Radiometry

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].

The study of light propagation, absorption and scattering in marine water. It connects spectral measurements to dissolved material, suspended particles and biological activity.

The illuminated region of a water body. In ecological usage its photosynthetically usable part is often identified with the euphotic zone; the operational light threshold must be stated.

The upper water interval with sufficient light for substantial photosynthesis. A common operational lower boundary is the euphotic depth at 1% of surface photosynthetically active radiation.

The depth of a chosen light threshold, commonly 1% of just-below-surface photosynthetically active radiation. It depends on spectral attenuation and is not a universal species-specific compensation depth.

The spectral distribution of light emerging from a water body, determined by absorption and scattering by water, dissolved material, particles and sometimes the bottom. It supports remote inference of biological and optical properties.

The logarithmic rate at which downwelling spectral irradiance decreases with depth. It depends on absorption, scattering and the angular light field, rather than equaling the absorption coefficient.

Inference of chlorophyll a concentration from measured optical signals. Simple blue/green ratios rely on assumptions about covarying absorbers and scatterers; coastal mixtures can violate those assumptions.

The rate per unit path length at which light is scattered into the backward hemisphere. It influences water-leaving radiance and must be distinguished from the absorption coefficient.

The ratio of water-leaving radiance to incident downwelling irradiance just above the surface. It depends on both absorption and backscattering and has units of inverse steradian.

An optically defined non-phytoplankton particle category, including detritus and suspended mineral material. Its absorption is often strongest in the blue; scattering can also strongly affect remote-sensing reflectance.

The light-absorbing fraction of dissolved organic material. Its visible absorption commonly decreases approximately exponentially from blue to red, and can confound chlorophyll retrieval.

Solar radiant power incident per unit area. Horizontal, beam-normal and tilted-plane irradiances are different quantities: for a direct beam on a horizontal plane, $I_h=I_n\sin \alpha$, where $\alpha$ is solar elevation.

Inferring properties of a distant object from measured signals, often electromagnetic radiation. Spatial sampling, spectral information and observation cadence must jointly resolve the physical process of interest. Received radiance alone need not uniquely separate temperature, emissivity and atmospheric attenuation.

Range measurement using transmitted laser light and its returned signal. For a pulse traveling a round trip in air, the elementary range is $r\simeq c\Delta t/2$, where $c$ is the propagation speed and $\Delta t$ the measured travel time. Sensor position, pointing and time synchronization place returns in a three-dimensional point cloud. Scanning lidar can record multiple returns from vegetation and the ground; a return is an observation of a reflecting surface, not automatically bare-earth elevation.

Lidar acquired from an aircraft with surveyed trajectory and orientation. Ground-return classification removes canopy and structure returns before constructing a bare-ground digital elevation model. Penetration through forest is sampling through canopy gaps, not unrestricted transmission through leaves. Areas with few or no ground returns require uncertainty assessment rather than treating interpolated heights as observations.^[ref]

Recovery of object positions and shape from photographs with overlapping views. Known or estimated camera geometry and matched image features constrain three-dimensional point positions. A metric reconstruction requires scale and a coordinate reference; image pixel size is not automatically the positional accuracy of the reconstructed surface. Stereo photogrammetry and structure-from-motion photogrammetry use related geometry with different acquisition and camera-estimation strategies.

An identifiable image feature or target with independently surveyed coordinates, used to georeference and constrain a photogrammetry model. A check point is withheld from the fit to assess reconstruction accuracy. Distribution across the site, accurate survey and consistent height datums matter as well as the number of points.

A photogrammetry workflow jointly estimating camera geometry and scene structure from many overlapping images, commonly followed by dense surface reconstruction. Ground control points or accurately surveyed camera positions establish scale and georeferencing. Independent check points test accuracy rather than participating in the fit. A fine image ground-sampling interval does not alone establish equal vertical accuracy.

Three-dimensional reconstruction using two or more overlapping photographs from different viewing positions. Corresponding image points define viewing directions whose intersection estimates the surface position. Surveyed camera positions and orientations, camera calibration and ground control points establish the coordinate geometry. Ordinary optical stereo models the visible surface, which can be forest canopy rather than bare ground.

