Mean & True Black Moon Lilith

Black Moon Lilith is usually defined as the apogee of the lunar orbit. I.e., the Moon moves around the Earth in an ellipse shaped orbit, and the apogee is the point where the Moon is farthest from Earth. The position of the apogee can be calculated as "Mean" or "True."

Mean and True Apogees

The Mean and True positions are based on two different estimations of the Moon's current orbit. Neither can exactly be said to be right or wrong, because at the moment you request the position of the Moon's apogee, the Moon is simply where it is, and projecting where it would be if it was at apogee is hypothetical whatever formula you use.

What the Moon Really Does

The Moon does not orbit about the centre of the Earth. The Earth and Moon work as a two-body system, both spinning around their common centre of gravity. This centre-of-gravity point, about which they both turn, is called the Earth-Moon Barycenter. The barycenter is within the Earth, about 4,700 km from its centre. Therefore the Earth wobbles around this off-centre point within itself, and the Moon also orbits the same point. You can see a good animation of it here.

A "pure," undisturbed orbit of a body about a point would be ellipse-shaped (see Kepler's work). In practice, the Moon experiences stronger gravitational pull from the Sun than it does from the Earth, and this disturbs its orbit a lot, as does pull from other bodies in the solar system. The result is a lunar orbit which looks like an ellipse drawn with a wavy, wobbly line.

The Mean Orbit

The Mean Lunar Orbit is an attempt to even out the wobbly line into a nice smooth ellipse, by averaging out the Moon's positions over time. Having done that, we can determine where the apogee of the mean orbit is, and that would be the Mean Lunar Apogee, or Mean Black Moon.

Swiss Ephemeris and others normally calculate this mean lunar orbit using the formula derived by Chapront, Chapront-Touzé and Francou of the Observatoire de Paris.

The "True" or Osculating Orbit

The "true" (a.k.a. osculating) orbit is a calculated, hypothetical orbit based on a momentary snapshot in time. You take the exact current position, speed, angle etc. of the Moon (i.e., the orbital elements), and apply Kepler's formula to that data. This calculates the path that a pure elliptical orbit continuing from that point would take.

Because the Moon wobbles around all over the place, and this orbit is based on a snapshot, the apogee of this orbit can be far away from the Mean Apogee. It can also be far from anywhere the apogee is likely to be in the near future. Yet this point is known as the True, or Osculating, Lunar Apogee -- True Black Moon.

Kepler's Laws of Planetary Motion are here.

Astrologers seem to like to argue about which point is best, but I think the thing is to understand what each of them are, so you can apply the astronomical symbolism to your own observations of how the points work in charts.

Other Notes

The point opposite Black Moon Lilith (i.e., the Lunar Perigee) is sometimes known as Priapus, "a minor rustic fertility god of purely phallic character."

Some astrologers view the Black Moon as being not the apogee of the elliptical orbit, but the empty focus of the ellipse.

I.e., the Earth is inside the Moon's elliptical orbit towards one end. The other end, where the Earth is not, is the empty focal point of the ellipse.

In practice, the Apogee and the Empty Focus can be thought of as being in the same place on the chart, but the difference in symbolism can be relevant.

Further Reading

Juan Revilla gives a thorough and detailed technical explanation here.