spheroidal surface of the Earth.

Of course, a spherical space station with the same surface gravity as the Earth would probably be much larger than the Earth, since it would be mostly hollow space. Thus a 100 kilometer high tower on such a space station would be even less noticeable than on Earth.

I saw an image of the Crucible online and it looked lie a tower on it had a height of at least a quarter of the diameter of the crucible.

Therefore, at a rough guess, and without making any sort of structural calculations, I would guesstimate that, like the Deathstar, the Crucible was "the size of a small moon".

You might be interested in Larry Niven's famous article "Bigger than Worlds" about the hypothetical construction of very large structures in outer space.

https://en.wikipedia.org/wiki/Bigger_Than_Worlds 9

https://www.isfdb.org/cgi-bin/title.cgi?133302[2]

https://www.scribd.com/document/100964566/Larry-Niven-Bigger-Than-Worlds-v1-0-Italics[1]

I think that Niven says that the largest hypothetical structures would require hypothetical undiscovered super strong materials to construct. But if the fictional Daleks had access to such fictional super strong materials, the Crucible could be much larger than I roughly estimated.