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.
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
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."
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.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."
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"
Sadly, if you read the person's own facebook page, you can only come to one conclusion: she's batshit crazy.
There's an idiot who's claiming that Curiosity is on Devon Island, in Canada's Arctic. Has it never occurred to her and those of like mind, that signals to and from Mars take from 4 to 24 minutes to arrive at their destination, depending on where Mars is, in its orbit, not instantaneously, as they would if the rover were in Canada? Strange how the conspiracists have overlooked that inconvenient little fact. Not. They're not even aware of such a time delay, which adds to their idiocy.
I followed that crazy person's Facebook links and eventually came to this:
So it's true all along. They never went to Mars and it was all filmed in the most uninhabited island in the world. That Mars Rat was actually an arctic lemming. NASA lied all along. Case Closed.
Other links on that page include:
Did Putin Just Out The World’s Ruling Class As Reptilians?
They Look Like People
Has NASA accidentally found an Ancient Sumerian Statue on the Surface of Mars?
Reasons Why Humans May Not Have Come From Earth
Alien drone ‘built by Martians’ is seen crashed on Mars
Update: Today I disconnected from Morningstar on FB. The torrent of hatred was just too much, and was obviously due to get even worse as the US election season wears on. Yesterday he posted a flagrantly anti-semitic rant from Jeff Rense. Enough.
So there probably won't be much Morningstar news on this blog from now on.
Those devious NASA illuminati shills, keeping details of Devon Island from us by publishing extensive material explaining how they use it! How are we meant to find out the Truth when all this information is indexed publicly behind one or even two mouse clicks? It's thanks to people like Person A that we know what's really going on!
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