The n-Category Café

Redoing the analysis for white/black vanes (in hard vacuum) seems simple:

Both sides are perfect Lambertian reflectors, so they radiate equally in all directions. For each photon, energy is proportional to momentum (E=hv and p=hv/c), so the momentum transferred depends only only radiated power, not on the wavelength of the light. Once the system reaches thermal equilibrium, the emitted power will exactly equal the incoming power.

So the fact that one side radiates in infrared and one in visible is irrelevant, and the mill will behave exactly as if both were white or both were black, i.e. it will not turn.

There’s something funny about this. Both of the accepted contributing factors — the direct transfer of heat energy to gas particles, and the increased pressure on the hotter side due to transfer of particles from the colder side — are what us physicists call ‘edge effects’, according to the Wikipedia quote above. So why doesn’t anybody sell a Super Light Mill, with the vanes full of holes?!

Jamie wrote:

Both of the accepted contributing factors — the direct transfer of heat energy to gas particles, and the increased pressure on the hotter side due to transfer of particles from the colder side — are what us physicists call ‘edge effects’, according to the Wikipedia quote above. So why doesn’t anybody sell a Super Light Mill, with the vanes full of holes?!

Ah, good. Having shot the usual story about the light mill in a perfect vacuum full of holes, let’s move on to the really interesting case, where there’s a bit of gas left in the bulb!

There’s a post from Physics Forums that addresses your question.

On December 13, 2006, wirelessguy wrote:

I think the Radiometer might be an excellent subject to bring up for a project, I just wouldn’t try to go answering any questions.

I recently read an article by Mike Ivsin and believe that he’s correct (well at least his questions seem to lead down the right path).

In short, he thinks that the latest theory regarding thermal transpiration happening within the Radiometer is indeed a myth. You can read more here:

http://www.hyperflight.com/mainleft.htm#aboutlight

I still don’t fully understand how the mill works, but if you think about it, the mill should NEVER rotate in reverse since either the black side would be pushing (heat pressure) or the white side pulling (lack of heat pressure).

Additionally, Scandurra has a patent which works on the theory that thermal creep happens around the ‘edges’ and that increasing the surface area of the edges (i.e. punching extra holes in the vanes) should speed up the mill.

I had a unit built for me (but I can’t swear to it’s accuracy). It moves significantly SLOWER than my identical reference mill, so at present I’m inclines to agree with Mike as far as thermal transpiration goes.

I’m no scientist so I’d love to hear more on the subject from people who can explain more than I can.

Unfortunately the website by Mike Ivsin has a high crackpot index. For example, he says that NASA scientists refused to test the existence of light pressure: “Over the years and into the present, government scientists publish the results of pretend physics from which they fund their wishful thinking.” He claims the bending of light that passes by the Sun is due to the solar corona — not general relativity. And, he has a book Quantum Pythagoreans that explains the truth.

However, Ivsin may be correct in his claim that a light mill put into a freezer runs backwards. I’ve never tried it myself, but the Wikipedia article claims the effect is real!

Unfortunately, this section gives no references. And, it claims that the reversed rotation is due to black-body radiation from the black sides of the vanes. Goldman’s article suggests that this radiation would be too weak.

Can anyone out there do the experiment? I had a light mill when I was a kid, but my mom threw it out.

Anyway… you should read Scandurra’s paper! He does a bunch of interesting calculations and concludes: “The force should be enhanced when the vane is perforated with a large number of small holes.”

Then he makes me skeptical by saying: “In particular nanotechnology could permit for an enhancement of the force by a factor of $10^7$ with respect to the typical order of magnitude. Such force could be observable at atmospheric pressure.” He also shows no sign of having carried out an experiment.

So, I think this issue is up for grabs.

Recently I got to watch one of these light mills as a salesman tried to convince me to purchase his company’s brand of “low E” windows. These windows are coated with a metallic oxide that is designed to let visible light through but to partially block both the UV and infrared. The salesman had a very bright lamp to imitate the sun, and the demonstration consisted of showing how much more slowly the light mill turned when the low E glass was interposed between it and the source. Does this effect have any bearing on the issue? What would happen in the reverse situation, if just infrared, or just UV, or a combination of those were used? Here is just about
everything you need to know about low E windows.

I agree with John that the soft vacuum case is the (very!) interesting one. However there’s a couple more things to say about the hard vacuum case.

1. Almost all successful attempts to measure light pressure in this way have succeeded by introducing an asymmetry, namely not shining an equal amount of light on both sides of the vane, accomplished in part by not spinning it but instead measuring displacement. Once one realizes that Crookes’s idea is badly flawed, it is natural to turn to asymmetric radiometers. Simply blasting light at one side of a mirror suspended from a long thread is a straightforward approach. A more delicate one is to put a mirror at each end of a horizontal rod suspended by a thread and blast light at each mirror (in opposite directions, thus developing a torque) periodically, with a period matching that of the natural oscillation of the setup so as to gradually build up oscillation, as when pumping a swing or a laser. Even better is to blast the opposite sides of the mirrors alternately with that period so as to keep both sides equally warm, eliminating the asymmetry in whatever warming effect might be resulting from the illumination. Doing this in a hard vacuum is considered mandatory for meaningful results here, though enough symmetry should compensate for a less-than-perfect vacuum. There seem to be a number of measurements incorporating some if not all of these ideas.

2. With the requisite scroll of enchantment you can find much interesting information about radiometers in Frank Crawford’s one-page 1985 AJP paper “Running Crooke’s radiometer backwards” here.

One error Crawford makes (in my opinion) is to echo Ditchburn and Britannica in claiming that “molecules in the residual gas that drift up against a black (hotter) side of a vane therefore get a stronger kick than from a white side, and the corresponding stronger recoils drive the rotation with the black sides receding.” My take on the situation is that this supposedly stronger kick is even weaker than the edge effects of Reynolds, Maxwell, and Einstein, and that the explanation has to lie elsewhere. However he does cite what one presumes are experiments he did with a radiometer in a fridge, which made it run backwards for 5-10 minutes until equilibrium was reached. He also warmed it up in an oven, let it settle into equilibrium, then took it out and observed the same behavior as in the fridge: backwards rotation.

None of that is hard-vacuum stuff; however Crawford also cites Blum and Roller, Physics Vol 2 (Holden-Day SF 1982) who claim someone was able to demonstrate the effect with a highly evacuated “standard” Crookes’s radiometer (the toy-store kind, one hopes, not the non-spinning kind already known to do the job), but says “I have not been able to find an original reference on this.” (Anyone know what this refers to?)

