Radiation pressure

Radiation pressure(also known aslight pressure) is mechanicalpressureexerted upon a surface due to the exchange ofmomentumbetween the object and theelectromagnetic field.This includes the momentum of light orelectromagnetic radiationof anywavelengththat isabsorbed,reflected,or otherwise emitted (e.g.black-body radiation) by matter on any scale (from macroscopic objects to dust particles to gas molecules).^[1]^[2]^[3]The associated force is called theradiation pressure force,or sometimes just theforce of light.

Force on a reflector results from reflecting the photon flux

The forces generated by radiation pressure are generally too small to be noticed under everyday circumstances; however, they are important in some physical processes and technologies. This particularly includes objects inouter space,where it is usually the main force acting on objects besides gravity, and where the net effect of a tiny force may have a large cumulative effect over long periods of time. For example, had the effects of the Sun's radiation pressure on the spacecraft of theVikingprogrambeen ignored, the spacecraft would have missed Mars orbit by about 15,000 km (9,300 mi).^[4]Radiation pressure from starlight is crucial in a number ofastrophysicalprocesses as well. The significance of radiation pressure increases rapidly at extremely high temperatures and can sometimes dwarf the usualgas pressure,for instance, instellar interiorsandthermonuclear weapons.Furthermore, large lasers operating in space have been suggested as a means of propelling sail craft inbeam-powered propulsion.

Radiation pressure forces are the bedrock oflaser technologyand the branches of science that rely heavily on lasers and otheroptical technologies.That includes, but is not limited to,biomicroscopy(where light is used to irradiate and observe microbes, cells, and molecules),quantum optics,andoptomechanics(where light is used to probe and control objects like atoms, qubits and macroscopic quantum objects). Direct applications of the radiation pressure force in these fields are, for example,laser cooling(the subject of the 1997Nobel Prize in Physics),^[5]quantum controlof macroscopic objects andatoms(2012 Nobel Prize in Physics),^[6]interferometry(2017 Nobel Prize in Physics)^[7]andoptical tweezers(2018 Nobel Prize in Physics).^[8]

Radiation pressure can equally well be accounted for by considering the momentum of a classical electromagnetic field or in terms of the momenta ofphotons,particles of light. The interaction of electromagnetic waves or photons with matter may involve an exchange ofmomentum.Due to the law ofconservation of momentum,any change in the total momentum of the waves or photons must involve an equal and opposite change in the momentum of the matter it interacted with (Newton's third law of motion), as is illustrated in the accompanying figure for the case of light being perfectly reflected by a surface. This transfer of momentum is the general explanation for what we term radiation pressure.

Discovery

Soviet rublecoin commemorating Lebedev's discovery

Johannes Keplerput forward the concept of radiation pressure in 1619 to explain the observation that a tail of acometalways points away from the Sun.^[9]

The assertion that light, aselectromagnetic radiation,has the property ofmomentumand thus exerts apressureupon any surface that is exposed to it was published byJames Clerk Maxwellin 1862, and proven experimentally by Russian physicistPyotr Lebedevin 1900^[10]and byErnest Fox NicholsandGordon Ferrie Hullin 1901.^[11]The pressure is very small, but can be detected by allowing the radiation to fall upon a delicately poised vane of reflective metal in aNichols radiometer(this should not be confused with theCrookes radiometer,whose characteristic motion isnotcaused by radiation pressure but by air flow caused by temperature differentials.)

Theory

Radiation pressure can be viewed as a consequence of theconservation of momentumgiven the momentum attributed to electromagnetic radiation. That momentum can be equally well calculated on the basis of electromagnetic theory or from the combined momenta of a stream of photons, giving identical results as is shown below.

Radiation pressure from momentum of an electromagnetic wave

According to Maxwell's theory of electromagnetism, an electromagnetic wave carries momentum. Momentum will be transferred to any surface it strikes that absorbs or reflects the radiation.

