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Virgin Galactic will take passengers aboard SpaceShipTwo as high as 65 miles above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the curvature of the earth through windows as large as 17 inches in diameter.

How much further into space would SpaceShipTwo have to travel to give passengers a view of the entire sphere of earth through one of these windows?

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What do you mean by "entire earth"? It's a round object, you can never see its entire surface. Even half of it, it's still not visible at once from any distance (although if you're far enough you see almost half). First clarify what you're asking. –  Florin Andrei Jan 30 '12 at 23:55
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As long as you don't have any mountains above the straight line between your eye and the 'ideal' horizon, you will be able to see 'the full extent', constrained only by the extent of your peripheral vision.

So isn't this question much more around what angle your peripheral vision covers? As soon as you can see to the horizon all round, when looking down, that surely meets the requirements?

Quick calculation done:

Remembered my geometry from 25 years ago:

  • SOHCAHTOA.
  • Right angled triangle.
  • One side 6371 miles.
  • Assuming peripheral vision = 135 degrees, acute angle 28 degrees.
  • Cos 28 = 6371/Hypoteneuse so Hyp = 7215.
  • Subtract 6371 = 844 miles up

this does assume peripheral vision is 135 degrees all round (left, right, up and down) - so feel free to update if that assumption is false

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Exactly my thought. According to en.wikipedia.org/wiki/Human_eye#Field_of_view it looks like the angle for a normal human eye is about 135 degrees or so. Can you calculate what that height corresponds to? –  FrankH Jan 31 '12 at 20:18
    
there you go - think my geometry is right –  Rory Alsop Jan 31 '12 at 20:41
    
Thanks! You already had my +1, maybe you will get more... –  FrankH Jan 31 '12 at 23:00
    
Fascinating! Space tourism has a ways to go. –  SamtheBrand Feb 2 '12 at 22:47
    
Question: 135 degrees corresponds to the angle one can see with or without moving one's eyes? –  SamtheBrand Feb 2 '12 at 22:50
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I'm going to assume that you want to see half of the Earth, as half of the Earth cannot be seen.

First of all, seeing 50% of the Earth isn't really possible, no matter how far away you get. So, I'm going to set as a goal that one can see 45% of the circumference of the Earth, as I doubt anyone would be able to tell the difference once one has gotten that far.

The size of the window doesn't really matter, as one could simply get closer to the window, and any such considerations go away. What does matter is the tangent angles seen from the observer of the Earth.

The tangent lines to the circle are at angles plus or minus $0.45\pi$. The slope of these lines will be equal to $-\cot\theta$, $x_1=r\cos\theta$, $y_1=r\sin\theta$, $x_1\times x+y_1\times y=r^2$. Solving for $y=0$, setting $r=6,371$ km, $x_1= 0.1564r$, $y_1=0.9877r$ will result in $40,735$ km. This is the distance as measured from the center of the Earth. For reference, the Geosynchronous orbit is $42,164$ km from Earth's center.

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Great answer. Curious why one can't see 50% of the earth at once, or at least 100% of the diameter of the earth. Also, is there a word for the area of a sphere viewable from a point in space? –  SamtheBrand Jan 31 '12 at 1:21
    
Also, in your answer how does 40,735km from the earth's core compare to 65 miles above the surface of the earth? –  SamtheBrand Jan 31 '12 at 1:24
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@SamTheBrand: 50% is impossible, unless you have eyes that are the diameter of the Earth apart. Basically, draw a circle, take the top and bottom of the circle, and draw lines that are perpendicular to the circle. They are parallel, so they never will intersect. As far as your second question, the radius of the earth is 6371 km, so that distance is 34,000 km, give or take, which is much much higher than the 65 mile, over 300 times higher. –  PearsonArtPhoto Jan 31 '12 at 1:27
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So, what you're saying is I'd have to get about 300 times higher than Virgin Galactic and about 1/11 of the way to the moon (at its average distance from Earth) to see the full extent of the visible earth at once? –  SamtheBrand Jan 31 '12 at 1:37
    
@SamTheBrand: It depends a bit on what you mean by the full extent, but yes. –  PearsonArtPhoto Jan 31 '12 at 1:40
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