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A solid angle measures how much of the surrounding directions a region occupies, as seen from a specified point $O$. Draw a sphere of radius $R$ centered at $O$ and mark where those directions meet its surface. If the marked patch has surface area $A_R$, thenA Euclidean similarity scales both $A_R$ and $R^2$ by the same factor, so the chosen radius does not affect the answer. On the unit two-sphere, the numerical value is simply the patch's surface area. Unlike an angle drawn in one plane, a solid angle measures coverage in two angular directions.

For example, the full sphere has surface area $4\pi R^2$ and hence solid angle $4\pi$ steradians. A hemisphere occupies $2\pi$ steradians. Three mutually perpendicular coordinate planes divide the directions into eight congruent pieces, so each octant occupies $4\pi /8=\pi /2$ steradians. A solid angle of one steradian therefore covers $1/(4\pi )\approx 7.96\%$ of all directions, rather than one full turn.

This makes a geometric question answer a collection question. For an ideal point emitter with isotropy, equal solid angles carry equal radiant flux. A detector accepting directions of total solid angle $\Omega$ intercepts the fraction $\Omega /(4\pi )$ of that emitted radiant flux, before losses. With direction-dependent radiant intensity $I$, the accepted radiant flux is instead $\int Id\Omega$. The solid angle of a disk below gives an exact example where the familiar inverse-square law is only a distant-source approximation.

Consider a flat circular disk of radius $a>0$, with its center a perpendicular distance $d>0$ from the viewpoint $O$. The line from $O$ to its center is normal to the disk. Its rim defines a right circular cone with $\tan \theta =a/d$ and $\cos \theta =d/\sqrt{d^2+a^2}$. The solid angle of a circular cone immediately givesFor $a=d$, this is the same $45^{\circ }$ half-opening example: about $1.8403$ steradians. Both the radius and distance enter; doubling both leaves the solid angle unchanged.

A useful local version applies to a small surface patch of area $dA$, at distance $r$ from $O$. If $\psi$ is the angle between its unit normal and the line to $O$, its unsigned geometric contribution isThe numerator is the projected area. For a visible patch, perpendicular projection followed by radial projection onto a small piece of a sphere gives this formula. Integrating counts the directions once provided no direction encounters multiple counted patches. For an opaque detector, use its unoccluded face. At grazing view, the projected area tends to zero.

For this on-axis disk, a ring of radius $\rho$ has $r=\sqrt{d^2+{\rho }^2}$ and $|\cos \psi |=d/r$. Its polar coordinates gives $dA=\rho d\rho d\varphi$, so the same answer follows directly from the surface integral:The changing distance and obliquity are both already accounted for; an extra detector-wide cosine factor would count the projection twice.

For $d\gg a$, the Taylor series of $(1+a^2/d^2{)}^{-1/2}$ givesThus the accepted fraction of an ideal point emitter's isotropic output is approximately $A/(4\pi d^2)$, where $A=\pi a^2$. This is the geometry behind the inverse-square law for a small detector. At fixed $a$, however, taking $d\to 0^{+}$ gives $\Omega \to 2\pi$, so the accepted fraction approaches $1/2$ rather than diverging.

For a chosen numerical example, let $a=1\mathrm{cm}$ and $d=5\mathrm{cm}$. The exact solid angle is approximately $0.122015$ steradians, giving an acceptance fraction $0.00970966$. An ideal point emitter with isotropy and total radiant flux $1\mathrm{mW}$ would therefore send about $9.71\mu \mathrm{W}$ onto the disk. The distant-source approximation gives $10.0\mu \mathrm{W}$, about $3.0\%$ high. This calculation assumes unobstructed propagation and counts incident radiant flux; photon absorption, detector response, and any optical attenuator require separate factors.

Incident radiant flux per unit area and unit wavelength or frequency. Integrating $E_{\lambda }$ over a wavelength interval gives the contribution to irradiance in that interval. Conversion between $E_{\lambda }$ and $E_{\nu }$ requires the absolute Jacobian determinant associated with $\nu =c/\lambda$, as for spectral radiance.

