Appendix

  What is a solid angle of 1 square degree ($1\Box^\circ$)? This is this the solid angle encompassed by taking 1 degree in one direction by 1 degree in the perpendicular direction. In other words,  

$$1\Box^\circ= \int d\Omega = \int_0^{1^{\circ}} \int_0^{1^{\circ}} d\theta d\phi.$$ (22)

If you simply integrate over the solid angle

$$\int d\Omega = \int_0^{1^{\circ}} \int_0^{1^{\circ}} \sin\theta\,d\theta d\phi$$

you will get the wrong answer. This solid angle is shown in Fig. 6.

  

Figure 6:                         Figure 7:  

The correct limits to use are

$$\int d\Omega = \int_0^{1^{\circ}} \int_{\frac{\pi}{2}}^{\frac{\pi}{2}+1^{\circ}} \sin\theta\,d\theta d\phi$$

which is shown in Fig. 7. Converting degrees to radians, radians. Now .So, we can rewrite the solid angle as

$$\int d\Omega = \int_0^{1^{\circ}}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}+1^{\circ}}\,d\theta d\phi.$$

Shifting the limits on $\theta$ by $\pi/2$, we see that this is the same as Equation (28).