This seems hard to believe given that the possibility of demonstrating the effect in a hard vacuum with a standard (spinning) Crookes’s radiometer would seem to have been completely dashed by the above discussions.

Nevertheless there still remain loopholes. In particular Crookes’s radiometer could work in a hard vacuum if the emissivity of either side is frequency dependent.

Suppose the black side is actually white at far infrared, around 1000 cm^-1 (10 microns, the peak radiation of a black body at 300 K, room temperature), thus blocking reradiation on the black side. This could happen if covered with an ordinary glass cover slip (as used with microscope slides), which is transparent at 10000 cm^-1 (1 micron) but I think opaque at 1000 cm^-1 (certainly if as thick as a slide itself).

Suppose further that the white side is actually black at far infrared, which could happen if the white pigment were anodized on, which should be thin enough to be transparent at 1000 cm^-1.

Then inbound light incident on both sides will be either reflected or reradiated, in both cases entirely from the side that looks white but is actually black at far IR. The net effect should be to drive the vane towards the black side.

It’s a long time since I wrote that article for the FAQ and it’s funny to see it being discussed again.

One thing about the article that was always lacking was any quantitative formula to back up the qualitative explanation given. What does the force on the vanes depend on? Does it depend on the area of the vanes, the perimeter, the thickness? How does it vary with pressure and temperature gradient? There should be some kind of formula. I always thought I (or someone else) would find it later and add it in, but that never happened, It’s a bit odd.

The theory that it is related to transpiration applied to the edges of the vane suggests that it would work better if the vanes had thicker edges and a longer perimeter, or even holes as suggested above, but I suspect that this is wrong otherwise they would already be made that way.

One thing I realised when I researched the article was that late 19th century physicists knew an awful lot about the kinetic theory of gases. Maxwell, Rayleigh and Crookes were real experts on it. It was the frontline of research at the time and they spent a lot of time thinking about it. I think they must have been right about their final explanation, but perhaps there are other ways of looking at it that would help us understand better.

I have just found out that if you search on google books for Maxwell’s paper “On stresses in rarefied gases arising from inequalities of temperature” you find a copy of it in his collected papers.

Maxwell gives a formula for the pressure due to variations of temperature. It is inversly proportional to density which is why the radiometer works for rarefied gasses. The pressure as (a symmetric tensor) depends on the second derivatives of the temperature. In a static situation the temperature satisifies the same equations as an electic potential. For a vane in the radiometer the temperature on the black side is higher than the ambient temperature so there is a temerature gradient out from the vane. Away from the edges the gradient is uniform so its second derivative is small. That is why they say that the force is around the edges of the vanes.

It should be possible to solve the equations given by Maxwell for some vane shapes (e.g. a disk with a hole in it) and work out the total force.

Why not go with the flow? Craft and polish thin vanes from unetched channel plate, blacken one side in a reducing atmosphere, etch the channels, and build the radiometer. If thin channel gas diffusion across a temp gradient is the boojum, that should be a real whizzer of a radiometer!

For smaller holes, cleverly blacken one side of Anopore filters without plugging the nm-diameter holes.

Phil, good call going to the source. The trouble I find with Google books however is that you only get glimpses (Google is good about the glimpses but evil about the omissions, so much for “intellectual property”), so I borrowed the Physics library’s copy. (I also looked for Einstein’s 1924 paper but they only had it in German which I’m painfully rusty on. The collected translated works of Einstein seem to stop at 1920, anyone know of translations of his later works?)

The Maxwell-Reynolds shoot-out is every bit as fascinating as that between De Morgan and William Hamilton (the Edinburgh one, not the Irish physicist) 30 years earlier. The editors’ introduction to this volume of Maxwell’s works (the third and last in a series, this one titled not “Vol 3” but rather “On `Avoiding All Personal Enquiries’ of Molecules”) devotes a dozen pages to JCM’s last paper and this episode with OR.

An important change that Maxwell made to Reynold’s edge-oriented account was to suggest that sliding induced forces via roughness of the surface. Reynold’s idea was that at this pressure, mean free path (on the order of 1 mm at 60 mTorr = 8 pascals) started to approach the scale of a vane, being about 10% of a 1 cm vane. Maxwell’s stated objection to this was that Reynold’s calculations showing the relevance of this effect were unreasonably complicated, and he proceeded to obtain his second-derivative approach in its place, which others have since complained (Sutherland 1896 for example) was also quite complicated.

However Maxwell may also have objected under his breath to Reynolds’ introduction of mean free path as being an irrelevant factor, though he appears not to have raised this explicitly. As hinted at by his “no personal enquiries” philosophy alluded to in the volume’s title, Maxwell appears to have an unswerving faith in the ideal gas laws, reminiscent of Einstein’s faith in the determinacy of physics—-one imagines Maxwell’s slogan being “God does not play bowls.”

Nontrivial mean free path invalidates the ideal gas laws by the following contradiction. In any temperature gradient away from a surface (i.e. with the isothermals parallel to the surface) molecules will arrive at the surface from an expected distance from the surface that is on the order of the mean free path. At the start of that trip one supposes the molecules to be at a lower temperature than molecules closer to the surface—-those closest to the surface are assumed in the ideal gas theory to be at the same temperature as the surface, but this will not hold for molecules further out.

It is these long-distance travelers that undermine the ideal gas laws. They arrive with an expected velocity less than the closer molecules, hence a lower temperature than the surface, and hence receive a nontrivial boost in velocity from their encounter with the surface, which speeds them up in proportion to the length of their journey (for any given direction of arrival). This is an actual kick (and I take back my earlier objection to Ditchburn and Britannica, not because they’re necessarily right but because I no longer have the argument I thought I had for their being wrong). This kick is not offset by an equal and opposite kick on the cold side of the vane because it is presumably closer to the temperature of the container; instead momentum is conserved with the help of the container, the wall nearer the hot side of the vane absorbing the reacting kick.

This should be contrasted with molecules arriving at a squishy virtual surface in the interior of a gas, where they typically are scattered. At such virtual surfaces the comings and goings of molecules obey the ideal gas laws due to scattering (it is a nice exercise to show that if all collisions are head-on the gas laws are extremely badly violated, by far worse than mere constant factors). Sufficiently hard and smooth surfaces however do not scatter but rather reflect, like a mirror. (Rough surfaces are in between, acting neither as mirrors nor as virtual surfaces in the interior of the gas; unless they act like virtual surfaces roughness does not save the ideal gas laws.) This does not make any difference in the absence of a temperature gradient but does (in the case of the standard radiometer) for temperature gradients of more than say a millidegree per mean free path.

