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P. Svarogich

(English editing by Ed Falis)

Principia of polyzodiacal astrology

2. The zodiac of massive heavenly body
2.1 Introduction.
2.2 Theoretical model of a zodiac for a born, situated in the vicinity of a heavenly body.
2.3 Choice of the reference point on the local equator (zodiac circle).
2.4 Examples of zodiacs.

2.4.1 Terrestrial zodiac.
2.4.2 The Lunar zodiac.

2.4.2.1 The Lunar nodes.

2.4.3 Solar and planetary zodiacs.

2.1 Introduction.

Modern development of the quantum theory of gravitation already allows us to construct the phenomenological model of a zodiac of a massive heavenly body in concordance with the notions of modern non-classical physics. We have in view the Unruh effect — the excitation of a detector moving with acceleration relative to a vacuum. It is possible to distinguish the acceleration caused by gravitational attraction from that connected with the non-inertiality of the reference system. Here we apply only the fundamental possibility of distinguishing the force of gravitational attraction from inertial force. So, there are no grounds to consider that the basis of astrological rules is gravitational interaction. Furthermore, recent developments in quantum field theory and in theories of "Grand Unification" leave little hope of discovering a new interaction that accounts for astrological rules.

The basis of our conception is a zodiac model for a born that is in the vicinity of a massive body. For an observer on the Earth’s surface three bodies predominate — the Sun, Earth and Moon — and correspondingly, their three zodiacs. We cannot however neglect planetary influences18). A zodiac is a dynamic construct in that it depends on the motion of a given born relative to a designated heavenly massive body. By born we mean a wholeness or person for which we study events.

The motion of an observer stationary on the Earth’s surface relative to the Earth’s centre, or to the Sun, is the simplest case due to its closeness (with a certain accuracy) to circular motion. The deflections of the angular parameters of the solar and terrestrial zodiacs due to the deviation of the observer’s motion from ideal circle reach only several angular minutes. It is exactly the relative insignificance of these deviations that has given a static character to their geometries and thus allowed ancient astronomers to formulate the concept of a zodiac using only elementary geometric constructions. In traditional astrological language the solar zodiac is the ordinary Zodiac, and the projection of the terrestrial zodiac onto the solar is a house system.

The motion of a born relative to the Moon is much less circular. The taking into account the deviation of this motion from circular changes the angular parameters of the lunar zodiac by degrees. It is likely that this characteristic is exactly what prevented a comprehensive quantitative description of this zodiac in antiquity, though some steps were taken in this direction. Examples are the lunar zodiac of Chinese astrologers and the draconic astrology of the West. The difference of these models from the exact construction is so great that it is impossible to produce quantitative calculations. The lunar zodiac is a dynamic construction, in which the velocities of planets on the zodiacal circle experience significant variations not through changes of their observable velocities in the sky, but due to the complex nature of the motion of an observer around the Moon.

In the model we elaborate an event as a qualitative change of a born occurs when certain group correlations (in the language of the group theory, or aspects in astrological language) arise among the elements of a zodiac. The concept of aspect - one of the central and very old notions of astrology - can be most precisely described by the language of algebraic group theory. The elements of a zodiac are completely determined by the motion of massive heavenly bodies and a born. Since the motion of heavenly bodies can be calculated with good accuracy for many years into the future, and that of a born can to a large extent be considered as fixed19), there results the possibility of forecasting events.

2.2 Theoretical model of a zodiac for a born, situated in the vicinity of a heavenly body.

We posit that a zodiac of a born is a way to describe the spatial anisotropy at the born’s location, which is generated by a massive body, relative to which the born moves20). Such an approach to describing spatial non-homogeneity is a construct exactly suitable for forecasting events. If a born is surrounded by several massive bodies, then it is possible to construct a corresponding number of zodiacs. When constructing the zodiac of one of these bodies, other massive bodies become its elements or planets (in ancient Greek sense of the word planhV ). The most interesting zodiacs are those of the closest and heaviest bodies. These zodiacs are the most stable due to the born’s dynamics governed by the gravitational fields of these bodies. We call such a massive body that generates a given zodiac its central body; we call the remaining bodies simply bodies or planets.

A zodiac as a dynamic structure is defined in general by three directions: 1) the vector of force acting on a born from the central body as a whole, 2) the velocity vector of the central body in the reference system of the born21) and 3) the axis of space anisotropy generated by the central body in the location of the born. The first two vectors determine the plane of a local equator (Fig. 1). When a born moves with acceleration relative to the central body, the vector of force of the central body and the axis of space anisotropy generated by it do not coincide. The angle between the plane of the local equator and the axis of anisotropy is called the dynamic angle. If the size of the central body is far less than the distance to it, then the direction to it in space can be used as its axis of anisotropy. That is, we use its position in space without correction for planetary and stellar aberration.

Thus, when studying the zodiac of a distant massive body, the angle between the central body’s vector of force and its axis of space anisotropy is the angle of the correction for aberration. This is due to the fact that such a distant body acts on a person for the most part only by gravitation. For a person on the Earth’s surface the correction for aberration for planets of the solar system is some dozens of angular seconds22). However, for the terrestrial zodiac the dynamic angle is much larger, being approximately equal to the latitude of the point on the Earth’s surface23).

