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For fixed-composition monatomic ideal gas plus equilibrium radiation pressure, specific internal energy is $u=3\mathcal{R}T/2+a_{\mathrm{r}}{T}^4/\rho$. An isentropic process satisfies $du=Pd\rho /{\rho }^2$, giving ${\Gamma }_3-1=(8-6\beta )/(24-21\beta )$ andThe stellar gas-pressure fraction must be allowed to change along the adiabat. The pure-radiation and pure-gas limits give ${\Gamma }_1={\Gamma }_2=4/3$ and $5/3$, respectively. Partial ionization or different gas heat capacities change these formulas.

Arthur Ashkin demonstrated that radiation pressure from a continuous laser could accelerate and trap microscopic particles. This made transfer of optical momentum a tool for controlling matter rather than merely a correction to optical measurements. Later, he and collaborators demonstrated a single-beam gradient-force optical trap: a strongly focused beam held dielectric particles in water. The optical tweezers arrangement balances the restoring effect of the intensity gradient against the forces tending to push the particle away.^[ref][ref]

A physicist who researched nonlinear optics and optical trapping at Bell Laboratories.^[ref]

Arthur Ashkin's 1970 first experiment suspended transparent latex spheres in water inside a $120\mu \mathrm{m}$ glass cell. A horizontal $514.5\mathrm{nm}$ argon laser beam, focused to a $6.2\mu \mathrm{m}$ radius, illuminated individual beads viewed through a microscope. An off-centre $2.68\mu \mathrm{m}$ bead moved toward the beam axis and downstream until reaching the cell wall; blocking the beam released it into Brownian motion. Transparent particles and liquid reduced competing thermal effects. This observed sequence separates transverse attraction from downstream radiation pressure: wall-assisted confinement is not free three-dimensional confinement by one beam.^[ref]

The beam-blocking comparison is particularly informative: persistence against a wall while illuminated and release when illumination stops test an optical cause. To infer a force from the downstream velocity, one additionally needs a linear drag model and the bead's distance from the wall; the observed motion alone does not calibrate an unrestricted harmonic optical trap.

Black-body radiation is the electromagnetic radiation in thermal equilibrium with an ideal blackbody. It can be described as a photon gas: a cavity whose walls are all at the same temperature contains radiation with the universal Planck's law spectrum. A small opening in such a cavity approximates a blackbody emitter because incoming radiation undergoes repeated absorption inside. The radiation's equilibrium spectrum depends on temperature, rather than the wall material. ^[ref]

There are three useful quantities to keep distinct: directional spectral radiance $B_{\nu }$ or $B_{\lambda }$, total energy density $u$ in the cavity, and outward radiant exitance $j^{\ast }$ from its opening. Planck's law supplies the first; integrals over frequency and directions supply the others. For isotropy, every direction has the same $B_{\nu }$, so the full $4\pi$ solid angle givesHere $\hslash =h/(2\pi )$ is the reduced Planck constant, and the radiation constant is $a_{\mathrm{rad}}\approx 7.565733250\times 10^{-16}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$. The explicit Boltzmann constant makes this an SI formula; adopting natural units with $k_B=1$ recovers $u={\pi }^2{T}^4/(15{\hslash }^3{c}^3)$.

For a short integral outline, substitute $x=h\nu /(k_{B}T)$ and useThe expansion is a geometric series with nonnegative terms; repeated integration by parts evaluates each integral, and $\zeta (4)={\pi }^4/90$ is a value of the Riemann zeta function. This gives the coefficient above.

The photon gas also exerts radiation pressure, $p=u/3$. The factor $1/3$ comes from the angular average of the squared directional component of photon momentum, $⟨{\cos }^2\theta ⟩=1/3$. The thermodynamic derivation of blackbody radiation pressure gives the same result. In contrast, outgoing power involves one projected-direction factor over an outward hemisphere, yielding radiant exitanceThis is the Stefan–Boltzmann law with unit emissivity; $\sigma$ is the Stefan-Boltzmann constant. The factor $\pi$ follows from Lambert's cosine law, rather than the $4\pi$ full-sphere factor used for energy density.

For example, an ideal $6000\mathrm{K}$ cavity has $u\approx 0.980519\mathrm{J}{\mathrm{m}}^{-3}$, radiation pressure $p\approx 0.326840\mathrm{Pa}$, and outward radiant exitance $j^{\ast }\approx 7.34881\times 10^7\mathrm{W}{\mathrm{m}}^{-2}$. A stipulated $3000\mathrm{K}$ cavity has one-sixteenth each value because all three scale as $T^4$. These describe stored energy, mechanical pressure, and outgoing power per area, respectively. In a closed equilibrium cavity, equal opposing radiation balances the directional power flow even though $u$ and $p$ are nonzero.

