“I returned day before yesterday,” he answered, while he leaned his arm on the keys, bringing forth a crash of discordant sound.
“Day before yesterday!” she repeated, aloud; and went on thinking to herself, “day before yesterday,” in a sort of an uncomprehending way. She had pictured him seeking her at the very first hour, and he had lived under the same sky since day before yesterday; while only by accident had he stumbled upon her. Mademoiselle must have lied when she said, “Poor fool, he loves you.”
- The Awakening, by Kate Chopin
In 12th grade English we read Kate Chopin’s The Awakening. I wasn’t a fan. For a supposedly feminist piece of literature the protagonist Edna is really not quite a model of empowerment. She emancipates herself and promptly wrecks her life pretty comprehensively. Most of the self-sufficient women I know have managed their lives quite adroitly. I am not a literary critic however, and I could be missing the point. But my point here is quite different. The point is that Edna is disappointed that Robert has been under the same sky for two days without visiting her.
Under the same sky? That always struck me as a rather beautiful turn of phrase. Stand outside and look to the horizon. Consider the horizon to be the edge of the dome of the sky and whatever is within your horizon is under the same sky as yourself. You might like to know just exactly how far that is. Literature wonders, science answers.
Call the radius of the earth R. Call your own height h. Call the line from your eyes to the horizon x. Let’s let Apollo 17 provide the image, with h greatly exaggerated for clarity.
We’re interested in the distance x; or more accurately we’re interested in the arc length of the earth just beneath that line. If the angle between the lines R and R + h is denoted by θ, we know that arc length (the distance to the horizon) is simply Rθ. And θ is by simple trig just
Therefore our answer, the ground distance for your beloved to be under the same sky (call it S) with you is R times the above quantity.
That’s an exact answer, but we need not be so precise. The height h is very small compared to the other parameter of interest, R. Therefore nothing prevents us from very accurately approximating the above expression by the first term in its series expansion about h = 0. After a little bit of work, I get:
This approximation is extraordinarily accurate for human-sized values of h. We can get the same thing by just using the Pythagorean theorem to find x and dropping second-order terms in h. This is what we should expect, since for small angles the arc length approaches x. Nothing is stopping us from expressing the quantity (2R)^(1/2) in appropriate units to make our life even simpler. With S in miles and h in feet, the expression most convenient for calculation is
If Edna is 5 feet tall (really this should be the height of her eyes), her horizon is about 2.74 miles away. If you are closer to her than that, you and she are under the same sky.
Complications? Yes, there are a few. The earth is not perfectly spherical. Both its global oblateness and its local roughness will affect the result. The atmosphere provides an additional distorting effect, and we have not considered the potential problem of how to characterise a person who is only partially visible above the horizon. But perhaps we have subjected literature to enough for one day.