Force on a reflector results from reflecting the photon flux

Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. Radiation pressure implies an interaction between electromagnetic radiation and bodies of various types, including clouds of particles or gases. The interactions can be absorption, reflection, or some of both (the common case). Bodies also emit radiation and thereby experience a resulting pressure.

The forces generated by radiation pressure are generally too small to be detected under everyday circumstances; however, they do play a crucial role in some settings, such as astronomy and astrodynamics. For example, had the effects of the sun's radiation pressure on the spacecraft of the Viking program been ignored, the spacecraft would have missed Mars orbit by about 15,000 kilometers.[1]

## Discovery

Johannes Kepler put forward the concept of radiation pressure back in 1619 to explain the observation that a tail of a comet always points away from the Sun.[2]

The assertion that light, as electromagnetic radiation, has the property of momentum and thus exerts a pressure upon any surface exposed to it was published by James Clerk Maxwell in 1862, and proven experimentally by Russian physicist Pyotr Lebedev in 1900[3] and by Ernest Fox Nichols and Gordon Ferrie Hull in 1901.[4] The pressure is very feeble, but can be detected by allowing the radiation to fall upon a delicately poised vane of reflective metal in a Nichols radiometer (this should not be confused with the Crookes radiometer, whose characteristic motion is not caused by radiation pressure but by impacting gas molecules).

## Theory

Radiation pressure can be analyzed as interactions by either electromagnetic waves or particles (photons). The waves and photons both have the property of momentum, which allows their interchangeability under classical conditions.

### Radiation pressure in classical electromagnetism: waves

Main article: Poynting vector

According to Maxwell's theory of electromagnetism, an electromagnetic wave carries momentum, which can be transferred to a reflecting or absorbing surface hit by the wave.

The energy flux (intensity) is expressed by the Poynting vector $\mathbf{S} = \mathbf{E}\times\mathbf{H}$, whose magnitude we denote by S. S divided by the square of the speed of light in free space is the density of the linear momentum of the electromagnetic field. The time-averaged intensity $\langle\mathbf{S}\rangle$ divided by the speed of light in free space is the radiation pressure exerted by an electromagnetic wave on the surface of a target, if the wave is completely absorbed:

$P_{absorb}=\frac{\langle S\rangle}{c} = \frac{E_f}{c}$   ( N·m−2 or Pa )

where P is pressure, Ef is energy flux (intensity) in W/m2, c is speed of light in vacuum.

If the absorbing surface is planar at an angle α to the radiation source, the intensity across the surface will be reduced:

$P_{absorb} = \frac{E_f}{c} \cos \alpha$   ( N·m−2 or Pa )

### Radiation pressure by particle model: photons

Electromagnetic radiation is quantized in particles called photons, the particle aspect of its wave–particle duality. Photons are best explained by quantum mechanics. Although photons are considered to be zero-rest mass particles, they have the properties of energy and momentum, thus exhibit the property of mass as they travel at light speed. The momentum of a photon is given by:

$p = \dfrac{h}{\lambda} = mc$

where p is momentum, h is Planck's constant, λ is wavelength, m is mass, and c is speed of light in vacuum. This expression shows the wave–particle duality.

$E = mc^2 = pc$

is the mass-energy relationship where E is the energy. Then

$p = \dfrac{E}{c}$

The generation of radiation pressure results from the momentum property of photons, specifically, changing the momentum when incident radiation strikes a surface. The surface exerts a force on the photons in changing their momentum by Newton's Second Law. A reactive force is applied to the body by Newton's Third Law.

The orientation of a reflector determines the component of momentum normal to its surface, and also affects the frontal area of the surface facing the energy source. Each factor contributes a cosine function, reducing the pressure on the surface.[5] The pressure experienced by a perfectly reflecting planar surface is then:

$P_{reflect} = \frac{2E_f}{c} \cos^2 \alpha$   ( N·m−2 or Pa )

where P is pressure, Ef is the energy flux (intensity) in W/m2, c is speed of light in vacuum, α is the angle between the surface normal and the incident radiation.[6]

Bodies radiate thermal energy according to their temperature. The emissions are electromagnetic radiation, and therefore have the properties of energy and momentum. The energy leaving a body tends to reduce its temperature. The momentum of the radiation causes a reactive force, expressed as a pressure across the radiating surface.

The Stefan–Boltzmann law describes the power radiated from a black body. The law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or emissive power) is directly proportional to the fourth power of the body's absolute temperature. The emissions from 'gray' bodies can be approximated by this law.

The emissions by other bodies are treated in an empirical manner, relying on in particular the coefficient of emission (emissivity), which is determined by measurements.

