How far away is the horizon?

James Concannon writes...

        The figure depicts an observer at height h on a planet of radius R. The observer's sight line to the horizon is the tangent to the planet, and therefore the distance to the horizon is given by d.

Data:

Say h is the eye level of a very tall man, to make it simple, call it 2 m
For planet Earth, R = 6,365,000 m

Calculation:

By Pythagoras's theorem, d² = (R + h)² - R²

(R + h)² = R² + h² + 2Rh
so  d² = R² + h² + 2Rh - R²
d² = h² + 2Rh
d = sqrt( h² + 2Rh )
h² + 2Rh = 4 + 25,460,000 and √25,460,004 = 5,046 m

So a very tall man on planet Earth sees the horizon at about 5.1 km. Of course that's assuming the planet is a perfect sphere, with no irregularities.

BY THE WAY: Phil Plait blogged this in January 2009, and ended with a table relating d to h for much larger values.

What about Mars?
The radius of Mars is 53% that of Earth, 3389 km or 3,389,000 m

So for Mars h² + 2Rh = 4 + 13,556,000 and √13,556,004 = 3,682 m
The Martian horizon for the same basketball player is  3.682 km away

So the distance of the horizon is not directly proportional to the relative sizes of the planets, but to the square root. In the case of Mars - Earth, √0.53 = 0.728

Why all this fuss?
        I was moved to contemplate this calculation by an egregious example of the modern phenomenon of "Facebook ignorance." I took part in a convo on Robert Morningstar's page, prompted by this excellent panning shot across Gale crater, which ends on a clear view of Sharp Mountain (sadly, Blogspot doesn't allow embedded video but here's the URL.)

        Mr. Morningstar is quite ignorant of astronomy and mathematics, and it seems that his disciples are even more so.  I see no reason to shame the ignoramus so I'll just call him or her "Person A."

Person A: "Excuse me. This is so funny, I have to giggle. Did you know, Mars is a very small planet. and it HAS NO FLAT PANORAMAS at all, having such tight curvature, there is only one flat place at the North Pole for a sprawling building with a parking lot. So all these flat-stereoscopic images are phony as hell."

James Concannon: "The diameter of Mars is 53% that of Earth. Strictly speaking, there are no flat places on either planet. It's just a matter of how far away the horizon is."

 

Person A: "No it isn't. It's less than that of our Moon, yes. However, every single distance and dimension that NASA publishes is a hoaxed, false figure that has no relationship whatever to reality . . . like everything else they do."

        So this is my answer to those who say "Leave Richard Hoagland and Mike Bara alone FGS. Their NASA hatred is harmless." It isn't harmless if it leads to opinions like that.

        However, "Person A" then went on to make me doubt whether he or she was serious. I think perhaps my leg was being pulled.

Person A: "Let me relate to you, what I have been able to find out. No, they didn't go to the Moon in Apollo 11 the way they said; they hitched a ride with an Annunaki cruiser. No, Challenger isn't blow up in 1986; that was hoaxed, and 6 of the 7 crew members are still alive. Now, NASA nuked Jupiter with a spent nuclear engine; they bombed our moon and penetrated its southern water reservoir in 2009. They sent out Rovers to both Mars and Venus without getting an Okay from the people who live there; and so on Mars one Rover crashed a cliffwall and killed a score of people in summer 2007; then we invaded and lost a rescue party, who were held for ransom and died in 2008. The latest Curiosity Rover was met with gunfire and had to be replaced immediately. No, NASA's doctrine that there's nobody out there, and the photoshopping they do of all asteroid, moon and planetary photos doesn't hide the fact of Life. You can laugh at me all you want and wish; but someday you will find out I have been correct about what NASA is up to : the same sort of military aggression that the USSA is up to on Earth : conquest and coercion"