The road to Hell is paved with good equations

Richard Scott Nokes has an interesting post over at the Unlocked Wordhoard, in which he takes a stab at calculating the distance between Milton’s Heaven and Hell. Actually, his student took the stab, and he only reported the results. To set the context, let’s have a look first at what Paradise Lost has to say:

Th’ infernal Serpent; he it was whose guile,
Stirred up with envy and revenge, deceived
The mother of mankind, what time his pride
Had cast him out from Heaven, with all his host
Of rebel Angels, by whose aid, aspiring
To set himself in glory above his peers,
He trusted to have equalled the Most High,
If he opposed, and with ambitious aim
Against the throne and monarchy of God,
Raised impious war in Heaven and battle proud,
With vain attempt. Him the Almighty Power
Hurled headlong flaming from th’ ethereal sky,
With hideous ruin and combustion, down
To bottomless perdition, there to dwell
In adamantine chains and penal fire,
Who durst defy th’ Omnipotent to arms.
Nine times the space that measures day and night
To mortal men, he, with his horrid crew,
Lay vanquished, rolling in the fiery gulf,
Confounded, though immortal.
[I.34–53]

So, as Nokes says, “Satan and the rebel angels fall for nine days through Chaos before landing in Hell.” His student calculated the result to be 25,920 miles. Now I am not going to object that this is an impossibly small distance — noting that the distance between the earth and the moon alone is some ten times as great. No, in the mythopoeic world of Milton’s Paradise Lost, I’m perfectly willing to accept that theological space is not necessarily the same as physical space (never mind that we are applying Newtonian mechanics — a tool of regular space — to the problem).

No, my concern is in some of the assumptions. “The math,” Nokes points out, “assumes that the terminal velocity of a falling angel is 176 ft/sec through a matrix of chaos.” Why assume this value? As we all know, terminal velocity depends on the shape, size, and orientation of the falling object, as well as on the viscosity or density of the material through which the object is falling. It depends also on the gravitational force being exerted on the object, which is around 9.8m/sec² — but only at sea-level on planet Earth. The gravitational force in other parts of space is completely difference; hence, terminal velocity in those regions also has different bounding parameters. It also depends on the initial force of God’s wrath (as Nokes mentions) — but was there torque involved? And if so, what is the radius of the circle described by God’s radius?

“Being that,” we are further told, “chaos probably has no air or fluid to resist movement through drag or friction (since it is a void), terminal velocity would be the same as initial velocity.” Well, I’m willing to accept that chaos might indeed be a vacuum, but if so, I don’t think this statement that the terminal velocity equals the initial velocity is true. In that case, there would be no acceleration. But we know that there is — unless we are talking about a perfect vacuum in which there are no objects to exert gravitations forces. But I would assume that Heaven and Hell both are quite gravitationally massive, wouldn’t you? In which case, velocity would converge asymptotically on the speed of light (without ever reaching it).

Let’s come at this from another angel — er, I mean, angle. According to our good friends at Wikipedia (because, let’s face it, it’s been fifteen years since my college physics classes), you can reduce terminal velocity to the following equation:

That’s if you can set aside buoyancy effects (which I think we can). Here, g is the acceleration due to gravity. As I said above, I think we’ve got to consider g a large, but unknown, constant. I think we can all agree it’s probably not the relatively weak 9.8m/sec² of our familiar Terra Cognita. The other part of the numerator of the fraction is m, the mass of the falling object. Would an angel have a mass much larger than that of a human being? Or perhaps, because they’re incorporeal by nature, much, much smaller? Hmm, let’s move on.

In the denominator, ρ is the density of the air or fluid through which the object is falling. The conventional wisdom might be that the “chaos” in question is rarified to the point of being a near-vacuum, a “void”; thus, ρ should approach 0. But on the other hand, a vacuum, even a very rarified near-vacuum, doesn’t sound very “chaotic”, does it? Perhaps ρ is actually very large, with a commotion of heavy molecules of air bouncing and colliding in every direction at all times.

Another factor, A, must be taken into account. This is the cross-sectional surface area of the falling body. I already broached the question of whether angels are light, since incorporeal; or heavy, since much greater than man. What about their size? Wouldn’t we have to think they’re very large indeed — at least as compared to people, or to a serpent — even if they are very light? Therefore, A is large. Er, but hang on a moment! How many angels was it that could dance on the head of pin? Maybe A is very small. Hmm.

Finally, we have Cd, the coefficient of drag. For a human being, oriented upright, the drag coefficient is in the neighborhood of around 1.0, roughly the same as for a simple cube; but for other shapes, and in different orientations, the coefficient may be higher or lower. Would an angel fall through space like a cube (1.0) or a sphere (0.47) — or perhaps more like a bullet (0.295) or even a Boeing 747 (0.031)? And would the angel twist and turn during his fall, resulting in a dynamic drag coefficient, changing with each gyration? Or would he clasp his hands to his chest and resolve himself to fall gracefully and without complaint, come what may? It all makes a difference to the calculations!

There are, alas, too many unknowns for us to arrive at any final answer. In the numerator, we think that the acceleration due to gravity should be very large, but the mass of the angel could be either very great or teensy-weensy. In the denominator, we think that the density of chaos is probably pretty great — or else it wouldn’t be called “chaos” — but we must acknowledge that it could be very little ( “void”). The cross-sectional surface area of the falling angel could be enormous, or infinitesimal. And the drag coefficient, well, who in the hell knows?

So, let’s see. Turn a few beads on the ol’ abacus. Putting this all together, it look’s like the answer is probably ... uh ... somewhere between a couple of beard-seconds and perhaps a megaparsec. I can live with the uncertainty. How about you?