Normally you would just look it up for your location--but I'm thinking of a peak where the horizon will be at least a mile lower and I would think that would have some effect.
Hmmm, good question. I am sure someone has worked it out, but I think light scatter is the difference here - if you can see light in the sky at height you can probably see it at sea level too. There's different levels of first light too - see here, it is all defined by degrees of the sun below horizon, so if you are 1 degree equivalent height above sea-level, then dawn will be 1 degree/rotational speed of earth sooner than at sea-level (at least that;s how I understand the maths).– bob1Oct 5, 2020 at 7:42
1Possibly related: Islamic prayer times etc adjusted for peope living high up in the world's tallest building– BohemianOct 5, 2020 at 23:14
I don't recall if it included "first light", but there are ephemeris websites that will adjust sunrise times for elevation. The one I had used requires a login now, but I knew exactly when and where the sun would appear at my specific chosen location. Thus, you might look at some of those to see if they meet your need.– topshotOct 21, 2020 at 13:03
@topshot The math presented in the answer I accepted shows that the difference isn't big enough to be of importance.– Loren PechtelOct 21, 2020 at 21:27
With some trig and simple approximations, for the case where the sun is rising directly to the east, and is rising in the direction perpendicular to the horizon, the result is
T = (24 hours)(1/2π)√(2h/R),
where h is your height above the spherical surface of the earth, R is the earth's radius, and T is the amount of time, in units of hours, by which sunrise comes early. This might be a reasonable model for cases like the eastern Sierra or Kilimanjaro, where you have a mountain range that is fairly prominent relative to low-lying terrain to the east. It might not make sense, e.g., for ranges that run east-west, such as the San Bernardinos or the Himalayas. Then there would be more chance of having other heights to your east that are not much lower than you, or maybe even higher.
Putting in numbers for Kilimanjaro, I get 10 minutes.
For a one-mile tower, the result is 5 minutes.
It is not normally true that the sun rises straight up and to the east, so the result will really only give a rough idea of how big the effect is. At extreme latitudes, the effect could be much bigger. If you're above the arctic circle, the sun could come up in the spring a week earlier because you're on the top of a mountain.
The experience I have had much more often is that the day before summit day, I'm camping in some valley near the peak, with tall mountains on all sides. Then it gets dark fairly early as the shadows sweep across my position, but I can still see the bright sky and the sun lighting up the peaks to the west.
I've had the reverse experience--descending in the pseudo-dark for hours, but not to the point of needing light. Oct 6, 2020 at 0:46
The question asks for first light but let's assume the time difference will be the same as for dawn. And also that the terrain is flat, apart from an observer in a tower 1 mile high, and the radius of the Earth is 3959 miles.
Here is an exaggerated geometry sketch showing a horizontal line in the direction of the sun for an observer at ground level, and a diagonal line showing where the sun will rise for an observer on the tower, which is tangential to the Earth's surface.
The difference is the angle A which is the same angle as subtended by two radii from the tangent points.
So we have cosine(A) = (3959 / 3960), giving A = 1.28 degrees.
It takes the sun 24 hours to move through an angle of 360 degrees, from this can be calculated that the sun will rise 5.12 minutes earlier (or set 5.12 minutes later) than for an observer on the ground.
So about 5 minutes.
Your formula works, but it's awkward because r/(r+i) is very close to 1 when i is small, so you get potential problems with rounding errors. That was the reason for doing the additional approximation that I did in my answer.– user2169Oct 5, 2020 at 19:38
That is well within the ability of today's calculators, without significant errors and I don't see how "additional approximation" lessens that. It's a simplified estimate anyway, as there are many other factors at play when "first light" is seen. I think it is poor form to criticise other answers, unless perhaps there is a serious error. Oct 5, 2020 at 20:42