Imaging in many narrow, usually adjacent wavelength bands. Relative to broad-band multispectral imaging, it can resolve narrower mineral absorption features in spectral reflectance, helping distinguish surface lithologies, at the cost of data volume and often reduced signal per band.

Removal or modeling of atmospheric absorption and scattering contributions to a measured signal to estimate surface or water-leaving optical quantities.

The Geostationary Operational Environmental Satellite series, used for frequent weather observations from geostationary orbits.

A European meteorological satellite series including geostationary orbit missions for repeated regional weather observations.

A European radar mission using Sun-synchronous orbits and supporting interferometric synthetic-aperture radar.

A European optical multispectral imaging mission using Sun-synchronous orbits for repeated observations of land and coastal areas.

A series of Earth-observation missions using repeated optical and multispectral imaging from near-polar Sun-synchronous orbits.

Imaging in several wavelength bands. Differences in spectral reflectance can distinguish surface materials, including vegetation and water.

An image formed by dividing corresponding calibrated pixels in two spectral bands. If their signals have a common multiplicative factor, $L_1=k{\rho }_1$ and $L_2=k{\rho }_2$, their ratio is ${\rho }_1/{\rho }_2$ and does not depend on $k$. This explains suppression of some common illumination effects and enhancement of spectral contrasts. Additive atmospheric radiance and small denominators prevent exact cancellation and can amplify errors. Normalized difference vegetation index is a related normalized spectral contrast.

The normalized contrast of red and near-infrared reflectances. Large positive values indicate a vegetation-like spectrum, but soil, atmosphere, shadows and canopy saturation affect its interpretation.

A displayed image assigning measured bands to colour channels that differ from ordinary visual perception. Assigning near-infrared to red is a common way to highlight vegetation.

The steep increase of healthy green vegetation reflectance between red absorption and the high near-infrared scattering plateau, typically around 0.68–0.75 micrometres.

Comparison of coherent radar phases from repeated observations to infer relative line-of-sight displacement. A complete deformation fringe represents half a radar wavelength after other phase contributions have been removed.

An image of the relative radar phase between co-registered coherent observations. A displacement changes the two-way path length, giving phase change proportional to $4\pi d_{\mathrm{LOS}}/{\lambda }_r$, with sign set by the adopted range convention. A full wrapped phase cycle therefore corresponds to half a radar wavelength of line-of-sight displacement. Orbital geometry, topography, atmospheric delay and scattering changes must be separated from deformation; a fringe is not uniquely an elevation contour or an earthquake displacement.^[ref]

A normalized measure of consistency of complex radar returns over a neighborhood. Low coherence makes interferometric phase unreliable and can arise from vegetation, surface changes, viewing differences or extreme displacement gradients.

The component of a displacement vector along a specified sensor–target viewing direction. Its sign must state whether motion toward or away from the sensor is positive.

The phase difference of complex radar returns. It includes deformation, topography, orbital geometry and propagation contributions; the displayed wrapped phase is defined modulo $2\pi$.

The fraction of transmitted light scattered sufficiently far from the forward direction under a specified measurement geometry. A transparent electrode can have high total transmittance yet blur an image if scattering is strong. State the collection angles and wavelength when comparing haze measurements.

The photon flux incident from all propagation directions, resolved by wavelength when needed for photodissociation. It is distinct from the net radiative flux through a horizontal surface; scattered and diffuse photons can contribute even when their net vertical flux cancels.

For an optical beam, time-integrated light intensity, giving incident energy per unit area.

Energy carried by incident photons per unit area. For monoenergetic photons, $\Psi =E_{\gamma }\Phi$, where $\Phi$ is photon fluence. With charged-particle equilibrium, $D=\Psi {\mu }_{\mathrm{en}}/\rho$ relates it to absorbed dose.

The number of photons incident per unit area in a specified geometry. It is distinct from a rate per unit time. For a monoenergetic beam, photon energy fluence is $\Psi =E_{\gamma }\Phi$.

A radiometer measures electromagnetic radiation through its received power. A radio radiometer can express that power as a noise temperature, allowing an unknown source to be compared with calibrated reference temperatures.

