Light has a pressure

Light carries momentum, and anything that absorbs or reflects it feels a force. The number for full sunlight is 4.5 micropascals — about the pressure a grain of sand exerts on the table it is sitting on, spread over a square metre — and for most of the objects a person encounters that is a rounding error on a rounding error.

For a small enough object it is not. The force from light grows with the object’s cross-sectional area while gravity grows with its volume, so the ratio between them goes as one over the size, without limit.

Where the momentum comes from #

Three separate arguments give the same relation, which is a good sign that it is not an artefact of any of them.

From electromagnetism. A wave with electric field $E$ and magnetic field $B$ carries energy at rate $S = EB/\mu_0$ per unit area and momentum density $S/c^2$. Maxwell derived the pressure on an absorbing surface, $p = S/c$, in 1873, and the result predates any notion of a photon.

From relativity. Anything carrying energy $E$ at speed $c$ has momentum $E/c$, because the energy–momentum relation $E^2 = (pc)^2 + (mc^2)^2$ with $m = 0$ leaves $E = pc$. Nothing about light in particular is used.

From the photon. Each photon carries $h\nu/c$ of momentum, and a beam is a stream of them. This is the version that was measured directly, by watching photons bounce off electrons and observing that the recoil obeys the ordinary conservation laws.

The photon’s momentum has been measured directly: X-rays scattering off electrons shift in wavelength by an amount depending on the scattering angle and nothing else, and the shift is exactly what conservation of momentum requires if the photon carries $h/\lambda$. Radiation pressure is that same momentum delivered by a great many photons instead of one, which is why the two subjects have the same constant in them.

The idea is older than any of the three. Kepler, watching comets in 1619, noticed that their tails point away from the Sun regardless of which way the comet is travelling, and proposed that something in sunlight was pushing the material outward. He was half right in a way that took three hundred years to sort out: the dust tail is pushed by radiation pressure and by exactly the size-dependent mechanism the hero figure draws, while the separate ion tail is driven by the outflowing solar wind, which is a stream of particles and not light at all. Two tails, two mechanisms, one correct intuition about the first.

How strong would it have to be to hold something up #

A useful way to feel the size of the effect is to ask what a beam could support against gravity.

The pressure has to exceed the weight per unit area, so the condition is $I/c > \sigma g$, with $\sigma$ the mass per unit area of the thing being lifted. Full sunlight gives $\sigma < 4.6\times10^{-7}$ kg/m². For aluminium at 2700 kg/m³, that is a sheet 0.17 nanometres thick — thinner than a single atom. Sunlight cannot levitate any material object on Earth, and not by a small margin.

Reverse the question and it becomes achievable. To levitate a one-micron sphere of density 1000 kg/m³, whose mass per unit area is about $10^{-3}$ kg/m², needs an intensity of $3\times10^{6}$ W/m² — a few watts focused into a spot a micron across, which is an ordinary laboratory laser. That is precisely the arithmetic that makes optical trapping possible for small particles and impossible for large ones, and it is the same mass-per-area quantity, arriving from the other end.

The pressure is a momentum flux, exactly as a gas’s is #

There is a satisfying parallel with the pressure of a gas, and it makes the factor-of-two question below obvious rather than mysterious.

Gas pressure is a rate of arrival — molecules strike a wall, each delivering a momentum, and the pressure is the momentum arriving per unit time per unit area. Radiation pressure is that statement with photons in place of molecules and $E/c$ in place of $mv$. The one structural difference is that the photons all arrive at the same speed and mostly from the same direction, so there is no thermal average to take, and the factor of a third that appears for an isotropic gas becomes a factor of one for a beam.

A molecule that sticks to the wall delivers $mv$. A molecule that bounces elastically delivers $2mv$, because its momentum is reversed rather than merely stopped. Photons are no different: a black surface takes $I/c$ and a mirror takes $2I/c$.

