Illustrated Physics
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Series

Radiation pressure — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

    Light has a pressure

    Sunlight pushes on a square metre with about the weight of a grain of sand, which sounds like a curiosity until the object being pushed is small enough. The demonstration in every school cupboard turns the wrong way, and the reason it does is more interesting than the effect it is supposed to show.

    part 1 · astrophysics
  2. The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals.

    The size the light cannot blow away

    Radiation pressure and gravity both fall as the inverse square of distance, so their ratio is a property of the grain and not of where it is. What follows is a band of sizes that get blown out — with a lower edge as well as an upper one — and a drag, on everything else, that is the same pressure read one order further in v/c.

    part 2 · astrophysics
  3. The brightest anything of a given mass can be. The Eddington luminosity against mass, with the main sequence drawn beside it. Radiation pushes outward on the electrons and gravity pulls inward on the protons, and both go as one over the distance squared — so the radius cancels out of the comparison entirely, checked here at three radii spanning four decades and coming out identical to 1e-20. What is left is a luminosity: L = 4πGMc/κ, which is 1.47 × 10³¹ watts per solar mass, or 3.8·10⁴ solar luminosities. Above it, radiation drives the outer layers away faster than gravity can hold them. The Sun is at 2.6e-5 of its own limit and in no danger; a star of 10 solar masses is at 1.2e-2; and a star of 100 is at 0.83, which is why the two lines converge at the top of the chart and why the most massive stars known are a few hundred solar masses rather than a few thousand. They do not fail to form for lack of gas; they blow away the gas that would have made them heavier, and once formed they shed mass continuously in a radiation-driven wind. The limit is the same expression for an accreting black hole, where it caps not the brightness but the rate at which mass can be taken on.

    The brightness a mass cannot exceed

    Light pushes outward on the electrons of a star and gravity pulls inward on its protons, and both forces fall off as one over the distance squared. The distance therefore cancels, and what is left is a limit on brightness rather than on size — 3.8 × 10⁴ times the Sun's luminosity for every solar mass, above which a body drives its own outer layers away.

    part 3 · astrophysics
  4. A force that points across the beam, not along it. The intensity of a beam focused to a waist of 0.5 µm, and the force it exerts on a 60 nm polystyrene sphere in water, across the beam. The force is not a pressure and does not point along the light: it is the pull on an induced dipole sitting in a non-uniform field, proportional to the gradient of the intensity rather than to the intensity, and it points up the gradient — towards the bright axis from either side. It vanishes exactly on the axis, which is what makes the axis a trap rather than a place. The well is 47 times kT deep at 100 milliwatts, which is why the particle stays: thermal motion explores it and does not escape it. The stiffness near the bottom is 3.070 piconewtons per micrometre, and a particle in a spring that stiff wanders 36 nanometres from centre.

    The light that pulls rather than pushes

    Radiation pressure is momentum arriving, and it points along the beam. A gradient of intensity does something else entirely — it polarises a particle and then pulls the induced dipole up the gradient — so a focused beam holds a particle at its waist against the push, and the force that does it is not a pressure at all.

    part 4 · astrophysics
  5. Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds.

    The friction made of light

    Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

    part 5 · astrophysics
  6. A hill that is always ahead of the atom. The light-shift potentials of the two ground sublevels of a spin-½ atom in two counter-propagating beams with crossed linear polarisations, over one and a half wavelengths, in units of the well depth. The polarisation of the light turns from σ+ to linear to σ− every quarter wavelength, and the two sublevels see sinusoidal potentials a quarter wavelength out of step. Optical pumping transfers the atom from one sublevel to the other fastest exactly where its own potential is highest — the fastest pumping and the hilltop coincide, which the figure checks — and that is the bottom of the other potential. The heavy line is an atom that starts with 3.3 well depths of kinetic energy: each time it reaches a hilltop it is pumped down by exactly one well depth, climbs the next hill, and is pumped down again, 3 times, until it no longer has the energy to reach a hilltop and is left oscillating in one well. The energy is carried off by the pumping photon, which leaves bluer than the light that drove it by the depth of the well.

    The limit that belonged to a simpler atom

    The theory of laser cooling predicted a floor, and the first careful measurement came in six times below it. Nothing was wrong with the measurement or the arithmetic. The floor belonged to an atom with one ground state, and real atoms have several — which lets the light build a hill in front of every atom, move it to the bottom before it can roll back, and repeat the trick until the atom is a few microkelvin from rest.

    part 6 · astrophysics

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