## Why Stuff Can Orbit, Part 4: On The L

Less way previously:

We were chatting about central forces. In these a small object — a satellite, a planet, a weight on a spring — is attracted to the center of the universe, called the origin. We’ve been studying this by looking at potential energy, a function that in this case depends only on how far the object is from the origin. But to find circular orbits, we can’t just look at the potential energy. We have to modify this potential energy to account for angular momentum. This essay I mean to discuss that angular momentum some.

Let me talk first about the potential energy. Mathematical physicists usually write this as a function named U or V. I’m using V. That’s what my professor used teaching this, back when I was an undergraduate several hundred thousand years ago. A central force, by definition, changes only with how far you are from the center. I’ve put the center at the origin, because I am not a madman. This lets me write the potential energy as V = V(r).

V(r) could, in principle, be anything. In practice, though, I am going to want it to be r raised to a power. That is, V(r) is equal to C rn. The ‘C’ here is a constant. It’s a scaling constant. The bigger a number it is the stronger the central force. The closer the number is to zero the weaker the force is. In standard units, gravity has a constant incredibly close to zero. This makes orbits very big things, which generally works out well for planets. In the mathematics of masses on springs, the constant is closer to middling little numbers like 1.

The ‘n’ here is a deceiver. It’s a constant number, yes, and it can be anything we want. But the use of ‘n’ as a symbol has connotations. Usually when a mathematician or a physicist writes ‘n’ it’s because she needs a whole number. Usually a positive whole number. Sometimes it’s negative. But we have a legitimate central force if ‘n’ is any real number: 2, -1, one-half, the square root of π, any of that is good. If you just write ‘n’ without explanation, the reader will probably think “integers”, possibly “counting numbers”. So it’s worth making explicit when this isn’t so. It’s bad form to surprise the reader with what kind of number you’re even talking about.

(Some number of essays on we’ll find out that the only values ‘n’ can have that are worth anything are -1, 2, and 7. And 7 isn’t all that good. But we aren’t supposed to know that yet.)

C rn isn’t the only kind of central force that could exist. Any function rule would do. But it’s enough. If we wanted a more complicated rule we could just add two, or three, or more potential energies together. This would give us $V(r) = C_1 r^{n_1} + C_2 r^{n_2}$, with C1 and C2 two possibly different numbers, and n1 and n2 two definitely different numbers. (If n1 and n2 were the same number then we should just add C1 and C2 together and stop using a more complicated expression than we need.) Remember that Newton’s Law of Motion about the sum of multiple forces being something vector something something direction? When we look at forces as potential energy functions, that law turns into just adding potential energies together. They’re well-behaved that way.

And if we can add these r-to-a-power potential energies together then we’ve got everything we need. Why? Polynomials. We can approximate most any potential energy that would actually happen with a big enough polynomial. Or at least a polynomial-like function. These r-to-a-power forces are a basis set for all the potential energies we’re likely to care about. Understand how to work with one and you understand how to work with them all.

Well, one exception. The logarithmic potential, V(r) = C log(r), is really interesting. And it has real-world applicability. It describes how strongly two vortices, two whirlpools, attract each other. You can write the logarithm as a polynomial. But logarithms are pretty well-behaved functions. You might be better off just doing that as a special case.

Still, at least to start with, we’ll stick with V(r) = C rn and you know what I mean by all those letters now. So I’m free to talk about angular momentum.

You’ve probably heard of momentum. It’s got something to do with movement, only sports teams and political campaigns are always gaining or losing it somehow. When we talk of that we’re talking of linear momentum. It describes how much mass is moving how fast in what direction. So it’s a vector, in three-dimensional space. Or two-dimensional space if you’re making the calculations easier. To find what the vector is, we make a list of every object that’s moving. We take its velocity — how fast it’s moving and in what direction — and multiply that by its mass. Mass is a single number, a scalar, and we’re always allowed to multiply a vector by a scalar. This gets us another vector. Once we’ve done that for everything that’s moving, we add all those product vectors together. We can always add vectors together. And this gives us a grand total vector, the linear momentum of the system.

