Solid angle in the context of Surface brightness

In astronomy, a deep field is an image of a portion of the sky taken with a very long exposure time, in order to detect and study faint objects. The depth of the field refers to the apparent magnitude or the flux of the faintest objects that can be detected in the image. Deep field observations usually cover a small angular area on the sky, because of the large amounts of telescope time required to reach faint flux limits. Deep fields are used primarily to study galaxy evolution and the cosmic evolution of active galactic nuclei, and to detect faint objects at high redshift. Numerous ground-based and space-based observatories have taken deep-field observations at wavelengths spanning radio to X-rays.

The first deep-field image to receive a great deal of public attention was the Hubble Deep Field, observed in 1995 with the WFPC2 camera on the Hubble Space Telescope. Other space telescopes that have obtained deep-field observations include the Chandra X-ray Observatory, the XMM-Newton Observatory, the Spitzer Space Telescope, and the James Webb Space Telescope.

Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted from, or is reflected from a particular area, and falls within a given solid angle.

The procedure for conversion from spectral radiance to luminance is standardized by the CIE and ISO.

In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye. The SI unit of luminous intensity is the candela (cd), an SI base unit.

In the field of heat transfer, intensity of radiation ${\displaystyle I}$ is a measure of the distribution of radiant heat flux per unit area and solid angle, in a particular direction, defined according to

In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, and to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre (W·sr·m). It is a directional quantity: the radiance of a surface depends on the direction from which it is being observed.

The related quantity spectral radiance is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength.

Candela (symbol: cd) is the SI unit of luminous intensity. It measures the luminous power per unit solid angle emitted in a particular direction. A common wax candle has a luminous intensity of roughly 1 cd.

The word candela is Latin for candle. The old name "candle" is still sometimes used, as in foot-candle and the modern definition of candlepower.

In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are directional quantities. The SI unit of radiant intensity is the watt per steradian (W/sr), while that of spectral intensity in frequency is the watt per steradian per hertz (W·sr·Hz) and that of spectral intensity in wavelength is the watt per steradian per metre (W·sr·m)—commonly the watt per steradian per nanometre (W·sr·nm). Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity in branches of physics other than radiometry. In radio-frequency engineering, radiant intensity is sometimes called radiation intensity.

The bidirectional reflectance distribution function (BRDF), symbol ${\displaystyle f_{\text{r}}(\omega _{\text{i}},\,\omega _{\text{r}})}$, is a function of four real variables that defines how light from a source is reflected off an opaque surface. It is employed in the optics of real-world light, in computer graphics algorithms, and in computer vision algorithms. The function takes an incoming light direction, ${\displaystyle \omega _{\text{i}}}$, and outgoing direction, ${\displaystyle \omega _{\text{r}}}$ (taken in a coordinate system where the surface normal ${\displaystyle \mathbf {n} }$ lies along the z-axis), and returns the ratio of reflected radiance exiting along ${\displaystyle \omega _{\text{r}}}$ to the irradiance incident on the surface from direction ${\displaystyle \omega _{\text{i}}}$. Each direction ${\displaystyle \omega }$ is itself parameterized by azimuth angle ${\displaystyle \phi }$ and zenith angle ${\displaystyle \theta }$, therefore the BRDF as a whole is a function of 4 variables. The BRDF has units sr, with steradians (sr) being a unit of solid angle.

The field of view (FOV) is the angular extent of the observable world that is seen at any given moment. In the case of optical instruments or sensors, it is a solid angle through which a detector is sensitive to electromagnetic radiation. It is further relevant in photography.

The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in the form of a circular cone can be projected onto a sphere from its centre, delineating a spherical cap where the cone intersects the sphere. The magnitude of the solid angle expressed in steradians is defined as the quotient of the surface area of the spherical cap and the square of the sphere's radius. This is analogous to the way a plane angle projected onto a circle delineates a circular arc on the circumference, whose length is proportional to the angle. Steradians can be used to measure a solid angle of any projected shape. The solid angle subtended is the same as that of a cone with the same projected area. A solid angle of one steradian subtends a cone aperture of approximately 1.144 radians or 65.54 degrees.

In the SI, solid angle is considered to be a dimensionless quantity, the ratio of the area projected onto a surrounding sphere and the square of the sphere's radius. This is the number of square radians in the solid angle. This means that the SI steradian is the number of square radians in a solid angle equal to one square radian, which of course is the number one. It is useful to distinguish between dimensionless quantities of a different kind, such as the radian (in the SI, a ratio of quantities of dimension length), so the symbol sr is used. For example, radiant intensity can be measured in watts per steradian (W⋅sr). The steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit.

A square degree (deg) is a non-SI unit measure of solid angle. Other denotations include sq. deg. and (°). Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere.

Analogous to one degree being equal to ⁠π/180⁠ radians, a square degree is equal to (⁠π/180⁠) steradians (sr), or about ⁠1/3283⁠ sr or about 3.046×10 sr.The whole sphere has a solid angle of 4πsr which is approximately 41253 deg:

The jansky (symbol Jy, plural janskys) is a non-SI unit of spectral flux density, or spectral irradiance, used especially in radio astronomy. It is equivalent to 10 watts per square metre per hertz.

The spectral flux density or monochromatic flux, S, of a source is the integral of the spectral radiance, B, over the source solid angle:$${\displaystyle S=\iint \limits _{\text{source}}B(\theta ,\phi )\,\mathrm {d} \Omega .}$$

The International Astronomical Union (IAU) designates 88 constellations of stars. In the table below, they are ranked by the solid angle that they subtend in the sky, measured in square degrees and millisteradians.

These solid angles depend on arbitrary boundaries between the constellations: the list below is based on constellation boundaries drawn up by Eugène Delporte in 1930 on behalf of the IAU and published in Délimitation scientifique des constellations (Cambridge University Press). Before Delporte's work, there was no standard list of the boundaries of each constellation.

Etendue or étendue (/ˌeɪtɒnˈduː/) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent, and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer where it is related to the view factor (or shape factor). It is a central concept in nonimaging optics. The term étendue comes from French, where it means "extent".

From the source point of view, etendue is the product of the area of the source and the solid angle that the system's entrance pupil subtends as seen from the source. Equivalently, from the system point of view, the etendue equals the area of the entrance pupil times the solid angle the source subtends as seen from the pupil. These definitions must be applied for infinitesimally small "elements" of area and solid angle, which must then be summed over both the source and the diaphragm as shown below. Etendue may be considered to be a volume in phase space.