THE SUN IS A MASS OF INCANDESCENT GAS: SDB explains that slightly less than half the earth is lit by the sun at any given time and near the end he gives a list of assumptions that he used to simplify the problem. Most of the assumptions are fine with me because they represent local earth conditions (altitude, weather, imperfect shape of the planet, &c.), but one of them sticks out: he treats the sun as a point light source rather than as a disk, and I think that incorporating this assumption changed his answer.

Think of the point on the equator on one side, just at dawn, and at the other side, just at dusk. They're the two points on the equator which are both lit which are furthest apart. The light arriving at those two points isn't parallel. Those two points plus the sun itself form an isosceles triangle whose base is the diameter of the earth (about 7900 miles) and height is the distance to the sun, about 93 million miles). Do the math and what you find is that there's a difference just shy of .005 degrees between the two, and that means that the sun is actually illuminating just slightly less than 50% of that circle. In order for it to be exactly 50%, those two light beams would have to be exactly parallel, and the light source would have to be infinitely far away.
I agree, except that the sun is a disk approximately 0.5 degrees across when viewed from the earth. This changes SDB's reasoning because the man standing at the sunset will be looking at one edge of the sun, while the man standing at the sunrise will be looking at the other, opposite edge, and those two edges will be 0.5 degrees apart.

The rays of light can be parallel for both viewers at the same time, or come in at an angle even greater than 90 degrees. In fact, slightly more than half the earth is illuminated by the sun at any given time, and this would be true no matter how small the sun appeared in our sky, as long as it's larger than a point.



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