A body that does not absorb all incident radiation (sometimes known as a gray body) emits less total energy than a black body and is characterized by an emissivity, $\varepsilon < 1$, so the emitted energy flux (intensity) is:

$E_f = \varepsilon\sigma T^{4}$   ( J·s−1·m−2 or W·m−2 )

where $\sigma$ is the Stefan–Boltzmann constant and $T$ is absolute temperature. The emissivity depends on the wavelength, $\varepsilon=\varepsilon(\lambda).$

The radiation pressure on an emitting surface by emitted radiation is then:

$P_{emission} = \frac {E_f}{c} = \frac {\varepsilon\sigma}{ c } T^{4}$   ( N·m−2 or Pa )

### Moderating factors

Several factors affect the radiation pressure on a body or a cloud of particles or gases. The most prominent are the surface reflectivity, absorptivity, and emissivity. The values of these parameters vary across the spectrum, so a representative value is typically used in calculations. Calculations are also affected by surface curvature and roughness on a wide range of scales. Rotation of a body can also be an important factor.

### Compression in a uniform radiation field

A body in a uniform radiation field (equal intensities from all directions) will experience a compressive pressure. It may be shown by electromagnetic theory, by quantum theory, or by thermodynamics, making no assumptions as to the nature of the radiation (other than isotropy), that the pressure against a surface exposed in a space traversed by radiation uniformly in all directions is equal to one third of the total radiant energy per unit volume within that space.[7][8][9][10]

Quantitatively, this can be expressed as [11]

$P_{compress} = \frac{u}{3} = \frac{4\sigma}{3c} T^4$   ( N·m−2 or Pa )

for a radiation energy density $u$ ( J·m−3 ). The second equality holds if we are considering uniform thermal radiation at a temperature $T$. There $\sigma$ is the Stefan–Boltzmann constant and $c$ is the speed of light in vacuum.

Solar radiation pressure is exerted by solar radiation on objects within the solar system. While it acts on all bodies within the system, the smaller bodies are most affected. All spacecraft experience the pressure.

Solar radiation pressure is calculated on an irradiance (solar constant or radiant flux) value of 1361 W/m2 at 1 AU, as revised in 2011.[12]

All stars have a spectral energy distribution that depends on their surface temperature. The distribution is approximately that of black-body radiation. This distribution is important in selecting reflector materials best suited for the application.

### Pressures of absorption and reflection

Solar radiation pressure is calculated from the solar constant. It varies inversely by the square of the distance from the sun. The pressure experienced by a perfectly absorbing planar surface that may be at an angle to the source is:

$P_{absorb} = \frac{W}{c R^2} \cos \alpha$   ( N·m−2 or Pa )
$P_{absorb} = \frac{4.54}{R^2} \cos \alpha$   ( μN·m−2 or μPa )

The pressure experienced by a perfectly reflecting planar surface is:

$P_{reflect} = \frac{2W}{c R^2} \cos^2 \alpha$   ( N·m−2 or Pa )
$P_{reflect} = \frac{9.08}{R^2} \cos^2 \alpha$   ( μN·m−2 or μPa )

where P is pressure, W is the solar constant ( W·m−2 ), c is speed of light in vacuum, R is solar distance in AU treated as a dimensionless number, and α is the angle between the surface normal and the incident radiation.[6][13]

Solar Distance µPa (µN/m2)
0.20 AU = close 227
0.39 AU = Mercury 60.6
0.72 AU = Venus 17.4
1.00 AU = Earth 9.08
1.52 AU = Mars 3.91
3.00 AU = asteroid 1.01
5.20 AU = Jupiter 0.34

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

Solar radiation pressure affects bodies throughout much of the Solar System. Small bodies are more affected than large because of their lower mass and inertial properties. 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.

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 facets.

At any particular time, some facets will be exposed to the Sun and some will be in shadow. Each surface exposed to the Sun will be reflecting, absorbing, and emitting radiation. Facets in shadow will be emitting radiation. The summation of pressures across all of the facets will define the net force and torque on the body. These can be calculated using the equations in the preceding sections.[6][13]

The Yarkovsky effect affects 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 will be more intense than that of the opposite face, resulting in a net force on the body that will affect its motion.

The YORP effect is 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.

The Poynting–Robertson effect applies 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 extremely small 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 slow 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 are able to produce marked effects upon minute particles like gas ions and electrons, and are important in the theory of electron emission from the Sun, of cometary material, 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 the outer rings of Saturn is significantly influenced by radiation pressure.

### Solar sails

Main article: Solar sail

Solar sailing, an experimental method of spacecraft propulsion, uses radiation pressure from the Sun as a motive force. The idea of interplanetary travel by light was mentioned by Jules Verne in From 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 radiated 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 the IKAROS project.

