Why is the amount of Mars that appears dark directly proportional to angle SME?
Let O be the center of Mars. Imagine we were to draw a line segment between the center of the sun and O. Let X be the point where this line segment intersects the surface of Mars. Imagine we were to draw a line segment between O and the center of Earth. Let Y be the point where this line segment intersects the surface of Mars. We can draw a big circle that goes through the points X and Y and all the way around Mars. Imagine we are on Earth looking at Mars and it appears partially illumined, then let Z be the point where the boundary line between the dark and light regions intersects the circle. And let W be the point where the other edge of the dark region of Mars intersects the circle. These points are indicated in figure below, where the illumined half of Mars is yellow and the dark half is gray. Note that the lower half of Mars only is visible from Earth. A full one half (180 degrees) of Mars is actually illuminated. The line segment from X to O is directly down the middle of this,
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