Editing Radiation pressure (section)

== Theory ==
{{See also|Electromagnetic radiation|Speed of light}}
Radiation pressure can be viewed as a consequence of the [[conservation of momentum]] given 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 ===
{{Main|Poynting vector}}
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 the [[Poynting vector]] {{nowrap|<math>\mathbf{S} = \mathbf{E} \times \mathbf{H}</math>}}, which is the [[cross product]] of the [[electric field]] vector ''E'' and the [[magnetic field]]'s auxiliary field vector (or ''[[Magnetic field#The H-field|magnetizing field]]'') ''H''.  The magnitude, denoted by ''S'', divided by the [[speed of light]] is the density of the linear momentum per unit area (pressure) of the electromagnetic field.  So, dimensionally, the Poynting vector is {{nowrap|1=''S'' = {{sfrac|power|area}} = {{sfrac|rate of doing work|area}} = {{sfrac|{{sfrac|Δ''F''|Δ''t''}} Δ''x''|area}}}}, which is the speed of light, {{nowrap|1=''c'' = Δ''x'' / Δ''t''}}, times pressure, {{nowrap|1=Δ''F'' / area}}.  That pressure is experienced as radiation pressure on the surface:
<math display="block"> P_\text{incident} = \frac{\langle S\rangle}{c} = \frac{I_f}{c}</math>
where <math>P</math> is pressure (usually in [[pascals]]), <math>I_f</math> is the incident [[irradiance]] (usually in W/m<sup>2</sup>) and <math>c</math> is the [[speed of light]] in vacuum.  Here, {{nowrap|{{sfrac|1|''c''}} ≈ {{val|3.34|u=N/GW}}}}.

If the surface is planar at an angle ''α'' to 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 of ''α'', resulting in a pressure:
<math display="block"> P_\text{incident} = \frac{I_f}{c} \cos^2 \alpha </math>

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.<ref name="Wright">{{citation | last = Wright | first = Jerome L. | date = 1992 | title = Space Sailing | publisher = Gordon and Breach Science Publishers}}</ref>

=== 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 is [[specularly 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:

<math display="block"> P_\text{emitted} = \frac{I_f}{c}</math>

thus ''doubling'' the net radiation pressure on the surface:

<math display="block"> P_\text{net} =   P_\text{incident} +  P_\text{emitted} = 2 \frac{I_f}{c}</math>

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 a [[Diffuse reflection|diffusely reflective]] surface, 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 emission ''in the direction normal to the surface'' ''I''<sub>e</sub>:
<math display="block"> P_\text{emitted} = \frac{I_\text{e}}{c}</math>

The emission can be from [[black-body radiation]] or 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 the [[Stefan–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 ===
{{See also|Photons|Momentum}}
Electromagnetic radiation can be [[wave–particle duality|viewed]] in terms of particles rather than waves; these particles are known as [[photons]]. 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:
<math display="block">
p = \dfrac{h}{\lambda} = \frac{E_p}{c},
</math>
where {{math|''p''}} is momentum, {{math|''h''}} is the [[Planck constant]], {{math|''λ''}} is [[wavelength]], and {{math|''c''}} is speed of light in vacuum. And {{math|''E<sub>p</sub>''}} is the energy of a single photon given by: 
<math display="block">
E_p = h \nu = \frac{h c}{\lambda}
</math>

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 {{math|''I<sub>f</sub>''}} over an area {{math|''A''}} has a power of  {{math|''I<sub>f</sub>A''}}, this implies a flux of {{math|''I<sub>f</sub>''/''E<sub>p</sub>''}} 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 the [[Electromagnetic stress–energy tensor#Algebraic properties|vanishing of the trace of the electromagnetic stress tensor]]: since this trace [[Stress–energy tensor#Stress–energy of a fluid in equilibrium|equals 3''P'' − ''u'']], we get
<math display="block">P = \frac{u}{3},</math>
where {{math|''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 in [[thermal equilibrium]] with its surroundings, at a temperature {{math|''T''}}: the body will be surrounded by a uniform radiation field described by the [[Planck law|Planck black-body radiation law]] and 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 total [[radiant energy]] per unit volume in the surrounding space.<ref>{{cite book | author = Shankar R. | title = Principles of Quantum Mechanics | edition = 2nd | url = https://www.fisica.net/mecanica-quantica/Shankar%20-%20Principles%20of%20quantum%20mechanics.pdf }}</ref><ref>{{cite book | last1 = Carroll | first1 = Bradley W | author2 = Dale A. Ostlie | title = An Introduction to Modern Astrophysics | edition = 2nd}}</ref><ref>{{cite book | last = Jackson | first = John David | year = 1999 | title = Classical Electrodynamics}}</ref><ref>Kardar, Mehran. "Statistical Physics of Particles".</ref>

By using [[Stefan–Boltzmann law]], this can be expressed as
<math display="block">P_\text{compress} = \frac{u}{3} = \frac{4\sigma}{3c} T^4,</math>
where <math>\sigma</math> is the [[Stefan–Boltzmann constant]].