Spectral luminous efficacy converts radiant flux at one wavelength to luminous flux. The candela convention fixes $K_{\mathrm{cd}}$ exactly at $540\times 10^{12}\mathrm{Hz}$, not at an exactly specified wavelength of $555\mathrm{nm}$. ^[ref]

In vacuum, that frequency corresponds to $c/(540\times 10^{12}\mathrm{Hz})\simeq 555.1712\mathrm{nm}$, using the speed of light. In standard air it is approximately $555.017\mathrm{nm}$, close to the $555\mathrm{nm}$ maximum of the photopic luminous efficiency function. Thus $K_m$ is slightly larger than $683\mathrm{lm}{\mathrm{W}}^{-1}$; using $K_m\simeq 683\mathrm{lm}{\mathrm{W}}^{-1}$ in ordinary calculations is a rounded convention. The exact anchor is $K({\lambda }_a)=683\mathrm{lm}{\mathrm{W}}^{-1}$. ^[ref]

Spectral radiance is radiance per unit frequency or wavelength interval. Its astronomical synonym specific intensity is usually written $I_{\nu }=L_{\nu }$ or $I_{\lambda }=L_{\lambda }$; this $I_{\nu }$ is a spectral radiance, distinct from the radiant intensity $I$ in radiometry. For positive small intervals,The two expressions refer to the same spectral slice, with the positive widths related through frequency $\nu$ and wavelength $\lambda$. For frequency measured in hertz, $\mathrm{Hz}={\mathrm{s}}^{-1}$, the units are $[L_{\nu }]=\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$. For wavelength measured in metres, $[L_{\lambda }]=\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}{\mathrm{m}}^{-1}$. Integrating over a spectral band gives its radiance, and then multiplying by the collecting projected area and accepted solid angle gives a small uniform beam's radiant flux. A spectral measurement therefore retains information that the band-integrated power alone loses.

For an ideal blackbody, Planck's law gives $L_{\nu }=B_{\nu }(T)$ and $L_{\lambda }=B_{\lambda }(T)$ at temperature $T$, independent of outgoing direction. Its existing frequency and wavelength peaks of Planck's law illustrate why the coordinate matters. The simpler flat-band calculation below isolates that same change of variables without the thermal physics.

Spectral radiant flux resolves radiant flux by wavelength or frequency:Its units are $\mathrm{W}{\mathrm{m}}^{-1}$ per metre of wavelength or $\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1}$ per hertz of frequency. A density quoted per nanometre must be integrated with wavelength intervals in nanometres; its numerical value is $10^{-9}$ times the per-metre density. The frequency and wavelength spectral radiance conversion applies equally to spectral radiant flux, through the change of variables formula:with $c$ the speed of light in vacuum and ${\Phi }_{e,\lambda }$ expressed per metre. The Jacobian determinant preserves integrated power rather than the height or location of an optical spectrum's peak.

For two explicitly synthetic optical spectra, let $x=\lambda /(1\mathrm{nm})$ and quote the following densities in $\mathrm{mW}\mathrm{n}{\mathrm{m}}^{-1}$:The triangle under $f_A$ and the rectangle under $f_B$ both have area $2\mathrm{mW}$: $200\times 0.020/2=200\times 0.010=2$. Their band radiant fluxes differ: $450$–$550\mathrm{nm}$ contains $1.5\mathrm{mW}$ of $A$, while $650$–$750\mathrm{nm}$ contains $1.0\mathrm{mW}$ of $B$. These are specified functions, not measured source spectra.

The dark-subtracted electric current of a linear photodiode isusing its spectral responsivity. The figure's stipulated $R$ gives $i_A=0.60\mathrm{mA}$ and $i_B=1.00\mathrm{mA}$ despite equal total radiant flux. Symmetry places $A$'s mean wavelength at $500\mathrm{nm}$ and $B$'s at $700\mathrm{nm}$; a linear $R$ averages to its value at that mean. A single electric current therefore cannot determine an unknown broadband radiant flux without additional spectral information or a sufficiently flat calibrated response.

