THE ASTRONOMICAL VARIANTS

OF THE LUNAR APOGEE - BLACK MOON

by Juan Antonio Revilla

I. Introduction The Black Moon / Lunar Apogee or "Empty Focus" is an essential element of astrological and astronomical symbolism. Its action is very powerful in every horoscope, but unfortunately it tends to be underestimated and there is great confusion among astrologers about how to calculate it. I will start by giving their definitions and categorization, without delving into their astrological meaning first. Astronomically, there are 3 types and 7 variants of "lunar apogee": the types are "osculating", "mean", and "natural". From the "osculating geocentric apogee" (1) are derived the "osculating topocentric apogee" (2), the "osculating topocentric perigee" (3), and the "osculating topocentric empty focus of the lunar orbit" (4). To these are added the so-called "interpolated" apogees, which Riyal calls "natural apogee" (5) and "natural perigee" (6). The "Mean" apogee (7) by definition excludes short-period fluctuations and therefore has no variants (see explanation further below).

II. The Mean Apogee



The "Mean Apogee" is the most popular of the alternative "Black Moons", mainly because it has been used for a much longer time than all the others, and also because often astrologers are not aware of the alternatives. It seems to have been introduced into astrological practice in France by Don Neroman (Pierre Rougi�, 1884-1953) in the early or mid 1930's. He apparently was also the first to call it "Black Moon".



Astronomically, this point corresponds to the apogee or perigee of the reference lunar orbit used by astronomers to construct lunar ephemerides. It moves very regularly in a perfectly circular orbit with a radius of 405,863 Km around the Earth/Moon barycenter, i.e., its positions are not geocentric, but the difference between its barycentric and geocentric position is never more than 0,40'. Despite what one may think in theory, though, the barycenter does not really have any effect in the calculation of the lunar apogee. Please consider the following: Astronomically, the concept of "mean apogee" (or of mean elements in general) excludes by definition any differences between geocentric and barycentric. These differences represent short-period fluctuations that have been already averaged-out in the "mean position". In other words, when talking about mean orbital elements and positions, the geocenter and the barycenter coincide, there is no difference between them. Therefore, there is no "barycentric" *mean* apogee. For this same reason, there really is no topocentric *mean* apogee, since the difference between geocentric and topocentric is a short-term fluctuation and is excluded in the definition of "mean", it has already been averaged out. (Of course one can always calculate barycentric and topocentric equivalents of a "mean" position, but they have no astronomical meaning.) Therefore, when using the MEAN apogee or perigee, the geocenter, barycenter, and topocenter all coincide, there is only one, not 3 positions. Likewise, the MEAN empty focus and the MEAN apogee will always be aligned, i.e., their longitude will always be the same. This sounds a little odd, but this is what the *mean* lunar elements represent: an average devoid of any real-world short-term fluctuations. In the case of the "true" or osculating lunar apogee (see below), one must keep in mind that it is always calculated from the geocentric position and velocity vectors of the Moon, therefore there is no barycentric equivalent. The periodic fluctuations between barycentric and geocentric positions do not have any effect on it, because the barycenter is never part of the equation when it is calculated. In other words: there is no "barycentric" true apogee or empty focus. The osculating apogee is already geocentric from the start, and it always coincides with the position of the empty focus of the lunar orbit. NOTE: I have to thank Alois Treindl for helping me see these points clearly after a discussion last February in the Riyal_Compute forum. After the discussion Alois added a clarifying note to the Swiss Ephemeris documentation and I added a corresponding explanation in my compilation "Variants of the Apogee" (Alois and I disagree, however, on the general significance of the osculating apogee). Seen geocentrically, the line of the apsides (i.e., from perigee to apogee across the orbit) is identical to the position of the "empty" or 2nd focus, sometimes called "kenofocus". This empty focus is an essential aspect of the Black Moon symbolism, and is the basis of the ideas and metaphors I developed in my Black Moon essay at my site.



The circular, extremely regular motion of the fictitious mean lunar apogee / Black Moon belongs to the world of solar symbolism. Such type of motion is alien to the lunar world and to the symbolism of the Black Moon. Although I respect the experience of astrologers who don't question the validity of this mathematical point, my mind, accustomed to find the astronomical symbolism reflected in the astrological symbolism, finds it impossible to identify this point with the "dark" world of lunar symbolism represented by the Black Moon.





III. Topocentric Positions



One of the many proofs that astrology does not deal with what really happens in the sky, besides the fact that the mean lunar apogee is completely fictitious and its motion has little to do with the real changes in the lunar orbit and in lunar motion, is to realize that the apparent topocentric position of the Moon (relative to the observer), which can differ by more than 1 degree from its geocentric position, is almost never used by astrologers, that almost invariably use the Moon's geocentric position.



The difference between the geocentric positions of the lunar apogee and the lunar empty or 2nd orbital focus reaches 6.4 degrees every 27 days; however, when we compute their topocentric positions, this difference (a result of "parallax") reaches a maximum of 7.9 degrees every 6 months. It gives rise to 3 topocentric variants of the Mean Black Moon, which can be labelled as:



- the topocentric equivalent of the geocentric position of the Mean Lunar Apogee

- the topocentric equivalent of the geocentric position of the Mean Lunar Perigee, and

- the topocentric equivalent of the geocentric position of the 2nd or empty focus of the mean lunar orbit



Topocentric positions are normally not used in Astrology, but the large difference of up to 7.9 degrees between the geocentric and the topocentric empty focus of the lunar orbit shows that of all points in an astrological chart, this is the one closest to the Earth, much closer than the Moon, giving a more personal perspective of basic or primitive lunar symbolism than anything else in a chart. The unicorn and Priapus-like or --as Axel Harvey calls it-- Charybdis-like symbolism of this point is enormous.





Recapitulating:



1-) the ordinary, traditional "Mean Black Moon" or mean lunar apogee is not geocentric but barycentric. It orbits the Earth/Moon barycenter, located in a straight line between the centers of the Earth and of the Moon about 1350 Km inside the Earth's crust (both the Earth and the Moon orbit this point once every 27 days).



2-) the Mean Apogee is a fictitious point that is used only as reference, it does not represent the true orbit of the Moon at a given point in time. Its orbit is a perfect circle with a radius of 405,863 Km, and its motion is almost completely linear or - symbolically-- "solar".



3-) if instead of the apogee we think of the mean empty (or second) focus of the lunar orbit, which is closer to the basic "Black Moon" symbolism, its barycentric position is the same as that of the mean apogee, but the radius of its orbit is 42,230 Km.



