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    Re: EoT Curve calculated by VB
    From: Frank Reed
    Date: 2021 Nov 24, 08:17 -0800

    There are short calculations and long calculations. For the purpose of a visual graphic, you can use a short calculation. Years ago Wikipedia was a goof place to find nice short formulae for this sort of thing quickly and easily, but most technical articles have become buried in unnecessary detail, and that certainly applies to section on calculation in the article on the Equation of Time. But if you look carefully, they do provide a short formula:
       Eq.T = −7.659 · sin(M) + 9.863 · sin(2M + 3.5932)
    where M is just a day count from Jan 4 (the approx. date of perihelion) converted to angle by multiplying by 2pi/365.25. For the level of approximation here, we can shorten this slightly to:
       Eq.T = −7.66 · sin(M) + 9.86 · sin(2M + 3.593),
    where M = 0.0172 · (dayofyear - 4). And that's it!
    Fast and easy.

    Example:
    What's the equation of time on November 4 (near maximum "Sun fast" for the year)?

    First, let's get the answer. Hit any almanac data app. I'll use the "Nautical Almanac app" on my web site. Here it is set up for Nev 4, 2021 with angles in decimal degrees. The equation of time is, by definition, the difference in time between the meridian passage (or longitude) of the Mean Sun and the True Sun. So all we have to do is look at the Sun's longitude (GHA) on any day at Greenwich noon (this can also be done at any other GMT/UT but we don't need that at the level of approximation required for a graphical presentation). At Greenwich noon on Nov 4, the GHA of the Sun is 4.11° to the nearest 0.01°. The Sun's speed in longitude is 1° every four minutes of time, so we multiply by four to convert to time: 16.44 minutes. That's the equation of time on Nov 4. Note that it's a value, not an equation in the modern sense. The old meaning of an equation was more or less equivalent to our word correction. The correction from mean time to apparent solar time is 16.44 minutes. You should also check for other nearby years. There are variations especially due to the leap year cycle, but 16.44 minutes is still close enough. And one further note on this "answer". This is by far the best way to calculate the equation of time. If you have a tool that calculates GHA Sun already, then just use that! The equation of time is not an independent quantity. That said, it's frequently convenient to have a short approximation, so let's continue...

    Next, let's calculate by the approximate equation above. We need the argument M [which, btw, is the "mean anomaly" of the Earth's orbital motion, but that is really not important --except to explain why it's "M" and not "x" :) ]. To get M, we start with the number of days since perihelion (dayofyear-4, that is). You can count this off various ways. For Nov 4 it's 304 days. Convert that to an angle by multiplying by 0.0172 which yields 5.229. Plug that into the short formula above. I find 16.48 minutes. That's close. The difference is 0.04 minutes or 2.4 seconds which is entirely insignificant for a visual representation.

    If you work other examples on other dates you will find larger errors, but, again, they are probably insignificant for a visual graphic. Also, this short formula will become less accurate after a few decades. You can worry about that... in a few decades!

    Frank Reed
    Clockwork Mapping / ReedNavigaton.com
    Conanicut Island USA

       
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