Fig.1 Geometry of the local zodiac of a born in the vicinity of a massive body.
e — unit vector of anisotropy generated by the central body at the point of the born,
A - unit east point vector (fundamental reference point) of the local zodiac of the born,
P — polar vector — the direction to the north pole of the local equatorial system of coordinates,
F — force acting on the born from the central body as a whole,
V — velocity of the central body in the proper non-rotating reference system of the born,
E — east point (fundamental reference point),
W — west point,
S — south point,
N — north point,
C — point of conjunction,
T — point of opposition,
M0 — projection of the central body on the celestial sphere,
jd — dynamic angle.

Let us define the local equatorial system of astronomical coordinates of a born with respect to a given central body. The equatorial plane of this coordinate system has been introduced above in the definition of the dynamic angle — it is the plane of a local equator, whereon lie two vectors: the force vector F acting on the born on the part of the central body, and its velocity vector V in the proper non-rotating reference system without correction for aberration. The polar angle n (also called the latitude) is measured from the equatorial plane in the positive direction toward the North Pole, and in the negative direction to the south. The vector product P= F V defines a vector directed to the north pole of this system of astronomical coordinates.

The azimuth angle m in the equatorial plane (also called the longitude) is measured in the positive direction (counter-clockwise if one looks from the north pole of the coordinate system) from the fundamental reference point on the local equator. It is fitting to label the fundamental reference point of the coordinate system as the east point of the central body. The vector A towards the east point of the central body is given by vector product A= e P, where e is the vector of space anisotropy of the central body acting on the born, and P is the previously-defined polar vector of the coordinate system.

For a central body whose size is many times smaller than the distance to it, it is appropriate to designate its space anisotropy vector as a unit vector e tangent to a purely spatial geodesic, connecting its centre of mass and the born. The vector of anisotropy generated by the Earth is practically coincides with the vector of gravitational attraction at a given point on its surface.

It is necessary to distinguish the notions of local equator of a central body and its local equatorial coordinate system as introduced in this paper etc, from such traditional astronomical notions as the celestial equator, first and second system of equatorial coordinates etc. For the definitions of the traditional astronomical concepts see [16].

At this point, the concept of a zodiac is almost elaborated. Let us designate massive bodies other than the central one as zodiac elements. Then a zodiac is a collection of longitudes of massive bodies surrounding the born expressed in zodiacal coordinates. The zodiacal coordinate system in which we describe the “position” of the central body and other massive bodies is a curvilinear coordinate system distinct from the system of spherical coordinates on the local equator. It is not a coordinate system in the usual sense, as some points on the sphere are simultaneously associated with three values of zodiacal longitude24). The zodiacal longitudinal coordinates converge to the spherical longitudinal coordinates of the local equatorial coordinate system as the dynamic angle approaches zero.

For a complete introduction of zodiacal longitude it is necessary to consider several mathematical notions. Zodiacal coordinates can be considered a generalization of spherical coordinates. Let us denote the zodiacal longitude by t. We may imagine it as a circumference of longitudes, called zodiacal circle in astrology. Let us move to the definition of the zodiacal longitudinal coordinate on the celestial sphere. For this it is necessary to construct a mapping of the sphere onto the circumference, and to introduce a distance function on the circumference as well as a reference point. Let us call this procedure of assigning a zodiacal longitudinal coordinate to a point on the celestial sphere a parameterisation, since it can be described by the movement of a great semicircle25) along the surface of the sphere.

To determine the zodiacal coordinate we restrict ourselves to a very narrow class of motions. We consider the celestial sphere as a sphere of unit radius with its centre in the location of the born. We identify the great circumference with distance function and reference point with the intersection of the celestial sphere and the local equatorial plane. The reason for this choice is that any point having zero spherical latitude has identical zodiacal and spherical longitudes.

We define a plane called the local horizon of anisotropy, or the local horizontal plane, as the plane perpendicular to the axis of anisotropy generated by the central body on the point of born. We define the local meridian plane as the plane perpendicular to the local horizontal and equatorial planes. The crossing points of the local horizontal and meridian great circles we call the north point and the south point. The north point is situated on the celestial sphere closer to the north pole of the local equatorial coordinate system of born.

The fundamental reference point on the local equator (zodiacal circle), or east point of the central body, is defined as that crossing of the local equator and the local horizontal plane the direction to which forms an obtuse central angle to the velocity vector of the central body in the non-rotating proper reference system of born. The opposite point it is called the west point. The crossing point of the line common to local meridian and equatorial planes with the celestial sphere, the direction to which from the centre of sphere forms an acute angle with the direction along the axis of anisotropy from the born to the central body, is called the point of conjunction, and its opposite point is called the point of opposition26).