4-) the apogee is measured from the vernal equinox (0 Aries) along the lunar orbit and not the cliptic, so it has a latitude that is a function of its distance from the node and can reach more than +- 5 degrees. This latitude produces an oscillation of +- 0,06' in the ecliptical longitude of the apogee.



5-) the transformation from barycentric to geocentric is never made in astrological practice. It results in an oscillation of +- 0,40' when the Black Moon is defined as the apogee, and of +- 6.4 degrees when it is defined as the empty focus of the orbit. In other words, the geocentric position of the apogee can differ by more than 6 degrees from the geocentric position of the empty focus of the orbit. They are the same only in the barycentric reference frame, not in the geocentric.



6-) if we are interested in the observer or topocentric --not the geocentric-- point of view, then the difference between the barycentric and the topocentric positions will periodically reach as much as 7.9 degrees.



7-) the distinction between the barycentric and the geocentric position applies also to the mean lunar node. The transformation from barycentric to geocentric produces an oscillation of +- 0,44' in the position of the node, although this distinction, like in the case of the apogee, is never made.



8-) when calculated geocentrically instead of barycentrically, the mean perigee and mean apogee will no longer be 180 degrees from each other. They will be different also from the topocentric point of view.



9-) the 3 frames of reference: barycentric, geocentric, and topocentric plus the 2 points of the axis (apogee and perigee, ascending and descending node) produce the following variations of the lunar apogee and node:



- ordinary mean apogee/perigee (barycentric)

- geocentric mean apogee (+- 0,40')

- geocentric mean perigee (+- 0,40')

- geocentric mean empty focus (+- 6.4 degrees)

- topocentric mean apogee (+-0,40')

- topocentric mean perigee (+- 0,40')

- topocentric mean empty focus (+- 7.9 degrees)



- ordinary mean ascending node (barycentric)

- geocentric mean ascending node (+-0,44')

- geocentric mean descending node (+-0,44)

- topocentric mean ascending node (+-0,44')

- topocentric mean descending node (+-0,44)



We have then no less than 7 variations of the *mean* Black Moon only, and 5 variations of the mean node. Their values will all be slightly different.





IV. Riyal's output



Here are the values calculated by Riyal 1.4 in the " Tables --> Astronomical Data " routine for the time I am writing this:



Mean Node = 28,01.3 Tau

--geocentric = 28,45 Tau

--descending = 27,18 Sco

--topocentric = 28,25 Tau

--descending = 27,37 Sco

Mean Apogee = 14,23.6 Tau

--geocentric = 15,00 Tau

--perigee = 13,44 Sco

--empty focus = 20,22 Tau

--topocentric = 14,53 Tau

--perigee = 13,50 Sco

--empty focus = 18,45 Tau

and a sample of part of the ephemerides routine display (" Special-->Generate Ephemerides-->Apsides and node-->Moon... "):



M.Bari M.Geoc M.Peri M.Foco

10 Jun 2003| 13Ta16 | 13Ta00 | 13Sc35 | 10Ta50 |

11 Jun 2003| 13Ta23 | 13Ta15 | 13Sc32 | 12Ta16 |

12 Jun 2003| 13Ta30 | 13Ta31 | 13Sc28 | 13Ta45 |

13 Jun 2003| 13Ta36 | 13Ta47 | 13Sc24 | 15Ta13 |

14 Jun 2003| 13Ta43 | 14Ta03 | 13Sc20 | 16Ta37 |

15 Jun 2003| 13Ta50 | 14Ta17 | 13Sc19 | 17Ta53 |

16 Jun 2003| 13Ta56 | 14Ta29 | 13Sc19 | 18Ta57 |

17 Jun 2003| 14Ta03 | 14Ta40 | 13Sc21 | 19Ta46 |

18 Jun 2003| 14Ta10 | 14Ta49 | 13Sc26 | 20Ta18 |

19 Jun 2003| 14Ta16 | 14Ta55 | 13Sc33 | 20Ta31 |

20 Jun 2003| 14Ta23 | 14Ta59 | 13Sc43 | 20Ta24 |



MBari = ordinary Mean Black Moon, reduced to the ecliptic

M.Geoc = geocentric mean apogee

M.Peri = geocentric mean perigee

M.Foco = geocentric mean empty or 2nd focus



Besides the comments of Alois Treindl in his source code for "Placalc" in the mid or late 80's, which he later reiterated in the Swiss Ephemeris, I have never seen anyone else making a distinction between the barycentric and the geocentric position of the mean lunar apogee (this distinction does not exist in the case of the true apogee, which is already geocentric). Since I am not familiar with the French "Lune Noire" literature, I don't know if that distinction is made in Europe.



Alois Treindl's commentary is what inspired me to investigate further the matter of "mean apogee positions". Unfortunately, he never went on to offer the calculations in his software (barycentric to geocentric and then geocentric to topocentric positions of the lunar node and apogee), so I have nothing against which to check Riyal's accuracy. The Swiss Ephemeris assumes the mean node and apogee as geocentric and then tranforms them to topocentric; the topocentric positions of the true or osculating node, apogee, perigee, and second focus --which do not need to be transformed from barycentric to geocentric-- agree with Riyal's.

NOTE: In early February 2005, I personally asked Alois Treindl why the distinction between the geocentric and the barycentric apogee, though mentioned in the Swiss Ephemeris documentation, is never made in the Swiss Ephemeris program itself. We had an exchange of emails on this subject in the "Riyal_compute" forum, the result of which was an important clarification. Alois wrote: "The whole concept of a mean orbits precludes consideration of such short term (monthly) fluctuations. In the temporal average, the EMB [Earth/Moon Barycenter] coincides with the geocenter... It is probably pointless to compute topocentric positions of mean points - a contradiction in itself. Don't do it, or don't expect meaningful results from it." He subsequently added the following note in the Swiss Ephemeris documentation: "[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. In addition, computing topocentric positions of mean elements is also meaningless and should not be done." As a result of this clarification, the conversion from barycentric to geocentric --and from geocentric to topocentric-- mean lunar node and apogee was immediatly removed from Riyal. . The "true" or osculating apogee is not affected by any of this.



V. The "Corrected" Apogee The erroneous "corrected" apogee used in Europe is based on a regular sinusoidal correction of 12.x degrees (the fraction apparently varies among different authors) applied to the mean apogee. This came as a result of efforts to find the "true" position when the astronomical theory necessary to calculate this "true" position had not been developed. The ability to calculate it was there, but the method of calculation, based on the position and velocity vectors of the Moon (the method used in Riyal), was not readily accessible to astrologers.



