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Notes for Astrologers:

Astrological Lilith or the Dark Moon is either the apogee (”aphelion”) of the lunar orbital ellipse or, according to some, its empty focal point. As seen from the geocenter, this makes no difference. Both of them are located in exactly the same direction. But the definition makes a difference for topocentric ephemerides.

The opposite point, the lunar perigee or orbital point closest to the Earth, is also known as Priapus. However, if Lilith is understood as the second focal point, an opposite point makes no sense, of course.

Originally, the term ”Dark Moon” stood for a hypothetical second body that was believed to move around the earth. There are still ephemerides circulating for such a body, but modern celestial mechanics clearly exclude the possibility of such an object. Later the term ”Dark Moon” was used for the lunar apogee.

The Swiss Ephemeris apogee differs from the ephemeris given by Joëlle de Gravelaine in her book ”Lilith, der schwarze Mond” (Astrodata 1990). The difference reaches several arc minutes. The mean apogee (or perigee) moves along the mean lunar orbit which has an inclination of 5 degrees. Therefore it has to be projected on the ecliptic. With de Gravelaine"s ephemeris, this was not taken into account. As a result of this projection, we also provide an ecliptic latitude of the apogee, which will be of importance if declinations are used.

There may be still another problem. The "first" focal point does not coincide with the geocenter but with the barycenter of the earth-moon-system. The difference is about 4700 km. If one took this into account, it would result in a monthly oscillation of the Black Moon. If one defines the Black Moon as the apogee, this oscillation would be about +/- 40 arc minutes. If one defines it as the second focus, the effect is a lot greater: +/- 6 degrees. However, we have neglected this effect.

[added by Alois 7-feb-2005, arising out of a discussion with Juan Revilla] The concept of "mean lunar orbit" means that short term. e.g. monthly, fluctuations must not be taken into account. In the temporal average, the EMB coincides with the geocenter. Therefore, when mean elements are computed, it is correct only to consider the geocenter, not the Earth-Moon Barycenter.

Computing topocentric positions of mean elements is also meaningless and should not be done.

As has been said above, the osculating lunar apogee (so-called "true Lilith") is a mathematical construct which assumes that the motion of the moon is a two-body problem. This solution is obviously too simplistic. Although Kepler ellipses are a good means to describe planetary orbits, they fail with the orbit of the moon, which is strongly perturbed by the gravitational pull of the sun. This solar perturbation results in gigantic monthly oscillations in the ephemeris of the osculating apsides (the amplitude is 30 degrees). These oscillations have to be considered an artifact of the insufficient model, they do not really show a motion of the apsides.

A more sensible solution seems to be an interpolation between the real passages of the moon through its apogees and perigees. It turns out that the motions of the lunar perigee and apogee form curves of different quality and the two points are usually not in opposition to each other. They are more or less opposite points only at times when the sun is in conjunction with one of them or at an angle of 90° from them. The amplitude of their oscillation about the mean position is 5 degrees for the apogee and 25 degrees for the perigee.

This solution has been called the "interpolated" or "realistic" apogee and perigee by Dieter Koch in his publications. Juan Revilla prefers to call them the "natural" apogee and perigee. Today, Dieter Koch would prefer the designation "natural". The designation "interpolated" is a bit misleading, because it associates something that astrologers used to do everyday in old days, when they still used to work with printed ephemerides and house tables.

The position of "True Lilith" is given in the "New International Ephemerides" (NIE, Editions St. Michel) and in Francis Santoni "Ephemerides de la lune noire vraie 1910-2010" (Editions St. Michel, 1993). Both Ephemerides coincide precisely.

The relation of this point to the mean apogee is not exactly of the same kind as the relation between the true node and the mean node. Like the "true" node, it can be considered as an osculating orbital element of the lunar motion. But there is an important difference: The apogee contains the concept of the ellipse, whereas the node can be defined without thinking of an ellipse. As has been shown above, the node can be derived from orbital planes or great circles, which is not possible with the apogee. Now ellipses are good as a description of planetary orbits because planetary orbits are close to a two-body problem. But they are not good for the lunar orbit which is strongly perturbed by the gravity of the Sun (three-body problem). The lunar orbit is far from being an ellipse!

The osculating apogee is "true" twice a month: when it is in exact conjunction with the Moon, the Moon is most distant from the earth; and when it is in exact opposition to the moon, the moon is closest to the earth. The motion in between those two points, is an oscillation with the period of a month. This oscillation is largely an artifact caused by the reduction of the Moon’s orbit to a two-body problem. The amplitude of the oscillation of the osculating apogee around the mean apogee is +/- 30 degrees, while the true apogee"s deviation from the mean one never exceeds 5 degrees.

Selena/White Moon

This is a ‘hypothetical’ second moon of the earth (or a third one, after the “Black Moon”) of obscure provenance. Many Russian astrologers use it. Its distance from the earth is more than 20 times the distance of the moon and it moves about the earth in 7 years. Its orbit is a perfect, unperturbed circle. Of course, the physical existence of such a body is not possible. The gravities of Sun, Earth, and Moon would strongly influence its orbit.

Я пользуюсь так называемой Томой, точкой, противоположной Друдж, то есть точкой противоположной истинному апогею лунной орбиты.

в первом посте всё описано, другое дело, что народ не прочёл Не знаю

Луна двигается вокруг Земли не по кругу, а по эллипсу. Самая ближняя к Земле точка лунной орбиты называется перигеем лунной орбиты, самая дальняя точка лунной орбиты называется апогеем... итп

Тома там нет совсем.