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 can be projected onto a sphere, defining 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 defines a circular arc on the circumference, whose length is proportional to the angle. Steradians can be used to measure a solid angle of any shape. The solid angle subtended is the same as that of a cone with the same projected area.

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⁻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.

The name steradian is derived from the Greek στερεός stereos 'solid' + radian.

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, sr = rad².

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