Question

Determine in steradians what portion of the sun’s direction is occupied by the earth.

Answer 1

The solid angle is the angle created in three-dimensional space that an object subtends at a point. Solid angle is a measure of how big an object appears to an observer when looking from that point. The symbol for solid angle is Ω (Greek letter omega).

A solid angle can be defined in terms of an area on the surface of a sphere which is centred at the vertex of the angle. This is the equivalent of defining a planar angle in terms of the length of arc on a circle subtended to the centre. Solid angle can also be defined as an angle formed by three or more planes intersecting at a common point (the vertex). The SI unit of solid angle is the steradian (sr). The solid angle of a complete sphere is 4π sr. The solid angle corresponding to the face of a cube measured at the centre is 2π/3 sr. One steradian is equal to (180/π)² square degrees.

First let us consider a reference circle and an ordinary angle. The angle θ can also be explained in terms of the arc. Let s be the length of the arc and r be the radius of the circle. The ordinary angle in radians is given by θ = (s/r). or, in degrees it is defined as θ = (360/2π)(s/r). Now assume a cone which intersects the sphere of radius R. Consider S be the area of surface subtended by the intersection of the sphere and the cone. The solid angle is defined Ω = (S/r²). This defines the solid angle in steradians.

If the surface covers the entire sphere then the number of steradians is 4π. If you know the solid angle Ω in steradians then you can easily calculate the corresponding area of the surface of any sphere from the expression S = R²Ω, where R is the radius of the sphere..

The Sun and Moon are about the same apparent size when seen from the earth. They occupy an area of 0.001% of the celestial hemisphere, which equates to a solid angle of approximately 6 × 10^-5 steradians.