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**Opacity** is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc. An **opaque** object is neither transparent (allowing all light to pass through) nor translucent (allowing some light to pass through). When light strikes an interface between two substances, in general some may be reflected, some absorbed, some scattered, and the rest transmitted (also see refraction). Reflection can be diffuse, for example light reflecting off a white wall, or specular, for example light reflecting off a mirror. An opaque substance transmits no light, and therefore reflects, scatters, or absorbs all of it. Other categories of visual appearance, related to the perception of regular or diffuse reflection and transmission of light, have been organized under the concept of cesia in an order system with three variables, including opacity, transparency and translucency among the involved aspects. Both mirrors and carbon black are opaque. Opacity depends on the frequency of the light being considered. For instance, some kinds of glass, while transparent in the visual range, are largely opaque to ultraviolet light. More extreme frequency-dependence is visible in the absorption lines of cold gases. Opacity can be quantified in many ways; for example, see the article mathematical descriptions of opacity.

Different processes can lead to opacity including absorption, reflection, and scattering.

Late Middle English opake, from Latin opacus 'darkened'. The current spelling (rare before the 19th century) has been influenced by the French form.

*Radiopacity* is preferentially used to describe opacity of X-rays. In modern medicine, radiodense substances are those that will not allow X-rays or similar radiation to pass. Radiographic imaging has been revolutionized by radiodense contrast media, which can be passed through the bloodstream, the gastrointestinal tract, or into the cerebral spinal fluid and utilized to highlight CT scan or X-ray images. Radiopacity is one of the key considerations in the design of various devices such as guidewires or stents that are used during radiological intervention. The radiopacity of a given endovascular device is important since it allows the device to be tracked during the interventional procedure.

The words "opacity" and "opaque" are often used as colloquial terms for objects or media with the properties described above. However, there is also a specific, quantitative definition of "opacity", used in astronomy, plasma physics, and other fields, given here.

In this use, "opacity" is another term for the mass attenuation coefficient (or, depending on context, mass absorption coefficient, the difference is described here) at a particular frequency of electromagnetic radiation.

More specifically, if a beam of light with frequency travels through a medium with opacity and mass density , both constant, then the intensity will be reduced with distance *x* according to the formula

where

*x*is the distance the light has traveled through the medium- is the intensity of light remaining at distance
*x* - is the initial intensity of light, at

For a given medium at a given frequency, the opacity has a numerical value that may range between 0 and infinity, with units of length^{2}/mass.

Opacity in air pollution work refers to the percentage of light blocked instead of the attenuation coefficient (aka extinction coefficient) and varies from 0% light blocked to 100% light blocked:

It is customary to define the average opacity, calculated using a certain weighting scheme. **Planck opacity** (also known as Planck-Mean-Absorption-Coefficient^{ [1] }) uses the normalized Planck black-body radiation energy density distribution, , as the weighting function, and averages directly:

where is the Stefan–Boltzmann constant.

**Rosseland opacity** (after Svein Rosseland), on the other hand, uses a temperature derivative of the Planck distribution, , as the weighting function, and averages ,

The photon mean free path is . The Rosseland opacity is derived in the diffusion approximation to the radiative transport equation. It is valid whenever the radiation field is isotropic over distances comparable to or less than a radiation mean free path, such as in local thermal equilibrium. In practice, the mean opacity for Thomson electron scattering is:

where is the hydrogen mass fraction. For nonrelativistic thermal bremsstrahlung, or free-free transitions, assuming solar metallicity, it is:^{ [2] }

The Rosseland mean attenuation coefficient is:^{ [3] }

Look up ** opacity (optics) ** in Wiktionary, the free dictionary.

In physics, **optical depth** or **optical thickness** is the natural logarithm of the ratio of incident to *transmitted* radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. **Spectral optical depth** or **spectral optical thickness** is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.

In physics, **Planck's law** describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.

The **Eddington luminosity**, also referred to as the **Eddington limit,** is the maximum luminosity a body can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. The state of balance is called hydrostatic equilibrium. When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers. Since most massive stars have luminosities far below the Eddington luminosity, their winds are mostly driven by the less intense line absorption. The Eddington limit is invoked to explain the observed luminosity of accreting black holes such as quasars.

