Steradian

steradian
steradian
	A graphical representation of two different steradians.; The sphere has radius r, and in this case the area A of the highlighted spherical cap is r2. The solid angle Ω equals [A/r2] sr which is 1 sr in this example. The entire sphere has a solid angle of 4π sr.
General information
Unit system	SI
Unit of	solid angle
Symbol	sr
Conversions
1 sr in ...	... is equal to ...
SI base units	1 m2/m2
square degrees	⁠1802/π2⁠ deg2; ≈ 3282.8 deg2

The 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. A solid angle in the form of a right circular cone projected onto a sphere, gives the area of a spherical cap on the surface, whereas a plane angle projected onto a circle, gives the length of a circular arc on the circumference. The area may be any shape. The magnitude of the solid angle expressed in steradians is defined as the quotient of the surface area and the square of the sphere's radius. The name is derived from the Greek στερεός stereos 'solid' + radian.

In the International System of units (Système Internationale d'unités or SI), the steradian is considered to be a dimensionless unit, the quotient of the area projected onto a surrounding sphere and the square of the sphere's radius. Both the numerator and denominator of this ratio have dimension length squared (i.e. L²/L² = 1, dimensionless). In the SI, "solid angle" is therefore the number of steradians in the physical solid angle: (solid angle)/sr. So the SI "steradian" is the number of physical steradians in one physical steradian. It is useful, however, 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 to indicate a physical solid angle. The steradian is defined as one square radian: sr = 1 rad². The SI "steradian" is therefore "sr" = (1 sr)/sr = 1. 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.

Definition

A steradian can be defined as the solid angle subtended at the centre of a unit sphere by a unit area (of any shape) 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]

A solid angle in the form of a circular cone is related to the area it cuts out of a sphere:

\Omega ={\frac {A}{r^{2}}}\ {\text{sr}}\,={\frac {2\pi h}{r}}\ {\text{sr}},

where

$Ω$ is the solid angle
$A$ is the surface area of the spherical cap, $2\pi rh$ ,
$r$ 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 $1/4 π \approx 0.07958$ of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is $4 π sr$ .

Other properties

The area of a spherical cap is $A = 2 πrh$ , where $h$ is the "height" of the cap. If $A = r 2$ , then ${\tfrac {h}{r}}={\tfrac {1}{2\pi }}$ . From this, one can compute the plane aperture angle $2 θ$ of the cross-section of a simple cone whose solid angle equals one steradian:

\theta =\arccos \left({\frac {r-h}{r}}\right)=\arccos \left(1-{\frac {h}{r}}\right)=\arccos \left(1-{\frac {1}{2\pi }}\right),

giving $θ \approx$ 0.572 rad or 32.77° and $2 θ \approx$ 1.144 rad or 65.54°.

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

\Omega =2\pi (1-\cos \theta ){\text{ sr}}=4\pi \sin ^{2}\left({\frac {\theta }{2}}\right){\text{ sr}}.

A steradian is also equal to ${\tfrac {1}{4\pi }}$ of a complete sphere (spat), to $\left({\tfrac {360^{\circ }}{2\pi }}\right)^{2}$ $\approx$ 3282.80635 square degrees, and to the spherical area of a polygon having an angle excess of 1 radian.^{[clarification needed]}

SI multiples

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

References

^ Stutzman, Warren L; Thiele, Gary A (2012-05-22). Antenna Theory and Design. John Wiley & Sons. ISBN 978-0-470-57664-9.
^ Woolard, Edgar (2012-12-02). Spherical Astronomy. Elsevier. 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)

External links

Media related to Steradian at Wikimedia Commons

[1] Stutzman, Warren L; Thiele, Gary A (2012-05-22). Antenna Theory and Design. John Wiley & Sons. ISBN 978-0-470-57664-9.

[2] Woolard, Edgar (2012-12-02). Spherical Astronomy. Elsevier. ISBN 978-0-323-14912-9.

[3] "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.

[4] Stephen M. Shafroth, James Christopher Austin, Accelerator-based Atomic Physics: Techniques and Applications, 1997, ISBN 1563964848, p. 333

[5] 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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v t e SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 revision of the SI System of units of measurement
Category

Definition

Other properties

SI multiples

See also

References

External links