For a stable linear radiometer, known cold and hot radiances give $B_S=B_C+(B_H-B_C)(V_S-V_C)/(V_H-V_C)$. Put $a=(B_H-B_C)/(V_H-V_C)$ and $b=-a(V_S-V_C)/(V_H-V_C)$. Then $\delta B_S=a\delta V_S+b\delta V_H-(a+b)\delta V_C$; all three measured voltages contribute to the retrieved uncertainty.

A noise temperature expresses received or internally added noise power on an equivalent temperature scale. For one matched receiving channel in the classical thermal limit, an ideal rectangular bandwidth $\Delta \nu$ gives $P=k_{\mathrm{B}}{T}_N\Delta \nu$, with Boltzmann constant $k_{\mathrm{B}}$. An equivalent noise temperature need not equal a component's physical temperature.

The system noise temperature includes all noise contributions referred to a specified receiver input. In a simple additive model with negligible intervening losses, $T_{\mathrm{sys}}=T_A+T_r$, where $T_A$ is the antenna temperature and $T_r$ the receiver noise temperature.

The receiver noise temperature represents noise added by the receiver as an equivalent noise temperature referred to its input. Separating it from the antenna temperature prevents internal receiver noise from being mistaken for radiation from the sky.

For a stable radiometer with affine output $S=g(T+T_r)$, let $g>0$ be the fixed response coefficient and $T_r$ the receiver noise temperature. Two reference loads with known equivalent input noise temperatures $T_c<T_h$ give outputs $S_c$ and $S_h$. Subtracting their outputs eliminates $T_r$; dividing source and reference differences eliminates $g$:This two-load radiometer calibration assumes unchanged response and receiver noise temperature between readings, with the loads and source referred to the same input. It also eliminates an additional constant output offset. Physical load temperatures must first be converted to the applicable equivalent input noise temperatures when the classical thermal approximation fails. The Penzias-Wilson microwave background experiment gives an original worked example.

The fraction of incident radiant flux returned by reflection, for specified wavelength, angle of incidence and collection directions.

A function relating reflected radiance in one direction to incident irradiance from another direction at a surface location. A local nonemissive, nonfluorescent optical reflection model integrates $f_r{L}_i\cos {\theta }_i$ over incident solid angle to obtain outgoing radiance. Transmission, wavelength conversion and nonlocal transport need additional models.

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.

The transfer of energy from electromagnetic radiation to matter. Reflection redirects the incident electromagnetic radiation, and transmission lets it pass through; absorption transfers some of its energy to the material instead.

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.

In geometrical optics, suppose radiation has a nonnegative radiance $L(\mathbf{x},\mathbf{s})$ whose directional contributions add as power. Here $\mathbf{s}$ is the actual propagation unit vector, $d\Omega$ is a solid angle element, and the collecting oriented surface $A$ has chosen unit normal $\mathbf{n}(\mathbf{x})$. Assume the following integrals exist. The outward and inward radiant fluxes areBoth quantities are nonnegative; their difference is the signed net radiant flux. The dot product supplies the cosine of the angle with the unit normal, hence the projected area. For incoming radiation the outward unit normal gives a negative dot product; the minus sign makes the collected power positive. Measuring the incoming angle from a source-facing unit normal instead gives the familiar positive $\cos \theta$. The units check is $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\times {\mathrm{m}}^2\times \mathrm{sr}=\mathrm{W}$.

Equivalently, the incoming surface integral is ${\Phi }_{-}={\int }_{A}EdA$ with local irradiance $E$; the emitted part of ${\Phi }_{+}$ is ${\int }_{A}MdA$ with radiant exitance $M$. For a small flat patch and a narrow accepted cone, nearly constant radiance and angle give $\Phi \simeq LA\cos \theta \Delta \Omega$. This approximation requires uniformity across the patch and cone. Direction-independent outgoing radiance over a whole hemisphere gives the different exact result ${\Phi }_{+}=\pi LA$, by Lambert's cosine law and the spherical coordinates calculation in radiant exitance. Replacing the cosine-weighted integral by the hemisphere's $2\pi$ solid angle would double the answer.^[ref]

These radiance formulas presume a description in additive directional power; they do not reconstruct arbitrary coherent wave interference from separate ray contributions. For general electromagnetic fields, the signed instantaneous power crossing an oriented surface is ${\int }_A\mathbf{S}\cdot \mathbf{n}dA$, using the Poynting vector $\mathbf{S}$; optical measurements commonly average over many wave periods. A zero net radiant flux can conceal equal nonzero incoming and outgoing radiant fluxes.