This nonzero momentum gain creates a real pressure on the vane above that in the gas near the vane. Now this is not to say that the gas can’t reorganize itself to relieve that surplus pressure, but rather that the ideal gas laws don’t show how that organization might happen. In other words one cannot dive right in and calculate the force on a vane based on this insight, but one must at least redo the dynamics in light of it. This Maxwell does not do, or even acknowledge (but then no one challenged him on this point at the time). This is the contradiction, i.e. inconsistency, I wanted to point out.

What is needed next here is to work out what the gas actually does to resolve this inconsistency. To that end I settled down with Schey’s “Div, grad, curl, and all that” to refresh my several decades old acquaintance, but I’m really rusty on it. If there’s someone on this list who’s already up to speed on “all that” please feel free to dive in with the appropriate equations describing this situation (large temperature gradient relative to the mean free path, in the vicinity of .01 degrees per mean free path or more, in the neighborhood of a hard smooth surface) so we can all play around with their solutions either empirically or analytically.

If this has already been done, e.g. for plasma physics, an obvious customer for such equations, where one can imagine encountering hundreds of degrees per mean free path, so much the better.

I had a terrible time getting to sleep last night (and for that matter staying asleep) because my brain refused to stop thinking about the problem.

In the morning I hoped I could write down the equations describing what was going on at the surface of the radiometer vane, with conservation of momentum and energy being the only two conservation laws I thought the problem ought to need. But they kept not making sense.

Suddenly it dawned on me: no wind! Molecules are busily diffusing all over the place, but none of them are blowing steadily in one direction, at least not in the manner of an ordinary wind. This means that with all the diffusion there is a third conservation principle: conservation of mass across any boundary.

This supplies the missing equation. I will show that when the mean free path is significant, the ratio of “purposeful” motion to Brownian motion rises to a level where a new law kicks in: temperature is inversely proportional to the square of the pressure.

Before proving this, let me convince you it’s true with a simple little experiment (well, it wasn’t simple to do but interpreting the results is simple). Plot the pressure and temperature of the earth’s atmosphere. This is already done for you in the nice chart at http://en.wikipedia.org/wiki/Image:Atmosphere_model.png from the Wikipedia article on Earth’s atmosphere. The critical point seems to be at an altitude of around 70-80 km, the onset of the thermosphere (appropriately named as we’ll see in a moment).

Below this point the temperature of the atmosphere approximates the effective temperature of the Earth, 248 K. (This is most readily obtained as follows. Start with the Earth’s albedo of .367, subtract that from 1 giving .633, take the square root of that to get .7956, multiply that by the radius .6955 megameters of the Sun giving .5533, divide that by the diameter 149.6 megameters of the Earth’s orbit giving .00185, take the square root of that to get .043, and multiply that by the Sun’s temperature of 5778 K to get 248.5 K. The often-seen figure of 255 K is due entirely to using the poor approximation of .3 for the albedo as you can check for yourself by using it in the above calculation, which is identically equivalent to other more complicated formulas for effective temperature of a planet.)

At sea level the atmosphere is warmed by the Earth’s molten core and perhaps other effects such as tidal friction. At 10 km those contributions vanish and the effective temperature is reached and even overshot slightly (one plausible explanation sometimes given being that vertical diffusion has to overcome gravity). Then it heats up again at 50 km, presumably because this is the region that burns off the incoming meteoric detritus and anthropogenic space junk (the latter less significant than the former to date). Above that it goes back to the effective temperature.

But then at 70 km it dives steeply below the effective temperature. I attribute this to the mean free path there. At sea level this is 70 nm, essentially a crowded cocktail party where even with a lot of energy it takes time to work your way across the room. At 70 km the pressure has dropped four orders of magnitude, bringing the mean free path up to .7 mm, readily visible to the naked eye and over a million times larger than an air molecule. The hitherto-confined kinetically dancing 244 K molecules can now start to stretch their legs, darting between levels of significantly different pressure (relative to before).

This new freedom is a release for them that has the effect of dropping the pressure more sharply than before. I’m guessing that at this level this sudden escape initially drops the temperature, much as CO2 escaping from a cylinder becomes dry ice.

But as the pressure continues to drop, long-distance travel between collisions starts to really dominate, at which point a new effect kicks in: temperature rises quadratically with decreasing pressure. You can see this effect very graphically in the right half of the graph, where the slope of the green line is very close to twice that of the blue line!

With the rapidly rising temperature the precipitous drop in pressure resulting from this new-found freedom of the molecules is offset by the pressure created by this temperature, which brings back some of the cocktail party atmosphere, but now more in the manner of an aerial dogfight than a living room as the molecules furiously crash into each other at much higher speed than at sea level.

Ok, so how does T grow as 1/P²? (Actually mP²/4dR to be precise where m = .02897 kg, the molar mass of air, d is number of degrees of freedom of an average air molecule, and R is the gas constant.) We can see this as follows.

Claim 1. The mass rate of flow of gas across any collision-free surface of unit area is P/2v, pressure divided by twice the velocity. This is because P = 2mv at such a surface, proved by making the surface solid and noting that the pressure must be double the rate mv of momentum bouncing off that surface (double because of the reversal).

By conservation of matter throughout the atmosphere, P/2v must be the same over the entire range of this effect, from 80 km onwards. This is the crucial point I’d been missing before. (Although matter is also conserved below this level, v is relatively independent of P, being as constant as T, which holds fairly steady when descending to ground while P increases under the increasing burden of the atmosphere.)

At this point we have an obvious inconsistency between the ideal gas law PV = nRT and the more fundamental E = dRT/2 where E is the energy of a mole of gas of temperature T and d is the number of degrees of freedom of one molecule of that gas. Guess which one loses? The ideal gas law only holds in the limit, and with molecules jumping more than a millimeter between collisions and the pressure dropping by a factor of two every five kilometers we’re now very far from that limit.

Now E = 1/2 mv², everywhere including the center of the earth. Eliminating E gives T = mv²/dR = mP²/4dR, definitely not at the center of the earth, but throughout the thermosphere and beyond. QED

Claim 2. Crookes’s radiometer works essentially like the earth’s atmosphere, despite being on so vastly smaller a scale.

If you bought Claim 1, the proof of Claim 2 should be pretty obvious. With the constants all clearcut from the formulas (assuming I didn’t forget one) it should be straightforward to calculate radiometer behavior analytically and check it against the known radiometer behavior, known for the past 135 years (more than twice my age).