At the moment t = + 0 of virtual time27) the moving great semicircle (the parameterizing semicircle) lies in the horizontal plane and passes through the east point. The straight line passing through the edges of this great semicircle is rotated by the angle –b (i.e. by the angle b clockwise) from the straight line connecting the north and south points as viewed from the point of opposition, which we consider by definition to be above the plane of the horizon. The angle b is given by equation28)
(1)
where j d – the dynamic angle, the analogue of latitude for the terrestrial zodiac – can differ from the geographic latitude by several angular minutes. The parameterizing semicircle, moving in a specified manner, returns to its initial position after 360 units of virtual time t. Zodiacal longitude, like sidereal time, can be measured in hours or in degrees as is done here. Each point on the celestial sphere is marked by the moment at which the parameterizing semicircle passes through it. We consider each such marking moment as the point’s zodiacal longitudinal coordinate.

The description of the zodiacal longitudinal parameterisation is not yet complete. We describe the motion of the great semicircle by the angle q(t ) between the polar axis connecting the north and south poles of the local equatorial coordinate system of a born and the parameterizing half-plane whose intersection with the celestial sphere yields the parameterizing semicircle. But this still does not give the exact orientation of the parameterizing semicircle. According to the second condition to complete the description given earlier, the crossing point of the parameterizing semicircle and the equator must have uniform motion along the equator29) (Fig. 2).

Therefore, when at the zodiacal positions 0 , 90 , 180 and 270 the parameterizing semicircle must correspondingly pass through the east point, the point of conjunction, the west point and the point of opposition. At the same time, the straight line that supports the parameterizing semicircle in the horizontal plane, and which is the instantaneous axis of rotation, moves in the positive direction (counter-clockwise as viewed from the north pole of the equatorial coordinate system) from the angle –b (t = + 0), through the north-south axis, and up to the angle + b as the parameterizing semicircle approaches the west point (t = 180– 0).

Moving further the supporting straight line discontinuously changes its position, again forming the angle – b with the north-south axis (t  = 180+ 0), and then moves again in the positive direction up to the angle + b(t = 360– 0). This motion ensures that the parameterizing half-planes in positions t and 180+t form a single plane. At the moment t = + 0 the angle between the parameterizing half-planes to the polar axis is equal to jd. For an arbitrary moment t, the angle q (t) is defined by the following expression, for
– 180<  t  <  + 180):

(2)
The sign of the angle q in the formula is conventional, and the resulting tangent with the specified sign is substituted into formula (3).

It is interesting to note that the plane with which the parameterizing semicircle coincides in the position t = + 90 passes through the south and north points and is identical to the plane passing through the points with spherical longitude m = + 90 in the local equatorial coordinate system of the born. So, when converting from spherical coordinates to zodiacal coordinates the central body, which always has a spherical longitude m = + 90 with respect to the fundamental reference point, does not change its longitude. The same is true for the majority of points with spherical longitude m = + 90. Only those points with spherical longitude m = + 270 and a latitude n lying in the range + 90– j d <  n  <  + 90 in the local equatorial coordinate system of the born will have a zodiacal longitude t = + 90. Similarly, points with spherical longitude m = + 90 and latitude n in the range – 90<  n <  – 90+ jd in the local equatorial coordinate system will have a zodiacal longitude t = + 270.

Fig. 2. Illustration of the parameterizing motion of the half-plane from the cusp of the 7th station to the end of the 12th station (the range of zodiacal longitude 180+0 to 360–0). The notation is that of Fig. 1. The angle b is given by (1). The great semicircles limited by the supporting straight lines NiSi, lying in the plane of the local horizon of anisotropy, represent the two-dimensional cusps of the stations of the local zodiac of a massive body. The great semicircle NS lies in the plane of the local meridian and represents the cusp of the 10th station. The straight lines NiSi are the axes of instantaneous rotation of the parameterizing semicircle in positions 30 (i–1) of zodiacal longitude. On the local equator ECWT, the spherical longitude counted from the east point E in usual angular measure is equal to the zodiacal longitude. Its (and the zodiacal longitude’s) values for the cusps of the stations are marked on the drawing at the crossing points of the two-dimensional cusps of the stations with the great circle of the local equator.

It is quite possible that unlike the other massive bodies there is no need to mark the central body on the zodiacal circle. Its presence is always implicitly indicated by the fundamental reference point, the point of conjunction, or possibly by all 4 cardinal stations of the east point: the east point itself, the point of conjunction, the west point and the point of opposition.

Unfortunately it is not clear how to define the latitudinal coordinate of the zodiacal coordinate system. We hope it is possible to derive the formula theoretically from the conditions of complex analyticity of the mapping of the pair of angular variables, longitude and latitude, ( m , n ) of the local equatorial coordinate system of a born onto the pair of angular variables ( ts ) of the corresponding zodiacal coordinate system, as its inference from experimental data could require a great deal of time.