This situation changed only after the publication of " Lunar Tables and Programs " in 1991, authored by Jean Chapront and Michelle Chapront-Touz�, astronomers at the Bureau des Longitudes and developers of the most modern and accurate lunar theory to this date, called " ELP-2000" (Riyal uses a truncated long-term version of it called "ELP2000-85", published by the same authors in 1988. I had been following the development of this theory since the authors' first publications in the early 1970's until its final working version introduced world-wide in 1984.



The tables of the trigonometric expansion of the mean lunar apogee to produce an accurate approximation of the true or osculating apogee, published in the above-mentioned book in 1991 and based on ELP2000-85, made evident that the "12.x" degree correction used by some astrologers until then (promoted by reputable thinkers such as Jean Carteret), was wrong from every point: one, because the real maximum difference between the true and the mean apogee was not 12-13 degrees but 30, two, because the "main solar perturbation term" (period=31.8 days) on which this correction was based was not "12.x" degrees but 15.4 degrees, and three, because the so called "correction" used by astrologers, and for which tables had been published, etc., was being applied in the opposite (wrong) direction.



These facts were not known even to astronomers in general before the 1991 book by the Chapronts. The "ignorance" here was for practical and historical reasons: the true or osculating lunar orbit was a factor that had not been a part of present-day lunar theory since it began to be developed in the late 1800's. Lunar theory was (and is to this date) based on a reference or idealized ellipse that establishes the so-called "Delaunay arguments" from which to build the trigonometric expansion of the 3 lunar coordinates: longitude, latitude, and distance. In this process, the mean reference perigee/apogee is used to form the arguments of the trigonometric terms, but the true instantaneous position of the apogee is never needed.



It may come as a surprise that an accurate theory of the true or osculating lunar apogee did not come until 1991. This may give an idea of the complexity of the Moon's motion and orbit in space, and the enormous difficulties that theoreticians of celestial mechanics had to face to develop a suitable theory for it. So it is no wonder that astrologers -- or even astronomers-- had an erroneous understanding of the real instantaneous motion and gravitational perturbations of the lunar apogee/perigee. There were theoretical developments and approximations, but nobody had tried the real numerical solution until the Chapronts published their results.



Nevertheless it is very common for astrologers to ignore or misunderstand the astronomical facts, and to this date there are still people working with this erroneous "corrected" apogee or Black Moon. The 12/13- degree gap that opens and closes periodically between the corrected (called "true" by its users adding to the confusion) and the mean position is even given special significance by some astrologers... interesting concept this gap... but based on something that is mathematically and astronomically erroneous or non-existent. Astrology has many examples of this: the Uranian planets, the Dark Moon, and in my opinion, the mean lunar apogee (more on this later).



One wonders, with so many options available (we still have to see the variants of the true or osculating apogee), if the Black Moon has meaning at all. Astrology is full of cases like this (e.g., house division, asteroids...). There is no easy answer. However, I think this question disappears when astrologers realize that Astrology is what astrologers do: work with more or less fancy and abstract mathematical points in the imagination. Astrology has very little or nothing to do with our relationship with "the sky out there" or with "the cosmos". If you realize this then the efficacy of imaginary points comes to light under a different perspective, one which has to do with cognitive patterns and structures in the human brain and not with astronomical events. It becomes a matter of individual idiosyncrasy the tools you choose to work with, and there is no fear or prejudice against tools that have no solid astronomical basis. Some people simply do not need that basis... however, I do, and I think that this basis is important in order to keep Astrology (or astrologers' minds) disciplined and "down to earth", i.e., to keep Astrology healthy.





V. The "True" or Osculating Apogee



Unlike the mean apogee where topocentric positions do not make much sense astronomically (and therefore, there is really never any difference between the mean empty focus and the mean apogee), the osculating or "true" apogee, by definition, represents the actual, real-world fluctuations of the lunar orbit, so calculating its topocentric equivalents makes sense astronomically. The word "osculating" is explained at the end of a long compilation of posts that you will find in my site discussing the astronomical definition of the Black Moon or lunar apogee:



http://www.expreso.co.cr/centaurs/blackmoon/apogee.html



This is the relevant part (written in June 2000):



[BEGIN QUOTE]

If you look in a Latin dictionary, you find:

--"OSCULATIO" = kiss, the action of kissing "OSCULOR, OSCULATUM SUM": to kiss, to caress, to pet.

The word is also in the Webster's:

--"osculant" = united by certain common characteristic "oscular" = pertaining to an osculum, pertaining to the mouth or kissing

--"osculate" = 1- to kiss; 2- to come into close contact or union; 3- (geometry, of a curve) to touch another curve or another part of the same curve so as to have the same tangent and curvature at the point of contact; 4- to bring into close contact or union; 5- to touch another curve or another part of the same curve in osculation; 6- (archaic) to kiss

--"osculating plane" = the plane containing the circle of curvature of a point on a given curve.

--"osculation" = 1- the act ok kissing; 2- a kiss; 3- close contact; 4- (geometry) the contact between two osculating curves or the like

--osculum" = a small mouthlike aperture as of a sponge.

The "moment of osculation" is only a brief moment: the next instant the "point of osculation" will have shifted in space...

[END QUOTE]



That is, the real orbit of an object --and in particular the orbit of the Moon-- is changing all the time due to the attraction of many or of several perturbing gravitational forces, so the moment of osculation is only an instant, it represents an "instantaneous orbit" that "kisses" the real orbit and then diverges as the real orbit is accelerated. It is like opening a momentary window to observe the orbit at that instant, knowing that it will change its appearance as soon as we close the window again, or like taking a picture that "freezes" the instantaneous reality of the orbit.



By "real orbit" is meant the orbital plane as it looks through time; it can be conceived as a collection or accumulation of these instants, of an endless series of instants describing its changes or oscillations, the slightly different shapes that the orbit assumes through time. Of course the word "real" used here is very relative, because it does not imply that the osculating orbit is not real.



Some people prefer to think arbitrarily that an osculating orbit --a perfectly defined keplerian ellipse--, does not correspond to "reality", forgetting the fact that the osculating ellipse is the accurate representation of an object's trajectory in space at a given moment. The Keplerian ellipse, i.e., the osculating orbit, describes the motion of the object at that moment of time.



This is exactly what we do in Astrology when we make a chart: we "freeze" artificially the movement of the celestial sphere and work only with that instantaneous picture .