Now this theory may well be all hogwash, since I just thought of it, and everything prior to it that I “just thought of” turned out not to work. However I must say I find very reassuring that in the thermosphere the temperature is found empirically to slope upwards at twice the rate the pressure is falling (on its log-linear scale).

If it holds up then a huge advantage of this explanation over those of Reynolds, Maxwell, and Einstein is that it can be derived in a high school physics class, with more students following than for some of the other laws. The burgeoning interest in climatology these days should make this eminently accessible phenomenon of strong interest to the general public.

(In 1876, at the height of the radiometer excitement, Maxwell was summoned by Queen Victoria to explain it to her. His letter to Robert Cay of 15 May 1876 indicates that he was noncommittal about the explanation. With a simple explanation he would presumably not have needed to be so coy.)

Incidentally this relationship at the top of any astronomical body’s atmosphere might provide a more satisfactory explanation of the long-standing mystery of the excess heat of the Sun’s corona than the (very new) explanation at http://www.sciencedaily.com/releases/2008/05/080529120710.htm , which has something of the flavor of the Reynold’s-Maxwell-Einstein explanation of the radiometer effect: “what else could it possibly be, all other explanations have been ruled out.” In this case it might be that both effects are at work; if they were equally strong that would be particularly interesting.

Correction to my previous message: the diameter of the Earth’s orbit is 299.2 megameters (within .3% of 300). The figure I gave of 149.6 is the radius. The other numbers are all correct, in particular the quotient of .5533 by the diameter is indeed .00185.

The more easily remembered approximations of .37 for albedo, .7 for the Sun’s radius, 300 for the diameter of the Earth’s orbit, and 5800 K for the temperature of the Sun, give 249.6 K, a lot closer to the correct answer of 248.5 K than the oft-mentioned 255 K.

A Latex’d version of the formula I used is at http://en.wikipedia.org/wiki/Talk:Effective_temperature . Its content is simply that the fourth power of the ratio of the planet’s effective temperature to that of the illuminating star equals the fraction of the sky (all 4π steradians of it) occupied by the Sun, corrected for the planet’s albedo.

I’m afraid this theory too is hogwash. I just realized that I was treating the temperature on the plot of Earth’s pressure and temperature as logarithmic, justifying the comparison with pressure. Unfortunately for my theory it is linear, making the fact that the slope is twice that of the pressure immaterial. The reason it is twice is the scientifically irrelevant one that the temperature scale was chosen to make the temperature fit on the same plot.

That temperature is linear with altitude in the thermosphere is very interesting, but unfortunately the proportionality of P and v does not predict it. Empirically P decreases exponentially with increasing T, but I have no explanation.

This relationship does however strongly violate the ideal gas law PV = nRT, so at least my point about it being violated remains valid.

I was so sold on the slope argument that I did not check the algebra carefully enough. Another error (much more minor): only 3 of the d degrees of freedom of a molecule contribute to its velocity. Hence T = mv²/3R, not mv²/dR, a small change in constant factor.

Also the occurrence of m in v = P/2m was lost when substituting for v in T = mv²/3R, which should have given P²/4dRm, though this too is only a constant factor off given that the mass flow m (in kg/s/m²) is constant over the whole atmosphere by the argument in the paragraph following my Claim 1. (Also since m is a mass rate, E wherever I used it denotes power, not energy.)

This 3R denominator is itself suspect outside the domain of applicability of the ideal gas laws, since we can no longer assume that all directions are equally likely. So T as a function of v could be as much as 3 times the above: T = mv²/R, if the motion were somehow constrained to be largely vertical.

Worse yet, if say solar radiation were to heat the molecules almost entirely in their vibrational and rotational modes (a very reasonable assumption given the heating mechanism involved), v could be arbitrarily small relative to √T. If furthermore the density of the gas decreased even faster than the pressure, the rise in T would not be contradicted by the rapidly falling P because the slow-moving but high-energy molecules would encounter the (virtual) surface of unit volume so infrequently as to develop very little pressure, even though the energetically vibrating molecules will give the (real) surface of any barometer quite a bang on those extremely rare occasions (due to the extremely low velocity) when they do touch it.

In this scenario the ideal gas law P = ρRT does not disappear altogether but instead morphs into the form P = α(h)ρRT where α(h) < 1 is a dilution factor depending on altitude h and reflecting the greatly reduced rate of arrival of molecules at the barometer’s surface due to their vanishingly small velocity. So we need α(h) = P/ρRT, i.e. something of the form α(h) = c^h/bh for suitable constants b > 0 and c < 1 deducible empirically from the atmosphere graph. α(h) gives the amount by which v must be decreased relative to normal gas behavior.

One relationship we can rely on is the simple one between P and the remaining mass of atmosphere above, which P supports. This should permit prediction of the density at h from the empirical graph, noting that with translation contributing far less energy than the other degrees of freedom the relationship between P and T must be diluted along the above lines.

Predicting all this analytically however seems like a big challenge. It seems clear that T must be computed from consideration of absorption of solar radiation without (immediate) reference to v. On the other hand the relationship between P and mass seems clear since P measures the remaining mass above. So that’s one reliable handle.

Well, enough rambling, time to check these and other handles to see which can be used to give a reliable model of what’s going on.

What I do continue to believe however is that the key to solving the radiometer riddle lies in a better understanding of the earth’s atmosphere. With more insight into the ways in which PV = nRT can break down at low pressure and high temperature gradient one stands more of a chance of understanding how the radiometer works. We have a lot more information about the atmosphere than the radiometer to check these kinds of theories against. This is not to say that the radiometer closely mimics the atmosphere but only that the same or similar breakdowns at low pressure seem like promising candidates for exploration.

I would be most surprised if the radiometer’s behavior could be described within the domain of applicability of the ideal gas laws.

I slept on this problem but the solution was so obvious it woke me up after two hours.

Normally the energy of a molecule is equally distributed among its degrees of freedom by frequent collisions. When the source of heating is radiation lines specific to a transition of a molecule from one combination of eigenvalues of rotation and vibrational modes to another, initially the molecule will acquire energy in those degrees of freedom.

Translation does not participate in this initial feeding frenzy because free motion has no frequency-specific resonances and molecules can only be pushed down, which gravity is already doing very nicely thank you.

