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2.7 Sidereal Ephemerides

  1. Sidereal Calculations

Sidereal astrology has a complicated history, and we (the developers of Swiss Ephemeris) are actually tropicalists. Any suggestions how we could improve our sidereal calculations are welcome!



      1. The problem of defining the zodiac

One of the main differences between the western and the eastern tradition of astrology is the definition of the zodiac. Western astrology uses the so-called tropical zodiac in which 0 Aries is defined by the vernal point (the celestial point where the sun stands at the beginning of spring). The tropical zodiac is a division of the ecliptic into 12 equal-sized zodiac signs of 30° each. Astrologers call these signs after constellations that are found along the ecliptic, although they are actually independent of these constellations. Due to the precession of the equinox, the vernal point and tropical Aries move through all constellations along the ecliptic, staying for roughly 2160 years in each one of them. Currently, the beginning of tropical Aries is located in the constellation of Pisces. In a few hundred years, it will enter Aquarius, which is the reason why the more impatient ones among us are already preparing for the “Age of Aquarius”.


There are also sidereal traditions of astrology, both a Hindu tradition and a western tradition, which derives itself from ancient Hellenistic and Babylonian astrology. They use a so-called sidereal zodiac, which consists of 12 equal-sized zodiac signs, too, but it is tied to some fixed reference point, i.e. usually some fixed star. These sidereal zodiac signs only roughly coincide with the sidereal zodiacal constellations, which are of variable size.
While the definition of the tropical zodiac is very obvious and never questioned, sidereal astrology has considerable problems in defining its zodiac. There are many different definitions of the sidereal zodiac that differ by several degrees from each other. At first glance, all of them look arbitrary, and there is no striking evidence – from a mere astronomical point of view – for anyone of them. However, a historical study shows at least that many of them are related to each other and the basic approaches aren’t so many.
Sidereal planetary positions are usually computed from tropicl positions using the equation:

sidereal_position = tropical_position – ayanamsha(t) ,

where ayanamsha is the difference between the two zodiacs at a given epoch. (Sanskrit ayanâmsha means ”part of a solar path (or half year)”; the Hindi form of the word is ayanamsa with an s instead of sh.)

The value of the ayanamsha of date is usually computed from the ayanamsha value at a particular start date (e.g. 1 Jan 1900) and the speed of the vernal point, the so-called precession rate in ecliptic longitude.
The zero point of the sidereal zodiac is therefore traditionally defined by the equation:

sidereal_Aries = tropical Aries + ayanamsha(t).
The Swiss Ephemeris offers about fourty different ayanamshas, but the user can also define his or her own ayanamsha.

      1. The Babylonian tradition and the Fagan/Bradley ayanamsha

There have been several attempts to calculate the zero point of the Babylonian ecliptic from cuneiform lunar and planetary tablets. Positions were given relative to some sidereally fixed reference point. The main problem in fixing the zero point is the inaccuracy of ancient observations. Around 1900 F.X. Kugler found that the Babylonian star positions fell into three groups:


Kugler ayanamshas:

9) ayanamsha = -3°22´, t0 = -100

10) ayanamsha = -4°46´, t0 = -100 Spica at 29 vi 26

11) ayanamsha = -5°37´, t0 = -100

(9 – 11 = Swiss Ephemeris ayanamsha numbers)
In 1958, Peter Huber reviewed the topic in the light of new material and found:
12) Huber Ayanamsha:

ayanamsha = -4°28´ +/- 20´, t0 = –100 Spica at 29 vi 07’59”

The standard deviation was 1°08’


(Note, this ayanamsha was corrected with SE version 2.05. A wrong value of -4°34’ had been taken over from Mercier, “Studies on the Transmission of Medieval Mathematical Astronomy”, IIb, p. 49.)
In 1977 Raymond Mercier noted that the zero point might have been defined as the ecliptic point that culminated simultaneously with the star eta Piscium (Al Pherg). For this possibility, we compute:
13) Eta Piscium ayanamsha:

ayanamsha = -5°04’46”, t0 = –129 Spica at 29 vi 21
Around 1950, Cyril Fagan, the founder of the modern western sidereal astrology, reintroduced the old Babylonian zodiac into astrology, placing the fixed star Spica near 29°00 Virgo. As a result of “rigorous statistical investigation” (astrological!) of solar and lunar ingress charts, Donald Bradley decided that the sidereal longitude of the vernal point must be computed from Spica at 29 vi 06'05" disregarding its proper motion. Fagan and Bradley defined their ”synetic vernal point” as:
0) Fagan/Bradley ayanamsha

ayanamsha = 24°02’31.36” for 1 Jan. 1950 with Spica at 29 vi 06'05" (without aberration)

(For the year –100, this ayanamsha places Spica at 29 vi 07’32”.)


The difference between P. Huber’s zodiac and the Fagan/Bradley ayanamsha is smaller than 1’.
According to a text by Fagan (found on the internet), Bradley ”once opined in print prior to "New Tool" that it made more sense to consider Aldebaran and Antares, at 15 degrees of their respective signs, as prime fiducials than it did to use Spica at 29 Virgo”. Such statements raise the question if the sidereal zodiac ought to be tied up to one of those stars.
For this possibility, Swiss Ephemeris gives an Aldebaran ayanamsha:
14) Aldebaran-Antares ayanamsha:

ayanamsha with Aldebaran at 15ta00’00” and Antares at 15sc00’17” around the year –100.
The difference between this ayanamsha and the Fagan/Bradley one is 1’06”.
In 2010, the astronomy historian John P. Britton made another investigation in cuneiform astronomical tablets and corrected Huber’s by a 7 arc minutes.
38) Britton ayanamsha:

ayanamsha = -3.2° +- 0.09° (= 5’24”); t0 = 1 Jan. 0, Spica at 29 vi 14’58”.

(For the year -100, this ayanamsha places Spica at 29 vi 15’02”.)


This ayanamsha deviates from the Fagan/Bradley aynamasha by 7 arc min.
Sources:

- Raymond Mercier, ”Studies in the Medieval Conception of Precession”,

in 'Archives Internationales d'Histoire des Sciences', (1976) 26:197-220 (part I), and (1977) 27:33-71 (part II)

- Cyril Fagan and Brigadier R.C. Firebrace, -Primer of Sidereal Astrology, Isabella, MO, USA 1971.

