In geometry, a solid angle (symbol: Ω) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr), which is equal to one square radian, sr = rad2. One steradian corresponds to one unit of area (of any shape) 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, 4 π {\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.
Definition and properties
The magnitude of an object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the apex, that the object covers. Giving the area of a segment of a unit sphere in steradians is analogous to giving the length of an arc of a unit circle in radians. Just as the magnitude of a plane angle in radians at the vertex of a circular sector is the ratio of the length of its arc to its radius, the magnitude of a solid angle in steradians is the ratio of the area covered on a sphere by an object to the square of the radius of the sphere. The formula for the magnitude of the solid angle in steradians is
Ω = A r 2 , {\displaystyle \Omega ={\frac {A}{r^{2}}},}
where A {\displaystyle A} is the area (of any shape) on the surface of the sphere and r {\displaystyle r} is the radius of the sphere. Solid angles are often used in astronomy, physics, and in particular astrophysics. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction.
The solid angle of a sphere measured from any point in its interior is 4π sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2π/3 sr. The solid angle subtended at the corner of a cube (an octant) or spanned by a spherical octant is π/2 sr, one-eighth of the solid angle of a sphere. Solid angles can also be measured in square degrees (1 sr = (180/π)2 square degrees), in square arc-minutes and square arc-seconds. It can also be expressed in fractions of the sphere (1 sr = 1/4π fractional area), also known as spat (1 sp = 4π sr). In spherical coordinates there is a formula for the differential,
d Ω = sin θ d θ d φ , {\displaystyle d\Omega =\sin \theta \,d\theta \,d\varphi ,}
where θ is the colatitude (angle from the North Pole) and φ is the longitude. The solid angle for an arbitrary oriented surface S subtended at a point P is equal to the solid angle of the projection of the surface S to the unit sphere with center P, which can be calculated as the surface integral:
Ω = ∬ S r ^ ⋅ n ^ r 2 d S = ∬ S sin θ d θ d φ , {\displaystyle \Omega =\iint _{S}{\frac {{\hat {r}}\cdot {\hat {n}}}{r^{2}}}\,dS\ =\iint _{S}\sin \theta \,d\theta \,d\varphi ,}
where r ^ = r → / r {\displaystyle {\hat {r}}={\vec {r}}/r} is the unit vector corresponding to r → {\displaystyle {\vec {r}}} , the position vector of an infinitesimal area of surface dS with respect to point P, and where n ^ {\displaystyle {\hat {n}}} represents the unit normal vector to dS. Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product r ^ ⋅ n ^ {\displaystyle {\hat {r}}\cdot {\hat {n}}} . Thus one can approximate the solid angle subtended by a small facet having flat surface area dS, orientation n ^ {\displaystyle {\hat {n}}} , and distance r from the viewer as:
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