To conclude this section, let us write down the formula for converting the longitudinal coordinate of a point on the sphere from its spherical equatorial coordinates into zodiacal coordinates. The zodiacal and spherical longitudes of a point are different whenever the point has non-zero spherical equatorial latitude. The zodiacal longitude of a point (given the condition that the reference point is set to the corresponding east point or fundamental reference point) is obtained with domain and range (-180<  t

 <  + 180 and – 180<  m <  + 180) by:
(3)
where ( m , n ) are the spherical longitude and latitude of the point in the local equatorial system; t is the longitude of planet in the zodiacal system; tg qis the tangent of the angle between the parameterizing half-plane in position tand the polar axis of the local equatorial coordinate system of a born with the sign conventionally defined by formula (2).

Before solving this equation, it is necessary to test an additional condition to define whether the point is above or below the horizon (the positions "below the horizon" include the central body).. (4)
The upper sign of inequality is used for a position below the horizon, 0<  t <  + 180; the lower for a position above the horizon, – 180<  t <  0.

2.3 Choice of the reference point on the local equator (zodiacal circle).

A zodiacal circle or zodiac in astrology is a circumference of zodiacal longitudes with a designated reference point and sensitive points located on it. The sensitive points can be the points with the longitudes of massive bodies as well as some other points. Once the identification of a circumference of zodiacal longitudes with the local equator of a born has been made, the points of massive bodies on the zodiacal circle can be considered as the projections of these bodies onto this equator in the zodiacal coordinate system.

Once we have an equatorial circle with sensitive points on it, we can choose its reference point. We have already seen one variant: the fundamental reference point (east point) of the central body. The experience of Uranian astrology, founded by Alfred Witte, shows that the intended facet of interpretation determines the choice of reference point. In Uranian astrology six systems of equal houses are used. In the language of the present work this means that the reference point is chosen from six points in addition to the fundamental reference point. The principle becomes clear in the context of the technique of discrete symmetries on a zodiacal circle, which was used actively by Alfred Witte and his adherents. The forgotten originator of this technique was Johannes Kepler.

If a sensitive point is sufficiently strong, astrological experience shows that it reveals itself through a number of additional points, symmetrically located in the corners of a regular N-agon inscribed in the zodiacal circle30). Let us call these points the Nth order family of the point, and designate the point as the first point of the family. The points of the Nth order family of the fundamental reference point (located 90 clockwise from the central body) are the cusps of the stations of the central body of the Nth order. For N=12, we call them simply the cusps of the stations31).

By the term station we understand an interval on the zodiacal circle from a station cusp to the cusp of the next station, in accordance with astrological tradition. When we talk about zodiacal sign cusps of the Nth order we mean a family of a reference point that does not coincide with the fundamental reference point.

As the number N increases the subsidiary points of a family become weaker. Uranian astrology demonstrates that the points of the 4th order family work for any point representing a massive body. The points chosen in astrology as reference points are so strong that the subsidiary points of their 12th order families are also noticeable. The Sabian symbols for degrees demonstrate that very strong points can generate a family of effective points even at the 360th order [20]. This also testifies in favour of the special significance of certain positive integers in the zodiac.

In astrology it is not only aspects of sensitive points to the station or house cusps that are important; the areas from one cusp (station, sign or house) to another are meaningful as well. In traditional astrology these areas are called houses when projecting the terrestrial zodiac onto the solar one (the sections between 2 subsequent house cusps32)), or signs of the Zodiac for the solar zodiac when the reference point is taken as the crossing of the celestial equator and the ecliptic. These sign sectors are considered to carry certain qualities that modify the expression of a sensitive point found within them.

We can note the dialectical character of this scheme, where symbolic motion in the positive direction along the zodiac circle describes a cycle of evolution resulting in an accumulation of characteristics that manifest as qualities when entering a new area. Traditional astrology makes the reference point absolute; nevertheless, its meaning is highly relative. Each time we choose a reference point we obtain a new zodiacal circle. The zodiacal circle is actually an ensemble of evolutionary cycles, an ensemble of ‘vernal equinoctial points’ and, in accordance with symmetry order, an ensemble of sets of phases of development.

By the notions of station, house and sign we imply one- and two-dimensional areas. By station, house or sign we mean the area between the border of a considered station (house, sign), called its cusp, and the cusp of the following station (house, sign). If this makes sense.

As a one-dimensional station (house, sign) we mean an arc on the zodiacal circle identified with the equator of a local zodiac. By definition, a station cusp is a point on a zodiacal circle whose longitude is equal to the part of the full circle , where n is the number of cusp, and N is the order of a family of the fundamental reference point.

We understand a two-dimensional station (house, sign) as an ensemble of points on the celestial sphere having the same longitudes on the considered zodiacal circle as the corresponding points of one-dimensional station, house or sign. The cusp of such a station (house, sign) is the great semicircle of points on the celestial sphere with zodiacal longitude equal to .

Astrology commonly uses both a fundamental reference point (the terrestrial zodiac), and a reference point fixed in the crossing point of a local equator with the local equator of the zodiac of another central body (the solar zodiac). The choice of a reference point in the crossing point of two equators testifies that this is a strongly manifesting point. The issue of choosing the reference point from the two crossing points is not clear. We can only state a preference for either point from natural science considerations. This leads us to the CPT-theorem of quantum field theory33) as a possible principle for resolution, though the specific choice remains unclear. The East point, or fundamental reference point, is a strong sensitive point in its own right, since it is connected with the singularity that occurs when crossing the horizon – the leap in the motion of the parameterizing semicircle34) [21].

