Longer Day on Burj Dubai, Mt.Everest and a Plane

This problem is one that I encountered when trekking in the Himalayas. If you are on a mountain with an unobstructed view of the horizon, then your day is extended by a few extra minutes that the Sun takes to drop below the horizon at sunset (or rise above, at sunrise). The geometry is fairly simple, but a few days ago I saw a quote attributed to the astronomer Neal de Grasse Tyson. I tried to verify it, and I did not get the same numbers.

So I checked online, and I got differing answers - which I reproduce below. As well as the simple derivation that I did. Have I got something wrong? Dunno!

a a)    Neal de Grasse Tyson, quoted by the Indian Express in the “Social Intelligence” column, 15^th June 2016:

“Indeed from atop Burj Khalifa in Dubai they get four extra minutes of daylight, two in the morning and two in the evening.”

The top of Burj Khalifa is at about 828 metres height.

b b)  Quote from the following website: assuming a plane flying at an altitude of 12 kms:

http://curious.astro.cornell.edu/about-us/161-our-solar-system/the-earth/day-night-cycle/189-how-do-sunrise-and-sunset-times-change-with-altitude-intermediate

“At most latitudes on the Earth, the effect of increased altitude is the same: it makes the Sun rise earlier and set later than it would at that same location from the ground. To make things simple, let's assume that you are in a plane over the ocean, at the equator at sunset. In that case, straightforward trigonometry indicates that at a typical commercial airplane altitude of 12000 metres, you can see an extra 2 degrees (emphasis added) "around" the Earth. Since the Earth moves around the Sun at a rate of a quarter of a degree a minute, it means that at this altitude, sunset occurs 8 minutes later than it would from the ground. The variation with altitude is approximately linear, (emphasis added) and so we conclude that sunset is later by 1 minute for every 1.5 kilometres in altitude, and that sunrise is earlier by the same amount.

Now, all of this is complicated somewhat by the fact that you don't stay in one place in a plane, but you travel in a given direction: if this direction is predominantly East or West, then the plane's motion will completely change the answer we got above (in particular, travelling West at sunset can lengthen the latter significantly in a commercial jet). So, the results above are valid in a plane if a) the plane is moving rather slowly (like a personal plane) or b) the plane is travelling in the North-South direction.”

c c)   Derived formulas:

 

Mean radius of Earth is R = 6371 kms. The horizon in the above figure is at the point that is tangent to a spherical Earth, so it makes a right angle to the radius (apologies for the figure!).

The angle to the horizon, from a height h above Mean Sea Level, is given by:

                                                cos (q) = R / (R+h)

Approximating for small angles:

                                               cos(q) = 1 – (q²/2) = 1/ [ 1 + (h/R)] = 1 – (h/R)

Thus:

                                                          q = (h/R)^1/2

Let h = 12 kms, and R = 6371 kms, so q = 0.0614 rads = 3.52 degs – instead of 2 degs according to the Cornell blog.

Burj Khalifa is at h = 0.828 kms, so q = 0.0161 rads = 0.924 degs.

Multiplying by 4 mins/deg,as mentioned above, (180° = 12 hours) we get 3.69 mins for Burj instead of 2 mins, and 14 mins for the plane at cruising altitude.

Also, the angle varies as the square root of the altitude – not directly proportional as the Cornell blog b) stated.

cd)   However, the following blog gives values for Burj Khalifa (828 m) and Mt.Everest (8,848 m):

These calculations are in better (but not perfect) agreement with the calculations in c):

http://www.kosherjava.com/2010/03/07/faq-how-much-earlier-is-sunrise-on-mount-everest-due-to-elevation/

“With a height of 828 m (2,717 ft), visible sunrise to someone standing on top of the crown of Burj (something unrealistic) on June 22nd would be at 5:24:56 AM versus 5:29:31 AM on sea level, a difference of 4 minutes and 35 seconds. 

With an elevation of 8,848 Meters (29, 029 feet), sunrise on Mt.Everest would be up to 15 minutes and 31 seconds earlier on Mount Everest than on sea level. The range of the effect is from 15 minutes and 31 seconds on June 22nd, to a “low” of 13 minutes 41 seconds earlier on March 18th.”

This website implies that it is a java-based app based on calculations made by Rabbi Harfenes – that are more detailed because they take both the date as well as the latitude and longitude into account.

The formulas given earlier for Mt.Everest yields: q = 3.02°, which means 12.1 mins. That is: kosherjava gives an answer for Mt.Everest that is 13% higher or 28% higher (depending on the date) than the answer from the formulas above.

So: what gives? How much would the latitude and longitude matter if the Earth is pretty much spherical (about 40 kms more radius at the equator than at the Poles) ?

de) Also see the following site, which gives a low value for Mt.Everest and also again states that the variation is linear with altitude:

http://heimhenge.com/skylights/2014/08/25/the-time-of-sunrise-and-sunset/

ef)    TimeAndDate.com also includes a handy calculator for the times of sunrise and sunset on any date for any location on the planet. As with their Day and Night Map, that calculator assumes a flat and unobstructed horizon. They also assume the observer is at the same elevation (measured from sea level) as the horizon. If you were observing from the top of a tall mountain, sunrise would happen slightly earlier, and sunset slightly later. But it would be a small correction — only about 6 minutes if observing from the top of Mt. Everest. The correction factor is: ΔT = ±1 minute per 1.5 km elevation. Again, assumed linear!

Summary: Extra time:

	Burj Dubai 828 m	Mt.Everest 8,848 m	Plane at 12 kms 12,000 m
Tyson	2 mins
Cornell*	33 secs	5 mins 54 secs	8 mins
Kosherjava	4 mins 35 secs	13 mins 41 secs to 15 mins 31 secs
TimeAndDate.com		~ 6 mins
Calculations	3 mins 41 sec	12 mins 6 secs	14 mins 5 secs

 *Cornell calculations mentioned only the plane; the other two I have calculated assuming that the 1 min per 1.5 km rule is correct – which I seriously doubt!