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The following equation is often used to describe the flux at a receiver:

$$F_\nu = \int I_\nu \cos(\theta)\,\mathrm d\Omega.$$

Where $I_\nu$ is the specific intensity.

Could somebody explain with pictures how $d\Omega$ and $\theta$ are defined as seen from the observer?

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$\theta$ is the angle between the normal vector $\mathbf{\hat{n}}$ of some infinitesimal surface $dA$. The solid angle is then given by $$d\Omega = \sin(\theta)\mathrm{d}\theta \mathrm{d}\phi,$$

where $\theta,\phi$ are just the regular spherical coordinates. We can always write this in this way, but the integral limits can become quite complicated in spherical coordinates, then one should use other coordinates.

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