## 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.

### The early universe

The photon epoch is a phase when the energy of the universe was dominated by photons, between 10 seconds and 380,000 years after the Big Bang.

### Galaxy formation and evolution

The process of galaxy formation and evolution began 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).

Early in the universe, galaxies were composed mostly of gas and dark matter. As a galaxy gained mass by accretion of smaller galaxies, the dark matter stayed mostly in the outer parts of the galaxy. The gas, however, contracted, causing the galaxy to rotate faster, until the result was a thin, rotating disk.

Astronomers do not currently know what process stopped the contraction. Theories of galaxy formation are not successful at producing the rotation speed and size of disk galaxies. It has been suggested that the radiation from bright newly formed stars, or from an active galactic nuclei, could have slowed the contraction of a forming disk. It has also been suggested that the dark matter halo could pull on galactic matter, stopping disk contraction.

### Clouds of dust and gases

The Pillars of Creation clouds within the Eagle Nebula shaped by radiation pressure and stellar winds.

The gravitational compression of 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 to star clusters. Radiation pressure from the member stars eventually disperses the clouds, which can have a profound effect on the evolution of the cluster.

Many open clusters are inherently unstable, with a small enough mass that the escape velocity of the system is lower than the average velocity of 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.

### Star formation

Star formation is the process by which dense regions within molecular clouds in interstellar space collapse to form stars. As a branch of astronomy, star formation includes the study of the interstellar medium and giant molecular clouds (GMC) as precursors to the star formation process, and the study of protostars and young stellar objects as its immediate products. Star formation theory, as well as accounting for the formation of a single star, must also account for the statistics of binary stars and the initial mass function.

### Stellar planetary systems

A protoplanetary disk with a cleared central region.

Planetary systems are generally believed to form as part of the same process that results in star formation. A protoplanetary disk forms by gravitational collapse of a molecular cloud, called a solar 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.

### Stellar interiors

In stellar interiors the temperatures are very high. Stellar models predict a temperature of 15 MK in the center of the Sun, and at the cores of supergiant stars 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.[14]

### Comets

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

Solar radiation pressure strongly affects comet tails. Solar heating causes gases to be released from the comet nucleus, which also carry away dust grains. Radiation pressure and solar wind then 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

Laser cooling is applied to cooling materials very close to absolute zero. Atoms traveling towards a laser light source perceive a doppler effect tuned 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.

Large lasers operating in space have been suggested as a means of propelling sail craft in beam-powered propulsion.

The reflection of a laser pulse from the surface of an elastic solid gives rise to various types of elastic waves that propagate inside the solid. The weakest waves are generally those that are generated by the radiation pressure acting during the reflection of the light. Recently, such light-pressure-induced elastic waves were observed inside an ultrahigh-reflectivity dielectric mirror.[15] These waves are the most basic fingerprint of a light-solid matter interaction on the macroscopic scale.

## References

1. ^ Eugene Hecht, "Optics", 4th edition
2. ^
3. ^ P. Lebedev, 1901, "Untersuchungen über die Druckkräfte des Lichtes", Annalen der Physik, 1901
4. ^ Nichols, E.F & Hull, G.F. (1903) The Pressure due to Radiation, The Astrophysical Journal,Vol.17 No.5, p.315-351
5. ^ T. Požar (2014), Oblique reflection of a laser pulse from a perfect elastic mirror. Optics Letter 39 (1), 48-51
6. ^ a b c Wright, Jerome L. (1992), Space Sailing, Gordon and Breach Science Publishers
7. ^ Shankar R., Principles of Quantum Mechanics, 2nd edition.
8. ^ Carroll, Bradley W. & Dale A. Ostlie, An Introduction to Modern Astrophysics, 2nd edition.
9. ^ Jackson, John David, (1999) Classical Electrodynamics.
10. ^ Kardar, Mehran. "Statistical Physics of Particles".
11. ^ Planck's law
12. ^ Kopp, G.; Lean, J. L. (2011). "A new, lower value of total solar irradiance: Evidence and climate significance". Geophysical Research Letters 38.
13. ^ a b Georgevic, R. M. (1973) "The Solar Radiation Pressure Forces and Torques Model", The Journal of the Astronautical Sciences, Vol. 27, No. 1, Jan–Feb. First known publication describing how solar radiation pressure creates forces and torques that affect spacecraft.
14. ^ Dale A. Ostlie and Bradley W. Carroll, An Introduction to Modern Astrophysics (2nd edition), page 341, Pearson, San Francisco, 2007
15. ^ T. Požar and J. Možina (2013), Measurement of elastic waves induced by the reflection of light. Physical Review Letters 111 (18), 185501