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For normal incidence in vacuum, a uniform beam delivers energy $IA\Delta t$ to an illuminated surface area $A$ during time $\Delta t$. A perfect absorber receives momentum $IA\Delta t/c$, so force divided by surface area gives $p_{\mathrm{abs}}=I/c$. A perfect stationary mirror sends the photons back with reversed momentum, so momentum conservation gives twice the normal transfer, $p_{\mathrm{refl}}=2I/c$. These are time-averaged pressures; the microscopic arrivals need not be uniform.

For a stipulated model light intensity $I=1000\mathrm{W}{\mathrm{m}}^{-2}$ and the exact SI speed of light $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$,These are computed predictions, not measured data or a universal sunlight value. If the beam fully covers a normally oriented surface area $A=0.010{\mathrm{m}}^2$, the intercepted radiant flux is $\Phi =IA=10\mathrm{W}$ and the absorber's force is $\Phi /c\simeq 33.356\mathrm{nN}$. Using the total source power would be wrong if only part of its beam reaches the receiver. The absorption prediction assumes complete photon absorption, while the reflection prediction assumes complete specular reflection; transmission, scattering of light and directed thermal photon emission require their own outgoing momentum accounting.

Measure the angle of incidence $\theta$ between the incoming path and the surface's normal line, with $0\le \theta <\pi /2$. There are two separate cosine factors. A fixed actual surface area $A$ intercepts only its projected area $A\cos \theta$, so $\Phi =IA\cos \theta$. Each incoming photon supplies normal momentum $(E_{\gamma }/c)\cos \theta$ on absorption, or twice that on specular reflection. Thus, for radiometric absorptance $a$ and reflected power fraction $R$,This model assumes transmitted light continues straight in the same medium and carries away its original momentum. It omits thermal recoil and assumes a stationary receiver. At $\theta =60^{\circ }$, fixed beam light intensity therefore gives one quarter of the normal-incidence pressure. For fixed intercepted radiant flux $\Phi$, however, the normal force has only one cosine factor. Equivalently, if the given quantity is receiving-surface irradiance $E_{\mathrm{inc}}=I\cos \theta$, then $p_{\perp }=(a+2R)E_{\mathrm{inc}}\cos \theta /c$. The distinction matters when interpreting an experiment's angle scan. ^[ref]

Absorption also transfers a tangential momentum component at oblique incidence. It contributes a tangential force, rather than normal pressure. Ideal specular reflection preserves the tangential momentum component, so it transfers only normal momentum.

For radiation in vacuum with isotropy, energy is distributed equally per solid angle: write $u_{\Omega }=u/(4\pi )$ for energy density per unit solid angle. Let $\theta$ be the angle between a ray's path and the wall normal. Its normal velocity $c\cos \theta$ determines the arrival rate, while normal momentum per energy is $\cos \theta /c$. Their product is ${\cos }^2\theta$. For a perfectly reflecting cavity wall,The leading two is the normal momentum reversal; the incoming hemisphere contains half of the field's directional energy density. This is also the normal component of the bulk pressure tensor, $P_{nn}={\int }_{4\pi }{u}_{\Omega }{\cos }^2\theta d\Omega =u/3$. Isotropy makes all three diagonal components equal, while their sum is $u$ because the squared direction components sum to one. The factor one third therefore depends on three-dimensional angular geometry, not on the shape of the blackbody radiation spectrum.

A perfectly absorbing wall exposed to this incoming hemisphere receives only $u/6$ from absorption. If it is in thermal equilibrium with the isotropic radiation, its outgoing emission supplies another $u/6$ of recoil, recovering $u/3$. A cold absorber before equilibrium is a different boundary condition. Similarly, the one-way incident energy flux is $cu/4$, whereas the net energy flux across a plane in the undisturbed isotropic field is zero. Dividing $cu/4$ by $c$ misses the extra normal-momentum projection and does not give the cavity pressure.