For sunlight at the Earth’s distance, $I = 1361$ W/m², so a black sail feels 4.5 µPa and a mirrored one 9.1 µPa. On a square kilometre of mirror that is nine newtons, which is the weight of a bag of sugar — and on a spacecraft weighing tens of kilograms, applied continuously for years with no fuel, it is a propulsion system.

The demonstration that shows the opposite #

Every school physics cupboard contains a Crookes radiometer: four vanes in a partly evacuated bulb, each with one face blackened and one silvered, mounted to spin freely. Put it in sunlight and it spins. It is presented, in an unfortunate number of textbooks, as a demonstration of radiation pressure.

It turns the wrong way.

The force is a gas effect. The blackened face runs a few degrees warmer than the silvered one, and at a pressure where the gas’s mean free path is comparable with the vane’s size — around a pascal, a hundred-thousandth of an atmosphere — a temperature difference along a surface drives a flow of gas along it. The reaction pushes the warm face backwards. The mechanism is thermal transpiration, worked out by Reynolds and Maxwell in the 1870s, and it is a few orders of magnitude larger than the radiation pressure in the same apparatus.

Two features of this are worth keeping. The first is that the direction of rotation is the whole evidence: a wrong-signed result is a much stronger refutation than a wrong-sized one, since no calibration error can flip a sign. The second is the pressure dependence — a light-driven device works better in a better vacuum, and this one stops. Crookes himself thought he had found radiation pressure; Reynolds and Maxwell corrected him within a few years, and the correct explanation has been available for a century and a half without dislodging the wrong caption.

There is a design lesson buried in the failure, and it generalises well beyond this apparatus. The radiometer’s vanes are mounted on a low-friction bearing precisely so that a tiny force can move them — and a bearing that admits a tiny force admits every tiny force, including ones nobody was thinking about. An instrument sensitive enough to see the effect of interest is by construction sensitive enough to see effects that are larger. The way out is not a better bearing but a discriminating test: change something that alters one mechanism and not the other, and see which way the answer moves. Here the pressure of the residual gas is that knob, and it settles the question in one sweep.

Radiation pressure was eventually measured in the laboratory, by Lebedev in 1901 and by Nichols and Hull in 1903, using torsion balances in a much harder vacuum, with the residual gas effect suppressed rather than exploited. The measurements agreed with Maxwell’s prediction to a few per cent.

Why the ratio to gravity does not depend on distance #

The hero figure makes a claim that is worth stating carefully, because it is the reason radiation pressure matters at all outside a laboratory.

For a grain of radius $a$ and density $\rho$ at distance $r$ from a source of luminosity $L$ and mass $M$:

$$\frac{F_{\text{rad}}}{F_{\text{grav}}} = \frac{L\,\pi a^2 / 4\pi r^2 c}{GM\cdot\tfrac43\pi a^3\rho / r^2} = \frac{3L}{16\pi G M c \rho a}.$$

The $r^2$ cancels. Both forces obey the same inverse-square falloff, for the same geometrical reason, so their ratio is a property of the grain alone — its size and what it is made of — and of the source’s luminosity-to-mass ratio.

There is one more variable hiding in the numerator, and it is the reason this ratio is worth writing down at all: the source’s luminosity divided by its mass. That combination, and not the luminosity alone, is what decides whether light or gravity wins for a grain of a given size — so two sources of very different size and brightness that happen to share a luminosity-to-mass ratio push the same grains away and hold the same ones. It is the closest thing this subject has to a similarity parameter, in the sense a dimensionless group is one elsewhere.

The quantity that decides everything is therefore mass per unit area. That is the single most useful way to think about radiation pressure: a beam pushes on an area and drags along a mass, so what matters is how much mass each square metre has to carry. A sail is a device for making that number as small as possible.

The pressure a thermal field exerts #

Directed light is the easy case. An enclosure full of thermal radiation exerts a pressure too, and it comes out as one third of the energy density:

$$p = \frac{u}{3} = \frac{aT^4}{3},$$

with the third arising because the radiation is isotropic and only one direction in three pushes on any given wall — the same averaging that gives a gas $p = \tfrac13 n m \overline{v^2}$.

