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I am studying about apparent and mean positions of Sun using quite old books: Astronomical algorithms (1991) and Explanatory supplement to the astronomical almanac (1992).

The question is about the mismatch between the equation of the equinoxes and the correction for a right ascension. It seems like these two are the same concept. However, their equations do not seem to be matched.

According to the p.116 of the second book, the equation of the equinoxes refers to the nutation in right ascension. It is defined as follows: $$ \Delta \psi \cos \epsilon $$ where $\Delta \psi$ is the nutation in longitude and $\epsilon$ is the true obliquity of the ecliptic (i.e., correction about the nutation is applied).

However, according to the p.120 of the same book, the first order correction $\Delta \alpha$ to the right ascension $\alpha$ is as follows: $$\Delta \alpha = \left ( \cos \epsilon + \sin \epsilon \sin \alpha \sin \delta \right ) \Delta \psi - \cos \alpha \tan \delta \Delta \epsilon$$ where $\delta$ is the declination.

Because $\epsilon$ is about 23.4 degrees and others are arbitrary values, $\Delta \alpha$ cannot be approximated as $ \Delta \psi \cos \epsilon $.

It seems like there is some assumptions that I am missing or there is a problem in my understanding. I would like to ask help for this problem.

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The equation of the equinoxes only applies to the origin, so it is assumed $ \delta = 0 $. So $\tan \delta = 0, \sin \delta = 0 $, which means your second equation simplifies to $\Delta\psi \cos \epsilon $.

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