(Skip this on a first reading, but in deference to this blog’s venue, the n-category cafe, it should be mentioned that this distinction between free and bound behavior is nicely modeled by Pontrjagin duality. According to yin-yang and Cartesian duality, the universe divides up equally into two dual parts. This duality was originally believed by Descartes and his followers to be noninteracting, though Descartes himself, perhaps mindful of Galileo’s recent shameful treatment, allowed an exception for humans, postulating the pineal gland as the seat of that interaction, a ploy that failed to keep his works off the Index Librorum Prohibitorum. Today we (well, some nuts among us) understand the interaction to be via inner product, first recognized in the setting of Hilbert space but subsequently observed in Chu spaces, a much more general setting having a couple of connections with n-categories, one of which is that John Baez and I gave an interleaved series of lectures on the two subjects to a summer school in Coimbra, Portugal in 1999. Inner product is nature’s pineal gland, overcoming the standard objection to dualism that it is a noninteracting and hence inoperative theory. The Pontrjagin duality of the real line with itself expresses free existence, while that of the integers and the unit circle with each other expresses bound existence, with integers modeling the particles and the unit circle their dual waves. The setting for Pontrjagin duality is the category Ab of abelian groups, more precisely topological such, more precisely yet locally compact such. Fields arise as monoids in Ab, and Hilbert space as a vector space arises as a module over a field, typically Q or sometimes R, with an inner product placing it midway in abstractness between a vector space and a coordinate space. That’ll be enough of that now.)

At terrestrial pressures, or in the case of the Sun, at pressures found below the corona, this newly acquired energy is quickly redistributed equitably among all the degrees, thereby feeding translation. The mechanism here is that when two almost stationary but violently vibrating molecules touch, the vibration drives them apart violently, converting vibrational energy into translational.

As the pressure drops, so does the frequency of collisions, starving translation. The knee of the curve describing translation starvation is where the ideal gas laws break down. Molecules slow down relative to the prevailing temperature. This reduces translational momentum, an irrelevancy given that collisions transfer energy whether or not translational momentum is present. More significantly it reduces collision rate, the mechanism underlying translation starvation. Even if you get nothing else out of this extended ramble, at least quote me on that.

A concomitant factor is the early bird getting the worm of radiation in the upper atmosphere. As the worms are eaten the late birds lower down are starved of lines, exponentially with altitude by the Beer-Lambert law. So actually two effects are at work here in governing the knee of the curve, pressure and line absorption, both exponential in altitude.

To summarize in one sentence, the principle here is that although each collision can transfer a lot of energy to translation, the collisions need to happen on a regular schedule for that transfer to be effective, which low pressure prevents, whose effect is magnified by the better access to radiation lines enjoyed by the higher molecules.

I conjecture that this principle is the common underlying mechanism for the radiometer effect, the thermosphere, and the solar corona. In all three it puts temperature out of whack with velocity, allowing the former to rise far beyond what the ideal gas laws permit. The early-bird effect may play less of a role with some of these effects than others, but certainly low pressure is key here.

This mechanism in hand, one can proceed with numerical computation of the effects, all of which have been studied sufficiently as to make this feasible, if not all bits equally easy.

The easiest bit should be the mechanics of velocity, pressure, and temperature. Below the knee (but not so far as to approach the liquid phase) this is just PV = nRT. Above it there needs to be a new law. It seems to me that the law PV = α(h)nRT should be the basic form, where α is a suitably parametrized dilution factor. Obviously h can’t be the fundamental parameter for α, which is going to depend rather on molecular mass and radiation parameters. The former is easy, the latter divides up into absorption strength and the early-bird effect, the exponential decay in the strength of absorption.

In view of the atmosphere’s behavior, GHG, dominantly CO2, may be the culprit behind the strange downwards pressure bulge peaking at 113 km. If true this would constitute the GHG bulge in atmospheric pressure. CO2 being 50% heavier than the average air molecule, its proportion in the atmosphere decreases at some point, whence the early-bird effect kicks in for GHG at a lower altitude than the homonuclear molecules. The bulk of the heat absorbed by GHG should be in a region where it is still quite dense relative to air, maybe not 300 ppm but say 100 ppm (pure guess, it may well be much less). In that region the gas law breaks down strongly for GHG but at best weakly for homonuclear. In the absence of the latter the CO2 molecules would slow down, but surrounded by fast-moving homonuclear molecules they are sitting ducks for collisions and hence participate in collisions at the expected rate of 1 in every 10,000 collisions (assuming 100 ppm GHG), which should keep them moving as fast as the homonuclear molecules.

Well, with that back-of-the-envelope calculation I confess I now don’t see how hot GHG in cold homonuclear gas drives things one way or the other, although empirically it looks like at lower altitudes it somehow lowers the pressure, adiabatically for the homonuclear molecules at least, which pushes down the temperature since the dominant homonuclear molecules still constitute an ideal gas. But at higher altitudes the heating effect becomes irresistible, presumably because this is the knee of the curve for the homonuclear molecules judging by the smoothness of the turn-around in temperature at 90 km.

Judging by the downward pressure bulge the GHG contribution to radiative heating may well peak at 113 km. Higher up, that effect dies down, just as the radiometer effect ultimately dies down with decreasing pressure. In other words I’m conjecturing (not particularly strongly for the moment since it all seems so iffy) that the downward pressure bulge in the atmosphere and the upward activity bulge in the radiometer have a similar origin even if not exactly the same; for one thing there is no obvious reason why any pressures should vary exponentially in the radiometer.

The Sun on the other hand should behave much more like the earth. A key difference however is that the early bird effect works the opposite way to Earth because the Sun is a radiation source, not a sink like the Earth. I have no idea how that change in sign might affect things, though it still seems clear that at some point the translation starvation effect will kick in somewhere in the solar corona, heating it above the expectation based on not taking that effect into account.

Regarding the Chapman-Enskog solution to the Boltzmann equation that John mentioned, I don’t see how it helps in the case where translational energy is vastly less than vibration in a given mode. The Burnett equations might be more relevant, but it seems to me that the effect is so extreme that the equations should be relatively straightforward, this breakdown of the gas laws being so gross. The Burnett equations may however describe the knee of the curve, whose location however can surely be pegged more simply from backwards extrapolation of its extensions on either side.

Well, absent more quantitative support this is probably enough qualitative theorizing for now. Not so much TMI as TMT. Back to bed.

Couple more corrections. First, I just noticed another misreading of the NRLMSISE plot at [[Earth’s atmosphere]]. The blue line I was calling pressure is actually mass density in g/cm³. Oops. Below the thermosphere (e.g. in the mesosphere or stratosphere) they’re linearly related to the extent T remains more or less constant, but they diverge badly in the thermosphere.

