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 socalled "interpolated" apogees, which Riyal calls
"natural apogee" (5) and "natural
perigee" (6). The "Mean" apogee (7) by definition excludes shortperiod
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 Rougié, 18841953) 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 shortperiod fluctuations that
have been already averagedout 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 shortterm 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
realworld shortterm 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 Priapuslike or as Axel Harvey
calls it Charybdislike 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 7feb2005, 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
EarthMoon 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
ChaprontTouzé, astronomers at the
Bureau des Longitudes and developers of the most modern and accurate
lunar theory to this date, called "
ELP2000" (Riyal uses a truncated longterm version of it called "ELP200085",
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 worldwide 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 abovementioned book in 1991 and based on ELP200085, 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 1213 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
presentday 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 socalled "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 nonexistent. 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, realworld 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 EarthMoon
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:
(20022003)
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:
(20022003)
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
112003 to 112004.:
You can see clearly in the graphic the main
15degree 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
mid2003. 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
José Asunción Silva was born November 27th, 1865,
in Bogotá. 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:
11Jan1891 0h 13,33 Leo
12Jan1891 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 Bogotá, were marked by the Black Moon, Venus, and
Pluto,
a fusion that perfectly describes the "Nocturne" the poet wrote.
José 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:
Nocturne III
José Asunción Silva (18651896)
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 snowwhite 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 tearfilled and somber nights!
VIII. Another Example
The case of José Asunción 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 pseudomythical
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 symbolman, 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 selfabsorption (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,
quasireligious 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 day2451545)/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 García in 1997 in an article published
in Spain ("Realidad y ficción
astronómica de Lilith, Cuadernos de Investigación astrológica
Mercurio3, 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
AstrologieWeltkongress 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 García 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:
Riyal is FREEWARE. It can be downloaded
here.
Juan Antonio Revilla
San José, Costa Rica
October 2003
