The vane that turns the wrong way

Light has a pressure described a measurement that is delicate because the force is so feeble, and it mentioned in passing a very famous false lead. William Crookes, in the 1870s, a chemist who had discovered thallium, was weighing things in a vacuum balance and found that warm objects seemed to weigh less. Working on what that could be, he made a little rotor with four vanes on a fine pivot in a glass bulb, with the air pumped out to a good level but not all of it, and each vane blackened on one side and shiny on the other. Put it in sunlight and it spins. He announced in 1874 that light had a mechanical force. James Clerk Maxwell, on his deathbed five years later, was among those writing up why that was not the case.

The toy is still sold, and for anyone looking at one, it is natural to suppose that photons are hitting vanes. The explanation is deeper than it seems, and a good deal less tidy than most short accounts of it suggest. It is one of the rare cases where the first two theories, each from excellent physicists, are wrong in an interesting way and where the correct picture needed Einstein to finish it.

Which way it runs #

First, the direction. If light’s own pressure were at work, a photon absorbed by a black face transfers its momentum $E/c$, and one reflected from a silver face transfers twice that. So the silver face should retreat. The rotor actually turns with the black face moving away. It runs the opposite way.

A real radiation-pressure radiometer does exist, and it needs a far better vacuum. The first one, by Lebedev in 1901 and Nichols and Hull in the same year, did spin the way light would push it, but only after gas effects had been carefully minimised. Crookes’s vanes spin at a speed that anyone can watch. In the sunshine of a window, radiation pressure is $I/c$, at about 500 W/m² nearly 1.7 µPa, a very small number on a vane with a mass of tens of milligrams on a pivot. Heat is another matter. A blackened mica vane will warm on its dark side by a kelvin or so. The gas does the rest.

The reasoning that fails #

The first explanation tried is the one found in most old textbooks. Gas molecules strike the warm face, leave faster and give it a greater recoil. It sounds right, and the reasoning does give a net force in the right direction. But Osborne Reynolds and Maxwell saw a problem. In a normal gas, the push that the molecules leaving a warm face apply gets balanced. Warm gas near the warm face has the same pressure as cool gas, simply less dense. In a dense gas, molecules striking the warm face collide with each other just above it, and they establish a layer with a density that falls as temperature rises. So the pressure is the same on the warm and the cool side, as hydrostatics requires. A steady gas cannot sustain a pressure difference across a thin plate that it can freely flow around.

So at ordinary pressures the face-on force vanishes. Yet at very low pressure, when molecules arrive straight from the walls without any collision near the vane, the reasoning does hold. Such molecules come in at the walls’ temperature, and they leave the warm face faster than they leave the cool face, and there is no layer to adjust. Then the net push is, up to accommodation factors, a known fraction of the pressure:

$$\Delta p = p\,\frac{\Delta T}{2T}.$$

That expression is linear in the gas pressure. At a low enough pressure, there is too little gas. The force falls in proportion, and so does the toy’s speed if it cannot overcome its pivot.

So there is a range, and it is set by the only lengths in the problem. How far a molecule gets calculated the mean free path: 67 nm at atmospheric pressure, and proportional to the inverse of pressure. It is therefore 6.7 mm at 1 Pa. When the free path is much smaller than the vane, the gas behaves as a continuum and the face-on force cancels. When much bigger, the free-molecule formula applies. The peak has to lie about where they meet: where the free path is about the size of the vane.

The chart has that shape: a straight rise at low pressure, a peak and a fall at high pressure. Its interpolation between the two limits is a simple one of the form used in engineering estimates, and not derived from anything deep. What does not depend on the interpolation is the size of the effect. Near the peak the gas force is hundreds of times the light force, so that a very weak temperature difference does more than the whole momentum of the sunlight, by a margin that puts the toy on the safe side of any likely error.

The force at the edge #

The high-pressure side has a further story. In a real bulb at about one pascal, a vane is not in the free-molecule regime and it is not in the continuum regime either. Reynolds worked on this in 1879, in a long paper that Maxwell refereed. He noticed that if a gas flows along a surface with a temperature gradient, something happens. Maxwell went on to calculate it in the same year, and he called it thermal creep.

