Monday, November 26, 2012

Sun Motion 3: Analemma

Equation of time and location

In the previous post, the equation of time is derived from the angle difference from the mean sun to the true sun, after projection of the true sun on the equator plane. This angle is then converted to a time difference by proportion with the (constant) Earth's rotation speed.

The equation of time is based on the position of the true sun on the celestial sphere, which is independent of the observer's location on Earth. The equation of time is the same irrespective of the position of the observer. So knowing the equation of time, an observer knows when to look up and record the altitude of the sun, and subsequently deducts the latitude of his location.

But reversing the problem, if an observer looks up at 12:00pm sharp, the azimuth of the sun (in local coordinates) may vary significantly depending on his latitude. So if you define the analemma as the position of the sun (azimuth, altitude) in the sky at 12:00pm every day of the year, you expect significantly different aspects for different latitudes.

Below I have plotted the analemma for a number of locations (only the latitude matters), taking into account the obliquity and eccentricity of the Earth's orbit, and using the (quite precise) Gregorian duration of the year (365.2425 days), going through 366 days from 20mar12 (included) to 20mar13 (included), the 2012 spring equinox being 20mar12 05h14mn, and the perihelion date being 2jan13 04h38mn.

Note that there is a slight difference between the position of the sun on the 20mar12 and 20mar13. This is due to the extra 0.2425 day (beyond 365) in the duration of the year. This offset would grow from 2013 to 2015, and then return to almost zero on 20mar2016, because 2016 is a leap year.

Why almost ?
Because (i) the duration of the year is not exactly 365.25 but 365.2425, and (ii) even beyond Gregorian calendar precision, there are second/third order influences on the duration of the year:
There are a number of small order variations. It is difficult to understand more than basic concepts about them and even more to put them in perspective as they have very different periods. This short video from the Casssiopeia project is an excellent tutorial. Anyway, in the long run, the Earth orbit, and consequently duration of the year, is chaotic, meaning it cannot be accurately predicted a long in time in advance (here, say some million years). The smallest (inevitable) measurement error would become a prevailing factor in the long run.
All this can safely be and is neglected here.


The nearer the equator, the higher up in the sky the sun and the more inadequate the cartesian plot y=altitude/x=azimuth is. The altitude=90° line on the plot is indeed a point in the sky, on this type of plot, for low latitude observation points (meaning nearing and below the Earth tilt ~23.45°), the upper part of the analemma will appear very wide in terms of azimuth. This is only due to the fact that the small change in position near the zenith corresponds to a large variation in azimuth. At the extreme, the azimuth instantly switches from 90° to 270° when the sun passes the zenith on its way from East to West. These distortions are independent of the equation of time. And for an observation point with latitude below 23.45° the plot becomes even hardly understandable as the sun's azimuth jumps by 180° while the altitude stays around 90°. And in the southern hemisphere beyond the tropic of Capricorn, the sun will appear about North at 12:00pm, so the plot is again difficult to read (unless the azimuth is plotted modulo 360°).
The cartesian plot is clearly not very satisfying.

Instead it is more appropriate to use another type of plot: A polar plot where the zenith is the center point (the pole), the azimuth the polar angle, and the altitude the distance to the center. In a word, this is a projection of the celestial sphere, visible from a ground observer, onto a 2D map.

It is also convenient to show the analemma for all hours (for which the sun is visible, or just below the horizon) and, for different dates the daily sun path from dawn to dusk. There are typically 3 types of twilights: geometrical (when the sun's altitude reaches 0°, or more precisely -0.833°; this is when the rim of the sun touches the horizon), civil (-6°), nautical (-12°), astronomical (-18°). On the plots below, the 21st of each month is shown. The dates on the way to the summer solstice (from 21dec to 21may) are dashed, the dates on the other way (from 21jun to 21nov) are full.

Below are the sunpaths for different locations, from the North pole to the South pole.
For Paris I show the equivalent in 3D, for better visualization of what the polar map tells.

As explained above the analemma can be defined as the sun path across the celestial sphere relative to the position of the mean sun. Another way to define it is the path of the location on Earth where the sun is at zenith at 12:00pm UTC, over one year. It is naturally in between the tropics of Cancer and Capricorn, and this path has the same loose eight shape.

You may notice such shape on old globes, as depicted below. The internet has brought and will no doubt continue to bring immense benefits for those who have access to it, but one sad minor consequence though, is that in the age of Google maps you never need a globe any longer, and the few remaining ones are stored as antics in museums or in the less busy rooms of libraries... A globe is such a beautiful, rich, stimulating object that this state of things is a pity...

The diagrams and animations in the 3 posts about sun motion were made with Mathematica 8.0.4. The notebook, the images and some tables (created with the notebook) are all in this github repo.