The usefulness of black-body radiation is this common reference: Planck's law, Wien's displacement law, and the Stefan–Boltzmann law consistently connect spectral measurements, temperature, and total emission. The three illustrative curves in Planck's law are ideal equilibrium spectra; applying them to a material requires considering its emissivity and whether thermal equilibrium is a suitable approximation.

An orbit about a planet. Stellar tides, radiation pressure and other perturbations can limit its stability; a sufficiently small semi-major axis compared with the Hill sphere permits a local approximately planet-centred Kepler orbit.

An orbit about a star. Dust on such an orbit can experience radiation pressure and Poynting–Robertson drag, in addition to gravitational perturbations.

A comet is a small astronomical body containing volatile material and solid grains, capable of becoming active when heated. Its comet nucleus is the compact reservoir. As volatile ice undergoes sublimation, escaping gas carries dust outward and can make a diffuse cometary coma and a comet tail.^[ref]

A cometary dust tail responds strongly to radiation pressure, whereas ionized gas couples to the solar wind and magnetic fields. The visible tail is therefore not simply an exhaust trail opposite the direction of travel. An inactive distant comet can lack a prominent tail. Its Kepler orbit, later planetary scattering and material loss together determine the observable history.

A cometary dust tail is the extended distribution of solid grains released by a comet. Radiation pressure sorts its grains by size: high-$\beta$ grains are displaced rapidly in the anti-stellar direction, while larger low-$\beta$ grains remain closer to the projected orbit and form a more curved trail.

A comet tail is an extended structure formed from material released by a comet. Its dust tail is shaped largely by radiation pressure, while an ion tail is shaped by the solar wind and magnetic field. The cometary dust tail is the solid-grain component.

A debris disk is a circumstellar population of solid bodies and dust whose observable fine grains are maintained mainly by collisions. Grains scatter starlight and reradiate absorbed energy as thermal radiation, producing an excess above the stellar spectrum, often at infrared wavelengths.

Small grains can be removed by radiation pressure, Poynting–Robertson drag and collisions on times shorter than the system age, requiring replenishment by larger parent bodies. A collisional cascade models that transfer through grain sizes. The light measures emitting cross-section more directly than the mass in large unseen bodies, so dust brightness alone does not determine the full reservoir.

A dust birth ring is a belt of parent planetesimals whose collisions supply dust. Radiation pressure gives the fragments different orbits even when their release velocities match those of their parents.

A dust halo is an extended, low-density distribution of grains around a more concentrated debris disk or dust birth ring. Collisions in the parent belt release particles whose radiation pressure response differs from that of the larger parent bodies. Small grains can be placed on much more eccentric or unbound trajectories, spreading the observable material beyond the region where it was produced.

An image of an extended halo therefore need not imply a comparably extended population of parent planetesimals. Its brightness depends on the grain cross-sections, scattering or emission properties and residence times. Distinguishing the dust-producing belt from the visible grain distribution is essential when using a debris disk to infer unseen planets or the underlying collision rate.

A dust tail is an extended distribution of grains released from a localized moving source. Radiation pressure changes the grains' orbits; their release times, grain properties and survival probabilities determine the spatial profile.

An electromagnetic field carries momentum as well as energy. In vacuum, its SI momentum density iswhere $\mathbf{S}=\mathbf{E}\times \mathbf{B}/{\mu }_0$ is the Poynting vector. Thus the field momentum in a region is ${\mathbf{P}}_{\mathrm{field}}=\int \mathbf{g}dV$. ^[ref]

For a directed vacuum plane wave, $\mathbf{S}=cu\hat{\mathbf{n}}$, so a pulse with total energy $U$ carries momentum $U\hat{\mathbf{n}}/c$. Complete absorption transfers that momentum to matter. Reversing the pulse at a stationary perfect mirror transfers twice as much normal momentum, explaining radiation pressure.

Conservation joins field momentum to the mechanical momentum of matter and momentum crossing the chosen boundary. Changing a field can therefore accelerate matter even when no material particle carries the momentum across the intervening space. Static configurations can also contain nonzero field momentum; the vacuum formula does not require radiation. ^[ref]

The entropy density of a photon gas is its entropy per physical volume, $s=S/V$. For homogeneous equilibrium vacuum radiation with two states of electromagnetic polarization and zero photon chemical potential,The first equality is the entropy density at zero chemical potential, obtained from the thermodynamic Euler relation $U=TS-pV+{\mu }_{\gamma }{N}_{\gamma }$ with internal energy $U=uV$. The second uses the photon gas equation of state, $u=a_R{T}^4$ and $p=u/3$. If an independently constrained photon number gives a nonzero chemical potential, the general entropy density is $(u+p-{\mu }_{\gamma }{n}_{\gamma })/T$ and $u=a_R{T}^4$ need not hold. This distinction explains why the absence of a conserved number matters for entropy as well as for the spectral shape. ^[ref]

