# Why is my approach to the equation of time off by a constant?

I'm trying to better understand the causes for the equation of time by deriving an approximation from first principles.

My naive approach, $EOT_{NAIVE}$, is to take the difference between the right ascension of the mean sun, $\alpha_M(t)$, and the right ascension of the "real" sun $$EOT_{NAIVE}(t)=\langle\dot{\alpha }\rangle\cdot(t-t_0)-\alpha(t)$$ where $\alpha(t)$ is simply the "actual" position of the sun at time $t$ from ephemeris data (e.g. either of the RA values reported by JPL HORIZONS, or a similar source) and where $\langle\dot{\alpha }\rangle = 24/365.242$ and $t_0=t:\alpha_M(t_0)=0$.

To confirm that this is about right, I compare my result with what I get using the USNO definition of GMST $$EOT_{GMST}(t)=GMST(t)-(t-t_{noon})-\alpha(t)$$ and with $EOT_{POLY}$, a standard polynomial expression from Dershowitz & Reingold as described in The Clock of the Long Now documentation.

But the values I get for $EOT_{NAIVE}$ (blue dashed line) are consistently about $7.4$ minutes greater than these reference methods (exactly $7.4537$ for $EOT_{GMST}$, red line; and within a couple seconds of that for $EOT_{POLY}$, gray outline) give:

Surprisingly, I can fix this by simply changing $t_0$ from the date, $t_E$, on which the most recent vernal equinox occurred (2012-03-20T05:14:33Z) — which I had assumed would give a "starting" RA of 0 — to 2012-03-22T02:20:32.41Z, which brings my approximation (blue line) exactly in register with the GMST approach (gray band), and within a couple seconds of the polynomial approach (red line), as can be seen by taking the difference between the GMST approach (gray band) and each of these approaches (note the change of scale, this figure essentially takes the difference between $EOT_{GMST}$ and each of the plots in the first figure above, and "zooms in" on the gray band):

Why should changing the date in this way "fix" my approximation? Why doesn't my approximation work with $t_0=t_E$? Why does it work perfectly with a different date? Are the approaches I'm using as references ($EOT_{GMST}$ and $EOT_{POLY}$) the right ones for this kind of exploration?

Note that the reported RA of the sun at 2012-03-22T02:20:32.41Z is $0.1146^h$, and that the fit between $EOT_{GMST}$ and $EOT_{NAIVE}$ using that date is quite good (this figure zooms in further on the gray band in the second figure above):

Note also that despite the documentation for the ephemeris I use claiming that "light time delay is not included", the values I get match those from JPL's "airless apparent right ascension and declination of the target center with respect to the Earth's true-equator and the meridian containing the Earth's true equinox-of-date. Corrected for light-time, the gravitational deflection of light, stellar aberration, precession and nutation."

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I deleted my answer, until I think further about the question. For the moment, I am not sure about it. – Eduardo Guerras Valera Dec 15 '12 at 1:21
I think I got it, after the night sleep. I will give you a nice explanation tomorrow at the computer. Now I am tapping at a tiny, annoying touchscreen. The Mean Sun and the Apparent Sun cannot coincide at the V. Equinox but rather later because of the higher orbital speed the Earth has had from the Perihelion. It is a more subtle issue. A complete explanation with drawings and geometric arguments follows tomorrow or on monday. – Eduardo Guerras Valera Dec 15 '12 at 10:12
@EduardoGuerras: A tip here is (of course) that solving $GMST(t)-(t-t_{noon})=0$ for $t$ gives 2012-03-22T02:20:32.5Z. But beyond that, I'm not sure. – raxacoricofallapatorius Dec 15 '12 at 23:25
@raxaco... : That trick seems to work because the RA variation of the Sun(s) is equivalent to ~1 deg per day, and the diurnal motion is ~1 deg every 4 minutes of time. Therefore, by changing that date in two days, you compensate adhoc for that 7.4 minutes (the RA change is retrograde). – Eduardo Guerras Valera Dec 16 '12 at 0:23
@EduardoGuerras: But it tells us that $t_0$ is not in fact $t_E$ (independent of my naive approximation), right? – raxacoricofallapatorius Dec 16 '12 at 0:26