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Chris P
10-08-2010, 08:36 PM
Is there a simple formula by which to calculate how far you can see from an airplane if you know how high up you are? For example, how high would you have to be above Nashville to see the Gulf of Mexico (ignoring haze, clouds, etc.)? Doing the proper math got too complicated too quickly.

Assuming that the Earth is flat and calculating the length of the sides of a triangle doesn't work because you could see almost infinitely far by looking just-less-than straight out. So we're faced with calculating the farthest point on a sphere you can see based on your altitude at 8 inches per mile curvature. Argh! I know I'm making this too hard. Any help?

benbradley
10-08-2010, 08:50 PM
Is there a simple formula by which to calculate how far you can see from an airplane if you know how high up you are? For example, how high would you have to be above Nashville to see the Gulf of Mexico (ignoring haze, clouds, etc.)? Doing the proper math got too complicated too quickly.

Assuming that the Earth is flat and calculating the length of the sides of a triangle doesn't work because you could see almost infinitely far by looking just-less-than straight out. So we're faced with calculating the farthest point on a sphere you can see based on your altitude at 8 inches per mile curvature. Argh! I know I'm making this too hard. Any help?
Yes, and as you might imagine, this is an age-old question and the math was surely first done long ago. This appears to have answers:

http://en.wikipedia.org/wiki/Horizon

RJK
10-08-2010, 09:14 PM
Here's a rough formula: aboard ship, our RADAR mast was 50 feet above the water, and the line-of-sight horizon was 20 miles. So for every 100 feet you can see 40 miles.

Based on that calculation, if the Gulf coast is 400 miles from Nashville, if you are 10,000 feet above the ground, you should be able to see something, but it would need to be pretty big to be seen from 400 miles away.

I can see the CN Tower in Toronto from my window. I just checked on Google Earth and it's about 38 miles away. On a really clear day, I can see buildings across Lake Ontario that are more than 50 miles away, but it would be stretching it to be able to discern objects from much farther than that.

Kenn
10-08-2010, 10:38 PM
Here's a calculator.

http://newton.ex.ac.uk/research/qsystems/people/sque/physics/horizon/

In practice, I think you'll find that the maximum line of site will depend on the level of particulate material in the atmosphere. It will also depend what you are looking at. Mountains are visible for much greater distances. That is because they stick up (and because they are less affected by atmospheric dust).

SirOtter
10-08-2010, 11:17 PM
I can see for miles and miles and miles and miles and miles... oh, yeah.

Chris P
10-08-2010, 11:26 PM
I can see for miles and miles and miles and miles and miles... oh, yeah.

That's because you look pretty tall but your heel's are high.

And thanks for your responses, everyone. So my character is fibbing when he says you can see the Great Lakes and the Gulf at the same time from a commercial airliner. He might, however, be able to see Lake Erie and the Atlantic from over central Pennsylvania....

RJK
10-09-2010, 10:07 PM
According to the formula in Kenn's link, my estimate was wayyy off. You'd have to be close to 105,000 feet high to see 400 miles. Commercial airliners don't fly that high.

Chris P
10-11-2010, 12:30 AM
According to the formula in Kenn's link, my estimate was wayyy off. You'd have to be close to 105,000 feet high to see 400 miles. Commercial airliners don't fly that high.

I think you would be correct if the earth was flat, but also because your boom and the distance to the point on the horizon forms a triangle the relationship isn't linear; the length of the opposite side (distance along the surface) will be a function of the tangent of the angle made from your boom to the point on the horizon, with the adjacent side being your altitude (height of the boom).

Also this: because the earth curves away from you at 8 inches per mile, that's a drop of 160 inches or 13.3 feet over 20 miles, bringing the horizon closer to you, and it deviates more the higher up you are and the farther out you look. It was figuring the curve and the changing angle of the observation point with perpendicular that was giving me fits.

Qbynewbie
10-11-2010, 12:49 AM
I fly a lot, in airliners and in my own plane. When I'm in my own plane, I'm usually fairly low and often only 3000' or 4000' AGL (above ground level). How far I can see from 3000' AGL is completely dependent on the visibility at the time. On some days, the reported visibility will be greater than 10 miles but it might actually be far more than that.

I've been at 3000' AGL above Queensbury and been able to see the Albany skyline silhouetted against the sky 45 miles away. I can also see the tops of mountains that I know are 75 miles away but they, too, are sticking up above ground level. On days with diminished visibility, I might be able to see for only a small number of miles or the visibility can be so bad that I won't fly at all.

An airliner that is flying a leg that is longer than an hour or so (where much of the flight would be just climbing to altitude and then descending again) will typically be at an altitude somewhere between 27,000' and 41,000'. From those altitudes, if the day is clear and there aren't clouds between you and the ground, then you can easily see for hundreds of miles but the human eye has trouble picking out small objects at that distance. In most planes, the pilot has the best view of all because he can see out both sides of the plane. At 41,000' on a severe clear day, I think it's entirely possible that a pilot could see Lake Ontario and the Atlantic Ocean simply by turning his head in one direction or the other.

As an aside, I was surprised when I started flying how very far away you can see large features from even relatively low altitudes. Airports are often one of the largest visible objects in a surrounding area and on good days you can see them from dozens of miles away from even just 5000'. That's a good thing because they make decent navigational references. But large lakes act the same way. I can see Saratoga Lake, which is about 25 miles from here, as soon as I'm up about 3000', give or take.

Qbynewbie
10-11-2010, 12:51 AM
That's because you look pretty tall but your heel's are high.

And thanks for your responses, everyone. So my character is fibbing when he says you can see the Great Lakes and the Gulf at the same time from a commercial airliner. He might, however, be able to see Lake Erie and the Atlantic from over central Pennsylvania....

I could be wrong, but I think that seeing the Great Lakes and the Gulf from an airliner would be indicative of having one-too-many glasses of booze. :)

Hallen
10-14-2010, 03:01 AM
Even on a severe clear day, you can't see 400 miles horisontally through the atmosphere. There will be too much haze/refraction of light. Taking off from Salem, Oregon, from about 5000ft in the middle of winter (cold, clear day, low humidity), I can see all the way from the Olympics to Mt. Shasta. The distance in either direction is about 300 miles. But, snow covered mountains are high contrast compared to an ocean. At those distances, the ocean will mostly blend into the ground -- everything turns to shades of blue because of refraction at those distances. Especially with the essentially flat terrain between Nashville and the gulf coast, you aren't going to pick out the gulf until you are pretty darned close.

Drachen Jager
10-14-2010, 03:09 AM
I could be wrong, but I think that seeing the Great Lakes and the Gulf from an airliner would be indicative of having one-too-many glasses of booze. :)

Or you're flying Virgin Galactic.