Consider an ideal parallel circular beam with uniform irradiance inside radius $b$, zero outside, and total radiant flux $P=2\mathrm{W}$. A centred circular collector of radius $a$, perpendicular to the light rays, interceptsFor $b=1\mathrm{cm}$ and $a=0.5\mathrm{cm}$, $E_b=20000/\pi \mathrm{W}{\mathrm{m}}^{-2}\simeq 6366.20\mathrm{W}{\mathrm{m}}^{-2}$ and the collected radiant flux is $0.50\mathrm{W}$. Expanding the beam to $b=2\mathrm{cm}$ while keeping $P$ fixed divides irradiance and the same collector's radiant flux by four: $E_b\simeq 1591.55\mathrm{W}{\mathrm{m}}^{-2}$ and ${\Phi }_{\mathrm{collected}}=0.125\mathrm{W}$. A collector covering either entire beam still receives $2\mathrm{W}$.

Tilting a flat collector of actual area $A$ by an angle $\theta$ to the beam direction gives ${\Phi }_{\mathrm{collected}}=E_{b}A\cos \theta$ if its entire projected area remains inside the uniform beam. At $60^{\circ }$ the small collector's radiant flux halves. A collector large enough to intercept the whole beam after tilting still receives $P$: the footprint on its surface grows as $1/\cos \theta$ and cancels the irradiance factor. This model ignores diffraction, beam-edge structure, and changes in absorption with incidence angle; it counts incident rather than absorbed power.

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.

An optical power meter makes the definition operational: collect a specified beam on a calibrated photodiode, measure its electric current, and infer incident radiant flux using spectral responsivity. In the manufacturer-authored Model 1830-R Series Optical Power Meters User's Manual by Newport Corporation, §5.5.1 specifies a concrete sequence: connect the detector and its calibration module, select watt mode and the source wavelength, block the beam and press ZERO, then unblock it and read the display. The stored background includes dark current and ambient light present during zeroing. For narrowband radiation,The electric current difference divided by spectral responsivity has units $\mathrm{A}/(\mathrm{A}{\mathrm{W}}^{-1})=\mathrm{W}$. For a stipulated $R=0.40\mathrm{A}{\mathrm{W}}^{-1}$ and difference $0.20\mathrm{mA}$, the inferred radiant flux is $0.50\mathrm{mW}$; these numbers illustrate the inference and are not reported measurements. The entire beam must hit the active area, and §5.6 calls for correct wavelength, background subtraction, and operation below detector saturation. If an optical attenuator is fitted, §5.4 requires the calibrated detector–attenuator pair and the matching meter setting.

The calibration itself must connect electric current to independently known incident power. NIST's Spectroradiometric Detector Measurements: Ultraviolet, Visible, and Near-Infrared Detectors for Spectral Power by Thomas C. Larason and others describes this chain. A cryogenic radiometer, an electrical substitution radiometer, compares temperature changes from absorbed light with those from known Joule heating (§5.2). Transfer and working detectors carry that watt scale to a spectral comparator (§5.1). A source, monochromator, focusing optics, shutter, beam splitter, and monitor photodiode illuminate a standard and test detector in turn; calibrated current-to-voltage electronics and voltmeters provide the readout (§§3.2.2–3.2.4, 4.3.1). At each wavelength, subtract blocked-beam signals and normalize each detector signal by the simultaneously sampled monitor signal. Comparison with the standard yields the test detector's spectral responsivity. NIST uses two working standards and alignment normal to the beam (§6.1.1).

Controls distinguish a smaller collected radiant flux from a faulty conversion: block and unblock the beam to check the baseline; check that its footprint fits the active area; verify the calibrated optical attenuator configuration; and test proportional response within the specified operating range. Source drift faster than monitor sampling is not removed. NIST discusses finite spectral bandwidth, stray light, temperature, and spatial response variations (§§3.1.1, 6.2, 7.2). A broadband optical spectrum requires the weighted integral in spectral radiant flux, rather than dividing by an arbitrary single spectral responsivity. These limitations explain why the meter's displayed watts need a stated collection boundary, spectral assumptions, and calibration conditions.

