### 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:

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:

- Precession of the equinoxes
- Chandler wobble, nutation
- Tidal acceleration
- Milankovitch cycles
- Length of day variation

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.