The explanation is pleasant. Think of molecules arriving at a point on a surface from above. Those from the side where the wall is hot carry more momentum along the wall than those from the cold side. They give the wall a net shear from cold to hot, and by reaction the gas flows from cold to hot along the wall, against the direction one would expect from anything else. It flows at a speed that is inversely proportional to pressure, and it needs no pressure difference. Reynolds observed the effect in porous plates. Gas seeps through them from the cool side to the warm side, a process now called thermal transpiration. The resulting pressure difference is a classical correction for accurate low-pressure gauges and related to what the gas that leaves is not the gas inside discussed.

Around the rim of a vane, where the black and silver sides meet, a sharp temperature gradient lies across a thin edge. Gas creeps round from the cold side to the hot side, and it gives the vane a push towards the cold side, in the direction opposite to the flow. So the black side retreats, again. Einstein in 1924 gave a neat estimate of the force from another point of view. He showed that it would be concentrated within a mean free path of the edge, so that it scales with the perimeter and not with the area. That matters to the design. A vane with a lot of edge per unit area should push better, and experiments have since shown that cutting slots in a vane does increase the force at some pressures. Newer numerical calculations of the full molecular dynamics, with Boltzmann solvers, have confirmed something like Einstein’s result and shown that these forces at the rim dominate at higher pressures. So the standard statement is that the toy is driven mostly by the free-molecule force at low pressures and by the edge effects at higher ones, with Reynolds, Maxwell and Einstein all having had hold of parts of it.

Why the toy does not care much what is in the bulb #

There is something else unexpected in the behaviour with pressure. Pumping a bulb out further should remove drag and it would seem to help. It does not, up to a point.

The explanation lies in the viscosity that does not care how much gas there is, which is the unexpected result of Maxwell that a gas’s viscosity does not change with pressure while it remains a fluid. At high pressure, therefore, drag is pressure-independent while drive falls, and speed tends to zero. In thin gas, both drive and drag are proportional to pressure and their ratio, the rotor’s terminal speed, is independent of it. Below some pressure the drive falls beneath the friction at the pivot and the rotor stops altogether. So a rotor’s speed curve has a plateau in the middle, with the plateau’s width set by the quality of the pivot. The numbers in this model are illustrative, as they depend on pivot friction, which any toy has in its own quantity. The shape is what matters, and it is what makes the object robust. A manufacturer does not have to pump it to a very exact value. A toy that lost a bit of vacuum in its seal would continue working until its pressure had risen tenfold.

The same model also explains the trick that catches many people out. If the toy is placed in a bright light and then in a cold room, and if the warm glass is left to cool, it may run backwards, as the vanes’ black sides now radiate and cool more quickly than the silver sides. The rotor is reading a temperature difference, not a beam of light. Hence, the toy will run when warmed by a hand or an electric ring or if cooled by ice, with no light on it at all.

Small things feel it in open air #

The peak’s dependence on size is simple and it leads somewhere interesting.

What holds for 1 cm at less than a pascal holds for a micrometre at a few thousandths of an atmosphere, and a tenth of a micrometre at about the standard pressure. So small particles and micromachined devices live in the regime where these forces are at their largest. In MEMS sensors, thermal forces can exceed the electrostatic forces that drive the device, and designers have to allow for them. Researchers have used this deliberately to make actuators and Knudsen pumps with no moving parts, in which thermal transpiration through a nano-porous material builds up a pressure across it.

In the upper atmosphere, particles near a tenth of a micrometre to a few micrometres can be pushed around by sunlight in the same way. It is called photophoresis and it was named by Felix Ehrenhaft in the early twentieth century. A sunlit soot aerosol warmed more on one side than the other will feel a force often larger than its weight, in the stratosphere and mesosphere. David Keith proposed engineered particles that could be lofted indefinitely by this force. It has also been suggested for how dust stays up in Mars’s thin atmosphere, and for how early planetary discs sorted dust by its conductivity. And in 2023 and thereafter experiments demonstrated levitation of very thin perforated plates by sunlight at low pressures, at the range of size and pressure the figure would predict, offering a means of flying in the mesosphere with no propulsion.

The test with a bulb on a thread #

A simple check on the whole class of explanation was available from the first. Hang the entire bulb on a thread, and watch it when the light comes on. If the vanes were being driven by something from outside the bulb, the sunlight, the rotor would speed up with nothing to oppose it. But if the force is between the vanes and the gas in the bulb, the gas and hence the glass must take up the opposite torque. As the vanes start, the bulb has to turn in the opposite sense on its thread, by an amount set by its moment of inertia, and it does. When the light is turned off, the rotor coasts and is slowed by drag on the bulb, which is carried along for a while in the rotor’s direction. Crookes himself hung a bulb in this way in 1876, and it showed the behaviour, and that should have been decisive. Angular momentum conservation does the work here that a calculation of forces could not, and it does not need to be told about a mechanism. All it says is that a source of torque lies inside the glass. It is a useful habit, and it recalls the push that needs nothing to push against: a thing that seems to be pushing itself has always handed something else an equal share.

