I can see for miles and miles and miles...
Recently, this question was broached in a conversation I had with a fellow backcountry skiing enthusiast: "How far can you see off the summit of Mt Rainier?"

Well this is obviously a very important question. Moreover, it's also an invitation to make another use of the fabulous Pythagorean Theorem.

The distance one can see from a high point is defined as follows. Draw a line from the high point that makes a tangent with the earth's surface (very well approximated by a circle). The distance from the high point to the point where the tangent touches the surface of the earth is how far the viewer can see. In the picture at right R is the earth's average radius, h is the height of the mountain on which the viewer stands, and the blue line is the tangent to the earth's surface from the top of the mountain.

Since the tangent is perpendicular to the radius, the Pythagorean Theorem gives us the solution to the problem quicker than the mountaineer on the summit can get his water bottle.

We can neglect the h2 term since it is very small. Mt Everest is the tallest mountain on our little planet; given Mt Everest's height (8,848m) and the Earth's mean radius (6,371km), neglecting the h2 term for Everest only gives an error of <3.5% of 1%, or <3.5 parts in 10,000. The error will be even smaller for all the smaller mountains.

The table gives the distance one can see from several northwest volcanos, Mt Everest and Denali (Mt McKinley).

MountainHeight(m)View(km)View(miles)
Mt Everest8,848336208
Denali6,195281174
Mt Rainier4,392237147
Mt Shasta4,317235146
Mt Baker3,285205127
Mt Hood3,426209130













Interesting aside: as the crow flies it is 163 miles from Orting, WA to Vancouver, BC so I don't think it's possible to see Vancouver (BC) from the summit of Mt Rainier. BUT I'M WRONG! Why?

Note that these results are not totally accurate because atmospheric refraction plays a role. The air is denser near sea level, so the effect is largest near it. Light travels ever-so-slightly slower in the denser air, which makes its trajectory bend toward the surface of the earth when looking down from a summit. Initially, I mistakenly thought this would make the viewing distance shorter because it bends the light ray down so that it touches the surface sooner. But what it really does is allows the viewer to look at a higher angle, so a light ray that wouldn't hit the surface of the earth at all gets bent and touches the surface over the horizon.

How about an interesting look at atmospheric refraction? The picture below illustrates the nature of the effect. The earth's spherical surface is blue. The straight red line shows the geometric estimate. The curved line shows the extended view provided by refraction.



This is definitely not the end of the story, though. Refraction depends on atmospheric conditions, which vary widely. Mirages due to layers of warm and cold air can swing the view in either direction. Here is a nice discussion of mirages, with good pictures and diagrams.

All these numbers assume the viewer is looking at sea level in the distance. Note that the range of view from high point to high point is extended. Just add the distance to the horizon view from each high point for the total. See the figure below.



I've now obtained personal testimony of seeing Rainier from farther than Vancouver in a sailboat. Adding a bit to the Pythagorean estimate would support this observation. Apparently, Mt St Helens is also visible from high points around Seattle.

Below is a graph of the distance one can view to flat ground at sea level from a given height, neglecting all atmospheric corrections. The highest point (far right edge) of the graph is Mt Everest.



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