- P. Huber, „Über den Nullpunkt der babylonischen Ekliptik“, in: Centaurus 1958, 5, pp. 192-208.

- John P. Britton, "Studies in Babylonian lunar theory: part III. The introduction of the uniform zodiac", in Arch. Hist. Exact. Sci. (2010)64:617-663, p. 630.



      1. The Hipparchan tradition



Raymond Mercier has shown that all of the ancient Greek and the medieval Arabic astronomical works located the zero point of the ecliptic somewhere between 10 and 22 arc minutes east of the star zeta Piscium. He is of the opinion that this definition goes back to the great Greek astronomer Hipparchus.
Mercier points out that according to Hipparchus’ star catalogue, the stars alpha Arietis, beta Arietis, zeta Piscium, and Spica are in a very precise alignment on a great circle which goes through that zero point near zeta Piscium. Moreover, this great circle was identical with the horizon once a day at Hipparchus’ geographical latitude of 36°. In other words, the zero point rose at the same time when the three mentioned stars in Aries and Pisces rose and when Spica set.
This would of course be a nice definition for the zero point, but unfortunately the stars were not really in such precise alignment. They were only assumed to be so.
Mercier gives the following ayanamshas for Hipparchus and Ptolemy (who used the same star catalogue as Hipparchus):
16) Hipparchus ayanamsha:

ayanamsha = -9°20’ 27 June –128 (jd 1674484) zePsc 29pi33’49” Hipparchus
(According to Mercier’s calculations, the Hipparchan zero point should have been between 12 and 22 arc min east of zePsc, but the Hipparchan ayanamsha, as given by Mercier, has actually the zero point 26’ east of zePsc. This comes from the fact that Mercier refers to the Hipparchan position of zeta Piscium, which was at least rounded to 10’, if correct at all.)
Using the information that Aries rose when Spica set at a geographical latitude of 36 degrees, the precise ayanamsha would be -8°58’13” for 27 June –128 (jd 1674484) and zePsc would be found at 29pi12’, which is too far from the place where it ought to be.
Mercier also discusses the old Indian precession models and zodiac point definitions. He notes that, in the Sûryasiddhânta, the star zeta Piscium (in Sanskrit Revatî) has almost the same position as in the Greek sidereal zodiac, i.e. 29°50’ in Pisces. On the other hand, however, Spica (in Sanskrit Citrâ) is given the longitude 30° Virgo. Unfortunately, these positions of Revatî and Citrâ/Spica are incompatible; either Spica or Revatî must be considered wrong.
Moreover, if the precession model of the Sûryasiddânta is used to compute an ayanamsha for the date of Hipparchus, it will turn out to be –9°14’01”, which is very close to the Hipparchan value. The same calculation can be done with the Âryasiddânta, and the ayanamsha for Hipparchos’ date will be –9°14’55”. For the Siddânta Shiromani the zero point turns out to be Revatî itself. By the way, this is also the zero point chosen by Copernicus! So, there is an astonishing agreement between Indian and Western traditions!
The same zero point near the star Revatî is also used by the so-called Ushâ-Shashî ayanamsha. It differs from the Hipparchan one by only 11 arc minutes.
4) Usha-Shashi ayanamsha:

ayanamsha = 18°39’39.46 1 Jan. 1900

zePsc (Revatî) 29pi50’ (today), 29pi45’ (Hipparchus’ epoch)


The Greek-Arabic-Hindu ayanamsha was zero around 560 AD. The tropical and the sidereal zero points were at exactly the same place.
In the year 556, under the Sassanian ruler Khusrau Anûshirwân, the astronomers of Persia met to correct their astronomical tables, the so-called Zîj al-Shâh. These tables are no longer extant, but they were the basis of later Arabic tables, the ones of al-Khwârizmî and the Toledan tables.
One of the most important cycles in Persian astronomy/astrology was the synodic cycle of Jupiter, which started and ended with the conjunctions of Jupiter with the Sun. This cycle happened to end in the year 564, and the conjunction of Jupiter with the Sun took place only one day after the spring equinox. And the spring equinox took place precisely 10 arcmin east of zePsc. This may be a mere coincidence from a present-day astronomical point of view, but for scientists of those days this was obviously the moment to redefine all astronomical data.

Mercier also shows that in the precession (trepidation) model used in that time and in other models used later by Arabic astronomers, precession was considered to be a phenomenon connected with “the movement of Jupiter, the calendar marker of the night sky, in its relation to the Sun, the time keeper of the daily sky”. Such theories were of course wrong, from the point of view of modern knowledge, but they show how important that date was considered to be.


After the Sassanian reform of astronomical tables, we have a new definition of the Greek-Arabic-Hindu sidereal zodiac (this is not explicitly stated by Mercier, however):
16) Sassanian ayanamsha:

ayanamsha = 0 18 Mar 564, 7:53:23 UT (jd /ET 1927135.8747793) Sassanian

zePsc 29pi49'59"


The same zero point then reappears with a precision of 1’ in the Toledan tables, the Khwârizmian tables, the Sûrya Siddhânta, and the Ushâ-Shashî ayanamsha.
Sources:

- Raymond Mercier, ”Studies in the Medieval Conception of Precession”,

in Archives Internationales d'Histoire des Sciences, (1976) 26:197-220 (part I), and (1977) 27:33-71 (part II)

      1. Suryasiddhanta and Aryabhata

The explanations above are mainly derived from the article by Mercier. However, it is possible to derive ayanamshas from ancient Indian works themselves.


The planetary theory of the main work of ancient Indian astronomy, the Suryasiddhanta, uses the so-called Kaliyuga era as its zero point, i. e. the 18th February 3102 BC, 0:00 local time at Ujjain, which is at geographic longitude of 75.7684565 east (Mahakala temple). This era is henceforth called “K0s”. This is also the zero date for the planetary theory of the ancient Indian astronomer Aryabhata, with the only difference that he reckons from sunrise of the same date instead of midnight. We call this Aryabhatan Kaliyuga era “K0a”.
Aryabhata mentioned that he was 23 years old when exactly 3600 years had elapsed since the beginning of the Kaliyuga era. If 3600 years with a year length as defined by the Aryabhata are counted from K0a, we arrive at the 21st March, 499 AD, 6:56:55.57 UT. At this point of time the mean Sun is assumed to have returned to the beginning of the sidereal zodiac, and we can try to derive an ayanamsha from this information. There are two possible solutions, though:
1. We can find the place of the mean Sun at that time using modern astronomical algorithms and define this point as the beginning of the sidereal zodiac.