Radiant intensity describes how a source distributes its radiant flux over directions. A small cone of solid angle $\Delta \Omega$ carries approximately $I\Delta \Omega$ when $I$ varies little across it; integrating $I$ over all emitting directions recovers the total radiant flux. For a source with isotropy emitting $4\pi \mathrm{W}$ over a full sphere, $I=1\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and a $0.01\mathrm{sr}$ cone carries $0.01\mathrm{W}$. A planar source satisfying Lambert's cosine law instead has direction-dependent radiant intensity, even though its radiance is direction-independent.

An ideal point emitter emits radiant flux from one location. It approximates a physical source when its emitting dimensions are small relative to the distances relevant to the calculation. With isotropy, its radiant intensity is $I=\Phi /(4\pi )$ for total radiant flux $\Phi$. An extended source generally requires integrating contributions from different emitting positions.

Radiance resolves both position and direction: it is radiant flux per projected area and per solid angle. For a surface element of area $dA$, and a small outgoing cone $d\Omega$ at angle $0\le \theta <\pi /2$ to its outward unit normal,Here $\mathbf{x}$ locates the patch and $\mathbf{s}$ is a unit vector along the outgoing direction. The factor $dA\cos \theta$ is the patch's apparent area viewed along that direction. The same local definition applies to transmitted or received radiation, choosing the positive projected area for the crossing direction. ^[ref].

For a planar patch of area $A$ with spatially uniform, direction-independent radiance $L$, integrating over the patch givesTake $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ and $A=0.020{\mathrm{m}}^2$. The radiant intensities are $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and $I(60^{\circ })=0.10\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. A sufficiently narrow $0.010\mathrm{sr}$ cone therefore carries approximately $2.0\mathrm{mW}$ or $1.0\mathrm{mW}$, respectively. Both views have the same radiance $L$: the oblique view presents half the projected area. This is Lambert's cosine law, not a loss of radiance with viewing angle.

Radiance is useful because it separates a source's local directional output from the geometry of an observation. Integrating over outgoing directions gives radiant exitance; integrating incoming radiance at a receiver gives irradiance. In the stationary vacuum setting of vacuum conservation of specific intensity, a resolved uniform source retains its radiance along each light ray, while the radiant flux received from the whole source can decrease as its apparent solid angle shrinks.

The projected area of a planar patch viewed from a direction is the area of its perpendicular shadow onto a plane normal to that direction. If $\mathbf{n}$ and $\mathbf{s}$ are its unit normal and viewing unit vector, then $A_{\perp }=A|\mathbf{n}\cdot \mathbf{s}|$ by the dot product. On the outward side, this is $A\cos \theta$. For a rectangle tilted around one edge, the width perpendicular to that edge contracts by $\cos \theta$ and the edge length stays fixed, proving the formula. At grazing view $A_{\perp }=0$; interpret the radiance definition by a limit rather than dividing by zero.

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.

In vacuum, $\nu =c/\lambda$, where $c$ is the speed of light. Increasing wavelength decreases frequency, so the ordinary signed differentials satisfy $d\nu =-c{\lambda }^{-2}d\lambda$. A spectral slice's positive contribution to radiance must be unchanged. The change of variables formula therefore gives the absolute Jacobian determinantThese formulas use $L_{\lambda }$ per metre and $L_{\nu }$ per hertz. The factor $c/{\nu }^2$ has units $\mathrm{m}\mathrm{s}=\mathrm{m}\mathrm{H}{\mathrm{z}}^{-1}$, which converts the former density into the latter. If ${\ell }_{\mathrm{nm}}$ denotes the numerical density per nanometre in $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$, its per-metre value is $10^9{\ell }_{\mathrm{nm}}$. Conversely, the per-nanometre value is $10^{-9}$ times the per-metre value. Likewise, a density per terahertz is $10^{12}$ times its per-hertz value because $1\mathrm{THz}=10^{12}\mathrm{Hz}$.