The other useful comparison is with the note too high for thin air. Sound stops propagating when a wavelength reaches the mean free path. The same ratio appears here, the Knudsen number, the free path over the length of interest, and the same regimes are present. In a gas at a Knudsen number well below one, collisions keep the molecules in local equilibrium, and all motion can be described with the Navier–Stokes equations plus small corrections. Above one, molecules fly from wall to wall. In between, which is where the vane operates best, neither description works, and that was what made the problem hard, as it remains hard for an engineer designing the flow in a vacuum pump. The gas that flows towards more of itself found a flow in the opposite direction to the concentration gradient, and it also came out of this regime, in that case by a difference in masses.

Most laboratory gas physics has taken care to stay out of the intermediate range. That Crookes, working on something else, stumbled on it with a glassblower’s bulb, and that its full explanation took half a century and the attention of Maxwell, Reynolds and Einstein, is a mark of how awkward it is. A toy can be built with a piece of mica and a good hand pump, and solving it from first principles still needs a computer. It is a modest demonstration that the difficulty of an explanation is no guide to the cost of the object.

There is a further reason to treat the speed limit that does not care how big the molecules are as part of the same family. A very thin gas is limited by rates at which molecules get to a surface, and no longer by the bulk properties of a fluid. So surface temperature and molecular speed come directly into the force. The radiometer’s force is what is left over when a pressure of about a pascal has been divided up by these speeds at surfaces of slightly different temperature.

What the figures simplify #

The whole of the pressure curve rests on a smooth interpolation between two limiting cases. It sets the peak by a rule of thumb, namely that a free path is equal to the vane size, whereas the correct calculation has different coefficients and requires a kinetic theory solver. So the height of the curve near its peak is not a prediction, and a measured curve would have a different shape in detail. Its low-pressure line uses perfect accommodation, in which a molecule leaves the surface with the surface’s temperature. Real surfaces have a coefficient that can be substantially below one, depending on the gas and how clean the surface is.

The temperature difference has been set at one kelvin and treated as given. In a real vane, it is itself the result of a balance among absorbed sunlight, thermal conduction through the mica and radiation and cooling by the gas, and it depends on pressure. At high gas pressures the gas conducts heat well and the faces of the vane stay nearly equal in temperature. Pressure thus enters in another way. The vane’s temperature difference falls as pressure rises, which further narrows the working range at the high-pressure side.

A two-faced flat plate is also a poor description of the flow’s geometry. Convection in the bulb, the wall heated by the light itself, and the interaction between the four vanes all matter in a real instrument, and at the higher pressures the gas is set moving in ways that a calculation at a single vane does not include. Since a Boltzmann calculation of a rotating, heated rotor in a bulb remains difficult, no full theory of the toy exists at every pressure.

Still open: how far thermal forces can be built up #

In a plate that hangs in the mesosphere, the question is how large a force can be obtained per unit of mass. Very thin structures, tens of nanometres across and perforated to allow thermal creep, have been levitated at low pressure by moderate light. An optimised device would lift a meaningful payload only if the same effects could be built up on a large area without tearing it. How these thin structures respond to a real atmosphere’s turbulence and ultraviolet light is unknown, and the mesosphere’s own conditions are hard to reproduce. The mesosphere is rather difficult to reach and is little studied, which makes such a platform interesting in itself. A related question at the other end of the scale is how much thermal creep sets the limit on the sensitivity of force measurements in vacuum, as in the search for gravity at short distances, where differences of a few millikelvin between surfaces at low pressure can produce forces larger than the effect being sought. Several experiments in this field have had to calibrate out what is essentially a radiometer.

A Crookes radiometer turns with its black faces retreating, opposite to what light’s pressure would do, driven by molecules leaving warm faces with more momentum at a push that peaks where the free path equals the vane, about 0.67 Pa, at about 335 times sunlight’s pressure per kelvin, with thermal creep around the edges carrying more of it at the higher pressures. The first explanation had the wrong physics, and so did the second. Each of them was right about a regime. The habit worth carrying is to ask what length decides which regime a thing is in, because the force that is real in a bulb is zero in a room, and what looks like one mechanism in a toy is really three.