Mel’s off work this week and we’re back from my family’s on the coast for the holidays and just lounging around. Mel’s watching TV and the old Star Trek episode “The Doomsday Machine” is on, featuring this planet-killer:
They mention it’s made of Neutronium and Mel complains that it would have too great a gravitational field for anything to be around it. I replied that, well duh, it destroys planets so that probably helps. She counters that the Enterprise gets far too close and I counter-argue that the Enterprise has no problem flying around Earth. She claims that much Neutronium would be far too massive for even the Enterprise to withstand.
Here comes the math!
Neutronium is the unspecified super-dense material that composes neutron stars, which Wikipedia says 5 mililitres weighs over 500 million metric tons (5×1012 kg). That works out to 1 cubic metre (1000 litres) weighing 1×1018 kg. Now I just have to find the volume of the Doomsday Machine!
The show mentions it being “several miles long”, but no solid details. The excellent Starship Dimensions site has it on its 10 meters-per-pixel page at a length of 271 pixels (2.7 km). Here is with the same scale Constitution Class ship:
Looks right!
Ok, according to that, the “mouth” has a radius of roughly 250m, so the overall volume is: 1/3 * 2700 * (pi * 2502) = 176,714,587 m3. Of course, the Doomsday Machine is hollow so we’ll have to subtract the volume of the inner area from that total to find the true volume of the “hull”. The radius of the inside of the mouth seems to be about 200m and it looks like the length of the hollow area would be around 2km, so we get: 1/3 * 2000 * (pi * 2002) = 83,775,804 m3.
Here’s where I fudge a bit to make things easier. Subtracting gets us pretty close to 100,000,000 cubic metres so I’m going to round it to that. The Doomsday Machine’s shape is pretty irregular so we’re just estimating anyway.
This means that the Doomsday Machine weighs approximately 1 x 1026kg. So how much is that? Well, the Earth weighs nearly 6×1024kg, so this weighs about 17x as much, which is the mass of Neptune!
Now we reach my physics knowledge limit. Neptune weighs the same, but because gravitation follows an inverse square law and though Neptune is much more massive than Earth its surface gravity is only 1.1g. What this suggests to me is that because the Doomsday Machine is far smaller you are able to get much closer to all that mass, so its gravitational pull is going to be enormous — much more than a simplified “oh it weighs 17 times as much so it has a pull of 17g”. It may very well be that Mel is correct but I’m going to need someone with more physics and calculus knowledge to finish this up for me.
Update: Ok, I think I’ve figured this out. Neptune’s surface gravity is 1.14g. That’s at a distance of 24,764 km from its centre of gravity. So let’s say the Enterprise is sitting out at distance of 10km from the Doomsday Machine, that’s 1/2476th of the distance from Neptune’s centre, so the gravitiational pull would be 24762 times as high. That’s the equivalent of nearly seven million Earth gravities. Standing on the “surface” of the Doomsday Machine would be much much higher — say the distance from the centre would be 1.35 km, you get 1/18343 of Neptune’s radius, so 183432 times as much gravity, which is 336,491,465 Neptunian gravities which is 383,600,270 standard gravities! Of course the Doomsday Machine isn’t a sphere so these numbers are all approximate, but the general order of magnitude is about right — the Enterprise has to resist several million standard gravities to even stay at a distance of 10km from the Doomsday Machine.
All of this assumes that the Doomsday Machine’s hull is solid neutronium and not just a thin shell! If the hull is only about a metre thick the total volume is only about 1.5 million m3, which weighs 1.5×1024 kg which is still 1/4 of Earth’s mass!
Recent Comments