Altitude vs. Latitude mad science:
Asked by Scott, a video producer from Brooklyn, NY:
I am trying to determine a rough correlate between altitude and latitude, to determine 'apparent latitude' of several ancient mountain observatories in China. Specifically in reference to mountain top star viewing of horizon events along the ecliptic (helical rising and setting of stars). For example: at Latitude 50 N. Pleiades set near the ecliptic about 15 minutes after the sun on the equinox of 2400 BC. At Lat. 35 N. [the Yellow River valley] how high up a mountain would one have to be in order to observe this event at Lat. 35 N. - if possible at all? The point of interest is in dating the received lore of certain events related to the creation of the Chinese calendar. Otherwise that 'conjunction' of the sun and the Pleiades would not be seen at lat. 35 N. until about 2000 BC (dates approximate for purposes of question). Looked at the other way: would say 10,000 feet of altitude provide a view of celestial events on the horizon otherwise only seen from the ground at Lat. 50 N.? To put it succinctly: How high up does one need to be to extend one's 'apparent' view of the east and western horizon to 'events' 5, 10, 15 degrees north?
This was a very interesting set of questions, and I spent too long trying to get a really thorough list of things to think about for you -- I apologize for the delay in answering.
It's fairly easy to get a formula for a spherical Earth with no atmosphere, but there are several tricky things to think about for a really accurate answer. I'll get to those later.
Let's start with the simplest part - a spherical Earth with no atmosphere.
The figure below (labeled 2-2 because I took it from a Celestial Navigation site?) shows an observer standing on the surface at altitude HE (which could be his height above the sea). The angle which the observer has to look down to see the Horizon is called the Dip angle by surveyors and celestial navigators.
Note that the observer's line of sight touches the surface (the Visible Horizon) at a tangent point. Since the Earth is assumed to be spherical, this line is perpendicular to a line from the center of the Earth to the tangent point. Some high school geometry, trigonometry, and algebra should convince you of the following:
1. The triangle is a right-angled triangle.
2. The angle at the center of the Earth is congruent to the Dip angle.
3. The dip angle can be determined from the following formula: Dip = arccos (R/(R+HE))
4. Inverting this for HE in terms of Dip gives: HE = R (sec (Dip) - 1)
Now, here's a bit of a twist. Imagine you are standing at altitude HE at some latitude lambda, but looking directly North. Your dip angle is described as above. Notice that your line of sight still touches the Earth at the Visible Horizon, or tangent point. Therefore, this line is equivalent to your geoidal horizon if you were standing at sea level at that higher latitude! In other words, at altitude HE, you can see farther North by lambda + dip, and will be able to see circumpolar stars with lower declinations.
OK, so how big is this effect? Well, it's pretty small. In fact, taking your examples of how high would you have to be to see 5, 10 and 15 degrees farther North, using a radius for a spherical Earth of 6371 km, you can use equation 4 to find out that you have to be VERY high (24, 98, and 225 km respectively). In other words, you would have to be in an U-2 or spaceship to see that far around! An altitude of 10,000 feet (3048 m) would give you a Dip of only 1.77 degrees.
When you look East or West, this simply changes the rise/set times by a matter of minutes.
You should also be aware of how sensitive these formulas are to errors in measured height. Try a few different heights, and you will see that arccos in this range is not very useful?
As we'll see, the dip angle is similar in size to some of the other complications.
A big effect comes from Atmospheric Refraction, which bends the path of the light from the stars as they pass through the atmosphere. This effect depends on the density of the air, so it is affected by the temperature of the various layers the light passes through. In general, the air gets less dense as you increase in altitude, so light will in general tend to curve towards the Earth's surface as it enters the atmosphere. Notice that this means you can actually see things that are below the Sensible Horizon of Fig. 2-2!
There are two important thing to note about refraction: first, it does not affect things directly overhead, but increases its effect as the viewed object approaches the horizon, and second, it depends on the state of the atmosphere between you and what you are observing. This means that your altitude will affect the size of the refraction, as will weather, atmospheric inversions, etc. etc. Pretty much a mess, if you are trying to be super accurate. How big is this effect? Well, at the worst it is about 35 seconds of arc, so that means that standing at sea level you actually see slightly over 180 degrees of sky! Note that the effect of refraction decreases with Height above Mean Sea Level and with the altitude (angle from vertical) of the observed object.
The next effect to think about is Geocentric Parallax which is related to the fact that the origin of your spherical coordinate system is the center of the Earth, but you are actually observing from several thousand kilometers away from that point, on the surface. Now, if you are observing the Moon, this effect is important since the distance between the centers of the Moon and the Earth is not too much larger than the radius of the Earth. However, when observing stars, the distance to the star is so much greater than the radius of the Earth that this effect is negligible and we can ignore it. For more info on this issue, see http://star-www.st-and.ac.uk/~fv/webnotes/chapt13.htm
The next thing to take into account is the Oblateness of the Earth, which refers to the fact that the Earth is not quite a sphere, but is shaped, to first order, like a slightly flattened ball. This means that for a given latitude, you will be able to see a little farther toward the nearest Pole and a little less towards the Equator than on a sphere. On Earth this effect is small because the oblateness is small, at its maximum at 45 latitude, about 12 seconds of arc, so it can be ignored here.
A much greater effect will be caused by Precession of the Equinoxes which is the change in the location of the celestial poles of rotation. Since you are dealing with dates up to 6,400 years in the past, this is about a quarter of the precessional cycle of 25,800 years, the stars were rotating about a point very different from today, somewhere in the constellation Draco. This changes the whole coordinate system for the stars, and is the largest factor by far -- make absolutely sure you are correcting for this.
The last complication that I can think of is the calendar system that you are relying on - as with the West, I am sure that the Chinese calendar was altered as observations and as political issues came and went. Since at some point you must make a transformation from a Chinese calendar to an astronomical one, this will be important.
I hope this gave you a good list of things to think about as you approach this problem. For more detail and some diagrams on many of these effects, I recommend this Celestial Navigation site.