And that’s conserved. If one part of the system starts moving slower it’s because other parts are moving faster, and vice-versa. In the real world momentum seems to evaporate. That’s because some of the stuff moving faster turns out to be air objects bumped into, or particles of the floor that get dragged along by friction, or other stuff we don’t care about. That momentum can seem to evaporate is what makes its use in talking about ports teams or political campaigns make sense. It also annoys people who want you to know they understand science words better than you. So please consider this my authorization to use “gaining” and “losing” momentum in this sense. Ignore complainers. They’re the people who complain the word “decimate” gets used to mean “destroy way more than ten percent of something”, even though that’s the least bad mutation of an English word’s meaning in three centuries.

Angular momentum is also a vector. It’s also conserved. We can calculate what that vector is by the same sort of process, that of calculating something on each object that’s spinning and adding it all up. In real applications it can seem to evaporate. But that’s also because the angular momentum is going into particles of air. Or it rubs off grease on the axle. Or it does other stuff we wish we didn’t have to deal with.

The calculation is a little harder to deal with. There’s three parts to a spinning thing. There’s the thing, and there’s how far it is from the axis it’s spinning around, and there’s how fast it’s spinning. So you need to know how fast it’s travelling in the direction perpendicular to the shortest line between the thing and the axis it’s spinning around. Its angular momentum is going to be as big as the mass times the distance from the axis times the perpendicular speed. It’s going to be pointing in whichever axis direction makes its movement counterclockwise. (Because that’s how physicists started working this out and it would be too much bother to change now.)

You might ask: wait, what about stuff like a wheel that’s spinning around its center? Or a ball being spun? That can’t be an angular momentum of zero? How do we work that out? The answer is: calculus. Also, we don’t need that. This central force problem I’ve framed so that we barely even need algebra for it.

See, we only have a single object that’s moving. That’s the planet or satellite or weight or whatever it is. It’s got some mass, the value of which we call ‘m’ because why make it any harder on ourselves. And it’s spinning around the origin. We’ve been using ‘r’ to mean the number describing how far it is from the origin. That’s the distance to the axis it’s spinning around. Its velocity — well, we don’t have any symbols to describe what that is yet. But you can imagine working that out. Or you trust that I have some clever mathematical-physics tool ready to introduce to work it out. I have, kind of. I’m going to ignore it altogether. For now.

The symbol we use for the total angular momentum in a system is $\vec{L}$. The little arrow above the symbol is one way to denote “this is a vector”. It’s a good scheme, what with arrows making people think of vectors and it being easy to write on a whiteboard. In books, sometimes, we make do just by putting the letter in boldface, L, which is easier for old-fashioned word processors to do. If we’re sure that the reader isn’t going to forget that L is this vector then we might stop highlighting the fact altogether. That’s even less work to do.

It’s going to be less work yet. Central force problems like this mean the object can move only in a two-dimensional plane. (If it didn’t, it wouldn’t conserve angular momentum: the direction of $\vec{L}$ would have to change. Sounds like magic, but trust me.) The angular momentum’s direction has to be perpendicular to that plane. If the object is spinning around on a sheet of paper, the angular momentum is pointing straight outward from the sheet of paper. It’s pointing toward you if the object is moving counterclockwise. It’s pointing away from you if the object is moving clockwise. What direction it’s pointing is locked in.

All we need to know is how big this angular momentum vector is, and whether it’s positive or negative. So we just care about this number. We can call it ‘L’, no arrow, no boldface, no nothing. It’s just a number, the same as is the mass ‘m’ or distance from the origin ‘r’ or any of our other variables.

If ‘L’ is zero, this means there’s no total angular momentum. This means the object can be moving directly out from the origin, or directly in. This is the only way that something can crash into the center. So if setting L to be zero doesn’t allow that then we know we did something wrong, somewhere. If ‘L’ isn’t zero, then the object can’t crash into the center. If it did we’d be losing angular momentum. The object’s mass times its distance from the center times its perpendicular speed would have to be some non-zero number, even when the distance was zero. We know better than to look for that.

You maybe wonder why we use ‘L’ of all letters for the angular momentum. I do. I don’t know. I haven’t found any sources that say why this letter. Linear momentum, which we represent with $\vec{p}$, I know. Or, well, I know the story every physicist says about it. p is the designated letter for linear momentum because we used to use the word “impetus”, as in “impulse”, to mean what we mean by momentum these days. And “p” is the first letter in “impetus” that isn’t needed for some more urgent purpose. (“m” is too good a fit for mass. “i” has to work both as an index and as that number which, squared, gives us -1. And for that matter, “e” we need for that exponentials stuff, and “t” is too good a fit for time.) That said, while everybody, everybody, repeats this, I don’t know the source. Perhaps it is true. I can imagine, say, Euler or Lagrange in their writing settling on “p” for momentum and everybody copying them. I just haven’t seen a primary citation showing this is so.