2.4 Examples of zodiacs.

2.4.1 Terrestrial zodiac.

What was written above about the structure and geometry of the zodiac of a massive body is very close to the structure and geometry of the terrestrial zodiac. The characteristic motion of a typical born — a person nearly always motionless with respect to the Earth’s surface (with the exclusion of a cosmonaut) — closely approaches an ideal rotation with respect to the terrestrial polar axis. That is, it is nearly a circular motion with a constant velocity. Therefore, although the geometry of the terrestrial zodiac is complicated due to its considerable dynamic angle, it is static. The longitudinal coordinates of the celestial sphere points, which are motionless with respect to an observer on the Earth surface, do not change with time35). This allowed a correct description of the geometry of the terrestrial zodiac, with sufficient accuracy, as early as the 17th century as the Placidus system of houses and the "mondial" (mundo) positions of planets in the houses [19] (by Ptolemy). It is easy to see that the instantaneous equator is a local equator of a born for the zodiac with the Earth as its central body. The plane of the dynamic angle is perpendicular to the plane of the equator, with high precision. This is a unique zodiac in that it has both a known geometry and a non-zero dynamic angle36).

For pinpoint accuracy calculations it is necessary to take into account three factors:

1. It is necessary to use the gravitational horizon rather than the usual gravity horizon (Fig. 3). Because the born is so close to the Earth the correction for aberration is many times less than for the Moon37). So for the vector of anisotropy it is possible to take the direction of the gravitational attraction by Earth. The usual horizon, defining the geographical or, taking into account the plumb deviation, the astronomical coordinates of a point on the Earth’s surface, is a plane perpendicular to the gravity force. This force is the sum of the gravitational attraction and of the inertial force generated by the rotation of a reference system motionless with respect to the Earth’s surface. To achieve a precision of several angular seconds it is sufficient to take into account the following correction to the geographic latitude:

, (5)
We add this correction to the geographical latitude of a point. Here j is the geographical latitude of place; w is the angular velocity of the Earth’s rotation expressed in radians per second; R is the radius of the Earth; g is free fall acceleration. The correction to the latitude calculated by this formula is expressed in radians. The use of the dynamic angle instead of the usual geographical latitude in the terrestrial zodiac will cause, for instance, a difference in the house cusp longitudes with respect to the usual determination of the terrestrial zodiac understood as a house system38).

Fig. 3. A section of the globe along the polar axis OP.

O —Earth’s centre. OQ — the plane of the terrestrial equator. N — the point of a born. R — the geocentric radius vector of the point N. jg — the geocentric latitude. j— the usual astronomical latitude defined as the angle between the plumb and the plane of equator. The direction of the plumb is the direction of the gravity force mg acting on the born at the point N. It is the sum of the gravitational attraction force e, which in this case practically complies with the axis of anisotropy, and of the inertial force i, whose value is defined by formula i =w2Rcosj . The surface of the terrestrial ellipsoid is perpendicular (with a certain accuracy) just to the plumb. The dynamic angle, or the gravitational latitude jd, can be obtained with a precision of several angular seconds by adding the correction given by formula (5) to the astronomical latitude j.

2. To achieve the maximum accuracy it is necessary to take into account the motion of the polar axis within the Earth’s body. The value of the latitude correction for this factor is about 1".

3. It is also necessary to take into account the plumb deviation as a correction to the latitude. The typical correction for flat country is about 1-2"; in the mountains it can reach several minutes of arc. However on the Earth’s surface, there are several regions of "anomalous gravitation" in geodesic terminology, where flat land has a considerable plumb deviation, reaching 10" and more. One such region is the area of Moscow and its suburbs. Indeed, one of the largest known leaps in plumb deviation is in the Kremlin, adjacent to the belfry of Ivan the Great39).

No additional peculiarities in the geometry of the terrestrial zodiac appear when passing into the circumpolar regions. However, areas exist on the celestial sphere above the north point and beneath the south point where points have three zodiacal longitudes. This occurs even in equatorial regions of the globe. In these areas equation (3) has 3 solutions for a single spherical coordinate pair ( m , n ) (Fig. 4). When approaching the polar region the size of these areas grows. At the latitude of Archangelsk and Reykjavik these areas are already possible for retrograde Venus40) and for the Moon. That is, Venus can be represented by three points on the circle of the terrestrial zodiac. In circumpolar regions, the Sun and all planets other than Pluto sometimes pass through these areas. Because Pluto moves close to the plane of the equator, it does not enter the multiplicity area. It is obvious that a planet enters the multiplicity area when it becomes non-descending (i.e. when it passes over the horizon near the north point). But it also enters into this area somewhat earlier, while still crossing below the horizon in its daily motion.