In thermodynamics and solid-state physics, the **Debye model** is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein photoelectron model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low-temperature dependence of the heat capacity of solids, which is proportional to – the **Debye T^{ 3} law**. Similarly to the Einstein photoelectron model, it recovers the Dulong–Petit law at high temperatures. Due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

In physics, the ** S-matrix** or

In mathematical physics, **Yang–Mills theory** is a gauge theory based on a special unitary group SU(*n*), or more generally any compact Lie group. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.

The **Einstein–Hilbert action** in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the (− + + +) metric signature, the gravitational part of the action is given as

In mathematics, the **Hankel transform** expresses any given function *f*(*r*) as the weighted sum of an infinite number of Bessel functions of the first kind *J _{ν}*(

In general relativity, the **metric tensor** is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.

**Einstein coefficients** are quantities describing the probability of absorption or emission of a photon by an atom or molecule. The Einstein *A* coefficients are related to the rate of spontaneous emission of light, and the Einstein *B* coefficients are related to the absorption and stimulated emission of light. Throughout this article, "light" refers to any electromagnetic radiation, not necessarily in the visible spectrum.

**Radiative transfer** is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The **equation of radiative transfer** describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media, with complex multiple scattering effects, numerical methods are required. The present article is largely focused on the condition of radiative equilibrium.

In physics and fluid mechanics, a **Blasius boundary layer** describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow, i.e. flows in which the plate is not parallel to the flow.

The **source function** is a characteristic of a stellar atmosphere, and in the case of no scattering of photons, describes the ratio of the emission coefficient to the absorption coefficient. It is a measure of how photons in a light beam are removed and replaced by new photons by the material it passes through. Its units in the cgs-system are erg s^{−1} cm^{−2} sr^{−1} Hz^{−1} and in SI are W m^{−2} sr^{−1} Hz^{−1}. The *source function* can be written

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

The **linear attenuation coefficient**, **attenuation coefficient**, or **narrow-beam attenuation coefficient** characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss. The SI unit of attenuation coefficient is the reciprocal metre (m^{−1}). **Extinction coefficient** is another term for this quantity, often used in meteorology and climatology. Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward *e*-folding distance of the original intensity as the energy of the intensity passes through a unit thickness of material, so that an attenuation coefficient of 1 m^{−1} means that after passing through 1 metre, the radiation will be reduced by a factor of *e*, and for material with a coefficient of 2 m^{−1}, it will be reduced twice by *e*, or *e*^{2}. Other measures may use a different factor than *e*, such as the *decadic attenuation coefficient* below. The **broad-beam attenuation coefficient** counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding. The *mass attenuation coefficient* is the attenuation coefficient normalized by the density of the material.

** f(R)** is a type of modified gravity theory which generalizes Einstein's general relativity.

The **Grey atmosphere** is a useful set of approximations made for radiative transfer applications in studies of stellar atmospheres based on the simplified notion that the absorption coefficient of matter within a star's atmosphere is constant—that is, unchanging—for all frequencies of the star's incident radiation.

**Kramers' opacity law** describes the opacity of a medium in terms of the ambient density and temperature, assuming that the opacity is dominated by bound-free absorption or free-free absorption. It is often used to model radiative transfer, particularly in stellar atmospheres. The relation is named after the Dutch physicist Hendrik Kramers, who first derived the form in 1923.

In fluid dynamics, the **Falkner–Skan boundary layer** describes the steady two-dimensional laminar boundary layer that forms on a wedge, i.e. flows in which the plate is not parallel to the flow. It is also representative of flow on a flat plate with an imposed pressure gradient along the plate length, a situation often encountered in wind tunnel flow. It is a generalization of the flat plate Blasius boundary layer in which the pressure gradient along the plate is zero.

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven **Ricci scalars** which consist of three real scalars , three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.

- ↑ Modest, Radiative Heat Transfer, ISBN 978-0-12386944-9
- ↑ Stuart L. Shapiro and Saul A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars" 1983, ISBN 0-471-87317-9.
- ↑ George B. Rybicki and Alan P. Lightman, "Radiative Processes in Astrophysics" 1979 ISBN 0-471-04815-1.

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