For a stipulated isotropic radiation energy density $u=6\mu \mathrm{J}{\mathrm{m}}^{-3}$, the bulk pressure is $2\mu \mathrm{Pa}$ regardless of its spectral shape. In thermal equilibrium, blackbody radiation supplies $u=a_{\mathrm{rad}}{T}^4$ with radiation constant $a_{\mathrm{rad}}=4\sigma /c$, so $p_{\mathrm{rad}}=a_{\mathrm{rad}}{T}^4/3$; Stefan–Boltzmann law defines $\sigma$. The equilibrium photon gas derives this same equation of state from its free energy.^[ref]

Radiation reaction is the influence on a source's motion of its own electromagnetic field, including the balance of radiated energy and momentum with changes of the near field. For sufficiently weak radiation loss, an averaged power balance can give the leading damping law without solving the full self-field dynamics. Replacing a constant-frequency radiation formula by its instantaneous slowly varying value requires an additional time-scale assumption; it is not an exact treatment of arbitrarily rapid motion changes.

A radiative-convective climate column applies a radiative-convective equilibrium model to the vertical thermal structure relevant to climate change. Radiative transfer determines heating and cooling; convective adjustment represents unresolved vertical redistribution when the chosen atmospheric lapse rate is too steep. Radiative equilibrium alone need not give a physically admissible lower-atmospheric profile.

In 1967, Syukuro Manabe and Richard T. Wetherald computed equilibrium with prescribed relative humidity, allowing atmospheric water vapor to change with temperature. Their average-cloud case warmed by $2.36\mathrm{K}$ for carbon dioxide increasing from 300 to 600 parts per million, versus $1.33\mathrm{K}$ with fixed absolute humidity. These are model outputs, not measured warming. The comparison isolates an important water-vapor feedback.^[ref]

An original two-temperature calculation exposes the bookkeeping. Let an opaque surface absorb solar irradiance $Q$ and emit blackbody thermal radiation $\sigma T_s^4$. A single isothermal atmospheric layer transmits all sunlight and absorbs a fraction $ϵ$ of surface infrared emission, emitting $ϵ\sigma T_a^4$ both upward and downward. This is a grey thermal-band emissivity, with $0<ϵ\le 1$. Let $H$ be upward nonradiative heat flux, and let $C_s,C_a>0$ be heat capacities per unit area. Then conservation of energy givesThe two equations sum toHere $L$ is outgoing longwave radiation. Both $H$ and downward atmospheric emission cancel internally: they redistribute energy without creating it. The factor two in atmospheric emission counts two faces. Dropping the downward face would spoil the surface budget. Each right-hand term has units $\mathrm{W}{\mathrm{m}}^{-2}$; each $C$ has units $\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

Choose $Q=240\mathrm{W}{\mathrm{m}}^{-2}$, $H=80\mathrm{W}{\mathrm{m}}^{-2}$ and $ϵ=1$. At equilibrium, $L=\sigma T_a^4=240\mathrm{W}{\mathrm{m}}^{-2}$ and $\sigma T_s^4=400\mathrm{W}{\mathrm{m}}^{-2}$, giving $T_a\simeq 255.06\mathrm{K}$ and $T_s\simeq 289.81\mathrm{K}$. Direct substitution gives surface balance $240+240-400-80=0$ and atmospheric balance $400-480+80=0$. These deliberately chosen fluxes illustrate the model; they are not an observed global budget or a reconstruction of the 1967 calculation. Prescribing $H$ illustrates heat transfer accounting but does not close a realistic convection law.

For fixed $Q,H,ϵ$, the positive-temperature equilibrium is locally stable. Put $a=4\sigma T_s^3>0$ and $b=4ϵ\sigma T_a^3>0$. The Jacobian matrix of the temperature equations has trace $-a/C_s-2b/C_a<0$ and determinant $ab(2-ϵ)/(C_s{C}_a)>0$. Thus both eigenvalues have negative real part. Temperature-dependent climate feedbacks alter this test. A single grey layer also omits the optical spectrum, cloud structure and the independent thermal behavior of the troposphere and stratosphere.