The orbit through time constitutes a series of oscillations around an average or "mean" slowly evolving keplerian ellipse. This would be the equivalent of the Mean Black Moon or barycentric mean lunar apogee. The osculating apogee represents the actual trajectory of the Moon as it actually is at a given instant. We can think of it as a ghost image that the Moon carries with it all the time. This ghost image represents a sort of ideal, an "ideal future" when the Moon is (or will be) at apogee, but it keeps changing or evolving as the Moon travels through space.



We can also think of the perigee, the north and south nodes, and the empty second focus of the orbit in the same way: they all represent idealized focal points or "directions" that are a result of the "psychic projections" of the Moon, they are "Moon ghosts" that the Moon always carries with it, that are part of the "lunar structure" of every individual. The Moon represents the present moment, the nodes, apsides, and empty focus represent the past and "look forward" psychic projections that give shape and structure to the lunar dynamics of a person's life. They are like the rooms, passages and corridors of a house (the different parts of the orbit) that become projections of the person who inhabits it (the Moon).



Some people think that an osculating orbit is something too artificial or unreal, and call the use of the osculating lunar apogee or Black Moon "nonsense", "close to nonsense", "makes no sense at all", etc. The main reasons usually given for this are 1-) the large difference (of up to 30 degrees) between the mean and osculating value of the lunar apogee, 2-) the fact that this osculating value "can travel to places where the Moon will never go" (see how amazingly significant this is in psychological and psychic terms!), and 3-) its erratic changes of direction and velocity, which for some it means it cannot be really called "a motion" at all.



But I have always insisted that it is precisely all that what makes the symbolism of the osculating apogee / Black Moon so powerful. It doesn't matter at all that its motion is erratic: it doesn't have to move like a planet because it is not a planet! ... this brings it symbolically closer to all the neglected psychical projections of the Moon, both positively and negatively. The osculating Black Moon, representing the constantly changing shape of the lunar orbit is a very good fit to the organic, instinctual nature of Black Moon symbolism.



NOTE: I discuss this symbolism in http://www.expreso.co.cr/centaurs/blackmoon/lilith.html





VI. Numeric Data



We saw that the Mean Black Moon or lunar apogee describes a circular orbit around the Earth/Moon barycenter (period=8.8 years), and how paradoxical it seems that such circular motion is used to represent the lunar apogee. The radius of this orbit depends on the value of the mean Earth-Moon distance, which the ELP2000 theory gives as 384,747.98 Km. This value, however, is derived directly from the Moon's mean sidereal motion and is barycentric, i.e., it is the semimajor axis of the Moon's orbit around the center of mass of the Earth and Moon.



In a publication of 1994 (Astronomy and Astrophysics, 282, p.663), the authors of ELP2000 provided for the first time to the consideration of astronomers true mean elements of the Moon comparable to those of the planets, and gave the value of the mean Earth/Moon distance as 383,397.77 Km (this is the quantity used by Riyal). Using the mean lunar eccentricity provided in the same publication, one can calculate the following:



- radius of the circular orbit of the mean apogee (Black Moon) around the Earth = 404,694 Km

- radius of the circular orbit of the mean perigee ("Priapus" in France) around the Earth = 362,102 Km

- distance between the 2 foci of the orbit (Earth and empty focus) = 42,592 Km



These distances, as far as the "Mean Black Moon" is concerned, are fixed, and represent concentric circles around the Earth. They are unmutable abstractions that represent the horizontal poles (called "line of the apsides") of a reference Lunar orbit circulating the Earth/Moon barycenter. In the real world, the apogee/perigee distances and the distance between the foci of the lunar orbit vary within a certain range:



- the apogee varies between 404,039 and 406,720 Km

- the perigee varies between 356,337 and 370,407 Km

- the distance of the 2nd focus varies between 34,506 and 49,841 Km



When one plots the true distance of the Moon in its cycle from apogee to perigee over a period of time against the distance of the osculating apogee, it becomes evident that the osculating apogee reaches distances that exceed those of the Moon, i.e., it "travels to places where the Moon can never go" (this is a phrase used by Alois Treindl in a post to alt.astrology.moderated). This is illustrated by a graphic done with Riyal's "Graphic Transits" routine :



(2002-2003)



You can see that most of the times when the Moon is at perigee the orbit stretches outward and the osculating apogee reaches distances of up to 415,000 Km, that the Moon will never (and can never) reach. Here is the complementary graphic, made with a modified Riyal in order to show the osculating perigee:



(2002-2003)



In this case, we can see the distance of the osculating perigee also stretching but much less, reaching minimums of about 352,000 Km, while the pattern is inverted: when the Moon is at apogee, the orbit stretches inward.



What we can conclude from this is that only the osculating apogee "goes to those places" that the Moon can never reach but which are nevertheless part of its osculating orbit, like the "ideals" or "ghosts" I mentioned before. The fluctuations shown here represent the real changes of the instantaneous lunar orbit, always matched by the expansion or contraction of the distance between the 2 foci, i.e, the empty focus and the Earth. The stretches or "ideals" of the osculating apogee are a reflection of the organic, "live" dialectical relationship between Earth and the empty focus of the Moon, the lunar ghost of the Earth.





VII. The oscillations



In classical planetary theory, every "real" orbit is seen as a series of periodic oscillations around a mean Keplerian orbit that changes slowly with time, this last secular change often being also an oscillation of very long period. This means that there is a real place in astronomy (and astrology) for "mean" or average values of orbital elements such as the lunar apogee. But normally the oscillations are of relatively small amplitude, as in the case of the lunar node (see below). It is a peculiarity of the lunar apogee that the oscillations (that is, the difference between the "true" or osculating value and the mean) can reach an amplitude of 30 degrees.



This large amplitude, according to some (e.g., the writers of the Swiss Ephemeris), implies that the osculating (or "oscillating" as is called sometimes by mistake) has no meaning. But as I have explained, it is exactly the opposite: this very large difference with respect to the mean value enhances its meaning, it makes the osculating lunar apogee --the True Black Moon, with its wild oscillations and changes of speed and direction-- more unique and powerful, the best representation there is of the "emotional accumulator" or reactor of primitive and organic lunar symbolism.



The changes in the shape of the lunar orbit reflected by the True Black Moon, precisely because they are swifter and more pronounced or irregular, resemble a living organism more than anything in Astrology!



I would like to illustrate numerically these changes, compared to the changes of the lunar node. This can be done with the tables in the 1991 book "Lunar Tables and Programs" by the Chapronts mentioned before. The tables allow to calculate the osculating or true node with an accuracy of 1.6 arcminutes and the osculating or true apogee with an accuracy of 29 arcminutes (0.5 degrees). This was the source of the first tables ever of the True Black Moon published in France in the early or mid 90's. (NOTE: Riyal does not use this method. It uses a different, more accurate procedure based on the instantaneous position and velocity vectors of the Moon).