(I don’t mean to sound too unnecessarily suspicious. But just because everyone agrees on the impetus-thus-p story doesn’t mean it’s so. I mean, every Star Trek fan or space historian will tell you that the first space shuttle would have been named Constitution until the Trekkies wrote in and got it renamed Enterprise. But the actual primary documentation that the shuttle would have been named Constitution is weak to nonexistent. I’ve come to the conclusion NASA had no plan in mind to name space shuttles until the Trekkies wrote in and got one named. I’ve done less poking around the impetus-thus-p story, in that I’ve really done none, but I do want it on record that I would like more proof.)

Anyway, “p” for momentum is well-established. So I would guess that when mathematical physicists needed a symbol for angular momentum they looked for letters close to “p”. When you get into more advanced corners of physics “q” gets called on to be position a lot. (Momentum and position, it turns out, are nearly-identical-twins mathematically. So making their symbols p and q offers aesthetic charm. Also great danger if you make one little slip with the pen.) “r” is called on for “radius” a lot. Looking on, “t” is going to be time.

On the other side of the alphabet, well, “o” is just inviting danger. “n” we need to count stuff. “m” is mass or we’re crazy. “l” might have just been the nearest we could get to “p” without intruding on a more urgently-needed symbol. (“s” we use a lot for parameters like length of an arc that work kind of like time but aren’t time.) And then shift to the capital letter, I expect, because a lowercase l looks like a “1”, to everybody’s certain doom.

The modified potential energy, then, is going to include the angular momentum L. At least, the amount of angular momentum. It’s also going to include the mass of the object moving, and the radius r that says how far the object is from the center. It will be:

$V_{eff}(r) = V(r) + \frac{L^2}{2 m r^2}$

V(r) was the original potential, whatever that was. The modifying term, with this square of the angular momentum and all that, I kind of hope you’ll just accept on my word. The L2 means that whether the angular momentum is positive or negative, the potential will grow very large as the radius gets small. If it didn’t, there might not be orbits at all. And if the angular momentum is zero, then the effective potential is the same original potential that let stuff crash into the center.

For the sort of r-to-a-power potentials I’ve been looking at, I get an effective potential of:

$V_{eff}(r) = C r^n + \frac{L^2}{2 m r^2}$

where n might be an integer. I’m going to pretend a while longer that it might not be, though. C is certainly some number, maybe positive, maybe negative.

If you pick some values for C, n, L, and m you can sketch this out. If you just want a feel for how this Veff looks it doesn’t much matter what values you pick. Changing values just changes the scale, that is, where a circular orbit might happen. It doesn’t change whether it happens. Picking some arbitrary numbers is a good way to get a feel for how this sort of problem works. It’s good practice.

Sketching will convince you there are energy minimums, where we can get circular orbits. It won’t say where to find them without some trial-and-error or building a model of this energy and seeing where a ball bearing dropped into it rolls to a stop. We can do this more efficiently.

## A Leap Day 2016 Mathematics A To Z: Yukawa Potential

Yeah, ‘Y’ is a lousy letter in the Mathematics Glossary. I have a half-dozen mathematics books on the shelf by my computer. Some is semi-popular stuff like Richard Courant and Herbert Robbins’s What Is Mathematics? (the Ian Stewart revision). Some is fairly technical stuff, by which I mean Hidetoshi Nishimori’s Statistical Physics of Spin Glasses and Information Processing. There’s just no ‘Y’ terms in any of them worth anything. But I can rope something into the field. For example …

## Yukawa Potential

When you as a physics undergraduate first take mechanics it’s mostly about very simple objects doing things according to one rule. The objects are usually these indivisible chunks. They’re either perfectly solid or they’re points, too tiny to have a surface area or volume that might mess things up. We draw them as circles or as blocks because they’re too hard to see on the paper or board otherwise. We spend a little time describing how they fall in a room. This lends itself to demonstrations in which the instructor drops a rubber ball. Then we go on to a mass on a spring hanging from the ceiling. Then to a mass on a spring hanging to another mass.