Fig. 4. part of the multiplicity area on the celestial sphere. This area is situated entirely above the horizon near the north point N. A similar area is situated beneath the horizon near the south point. MN is part of the great circle of the local meridian plane. ENW is the horizon great circle. For this example, suppose this is the terrestrial zodiac. Arcs R1R2 and S1S2 are the ascensional paths of planets for a born near the polar circle. R1R2 is the path of a non-descending planet; S1S2 is the path of a planet crossing the horizon but still falling into the multiplicity zone. P is a point on the path R1R2, whose zodiacal longitudinal arcs t 1, t 2 and t 3 are shown.

Let us describe the longitudinal dynamics of a star or planet on the terrestrial zodiacal circle for a single sidereal day, when it is at a point on the celestial sphere that goes under the horizon for a very short interval of time yet still enters into the multiplicity area (Fig. 5). As an example we may take the Sun for a point on the globe near the polar circle and on a day not long before or after the summer solstice. For the simplicity we ignore any proper motion of the Sun in ecliptic longitude for the day under consideration.

Fig. 5. Dynamics during a sidereal day of the zodiacal longitude coordinate of a point that is motionless with respect to the 2nd system of equatorial coordinates.

Terrestrial zodiac station cusps of a born in the vicinity of the polar circle are marked with large Arabic numerals. The point we consider falls into the multiplicity zone, despite crossing the horizon in its daily path. The small Arabic numerals designate zodiacal longitudes for the following moments.

1 — At astronomical noon (assuming the point we consider to be the Sun).

2 — A point on the border of the multiplicity area on the celestial sphere, within the 7th station. When the point crosses this border on the terrestrial zodiacal circle, two additional solutions to formula (3) appear near the antiscia point of the first solution, within the 12th station.

3 — The point crosses the horizon. Two solutions in the 7th and 12th stations disappear following their conjunctions with 1st and 7th cusps. The solution in 10th station disappears as well; a single solution 3" opposite it near the cusp of 4th station appears.

4 — The single solution 4' disappears near the cusp of 4th station when the point crosses the horizon, and three solutions 4" appear.

5 — The point leaves the multiplicity zone. The second and third solutions disappear after merging in the 7th station.

6 — A new astronomical noon.

We begin with an astronomical noon when Sun is on the celestial meridian conjunct the MC. In the terminology of this work, the Sun is conjunct the cusp of the10th terrestrial station. As it approaches the north point, the retrograde motion of the Sun through 7th station slows, and at a specific moment two additional solutions appear within the 12th terrestrial station. They move in opposite directions. The second solution moves quickly toward the 10th station. The third solution moves slowly toward the 1st station, reflecting the motion of the first solution near the cusp of the 7th station.

The Sun crosses the horizon before passing the north point. At the time of sunset the first and third solutions simultaneously conjunct the cusps of the 7th and 1st stations respectively and disappear. The second solution also disappears, replaced by a solution exactly opposite it. At the time of sunset the longitude of second solution is greater than 270. If, for example, its longitude is 275, then the single solution immediately after sunset is the longitude 95 = 90+ ( 275– 270). The longitude of this solution decreases with time. At the moment when Sun passes under the north point the longitude of this solution is 90 - the Sun is conjunct the cusp of the 4th terrestrial station.

Continuing with this example, when the terrestrial zodiac longitude of the Sun becomes equal to 85 = 90– ( 275– 270), the Sun again crosses the horizon line as it rises. The solution with terrestrial zodiac longitude 85 disappears and is replaced by the solution with the longitude 265 = 270– ( 275– 270). We consider it a second solution as the Sun once again falls into the multiplicity zone after it crosses the horizon. We consider the solution near the cusp of 1st station as the first solution. The third solution is a point near the cusp of 7th station. As the Sun rises almost tangentially to the horizon in its movement away from the north point, the second and third solutions approach each other with increasing velocity and disappear as they coincide. At this point we have again the usual situation of the rising Sun that moves toward the cusp of 12th terrestrial station.

The central body of the terrestrial zodiac always has a zodiacal longitude t= + 90. Within the framework of traditional terminology it is possible to say that the Earth is always conjunct the cusp of 4th house or the IC. Since by tradition one projects the terrestrial zodiac with the fundamental reference point onto the solar zodiac, the Earth is not represented as a celestial body in the system of astrological calculations, but is instead implicitly included in the system through house interpretation. However the point of the Earth on the solar zodiac will not coincide with the cusp of 4th house. So its consideration on the solar zodiacal circle promises to be of interest.

The point identified as the Ascendant in traditional astrology is the projection of the fundamental reference point, or east point, of the terrestrial zodiac onto the solar zodiac. The reference point coinciding with the crossing point of the terrestrial and solar equatorial circles (the equator and ecliptic in traditional terminology) is also used implicitly in traditional astrology. This reference point is required for a well-founded formulation of the methods of progressions and directions, as presented below. Here the words ‘directions’ and ‘progressions’ are used in the traditional sense. In this work the meaning of these words will be given by exact definitions. The newly corrected definitions of traditional symbolical methods will have names of the form solar-terrestrial progressions and so on.