The ratio of reflected to incident radiant flux as a function of wavelength, under specified geometrical and illumination conditions. Unlike a perceived colour, spectral reflectance describes a physical response; perceived colour also depends on the incident spectrum and the observer.

For a photodiode in its linear regime, spectral responsivity is the dark-subtracted electric current per incident radiant flux in a narrow band at wavelength $\lambda$, measured in $\mathrm{A}{\mathrm{W}}^{-1}$. More generally a photodetector's readout can be electrical voltage or another quantity, with corresponding units. A broad-band readout weights the incoming optical spectrum by the detector's spectral responsivity; it is not the unweighted total incident power. ^[ref].

The Stefan–Boltzmann law gives the total hemispherical radiant exitance of an ideal blackbody at absolute temperature $T$:Here $\sigma$ is the Stefan-Boltzmann constant. Total means integrated over all wavelengths; hemispherical means integrated over all outward directions with the projected area weighting. Thus $M_b$ is emitted power per actual emitting area, rather than directional radiance or spectral radiance. Absolute temperature must be in kelvin; raising a Celsius reading to the fourth power gives the wrong result. A uniform surface of area $A$ emits radiant flux $P_{\mathrm{emit}}=A{M}_b$. A real surface instead has $M=ϵ(T)\sigma T^4$, where $ϵ(T)$ is its total hemispherical emissivity. The older symbol $F$ in the formula above denotes this emitted radiant exitance.^[ref]

For an ideal blackbody, doubling absolute temperature multiplies radiant exitance by $16$. With the approximate Stefan-Boltzmann constant $5.670374419\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$, the radiant exitances at $300\mathrm{K}$ and $600\mathrm{K}$ are approximately $459.300$ and $7348.805\mathrm{W}{\mathrm{m}}^{-2}$. This steep response explains why thermal radiation becomes so consequential at high temperature. It also enables heat transfer through a vacuum, where thermal conduction through a gas and convection are absent.

A short derivation integrates Planck's law in its frequency form. For a blackbody, direction-independent spectral radiance is $B_{\nu }(T)=2h{\nu }^3/[c^2(e^{h\nu /(k_{B}T)}-1)]$, where $h$ is the Planck constant, $k_B$ the Boltzmann constant and $c$ the speed of light. The outward hemisphere contributes $\int \cos \theta d\Omega =\pi$, soExpanding $(e^x-1{)}^{-1}={\sum }_{n=1}^{\infty }{e}^{-nx}$, nonnegative terms allow termwise integration. Repeated integration by parts gives ${\int }_0^{\infty }{x}^3{e}^{-nx}dx=6/n^4$; the Riemann zeta function value $\zeta (4)={\pi }^4/90$ gives the remaining integral as ${\pi }^4/15$. Hence $M_b=\sigma T^4$ with the coefficient in Stefan-Boltzmann constant. The angular integral is essential: omitting it confuses radiance with radiant exitance.

Transmittance is the fraction of incident radiant flux transmitted through a specified object or boundary. Specify wavelength, angle of incidence, electromagnetic polarization and which outgoing light is collected.

Ultraviolet is electromagnetic radiation with wavelengths shorter than the violet end of ordinarily visible light. The CIE convention uses $100$–$400\mathrm{nm}$, subdivided into UV-A at $315$–$400\mathrm{nm}$, UV-B at $280$–$315\mathrm{nm}$ and UV-C at $100$–$280\mathrm{nm}$. These divisions name intervals; they do not mark universal changes in every material's response. The boundary with visible light is not sharp, and other fields use near-, far-, vacuum- and extreme-ultraviolet divisions. ^[ref]. For example, NASA's Solar Dynamics Observatory observes extreme-ultraviolet bands below $100\mathrm{nm}$; the CIE interval is therefore not the full extent of astronomical ultraviolet usage. ^[ref].

