Note: You will need pencil and paper for this blog.
Is it possible to predict how high the sun rises on a given day? If so, can we find this information for State College, PA?
Consider the Earth during an equinox.
Recall that the sun is directly overhead of the equator at this time. To represent this, draw a circle with a light ray coming from the right pointing towards the center of the circle. Now, assuming the sun’s light rays are parallel to each other (in reality, they are not so because the sun is spherical, but Earth’s distance from the sun mitigates this effect), draw another light ray parallel to the one already drawn that points above the center of the circle. Draw a tangent line where the second light ray strikes the circle’s edge. The angle between the second light ray and tangent line approximates how high the sun rises at that location on the circle’s edge during an equinox. For the sake of mathematical discussion, call this angle θ.
Extend the first light ray to the center of the circle. Using an appropriate radius, connect the location where the second light ray intersects the circle to the center of that circle as well. The angle between these segments is just the location’s latitude. We define that as the angle ϕ.
Now extend the tangent line until it intersects the first light ray. Notice the triangle formed by this extended line, the first light ray, and the radius that connects the center of the circle and intersection of the second light ray with the circle is right because the tangent to a circle at a point is always perpendicular to the radius which connects that point to the circle’s center.
Two of the angles in this triangle are fairly obvious—ϕ and 90°. It can be shown by the alternating interior angle theorem of parallel lines that the third angle is θ. These three angles must add to 180°.
Thus,
θ + ϕ + 90° = 180°
Therefore,
θ = 90° – ϕ
During an equinox at State College (latitude of 40.8°N), we expect the highest solar zenith angle to be 49.2°.
What if we wish to find the maximum solar zenith angle at any time of the year? Using the same methods presented here, I invite you to come up with your own solution.
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