Consider the momentum transferred to a perfectly absorbing (black) surface. The energy flux (irradiance) of a plane wave is calculated using thePoynting vector $\mathbf {S} =\mathbf {E} \times \mathbf {H}$ ,which is thecross productof theelectric fieldvectorEand themagnetic field's auxiliary field vector (ormagnetizing field)H.The magnitude, denoted byS,divided by thespeed of lightis the density of the linear momentum per unit area (pressure) of the electromagnetic field. So, dimensionally, the Poynting vector isS=⁠power/area⁠=⁠rate of doing work/area⁠=⁠⁠ΔF/Δt⁠Δx/area⁠,which is the speed of light,c= Δx/ Δt,times pressure,ΔF/ area.That pressure is experienced as radiation pressure on the surface: $P_{\text{incident}}={\frac {\langle S\rangle }{c}}={\frac {I_{f}}{c}}$ where $P$ is pressure (usually inpascals), $I_{f}$ is the incidentirradiance(usually in W/m²) and $c$ is thespeed of lightin vacuum. Here,⁠1/c⁠≈3.34 N/GW.

If the surface is planar at an angleαto the incident wave, the intensity across the surface will be geometrically reduced by the cosine of that angle and the component of the radiation force against the surface will also be reduced by the cosine ofα,resulting in a pressure: $P_{\text{incident}}={\frac {I_{f}}{c}}\cos ^{2}\ Alpha$

The momentum from the incident wave is in the same direction of that wave. But only the component of that momentum normal to the surface contributes to the pressure on the surface, as given above. The component of that force tangent to the surface is not called pressure.^[12]

Radiation pressure from reflection

The above treatment for an incident wave accounts for the radiation pressure experienced by a black (totally absorbing) body. If the wave isspecularly reflected,then the recoil due to the reflected wave will further contribute to the radiation pressure. In the case of a perfect reflector, this pressure will be identical to the pressure caused by the incident wave:

$P_{\text{emitted}}={\frac {I_{f}}{c}}$

thusdoublingthe net radiation pressure on the surface:

$P_{\text{net}}=P_{\text{incident}}+P_{\text{emitted}}=2{\frac {I_{f}}{c}}$

For a partially reflective surface, the second term must be multiplied by the reflectivity (also known as reflection coefficient of intensity), so that the increase is less than double. For adiffusely reflectivesurface, the details of the reflection and geometry must be taken into account, again resulting in an increased net radiation pressure of less than double.

Radiation pressure by emission

Just as a wave reflected from a body contributes to the net radiation pressure experienced, a body that emits radiation of its own (rather than reflected) obtains a radiation pressure again given by the irradiance of that emissionin the direction normal to the surfaceI_e: $P_{\text{emitted}}={\frac {I_{\text{e}}}{c}}$

The emission can be fromblack-body radiationor any other radiative mechanism. Since all materials emit black-body radiation (unless they are totally reflective or at absolute zero), this source for radiation pressure is ubiquitous but usually tiny. However, because black-body radiation increases rapidly with temperature (as the fourth power of temperature, given by theStefan–Boltzmann law), radiation pressure due to the temperature of a very hot object (or due to incoming black-body radiation from similarly hot surroundings) can become significant. This is important in stellar interiors.

Radiation pressure in terms of photons

Electromagnetic radiation can beviewedin terms of particles rather than waves; these particles are known asphotons.Photons do not have a rest-mass; however, photons are never at rest (they move at the speed of light) and acquire a momentum nonetheless which is given by: $p={\dfrac {h}{\lambda }}={\frac {E_{p}}{c}},$ where $p$ is momentum, $h$ is thePlanck constant, $λ$ iswavelength,and $c$ is speed of light in vacuum. And $E p$ is the energy of a single photon given by: $E_{p}=h\nu ={\frac {hc}{\lambda }}$

The radiation pressure again can be seen as the transfer of each photon's momentum to the opaque surface, plus the momentum due to a (possible) recoil photon for a (partially) reflecting surface. Since an incident wave of irradiance $I f$ over an area $A$ has a power of $I f A$ ,this implies a flux of $I f / E p$ photons per second per unit area striking the surface. Combining this with the above expression for the momentum of a single photon, results in the same relationships between irradiance and radiation pressure described above using classical electromagnetics. And again, reflected or otherwise emitted photons will contribute to the net radiation pressure identically.

