The size the light cannot blow away

Light carries momentum, so it pushes. Whether it pushes an object away from a star depends on how the push compares with the star’s gravity, and the comparison has a property that is not shared by many comparisons in physics: it does not depend on where the object is.

Two things follow. The blow-out threshold is a band rather than a boundary — and light does not only push, since a beam with a gradient across it pulls a small grain up the gradient instead. And everything outside the band stays — where “stays” turns out to mean something other than remaining in place.

Both statements are ordinary consequences of two lines of arithmetic, and both are the opposite of what the usual account of radiation pressure suggests. The usual account has a threshold size below which grains are expelled, and a competition between two forces that presumably resolves differently in different places — with the pressure light exerts read as a push and nothing else. Neither survives being written out.

The two inverse squares #

The radiation force on a spherical grain of radius $a$ is the momentum flux times the cross-section it presents:

$$F_{\text{rad}} = \frac{L}{4\pi r^2 c}\,\pi a^2 Q_{\text{pr}},$$

and the gravitational force on the same grain is

$$F_{\text{grav}} = \frac{GM}{r^2}\cdot\frac{4}{3}\pi a^3\rho.$$

Both fall as the inverse square for one reason, and it is geometric rather than physical. The same rays cross shells whose area grows as $r^2$, so their density falls as $1/r^2$ — and that applies to the flux of light and to the flux of the gravitational field alike. Push the luminosity up until the first beats the second everywhere and there is a brightness a mass cannot exceed. The two forces are computed from two different fluxes with the same geometric dilution, which is what makes their ratio blind to distance and turns the whole question into one about the grain.

Dividing, the $r^2$ cancels:

$$\beta = \frac{F_{\text{rad}}}{F_{\text{grav}}} = \frac{3LQ_{\text{pr}}}{16\pi GMc\,\rho a}.$$

Nothing on the right is a position. A grain of a given material and size has the same $\beta$ at Mercury’s orbit and at Neptune’s and in the Oort cloud, and if the light wins it wins everywhere.

The threshold that matters is not $\beta = 1$ but $\beta = \tfrac12$. A grain released from a circular orbit — chipped off a parent body that was orbiting — has the orbital speed of that body, and with the effective central force reduced by a factor $(1-\beta)$ it is on an unbound orbit as soon as $\beta$ exceeds a half. Grains are not released from rest, so a half is the number.

The distance-independence deserves one more sentence, because it is more unusual than it looks. Almost every competition in physics has a crossover: two forces with different distance dependences are equal somewhere, and which one dominates is a statement about where the object is. Here there is no crossover at any distance, and the reason is that both quantities are fluxes from the same point diluted by the same geometry. The consequence is that the question “is this grain bound?” has an answer that can be printed on the grain, and that a system’s dust population is sorted by size rather than by radius — which is a very different kind of prediction from the usual one.

The efficiency, which is not one #

$Q_{\text{pr}}$ is the ratio of the momentum a grain actually takes out of a beam to the momentum crossing its geometric cross-section. For a grain much larger than the wavelength it is about one — the grain casts a shadow and takes the momentum in it. For a grain much smaller than the wavelength it is nowhere near one.

The scattering efficiency from the full Mie series falls steeply below a size parameter of about one — Rayleigh’s regime, where it goes as the fourth power of the size and is the reason the sky is blue — and settles toward a value of order one above it, with resonances on the way. Everything in this essay’s hero figure is that behaviour smoothed, and the steep part at small sizes is what puts a floor under the grain radius.

So $\beta \propto Q/a$ has two regimes. Above the wavelength, $Q$ is flat and $\beta$ rises as grains get smaller. Below it, $Q$ falls as $a^4$ and $\beta$ falls as $a^3$. In between is a maximum, and for silicate grains in sunlight it is at about 115 nm, where $\beta$ reaches 1.87.