At ordinary temperatures the number is hopeless: $2\times10^{-6}$ Pa at 300 K. At $10^7$ K it is $2.5\times10^{11}$ Pa. There is a temperature, for any given gas density, above which the radiation is holding the container open rather than the gas.

The fourth power is what makes this a threshold rather than a gradient. Doubling the temperature multiplies the gas pressure by two and the radiation pressure by sixteen, so the two swap places over a narrow range, and any system hot enough for radiation pressure to matter at all is usually a system where it dominates.

The force that pulls toward the bright part #

Everything above is about a push along the beam. There is a second radiation force with a different direction, and it is the one that turned light into a laboratory instrument.

A transparent particle in a beam whose intensity varies across its width — a focused laser, for instance — refracts the light passing through it. Refraction changes the light’s direction, so it changes the light’s momentum, so the particle receives the opposite change. For a particle sitting off the axis of a bright spot, the bending is asymmetric and the reaction pulls it toward the bright region: a restoring force, proportional to the gradient of the intensity rather than to the intensity itself.

That gradient force competes with the ordinary push along the beam, and for a tightly enough focused spot it wins in all three directions. The result is an optical trap: a single particle, a bacterium or a single atom held in free space by nothing but light, with a stiffness that can be calibrated and used to measure forces down to femtonewtons. The technique measures the force a single molecular motor exerts as it walks along a filament, which is a chain of inference from Maxwell’s 1873 pressure to a biological measurement.

The distinction between the two forces is worth keeping crisp: the scattering force is a flux of momentum and pushes along the beam; the gradient force is a rearrangement of momentum and pulls toward the maximum. A well-made trap is one where the second exceeds the first, which is why the focus has to be so tight.

The tail that sorts the dust by size #

Kepler’s observation was mentioned at the start and deserves finishing, because a comet’s tail turns out to be a direct read-out of the hero figure’s ratio.

Call that ratio $\beta$ — radiation force divided by gravitational force on a grain, the quantity that depends on the grain and not on where it is. A grain released from a comet’s nucleus is then in orbit around a Sun whose effective gravity has been weakened by a factor $(1-\beta)$. Grains with $\beta$ above one feel a net outward force and leave on hyperbolic paths; grains with small $\beta$ stay on nearly the comet’s own orbit; and everything in between follows a trajectory in between.

So the dust released at any one moment fans out, sorted by size, with the smallest grains furthest from the comet’s path. The tail is broad and curved because it is the accumulated fan from dust released over weeks — and its precise shape encodes both the size distribution of the grains and the history of when they were released. Reconstructing those two from a photograph is a standard technique, and it is how the grain sizes in a comet are known without going there.

The second tail works on an entirely different principle, and the two are easy to tell apart once the distinction is known. The ion tail is straight, narrow and blue: it is molecules ionised by sunlight, which are then caught by the magnetic field carried in the solar wind and swept radially away from the Sun at several hundred kilometres a second. It points almost exactly anti-sunward, with a small aberration from the comet’s own motion, and it can disconnect and reform when the wind’s field reverses.

That aberration is worth a sentence of its own, because it was used as an instrument. Biermann pointed out in 1951 that the ion tails’ direction and their rapid changes required a continuous outflow of particles from the Sun at high speed — not radiation, which travels too straight, and not a static corona. The solar wind was inferred from comet tails, named by Parker in 1958, and detected by spacecraft in 1962.

So Kepler’s guess splits cleanly. One tail is radiation pressure, acting on solid grains, sorted by exactly the mass-per-area argument this essay is about. The other is a stream of particles, discovered because it made a tail point the wrong way by a few degrees.

The push that comes from a rotating body’s own warmth #

There is a radiation force that does not come from the light arriving at all. It comes from the light leaving, and it is now precise enough to matter for planetary defence.