The largest terms of the lunar node (i.e., the first "perturbations" or periodic oscillation that force the node to deviate from the mean value) look like this:



a-) 1,30' period = 173 days

b-) 0,09' period = 1 year

c-) 0,07' period = 14.8 days

d-) 0,07' period = 13.6 days

e-) 0,05' period = 3 years...



... up to 22 terms in the book. They are all functions of a combination of the Sun, the mean reference barycentric lunar orbit, and the Earth/Moon barycenter. You can see that the largest oscillation is moderately small (one degree and a half), and that the second largest is only a small fraction of the first. Now see the difference in the case of the apogee (it is the perigee in the book, but they are interchangeable simply adding 180 degrees to the result):



a-) 15,27' period = 31.8 days

b-) 9,28' period = 205.9 days

c-) 2,43' period = 27.5 days

d-) 2,36' period = 37.6 days

e-) 2,05' period = 15.9 days

f-) 1,29' period = 9.6 days



... up to 58 terms. They are too functions of a combination of the Sun, the mean reference barycentric lunar orbit, and the Earth/Moon barycenter. The oscillations are much larger, and many more terms are required to calculate the position of the osculating apogee from its reference mean value. (The first term, taken as 12.3 degrees instead of 15.4, applied in the opposite direction, and arbitrarily ignoring all the other terms, is the origin of the "corrected" apogee still used in Europe).



To illustrate the scale of the oscillations, here is a graphic done with Riyal's "Graphic transits" routine that plots the longitude of the true osculating node and the true osculating apogee for a period of 1 year from 1-1-2003 to 1-1-2004.:







You can see clearly in the graphic the main 15-degree monthly (31.8 days) oscillation. The graphic is also showing a series of conjunctions between the transiting True Black Moon and the transiting True Node beginning in mid-2003. No less than 13 conjunctions can be seen. These can be calculated very accurately with Riyal's "Special --> Phenomena (search)" routine:



Node/Apogee 29Ta25 29Ta25 2452794.6657 4/ 6/2003 3h58.6

Node/Apogee 29Ta25 29Ta25 2452804.2594 13/ 6/2003 18h13.5

Node/Apogee 28Ta38 28Ta38 2452823.9266 3/ 7/2003 10h14.4

Node/Apogee 27Ta50 27Ta50 2452837.3294 16/ 7/2003 19h54.3

Node/Apogee 26Ta15 26Ta15 2452856.0912 4/ 8/2003 14h11.3

Node/Apogee 24Ta52 24Ta52 2452868.7514 17/ 8/2003 6h02.1

Node/Apogee 23Ta04 23Ta04 2452890.6715 8/ 9/2003 4h06.9

Node/Apogee 22Ta04 22Ta04 2452903.3662 20/ 9/2003 20h47.4

Node/Apogee 20Ta52 20Ta52 2452921.2558 8/10/2003 18h08.4

Node/Apogee 20Ta31 20Ta31 2452936.6596 24/10/2003 3h49.9

Node/Apogee 20Ta27 20Ta27 2452950.2213 6/11/2003 17h18.6

Node/Apogee 20Ta22 20Ta22 2452972.1431 28/11/2003 15h26.1

Node/Apogee 20Ta23 20Ta23 2452977.2176 3/12/2003 17h13.4



In a period 6 months from June 2003 to December, the True Black Moon made 14 conjunctions with the lunar node. I have never understood why some people, seeing this, think that using the True Black Moon is "nonsense". I think it is fantastic! Its transits are really obsessive/compulsive, like the symbolism attributed to it! (NOTE: it is common to see it transiting a natal point 19 or 20 times during a year).





VII. A Brief Example



Jos� Asunci�n Silva was born November 27th, 1865, in Bogot�. His most famous poem, "Nocturne III" or "One Night", was dedicated to his sister Elvira, 5 years younger than him, who died of pneumonia on January 11, 1891, at the age of 21. By the way he expressed himself, this poem gave origin to many commentaries about a possible incestuous relationship between them. Personally, I think this is unjustified, although it is obvious, from what I read from his biographers, that her death affected him very deeply (all the biographical material is taken from the Web)



"One Night" shows a tremendous obsessive edypical erotism, of majestic beauty and musicality. It has the "wolf" atmosphere of a song to the night and the expansion of the "wings of death" moved by amorous passion. I believe this can be seen in the following position at his birth (I am using 18h GMT):



Venus = 13,58 Scorpio

Pluto = 12,49 Taurus



But most interesting is the position of the True or osculating Black Moon -- the queen of this type of symbolism-- the day her sister died:



11-Jan-1891 0h 13,33 Leo

12-Jan-1891 0h 11,48 Leo



Corrected for precession, the position on January 11th is 12 Leo. The last hours and agony of her beautiful young sister, whose death caused an impression throughout Bogot�, were marked by the Black Moon, Venus, and Pluto, a fusion that perfectly describes the "Nocturne" the poet wrote.



Jos� A. Silva shot himself in the heart in the night of 23 and 24th of May, 1896, when he was 30, overwhelmed by the economic bankruptcy of his family, a responsibility that he had assumed fully, and by the loss during a storm at sea of his last unedited writings. The shipwreck happened off the Venezuelan coast on January 28, 1895. If we take as reference that day at 18h GMT:



Mercury = 20,56 Aquarius

Venus = 22,19 Aquarius

True Black Moon = 21,28 Aquarius



We find a very good description of his literary treasure being "swallowed" by the sea, getting lost forever, from which the poet never recovered. The day of his death, a little more than 1 year later, we find:



Venus = 20,31 Taurus

Uranus = 21,35 Aquarius

Nessus death of Elvira = 21,52 Aquarius





NOTE: the complete material (written in July 2000) and the original poem in Spanish can be found at:



http://www.expreso.co.cr/centaurs/blackmoon/nocturno3.html





Nocturne III



Jos� Asunci�n Silva (1865-1896)

Translation by Luis Zalamea Borda





It was evening,

a night filled with perfumes, whispers, and the music of bird' wings;

A night

when fantastic glowworms flickered in the nuptial, humid shadows,

at my side, ever so slowly, close to me, listless and silent

as if prey to premonition of the most stinging pain

that inflamed the deep secret of your fibers,

over the path filled with flowers that stretched across the plain,

you were walking;

and the full moon

in the sky, so infinite, so unfathomable, spread its light.

And your shadow,

lean and languid,

and my shadow,

by the moon's rays silhouetted

on the path's sorrowful gravel,

were united

and were one,

but one long and lonely shadow,

but one long and lonely shadow,

but one long and lonely shadow...

