Illustrated Physics
physics, drawn
Concept

Radiation pressure — where it appears

The force light exerts on what absorbs or reflects it, equal to the momentum it delivers per unit time per unit area. It is the intensity divided by the speed of light for an absorber and twice that for a mirror, which is a few micropascals in full sunlight.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

astrophysics · Radiation pressure 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.

astrophysics · Radiation pressure 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.

astrophysics · Radiation pressure Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up.

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

thermodynamics · Blackbody 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.

astrophysics · Radiation pressure A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation 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.

astrophysics · Radiation pressure 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.

astrophysics · Radiation pressure What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry.

The rocket that leaves its fuel at home

A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

relativity · Mass-energy The bath does push back, by an unmeasurable amount. The retarding force on a perfectly absorbing body moving through isotropic radiation at 2.725 kelvin, against its speed, for three areas. Moving through a bath of radiation is not free: the radiation arriving from ahead is blue-shifted and more intense and the radiation from behind is red-shifted and weaker, so the body absorbs more momentum from the front than from the back and decelerates. A square metre at half the speed of light feels 3.7e-14 newtons. That is the reason a preferred frame exists without relativity being violated: the laws are the same in every frame and the radiation is not — it is a physical system with a state, and its state picks out the frame in which it is isotropic, exactly as a body of water does.

The bath that pushes back

Moving through a bath of radiation is not free. The light arriving from ahead is blue-shifted and more intense and the light from behind is weaker, so a body absorbs more momentum from the front than from the back and slows down. That drag picks out the frame in which the radiation is isotropic — without violating relativity, because the laws are the same in every frame and the radiation is not.

relativity · Relativistic thermodynamics

Named alongside it

The objects these essays reach for when they reach for this one.

BlackbodyDissipationBrownian motionCosmic microwave backgroundCross-sectionDoppler limitThe inverse-square lawLaser coolingMomentumMomentum conservationOptical molassesPhoton momentum

All concepts