How close must you be to the North Pole to see it from a plane at an altitude of 10,000 meters?

Question

Please account for these factors:

1. The radius of the earth is less at the poles.

2. The flattening effect of the earth's rotational spin also makes the earth's surface at the poles flatter resulting in a horizon further away.

3. The refraction of light through the earth's atmosphere allows me to see slightly below the geometric edge of the horizon.

4. You are not directly above the North Pole, you are actually a short distance away. What is the maximum distance from the North Pole you can be (at an altitude of 10,000 meters) and still see it? This will not be the exact same number as if you were above the pole since the flattest part of the planet is on the horizon, not straight down.

I was in an airplane which flew nearly over the north pole, but not quite. I found out when the plane was at it's closest point, pointed my camera out the window due north, and took a few shots. I simply want to find out if I have a picture of the north pole or not!

Answer 1

Did You follow the link in the fist answer? It is nicely explained there and I'll not repeat why the distance (D) to the horizon from a point at height H over the Earth surface is given by the expression:
D = sqrt(2RH + H^2) /R - Earth radius, approx. 6370 km/
That assumes the Earth being a perfect sphere, but in fact, as You correctly mention, the polar radius is some 20 kilometers less than the equatorial one and the refraction also counts. but the influence of both these factors is almost negligible regarding Your question.
So in approximate calculation we can go even further and eliminate the relatively small term H^2 above (comparison shows that 2RH is substantially larger if H is several kilometers, like a mountain summit height or a flight height!). Thus we obtain the following approximate formula:
D ≈ sqrt(2RH) ≈ sqrt(12740*H) ≈ 113*sqrt(H) /all measured in kilometers; H relatively small/
Its precision is quite sufficient for practical needs, so, taking H = 10 we obtain approximately 357 kilometers.
If Your plane has flown some 350 kilometers near the North Pole, You can be sure You have it on Your photos!
But if You insist to take into account the deviation of the perfect spherical shape of the Earth ellipsoid and the light refraction, the question becomes much more difficult and requires additional data (refraction coefficients etc.)

Answer 2

A flat plane touching a sphere at a single point is said to be tangent to the sphere.
Let the sphere be the earth and the point be the north pole.
An aircraft 10km in altitude and in the said plane will be at a
latitude of 86.8 degrees north, or 3.2 degrees from the north pole.
(180/pi)*acos(pr/(pr+10)) = 3.2117 degrees
Where pr is the polar radius of the earth or 6356.912 kilometers.
This corresponds to roughly 357 kilometers from the pole.
3.2117*2*pi*((er+pr)/2)/360 = 356.937 Km.
Where er is the equatorial radius of the earth or 6378.388 Km and (er+pr)/2 is the average radius of the earth.

Your ability to see, or take a picture, of the pole improves if you were closer than this, then you would be looking down toward the pole instead of straight at it in the tangent plane on the horizon.

Answer 3

The Northpole is fairly featureless. Take you picture to your friends and say, "This is the North Pole." Then show them the plane itinerary. And enjoy watching them be all impressed.
POOF!! Instant pic of the north pole, even if you missed by a mile or 5. NO ONE CAN TELL!

Answer 4

http://calgary.rasc.ca/horizon.htm