Solid angle

In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, ${\displaystyle 4\pi }$. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.


A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle (and therefore apparent size). This is evident during a solar eclipse.

Solid angles in arbitrary dimensions[edit]

The solid angle subtended by the complete (d − 1)-dimensional spherical surface of the unit sphere in d-dimensional Euclidean space can be defined in any number of dimensions d. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula $${\displaystyle \Omega _{d}={\frac {2\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}},}$$ where Γ is the gamma function. When d is an integer, the gamma function can be computed explicitly.^[10] It follows that $${\displaystyle \Omega _{d}={\begin{cases}{\frac {1}{\left({\frac {d}{2}}-1\right)!}}2\pi ^{\frac {d}{2}}\ &d{\text{ even}}\\{\frac {\left({\frac {1}{2}}\left(d-1\right)\right)!}{(d-1)!}}2^{d}\pi ^{{\frac {1}{2}}(d-1)}\ &d{\text{ odd}}.\end{cases}}}$$


This gives the expected results of 4π steradians for the 3D sphere bounded by a surface of area 4πr² and 2π radians for the 2D circle bounded by a circumference of length 2πr. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval [−r, r] and this is bounded by two limiting points.


The counterpart to the vector formula in arbitrary dimension was derived by Aomoto^[11][12] and independently by Ribando.^[13] It expresses them as an infinite multivariate Taylor series: $${\displaystyle \Omega =\Omega _{d}{\frac {\left|\det(V)\right|}{(4\pi )^{d/2}}}\sum _{{\vec {a}}\in \mathbb {N} _{0}^{\binom {d}{2}}}\left[{\frac {(-2)^{\sum _{i<j}a_{ij}}}{\prod _{i<j}a_{ij}!}}\prod _{i}\Gamma \left({\frac {1+\sum _{m\neq i}a_{im}}{2}}\right)\right]{\vec {\alpha }}^{\vec {a}}.}$$ Given d unit vectors ${\displaystyle {\vec {v}}_{i}}$ defining the angle, let V denote the matrix formed by combining them so the ith column is ${\displaystyle {\vec {v}}_{i}}$, and ${\displaystyle \alpha _{ij}={\vec {v}}_{i}\cdot {\vec {v}}_{j}=\alpha _{ji},\alpha _{ii}=1}$. The variables ${\displaystyle \alpha _{ij},1\leq i<j\leq d}$ form a multivariable ${\displaystyle {\vec {\alpha }}=(\alpha _{12},\dotsc ,\alpha _{1d},\alpha _{23},\dotsc ,\alpha _{d-1,d})\in \mathbb {R} ^{\binom {d}{2}}}$. For a "congruent" integer multiexponent ${\displaystyle {\vec {a}}=(a_{12},\dotsc ,a_{1d},a_{23},\dotsc ,a_{d-1,d})\in \mathbb {N} _{0}^{\binom {d}{2}},}$ define ${\textstyle {\vec {\alpha }}^{\vec {a}}=\prod \alpha _{ij}^{a_{ij}}}$. Note that here ${\displaystyle \mathbb {N} _{0}}$ = non-negative integers, or natural numbers beginning with 0. The notation ${\displaystyle \alpha _{ji}}$ for ${\displaystyle j>i}$ means the variable ${\displaystyle \alpha _{ij}}$, similarly for the exponents ${\displaystyle a_{ji}}$. Hence, the term ${\textstyle \sum _{m\neq l}a_{lm}}$ means the sum over all terms in ${\displaystyle {\vec {a}}}$ in which l appears as either the first or second index. Where this series converges, it converges to the solid angle defined by the vectors.