So, a couple months ago, I looked up the formula to find the Roche limit of a given body of mass to its satellite, and asked what was perhaps the most important question of our time:
"If the Earth were to be caught in a decaying orbit around a Jupiter-sized glob of osmium (the densest natural element), how big would it be in the sky when cities started floating into space?"
And now, finally, I have the tools to answer it!
Okay, for the non-physics majors out there, let me briefly explain the Roche limit. We experience this phenomenon called "weight" (and some of us more than others) due to the Earth's massive size, gravitationally pulling us toward its center. So now, imagine you're standing on the Moon, where you'd weigh six times less:
The Moon is pulling you toward it with a certain strength, and you're too far away from the Earth for it to be able to pull you. But, as the Earth gets closer and closer to the Moon, you'll start feeling a gravitational pull from both the Moon and the Earth:
At some point, Earth's pull on you would be stronger than the Moon's, and you'd start drifting into the sky toward Earth... along with big chunks of the Moon.
So! What if the Earth were orbiting a Jupiter-sized glob of osmium? How far away would the Earth be before cities started floating into the sky?
Roche limit = (the radius of the osmium glob) x (2(density of osmium / density of Earth))^(1/3)
Roche limit = (about 69,000 km) x (2(22.59/5.15))^(1/3)
Roche limit = 139,096.4333 kilometers
Now, that's the distance from the center of the Earth to the center of the osmium glob, so we'd have to subtract the radius of the Earth (6378.1 km) to get:
And all we have to do is apply the law of sines, grab a TI-83, and we've got the answer!
sin(90) / 132,718.333 = sin(ß) / 69,000
Sin(ß) = 69,000sin(90) / 132,718.333
Sin-1(69,000sin(90) / 132,718.333) = ß
ß = 31.32541187, or approximately 31.33
Since the angle at which we see the glob of osmium is 2ß, we get 62.66.
So, out of 180 degrees (horizon to horizon), the glob would be filling up 62.66 degrees of it, or about one third. Which, as rendered by my artistically inclined brother, means it would looks something like this: