# An integral for the Earth's insolation

Consider the function $$[-\pi/2,\pi/2] \ni \theta \mapsto s_\beta(\theta) = \int_0^{2\pi} \sqrt{ 1 - \left(\cos \theta \sin \beta \cos \gamma - \sin \theta \cos \beta \right)^2} \, d \gamma$$ for $\beta \in [0, \pi/2]$. This function is proportional to the average insolation (amount of solar energy) that reaches the top of the atmosphere of a planet with obliquity (axial tilt) $\beta$, at latitude $\theta$. So $\theta = 0$ at the equator and $\theta = \pi/2$ at the north pole. If $\beta = 0$, the planet's axis of rotation is perpendicular to its ecliptic plane, and if $\beta = \pi/2$, this axis is parallel to the plane.

Numerical evidence suggests that $s_\beta$ is an even function of $\theta$.

I am looking for a proof.

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The integrand is invariant under $\theta \to -\theta$ and $\gamma \to \gamma + \pi$. – achille hui Feb 1 '13 at 21:48
@achillehui That looks like an answer to me. – Alex Becker Feb 1 '13 at 21:51
Looks correct to me too. Please turn it into an answer so I can accept it. – Hans Engler Feb 1 '13 at 22:01

The integrand is invariant under $\theta \to -\theta$ and $\gamma \to \gamma + \pi$.