That gives the band. Grains between 48 and 574 nanometres have $\beta > \tfrac12$ and leave; grains larger stay because the light has too little cross-section to work on; grains smaller stay because they barely interact with the light at all. A particle a tenth the size of the wavelength of light is essentially transparent to it.

There is a satisfying reversal in that. The naive picture has small grains as the ones most easily swept away, and the correct picture has the smallest grains as among the most secure — more secure than grains ten times their size. The reason is that being small has two opposite consequences, and they win at different scales: less mass to move, which helps the light, and less interaction with the light, which does not. Below the wavelength the second wins decisively, because it carries a fourth power against the first’s third.

The areal density that cancels the Sun #

The distance-independence has a use that turns the whole argument into a specification, and it is the cleanest statement of what a solar sail has to be.

Rewrite the balance for a flat sheet rather than a sphere. The radiation force per unit area is $L/4\pi r^2c$, doubled if the sheet reflects rather than absorbs; the gravitational force per unit area is $GM$ times the areal density $\sigma$ over $r^2$. Setting them equal, the $r^2$ cancels as before and what is left is a mass per unit area:

$$\sigma^* = \frac{L}{2\pi GMc} = 1.53\ \text{grams per square metre}.$$

That is a constant of the solar system. A perfectly reflecting sheet lighter than a gram and a half per square metre is pushed outward harder than the Sun pulls it, anywhere at all; one heavier is not, anywhere at all. There is no orbit at which the answer changes, no distance at which a heavy sail becomes light, and no manoeuvre that alters it.

The number is a hard target and nothing built has met it. The sails flown so far — a few tens of metres across, of aluminised polymer a couple of micrometres thick, with booms and attachments — come out at something like ten grams per square metre once everything is counted, so their ratio is around a tenth and the Sun’s gravity wins comfortably. That is not a failure: a sail with $\beta$ of a tenth still accelerates continuously and for free, and continuous free acceleration is worth a great deal even when it is small.

What the constant does is set what the ambition costs. Reaching $\beta$ above one — a sail that could leave the solar system on sunlight alone, from rest — requires the whole structure, film, reflective coating, booms, tethers and payload, to average under a gram and a half per square metre. The film alone is achievable; the film plus everything else is the problem, and it is the reason the interesting proposals all involve sails without booms.

The same force, one order further #

A grain that is not expelled is not therefore left alone. The radiation pressure calculated above is the zeroth-order term in the grain’s own speed, and the first-order term is a drag.

Aberration is the change in the apparent direction of radiation between two frames in relative motion, and for a dust grain the relevant version is the same transformation at $v/c \approx 10^{-4}$. That tilts the apparent direction of the Sun by about twenty arcseconds, so the radiation arrives slightly ahead of the grain’s motion rather than radially — and a force with a small component opposing the orbital velocity is a force that removes angular momentum. Twenty arcseconds is the whole mechanism.

In the grain’s frame the starlight does not arrive radially. It arrives from a direction tilted forward by $v/c$, so it delivers momentum with a small component opposing the grain’s orbital motion. The grain absorbs that momentum, and then re-emits the energy as thermal radiation — isotropically in its own frame, so carrying away no net momentum in that frame. The tangential kick is therefore never given back, and the grain loses angular momentum steadily.

The Doppler factor is the other half of the same transformation, and the two accounts have to agree. Describing the drag as an aberration of the incoming light and describing it as an anisotropic Doppler shift of the outgoing thermal emission are the same calculation done in two frames; they give the same answer, which is the usual test that a $v/c$ argument has been done correctly rather than plausibly.

It is worth being careful about why the re-emission matters, because the argument is often given without it and then does not work. If the grain simply scattered the light elastically, sending it back out in a pattern fixed in the star’s frame, there would be no net tangential force at all — what came in tilted would go out tilted, and the momentum would balance. The drag exists because the grain absorbs the energy, thermalises it, and radiates it again in a pattern that is isotropic in the grain’s frame. Isotropic in one frame is forward-beamed in another, so the outgoing radiation carries away net forward momentum, and the grain is slowed. Written that way the effect is a recoil from the grain’s own thermal emission rather than a push from the star, and the two descriptions give the same number.