Then we go onto two things sliding on a surface and colliding, which would really lend itself to bouncing pool balls against one another. Instead we use smaller solid balls. Sometimes those “Newton’s Cradle” things with the five balls that dangle from wires and just barely touch each other. They give a good reason to start talking about vectors. I mean positional vectors, the ones that say “stuff moving this much in this direction”. Normal vectors, that is. Then we get into stars and planets and moons attracting each other by gravity. And then we get into the stuff that really needs calculus. The earlier stuff is helped by it, yes. It’s just by this point we can’t do without.

The “things colliding” and “balls dropped in a room” are the odd cases in this. Most of the interesting stuff in an introduction to mechanics course is about things attracting, or repelling, other things. And, particularly, they’re particles that interact by “central forces”. Their attraction or repulsion is along the line that connects the two particles. (Impossible for a force to do otherwise? Just wait until Intro to Mechanics II, when magnetism gets in the game. After that, somewhere in a fluid dynamics course, you’ll see how a vortex interacts with another vortex.) The potential energies for these all vary with distance between the points.

Yeah, they also depend on the mass, or charge, or some kind of strength-constant for the points. They also depend on some universal constant for the strength of the interacting force. But those are, well, constant. If you move the particles closer together or farther apart the potential changes just by how much you moved them, nothing else.

Particles hooked together by a spring have a potential that looks like $\frac{1}{2}k r^2$. Here ‘r’ is how far the particles are from each other. ‘k’ is the spring constant; it’s just how strong the spring is. The one-half makes some other stuff neater. It doesn’t do anything much for us here. A particle attracted by another gravitationally has a potential that looks like $-G M \frac{1}{r}$. Again ‘r’ is how far the particles are from each other. ‘G’ is the gravitational constant of the universe. ‘M’ is the mass of the other particle. (The particle’s own mass doesn’t enter into it.) The electric potential looks like the gravitational potential but we have different symbols for stuff besides the $\frac{1}{r}$ bit.

The spring potential and the gravitational/electric potential have an interesting property. You can have “closed orbits” with a pair of them. You can set a particle orbiting another and, with time, get back to exactly the original positions and velocities. (Three or more particles you’re not guaranteed of anything.) The curious thing is this doesn’t always happen for potentials that look like “something or other times r to a power”. In fact, it never happens, except for the spring potential, the gravitational/electric potential, and — peculiarly — for the potential $k r^7$. ‘k’ doesn’t mean anything there, and we don’t put a one-seventh or anything out front for convenience, because nobody knows anything that needs anything like that, ever. We can have stable orbits, ones that stay within a minimum and a maximum radius, for a potential $k r^n$ whenever n is larger than -2, at least. And that’s it, for potentials that are nothing but r-to-a-power.

Ah, but does the potential have to be r-to-a-power? And here we see Dr Hideki Yukawa’s potential energy. Like these springs and gravitational/electric potentials, it varies only with the distance between particles. Its strength isn’t just the radius to a power, though. It uses a more complicated expression:

$-K \frac{e^{-br}}{r}$

Here ‘K’ is a scaling constant for the strength of the whole force. It’s the kind of thing we have ‘G M’ for in the gravitational potential, or ‘k’ in the spring potential. The ‘b’ is a second kind of scaling. And that a kind of range. A range of what? It’ll help to look at this potential rewritten a little. It’s the same as $-\left(K \frac{1}{r}\right) \cdot \left(e^{-br}\right)$. That’s the gravitational/electric potential, times e-br. That’s a number that will be very large as r is small, but will drop to zero surprisingly quickly as r gets larger. How quickly will depend on b. The larger a number b is, the faster this drops to zero. The smaller a number b is, the slower this drops to zero. And if b is equal to zero, then e-br is equal to 1, and we have the gravitational/electric potential all over again.

Yukawa introduced this potential to physics in the 1930s. He was trying to model the forces which keep an atom’s nucleus together. It represents the potential we expect from particles that attract one another by exchanging some particles with a rest mass. This rest mass is hidden within that number ‘b’ there. If the rest mass is zero, the particles are exchanging something like light, and that’s just what we expect for the electric potential. For the gravitational potential … um. It’s complicated. It’s one of the reasons why we expect that gravitons, if they exist, have zero rest mass. But we don’t know that gravitons exist. We have a lot of trouble making theoretical gravitons and quantum mechanics work together. I’d rather be skeptical of the things until we need them.