The radiative efficiency is the escaping radiative power divided by an adopted accreted rest-mass energy rate. Its meaning depends on where the mass rate is measured and which emission is counted. For efficient Newtonian accretion onto a stellar surface it scales as $GM/(R_{\ast }{c}^2)$, whereas advection or outflows can reduce luminosity for the same supplied mass.

Transfer of energy by thermal radiation. A diffuse grey surface of area $A$ and emissivity $ϵ$ completely enclosed by isothermal black surroundings has net outward power $ϵ\sigma A(T_s^4-T_e^4)$. This subtracts absorbed environmental radiation from emitted radiation. Other geometries require exchange factors; a gross emitted power is not a net cooling rate.

Depositing supernova energy into dense unresolved gas can radiate it away before a numerically resolved expansion develops. Equal injected energy therefore need not imply equal outflow momentum or heating at different resolution. A feedback model should represent the coupling and cooling phase it intends to approximate, and comparisons should include gas recycling and the resulting stellar and circumgalactic properties.

Section 62.1 distinguishes inner bremsstrahlung from structure-dependent radiative pion decay and their interference. Equations (62.3)–(62.4) give the double differential decay width in terms of the scaled photon and charged-lepton energies and the squared lepton-to-meson mass ratio.

A charged pion decays to a charged antilepton, its associated neutrino, and a photon. Inner bremsstrahlung from charged external states interferes with structure-dependent radiation. An experimentally resolved photon needs an energy cut: the real-radiation rate alone diverges in the soft limit, whereas a suitably inclusive combination with virtual radiative corrections is finite.^[ref]

A shock wave with significant radiation losses in its downstream evolution. Often a narrow dissipative jump initially heats the gas according to adiabatic Rankine-Hugoniot conditions, followed by a much longer radiative cooling layer. The gas's specific entropy increases at the jump and may then fall as radiation carries entropy away. If both distant states have equal temperature, the combined mass and momentum balances give an isothermal shock; the gas's energy flux is not conserved across the whole radiating region.

A radiofrequency field induces transitions from trapped weak-field-seeking magnetic sublevels to untrapped states at a resonance surface. Atoms with sufficient energy reach that surface and leave. Sweeping the frequency changes the energy cutoff without equating every lost atom with the sample-average energy.

When the radiofrequency drive is much faster than the Paul trap secular motion and Paul trap micromotion is small, averaging the oscillatory force produces the displayed effective potential. The electric field here is its peak oscillating amplitude. It is an energy, not a static electrostatic potential, so confinement does not contradict the static electrostatic no-trapping theorem. Excess Paul trap micromotion from displacement from the radiofrequency null is distinct from the intended slow Paul trap secular motion.

Radiometry measures the energy and power carried by electromagnetic radiation, including light. It asks how much radiation a source supplies, where it travels, and how much reaches a receiving surface. The measurements are physical rather than weighted by the sensitivity of the human eye: equal radiant fluxes at different wavelengths need not look equally bright. ^[ref].

The five quantities below answer different questions about the same radiation. The International System of Units makes the distinctions visible: a watt is a joule per second, while a steradian measures solid angle. ^[ref].

Resolving wavelength or frequency adds a further “per unit interval”, as in spectral radiance. The notation is consistent with NIST's table of spectral quantities. In particular, radiant flux is power, whereas the existing astronomical radiative flux is power per area. ^[ref].

This bookkeeping makes an optical camera's collected signal calculable from a source's radiance and the geometry of the receiving aperture. It also lets radiation measurements from different instruments be compared. For example, NIST describes separate calibrations of spectral radiance and spectral irradiance for instruments used to observe Earth. ^[ref].

