And not just casually dropping it a little bit onto someone’s head, like in all those cartoons. No, this is a drop of mythic proportions.
In Hesiod’s Theogony, it’s stated that Tartarus is as far below earth as earth is below heaven; a bronze anvil will take nine days to fall to earth from heaven and, if dropped again, nine to fall to Tartarus. This is problematic on a number of levels.
Here’s what happens if you try the anvil thing on or around Earth, using mostly real physics.
You pick a starting point high enough for the fall to take nine days. This is going to be significantly higher than when we dropped the Enterprise, since that only took a couple days at most. We’ll be dropping the anvil from a lot more. I haven’t estimated the numbers (I wouldn’t have the equation to find a precise answer anyway), but I’d bet that it’s easily high enough to be outside the Moon’s orbit and probably a lot further out than the human distance record on top of that. Anyway, we’re dropping it from really high.
Nine days later (minus a few fractions of a second), it’s falling at several thousand kilometers per second. It punches through the earth’s atmosphere and probably vaporizes. Hopefully. If not, it hits the surface at precisely the nine-day mark and Bad Things Happen.
If we took Hesiod literally, it would mean that Tartarus and heaven are both at the same height (plus or minus the diameter of the earth), because if we let that anvil fall through the earth (through a strategically placed hole painted on the sidewalk?), it’d come out the other side and start going upward to nearly the same height as it started from. He did say it’s the same distance. And things don’t fall up, so let’s see what happens when we drop it and leave it for nine days.
To find the depth of Tartarus, we’ll have to assume there’s a frictionless hole deep enough to drop the anvil down. Either that or it’s a special divine lump of bronze endowed with the ability to pass through regular matter and affected only by gravity. For our purposes, those will act the same.
If you drop that down, it’s going to go straight through, accelerating until it reaches the center of earth. Then it’ll keep falling, but slowing down because gravity is pulling it the other way. When it reaches the other surface, it’ll stop and start falling back. One direction takes a bit over 42 minutes.
(Side note because it’s really cool: It’s the same amount of time for any frictionless straight line through Earth, no matter what angle it’s at. The math is a bit beyond me, but that doesn’t make the fact less cool.)
The anvil is falling from one side of the earth to the other, repeatedly, every 2530.3 seconds, for nine days. It traverses the earth 307.3 times, so that after precisely nine days it ends up 2011 km deep, as measured from the other side of the world. So it’s most of the way through the earth’s mantle, on the other side of the core. Of course, it hasn’t “landed” in any way, but that’s where it is when it hits the nine-day mark so that’s where Tartarus is. Even if it’s nowhere near as far down as the heavens are up.
OK, so that didn’t fit with Hesiod, like, at all.
But gravity doesn’t work the same way in mythology! Everything falls at its own natural speed, and “inverse square” doesn’t even mean anything. So we can just figure out the natural speed for a bronze anvil to fall at, and ignore all this Newtonian silliness.
Finding that is probably impossible. What we can do is find how fast an anvil would actually fall, and assume that that’s the number the Greeks would say is its natural speed. So, we just have to use the awesome power of the Internet to find terminal velocity.
Getting representative statistics from an online anvil store (of course there’s an online anvil store), your classic anvil would weigh about 167 lbs and has a cross-sectional area of 102 square inches. Those anvils are made of steel, and bronze has more variation in its density. But the range is about right.
Those numbers can get plugged into a handy terminal velocity calculator, and it spits out a number of 84.7 m/s, about the speed of a high-speed train. Since it’s falling for nine days, or 9*24*3600 seconds, it started at 65,800 km above the surface. And so that’s the height of heaven and the depth of Tartarus.
On the real Earth, those would be up in space and down…in space. The distance is about five times the diameter of Earth. But we don’t need to worry about that future stuff right now; it’s 800 B.C.
One last mental image: The anvil falls for over a week, and then flies directly into the bottleneck of the underground prison in the deepest pit on Earth. There was probably a red bulls-eye painted around it. Then it falls for another nine days, where it lands directly on the head of the most hated enemy of the gods, Wile E. Coyote. Tell me you weren’t picturing that all along.