Still, the Yukawa potential is an interesting mathematical creature even if we ignore its important role in modern physics. When I took my Introduction to Mechanics final one of the exam problems was deriving the equivalent of Kepler’s Laws of Motion for the Yukawa Potential. I thought then it was a brilliant problem. I still do. It struck me while writing this that I don’t remember whether it allows for closed orbits, except when b is zero. I’m a bit afraid to try to work out whether it does, lest I learn that I can’t follow the reasoning for that anymore. That would be a terrible thing to learn.

## The Set Tour, Part 10: Lots of Spheres

The next exhibit on the Set Tour here builds on a couple of the previous ones. First is the set Sn, that is, the surface of a hypersphere in n+1 dimensions. Second is Bn, the ball — the interior — of a hypersphere in n dimensions. Yeah, it bugs me too that Sn isn’t the surface of Bn. But it’d be too much work to change things now. The third has lurked implicitly since all the way back to Rn, a set of n real numbers for which the ordering of the numbers matters. (That is, that the set of numbers 2, 3 probably means something different than the set 3, 2.) And fourth is a bit of writing we picked up with matrices. The selection is also dubiously relevant to my own thesis from back in the day.

## Sn x m and Bn x m

Here ‘n’ and ‘m’ are whole numbers, and I’m not saying which ones because I don’t need to tie myself down. Just as with Rn and with matrices this is a whole family of sets. Each different pair of n and m gives us a different set Sn x m or Bn x m, but they’ll all look quite similar.

The multiplication symbol here is a kind of multiplication, just as it was in matrices. That kind is called a “direct product”. What we mean by Sn x m is that we have a collection of items. We have the number m of them. Each one of those items is in Sn. That’s the surface of the hypersphere in n+1 dimensions. And we want to keep track of the order of things; we can’t swap items around and suppose they mean the same thing.

So suppose I write S2 x 7. This is an ordered collection of seven items, every one of which is on the surface of a three-dimensional sphere. That is, it’s the location of seven spots on the surface of the Earth. S2 x 8 offers similar prospects for talking about the location of eight spots.

With that written out, you should have a guess what Bn x m means. Your guess is correct. It’s a collection of m things, each of them within the interior of the n-dimensional ball.

Now the dubious relevance to my thesis. My problem was modeling a specific layer of planetary atmospheres. The model used for this was to pretend the atmosphere was made up of some large number of vortices, of whirlpools. Just like you see in the water when you slide your hand through the water and watch the little whirlpools behind you. The winds could be worked out as the sum of the winds produced by all these little vortices.

In the model, each of these vortices was confined to a single distance from the center of the planet. That’s close enough to true for planetary atmospheres. A layer in the atmosphere is not thick at all, compared to the planet. So every one of these vortices could be represented as a point in S2, the surface of a three-dimensional sphere. There would be some large number of these points. Most of my work used a nice round 256 points. So my model of a planetary atmosphere represented the system as a point in the domain S2 x 256. I was particularly interested in the energy of this set of 256 vortices. That was a function which had, as its domain, S2 x 256, and as range, the real numbers R.

But the connection to my actual work is dubious. I was doing numerical work, for the most part. I don’t think my advisor or I ever wrote S2 x 256 or anything like that when working out what I ought to do, much less what I actually did. Had I done a more analytic thesis I’d surely have needed to name this set. But I didn’t. It was lurking there behind my work nevertheless.

The energy of this system of vortices looked a lot like the potential energy for a bunch of planets attracting each other gravitationally, or like point charges repelling each other electrically. We work it out by looking at each pair of vortices. Work out the potential energy of those two vortices being that strong and that far apart. We call that a pairwise interaction. Then add up all the pairwise interactions. That’s it. [1] The pairwise interaction is stronger as each vortex is stronger; it gets weaker as the vortices get farther apart.

In gravity or electricity problems the strength falls off as the reciprocal of the distance between points. In vortices, the strength falls off as minus one times the logarithm of the distance between points. That’s a difference, and it meant that a lot of analytical results known for electric charges didn’t apply to my problem exactly. That was all right. I didn’t need many. But it does mean that I was fibbing up above, when I said I was working with S2 x 256. Pause a moment. Do you see what the fib was?

I’ll put what would otherwise be a footnote here so folks have a harder time reading right through to the answer.

[1] Physics majors may be saying something like: “wait, I see how this would be the potential energy of these 256 vortices, but where’s the kinetic energy?” The answer is, there is none. It’s all potential energy. The dynamics of point vortices are weird. I didn’t have enough grounding in mechanics when I went into them.