For a concrete stipulated spectrum, let $L_{\lambda }=C=10^9\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}{\mathrm{m}}^{-1}$ between ${\lambda }_1=500\mathrm{nm}$ and ${\lambda }_2=600\mathrm{nm}$, and zero outside. This is exactly $1\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$ throughout the band. Using the exact International System of Units value $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, the corresponding frequency bounds areThe transformed spectral radiance is $L_{\nu }=Cc/{\nu }^2$ on this interval. At ${\nu }_1$ it is approximately $1.200830743\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$, and at ${\nu }_2$ it is $8.339102380\times 10^{-13}$ in the same units. These are respectively $1.200830743$ and $0.833910238$ per terahertz. Thus a flat wavelength density becomes a decreasing frequency density; relabelling the horizontal axis alone would miss both the shape and the units. ^[ref].

The equality of band radiances has a short exact proof:The reversed bounds and the Jacobian determinant preserve the same spectral measure. Each half-band, $500$–$550\mathrm{nm}$ and $550$–$600\mathrm{nm}$, carries $50\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, although the corresponding frequency widths are about $54.508$ and $45.423\mathrm{THz}$.

A spectrum's peak is consequently a statement about a particular density and coordinate. The frequency and wavelength peaks of Planck's law differ for precisely this reason. Integrated radiance over a specified physical band supplies a comparison that survives the change of variables.

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.

Limb darkening is the decrease of emergent specific intensity toward the apparent edge of a stellar disc. A transit depth then depends on the intensity sampled by the planet as well as its projected area; the uniform-disc area ratio is only an approximation.

For a spherical star of radius $R_{\star }$, let $r$ be distance from the centre of its projected disk and let $\theta$ be the angle between the outward unit normal and the line of sight. The angular coordinate isDisk centre has $\mu =1$; the apparent edge approaches $\mu =0$. A resolved image measures specific intensity in each direction. Limb darkening means that this specific intensity itself falls toward the edge; the projected area factor used when collecting radiant flux is a separate geometric effect.

A simple explanation assumes a horizontally uniform plane-parallel atmosphere, a sufficiently deep optically thick medium, negligible scattering of light, and thermal absorption and emission in local thermodynamic equilibrium. Let ${\tau }_{\nu }$ be inward vertical optical depth at fixed frequency $\nu$. An oblique light ray has line of sight optical depth ${\tau }_{\nu }/\mu$, so it encounters the same attenuation at a smaller vertical optical depth. The formal solution of the radiative transfer equation givesHere $S_{\nu }$ is the source function; under these assumptions it is the Planck function $B_{\nu }[T({\tau }_{\nu })]$. If temperature increases inward, an oblique view preferentially samples cooler layers and hence lower specific intensity. ^[ref]

The weight $e^{-{\tau }_{\nu }/\mu }/\mu$ is an exponential distribution in vertical optical depth, with expected value $\mu$ and standard deviation $\mu$. Thus “seeing down to optical depth one” refers to the line of sight, and represents a broad sampling of layers rather than a sharp emitting surface. For a source function that is linear throughout the atmosphere, the Eddington-Barbier relation is exact:This follows by integrating the constant and linear terms separately. A constant source function instead gives constant specific intensity, as in the blackbody limit of an isothermal atmosphere. An outward-rising temperature or important scattering of light changes the argument. Close enough to the actual edge, a plane-parallel atmosphere also loses validity because of curvature. Neither one limb-darkening coefficient nor a universal dependence on wavelength follows from this simple model.

The linear limb-darkening law parametrizes angular specific intensity by a stipulated limb-darkening coefficient $u$:Here $I$ denotes specific intensity at one frequency, or radiance integrated over a specified interval of frequency or wavelength; it does not denote radiant intensity. Within this stipulated range, $u=0$ gives a uniform disk and $I(0)/I_c=1-u$. The linear limb-darkening law is linear in $\mu$, while its radial profile $I(r)=I_c[1-u+u\sqrt{1-(r/R_{\star }{)}^2}]$ is curved. In the linear source function example above, $u=b_{\nu }/(a_{\nu }+b_{\nu })$ when $a_{\nu },b_{\nu }\ge 0$ and $a_{\nu }+b_{\nu }>0$.

For the illustrative choice $u=0.6$, the specific intensity at $r/R_{\star }=0$, $0.8$, and the limiting edge is respectively $I_c$, $0.76I_c$, and $0.4I_c$. These are synthetic values, not measurements of the Sun. Real optical spectra and stellar atmospheres need not obey a linear limb-darkening law; more flexible profiles can be necessary for accurate exoplanet transit photometry. ^[ref]

The parameter $u$ in the linear limb-darkening law specifies its contrast between disk centre and the limiting edge. Its interpretation requires the chosen law and interval of frequency or wavelength. A limb-darkening coefficient fitted to an exoplanet transit may trade off against the transit impact parameter and planet radius; it is not a direct measurement of one layer's temperature.

