Steradian

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Steradian
Steradian.svg
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 systemSI derived unit
Unit ofSolid angle
Symbolsr 

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[edit]

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[edit]

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

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[edit]

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[edit]

Notes[edit]

References[edit]

  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)