Question: Me and my buds do a lot of mountain camping. Last time we were at the top of Humphrey’s Peak near Flagstaff we got into a discussion about whether the day lasts longer on a mountain. Seems like it should, because you must see an earlier sunrise and a later sunset when you’re up that high. But we had no idea just how much longer. I’m sure there’s a way to work that out. Any ideas? — JS, Phoenix, AZ
Answer: You are correct. All things being equal, the day does last slightly longer on a mountaintop. And I say “all things being equal” because, unless you’re on a lone peak, there’s usually other mountains surrounding you that will delay sunrise or hasten sunset.
The photo above was taken after sunset, as you can see from the local terrain. But at the tops of those cumulonimbus clouds the Sun has yet to set. I’d estimate those clouds top-out around 12,000 m (39,000 ft), about average for cumulonimbus. For comparison, Humphrey’s Peak has a height of 3852 m (12,637 feet).
It’s quite easy to calculate how much longer the daylight lasts on a mountaintop. I’ll use Humphrey’s Peak for my example. The graphic below shows the geometry. For clarity, the height of the mountain (H) is exaggerated compared to the size of the Earth.
The average radius of the Earth is R = 6371 km. We’ll start by using the Pythagorean Theorem to calculate D, the distance to the horizon as measured from the summit:
D = √[(6371 km + 3.852 km)2 – (6371 km)2] = 222 km
For an average-height person standing at sea level, the distance to the horizon is only 4.6 km. So with 222 km distance the sunrise must be earlier, and the sunset later. How much longer depends on angle θ, which is how many degrees below horizontal you can see:
θ = tan-1(222 km/6371 km) = 2.0°
Since the Sun moves through the sky at 15°/hour the day will be lengthened by:
ΔT = 2×2°/(15°/hour) = 0.27 hours = 16 minutes
And if we define “daytime” as starting when the Sun is totally above the horizon, and ending when it first touches the horizon, then another effect adds even more daytime: atmospheric refraction. When you “see” the Sun sitting right on the horizon, it can actually be as much as 0.5° below the horizon depending on atmospheric conditions. Since the Sun itself has an apparent diameter of 0.5°, this adds another:
ΔT = 2(0.5°)/(15°/hour) = 0.07 hours = 4 minutes
of daylight. And refraction happens whether you’re on a mountaintop or at sea level, but you must have an unimpeded view of the horizon for this effect to matter.
Not like you’re going to notice these slight differences. There’s a usable amount of light well before dawn and after dusk — at least between astronomical dawn and dusk. So no need to calculate ΔT for your next outing … you’ll know when it’s time to pitch camp or brew the coffee whatever your altitude.
Next Week in Sky Lights ⇒ How the Jet Stream Affects Weather