Compression in a uniform radiation field

In general, the pressure of electromagnetic waves can be obtained from thevanishing of the trace of the electromagnetic stress tensor:since this traceequals 3P−u,we get $P={\frac {u}{3}},$ where $u$ is the radiation energy per unit volume.

This can also be shown in the specific case of the pressure exerted on surfaces of a body inthermal equilibriumwith its surroundings, at a temperature $T$ :the body will be surrounded by a uniform radiation field described by thePlanck black-body radiation lawand will experience a compressive pressure due to that impinging radiation, its reflection, and its own black-body emission. From that it can be shown that the resulting pressure is equal to one third of the totalradiant energyper unit volume in the surrounding space.^[13]^[14]^[15]^[16]

By usingStefan–Boltzmann law,this can be expressed as $P_{\text{compress}}={\frac {u}{3}}={\frac {4\sigma }{3c}}T^{4},$ where $\sigma$ is theStefan–Boltzmann constant.

Solar radiation pressure

Solar radiation pressure is due to the Sun's radiation at closer distances, thus especially within theSolar System.(The radiation pressure of sunlight on Earth is very small: it is equivalent to that exerted by the weight of about a milligram on an area of 1 square metre, or 10 μN/m².)^{[citation needed]}While it acts on all objects, its net effect is generally greater on smaller bodies, since they have a larger ratio of surface area to mass. All spacecraft experience such a pressure, except when they are behind the shadow of a largerorbiting body.

Solar radiation pressure on objects near the Earth may be calculated using the Sun'sirradianceat 1AU,known as thesolar constant,orG_SC,whose value is set at 1361W/m²as of 2011.^[17]

All stars have aspectral energy distributionthat depends on their surface temperature. The distribution is approximately that ofblack-body radiation.This distribution must be taken into account when calculating the radiation pressure or identifying reflector materials for optimizing asolar sail,for instance.

Momentary or hours long solar pressures can indeed escalate due to release ofsolar flaresandcoronal mass ejections,but effects remain essentially immeasureable in relation to Earth's orbit. However these pressures persist over eons, such that cumulatively having produced a measureable movement on the Earth-Moon system's orbit.

Pressures of absorption and reflection

Solar radiation pressure at the Earth's distance from the Sun, may be calculated by dividing thesolar constantG_SC(above) by thespeed of lightc.For an absorbing sheet facing the Sun, this is simply:^[18] $P={\frac {G_{\text{SC}}}{c}}\approx 4.5\cdot 10^{-6}~{\text{Pa}}=4.5~\mu {\text{Pa}}.$

This result is inpascals,equivalent to N/m²(newtonsper square meter). For a sheet at an angleαto the Sun, the effective areaAof a sheet is reduced by a geometrical factor resulting in a forcein the direction of the sunlightof: $F={\frac {G_{\text{SC}}}{c}}(A\cos \ Alpha ).$

To find the component of this force normal to the surface, another cosine factor must be applied resulting in a pressurePon the surface of: $P={\frac {F\cos \ Alpha }{A}}={\frac {G_{\text{SC}}}{c}}\cos ^{2}\ Alpha.$

Note, however, that in order to account for the net effect of solar radiation on a spacecraft for instance, one would need to consider thetotalforce (in the direction away from the Sun) given by the preceding equation, rather than just the component normal to the surface that we identify as "pressure".

