steradian | |
---|---|

General information | |

Unit system | SI |

Unit of | solid angle |

Symbol | sr |

Conversions | |

1 sr in ... | ... is equal to ... |

SI base units | 1 m^{2}/m^{2} |

The steradian and radian on physics **steradian** (symbol: **sr**) or **square radian**^{ [1] }^{ [2] } 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. Whereas an angle in radians, projected onto a circle, gives a *length* on the circumference, a solid angle in steradians, projected onto a sphere, gives an *area* on the surface. The name is derived from the Greek στερεός*stereos* 'solid' + radian.

The steradian, like the radian, is a dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. Both the numerator and denominator of this ratio have dimension length squared (i.e. L^{2}/L^{2} = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol "sr" is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian (W⋅sr^{−1}). 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 steradian can be defined as the solid angle subtended at the centre of a unit sphere by a circular unit area on its surface. For a general sphere of radius *r*, any portion of its surface with area *A* = *r*^{2} subtends one steradian at its centre.^{ [3] }

The solid angle is related to the area it cuts out of a sphere:

where

- Ω is the solid angle
- A is the surface area of the spherical cap, ,
- r is the radius of the sphere,
- h is the height of the cap, and
- sr is the unit, steradian.

Because the surface area *A* of a sphere is 4*πr*^{2}, the definition implies that a sphere subtends 4*π* steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4π (≈ 0.07958) of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4*π* sr.

If *A* = *r*^{2}, it corresponds to the area of a spherical cap (*A* = 2*πrh*) (where *h* stands for the "height" of the cap) and the relationship *h*/*r* = 1/2*π* holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2*θ*, with *θ* given by:

This angle corresponds to the plane aperture angle of 2*θ* ≈ 1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to 1/4*π* of a complete sphere, or to (180°/*π*)^{2}_{} ≈ 3282.80635 square degrees.

The solid angle of a cone whose cross-section subtends the angle 2*θ* is:

Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams.^{ [4] }^{ [5] } Other multiples are rarely used.

In physics, the **cross section** is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted *σ* (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

The **radian**, denoted by the symbol **rad**, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit. The radian is defined in the SI as being a dimensionless unit, with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, an ** n-sphere** or a

In optics, **Lambert's cosine law** says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle *θ* between the direction of the incident light and the surface normal; I = I_{0}cos(*θ*). The law is also known as the **cosine emission law** or **Lambert's emission law**. It is named after Johann Heinrich Lambert, from his *Photometria*, published in 1760.

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle at that point.

In geometry, a **circular segment**, also known as a **disk segment**, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by a circular arc and by the circular chord connecting the endpoints of the arc.

A **central angle** is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called *spherical distance*.

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. It is a *directional* quantity: the radiance of a surface depends on the direction from which it is being observed.

In geometry, a **spherical cap** or **spherical dome** is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a *hemisphere*.

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, while that of spectral intensity in frequency is the watt per steradian per hertz and that of spectral intensity in wavelength is the watt per steradian per metre —commonly the watt per steradian per nanometre. 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**.

**Etendue** or **étendue** 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. It is a central concept in nonimaging optics.

In classical mechanics, the **shell theorem** gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

A **square degree** (**deg ^{2}**) is a non-SI unit measure of solid angle. Other denotations include

In electromagnetics, **directivity** is a parameter of an antenna or optical system which measures the degree to which the radiation emitted is concentrated in a single direction. It is the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. Therefore, the directivity of a hypothetical isotropic radiator is 1, or 0 dBi.

A **golden triangle**, also called a **sublime triangle**, is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:

A **circular arc** is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the **minor arc**, subtends an angle at the centre of the circle that is less than π radians ; and the other arc, the **major arc**, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a *chord* of a circle. If the length of an arc is exactly half of the circle, it is known as a *semicircular arc*.

The **goat problem** is either of two related problems in recreational mathematics involving at least figuratively a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing the exterior of a circular area.

In the field of heat transfer, **intensity of radiation** is a measure of the distribution of radiant heat flux per unit area and solid angle, in a particular direction, defined according to

In geometry, a **spherical sector**, also known as a **spherical cone**, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap. It is the three-dimensional analogue of the sector of a circle.

- ↑ Stutzman, Warren L; Thiele, Gary A (2012-05-22).
*Antenna Theory and Design*. ISBN 978-0-470-57664-9. - ↑ Woolard, Edgar (2012-12-02).
*Spherical Astronomy*. ISBN 978-0-323-14912-9. - ↑ "Steradian",
*McGraw-Hill Dictionary of Scientific and Technical Terms*, fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997. ISBN 0-07-052433-5. - ↑ Stephen M. Shafroth, James Christopher Austin,
*Accelerator-based Atomic Physics: Techniques and Applications*, 1997, ISBN 1563964848, p. 333 - ↑ R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer"
*IRE Transactions on Antennas and Propagation***9**:1:22-30 (1961)

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