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steradian

 
Dictionary: ste·ra·di·an   (stĭ-rā'dē-ən) pronunciation
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n. (Abbr. sr)
A unit of measure equal to the solid angle subtended at the center of a sphere by an area on the surface of the sphere that is equal to the radius squared: The total solid angle of a sphere is 4π steradians.

[STE(REO)- + RADIAN.]


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Chemistry Dictionary: steradian
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Home > Library > Science > Chemistry Dictionary

Symbol sr. The dimensionless (supplementary) SI unit of solid angle equal to the solid angle that encloses a surface on a sphere equal to the square of the radius of the sphere.


Measures and Units: steradian
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sterad

solid angle. Symbol sr. (Metric) For the apex angle of a cone of any shape whose sides are radii of a sphere, 1 sr is the angle subtended by a section of surface of area equal to the square of the radius. Quantitatively defined for the apex angle of any such cone as the ratio of the circumferential surface area enclosed by the cone to the square of the radial length. Since the surface area of a sphere is 4π times the square of the radius, a complete sphere equals 4π sr = 12.566 37~ sr; hence 1 sr = 1/4π sphere = 0.079 577 47~ sphere. The triangular cone defined by a Pole on Earth and one radian (57.3~ °) of longitude along the Equator has central angle equal to 1 sr.

Creating an equivalent base unit in the SI, giving dimensionality to the plane angle, has been proposed.
[Eder W. E. Metrologia Vol. 18, 1-12 (1982)]
[Eder W. E. Metrologia Vol. 19, 1-8 (1983)]
[Torrens A. B. Metrologia Vol. 22, 1-7 (1986)]
[Torrens A. B. Metrologia Vol. 23, 57-8 (1986)] (See also spat.) However, the steradian of the SI, once a supplementary unit, has been a dimensionless derived unit since 1980.
[Giacomo P. Metrologia Vol. 17, 69-74 (1981)]

Unlike plane angle, where degree offers a scheme of essentially rational number values, there is no rational scheme for solid angle other than as fractions of the sphere.

For history, See radian.

Wikipedia: Steradian
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A graphical representation of 1 steradian.

The steradian (symbol: sr) is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane. The name is derived from the Greek stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is dimensionless because 1 sr = m2·m−2 = 1. It is useful, however, to distinguish between dimensionless quantities of different nature, so in practice the symbol "sr" is used where appropriate, rather than the derived unit "1" or no unit at all. For example, radiant intensity can be measured in watts per steradian (W·sr−1).

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Definition

Section of cone (1) and spherical cap (2) inside a sphere

A steradian is defined as the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere whose area, A, equals r2.[1]

Since A = r2, it corresponds to the area of a spherical cap (A = 2πrh), and the relationship \frac{h}{r}=\frac{1}{2\pi} holds. Therefore, the solid angle of the simple cone subtending an angle θ is given by:

\begin{align} \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} \mbox{ or } 32.77^\circ \end{align}

This angle corresponds to an apex angle of 2θ ≈ 1.144 rad or 65.54°.

Because the surface area of this sphere is 4πr2, the definition implies that a sphere measures 4π ≈ 12.56637 steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr. A steradian can also be called a squared radian.

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 steradian was formerly an SI supplementary unit, but this category was abolished from the SI in 1995 and the steradian is now considered an SI derived unit.

Analogue to radians

In two dimensions, the angle in radians is related to the arc length it cuts out:

\theta = \frac{l}{r} \,
where
l is arc length, and
r is the radius of the circle.

Now in three dimensions, the solid angle in steradians is related to the area it cuts out:

\Omega = \frac{S}{r^2} \,
where
S is the surface area, and
r is the radius of the sphere.

SI multiples

Steradians only go up to 4π ≈ 12.56637, so the large multiples are not usable for the base unit, but could show up in such things as rate of coverage of solid angle, for example.

Multiple Name Symbol
101 decasteradian dasr
100 steradian sr
10−1 decisteradian dsr
10−2 centisteradian csr
10−3 millisteradian msr
10−6 microsteradian µsr
10−9 nanosteradian nsr
10−12 picosteradian psr
10−15 femtosteradian fsr
10−18 attosteradian asr
10−21 zeptosteradian zsr
10−24 yoctosteradian ysr

See also

References

  1. ^ "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.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.  Read more
Chemistry Dictionary. A Dictionary of Chemistry. Sixth Edition. Copyright © Market House Books Ltd, 2008. All rights reserved.  Read more
Measures and Units. A Dictionary of Weights, Measures, and Units. Copyright © Donald Fenna 2002, 2004. All rights reserved.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Steradian".  Read more