Solid Angle

Radians

You should (hopefully) be familiar with what Radians are, but just as a reminder:

The important thing is that the arc length \(s = \alpha r\) (if the units of length for \(r\) and \(s\) are the same), i.e. we can find \(\alpha\) in radians using:

\[\alpha = \frac{s}{r}\]

Definition

We can define something similar for areas on a sphere:

such that the Area \(A = \omega r^2\) (if the units of length squared for \(r^2\) and \(A\) are the same), that is:

\[\omega = \frac{A}{r^2}\]

Where \(\omega\) is the solid angle subtended by the area. Solid angle can also be represented by \(\Omega\) instead.

If A is a circle of radius \(R\):

\[\omega = \pi\left(\frac{R}{r}\right)^2\]

For a cone

The solid angle of a cone with opening angle \(\theta\) is:

\[\Omega = 2\pi(1-\cos\theta) \approx \pi\theta^2\]

The approximation is more accurate when \(\theta\) is very small (which for astronomical objects this is usually the case). When \(\theta\) is small, \(\cos\theta \approx 1-\frac{\theta^2}{2}\)

Integration

If for some reason you need to integrate to find it, you can use:

\[d\Omega = \sin\theta d\theta d\phi\]