Place two equal point masses of $1\mathrm{kg}$ each on opposite sides of a fixed origin, at equal radius $r$, and let them share a signed angular velocity $\omega =\dot{\theta }$ in a plane. Their total mass is $M=2\mathrm{kg}$ and their moment of inertia about the perpendicular axis is $I=M{r}^2$. Radial pulling forces have zero torque about that origin, soTake an initial radius $r_i=1\mathrm{m}$ and angular velocity ${\omega }_i=1rad/s$, and a final radius $r_f=0.5\mathrm{m}$. ThenandThese are endpoint states with ${\dot{r}}_i={\dot{r}}_f=0$. During the pull, the full kinetic energy also contains radial motion: $K=M{\dot{r}}^2/2+K_{\mathrm{rot}}$. Assume the pulling apparatus supplies all net work to the two point masses; the work-energy theorem then fixes its net work at $+3\mathrm{J}$, including any radial acceleration and braking. Each mass spirals inward during the pull, allowing its inward force to do positive work. Conservation of angular momentum therefore predicts the final angular velocity without requiring kinetic energy conservation. For an apparatus included in a larger system, its stored energy supplies the increase consistently with conservation of energy. OpenStax uses the same mechanism to explain a skater pulling her arms inward. ^[ref]

On a nonradial zero-energy Kepler orbit, conserved specific angular momentum $h$ gives periapsis $r_p=h^2/(2GM)$ and ${\dot{r}}^2=2GM(r-r_p)/r^2$. Integrating with $s=\sqrt{r-r_p}$ yields $t-t_p=\pm \sqrt{2/(GM)}(r+2r_p)\sqrt{r-r_p}/3$. Squaring gives the displayed version of the Barker equation, valid on both incoming and outgoing branches. The time origin $t_p$ is passage through periapsis, not an arbitrary point on the orbit.

A logarithmic comparison of specific power against specific energy for storage or conversion devices. With consistent mass and energy units, constant-duration lines satisfy $p_s=e_s/t$. The plotted capability depends on operating conditions and mass convention.

Chandrasekhara Venkata Raman and Kariamanikkam Srinivasa Krishnan jointly reported the effect in 1928. Their letter describes light from the Sun concentrated by an $18\mathrm{cm}$-aperture, $230\mathrm{cm}$-focal length telescope objective and a second optical lens onto a liquid purified by repeated vacuum distillation. With blue-violet and yellow-green optical filters together in the incident beam, the visible track disappeared; moving the yellow-green optical filter between liquid and observer restored it. The complementary arrangement rejected unchanged-colour scattering of light while admitting newly generated light. Persistence after purification and strong wave polarization in many liquids argued against ordinary fluorescent contamination.^[ref]

In his 1930 lecture, Chandrasekhara Venkata Raman describes a cobalt-glass optical filter selecting indigo-violet light from the Sun, with a separated blue-green band in the scattered optical spectrum. Substituting a quartz mercury arc and condenser gave sharp new spectral lines. Small Hilger quartz spectrographs recorded them photographically; Kariamanikkam Srinivasa Krishnan obtained more precise records with a larger instrument and established violet-displaced lines. The optical prism disperses wavelengths for spatial readout; it does not create the Raman shift. The diagram gives functional optical paths rather than a reconstruction of a specific Hilger instrument.^[ref]

Their May 1928 report strengthens the comparison: filtered mercury illumination of toluene produced extra spectral lines. Removing the $4358\text{A˚}$ incident line removed its associated longer-wavelength group; a quinine filter removing the $4047$, $4078$ and $4109\text{A˚}$ group removed a different shifted group while preserving the one associated with $4358\text{A˚}$. These are actual selective-excitation controls, rather than inferred colours.^[ref]

The inference has limits. A complementary-filter test alone establishes newly admitted wavelengths, not their precise spacing or a unique microscopic mechanism. Fluorescence can also survive purification when intrinsic to the liquid, and fluorescence can be polarized. Useful distinctions therefore combine resolved spectral lines, excitation dependence, wave polarization and sample preparation. A proposed modern extension is to change a single incident frequency: a molecular Raman shift should follow the same internal energy gap, whereas ordinary relaxed molecular fluorescence usually retains approximately its emission-band positions. This is a diagnostic under stated conditions, not a universal theorem about every fluorescent material or resonant scattering process. Raman scattering need not involve a populated, relaxed electronic excited state as ordinary molecular fluorescence does; neither a redder photon nor the word Stokes identifies the mechanism by itself.