A short first law of thermodynamics check, for neighboring equilibrium states with ${\mu }_{\gamma }=0$, isThe thermodynamic Euler relation fixes the additive normalization. Equivalently, this radiation entropy tends to zero with temperature at fixed volume; no contribution from the containing walls is included. Interpreting $TdS$ as transferred heat and $pdV$ as boundary work requires a reversible thermodynamic process. For a reversible thermodynamic process that is also an adiabatic process, there is no entropy production or entropy exchange, soAn irreversible adiabatic process can produce entropy even without heat exchange, so the word adiabatic alone is insufficient to infer $T^{3}V=\text{constant}$.

For an analytic example, expand an initially equilibrated photon gas reversibly from $V_0$ to $8V_0$, keeping only the same two photon states of electromagnetic polarization. Its temperature halves, its energy density becomes $u_0/16$, its entropy density of a photon gas becomes $s_0/8$, and its total entropy stays $S_0$. The internal energy halves rather than remaining fixed: the missing half leaves as work against radiation pressure,This is a stipulated reversible radiation model, not a report of a piston experiment. Maintaining a real cavity along this path requires controlling wall heat exchange and including the wall heat capacity in any actual apparatus balance.

For a computed scale at stipulated $T=2.725\mathrm{K}$, $s\simeq 2.04122\times 10^{-14}\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$. Combining this with the photon number density givesThis is the total equilibrium radiation entropy divided by its mean photon number, not a separately assigned entropy carried by each labeled particle. During the ideal expansion, $n_{\gamma }\propto T^3$ and $T^{3}V$ is fixed, so the mean number $N_{\gamma }=n_{\gamma }V$ also stays constant. This follows from the path; it was not imposed as the independent number constraint used to establish chemical potential.

The beautiful connection is that expansion lowers each typical photon energy and dilutes the radiation in just the proportions needed to retain its equilibrium spectral shape and total entropy. For the cosmic microwave background, homogeneous expansion of the universe has physical volume proportional to the cube of the scale factor; fixed-species cosmological entropy conservation therefore gives $T\propto a^{-1}$. This describes preservation of an already established Planck photon distribution and connects the FIRAS spectral observation in photon gas to an entropy prediction; FIRAS did not perform a direct entropy measurement. When other equilibrated species disappear and transfer energy into radiation, use the effective number of relativistic entropy degrees of freedom, $g_{\ast s}$: the conserved quantity for the combined sector is $g_{\ast s}{T}^{3}V$, under the corresponding equilibrium and no-entropy production assumptions. A changing species population, separate decoupled sectors or entropy production requires an appropriate entropy balance rather than the pure photon gas formula. ^[ref]

The Finson–Probstein model follows cometary dust after release with negligible ejection speed under stellar gravity reduced by radiation pressure. A grid of synchrones and syndynes organizes the resulting dust-tail morphology.

For a blackbody grain with geometric cross-section $\sigma$, mass $m$ and stellar-frame velocity $\mathbf{v}$, define the radiation-pressure coefficient $\beta =L_{\star }\sigma /(4\pi cG{M}_{\star }m)$. To first order in $v/c$, steady absorption and isotropic rest-frame reradiation giveThe relative incoming flux provides $1-\dot{r}/c$; photon reradiation carries stellar-frame momentum proportional to $\mathbf{v}/c$. Subtracting static radiation pressure gives radial drag $-2m\beta G{M}_{\star }\dot{r}/(c{r}^2)$ and transverse drag $-m\beta G{M}_{\star }{v}_{\theta }/(c{r}^2)$. Non-blackbody grains require their actual radiation-pressure efficiency.

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.

Joseph A. Burns coauthored a study of radiation pressure and radiation-associated orbital drag on small Solar-System particles. Radiation pressure and radiation-associated orbital drag connect the microscopic exchange of photon momentum with long-term dust dynamics.^[ref]

Laser cooling uses laser light to reduce the random velocities of chemical atoms, and therefore their translational temperature. The striking ingredient is selective photon absorption: an atom moving towards a beam sees a Doppler shift that can make it absorb more strongly. Each absorbed photon transfers momentum along the beam; repeated absorption and spontaneous emission can then remove kinetic energy. Doppler cooling makes this selection with the atom's motion, while sub-Doppler cooling also exploits its internal quantum states.

Cooling makes chemical atoms move slowly enough for longer interrogation in atomic clocks and atom interferometry, and easier loading into an optical trap. The same radiation pressure that pushes an atom can cool it: with suitably tuned opposing beams, whichever direction the atom moves, it receives the stronger push against that motion.