Second, my use of the effective temperature of the Earth as a predictor for temperature below the thermosphere could be improved. For one thing, at 10 km the factor of 1-A correcting for albedo is not applicable since at that altitude most of the contribution to albedo is lower down and hence not blocking the Sun. Second, the concept of effective temperature of a whole planet should not be applied to objects smaller than the planet and near its surface, since unlike a whole planet they don’t have the night sky on their “dark side” (away from the Sun). Both factors predict a higher temperature for small objects near the planet than the effective temperature of the planet.

Since my previous post I’ve been reading up on aeronomy, aka atmospheric physics. Although I haven’t absorbed their picture fully yet I get the impression they have a good handle on why the thermosphere behaves in the odd way I described, and it seems for essentially my reasons if I’ve understood theirs. On the other hand I haven’t seen them point out that PV = nRT breaks down badly at an altitude of 150 km, so I’m not sure what that means – either I’m misunderstanding something or they’re not seeing any need to point this out. Meanwhile you can find no end of people willing to swear on a stack of bibles that PV = nrT is accurate at low pressure and high temperatures, so something’s wrong somewhere. At some point I may try writing up my explanation of what’s going on in the thermosphere and bounce it off people for their reactions. But not until next week at the earliest–I’m giving a paper at an algebra conference in Denver this week.

I’m assuming the solar corona theorists have already incorporated the known theoretical basis for the Earth’s thermosphere into their (low) prediction of the Sun’s temperature in the corona, and that I don’t have anything to add to their understanding. The anomaly there presumably lies elsewhere, e.g. the June article I mentioned earlier.

Meanwhile I remain convinced that the radiometer effect is essentially the anomalous behavior of the thermosphere in a bottle.

Following up on the last paragraph of my previous post, here’s my explanation of the radiometer effect, based on the present understanding of the thermosphere.

It is well known to atmosphere scientists that the thermosphere violates the equipartition theorem in an ideal gas in favor of bound (vibrational and rotational) modes over translational. In this condition there exists at least one bound degree of freedom for which the contribution kT/2 to the net energy of the molecule greatly exceeds the 3kT’/2 contribution of translation (3 assuming no directional bias in translation, less if there is bias). That is, the molecules are vibrationally hot (temperature T) but translationally cold (temperature T’). Translation contributes twice (multiplicatively) to pressure, once via rate of flow of molecules and once via translational energy (representable as p²/2m where p is the momentum mv) as well as vibrational exchanged in a collision. The energy exchanged can be coupled to the receiving particle both within and between modes. That is, energy exchange at a collision is conserved but can come from either bound or free modes and become either bound or free energy in the receiving molecule.

In this imbalanced condition, energy transfer is accomplished at the normal rate per collision (with the donor molecule contributing largely bound energy, which can become either bound or free), but the rate of collisions is drastically reduced. The upshot is that the apparent pressure is decreased in proportion to the rate reduction. Contrast this with the balanced condition, where the energy transferred per collision remains the same (though a more equitable proportion of it incidentally comes from the translation modes) but the collision rate is much higher, thereby increasing the pressure in proportion to the rate.

The radiometer effect can now be explained as an imbalance gradient between a cold surface and a hot, with the cold end more imbalanced than the hot. The total energy per molecule remains the same along this gradient, but the proportion of translational energy is greater at the hot end. This gradient is created initially by the hot surface contributing more energy than the cold, which is distributed equitably between vibration and translation in the molecules. The effect then runs away as the faster molecules at the hot end meet the hot surface faster than the slower ones at the other end meet the cold surface, suggesting that the gradient may not be linear and that in the middle the imbalance is closer to that of the cold end.

The equilibrium condition for this setup is that all molecules have the same average energy, and exchange the same amount of energy per collision. But due to the imbalance gradient these transfers slow down along the gradient from hot to cold, resulting in a steady dilution of the translational energy along that gradient.

The condition for this imbalance in the radiometer is that the incident radiation include the requisite frequencies to excite the vibrational modes, and that the pressure be low enough that the translation modes are starved of this energy due to the low rate of collisions.

When the pressure is too low there is not enough mass for this effect to create a sufficient pressure difference between the hot and cold sides of the vanes.

Among the more challenging prerequisites for a quantitative analysis of this effect are the line strengths of the gas and the proportion of those frequencies in the incident radiation. Recent tools developed for estimating line driving effects in planetary gases, e.g. http://www.cosis.net/abstracts/EGU2008/06613/EGU2008-A-06613-1.pdf?PHPSESSID= , should help.

In ordinary operation the same radiation is used to excite the gas molecules and heat the black side of the vanes. One way of confirming this explanation experimentally would be to precharge a radiometer with CO2 so that after evacuation the residual gas is CO2 rather than air. CO2 has a family of lines in the vicinity of 2300 cm^-1 (4.3 μm) associated with its bending mode and indexed by rotation numbers, the totality of which is far more strongly absorbing than any corresponding family for either O2 or N2. Ordinary light should therefore drive such a radiometer much more strongly, while infrared boosted at that neighborhood should drive it more strongly yet. An ordinary CO2 laser at 945 cm^-1 (10.6 μm) would excite the rather weaker lines at that wavelength in a CO2-charged radiometer, but the effect should still be much stronger than with an air-charged radiometer under white light.

(Note that glass strongly absorbs far IR, necessitating a strong source, but not so strong as to overheat and break the glass. An alternative is to use N2 and/or O2; although its absorption lines are weaker they are in a region where glass is transparent, permitting a weaker source for the lines.)

Illumination of the radiometer with two independently controlled radiation sources, a low-power narrow-band one for exciting the gas and a higher-power one with all but the lines of the other for warming the vanes, should allow observation of the interplay of these two components. The effect should drop by much more than a factor of two when either source is interrupted.

If the effect has some other origin such as the Reynolds-Maxwell-Einstein edge effects, no such strong dependence on choice of gas or exciting frequencies should be observed.

If my explanation is correct this will permit a variant of the radiometer serving to illustrate line-specific absorption by gases, of interest to students of both general physics and climatology.

My explanation neglected to stress the point that the hot side of the vane is driven by the greater pressure. The reason the pressure is not distributed uniformly in the radiometer is that the ideal gas law PV = nRT now has the form PV = αRT where α ≤ 1 is the dilution of pressure attributable to the imbalance. What is distributed uniformly is energy; in an ideal gas pressure is proportional to energy and hence is just as uniformly distributed as energy. An imbalance gradient permits a pressure gradient without creating an energy gradient.