For a circular stellar disk whose specific intensity is independent of azimuth, its mean specific intensity over projected area isThe polar coordinates element is $d{A}_{\mathrm{proj}}=2\pi rdr$ after angular integration, and $rdr=-R_{\star }^2\mu d\mu$. Substitution of the linear limb-darkening law givesFor a distant observer at distance $D\gg R_{\star }$ receiving light normally to the detecting plane, the observed radiative flux isIt is spectral irradiance, with units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{H}{\mathrm{z}}^{-1}$ when $I_{\nu }$ is per hertz. The local outward radiative flux per actual surface area is instead $F_{\nu }=2\pi {\int }_0^1{I}_{\nu }(\mu )\mu d\mu =\pi {\overline{I}}_{\nu }$, a spectral radiant exitance. In a spherical model with the same stellar atmosphere everywhere, the luminosity is $L_{\nu }=4\pi R_{\star }^2{F}_{\nu }$ and $f_{\nu }=L_{\nu }/(4\pi D^2)$, consistent with the inverse-square law. Compare disks at fixed $I_c$ or fixed total observed radiative flux explicitly: increasing $u$ lowers the total in the first comparison, while the second requires a larger $I_c$.

A dark, opaque circular planet of radius $R_p$ blocks specific intensity over its overlap with the projected star. With $k=R_p/R_{\star }$ and no transit dilution, the instantaneous exoplanet transit depth is exactlyWhen $k\ll 1$, the entire planetary disk is inside the stellar disk, and specific intensity varies little across that disk, small-planet transit weighting reduces toHere $z$ is the projected centre separation divided by $R_{\star }$. The numerator samples the local specific intensity; the denominator normalizes by the whole star's mean specific intensity. The small-size argument and the distinction between full overlap and ingress are explicit in the original analytic treatment. ^[ref]

For a short approximately straight crossing at constant transverse velocity $v_{\perp }$, write $x=v_{\perp }t/R_{\star }$ with $t=0$ at mid-transit, and let $b$ be the transit impact parameter. Then $z^2=x^2+b^2$ and ${\mu }_p=\sqrt{1-b^2-x^2}$. With stipulated $u=0.6$ and $k=0.05$, a central crossing has mid-transit exoplanet transit depth approximately $0.003125$, whereas $b=0.8$ gives $0.002375$. A uniform disk would give $k^2=0.0025$ for either fully overlapping crossing. The brighter central chord produces a deeper dip and the off-centre chord a shorter crossing at the same $v_{\perp }$.

At fixed $z<1$ away from the edge, a two-dimensional Taylor's theorem expansion over the symmetric planetary disk cancels the first derivatives; the first correction to the blocked radiant flux is of order $k^4$ if the second derivatives remain bounded. This error estimate is not uniform at the edge, where the radial derivative of the linear limb-darkening law diverges. Ingress, egress, a grazing exoplanet transit, and a crossing of a starspot require the overlap integral or a suitable more complete model. Inferring $k=\sqrt{\delta }$ without the local-to-mean specific intensity ratio can bias the estimated planet radius.

A motivating measurement used the McMath solar optical telescope at Kitt Peak. The Sun's image drifted across a $10\mathrm{mm}$-high, $0.2\mathrm{mm}$-wide optical slit; a double-pass spectrograph selected wavelength intervals from $0.0015\mathrm{nm}$ wide at $303\mathrm{nm}$ to $0.007\mathrm{nm}$ at $1099\mathrm{nm}$. An EMI 9750 QB photodetector or silicon photodiode supplied a digitized signal: $1\mathrm{kHz}$ samples were summed into 4096 positions per crossing, including dark and off-limb sky readings. ^[ref]

Controls included daily realignment of optical telescope mirrors, wavelength settings between spectral lines insensitive to the rotational Doppler shift, selection of detectors without detectable hysteresis, and measured electronic nonlinearity corrections below $0.1\%$ of maximum signal. Limb-position calibration combined centres of gravity of the edge derivative profiles with almanac semidiameters; the preliminary edge shifts were consistent with astronomical seeing of approximately $5$ arcseconds full width at half maximum. ^[ref]

The decreasing centre-to-limb readout gives a specific intensity profile, constraining stratification of the photosphere under a radiative transfer model. Astronomical seeing, instrumental scattering of light, and solar granulation limit this inference; brightness alone does not uniquely determine temperature versus geometric height. These observations need not follow the stipulated linear limb-darkening law.