The CIE ultraviolet interval corresponds to frequencies about $7.49\times 10^{14}$–$3.00\times 10^{15}\mathrm{Hz}$ and photon energies $4.97\times 10^{-19}$–$1.99\times 10^{-18}\mathrm{J}$, using vacuum wavelength and $E=hc/\lambda$ with Planck constant $h$ and speed of light $c$. Shorter wavelength means larger energy per photon; larger irradiance means more incident power per area. These are different comparisons. At the same incident power, a $254\mathrm{nm}$ ultraviolet beam has fewer photons per second than a $940\mathrm{nm}$ infrared beam, even though each ultraviolet photon has more energy.

A common ultraviolet mechanism is photon absorption promoting an electron from a ground state to an excited state of a chemical atom or molecule. By contrast, many infrared absorption bands involve stretching or bending chemical bonds. For either mechanism, both the energy difference and the coupling to the incident electromagnetic field matter. One concrete example is the $217\mathrm{nm}$ absorption discussed for butadiene: an electron moves between molecular orbitals, and the position of the absorption reveals their energy separation. Ultraviolet absorption can therefore probe molecular orbitals, while infrared spectroscopy probes many molecular motions. ^[ref].

Subsequent chemical reactions depend on the absorbing chemical substance and its environment; the label ultraviolet alone does not predict a reaction rate. At sufficiently short wavelength, a photon can cause photoionization if its energy exceeds the relevant ionization energy. There is no single wavelength that separates all ionizing from all nonionizing behavior for every chemical substance. Likewise, eventual heating after photon absorption is possible in both ultraviolet and infrared. The distinction is useful because different optical spectra expose different processes, not because the two ranges obey different laws of electromagnetism.

Ritter's 1801 observation used a chemical readout rather than a thermometer. In Caltech/IPAC's account, sunlight passed through a glass optical prism and silver chloride was placed in different parts of the optical spectrum. Darkening was weak toward red, stronger toward violet, and pronounced beyond the visible violet edge. The measured evidence was a change in the chemical substance's appearance at a spatial location with no visible color, establishing radiation beyond the eye's ordinary range. It was not a calibrated measurement of a universal ultraviolet optical spectrum. ^[ref].

A controlled modern interpretation of such a chemical test would compare equal exposure durations and matched fresh samples, keep an unexposed sample in darkness, monitor temperature, and distinguish scattered visible light from the intended short-wavelength beam with characterized filtering. These are proposed controls, not an assertion that Ritter performed them. Unequal sample preparation or heating can change a reaction rate. The optical prism, terrestrial atmosphere and sample response restrict which wavelengths the test detects. Darkening is a detector-weighted response, so its spatial maximum is not automatically the maximum of incoming radiant flux or of photon energy.

Thomas Larason and Jeanne Houston's NIST report documents a quantitative ultraviolet detector comparison. An argon source feeds a double-diffraction grating spectrometer; a shutter and mirrors deliver the selected band to a working-standard or test photodiode, successively at the same position. A beam splitter supplies a monitor photodiode. Amplifiers and voltmeters record detector and monitor simultaneously; shutter-closed readings provide dark baselines. The $200$–$500\mathrm{nm}$ scan, typically with $4\mathrm{nm}$ bandpass, measures spectral responsivity. Monitor normalization suppresses source drift; known spectral lines calibrate wavelength. The report assesses temperature effects and unwanted light outside the band, weighted by detector response. The latter need not cancel between different spectral responsivities. ^[ref].

This determines electrical response without visual perception. Bandpass and spatial nonuniformity limit the calibration; aging and changed temperature can affect later use. The scan includes visible light and ultraviolet; no response below $200\mathrm{nm}$ follows from it.

For a narrow beam, the idealized substitution calculation isHere $R$ is spectral responsivity; $I$ and $M$ are dark-subtracted detector and monitor electric currents. This follows by writing $I=RP$ for each detector and using the monitor to track changes in radiant flux. It assumes linear response, unchanged beam splitting and collection geometry, and a sufficiently narrow band. For an illustrative standard with $R_{\mathrm{std}}=0.15\mathrm{A}{\mathrm{W}}^{-1}$, ratios $I_{\mathrm{std}}/M_{\mathrm{std}}=2.0$ and $I_{\mathrm{test}}/M_{\mathrm{test}}=1.6$ give $R_{\mathrm{test}}=0.12\mathrm{A}{\mathrm{W}}^{-1}$. These invented arithmetic inputs illustrate the method; they are not NIST measurements.

