Angular Quantities

In astronomy, angular quantities are generally expressed in sexagesimal units. The main units of measure of plane and solid angles are the following:

$$1~\textrm{degree}=1~\textrm{deg}~,$$

$$1~\textrm{second~of~arc}=1~\textrm{arcsec}=1~\textrm{as}=\frac{1}{3600}~\textrm{deg}=10^3~\textrm{mas}=10^6~\mu\textrm{as}~,$$

$$1~\textrm{radian}=1~\textrm{rad}=\frac{180}{\pi}~\textrm{deg}=\frac{648000}{\pi}~\textrm{arcsec}~,$$

$$1~\textrm{square~degree}=1~\textrm{deg}^2~,$$

$$1~\textrm{steradian}=1~\textrm{sterad}=1~\textrm{sr}=\frac{32400}{\pi^2}~\textrm{deg}^2=\frac{4.2\cdot10^{11}}{\pi^2}~\textrm{arcsec}^2~.$$

The whole sky spans a solid angle

$$\Omega_{sky}=4\,\pi~\textrm{sterad}=\frac{129600}{\pi}~\textrm{deg}^2=41253~\textrm{deg}^2=\frac{1.68\cdot10^{12}}{\pi}~\textrm{arcsec}^2~,$$

while the sky region where the absolute value of the Galactic (or Ecliptic, for that matter) latitude $b$ is smaller than a given value $\phi$ measures

$$\Omega\left(\vert b\vert<\phi\right)=4\,\pi\,\sin\phi~~~\textrm{[sr]}~.$$