The 1930 Nobel Prize in Physics went to Chandrasekhara Venkata Raman alone for his work on scattering of light and the discovery of the effect. That award allocation should be distinguished from the joint experimental report with Kariamanikkam Srinivasa Krishnan.^[ref]

Raman scattering is inelastic scattering of light in which the outgoing photon has a different frequency because matter gains or loses internal energy. Molecular vibration is a particularly useful recipient: a visible photon can reveal an internal energy spacing normally associated with infrared frequencies. Rayleigh scattering supplies the much stronger, approximately unchanged-frequency reference. The distinction concerns the transferred energy, rather than merely a change of direction or the colour of the illuminating beam.

The elementary bookkeeping is conservation of energy:Here $h$ is the Planck constant, ${\nu }_0$ and ${\nu }_s$ are the incident and scattered photon frequencies, and $E_i,E_f$ are initial and final internal molecular energies. This expression neglects translational recoil and Doppler broadening; those can matter in sufficiently precise gas measurements. A positive internal energy change gives Stokes Raman scattering, and a negative one gives Anti-Stokes Raman scattering. The energy difference determines a possible line position, but it does not by itself establish that a transition is optically allowed or predict its strength.

The spontaneous molecular process developed here has a connection to the Raman transition used in atomic measurement methods: two photon interactions transfer an internal energy difference. Coherently driving that process with two applied light fields is a different experimental regime from collecting weak spontaneous Raman scattering beside an illuminating beam.

The signed Raman shift subtracts scattered from incident spectroscopic wavenumber:Here $c$ is the speed of light, and both wavelengths are in vacuum. Positive shifts denote Stokes Raman scattering; negative shifts denote Anti-Stokes Raman scattering. Some plots reverse the horizontal direction or report separate unsigned magnitudes, so check the convention. Spectroscopic wavenumber has no $2\pi$ factor, unlike the common angular wavenumber $k=2\pi /\lambda$. A shift of $1000\mathrm{c}{\mathrm{m}}^{-1}$ means $100000{\mathrm{m}}^{-1}$, not a frequency of $1000\mathrm{Hz}$.

For a one-quantum change of molecular vibration with ordinary frequency ${\nu }_v$, the magnitude is ${\tilde{\nu }}_v={\nu }_v/c$. This makes Raman spectroscopy attractive: the same molecular energy spacing can be read using different visible excitation wavelengths. Equal and opposite Raman shifts are not equal and opposite changes measured in nanometres, because wavelength is the reciprocal of spectroscopic wavenumber.

Raman spectroscopy measures shifted scattering of light to infer internal energy spacings and the molecular response that makes them visible. In a modern illustrative arrangement, narrowband laser illumination replaces the historical arc, an optical filter rejects the strong excitation line, and a spectrograph resolves the weak shifted spectral lines. The illumination, rejection and spectral dispersion are separate jobs. Increasing rejection can conceal small Raman shifts close to the excitation line.

A proposed measurement should record the excitation wavelength, a dark signal, a blank container and the liquid, then compare calibrated line positions and integrated signals under the same collection conditions. Measure optical filter transmission and detector response before using the Thermal Raman line ratio for temperature. Repeat at lower incident power to look for heating or altered sample response. These are suggested checks, not an assertion that the 1928 apparatus implemented modern calibration procedures.

Its appeal is the conversion of internal molecular energy differences into visible spectral lines: changing the excitation can move the whole pattern while preserving characteristic Raman shifts. Combined with vibrational infrared activity, Raman vibrational activity supplies complementary information about molecular vibration. A line pattern can constrain a material's identity or environment, but overlapping bands, fluorescence backgrounds and the finite spectral resolution element prevent an unqualified identification from one peak alone.