Steradian
| Steradian | |
|---|---|
 A graphical representation of 1 steradian. The sphere has radius r, and in this case the area A of the highlighted surface patch 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 4πsr. | |
| General information | |
| Unit system | SI derived unit |
| Unit of | Solid angle |
| Symbol | sr |
The steradian (symbol: sr) or square radian[1][2] is the SI unit of solid angle. 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 area subtended and the square of its distance from the center: 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 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.
Definition
A steradian can be defined as the solid angle subtended at the center 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 = r2 subtends one steradian at its center.[3]
The solid angle is related to the area it cuts out of a sphere:
- where
- A is the surface area of the spherical cap, ,
- r is the radius of the sphere, and
- sr is the unit, steradian.
Because the surface area A of a sphere is 4πr2, the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its center. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.
Other properties
If A = r2, it corresponds to the area of a spherical cap (A = 2πrh) (where h stands for the "height" of the cap) and the relationship h/r = 1/2π 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 2θ, with θ given by:
This angle corresponds to the plane aperture angle of 2θ ≈ 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 1/4π 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 2θ is:
- .
SI multiples
Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams.[4][5] Other multiples are rarely used.
Solid angles over 4π steradians—the solid angle of a full Euclidean sphere—are rarely encountered.
See also
Notes
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)