Tonight,

desolate; my soul

by your death so bitterly pained and anguished,

torn from you by time, distance and the grave

upon that infinite blackness

where our voice cannot be heard,

lone and mute,

on the path I kept on walking...

And dogs braying at the moon came to my ears,

at the pale face of the moon,

and the croaking of the frogs.

I felt cold; the same chill that in your chamber

numbed your precious cheeks, hands and brow

amidst the snow-white linens

of the funereal shroud.

It was frost out of the tomb, it was the ice of the dead,

and the chillness of the void...

And my shadow,

sketched out by the paleness of the moon,

walked alone

walked alone,

walked alone upon the prairie;

and your shadow, lean and graceful,

pure and languid,

as in that warm spring evening long ago,

as in that night filled with perfumes, whispers and the music of bird' wings,

approached me and walked with mine,

approached me and walked with mine,

approached me and walked whit mine... Oh embraced shadows!

Oh the shadows of the bodies mingling with the shadows of the souls!

Oh shadows that search each other in tear-filled and somber nights!











VIII. Another Example



The case of Jos� Asunci�n Silva exemplifies the emotional, psychological, oniric, dark, and erotic world to which the osculating Black Moon belongs; these are the traditional associations, which have originated in French and British interpretations of that ancient mysterious pseudo-mythical character "Lilith" of Hebrew and Babylonian folklore.



Because the Black Moon is usually associated with Lilith, its traditional interpretations unfortunately tend to ignore the sociological and collective aspects of its symbolism. This social and community aspect can be understood if we learn to disentangle the character "Lilith" from the Black Moon, and consider how the "Great Mother" --as explained by the Jungian school-- describes practically all the traditional Black Moon associations. Knowing this, it is easier to understand the collective manifestation of its primitive symbolism: our "ancestral" relationships, and particularly, our relationship with "Mother Earth".



The case of the Mexican revolutionary Emiliano Zapata is an excellent example of this level of expression of the Black Moon, normally neglected by astrologers "possessed" by the demonic and magical character Lilith. The sketchy material that follows is based on a large compilation in Spanish you can find in my site:



http://www.expreso.co.cr/centaurs/blackmoon/zapata.html



The war of Emiliano Zapata was a war of agrarian reivindication, with roots in ancient "mother earth" archetypes. In a matter of months, having been called by the leaders of his village because they needed "someone who could put his pants on" to fight the unscrupulous usurpation by the great land owners of the land they needed to survive, the young man of 31 had become "General Zapata", the living symbol of a religious or mystical utopia whom everybody followed with fervor.



He was born in Anenecuilco (18n46/98w59 ), state of Morelos, the night of Aug 8 1879 (the time appears in a fictionalized account of his life... it may be an invention):



Sun = 16,03 Leo (calculated for sunset)

True Black Moon = 15,40 Taurus



His Sun in the middle of Leo making a square to the Apogee/Perigee axis is a good description of the social and historical phenomenon called Emiliano Zapata, the symbol-man, his very strong individuality absorbed in an impossible and perennial struggle for the reivindication of man's primordial relationship with the earth.



He was betrayed and assassinated on April 10 1919 between 2:10 and 2:15 in the afternoon (Chinameca 18n37/99w00):



Sun = 19,56 Aries

True Black Moon (geocentric) = 21,29 Libra

True Perigee (topocentric) = 20,57 Aries



NOTE: although the orb at death is 94' (applying) geocentrically, it is made more significant because they were in square at birth. The Sun conjunct the topocentric perigee has a smaller orb (61')



In my essay on Black Moon symbolism I explained that this point carries the instinctive energies, including atavistic wisdom and clairvoyance, which in this case are related to the feelings peasants have for the earth. The most brilliant explanation of this dimension of the personality and struggle of Zapata, particularly in the last period before he was murdered, is found in a book by Alfredo Krauze "Biografia del Poder: Emiliano Zapata" (Mexico, FCE, 1987) :



<<Zapata didn't fight for "the little lands" --as Villa used to say-- but for Mother Earth, and from Her. His struggle takes roots because his struggle is roots. This is why none of his alliances remains. Zapata doesn't want to go anywhere: he wants to remain. His purpose is not to open the doors of progress... but to close them: to reconstruct the mythical map of a human ecological system where each tree and each hill were there with a purpose; a world alien to any dynamism that is not the vital dialog with the earth.>>



Here, I believe, lies one of the most fundamental and most neglected aspects of Black Moon symbolism. The isolation and self-absorption (the "Unicorn" aspect) of the Black Moon can be seen here (continue quoting Enrique Krauze):



<<Zapata doesn't come out of his land because he doesn't know and fears "the other": the central power is always perceived as an intruder, as "a prying nest of traitors and the greedy". His vision is not active or voluntaristic, like that of all religiosities marked by the father, but passive and animistic, marked by the mother. His war of resistance exhausts itself. During the truce of 1915, instead of gaining strength outwards, he goes inward in a search of the lost order, to the point of wanting to rebuild it with the memory of the elders. It is not a productive map what he is after, it is the bosom of Mother Earth and its constellation of symbols.>>



These 2 paragraphs concentrate the meaning of the Black Moon better than most explanations I have seen, and bring to light the social aspect of it which is so consistently neglected.



The film "Viva Zapata" (1952), directed by Elia Kazan and written by John Steinbeck, focuses on Zapata's personality and "passion": the mysterious, quasi-religious phenomenon of the apparition of a leader in moments of crisis, who becomes the focus of powerful collective feelings that give to his life a mythical character.



Thanks to the combined vision and talent of Elia Kazan and John Steinbeck, the film is free from modern psychologism and portrays men integrated to the land and to their social and historical milieu: there is no difference here between man and history.



The end of the film shows the white stallion of Zapata fleeing unharmed from the ambush, and is seen running proudly and free. The completely intangible character of the Black Moon, unreachable, oniric, ancestral and solitary, fits well a symbol like this, i.e., the mythical stallion is Zapata himself. Zapata was called "the purest of all revolutionaries", so passionately he strove to remain loyal to his cause, to his dear Plan of Ayala, and to this end he never accepted compromises of any kind with anybody, something that ultimately was the cause of his destruction.