2. Since Aryabhata believed that the zodiac began at the vernal point, we can take the vernal point of this date as the zero point.


The same calculations can be done based on K0s and the year length of the Suryasiddhanta. The resulting date of Kali 3600 is the same day but about half an hour later: 7:30:31.57 UT.
Algorithms for the mean Sun were taken from: Simon et alii, “Numerical expressions for precession formulae and mean elements for the Moon and the planets”, in: Astron. Astrophys. 282,663-683 (1994).
Suryasiddhanta/equinox ayanamshas with zero year 499 CE

21) ayanamsha = 0 21 Mar 499, 7:30:31.57 UT = noon at Ujjain, 75.7684565 E.

Based on Suryasiddhanta: ingress of mean Sun into Aries

at point of mean equinox of date.

22) ayanamsha = -0.21463395 Based on Suryasiddhanta again, but assuming ingress of mean Sun

into Aries at true position of mean Sun at the same epoch


Aryabhata/equinox ayanamshas with zero year 499 CE

23) ayanamsha = 0 21 Mar 499, 6:56:55.57 UT = noon at Ujjain, 75.7684565 E.

Based on Aryabhata, ingress of mean Sun into Aries

at point of mean equinox of date.

24) ayanamsha = -0.23763238 Based on Aryabhata again, but assuming ingress of mean Sun

into Aries at true position of mean Sun at the same epoch


According to Govindasvamin (850 n. Chr.), Aryabhata and his disciples taught that the vernal point was at the beginning of sidereal Aries in the year 522 AD (= Shaka 444). This tradition probably goes back to an erroneous interpretation of Aryabhata's above-mentioned statement that he was 23 years old when 3600 had elapsed after the beginning of the Kaliyuga. For the sake of completeness, we therefore add the following ayanamsha:
37) Aryabhata/equinox ayanamsha with zero year 522 CE

ayanamsha = 0 21.3.522, 5:46:44 UT
Sources:

- Surya-Siddhanta: A Text Book of Hindu Astronomy by Ebenezer Burgess, ed. Phanindralal Gangooly (1989/1997) with a 45-page commentary by P. C. Sengupta (1935).



- D. Pingree, "Precession and Trepidation in Indian Astronomy", in JHA iii (1972), pp. 28f.

      1. The Spica/Citra tradition and the Lahiri ayanamsha


There is another ayanamsha tradition that assumes the star Spica (in Sanskrit Citra) at 0° Libra. This ayanamsha is the most common one in modern Hindu astrology. It was first proposed by the astronomy historian S. B. Dixit (also written Dikshit), who in 1896 published his important work History of Indian Astronomy (= Bharatiya Jyotih Shastra; bibliographical details further below). Dixit came to the conclusion that, given the prominence that Vedic religion gave to the cardinal points of the tropical year, the Indian calendar, which is based on the zodiac, should be reformed and no longer be calculated relative to the sidereal, but to the tropical zodiac. However, if such a reform could not be brought about due to the rigid conservatism of contemporary Vedic culture, then the ayanamsha should be chosen in such a way that the sidereal zero point would be in opposition to Spica. In this way, it would be in accordance with Grahalaghava, a work by the 16th century astronomer Ganeśa Daivajña that was still used in the 20th century by Indian calendar makers. (op. cit., Part II, p. 323ff.). This view was taken over by the Indian Calendar Reform Committee on the occasion of the Indian calendar reform in 1956, when the ayanamsha based on the star Spica/Citra was declared the Indian standard. This standard is mandatory not only for astrology but also for astronomical ephemerides and almanacs and calendars published in India. The ayanamsha based on the star Spica/Citra became known as “Lahiri ayanamsha”. It was named after the Calcuttan astronomer and astrologer Nirmala Chandra Lahiri, who was a member of the Reform Committee.
However, as has been said, it was Dixit who first propagated this solution to the ayanamsha problem. In addition, the Suryasiddhanta, the most important work of ancient Hindu astronomy, which was written in the first centuries AD, but reworked several times, already assumes Spica/Citra at 180° (although this statement has caused a lot of controversy because it is in contradiction with the positions of other stars, in particular with zeta Piscium/Revati at 359°50‘). Finally yet importantly, the same ayanamsha seems to have existed in Babylon and Greece, as well. While the information given above in the chapters about the Babylonian and the Hipparchan traditions are based on analyses of old star catalogues and planetary theories, a study by Nick Kollerstrom of 22 ancient Greek and 5 Babylonian birth charts seems to prove that they fit better with Spica at 0 Libra (= Lahiri), than with Aldebaran at 15 Taurus and Spica at 29 Virgo (= Fagan/Bradley).
The standard definition of the Indian ayanamsha (“Lahiri” ayanamsha) was originally introduced in 1955 by the Indian Calendar Reform Committee (23°15' 00" on the 21 March 1956, 0:00 Ephemeris Time). The definition was corrected in Indian Astronomical Ephemeris 1989, page 556, footnote:

"According to new determination of the location of equinox this initial value has been revised to and used in computing the mean ayanamsha with effect from 1985'."

The mention of “mean ayanamsha” is misleading though. The value 23°15' 00".658 is true ayanamsha, i. e. it includes nutation and is relative to the true equinox of date.
1) Lahiri ayanamsha

ayanamsha = 23°15' 00".658 21 March 1956, 0:00 TDT Lahiri, Spica roughly at 0 Libra