Radiant exitance is total outward emitted radiant flux per actual emitting area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Integrate radiance over the outward hemisphere, keeping the projected area factor:If the radiance is direction-independent, the spherical coordinates element is $d\Omega =\sin \theta d\theta d\phi$, soThe weighted solid angle is $\pi \mathrm{sr}$, although the whole hemisphere covers $2\pi \mathrm{sr}$. Directions near grazing contribute less because of the projected area. This geometric factor is the bridge between Lambert's cosine law and the total output of a diffuse patch. ^[ref].

For the uniform patch with $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, radiant exitance is $M=10\pi \mathrm{W}{\mathrm{m}}^{-2}$. Its area $0.020{\mathrm{m}}^2$ gives total radiant flux $\Phi =MA=0.20\pi \mathrm{W}\approx 0.628319\mathrm{W}$ over the outward hemisphere. This total differs from its radiant intensity $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$, which describes only one direction.

For an ideal blackbody, integrate Planck's law over frequency as well: $M=M_e=j^{\ast }=\sigma T^4$ by the Stefan–Boltzmann law. An ideal $6000\mathrm{K}$ emitting patch of area $1\mathrm{m}{\mathrm{m}}^2$ emits about $73.4881\mathrm{W}$ outward. Irradiance counts incoming radiation separately; for purely emitted output the net outward exchange per area is $M-E$. Reflected output must also be included if the surface reflects incident radiation.

Radiant power leaving a surface per surface area and per unit frequency or wavelength. Isotropic cavity radiation obeys $M_{\nu }=c{U}_{\nu }/4$; the factor $1/4$ combines the outward hemisphere and projected area.

Irradiance measures incident radiant flux per actual receiving area. Its units $\mathrm{W}{\mathrm{m}}^{-2}$ match radiant exitance, but it counts the incoming side of the radiation exchange. Measure incoming angles from the unit normal pointing toward the incident hemisphere. ThenFor $2\mathrm{W}$ distributed uniformly over $0.010{\mathrm{m}}^2$, $E=200\mathrm{W}{\mathrm{m}}^{-2}$. If the receiving plane is tilted $60^{\circ }$ in the same broad uniform parallel beam, the incident power per actual area is halved to $100\mathrm{W}{\mathrm{m}}^{-2}$. This receiving-surface cosine factor is geometric and does not assume Lambertian reflectance. Together, irradiance and radiant exitance describe how radiation supplies and removes energy at a surface, while radiance preserves the directional information.

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.

A parallel beam incident at elevation $\alpha$ has horizontal irradiance $F_h=F_{\perp }\sin \alpha$. Its path optical depth in a plane-parallel atmosphere is ${\tau }_v/\sin \alpha$. Confusing these projections changes both attenuation and intercepted power.

Radiometric absorptance is the fraction of incident radiant flux converted to heat in the specified receiver: $P_{\mathrm{abs}}=a{P}_{\mathrm{in}}$, with $0\le a\le 1$ for a passive receiver. Specify the entrance boundary, wavelength, incidence angle and receiver temperature. Radiation reflected or transmitted out of that receiver is not absorbed power. A cavity can improve radiometric absorptance by giving light repeated opportunities for wave absorption; a dark coating alone does not establish $a=1$.

For broadband light, spectral radiant flux matters: $P_{\mathrm{abs}}=\int a(\lambda ){\Phi }_{e,\lambda }(\lambda )d\lambda$. A single effective radiometric absorptance therefore depends on the incident optical spectrum as well as on the receiver. If the stated incident boundary lies before a window of known power transmission ${\tau }_w$, the receiver absorbs $a{\tau }_w{P}_{\mathrm{before}}$ when other optical losses are absent. Window loss and receiver radiometric absorptance are separate corrections.