2.4.2 The Lunar zodiac.

It is likely that the lunar zodiac is the most interesting consequence of the theory of the dynamical origin of zodiac presented in this work. While ancient astrologers suspected the existence of the lunar zodiac, they had no mathematical means upon which to construct it quantitatively. It was then only possible to construct it on the basis of a static geometry, mimicking the geometry of the solar zodiac. While the traditional solar zodiac differs from its exact dynamic variant only by several angular minutes, the statically constructed lunar zodiac differs strikingly from its dynamic prototype. Because of this its static construction cannot have forecasting power, though it may work in a descriptive way. We mean, for example, the 28 stations of the Moon in Chinese astrology [18].

As in the case of the solar zodiac we can choose more than one reference point. In addition to the fundamental reference point it is possible to choose the crossing of the moon’s local equator with the local equator of either the solar or the terrestrial zodiac (i.e. set the reference point in the solar or terrestrial node).

Note an additional peculiarity of the lunar zodiac. The inclination of the moon’s orbit to the ecliptic is only about 6. This means that the projections onto the solar zodiacal circle of the 12th order family of the reference point will remain practically 30 from each other. In astrological language these points can be called the cusps of the lunar houses. If we choose the reference point on the lunar zodiac to be the solar node, the projections of the cusps of the lunar signs onto the solar zodiac will be close to the cusps of the signs of the solar zodiac with the reference point in the moon’s node (draconic astrology). With this choice of reference points on the local solar and lunar zodiacs it will be difficult to differentiate the interpretations of the solar and lunar zodiacs.

The lunar zodiac is likely the best case for testing the concept of a zodiac of a massive body. The aberration angle of the Moon for a born located on the Earth’s surface does not much exceed 1 angular second. Therefore the geometry of the lunar zodiac is a common spherical geometry. But the local equatorial plane of the lunar zodiac constantly oscillates with considerable amplitude. The rotation of a born around the Earth’s polar axis strongly deforms his circular motion around the Moon. This is because the Moon’s linear orbital velocity around the Earth is only a little greater than a born’s linear rotational velocity around the Earth’s polar axis.

2.4.2.1 The Lunar nodes.

When interpreting a natal chart considerable attention is given to the position of the Moon’s nodes in the signs and houses. In general, little or no attention is paid to the exact position of the node, for example, to an exact conjunction with a house cusp. If we consider the crossing point of the ecliptic with the equator (i.e. in the terminology of this work, the crossing of the equators of the terrestrial and solar zodiacs), the vernal and autumnal equinoctial points, as very important, then the crossing point of the ecliptic with the equator of the local lunar zodiac should also be of considerable importance. These two points of intersection (called here the local lunar nodes) are located at some distance from the traditional lunar nodes (whether the mean or the true nodes). Most likely it is these local lunar nodes that should be interpreted in the spirit of the north and south lunar nodes, as actively used in modern astrological consulting practice. When the local lunar nodes exactly conjoin a house cusp we would expect striking phenomena.

2.4.3 Solar and planetary zodiacs.

The solar zodiac was the first to be well understood. Its geometry is both simple and stationary — more precisely, nearly stationary. By this, we mean that a point on the surface of the earth follows a nearly circular path in its movement around the Sun. The plane formed by the velocity vector of a point on the Earth’s surface with respect to the Sun, and the vector pointing to the Sun with account for aberration41) (i.e. plane of the local solar equator), is oscillating with a period of one day and amplitude up to several arc minutes relative to the ecliptic. Remember that the ecliptic is a plane formed by the motion of the Earth-Moon barycentre42) around the Sun. At moderate latitudes this can result in a difference in the longitudinal coordinate of a planet on the solar zodiac of up to 1-2' compared to its ecliptic longitude. Here it is essential to apply the principle of locality: the events for a born with a given birthday and location on the Earth’s surface cannot be defined by a dynamic construction determined by the barycentre of the Earth-Moon system.

Planetary zodiacs have an additional important characteristic. At those moments when the central body changes its motion from direct to retrograde or vice versa, the plane of the local equator turns by 180. The longitudinal zodiacal positions of other planets on the considered zodiac experience significant displacements as a result. The accuracy of the calculation of the local equatorial plane orientation falls by several orders of magnitude. We have not yet analysed the implications in detail.