The Lahiri standard position of Spica is 179°59’04 in the year 2000, and 179°59’08 in 1900. In the year 285, when the star was conjunct the autumnal equinox, its position was 180°00’16. It was only in the year 667 AD that its position was exactly 180°. The motion of the star is partly caused by its proper motion and partly by the so-called planetary precession, which causes very slow changes in the orientation of the ecliptic plane. But what method exactly was used to define this ayanamsha? According to the Indian pundit A.K. Kaul, an expert in Hindu calendar and astrology, Lahiri wanted to place the star at 180°, but at the same time arrive at an ayanamsha that was in agreement with the Grahalaghava, an important work for traditional Hindu calendar calculation that was written in the 16th century. (e-mail from Mr. Kaul to Dieter Koch on 1 March 2013)
Swiss Ephemeris versions below 1.78.01, had a slightly different definition of the Lahiri ayanamsha that had been taken from Robert Hand's astrological software Nova. The correction amounts to 0.01 arc sec.
In 1967, 12 years after the standard definition of the Lahiri ayanamsha had been published by the Calendar Reform Committee, Lahiri published another Citra ayanamsha in his Bengali book Panchanga Darpan. There, the value of “mean ayanamsha” is given as 22°26’45”.50 in 1900, whereas the official value is 22°27’37”.76. The intention behind this modification is obvious. With the standard Lahiri ayanamsha, the position of Spica was “wrong”, i.e. it deviated from 180° by almost an arc minute. Lahir obviously wanted to place the star exactly at 180° for recent years. It therefore seems that Lahiri did not follow the Indian standard himself but was of the opinion that Spica had to be at exactly 180° (true chitrapaksha ayanamsha). The Swiss Ephemeris does not support this updated Lahiri ayanamsha. Users who want to follow Lahiri’s real intention are advised to use the True Chitrapaksha ayanamsha (No. 27, see below).
Many thanks to Vinay Jha, Narasimha Rao, and Avtar Krishen Kaul for their help in our attempt to understand this complicated matter.
Additional Citra/Spica ayanamshas:
The Suryasiddhanta gives the position of Spica/Citra as 180° in polar longitude (ecliptic longitude, but projection on meridian lines). From this, the following Ayanamsha can be derived:
26) Ayanamsha having Spica/Citra at polar longitude 180° in 499 CE

ayanamsha = 2.11070444 21 Mar 499, 7:30:31.57 UT = noon at Ujjain, 75.7684565 E.

Citra/Spica at polar ecliptic longitude 180°.


Usually ayanamshas are defined by an epoch and an initial ayanamsha offset. However, if one wants to make sure that a particular fixed star always remains at a precise position, e. g. Spica at 180°, it does not work this way. The correct procedure for this to work is to calculate the tropical position of Spica for the date and subtract it from the tropical position of the planet:
27) True chitrapaksha ayanamsha

Spica is always exactly at 180° or 0° Libra in ecliptic longitude (not polar!).


The Suryasiddhanta also mentions that Revati/zeta-Piscium is exactly at 359°50’ in polar ecliptic longitude (projection onto the ecliptic along meridians). Therefore the following two ayanamshas were added:
25) Ayanamsha having Revati/zeta Piscium at polar longitude 359°50’ in 499 CE

ayanamsha = -0.79167046 21 Mar 499, 7:30:31.57 UT = noon at Ujjain, 75.7684565 E.

Revati/zePsc at polar ecliptic longitude 359°50’


28) True Revati ayanamsha

Revati/zePsc is always exactly at longitude 359°50’ (not polar!).


(Note, this was incorrectly implemented in SE 2.00 – SE 2.04. The Position of Revati was 0°. Only from SE 2.05 on, this ayanamsha is correct.)
Siddhantas usually assume the star Pushya (delta Cancri = Asellus Australis) at 106°. PVR Narasimha Rao believes this star to be the best anchor point for a sidereal zodiac. In the Kalapurusha theory, which assigns zodiac signs to parts of the human body, the sign of Cancer is assigned to the heart, and according to Vedic spiritual literature, the root of human existence is in the heart. Mr. Narasimha Rao therefore proposed the following ayanamsha:
29) True Pushya paksha ayanamsha

Pushya/deCnC is always exactly at longitude 106°.


Another ayanamsha close to the Lahiri ayanamsha is named after the Indian astrologer K.S. Krishnamurti (1908-1972).
5) Krishnamurti ayanamsha

ayanamsha = 22.363889, t0 = 1 Jan 1900, Spica at 180° 4'51.
Sources:

- Burgess, E., The Surya Siddanta. A Text-book of Hindu Astronomy, Delhi, 2000 (MLBD).

- Dikshit, S(ankara) B(alkrishna), Bharatiya Jyotish Sastra (History of Indian Astronomy) (Tr. from Marathi), Govt. of India, 1969, part I & II.

- Kollerstrom, Nick, „The Star Zodiac of Antiquity“, in: Culture & Cosmos, Vol. 1, No.2, 1997).

- Lahiri, N. C., Panchanga Darpan (in Bengali), Calcutta, 1967 (Astro Research Bureau).

- Lahiri, N. C., Tables of the Sun, Calcutta, 1952 (Astro Research Bureau).

- Saha, M. N., and Lahiri, N. C., Report of the Calendar Reform Committee, C.S.I.R., New Delhi, 1955.

- The Indian astronomical ephemeris for the year 1989, Delhi (Positional Astronomy Centre, India Meteorological Department)

- P.V.R. Narasimha Rao, "Introducing Pushya-paksha Ayanamsa" (2013), http://www.vedicastrologer.org/articles/pp_ayanamsa.pdf.



      1. The sidereal zodiac and the Galactic Center

The definition of the tropical zodiac is very simple and convincing. It starts at one of the two intersection points of the ecliptic and the celestial equator. Similarly, the definition for the house circle which is said to be an analogy of the zodiac, is very simple. It starts at one of the two intersection points of the ecliptic and the local horizon. Unfortunately, sidereal traditions do not provide such a simple definition for the sidereal zodiac. The sidereal zodiac is always fixed at some anchor star such as Citra (Spica), Revati (zeta Piscium), or Aldebaran and Antares.