The solar constant is defined for the Sun's radiation at the distance to the Earth, also known as oneastronomical unit(au). Consequently, at a distance ofRastronomical units (Rthus being dimensionless), applying theinverse-square law,we would find: $P={\frac {G_{\text{SC}}}{cR^{2}}}\cos ^{2}\ Alpha.$

Finally, considering not an absorbing but a perfectly reflecting surface, the pressure isdoubleddue to the reflected wave, resulting in: $P=2{\frac {G_{\text{SC}}}{cR^{2}}}\cos ^{2}\ Alpha.$

Note that unlike the case of an absorbing material, the resulting force on a reflecting body is given exactly by this pressure acting normal to the surface, with the tangential forces from the incident and reflecting waves canceling each other. In practice, materials are neither totally reflecting nor totally absorbing, so the resulting force will be a weighted average of the forces calculated using these formulas.

Calculated solar radiation pressure on perfect reflector at normal incidence (α= 0)
Distance from Sun	Radiation pressure in μPa (μN/m²)
0.20 au	227
0.39 au (Mercury)	59.7
0.72 au (Venus)	17.5
1.00 au (Earth)	9.08
1.52 au (Mars)	3.93
3.00 au (typicalasteroid)	1.01
5.20 au (Jupiter)	0.34

Radiation pressure perturbations

Solar radiation pressure is a source oforbital perturbations.It significantly affects the orbits and trajectories of small bodies including all spacecraft.

Solar radiation pressure affects bodies throughout much of the Solar System. Small bodies are more affected than large ones because of their lower mass relative to their surface area. Spacecraft are affected along with natural bodies (comets, asteroids, dust grains, gas molecules).

The radiation pressure results in forces and torques on the bodies that can change their translational and rotational motions. Translational changes affect the orbits of the bodies. Rotational rates may increase or decrease. Loosely aggregated bodies may break apart under high rotation rates. Dust grains can either leave the Solar System or spiral into the Sun.^[19]

A whole body is typically composed of numerous surfaces that have different orientations on the body. The facets may be flat or curved. They will have different areas. They may have optical properties differing from other aspects.

At any particular time, some facets are exposed to the Sun, and some are in shadow. Each surface exposed to the Sun is reflecting, absorbing, and emitting radiation. Facets in shadow are emitting radiation. The summation of pressures across all of the facets defines the net force and torque on the body. These can be calculated using the equations in the preceding sections.^[12]^[18]

TheYarkovsky effectaffects the translation of a small body. It results from a face leaving solar exposure being at a higher temperature than a face approaching solar exposure. The radiation emitted from the warmer face is more intense than that of the opposite face, resulting in a net force on the body that affects its motion.^[20]

TheYORP effectis a collection of effects expanding upon the earlier concept of the Yarkovsky effect, but of a similar nature. It affects the spin properties of bodies.^{[citation needed]}

ThePoynting–Robertson effectapplies to grain-size particles. From the perspective of a grain of dust circling the Sun, the Sun's radiation appears to be coming from a slightly forward direction (aberration of light). Therefore, the absorption of this radiation leads to a force with a component against the direction of movement. (The angle of aberration is tiny, since the radiation is moving at the speed of light, while the dust grain is moving many orders of magnitude slower than that.) The result is a gradual spiral of dust grains into the Sun. Over long periods of time, this effect cleans out much of the dust in the Solar System.

While rather small in comparison to other forces, the radiation pressure force is inexorable. Over long periods of time, the net effect of the force is substantial. Such feeble pressures can produce marked effects upon minute particles likegas ionsandelectrons,and are essential in the theory of electron emission from the Sun, ofcometarymaterial, and so on.

Because the ratio of surface area to volume (and thus mass) increases with decreasing particle size, dusty (micrometre-size) particles are susceptible to radiation pressure even in the outer Solar System. For example, the evolution of theouter rings of Saturnis significantly influenced by radiation pressure.