The resulting inspiral time is

$$t_{\text{PR}} = \frac{c\,r^2}{4GM\beta} \approx \frac{400}{\beta}\left(\frac{r}{\text{AU}}\right)^2 \ \text{years},$$

which is short.

The dependence on distance is worth noting because it is the opposite of the blow-out criterion’s. The expulsion question has no distance in it at all; the drag question has $r^2$, so a grain at ten times the distance survives a hundred times as long. That is what makes the two effects sort a dust population in different directions: the light removes a band of sizes from everywhere at once, and the drag removes everything else, slowly, from the inside out.

Between the two turn-ups is a band of sizes that clears fastest, and it sits just outside the blow-out band. Grains slightly too large to be expelled are the ones the drag removes soonest — so a dust population is eaten from both sides of one narrow size range, with the very small and the very large surviving longest.

There is a second consequence of the drag that is easy to miss. It removes angular momentum without removing much energy, so the grain does not fall in — it spirals, staying on a nearly circular orbit that shrinks. That distinction matters because a spiralling grain passes slowly through every radius on its way in, spending longest at the largest radii, and a population being fed from outside therefore reaches a steady state whose surface density profile is set by the drag rather than by where the dust came from.

What that implies about any particular cloud of dust — whether it must be resupplied, and from what — is a question about that system rather than about this force, and belongs to the collection that owns the sky.

The same recoil on a rotating rock #

The drag above works by a body absorbing radiation and re-emitting it isotropically in its own frame. Relax the word “isotropically” and the same mechanism produces an effect on bodies far too large for radiation pressure to touch.

A rotating body heated on one side does not re-radiate symmetrically. It has thermal inertia, so it carries some of the absorbed heat round with it before radiating, and the hottest part of its surface is not the part facing the Sun but the part somewhat past local noon — afternoon rather than midday. The thermal emission is therefore strongest in a direction that is not radial, and the recoil from it has a component along or against the orbital motion depending on which way the body spins.

The result is a steady change in the orbit’s size: outward for a body rotating one way, inward for one rotating the other. It is minute — for a kilometre-sized asteroid it amounts to a few hundred metres of semi-major axis a year — and it acts for ever, so over millions of years it moves bodies across the asteroid belt and into the resonances that deliver them to the inner solar system. A substantial part of the account of where meteorites come from rests on it.

Two things make it worth putting beside the drag in this essay. It is the same physics — a recoil from a body’s own thermal emission, computed in the body’s frame — with the isotropy assumption dropped, so it is the drag’s first correction rather than a separate effect. And it has the opposite property to everything else here: its sign depends on the body’s rotation, which is a property nothing else in this essay depends on, so it can move a body outward as well as inward.

The same anisotropy applied to torque rather than force changes the spin instead of the orbit, which over long times can drive a small body’s rotation to the point where it flies apart. Both are one mechanism read for two quantities, and both are radiation pressure taken seriously about where the radiation actually leaves from.

Poynting’s derivation, which was made of the wrong stuff #

The drag is named for Poynting and Robertson, and the gap between them is thirty-four years and a change of the underlying physics.

Poynting worked it out in 1903, in the framework everybody had then, which included a luminiferous aether the light was supposed to be a disturbance of. The result he obtained — a tangential drag on an absorbing body in orbit, of the size given above — is the right one, and the reasoning he reached it by is built on a substance nobody believes in.

Robertson redid it in 1937 with special relativity, where the aberration and the Doppler shift are consequences of the transformation between frames rather than of motion through a medium, and got the same answer. The name carries both because the physics is Robertson’s and the effect is Poynting’s.

