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Towards the beginning of Independence Day, one of the officers reviewing scans of the alien mother ship says about it:

We estimate it has a diameter of over five hundred and fifty kilometers and a mass roughly one fourth the size of our moon.

This has always bothered me, because the moon is has a mass of 7.3x1022 kg, meaning the alien mother ship would have had a mass of about 1.8x1022 kg = 1.8x1025 grams.

Meanwhile, a 550-diameter ball has a volume of 4/3 x 3.14 x (275 km)3 = 8.7x107 km3 = 8.7x1022 cm3

This means the ship has an average density of 212 g/cm3, which is five times denser than if it was made of solid osmium (the densest element known to mankind)! Not only that, but shots of the mother ship clearly show it's not spherical, and later in the movie we see that much of the interior is empty. So whatever material the ship is made out of must be even denser. Even assuming that the 550km figure is a significant underestimate (he did say "over 550", after all), this is an incredibly dense ship!

This raises all kinds of questions regarding the ship's composition, structural integrity, and gravitational pull, as discussed in this paper.

I'm sure this was just a casual oversight by the writers, but what in-universe explanation might there be for the mother ship's incredibly high density? Does the movie or its sequel give any clues as to what the ship could be made out of in order to be so dense, and how it holds its shape without collapsing into a ball?

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    The aliens seem to have access to artificial gravity, which is how they can all stand upright on the floor of the interior. I'm not sure how they were estimating mass, but if it were based at all on gravitational effects, then the artificial gravity likely threw off our crude instruments. – user31178 Dec 22 '16 at 6:43
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We are only told the mass of the alien ship when it first appears; nobody double checks the math. My guess is that we were not detecting the alien ship itself, but its warp bubble.

Many warp drive systems depend on absolutely massive amounts of... well, mass, to warp space around the ship to allow the ship to reach a destination faster than it would without warping the space. Wormholes, black holes, Alcubierre drive: all these combine incredible masses with small external areas.

So, my handwaving armchair understanding of current "Faster Than Light" drives would lead me to believe that initial scans of the alien ship decelerating from Warp Drive would be incredibly over-inflated as it decelerates from interstellar travel. Where did this mass come from? Where does it go? To explain that would require higher handwaving-fu than I have. Alien technobabblum.

It's also possible that the ship itself never created or destroyed (or misplaced) mass, but the act of decelerating itself distorted whatever sensors we're using to detect its mass with a "lensing" effect which inflated the numbers.

But this would explain both the incredible initial masses detected, and the fact that an object 1/4 the mass of the moon was able to achieve low earth orbit without gravitationally screwing up the entire Earth orbital system. Remember, the satellite which crashes into the ship orbited into it. It wasn't pulled from a regular orbit into a heavy gravity well (as this would have thrown off its signal lock, which would have been easy to detect), it simply smashed into the ship and disappeared.

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