As a consequence of light pressure,Einstein^[21]in 1909 predicted the existence of "radiation friction", which would oppose the movement of matter. He wrote: "radiation will exert pressure on both sides of the plate. The forces of pressure exerted on the two sides are equal if the plate is at rest. However, if it is in motion, more radiation will be reflected on the surface that is ahead during the motion (front surface) than on the back surface. The backward acting force of pressure exerted on the front surface is thus larger than the force of pressure acting on the back. Hence, as the resultant of the two forces, there remains a force that counteracts the motion of the plate and that increases with the velocity of the plate. We will call this resultant 'radiation friction' in brief."

Solar sails

Solar sailing, an experimental method ofspacecraft propulsion,uses radiation pressure from the Sun as a motive force. The idea of interplanetary travel by light was mentioned byJules Vernein his 1865 novelFrom the Earth to the Moon.

A sail reflects about 90% of the incident radiation. The 10% that is absorbed is radiated away from both surfaces, with the proportion emitted from the unlit surface depending on the thermal conductivity of the sail. A sail has curvature, surface irregularities, and other minor factors that affect its performance.

The Japan Aerospace Exploration Agency (JAXA) has successfully unfurled a solar sail in space, which has already succeeded in propelling its payload with theIKAROSproject.

Cosmic effects of radiation pressure

Radiation pressure has had a major effect on the development of the cosmos, from the birth of the universe to ongoing formation of stars and shaping of clouds of dust and gasses on a wide range of scales.^[22]

Early universe

Thephoton epochis a phase when the energy of the universe was dominated by photons, between 10 seconds and 380,000 years after theBig Bang.^[23]

Galaxy formation and evolution

ThePillars of Creationclouds within theEagle Nebulashaped by radiation pressure and stellar winds.

The process ofgalaxy formation and evolutionbegan early in the history of the cosmos. Observations of the early universe strongly suggest that objects grew from bottom-up (i.e., smaller objects merging to form larger ones). As stars are thereby formed and become sources of electromagnetic radiation, radiation pressure from the stars becomes a factor in the dynamics of remainingcircumstellarmaterial.^[24]

Clouds of dust and gases

Thegravitational compressionof clouds of dust and gases is strongly influenced by radiation pressure, especially when the condensations lead to star births. The larger young stars forming within the compressed clouds emit intense levels of radiation that shift the clouds, causing either dispersion or condensations in nearby regions, which influences birth rates in those nearby regions.

Clusters of stars

Stars predominantly form in regions of large clouds of dust and gases, giving rise tostar clusters.Radiation pressure from the member stars eventually disperses the clouds, which can have a profound effect on the evolution of the cluster.

Manyopen clustersare inherently unstable, with a small enough mass that theescape velocityof the system is lower than the averagevelocityof the constituent stars. These clusters will rapidly disperse within a few million years. In many cases, the stripping away of the gas from which the cluster formed by the radiation pressure of the hot young stars reduces the cluster mass enough to allow rapid dispersal.

A protoplanetary disk with a cleared central region (artist's conception).

Star formation

Star formationis the process by which dense regions withinmolecular cloudsininterstellar spacecollapse to formstars.As a branch ofastronomy,star formation includes the study of theinterstellar mediumandgiant molecular clouds(GMC) as precursors to the star formation process, and the study ofprotostarsandyoung stellar objectsas its immediate products. Star formation theory, as well as accounting for the formation of a single star, must also account for the statistics ofbinary starsand theinitial mass function.

Stellar planetary systems

Planetary systemsare generally believed to form as part of the same process that results instar formation.Aprotoplanetary diskforms by gravitational collapse of amolecular cloud,called asolar nebula,and then evolves into a planetary system by collisions and gravitational capture. Radiation pressure can clear a region in the immediate vicinity of the star. As the formation process continues, radiation pressure continues to play a role in affecting the distribution of matter. In particular, dust and grains can spiral into the star or escape the stellar system under the action of radiation pressure.

Comet Hale–Bopp(C/1995 O1). Radiation pressure and solar wind effects on the dust and gas tails are clearly seen.