An independent photon-counting model connects spectral responsivity to quantum efficiency. If $\eta (\lambda )$ is the number of collected electrons per incident photon, there is no internal multiplication, and $q$ is the elementary charge, thenAt illustrative constant $\eta =0.60$ and $P=1\mu \mathrm{W}$, this gives about $0.123\mu \mathrm{A}$ at $254\mathrm{nm}$ and $0.455\mu \mathrm{A}$ at $940\mathrm{nm}$. The ultraviolet readout is smaller in this model because there are fewer incident photons, not because their individual energies are smaller. Real quantum efficiency depends on wavelength, windows, absorption and carrier collection; a detector's band gap and construction determine its usable range. Constant $\eta$ across these two wavelengths is an analytical comparison, not a claimed specification of a real photodiode.

The curve is the calculated relation $E=hc/\lambda$, and the shaded ranges use the stated CIE conventions. The lower diagram abstracts the detector comparison rather than reproducing an instrument drawing. Together with the infrared heat-balance illustration, it shows why a calibrated detector response, rather than what looks bright to an observer, is the bridge between invisible radiation and a quantitative measurement.

In a transparent, stationary vacuum with Euclidean geometry and observers at rest in the same frame, the radiative transfer equation has zero absorption and emission, so $d{I}_{\nu }/ds=0$ along a light ray. The same specific intensity reaches the observer without geometric dilution. More generally, $I_{\nu }/{\nu }^3$ is a Lorentz invariant; collisionless propagation preserves this ratio along a ray even with gravitational or cosmological redshift, rather than preserving $I_{\nu }$ at shifted frequencies. ^[ref].

The elementary geometry is instructive. Two facing surface elements separated by distance $s$ exchange spectral radiant fluxThe receiver subtends $d{\Omega }_2=\cos {\theta }_{2}d{A}_2/s^2$ at the source. Describing that same traveling beam at the receiver instead uses $d{\Omega }_1=\cos {\theta }_{1}d{A}_1/s^2$. Both descriptions have identical geometric factors, so conservation of its radiant flux gives equal specific intensities at the two ends. This compares the same propagation direction at both ends, not emission in the opposite direction.

A uniform source remains equally bright per resolved direction under these assumptions, but occupies a smaller apparent solid angle farther away. For a small source of projected area $A_{\perp }$ at distance $d$ and a small receiver facing it, irradiance is approximately $E=L{\Omega }_{\mathrm{source}}\simeq L{A}_{\perp }/d^2$. The radiant flux intercepted by receiver area $A_d$ is approximately $E{A}_d$. With $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, $A_{\perp }=0.020{\mathrm{m}}^2$ and $A_d=10^{-4}{\mathrm{m}}^2$, distances $10$ and $20\mathrm{m}$ give $E\simeq 0.002$ and $0.0005\mathrm{W}{\mathrm{m}}^{-2}$, and received radiant fluxes of $0.20$ and $0.050\mu \mathrm{W}$. The inverse-square law concerns collected radiation from the whole source; $L$ remains $10$ at either distance.

The view factor $F_{i\to j}$ is the fraction of uniformly distributed, diffuse outgoing radiant flux from surface $i$ that directly reaches surface $j$. It depends on geometry, including obstruction, rather than on temperature or emissivity. All destinations in a closed enclosure have fractions summing to one. For a convex body fully enclosed by walls, $F_{\mathrm{body}\to \mathrm{walls}}=1$; the reverse view factor is generally smaller. Reciprocity gives $A_i{F}_{i\to j}=A_j{F}_{j\to i}$. Two unobstructed spheres with outer radius ten times the inner radius have $F_{\mathrm{inner}\to \mathrm{outer}}=1$ and $F_{\mathrm{outer}\to \mathrm{inner}}=1/100$. Do not multiply both directions by an assumed identical view factor.^[ref]