Solid angle
From Wikipedia, the free encyclopedia
The solid angle, Ω, that an object subtends at a point is a measure of how big that object appears to an observer at that point. For instance, a small object nearby could subtend the same solid angle as a large object far away. The solid angle is proportional to the surface area, S, of a projection of that object onto a sphere centered at that point, divided by the square of the sphere's radius, R. (Symbolically, Ω = k S/R², where k is the proportionality constant.) A solid angle is related to the surface area of a sphere in the same way an ordinary angle is related to the circumference of a circle.
If the proportionality constant is chosen to be 1, the units of solid angle will be the SI steradian (abbreviated sr). Thus the solid angle of a sphere measured at its center is 4π sr, and the solid angle subtended at the center of a cube by one of its sides is one-sixth of that, or 2π/3 sr. Solid angles can also be measured (for k = (180/π)²) in square degrees or (for k = 1/4π) in fractions of the sphere (i.e., fractional area).
One way to determine the fractional area subtended by a spherical surface is to divide the area of that surface by the entire surface area of the sphere. The fractional area can then be converted to steradian or square degree measurements by the following formulae:
- To obtain the solid angle in steradians, multiply the fractional area by 4π.
- To obtain the solid angle in square degrees, multiply the fractional area by 4π × (180/π)², which is equal to 129600/π.
Contents |
[edit] Practical applications
- Defining luminous intensity and luminance
- Calculating spherical excess E of a spherical triangle
- The calculation of potentials by using the Boundary Element Method (BEM)
- Evaluating the size of ligands in metal complexes, see ligand cone angle.
- Calculating the electric field and magnetic field strength around charge distributions.
[edit] Solid angles for common objects
[edit] Triangle on a sphere
An efficient algorithm for calculating the solid angle Ω subtended by a triangle with vertices A, B and C, as seen from the origin has been given by Oosterom and Strackee (IEEE Trans. Biom. Eng., Vol BME-30, No 2, 1983):
![\tan \left( \frac{1}{2} \Omega \right) = \frac{[\vec a \vec b \vec c]}{ abc + (\vec a \cdot \vec b)c + (\vec a \cdot \vec c)b + (\vec b \cdot \vec c)a}](../../../math/7/a/8/7a871c13de0147d59f7b5229c78c26ae.png)
,
where:
![[\vec a \vec b \vec c]](../../../math/c/0/a/c0a886cd46e31c60a95c37058b281294.png)
denotes the determinant of the matrix that results when writing the vectors together in a row, e.g. 
and so on--this is also equivalent to the scalar triple product of the three vectors;
is the vector representation of point A, while a is the module of that vector (the origin-point distance);
denotes the scalar product.
[edit] Cone, spherical cap, hemisphere
The area of a spherical cap on a unit sphere is the solid angle of a cone with apex angle a, that is, 
.
(The above result is found by computing the following double integral using the unit surface element in spherical polars):
![\int_0^{2\pi} \int_0^{\theta} \sin \theta \ d \theta \ d \phi = 2\pi\int_0^{\theta} \sin \theta \ d \theta = 2\pi\left[ -\cos \theta \right]_0^{\theta} \ = 2\pi\left(1 -\cos \theta \right)](../../../math/5/7/7/5779448cfaba7f4f9d9fceab0aaa3aba.png)
When a = π, (θ = π/2), the spherical cap becomes a hemisphere having a solid angle 2π.
[edit] Pyramid
The solid angle of a four-sided right rectangular pyramid with apex angles a and b (measured to the faces of the pyramid) is 
.
[edit] Latitude-longitude rectangle
The solid angle of a latitude-longitude rectangle on a globe is 
, where φN and φS are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and θE and θW are east and west lines of longitude (where the angle in radians increases eastward).[1] Mathematically, this represents an arc of angle φN − φS swept around a sphere by θE − θW radians. When longitude spans 2π radians and latitude spans π radians, the solid angle is that of a sphere.
A latitude-longitude rectangle should not be confused with the solid angle of a rectangular pyramid. A rectangular pyramid intersects the surface of a sphere such that all four sides of the pyramid intersect the sphere's surface in great circle arcs. For a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.
[edit] Sun and Moon
The Sun and Moon are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere or about 6×10-5 steradian.[2]
[edit] Solid angle in arbitrary dimension
The volume of the unit sphere can be defined in any dimension. One often needs this solid angle factor in calculations with spherical symmetry.
Where Γ is the Gamma function.