The fact that the story of the horse as it appears in the film is authentic (his name was "As de Oros" -- Ace of Diamonds), is a beautiful expression of the mystery that was the phenomenon of Zapata and of men like him: in spite of all their human deficiencies, an elemental historical or cosmic force seems to be driving them, and in the end there is no difference between the man and the myth, and their figure becomes archetypal even before they die, expanding into intemporality after death



Now consider the following astrological fact:



Elia Kazan was born in Istabul 7 Sept1909. At 12h GMT:



Sun = 14,12 Virgo

True Black Moon = 14,35 Virgo



John Steinbek, who wrote the script and was to receive the Nobel Prize for literature in 1962, was born 27 Feb 1902 in Salinas, California, at 3 PM PST (+8h):



Sun = 8,26 Pisces

True Black Moon = 6,48 Sagittarius



A square (orb=1.6) like the case of Zapata.



[NOTE: the symbol of the white horse is examined in detail in my original compilation with respect especially to Asbolus , which is exactly conjunct the Sun of Zapata when John Steinbeck was born]



For a further examination of the social role of the Black Moon, particularly its relationship to the figure of the Virgin Mary in the Catholic Church, see my latest study on the Second Vatican Council. For another case from the psychological point of view, see my study on Tchaikovsky.





IX. The natural or "interpolated" apogee



If one compares the actual position of the Moon when it is at its apogee every 27 days, with the position of the "mean" apogee or Black Moon, the difference is never more than 5 degrees (actually -5.4 to +5.7), and the maximum is reached every 206 days. This has suggested to some people that the "true" position of the apogee must therefore describe a very smooth curve with an amplitude of 5 degrees only, in contrast to the very large 30 degree curve of the osculating apogee, which they describe as "unrealistic".



The Swiss Ephemeris documentation mentions a proposal made by Henry Gouchon in "Dictionnaire Astrologique, Paris 1992", based on a curve with an amplitude of 5 degrees. This solution is said to be "the most realistic of all so far". It is also explained that the actual curve of the deviation between this Moon position at every apogee and the position of the mean apogee is not exactly a sine, and that Dieter Koch published a table in "Meridian" in 1995 <<that pays regard to the fact that the motion does not precisely have the shape of a sine>>.



In the long (and old) compilation of posts I wrote on the calculation of the Black Moon in my site, you will find those quotes and also a numerical formula I devised that demonstrates this (written 25 Nov 1999):



[begin quote]

I calculated the times of all Lunar apogees from 2000 to 2010, a total of 136 apogees. I then calculate the positions of the Moon and of the mean apogee/Black Moon at those times, and the difference between the two. The difference oscillates between -5.4 to +5.7 degrees. When this difference is plotted on a graph, one sees a clear cycle with a period of 206 days. This is the period of the difference between the longitude of the Sun and the longitude of the mean apogee, and it shows that the main deviation in the longitude of the apogee is caused directly by the Sun.



Let's call the Sun/Apogee difference "A"



A = 197.1132 +31931.756*T +0.0106*T^2 (degrees)



where T is centuries from J2000

T = (Julian day-2451545)/36525



One can reduce the difference mentioned above (-5.4 to +5.7) to 1/4 or 1/5 of it, by adding the following correction to the mean apogee:



-4.7 degrees * sine of (2 * A)



With this correction the errors will be less than 1 degree, and the maximum will be 2 degrees or less.

[end quote]



If the difference were a perfect sine, the above formula would give the exact deviation of the position of the Moon at apogee and the position of the mean apogee. The remaining 1/4 or 1/5 means that the sine curve with a period of 206 days described will approximate the deviation with an error of 20 or 25%.



The main idea of this approach is to observe the position of the Moon when it is at apogee every "anomalistic" month (27 days). Its proponents use the positions of the Moon when it is at apogee as "the true apogee", and to find where this "true apogee" is at other moments when the Moon is at any other point of its orbit away from apogee, they use a numerical interpolation formula.



Because this way of understanding the lunar apogee is based on the actual occurrences of lunar apogees in the natural world, I think the word "natural" describes it well. This is why this variant of the Apogee or Black Moon is called "natural" in Riyal. It is the most recent version of how to calculate the Black Moon.



When the apogee and the perigee are calculated this way, they are no longer an axis, and the difference between the perigee and its mean position can reach 25 degrees instead of only 5 degrees as in the case of the apogee.



This approach was described by Miguel Garc�a in 1997 in an article published in Spain (" Realidad y ficci�n astron�mica de Lilith, Cuadernos de Investigaci�n astrol�gica Mercurio-3, n� 6 ), who implemented its calculation in his software "Armon" (1997). It is also the approach of Dieter Koch, who together with Bernhard Rindgen published "Lilith und Priapus, die Schalen des Menschen" (Frankfurt 2000) , with ephemerides of Lilith (the apogee) and Priapus (the perigee) from 1900 to 2010. Dieter's work was awarded as <<the best astrological research result in 2000>> by the Internationalen Astrologie-Weltkongress 2000 in Luzern.*



* information by Robert von Heeren.



Both the natural and the osculating apogee coincide at the time when the Moon reaches its apogee. One can say, therefore, that both are "true" only at that time, while at any other time they are an approximation. But there is an important difference: the osculating apogee is calculated rigorously from the geometry of an ellipse, while the natural apogee dismisses completely the idea of an ellipse (and of geometry), something evident in the fact that the apogee and the perigee do not form an axis.



Paradoxically, even though the proponents of the natural apogee reject the osculating value because of its very large divergence from the mean, the natural perigee can be almost as far away from the mean as the osculating value.



X. Riyal's output (b)



Instead of working with geometric projections and orbital planes as in everything else in Astrology, the natural apogee is calculated through a series of observations or "actual happenings" to which is applied a numerical approximation to find the intermediate value. Because it is only a numerical approximation, its position is not included in the Swiss Ephemeris, and we must use other sources in order to calculate it and compare the positions given by Riyal.