Stellar interiors

Instellarinteriors the temperatures are very high. Stellar models predict a temperature of 15 MK in the center of theSun,and at the cores ofsupergiantstars the temperature may exceed 1 GK. As the radiation pressure scales as the fourth power of the temperature, it becomes important at these high temperatures. In the Sun, radiation pressure is still quite small when compared to the gas pressure. In the heaviest non-degenerate stars, radiation pressure is the dominant pressure component.^[25]

Comets

Solar radiation pressure strongly affectscomet tails.Solar heating causes gases to be released from thecomet nucleus,which also carry away dust grains. Radiation pressure andsolar windthen drive the dust and gases away from the Sun's direction. The gases form a generally straight tail, while slower moving dust particles create a broader, curving tail.

Laser applications of radiation pressure

Optical tweezers

Laserscan be used as a source of monochromatic light with wavelength $\lambda$ .With a set of lenses, one can focus the laser beam to a point that is $\lambda$ in diameter (or $r=\lambda /2$ ).

The radiation pressure of aP= 30 mW laser withλ= 1064 nm can therefore be computed as follows.

Area: $A=\pi \left({\frac {\lambda }{2}}\right)^{2}\approx 10^{-12}{\text{ m}}^{2},$

force: $F={\frac {P}{c}}={\frac {30{\text{ mW}}}{299792458{\text{ m/s}}}}\approx 10^{-10}{\text{ N}},$

pressure: $p={\frac {F}{A}}\approx {\frac {10^{-10}{\text{ N}}}{10^{-12}{\text{ m}}^{2}}}=100{\text{ Pa}}.$

This is used to trap or levitate particles inoptical tweezers.

Light–matter interactions

In this optomechanical cavity, light is trapped and enhanced between two mirrors. One of the mirrors is attached to a spring and can move. The radiation pressure force of the light circulating in the cavity can damp or amplify the oscillation of the mirror on the spring.

The reflection of a laser pulse from the surface of an elastic solid can give rise to various types of elastic waves that propagate inside the solid or liquid. In other words, the light can excite and/or amplify motion of, and in, materials. This is the subject of study in the field of optomechanics. The weakest waves are generally those that are generated by the radiation pressure acting during the reflection of the light. Such light-pressure-induced elastic waves have for example observed inside an ultrahigh-reflectivitydielectric mirror.^[26]These waves are the most basic fingerprint of a light-solid matter interaction on the macroscopic scale.^[27]In the field ofcavityoptomechanics, light is trapped and resonantly enhanced inoptical cavities,for example between mirrors. This serves the purpose of gravely enhancing thepowerof the light, and the radiation pressure it can exert on objects and materials. Optical control (that is, manipulation of the motion) of a plethora of objects has been realized: from kilometers long beams (such as in theLIGO interferometer)^[28]to clouds of atoms,^[29]and frommicro-engineered trampolines^[30]tosuperfluids.^[31]^[32]

In this optomechanical system, the radiation pressure force is leveraged to detect a singleprotein molecule.Laser light interacts with aglass sphere:the radiation pressure force causes it to vibrate. The presence of a single molecule on the sphere disturbs that (thermal) vibration, and the disturbance in the sphere's motion can be detected in theoscillator spectrumon the left.^[33]

Opposite to exciting or amplifying motion, light can also damp the motion of objects.Laser coolingis a method of cooling materials very close toabsolute zeroby converting some of material's motional energy into light.Kinetic energyandthermal energyof the material are synonyms here, because they represent the energy associated withBrownian motionof the material. Atoms traveling towards a laser light source perceive adoppler effecttuned to the absorption frequency of the target element. The radiation pressure on the atom slows movement in a particular direction until the Doppler effect moves out of the frequency range of the element, causing an overall cooling effect.^[34]

An other active research area of laser–matter interaction is the radiation pressure acceleration of ions or protons from thin–foil targets.^[35]High ion energy beams can be generated for medical applications (for example in ion beam therapy^[36]) by the radiation pressure of short laser pulses on ultra-thin foils.

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