Unfortunately, nobody can tell why any of these stars should be so important that it could be used as an anchor point for the zodiac. In addition, all these solutions are unattractive in that the fixed stars actually are not fixed

forever, but have a small proper motion which over a long period of time such as several millennia, can result in a considerable change in position. While it is possible to tie the zodiac to the star Spica in a way that it remains at 0° Libra for all times, all other stars would change their positions relative to Spica and relative to this zodiac and would not be fixed at all. The appearance of the sky changes over long periods of time. In 100’000 years, the



constellation will look very different from now, and the nakshatras (lunar mansions) will get confused. For this reason, a zodiac defined by positions of stars is unfortunately not able to provide an everlasting reference frame.
For such or also other reasons, some astrologers (Raymond Mardyks, Ernst Wilhelm, Rafael Gil Brand, Nick Anthony Fiorenza) have tried to define the sidereal zodiac using either the galactic centre or the node of the galactic equator with the ecliptic. It is obvious that this kind of solution, which would not depend on the position of a single star anymore, could provide a philosophically meaningful and very stable definition of the zodiac. Fixed stars would be allowed to change their positions over very long periods of time, but the zodiac could still be considered fixed and “sidereal”.
The Swiss astrologer Bruno Huber has pointed out that everytime the Galactic Center enters the next tropical sign the vernal point enters the previous sidereal sign. E.g., around the time the vernal point will enter Aquarius (at the beginning of the so-called Age of Aquarius), the Galactic Center will enter from Sagittarius into Capricorn. Huber also notes that the ruler of the tropical sign of the Galactic Center is always the same as the ruler of the sidereal sign of the vernal point (at the moment Jupiter, will be Saturn in a few hundred years).
A correction of the Fagan ayanamsha by about 2 degrees or a correction of the Lahiri ayanamsha by 3 degrees would place the Galactic Center at 0 Sagittarius. Astrologically, this would obviously make some sense. Therefore, we added an ayanamsha fixed at the Galactic Center in 1999 in Swiss Ephemeris 1.50, when we introduced sidereal ephemerides (suggestion by D. Koch, without any astrological background):
17) Galactic Center at 0 Sagittarius (and the beginning of nakshatra Mula)
A different solution was proposed by the American astrologer Ernst Wilhelm in 2004. He projects the galactic centre on the ecliptic in polar projection, i.e. along a great circle that passes through the celestial north pole (in Sanskrit dhruva) and the galactic centre. The point at which this great circle cuts the ecliptic is defined as the middle of the nakshatra Mula, which corresponds to sidereal 6°40’ Sagittarius.
36) Dhruva Galactic Center Middle Mula Ayanamsha (Ernst Wilhelm)
For Hindu astrologers who follow a tradition oriented towards the star Revati (ζ Piscium), this solution may be particularly interesting because when the galactic centre is in the middle of Mula, then Revati is almost exactly at the position it has in Suryasiddhanta, namely 29°50 Pisces. Also interesting in this context is the fact that the meaning of the Sanskrit word mūlam is “root, origin”. Mula may have been the first of the 27 nakshatras in very ancient times, before the Vedic nakshatra circle and the Hellenistic zodiac were conflated and Ashvini, which begins at 0° Aries, became the first nakshatra.
Sources:

- https://groups.yahoo.com/neo/groups/StudyingKala/conversations/topics/14656

- private communication with D. Koch
Another ayanamsha based on the galactic centre was proposed by the German-Spanish astrologer Rafael Gil Brand. Gil Brand places the galactic centre at the golden section between 0° Scorpion and 0° Aquarius. The

axis between 0° Leo and 0° Aquarius is the axis of the astrological ruler system.


30) Galactic Centre in the Golden Section Scorpio/Aquarius (Rafael Gil Brand)
This ayanamsha is very close to the ayanamsha of the important Hindu astrologer B.V. Raman. (see below)
Sources:

- Rafael Gil Brand, Himmlische Matrix. Die Bedeutung der Würden in der Astrologie, Mössingen (Chiron), 2014.

- Rafael Gil Brand, "Umrechnung von tropischen in siderische Positionen", http://www.astrologie-zentrum.net/index.php/8-siderischer-tierkreis/5-umrechnung
The other possibility – in analogy with the tropical ecliptic and the house circle – would be to start the sidereal ecliptic at the intersection point of the ecliptic and the galactic plane. At present, this point is located near 0 Capricorn. However, defining this point as sidereal 0 Aries would mean to break completely with the tradition, because it is far away from the traditional sidereal zero points.

      1. The sidereal zodiac and the Galactic Equator

Other ways to define a sidereal zodiac without any dependence of anchor stars use the intersection point (or node) of the galactic equator with the ecliptic. This point has been used with the following ayanamshas:


34) Skydram Ayanamsha (Raymond Mardyks)

(also known as Galactic Alignment Ayanamsha)


This ayanamsha was proposed in 1991 by the American astrologer Raymond Mardyks. It had the value 30° on the autumn equinox 1998. Consequently, the node (intersection point) of the galactic equator with the ecliptic was very close to sidereal 0° Sagittarius on the same date, and there was an interesting “cosmic alignment”: The galactic pole pointed exactly towards the autumnal equinoctial point, and the galactic-ecliptic node coincided with the winter solstitial point (tropical 0° Capricorn).
Mardyks' calculation is based on the galactic coordinate system that was defined by the International Astronomical Union in 1958.
Source:

- Raymond Mardyks, “When Stars Touch the Earth”, in: The Mountain Astrologer Aug./Sept. 1991, pp. 1-4 and 47-48.

- Private communication between R. Mardyks and D. Koch in April 2016.
31) Ayanamsha based on the Galactic Equator IAU 1958
This is a variation of Mardyks' Skydram or "Galactic Alignment" ayanamsha, where the galactic equator cuts the ecliptic at exactly 0° Sagittarius. This ayanamsha differs from the Skydram ayanamsha by only 19 arc seconds.
32) Galactic Equator (Node) at 0° Sagittarius
The last two ayanamshas are based on a slightly outdated position of the galactic pole that was determined in 1958. According to more recent observations and calculations from the year 2010, the galactic node with the

ecliptic shifts by 3'11", and the "Galactic Alignment" is preponed to 1994. The galactic node is fixed exactly at sidereal 0° Sagittarius.


Source:

Liu/Zhu/Zhang, „Reconsidering the galactic coordinate system“, Astronomy & Astrophysics No. AA2010, Oct. 2010, p. 8.


33) Ardra Galactic Plane Ayanamsha

(= Galactic equator cuts ecliptic in the middle of Mula and the beginning of Ardra)
With this ayanamsha, the galactic equator cuts the ecliptic exactly in the middle of the nakshatra Mula. This means that the Milky Way passes through the middle of this lunar mansion. Here again, it is interesting that the Sanskrit word mūlam means "root, origin", and it seems that the circle of the lunar mansions originally began with this nakshatra. On the opposite side, the galactic equator cuts the ecliptic exactly at the beginning of the nakshatra Ārdrā ("the moist, green, succulent one", feminine).
This ayanamsha was introduced by the American astrologer Ernst Wilhelm in 2004. He used a calculation of the galactic node by D. Koch from the year 2001, which had a small error of 2 arc seconds. The current implementation of this ayanamsha is based on a new position of the Galactic pole found by Chinese

astronomers in 2010.