Here’s a more practical experiment. Illuminate a standard radiometer with an ultraviolet lamp and an infrared lamp. These serve to warm respectively the oxygen component of the residual air and the vanes. Shutting off either should stop the radiometer, instantly for the ultraviolet light since the gas has negligible thermal mass, more slowly for the infrared light since the vanes will radiate their heat away relatively slowly while the rarefied gas provides negligible convective cooling.

This depends on oxygen’s absorption being strongest in the ultraviolet, making it the principal contributor to the effect. I would not expect nitrogen to contribute as strongly, making the wavelengths of the oxygen lines the critical frequency-dependent aspect.

Another application is spectrophotometry. If my theory is right, different gases in a radiometer should respond differently to different spectra. There’s a wide range of gases and mixtures thereof, allowing the design of radiometers tuned to pick out specific spectra. This would be a cruder instrument than a real spectrophotometer, but at the price of an incandescent light bulb (which is no easier to make than a radiometer) you get at least what you pay for.

This prompts the interesting question, what is the resolving power of radiometers in the following sense? Given two spectra, does there exist a radiometer, or small set of radiometers if necessary, capable of distinguishing them reliably? Any given criterion for reliability should create a metric d on the space of all spectra and a real r such that two spectra s,t can be reliably distinguished by some radiometer (or set thereof when that’s allowed) if and only
d(s,t) ≤ r. A typical application would be go-no-go spectrophotometry, deciding for example whether a given spectrum was close enough to a desired spectrum by that metric. Not every metric need arise in this way, and you may have to settle for a less desirable metric than you’d like.

This is primarily applicable to 2000 cm^-1 and above (5 μm and below) as ordinary glass is opaque to far infrared light, and transparent material such as zinc selenide would be prohibitively expensive, defeating the economic benefit of the radiometer.

Incidentally if my writing seems to have drifted into a 19th century style it’s from reading too many 19th century papers about radiometers. :)

This applet is a very simple toy model that might be useful for exploring some aspects of this. It doesn’t pretend to be anything like a detailed, faithful simulation of all the actual physics, but people might enjoy playing around with it and seeing if anything relevant can be gleaned from it.

We’re having an interesting dispute regarding Greg Egan’s simulation of the Crookes radiometer. Phil Gibbs wants to make the simulation more realistic and therefore more complicated. Greg doubts the need to add these extra complications.

The discussion is getting ridiculously nested, since Phil isn’t doing what it takes to prevent this — you need to click ‘Reply to this’, not on the post you’re actually replying to, but to the one it replied to! Sneaky, huh? So, I thought I’d try to move this discussion down here.

Now, there’s a real factual issue in dispute here: does the rotor spin just because of the initial ‘kick’ that it gets before it reaches a steady state, or not? I think Greg’s experiment shows that it’s not just an initial kick effect. But we can continue to argue about this…

On the other hand, there’s also a methodological issue involved. Are we trying to simulate a Crookes radiometer well enough so we feel sure we know how that gadget works? Or should we be happy if we understand how Greg’s simulation does what it does?

I’ve already argued for the latter. Perhaps it’s because I’m a mathematical physicist, but when I see a mysterious effect, I like to start by studying the simplest situation in which this mysterious effect occurs. After all, if we can’t understand a simplified model, how can we possibly understand something more complicated? Understanding of the simplified model is no guarantee that we’ll understand the full reality with all its complexities, but it’s hard to see how it could hurt.

(And, being a mathematical physicist, I generally let someone else tackle the more complicated aspects of a problem after I’ve done the easy stuff.)

Since Greg’s simulation already displays a counterintuitive effect, I’m eager to try to understand that, regardless of how much his model resembles a full-fledged 3d Crookes radiometer with all its complexities.

In fact, I would lean further towards simplification. If I were to seriously study this problem, I would probably want to tackle a radiometer with two vanes instead of four. I would also reduce the diameter of the central ‘spoke’ to zero.

Along these lines, I also enjoy taking Greg’s applet and setting the rotor’s outer radius to 100% and the inner radius to 10% — currently the minimum value, since the ‘spoke’ has nonzero diameter. This divides the radiometer into four chambers: particles can’t get from one chamber to another. In this situation, it doesn’t spin: it just wiggles back and forth randomly. It would be nice to prove this must happen.

It’s also interesting to ponder the flow lines in this situation. Should there be some organized ‘circulation’ of particles in this situation, or not?

To get the radiometer to spin, we need to shrink the rotor’s outer radius. Then we get a counterflow of particles around the outside, balancing the angular momentum of the spinning vanes.

That much is reasonably obvious. The real question, of course, is why the steady state situation has the radiometer spinning instead of at rest.

There’s nothing inherently paradoxical about this, since we’ve got a dissipative system. If you put a pan of water on a very slightly warm burner, the water won’t circulate — heat transfer occurs only by conduction, not convection. But if you turn up the heat, a phase transition occurs at a certain point, and the water starts to circulate, forming so-called Bérnard cells. So, depending on the parameters, dissipative systems can either display ‘circulation’ or not. The question is just how this works for the radiometer.

I haven’t said anything very substantial here, but at least it was fun saying it. I actually do think the last century’s work on nonequilibrium thermodynamics and dissipative ystems is one advantage we have over the greats like Maxwell and Einstein when it comes to trying to explain this gadget. (The ability to simulate things is another.)

For example, we could plot the steady-state angular momentum of the rotor as function of its outer radius, and see if it drops off smoothly as this radius approaches 100%, or whether it drops off sharply at some point, which might indicate a kind of phase transition.

Sorry I disappeared there. I’m taking off for the UK in a couple of days and have been busy working on related stuff.

Phil, you really hit the spot with your mention of the book by Loeb. Section 84 is an encyclopedically complete history of the previous three-quarters of a century of work on the radiometric effect. A number of experiments support Reynold’s, Maxwell’s, and Einstein’s theoretical calculations that it has to be an edge effect. My intuition was wrong that edge effects wouldn’t be strong enough and that the explanation must lie elsewhere.

I think my intuition let me down when I tried to generalize my experience with 1 atmosphere down to 10^-4 atmospheres. There is so little resistance to motion down there that far less force is needed to spin the vanes around at high speed than would be required at normal air pressure, helped by the glass bearing seated on a very sharp needle point, a cheap but effective low-friction bearing.