That is a pattern worth noticing rather than treating as an anecdote. A calculation that turns on the relative motion between a body and the light reaching it does not much care what the light is made of, so long as it travels at a definite speed and carries momentum. Poynting’s aether supplied a definite speed and a momentum flux, which was enough; what it also supplied — a preferred frame — did not enter the answer, because the answer depends only on the difference between two frames’ descriptions and not on which of them is at rest. Several results from the aether era survive for the same reason, and it is a fair test of a piece of nineteenth-century physics to ask whether its conclusion mentions the aether or only uses it.

Where the model stops #

The grain is a sphere of one material. Real dust is fluffy, aggregated and porous, which changes both the cross-section it presents and the density that appears in the denominator, generally in opposite directions. A very porous aggregate can have a $\beta$ several times a compact grain’s of the same mass.

The efficiency is an interpolation. The hero figure uses a smooth form with the two limits right and no resonances; the exact Mie series has structure near a size parameter of one, exactly where the band’s edges are. The band’s existence and its rough position are robust; its edges to two significant figures are not.

The star is a point. Close to a stellar surface the star subtends a large angle, the radiation is not radial, and the geometric factor in $\beta$ changes — which matters for grains within a few stellar radii and not at all for anything in an ordinary planetary system.

The light is taken as monochromatic. A star’s spectrum is broad, and $Q_{\text{pr}}$ has to be averaged over it weighted by the flux — which for a hot star pushes the relevant wavelength down and moves the band with it. A grain that is blown out of a system around an A star can be retained around an M dwarf, at the same size.

The drag calculation assumes the orbit stays nearly circular. It does, for the reason that the timescale above is enormously longer than an orbital period — millions of orbits — so the spiral is adiabatic and eccentricity is damped rather than excited. An impulsive process, such as a collision or a close passage, breaks that and the subsequent evolution is a different calculation.

And nothing here is a plasma effect. A charged grain in a magnetised wind feels a Lorentz force too, and for the smallest grains — the ones the efficiency argument says the light cannot touch — that force is likely to dominate everything on this page.

The Rayleigh spectrum’s fourth power is the same fourth power that makes the efficiency collapse for small grains, and putting the two side by side makes the connection explicit: the reason the sky is blue and the reason a nanometre-scale grain cannot be blown out of a solar system are one exponent, applied to the same quantity, at two ends of one curve.

One further limit is worth stating because it is the reason all of these numbers are for the solar system. Everything scales with the star’s luminosity-to-mass ratio, which appears in $\beta$ as $L/M$ and nowhere else. A star ten times the Sun’s mass has roughly a thousand times its luminosity, so $L/M$ is a hundred times larger, every $\beta$ is a hundred times larger, and the blow-out band widens enormously in both directions — around a hot enough star, essentially all dust is expelled. The whole of this essay is one ratio evaluated for one star, and reading it for another is a matter of multiplying by that star’s $L/M$ in solar units.

What the pictures cannot show #

The figures draw a force ratio and a timescale, and both are computed for a grain in a circular orbit around a point source in an otherwise empty system. Real dust is in a population with collisions between its members, and a collision resets everything — the fragments have new sizes, new $\beta$ values and new orbits. Nothing in a plot against grain radius can show a process that changes the radius.

Nor can any figure show the grain’s temperature, which is what fixes the re-emission the drag argument depends on. The whole calculation assumes the grain reaches thermal equilibrium with the starlight quickly compared with an orbit, which is true for grains above about a nanometre and false for smaller ones, where a single absorbed photon can spike the temperature and the emission is not steady at all.

Where the ladder goes next #

This ladder began with light having a pressure, where the two inverse squares cancel and a single blow-out size follows. This rung puts the efficiency into that ratio and reads the same force one order further in $v/c$. The rungs above it: the sail as an engineering object, and the mass-per-area limit that decides whether one can be built; optical tweezers, where a focused beam’s gradient rather than its flow supplies a restoring force; 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.

The habit worth carrying away is about what cancels. When two effects share a geometric dilution, their ratio becomes a property of the object rather than of the situation — and a ratio that is scale-free is worth looking for, because it converts a question about a system into a question about a material.