      1. Other ayanamshas



35) True Mula Ayanamsha (K. Chandra Hari)
With this ayanamsha, the star Mula (λ Scorpionis) is assumed at 0° Sagittarius.
The Indian astrologer Chandra Hari is of the opinion that the lunar mansion Mula corresponds to the Muladhara Chakra. He refers to the doctrine of the Kalapurusha which assigns the 12 zodiac signs to parts of the human body. The initial point of Aries is considered to correspond to the crown and Pisces to the feet of the cosmic human being. In addition, Chandra Hari notes that Mula has the advantage to be located near the galactic centre and to have “no proper motion”. This ayanamsha is very close to the Fagan/Bradley ayanamsha. Chandra Hari believes it defines the original Babylonian zodiac.
(In reality, however, the star Mula (λ Scorpionis) has a small proper motion, too. As has been stated, the position of the galactic centre was not known to the ancient peoples. However, they were aware of the fact that the Milky Way crossed the ecliptic in this region of the sky.)
Sources:

- K. Chandra Hari, "On the Origin of Siderial Zodiac and Astronomy", in: Indian Journal of History of Science, 33(4) 1998.

- Chandra Hari, "Ayanāṃśa", xa.yimg.com/kq/groups/26252194/1355927039/name/Ayanamsa.PDF .

- http://www.indiadivine.org/content/topic/1229109-true-ayanamsa-views-of-chandra-hari/


The following ayanamshas were provided by Graham Dawson (”Solar Fire”), who had taken them over from Robert Hand’s Program ”Nova”. Some were also contributed by David Cochrane. Explanations by D. Koch:
2) De Luce Ayanamsha
This ayanamsha was proposed by the American astrologer Robert DeLuce (1877-1964). It is fixed at the birth of Jesus, theoretically at 1 January 1 AD. However, DeLuce de facto used an ayanamsha of 26°24'47 in the year 1900, which corresponds to 4 June 1 BC as zero ayanamsha date.
DeLuce believes that this ayanamsha was also used in ancient India. He draws this conclusion from the fact that the important ancient Indian astrologer Varahamihira, who assumed the solstices on the ingresses of the Sun into

sidereal Cancer and Capricorn, allegedly lived in the 1st century BC. This dating of Varahamihira has recently become popular under the influence of Hindu nationalist ideology (Hindutva). However, historically, it cannot be

maintained. Varahamihira lived and wrote in the 6th century AD.
Sources:

- Robert DeLuce, Constellational Astrology According to the Hindu System, Los Angeles, 1963, p. 5.


4) Raman Ayanamsha
This ayanamsha was used by the great Indian astrologer Bangalore Venkata Raman (1912-1998). It is based on a statement by the medieval astronomer Bhaskara II (1184-1185), who assumed an ayanamsha of 11° in the year 1183 (according to Information given by Chandra Hari, unfortunately without giving his source). Raman himself mentioned the year 389 CE as year of zero ayanamsha in his book Hindu Predictive Astrology, pp. 378-379.
Although this ayanamsha is very close to the galactic ayanamsha of Gil Brand, Raman apparently did not think of the possibility to define the zodiac using the galactic centre.
Sources:

- Chandra Hari, "Ayanāṃśa", xa.yimg.com/kq/groups/26252194/1355927039/name/Ayanamsa.PDF

- B.V. Raman, Hindu Predictive Astrology, pp. 378-379.
7) Shri Yukteshwar Ayanamsha
This ayanamsha was allegedly recommended by Swami Shri Yukteshwar Giri (1855-1936). We have taken over its definition from Graham Dawson. However, the definition given by Yukteshwar himself in the introduction of his work The Holy Science is a confusing. According to his “astronomical reference books”, the ayanamsha on the spring equinox 1894 was 20°54’36”. At the same time he believed that this was the distance of the spring equinox from the star Revati, which he put at the initial point of Aries. However, this is wrong, because on that date, Revati was actually 18°24’ away from the vernal point. The error is explained from the fact that Yukteshwar used the zero ayanamsha year 499 CE and an inaccurate Suryasiddhantic precession rate of 360°/24’000 years = 54 arcsec/year. Moreover, Yukteshwar is wrong in assigning the above-mentioned ayanamsha value to the year 1894; in reality it applies to 1893.
Since Yukteshwar’s precession rate is wrong by 4” per year, astro.com cannot reproduce his horoscopes accurately for epochs far from 1900. In 2000, the difference amounts to 6’40”.
Although this ayanamsha differs only a few arc seconds from the galactic ayanamsha of Gil Brand, Yukteshwar obviously did not intend to define the zodiac using the galactic centre. He actually intended a Revati-oriented

ayanamsha, but committed the above-mentioned errors in his calculation.


Source:

- Swami Sri Yukteswar, The Holy Science, 1949, Yogoda Satsanga Society of India, p. xx.


8) JN Bhasin Ayanamsha
This ayanamsha was used by the Indian astrologer J.N. Bhasin (1908-1983).
6) Djwhal Khul Ayanamsha
This ayanamsha is based on the assumption that the Age of Aquarius will begin in the year 2117. This assumption is maintained by a theosophical society called Ageless Wisdom, and bases itself on a channelled message given in 1940 by a certain spiritual master called Djwhal Khul.
Graham Dawson commented it as follows (E-mail to Alois Treindl of 12 July 1999): ”The "Djwhal Khul" ayanamsha originates from information in an article in the Journal of Esoteric Psychology, Volume 12, No 2, pp91-95, Fall 1998-1999 publ. Seven Ray Institute). It is based on an inference that the Age of Aquarius starts in the year 2117. I decided to use the 1st of July simply to minimise the possible error given that an exact date is not given.”
Sources:

- Philipp Lindsay, “The Beginning of the Age of Aquarius: 2,117 A.D.”, http://esotericastrologer.org/newsletters/the-age-of-aquarius-ray-and-zodiac-cycles/

- Esoteric Psychology, Volume 12, No 2, pp91-95, Fall 1998-1999 publ. Seven Ray Institute