Changing the subject (should I be changing the thread here? I don’t know the convention), here’s another question, not much further removed from n-categories than the radiometer one. If from a point Z in the plane you draw an expanding equiangular spiral of angle α (defined so that α=0 gives a circle) about a point O in one direction, and its tangent at Z in the other, they eventually meet, say at A. Let φ be the angle ZOA. Using the sine rule for triangles it’s not hard to show that φ is related to α by

cos(α) = cos(φ−α)*exp((2π−φ)tan(α))

For the physics application I have in mind, actually just the geometry of moving slices of pie around within a pie plate, it is natural to ask how α depends on φ.

Clearly α tends to zero with φ, and almost as clearly tends to π/2 as φ tends to π (I described the spiral starting from Z rather than A to make this clearer).

Rewriting the above relationship as

α = atan(ln(cos(α)/cos(φ-α))/(2π-φ))

doesn’t give a closed form for α, since α also appears on the right. However if α is initialized to φ²/4π this formula when iterated converges to the answer.

Some numerical experimentation with bc with scale=200 and φ set to various powers of 1/10 reveals that

α = φ²/4π + φ⁴/24π - φ⁵/24π² + .003872729996832391029424035567…φ⁶ + O(φ⁷).

Question: what is .003872729996832391029424035567…?

Trying to match this up to some formula involving powers of π and rationals has me completely stumped.

There is now a (still puny) entry on light mills at the $n$Lab.

In the Pioneer anomaly the acceleration is changed from calculated 7.8e-10 m/ss but adjusted to 8.74m/ss to account for radio transmission of 5 W on a vehicle mass of 374kg (IIRC). This happened in a vacuum and a 10% change is critically important. This discussion inre light mills raises the question as to whether this radiation force can be taken for granted. The answer appears to be yes and no.

I hesitate to jump into this fascinating discussion among you professionals with a layman’s confused ideas, but I can’t resist. I would like to carry to the extreme John Baez’s suggestion that we simplify the problem. Let’s make the light mill one-dimensional with only one particle and one vane, with the motion of the particle and vane restricted to a circular track. I can think of two ways to define the particle-vane interaction that will give a working mill.

First, assume that the vane is totally absorbing on the cold side, so that a collision of the particle with the vane’s cold side is completely inelastic (all the particle’s kinetic energy is converted into the vane’s thermal energy). Now imagine that we have a pump that will move both heat and particles from the cold side to the hot side (does this sound a wee bit like transpiration at the edges?). Initially, imagine the lone particle as stuck to the motionless hot side. In due course the particle will gain enough internal energy from the hot side’s thermal vibrations to exceed its binding energy with the vane. At that point it will pop off the surface and give a slight recoil kick to the the vane. The particle wizes around the track and the vane moves slowly in the reverse direction. They collide, the particle sticks and all macro-motion stops (all kinetic energy being converted into thermal energy). The pump moves the particle and the thermal energy it left behind over to the hot side, and the cycle repeats. Notice that averaged over time we have steady-state motion with no initial transient.

For the second scenario, assume the vane is a one-way mirror, totally transparent when seen from the “cold” side and totally opaque when seen from the “hot” side. As before, start with the particle stuck to the hot side (actually, both sides can be at the same temperature in this scenario). As before, the particle acquires energy and pop off with a recoil kick. This leaves the vane moving slowly around one direction while the particle wizes quickly around in the reverse direction. There is no furhter interaction between the particle and the vane because of the transparency. In this scenario the initial kick is the whole story.

Perhaps I have simplified the problem so far as to beg the question; but, I would like to hear what others think.

When I was a kid I had a Crookes radiometer. As I recall, it would initially spin rapidly under a strong light; however, after a minute or so it would slow down, but not stop. So, I wouldn’t be surprised if the motion has both transient and steady state components. Then again, it has been over 50 years since I played with the thing.

John Baez - I love your blog. What amazing moxie you have to even imagine that you can usefully convey ideas as subtle as n-categories to an interested lay audience.

srwenner

Change the shape of the vacuum and you’ll work this out. The sphere’s circular equator compliments the blade rotation along itself, in either direction. All else being equal (including the vacuum’s imperfection), replace the spherical container with perhaps a cube or an irregular volume, and notice the blades dramatically slow down because photons and/or rarefied gas bouncing off the white side will now achieve only incoherent momentum. But here’s the real mystery: Which volume shape(s) will cause a total halting of the blades?

Bailing into this way above my paygrade discussion with my diffeq math and physics 181 background I have a question for the assembled geniuses here:
Do we have a predictive equation for the radiometer? And if so what is it?

We can put the “thermal transpiration at the vane edges” explanation to the test in a simple desktop experiment. It fails.

From John Baez/Philip Gibbs popular explanation:

By “thermal transpiration”, Reynolds meant the flow of gas through porous plates caused by a temperature difference on the two sides of the plates… The vanes of a radiometer are not porous. To explain the radiometer, therefore, one must focus attention not on the faces of the vanes, but on their edges.

Green laser pointers emit ~10mW of light, plenty to cause a toy radiometer to spin rapidly. If the “edges are like pores” explanation is valid, shining a laser on a radiometer vane at its edge will propel it, but shining the laser on the center of the vane, not heating the edge, will not propel it.

In fact, the radiometer is propelled equally with a vane heated edge or center.

We’ve also heard from Jamie who reports he constructed a radiometer with a vane full of holes, and it turned slower. The “edges are like pores” explanation would say greater perimeter should provide greater propulsion. That’s a difficult experiment because radiometers are finicky; we have to worry that the swiss-cheese radiometer was a dud for some other reason.

To see if heat from a laser dot could somehow be conducted to the vane edges by a tricky layered vane construction, I broke open a toy radiometer. The vanes are thin cardstock (paper), about 0.2mm thick and 16mm square. Interestingly, the black side is not paint or ink, it’s lampblack or some similar powder that rubs off easily. Lampblack is the traditional radiometer black; yet, modern manufacturers probably wouldn’t go to the trouble if it didn’t matter.

Traditional construction also calls for mica vanes, which is not what we find in today’s toys. Mica is fully non-porous. Cardstock is not. It’s hard to believe that the minimal porosity of cardstock could be important, but my laser experiment doesn’t preclude this possibility.

I was just wandering through a 99percentinvisible article about the unintended sounds that buildings make, and happened therein upon This youtube video which purports to include what I can only describe as an Sound Mill. And it seems to me that this creature suffers the same inexplicability as the Light Mill. I certainly won’t venture to suggest that the vanes of a Light Mill are resonant in any useful way, but I can’t explain how the Sound mill gets a torque separation out of centrally-symmetric excitation, without invoking the viscosity of air (something like the jellyfish motor). And even if I do that, I don’t know how to get the right signum. (does air become more or less viscose when adiabatically heated?)