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Dear all,

In reference to our previous posts about Equinox definition, especially this excellent one by Herbert Prinz, it might be interesting to further investigate why and to what extent the Sun Ecliptic Latitude may and will vary.

First of all, for a given body its Ecliptic Latitude reckoned in either Mean Ecliptic of Date or True Ecliptic of Date ( = Mean Ecliptic of Date "modified" to take Nutation into account) has exactly the same numerical value. When performed in Ecliptic coordinates Nutation modifies only the Ecliptic Longitudes with the "ΔΨ" series being a mere zero offset. Things could have been done differently but that is the historical choice made by Astronomers then.

With the Moon / Earth mass ratio being equal to .0123 (easy to remember) - i.e. Earth / Moon mass ratio equal to 81.3 - if we assume Moon HP equal to 1° ( or 1/57.3 Radian), then the Earth-Moon Gravity Center is at 57.3 / (81.3 + 1) = 0.696 Earth Radius from the Earth Center. Such distance is seen from the Sun at 8.8" * 0.696 = 6.13". Anytime the Moon Latitude is equal to 1°, the Earth-Moon CG is seen from the Sun at an angle of 6.13" * sin 1° = 0.11", a value quite close from the 1/32,000 ° / degree of Moon Latitude quoted by Herbert Prinz here.

Herbert Prinz also quotes : (give or take 0.2") .

Additional insight here can be found in Simon Newcomb's most celebrated Tables of the Motion of the Earth (Astronomical Papers Vol VI - 1898).

Newcomb computes his coordinates in the Mean Ecliptic of date, and at page 18 he gives the effect of the Moon Latitude onto the Sun Latitude (hint : he uses decimal logarithms for one constant) equivalent to the approximate computation given just here-above.

But in addition and on page 17 Newcomb gives the effect the Perturbations of the Earth Latitude caused by Venus, Mars, Jupiter and Saturn. The biggest Venus perturbation is 0.21" and the biggest Jupiter perturbation is 0.17", and the cumulated effect of all these planetary perturbations in Latitude stays within +/- 0.4".

Hence, cumulated with the Moon "Perturbation" (which is mainly a matter and a result of actual Earth / Moon CG position), the planetary perturbations also affect the Sun Ecliptic Latitude altogether within maximum limits of +/- 1" at our current times.

Some 2,000 years ago, the same held true but the maximum amplitude of the Sun Ecliptic Latitude was close to +/- 1.4".

Physically these numbers confirm that the Moon orbit is bound to the Mean Equator of Date.

This gives us a simple and very efficient independent check of current Planetary Theories / Numerical Integrations in the past or into future.

Since all new planetary Theories and Numerical integrations are referred to Ecliptic 2000.0, for remotes dates in the past or into future (e.g. a few millennia around Year 2000.0) when computing the Sun Position in the Mean Ecliptic of Date with use of a consistent and compatible Precession Theory, one should observe that its Latitude referred to the Ecliptic of Date varies within extreme values centered on 0.0" Latitude, typically +/- 1.0 " at our current epoch, +/- 1.4" some 2.000 years ago and +/- 2" some 3.500 years ago.

Antoine M. "Kermit" Couëtte