Eyes on The Size

An excellent Fall day, perfect for a brisk walk around the park’s goose-governed lake. Suddenly there’s a goose-like yawp behind me. “Hey Moire, wait up, I got a question!”

“Afternoon, Mr Feder. What’s your question today?”

“You know how the Moon’s huge just after it gets over the horizon but then it gets small? How do they make it do that?”

“Well, ‘they’ is you, Mr Feder, except that nothing physically changes.”

“Whaddaya mean, I seen it change size every time there’s a full moon.”

“That’s what it looks like, but think it through. We’re here in the Midwest, two hours away from your folks back home in Fort Lee. Back when you lived there, did the Moon ever suddenly grow and then shrink when it was two hours up into the sky?”

“Um, no, just at the horizon. So you’re saying it’s one of them optical delusions?”

“Something like that. Here, I’ve got a video on Old Reliable. See how the disk stays the same size but it looks bigger in comparison to the railroad tracks? Your brain expects the tracks to be parallel lines despite the perspective, right, so it compensates by thinking the Moon must be wider when it’s next to them. In the real world you’ve looking at the Moon past trees or buildings, but the false perspective principle applies whether the horizon’s relatively close or far away.”

“Whaddaya mean, close or far horizon? It’s the edge of how far I can see and that’s always the same.”

“Oh, hardly, Mr Feder. You ever visit the Empire State Building’s observation deck?”

“Sure.”

“How about deep-sea fishing, out of sight of land?”

“Aw, that’s a blast, when you hook one of those big guys and you’re –“

“I’m sure you enjoyed it, but did you look around while you were waiting for a strike?”

“Yeah, nothin’ else to do but yammer and drink beer.”

“Mm-hm. So could you see as far from the boat’s deck as you could from the building’s deck?”

“Hey, you’re right. A lot farther from high up. They say on a clear day you can see 80 miles from the Empire State Building — nowhere near that from the boat, believe me. ‘S why they put those decks up there, I guess. How far up do I gotta be to see the whole world, I wonder.”

“Quick answer is, infinitely far away.”

“Wait, those astronauts got that ‘Blue Marble’ picture from the Moon and it showed the whole day side.”

“Take a closer look someday. It shows Antarctica but essentially nothing north of the 45th parallel. The limit’s set by the points on the planet where lines from your eye just graze the planet’s surface. The astronauts in this LEM, for instance, are about an Earth-radius away. They’d be able to see the Atlantic Ocean and a little bit of Brazil, but neither of the poles and no part of the USA.”

“Gimme a sec … yeah, I see how that works. So that ‘how high up you are‘ thing keeps going all the way out into space. There’s probably some complicated formula for it, right?”

“Not that complicated, just d=(h²+2Rh), where h is your height above the surface and R is the radius of the planet you’re looking at. Plug in the numbers and d gives you your distance to the horizon. For that LEM, for example, h is one Earth radius and R is one radius, so those straight lines are 3=1.73 Earth radii long.”

“How about the line on top of the ocean?”

“That’s a little more complicated.” <more tapping on Old Reliable> “Says here that line stretches exactly one-third of the Earth’s circumference.”

“You can do that with other planets?”

“Sure. Mars, for instance. It has the tallest volcano in the Solar System, Olympus Mons. Depending on where you’re measuring from it’s about 22 kilometers high. I’ll put that into the formula with Mars’ radius, 3389 kilometers, and … OK, if you’re standing on top, your horizon is 387 kilometers away. That’s like looking halfway across France. Mars’ big canyon Vallis Marinaris has 7-kilometer cliffs. There are places where the opposite wall is way beyond the cliff-top’s 96-mile horizon.”

“That beats the Empire State Building.”

~~ Rich Olcott

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