astronomy - When is the right ascension of the mean sun 0?

I understand that the right ascension of the mean sun changes (at least over a specified period) by a constant rate, but where is it zero? I had naively assumed that it would be zero at the most recent vernal equinox, but when I try to calculate the equation of time using this assumption and true sun positions, all my values are about 7.5 minutes larger than they should be.

When (at what date and UT time) is the right ascension of the mean sun 0? And why?

Answer

No, the right ascension of the mean Sun is NOT zero at the vernal equinox. It is in fact nearly identical to the ecliptic longitude of the mean Sun (the difference is due to UT vs ephemeris time), and this is defined such that it coincides with the ecliptic longitude of the apparent Sun when the Earth is at perihelion. So that should be the starting time to calculate the equation of (ephemeris) time.

Using an ephemerides site like this one, one can check that the Earth was at perihelion on Jan 2 2013, at 4:37 UT, and will return to perihelion on Jan 4 2014, at 11:58 UT. The difference between these is called an anomalistic year. The same site shows that on Jan 2 2013, at 4:37 UT, the Sun had an apparent right ascension

$$ \alpha_p = 18^\text{h}51^\text{m}56^\text{s}, $$

and the corresponding ecliptic longitude is

$$ \lambda_p = \tan^{-1}(\tan\alpha_p/\cos\varepsilon) = 18^\text{h}47^\text{m}46^\text{s}, $$ where $\varepsilon=23^\circ 26' 21.4''$ is the obliquity of the ecliptic. The equation of ephemeris time is then $$ \Delta t = M + \lambda_p - \alpha, $$ where $M = 2\pi(t-t_p)/t_Y$ is the mean anomaly, $t_p$ is the moment of perihelion and $t_Y$ is the length of the anomalistic year. The quantity $M + \lambda_p$ is the ecliptic longitude of the mean Sun. The apparent right ascension $\alpha$ can be calculated from $$ \begin{align} M &= E - e\sin E,\\ v &= 2\tan^{-1}\left[\sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}\right],\\ \lambda &= v + \lambda_p,\\ \alpha &= \tan^{-1}(\tan\lambda\cos\varepsilon), \end{align} $$ with $E,e,v,\lambda$ the eccentric anomaly, orbital eccentricity, true anomaly and apparent ecliptic longitude (see also wiki). This should give you an approximation of the equation of time, accurate to a few seconds. If you want a really detailed calculation, have a look at this paper.