Footnotes
18. It may be necessary to mark the positions of the local planetary nodes on the circles of solar, terrestrial and lunar zodiacs. By local planetary nodes we understand the longitudinal positions of the two crossing points of the equator of the corresponding planetary zodiac with the local equator of the zodiac under consideration.
19. Provided he does not travel to a nearby planet of the solar system.
20. The usual physical notions of length, time, simultaneity etc. are given in the reference system accompanying born in accordance with the general theory of relativity.
21. The velocity vector of the central body (without aberration) in the non-rotating proper reference system of the born.
22. To be fair it is necessary to note that there is no sufficient experimental basis for the introduction of the notion of the axis of space anisotropy different from the direction of attraction by the gravitational field. However it seems to us that this follows by necessity from the conceptual basis of astrological rules and the impossibility to explain the mechanism of astrological calculations on the basis of some known interaction, including gravitation. The experimental aspect of the problem will be discussed below.
23. Instead of the astronomical horizon it is necessary to take the gravitational one, i.e. the plane perpendicular to the force of gravitational attraction (force of gravity minus inertial force).
24. In topology, such a mapping is called a triple covering of the sphere.
25. One half of a great circle. A great circle is a section of a sphere made by a plane going through its center.
26. Projections of these points of the terrestrial zodiacal circle onto the solar zodiacal circle are the IC and MC respectively. It is suitable to visualize for this construction that the born is on the Earth’s surface at a middle latitude in the center of the celestial sphere. The local meridian plane of the terrestrial zodiac will practically comply with the plane of celestial meridian. The local equatorial plane will coincide with the plane of the instantaneous equator. The local horizontal plane will coincide with the plane of the gravitational horizon. The difference between the true horizon of anisotropy and the gravitational horizon will be extremely small. Here one should not muddle the gravitational horizon with the plane perpendicular to the plumb line. The plumb gives a direction of the gravity force, which is the sum of the gravitational force and the inertial force. The latter force appears because of the rotation of the reference system motionless with respect to the Earth’s surface.
27. It is convenient to represent the main parameter of this longitudinal parameterization of points on the celestial sphere as a virtual time, and the procedure of parameterization as a motion of the great semicircle with this virtual time.
28. When j d=0 the zodiacal coordinate system coincides with the spherical equatorial coordinate system. A nonzero value of j d skews the zodiacal coordinate system, eventually converting it into a triple covering of the celestial sphere. For now the characteristics of this distortion is known only for the case when the plane of dynamic angle is perpendicular to the plane of local equator (owing to the known geometry of the terrestrial zodiac). For the solar and planetary zodiacs the aberration plane coinciding with the plane of the dynamic angle practically complies with the plane of equator and the aberration angle is on the order of tens of angular seconds - a noticeable value if we suppose that the distortion can linearly depend on the aberration angle for this orientation. It is hardly possible that the geometry of the distortion of the zodiacal coordinate system relative to the spherical coordinate system is determined only by the value of the angle between the plane of local equator and the vector of anisotropy, rather then the orientation of the aberration angle. As a result, when constructing the algorithm for calculation of the local solar zodiac for a computer program, we have preferred to put j d equal to zero (i.e. to use as the zodiacal coordinates the corresponding spherical coordinates) and to analyze the errors appearing in the process of practical calculations. For the moon zodiac we can consider the dynamic angle to be equal to zero, since the correction for planetary and stellar aberration is less than an angular second.
29. The angle between the polar axis and the parameterizing half-plane plus uniform motion of the crossing point of the parameterizing semicircle along the equator do not uniquely fix the motion. For each t two different positions satisfying these conditions are possible. The choice of solution is defined by the continuity of the parameterising motion (except for the leaps of the supporting straight line in the plane of the horizon for zodiacal longitudes 0 and 180) and by the character of motion described above for the supporting straight line in the plane of horizon.
30. In the language of group theory this can be formulated: the sensitive points of the Nth order family of a given point can be generated from the given point by the action of the regular representation of the cyclic subgroup of the Nth order of the one-parametric group of rotations.
31. It is important for the number of stations to be divisible by 4, so that one of the station cusps coincides with the central body (point of conjunction). We introduced the notion of station to avoid confusion with the notion of house. By “house” (more exactly house cusp) we understand a projection of a station cusp onto another zodiac, as it is accepted in traditional medieval astrology when projecting the mondial (terrestrial) zodiac onto the solar one.
32. Near the polar circle and within it the projections of the station cusps of the terrestrial zodiac onto the solar zodiac (ecliptic) cease to follow one another in order. In this situation it is impossible to consider a house as an area from its cusp to the next one.
33. In a certain sense the CPT theorem can be proved within the framework of the special theory of relativity, i.e. remaining within the framework of classical description.
34. Here it is worthwhile to note that the axis opposition-conjunction contains another singularity — one family of parameterizing semicircles is changed for another.
35. The longitudinal zodiacal parameterization of the terrestrial zodiac is stationary in the first equatorial and horizontal astronomical coordinate systems.
36. Therefore such constructions as solar and planetary zodiacs, for which the plane of the aberration angle is close to the plane of equator and is on the order of tens of angular seconds, as concerns calculations of pinpoint accuracy (near 1"), should be subjected to study for the purpose of reconstruction of the geometry of zodiacs with a free orientation of the dynamic angle.
37. For the Moon it is a part of angular second.
38. For Moscow the correction to the geographic latitude is approximately equal to 5'30".
39. So to achieve an accuracy of 1" it is necessary to use maps of the plumb deviation.
40. In retrograde motion the ecliptic latitude of Venus can reach 8.
41. In a strict sense it is necessary to take the vector tangent to the light geodesic, going from the center of the Sun to the considered point on the Earth surface for this direction.
42. Center of mass.
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