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

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. Whereas an angle in radians, projected onto a circle, gives a length of a circular arc on the circumference, a solid angle in steradians, projected onto a sphere, gives the area of a spherical cap on the surface. The name is derived from the Greek στερεός stereos 'solid' + radian.

The steradian 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. L²/L² = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different kind, such as the radian (a ratio of quantities of dimension length), 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.

Solid angle of countries and other entities relative to the Earth.

Definition

A steradian can be defined as the solid angle subtended at the centre of a unit sphere by a 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:

\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 1/4π (≈ 0.07958) of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is $4 π sr$ .

Other properties

Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere

If $A = r 2$ , it corresponds to the area of a spherical cap ( $A = 2 πrh$ , where $h$ is the "height" of the cap) and the relationship ${\tfrac {h}{r}}={\tfrac {1}{2\pi }}$ 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:

{\begin{aligned}\theta &=\arccos \left({\frac {r-h}{r}}\right)\\&=\arccos \left(1-{\frac {h}{r}}\right)\\&=\arccos \left(1-{\frac {1}{2\pi }}\right)\approx 0.572{\text{ rad, or }}32.77^{\circ }.\end{aligned}}

This angle corresponds to the plane aperture angle of $2 θ \approx$ 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 ${\tfrac {1}{4\pi }}$ of a complete sphere, or to $\left({\tfrac {180^{\circ }}{\pi }}\right)^{2}\approx$ 3282.80635 square degrees.

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

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

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. 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)

External links

Media related to Steradian at Wikimedia Commons

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

[2] Woolard, Edgar (2012-12-02). Spherical Astronomy. 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 2005–2019 definition 2019 redefinition Systems of measurement
Category

Definition

Other properties

SI multiples

See also

References

External links