That’s all to the footnote.

Here’s where the fib comes in. If I’m really picking sets of vortices from all of the set S2 x 256, then, can two of them be in the exact same place? Sure they can. Why couldn’t they? For precedent, consider R3. In the three-dimensional vectors I can have the first and third numbers “overlap” and have the same value: (1, 2, 1) is a perfectly good vector. Why would that be different for an ordered set of points on the surface of the sphere? Why can’t vortex 1 and vortex 3 happen to have the same value in S2?

The problem is if two vortices were in the exact same position then the energy would be infinitely large. That’s not unique to vortices. It would be true for masses and gravity, or electric charges, if they were brought perfectly on top of each other. Infinitely large energies are a problem. We really don’t want to deal with them.

We could deal with this by pretending it doesn’t happen. Imagine if you dropped 256 poker chips across the whole surface of the Earth. Would you expect any two to be on top of each other? Would you expect two to be exactly and perfectly on top of each other, neither one even slightly overhanging the other? That’s so unlikely you could safely ignore it, for the same reason you could ignore the chance you’ll toss a coin and have it come up tails 56 times in a row.

And if you were interested in modeling the vortices moving it would be incredibly unlikely to have one vortex collide with another. They’d circle around each other, very fast, almost certainly. So ignoring the problem is defensible in this case.

Or we could be proper and responsible and say, “no overlaps” and “no collisions”. We would define some set that represents “all the possible overlaps and arrangements that give us a collision”. Then we’d say we’re looking at S2 x 256 except for those. I don’t think there’s a standard convention for “all the possible overlaps and collisions”, but Ω is a reasonable choice. Then our domain would be S2 x 256 \ Ω. The backslash means “except for the stuff after this”. This might seem unsatisfying. We don’t explicitly say what combinations we’re excluding. But go ahead and try listing all the combinations that would produce trouble. Try something simple, like S2 x 4. This is why we hide all the complicated stuff under a couple ordinary sentences.

It’s not hard to describe “no overlaps” mathematically. (You would say something like “vortex number j and vortex number k are not at the same position”, with maybe a rider of “unless j and k are the same number”. Or you’d put it in symbols that mean the same thing.) “No collisions” is harder. For gravity or electric charge problems we can describe at least some of them. And I realize now I’m not sure if there is an easy way to describe vortices that collide. I have difficulty imagining how they might, since vortices that are close to one another are pushing each other sideways quite intently. I don’t think that I can say they can’t, though. Not without more thought.

## One Way To Fall Over

[ Huh. My statistics page says that someone came to me yesterday looking for the “mathematics behind rap music”. I don’t doubt there is such mathematics, but I’ve never written anything approaching it. I admit that despite the long intertwining of mathematics and music, and my own childhood of playing a three-quarter size violin in a way that must be characterized as “technically playing”, I don’t know anything nontrivial about the mathematics of any music. So, whoever was searching for that, I’m sorry to have disappointed you. ]

Now, let me try my first guess at saying whether it’s easier to tip the cube over by pushing along the middle of the edge or by pushing at the corner. I laid out the ground rules, and particularly, the symbols used for the size of the box (it’s of length $a$) and how far the center of mass (the dead center of the box) is from the edges and the corners last time around. Here’s my first thought about what has to be done to tip the box over: we have to make the box pivot on some point — along one edge, if we’re pushing on the edge; along one corner, if we’re pushing on the corner — and so make it start to roll. If we can raise the center of mass above the pivot then we can drop the box back down with some other face to the floor, which has to count as tipping the box over. If we don’t raise the center of mass we aren’t tipping the box at all, we’re just shoving it.

## Tipping The Toy

My brother phoned to remind me how much more generally nervous I should be about things, as well as to ask my opinion in an utterly pointless dispute he was having with his significant other. The dispute was over no stakes whatsoever and had no consequences of any practical value so I can see why it’d call for an outside expert. It’s more one of physics, but I did major in physics long ago, and it’s easier to treat mathematically anyway, and it was interesting enough that I spent the rest of the night working it out and I’m still not positive I’m unambiguously right. I could probably find out for certain with some simple experiments, but that would be precariously near trying, and so is right out. Let me set up the problem, though, since it’s interesting and should offer room for people to argue I’m completely wrong.