Riyal gives the position of the natural apogee and perigee in "Tables --> Astronomical Data", and it is possible to construct an ephemerides of them. The ephemerides (" Special-->Generate Ephemerides-->Apsides and Nodes-->Moon ") will show their positions together with the other variants. Here is a sample output:



Oscu MBari MGeo MPeri MFoco nApo nPeri

1 Jan 2003| 17Ta27 | 25Ar30 | 26Ar00 | 24Li57 | 29Ar53 | 23Ar12 | 16Sc07 |

2 Jan 2003| 16Ta47 | 25Ar37 | 26Ar12 | 24Li58 | 0Ta52 | 23Ar25 | 15Sc01 |

3 Jan 2003| 15Ta26 | 25Ar44 | 26Ar22 | 25Li01 | 1Ta36 | 23Ar39 | 13Sc51 |

4 Jan 2003| 13Ta11 | 25Ar50 | 26Ar30 | 25Li06 | 2Ta03 | 23Ar53 | 12Sc38 |

5 Jan 2003| 10Ta07 | 25Ar57 | 26Ar35 | 25Li14 | 2Ta10 | 24Ar07 | 11Sc22 |

6 Jan 2003| 6Ta39 | 26Ar04 | 26Ar39 | 25Li24 | 1Ta58 | 24Ar21 | 10Sc04 |

7 Jan 2003| 3Ta17 | 26Ar10 | 26Ar41 | 25Li36 | 1Ta25 | 24Ar35 | 8Sc44 |

8 Jan 2003| 0Ta29 | 26Ar17 | 26Ar42 | 25Li50 | 0Ta35 | 24Ar49 | 7Sc21 |

9 Jan 2003| 28Ar21 | 26Ar24 | 26Ar41 | 26Li04 | 29Ar29 | 25Ar03 | 5Sc58 |

10 Jan 2003| 26Ar49 | 26Ar30 | 26Ar40 | 26Li20 | 28Ar12 | 25Ar17 | 4Sc32 |



The quantities are:

Oscu = geocentric osculating or "true" apogee

MBari = mean barycentric apogee

MGeo = mean geocentric apogee

MPeri = mean geocentric perigee

MFoco = mean geocentric kenofocus or empty focus

nApo = natural apogee

nPeri = natural perigee



To compare Riyal's accuracy (remember, in this case the positions can only be approximate by definition, especially in the case of the perigee), we will use sample outputs from the program "Armon 1.0" (1997) by Miguel Garc�a and "Ceres 1.17" (2001), by Dieter Koch. The sample is calculated for the 1st day of each month at 0h U.T.:



Table 1.- natural or "interpolated" apogee:



Armon Riyal Ceres

1 Jan 2000| 19Sa47 | 19Sa11 | 19Sa11 |

1 Feb 2000| 22Sa23 | 22Sa12 | 22Sa13 |

1 Mar 2000| 27Sa50 | 28Sa17 | 28Sa17 |

1 Apr 2000| 4Cp24 | 5Cp06 | 5Cp05 |

1 May 2000| 10Cp19 | 10Cp49 | 10Cp48 |

1 Jun 2000| 14Cp57 | 14Cp37 | 14Cp38 |

1 Jul 2000| 14Cp34 | 14Cp13 | 14Cp23 |

1 Aug 2000| 12Cp49 | 13Cp05 | 13Cp02 |

1 Sep 2000| 16Cp22 | 15Cp57 | 15Cp57 |

1 Oct 2000| 22Cp02 | 21Cp18 | 21Cp18 |

1 Nov 2000| 28Cp38 | 27Cp58 | 27Cp58 |

1 Dec 2000| 4Aq31 | 4Aq32 | 4Aq32 |



As you can see, the positions of Riyal are almost the same as those of Ceres. The most probable reason is that we are using the same algorithm. Originally, it was Dieter Koch who gave me the suggestion about how to calculate it when I implemented it in Riyal in November of 1999. The perigee shows larger discrepancies...



Table 2.- natural or "interpolated" perigee:



Armon Riyal Ceres

1 Jan 2000| 1Ca23 | 1Ca35 | 1Ca37 |

1 Feb 2000| 16Ca38 | 17Ca44 | 17Ca23 |

1 Mar 2000| 22Ca41 | 24Ca02 | 24Ca05 |

1 Apr 2000| 17Ge54 | 23Ge53 | 19Ge45 |

1 May 2000| 14Ge01 | 13Ge50 | 14Ge53 |

1 Jun 2000| 26Ge14 | 26Ge52 | 26Ge49 |

1 Jul 2000| 11Ca37 | 11Ca39 | 11Ca38 |

1 Aug 2000| 27Ca35 | 27Ca14 | 27Ca16 |

1 Sep 2000| 11Le31 | 11Le26 | 10Le40 |

1 Oct 2000| 15Le17 | 10Le39 | 14Le00 |

1 Nov 2000| 8Ca13 | 10Ca43 | 6Ca39 |

1 Dec 2000| 8Ca47 | 6Ca41 | 7Ca51 |



In this case, the discrepancies are larger between the 3 programs, probably because different algorithms are being used. (Details of Riyal's algorithm are given in the program's documentation.). Since there is no way of obtaining high accuracy in this case, it is very difficult to know which positions are more accurate or "correct".



The proponents of the natural apogee and perigee consistently disqualify the use of the osculating ellipse; however, this "interpolated" approach to the lunar apogee or Black Moon, based on past and future coordinate points instead of instantaneous positions and geometrical projections, represents a mixture of temporal planes and contradicts how all other radical astronomical points in an astrological chart are calculated.





XI. Constructing Tabular Black Moon Ephemerides



Riyal normally produces 2 different Black Moon ephemerides. Details of the calculations and of their accuracy are found in the program's documentation. The tables can be made for any period of time within the program's range (-4700 to +9000) and for any amount of days or fractions of a day as the tabular interval, and if needed, the output can be sent to a file to produce an independent tabular ephemeris in text form.



The first type has already been illustrated, and is the "apsides and node" ephemeris option, that shows all the different versions of the Black Moon together in columns (see sample in section X above). The second type of Black Moon ephemerides is constructed when one chooses an ephemeris of the Apogee directly through the "one body only" option. This will tabulate either the osculating or the mean apogee/Black Moon, depending on how the user has configured the program, and looks like this:



APOGEE lon lat dec distance velocity



1 Jan 2003| 17Ta26.6 1s53 15n14 403541 km -0�29'

2 Jan 2003| 16Ta47.2 1s56 15n00 405737 km -0�59'

3 Jan 2003| 15Ta26.2 2s02 14n31 406999 km -1�48'

4 Jan 2003| 13Ta11.0 2s13 13n41 407345 km -2�41'

5 Jan 2003| 10Ta07.1 2s27 12n31 407018 km -3�19'

6 Jan 2003| 6Ta38.5 2s43 11n10 406353 km -3�28'





When tabular ephemerides are not needed, but simply the positions for one single instant of time or one astrological chart, Riyal will detail all the variations in the routine "Astronomical Data", or by pressing "F2". Here is a sample screen (actual size 800x600) of the output in this case:







When tabular ephemerides are not needed, but simply the positions for one single instant of time or one astrological chart, Riyal will detail all the variations in the routine "Astronomical Data", or by pressing "F2". Here is a sample screen (actual size 800x600) of the output in this case:

Riyal is FREEWARE. It can be downloaded here.



Juan Antonio Revilla

San Jos�, Costa Rica

October 2003



