Import standard modules:
In [ ]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML
HTML('../style/course.css') #apply general CSS
Import section specific modules:
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pass
Recall that in two dimensions the plane angle $\theta$ subtended by a circular arc $l$ with radius $r$ is defined as $\frac{l}{r}$. In the same way the solid angle $\Omega$ subtended by a portion of a sphere's surface area $A$ with a radius $r$ is $\frac{A}{r^2}$. These definitions are illustrated in Fig. 2.12.1 ⤵ .
The plane angle is measured in radians (rad), while the solid angle is measured in steradians (sr).
Simple Angle Conversions
• 1 degree ($^{\circ}$) = 60 arcminutes ($'$) = 3600 arcseconds ($''$), where 360$^{\circ}$ = circle.
• 1 radian (rad) = 57.296$^{\circ}$ = 206265$''$, where 2$\pi$ rad = circle.
• 1 hour (h) = 60 minutes = 3600 seconds, where 24 h = circle
• 1 h = 15$^{\circ}$
• 1 steradian (sr) = 1 rad$^2$ = 3282.8 deg$^2$ = 4.25 x $10^{10}$ arcsec$^2$, where 4$\pi^2$ sr = sphere
The solid angle subtended by the surface area of an entire sphere with a radius of $r$ can be computed as follows: \begin{equation} \Omega_{\textrm{spere}}=\frac{4\pi r^2}{r^2}=4\pi~\textrm{sr}. \end{equation}
The spherical coordinates $(\rho,\theta,\phi)$ of a point $P$ are illustrated in Fig. 2.12.2 ⤵ .
The Cartesian coordinates $(x,y,z)$ of $P$ can be obtained from its spherical coordinates $(\rho,\theta,\phi)$ via:
Relation between spherical and Cartesian coordinates
\begin{eqnarray}
x &=& \rho \sin \theta \cos \phi\\
y &=& \rho \sin \theta \sin \phi \nonumber \\
z &=& \rho \cos \phi \nonumber
\end{eqnarray}
If we fix $\rho$ and $\theta$ and we vary $\phi$, $P$ maps out the red circle in Fig. 2.12.3 ⤵ . The radius of the red circle is equal to $\rho \sin \theta$. If the azimuth angle changes by $d\phi$, the arc length along the red circle changes by $\rho\sin\theta d\phi$. If $\rho$ and $\phi$ are fixed and $\theta$ is varied, $P$ traverses out the blue circle. If the polar angle changes by $d\theta$, the arc length along the blue circle changes by $\rho d\theta$. The infinitesimal area $dS$ depicted in Fig. 2.12.3 ⤵ is therefore equal to
\begin{equation} dS = \rho^2 \sin \theta d\theta d\phi. \end{equation}It now follows from the definition of the solid angle that the infinitesimal solid angle element occupied by the surface element $dS$ is equal to (see Gaskinetic theory ⤴)
\begin{equation} d\Omega = \frac{\rho^2 \sin \theta d\theta d\phi}{\rho^2}=\sin \theta d\theta d\phi. \end{equation}</p>