A rotating asteroid absorbs sunlight on its day side and re-radiates it as heat. If the body had no thermal inertia the hottest point would be the one directly under the Sun, the re-emission would be symmetric about the Sun-asteroid line, and the recoil would simply add to the ordinary outward push. Real rock does have thermal inertia, so the surface takes time to warm and time to cool, and the hottest point lags — it is the asteroid’s mid-afternoon, not its noon.

Re-emission from a lagging hot spot is not aligned with the incoming light. The recoil therefore has a component along the orbit, and its sign depends on which way the body spins: a prograde rotator is pushed forward and spirals outward, a retrograde one is pushed backward and spirals inward.

This is the Yarkovsky effect, proposed by a Polish engineer around 1900 in a pamphlet that was subsequently lost, remembered only because Öpik had read it. The force is minute — for a kilometre-sized body it changes the orbital radius by a fraction of a kilometre a year — and it accumulates.

It has been measured. Radar ranging to the asteroid Golevka over twelve years showed it displaced by about fifteen kilometres from where a purely gravitational orbit put it, in the direction and by the amount its measured spin and size predict. The drift of Bennu is now known to a few metres per year, and it is the dominant uncertainty in where that object will be a century and a half from now — a hazard calculation in which a thermal recoil outweighs every gravitational term that is not the Sun’s.

The same asymmetry applied to torque rather than to force is called YORP, and it spins irregular bodies up or down over millions of years. It has been measured too, on more than one object, and it accounts for things a purely gravitational history cannot: rubble piles spun to the point of shedding material, binary asteroids formed by fission, and the observed excess of very fast and very slow rotators.

The large consequence is one of supply. Main-belt asteroids drift slowly under Yarkovsky until they reach an orbital resonance with Jupiter, which then throws them onto planet-crossing orbits. Almost every near-Earth asteroid and almost every meteorite in a collection got there by this route — moved out of the belt, over millions of years, by the recoil of its own afternoon.

What it costs, and where the model stops #

The perfect-absorber and perfect-mirror cases are bounds. A real surface has a reflectivity between zero and one, and the pressure lies between $I/c$ and $2I/c$ in proportion. A surface that scatters diffusely is a third case again, since the returned momentum is spread over a hemisphere rather than reversed, giving a factor of about 1.4 rather than 2.

Incidence angle matters twice. A surface tilted at $\theta$ intercepts a beam over a smaller projected area, by $\cos\theta$, and the reflected momentum leaves at an angle, contributing another $\cos\theta$. So the normal force goes as $\cos^2\theta$ and there is a tangential component as well — which is what makes a sail steerable and what makes the calculation of its thrust a geometry problem rather than an arithmetic one.

A grain that absorbs and re-emits is not simply pushed along the beam. It absorbs momentum from one direction and re-radiates it in its own frame, and to an observer for whom the grain is moving the re-radiation is asymmetric. The result is a slow drag on any orbiting grain — Poynting–Robertson drag — which is a factor of $v/c$ smaller than the direct push and, over long enough times, the effect that decides where small grains end up.

Nothing here treats the grain as anything but a disc. A particle comparable in size with the wavelength does not have a cross-section equal to its geometrical area: it can be larger or smaller depending on the size parameter, which is the same scattering physics that decides the colour of the sky. The crossover radius computed in the figures is therefore an order-of-magnitude statement, and the honest version of the calculation carries a wavelength-dependent efficiency factor.

The ladder from here #

Later rungs on this anchor: the sail as an engineering object, and the mass-per-area limit that decides whether one can be built; Poynting–Robertson drag derived properly, as a $v/c$ correction with a sign; optical tweezers, in which a focused beam’s gradient rather than its flow supplies a restoring force, and single cells and single atoms are held in one; laser cooling, where the momentum is delivered selectively by exploiting the Doppler shift; and radiation pressure in an enclosure, where $aT^4/3$ meets the gas law and the two swap dominance.

The neighbouring ladders are the photon’s momentum, which is this pressure with one photon in it, and the kinetic theory of gases, whose derivation of pressure from arriving momentum is the same argument with mass instead of $E/c$.