      1. Conclusions

We have found that there are basically five definitions, not counting the manifold variations:

1. the Babylonian zodiac with Spica at 29 Virgo or Aldebaran at 15 Taurus:

a) Fagan/Bradley b) refined with Aldebaran at 15 Tau, c) P. Huber, d) J.P. Britton

2. the Greek-Hindu-Arabic zodiac with the zero point between 10 and 20’ east of zeta Piscium:

a) Hipparchus, b) Ushâshashî, c) Sassanian, d) true Revati ayanamsha

3. the Hindu astrological zodiac with Spica at 0 Libra

a) Lahiri

4. ayanamshas based on the Kaliyuga year 3600 or the 23rd year of life of Aryabhata

5. galactic ayanamshas based on the position of the galactic centre or the galactic nodes (= intersection points of the galactic equator with the ecliptic)


1) is historically the oldest one, but we are not sure about its precise astronomical definition. It could have been Aldebaran at 15 Taurus and Antares at 15 Scorpio.


      1. In search of correct algorithms

A second problem in sidereal astrology – after the definition of the zero point – is the precession algorithm to be applied. We can think of five possibilities:


1) the traditional algorithm (implemented in Swiss Ephemeris as default mode)
In all software known to us, sidereal planetary positions are computed from the following equation:

sidereal_position = tropical_position – ayanamsha,

The ayanamhsa is computed from the ayanamsha(t0) at a starting date (e.g. 1 Jan 1900) and the speed of the vernal point, the so-called precession rate.


This algorithm is unfortunately too simple. At best, it can be considered an approximation. The precession of the equinox is not only a matter of ecliptical longitude, but is a more complex phenomenon. It has two components:
a) The soli-lunar precession: The combined gravitational pull of the Sun and the Moon on the equatorial bulge of the earth causes the earth to spin like a top. As a result of this movement, the vernal point moves around the ecliptic with a speed of about 50” per year. This cycle has a period of about 26000 years.
b) The planetary precession: The earth orbit itself is not fixed. The gravitational influence from the planets causes it to wobble. As a result, the obliquity of the ecliptic currently decreases by 47” per century, and this change has an influence on the position of the vernal point, too.
(Note, the rotation pole of the earth is very stable, it the equator keeps an almost constant angle relative to the ecliptic of a fixed date, with a change of only a couple of 0.06” cty-2.)
Because the ecliptic is not fixed, it is not completely correct to subtract an ayanamsha from the tropical position in order to get a sidereal position. Let us take, e.g., the Fagan/Bradley ayanamsha, which is defined by:

ayanamsha = 24°02’31.36” + d(t)

24°02’31.36” is the value of the ayanamsha on 1 Jan 1950. It is obviously measured on the ecliptic of 1950.



d(t) is the distance of the vernal point at epoch t from the position of the vernal point on 1 Jan 1950. However, the whole ayanamsha is subtracted from a planetary position which is referred to the ecliptic of the epoch t. This does not make sense. The ecliptic of the epoch t0 and the epoch t are not exactly the same plane.
As a result, objects that do not move sidereally, still do seem to move. If we compute its precise tropical position for several dates and then subtract the Fagan/Bradley ayanamsha for the same dates in order to get its sidereal position, these positions will all be different. This can be considerable over long periods of time:
Long-term ephemeris of some fictitious star near the ecliptic that has no proper motion:

Date Longitude Latitude

01.01.-12000 335°16'55.2211 0°55'48.9448

01.01.-11000 335°16'54.9139 0°47'55.3635

01.01.-10000 335°16'46.5976 0°40'31.4551

01.01.-9000 335°16'32.6822 0°33'40.6511

01.01.-8000 335°16'16.2249 0°27'23.8494

01.01.-7000 335°16' 0.1841 0°21'41.0200

01.01.-6000 335°15'46.8390 0°16'32.9298

01.01.-5000 335°15'37.4554 0°12' 1.7396

01.01.-4000 335°15'32.2252 0° 8'10.3657

01.01.-3000 335°15'30.4535 0° 5' 1.3407

01.01.-2000 335°15'30.9235 0° 2'35.9871

01.01.-1000 335°15'32.3268 0° 0'54.2786

01.01.0 335°15'33.6425 -0° 0' 4.7450

01.01.1000 335°15'34.3645 -0° 0'22.4060

01.01.2000 335°15'34.5249 -0° 0' 0.0196

01.01.3000 335°15'34.5216 0° 1' 1.1573

Long-term ephemeris of some fictitious star with high ecliptic latitude and no proper motion:



Date Longitude Latitude

01.01.-12000 25°48'34.9812 58°55'17.4484

01.01.-11000 25°33'30.5709 58°53'56.6536

01.01.-10000 25°18'18.1058 58°53'20.5302

01.01.-9000 25° 3' 9.2517 58°53'26.8693

01.01.-8000 24°48'12.6320 58°54'12.3747

01.01.-7000 24°33'33.6267 58°55'34.7330

01.01.-6000 24°19'16.3325 58°57'33.3978

01.01.-5000 24° 5'25.4844 59° 0' 8.8842

01.01.-4000 23°52' 6.9457 59° 3'21.4346

01.01.-3000 23°39'26.8689 59° 7'10.0515

01.01.-2000 23°27'30.5098 59°11'32.3495

01.01.-1000 23°16'21.6081 59°16'25.0618

01.01.0 23° 6' 2.6324 59°21'44.7241

01.01.1000 22°56'35.5649 59°27'28.1195

01.01.2000 22°48' 2.6254 59°33'32.3371

01.01.3000 22°40'26.4786 59°39'54.5816
Exactly the same kind of thing happens to sidereal planetary positions, if one calculates them in the traditional way. The “fixed zodiac” is not really fixed.
The wobbling of the ecliptic plane also influences ayanamshas that are referred to the nodes of the galactic equator with the ecliptic.
2) fixed-star-bound ecliptic (implemented in Swiss Ephemeris for some selected stars)
One could use a stellar object as an anchor for the sidereal zodiac, and make sure that a particular stellar object is always at a certain position on the ecliptic of date. E.g. one might want to have Spica always at 0 Libra or the Galactic Center always at 0 Sagittarius. There is nothing against this method from a geometrical point of view. But it has to be noted, that this system is not really fixed either, because it is still based on the moving ecliptic, and moreover the fixed stars have a small proper motion, as well. Thus, if Spica is assumed as a fixed point, then all other stars will move even faster, because the proper motion of Spica will be added to their own proper motions. Note, the Galactic Centre (Sgr A*) is not really fixed either, but has a small apparent motion that reflects the motion of the Sun about it.
This solution has been implemented for the following stars and fixed postions:

Spica/Citra at 180°

Revati (zeta Piscium) at 359°50’

Pushya (Asellus Australis) at 106° (PVR Narasimha Rao)

Mula (lambda Scorpionis) at 240° (Chandra Hari)

Galactic centre at golden section between 0° Sco and 0° Aqu (R. Gil Brand)

Polar longitude of galactic centre in the middle of nakshatra Mula (E. Wilhelm)
With Swiss Ephemeris versions before 2.05, the apparent position of the star relative to the mean ecliptic plane of date was used as the reference point of the zodiac. E.g. with the True Chitra ayanamsha, the star Chitra/Spica had the apparent position 180° exactly. However, the true position was slightly different. Since version 2.05, the star is always exactly at 180°, not only its apparent, but also its true position.
A special case are the ayanamshas that are based on the galactic node (the intersection of the galactic equator with the mean ecliptic of date). These ayanamshas include:

Galactic equator (IAU 1958)



Galactic equator true/modern

Galactic equator in middle of Mula

(Note, the Mardyks ayanamsha, although derived from the galactic equator, does not work like this. It is calculated using the method described above under 1).)


The node is calculated from the true position of the galactic pole, not the apparent one. As a result, if the position of the galactic pole is calculated with the ayanamsha that has the galactic node at 0° Sagittarius, then the true position of the pole is exactly at sidereal 150°, but its apparent position is slightly different from that.
3) projection onto the ecliptic of t0 (implemented in Swiss Ephemeris as an option)
Another possibility would be to project the planets onto the reference ecliptic of the ayanamsha – for Fagan/Bradley, e.g., this would be the ecliptic of 1950 – by a correct coordinate transformation and then subtract 24.042°, the initial value of the ayanamsha.
If we follow this method, the effect described above under 1) (traditional ayanamsha method) will not occur, and an object that has no proper motion will keep its position forever.
This method is geometrically more correct than the traditional one, but still has a problem. For, if we want to refer everything to the ecliptic of some initial date t0, we will have to choose that date very carefully. Its ecliptic ought to be of special importance. The ecliptic of 1950 or the one of 1900 are obviously meaningless and not suitable as a reference plane. So, how about some historical date on which the tropical and the sidereal zero point coincided? Although this may be considered as a kind of cosmic anniversary (the Sassanians did so), the ecliptic plane of that time does not have an “eternal” value. It is different from the ecliptic plane of the previous anniversary around the year 26000 BC, and it is also different from the ecliptic plane of the next cosmic anniversary around the year 26000 AD.
This algorithm is supported by the Swiss Ephemeris, too. However, it must not be used with the Fagan/Bradley definition or with other definitions that were calibrated with the traditional method of ayanamsha subtraction. It can be used for computations of the following kind:

a) Astronomers may want to calculate positions referred to a standard equinox like J2000, B1950, or B1900, or to any other equinox.

b) Astrologers may be interested in the calculation of precession-corrected transits. See explanations in the next chapter.

c) The algorithm can be applied to any user-defined sidereal mode, if the ecliptic of its reference date is considered to be astrologically significant.

d) The algorithm makes the problems of the traditional method visible. It shows the dimensions of the inherent inaccuracy of the traditional method. (Calculate some star position using the traditional method and compare it to the same star’s position if calculated using this method.)
For the planets and for centuries close to t0, the difference from the traditional procedure will be only a few arc seconds in longitude. Note that the Sun will have an ecliptical latitude of several arc minutes after a few centuries.
For the lunar nodes, the procedure is as follows:

Because the lunar nodes have to do with eclipses, they are actually points on the ecliptic of date, i.e. on the tropical zodiac. Therefore, we first compute the nodes as points on the ecliptic of date and then project them onto the sidereal zodiac. This procedure is very close to the traditional method of computing sidereal positions (a matter of arc seconds). However, the nodes will have a latitude of a couple of arc minutes.


For the axes and houses, we compute the points where the horizon or the house lines intersect with the sidereal plane of the zodiac, not with the ecliptic of date. Here, there are greater deviations from the traditional procedure. If t is 2000 years from t0, the difference between the ascendant positions might well be 1/2 degree.
4) The long-term mean Earth-Sun plane (not implemented in Swiss Ephemeris)
To avoid the problem of choice of a reference ecliptic, one could use a kind of ”average ecliptic”. The mechanism of the planetary precession mentioned above works in a similar way as the mechanism of the luni-solar precession. The motion of the earth orbit can be compared to a spinning top, with the earth mass equally distributed around the whole orbit. The other planets, especially Venus and Jupiter, cause it to move around an average position. But unfortunately, this “long-term mean Earth-Sun plane” is not really stable either, and therefore not suitable as a fixed reference frame.
The period of this cycle is about 75000 years. The angle between the long-term mean plane and the ecliptic of date is currently about 1°33’, but it varies considerably. (This cycle must not be confused with the period between two maxima of the ecliptic obliquity, which is about 40000 years and often mentioned in the context of planetary precession. This is the time it takes the vernal point to return to the node of the ecliptic (its rotation point), and therefore the oscillation period of the ecliptic obliquity.)
5) The solar system rotation plane (implemented in Swiss Ephemeris as an option)
The solar system as a whole has a rotation axis, too, and its equator is quite close to the ecliptic, with an inclination of 1°34’44” against the ecliptic of the year 2000. This plane is extremely stable and probably the only convincing candidate for a fixed zodiac plane.
This method avoids the problem of method 3). No particular ecliptic has to be chosen as a reference plane. The only remaining problem is the choice of the zero point.
It does not make much sense to use this algorithm for predefined sidereal modes. One can use this algorithm for user-defined ayanamshas.


      1. More benefits from our new sidereal algorithms: standard equinoxes and precession-corrected transits

Method no. 3, the transformation to the ecliptic of t0, opens two more possibilities:

You can compute positions referred to any equinox, 2000, 1950, 1900, or whatever you want. This is sometimes useful when Swiss Ephemeris data ought to be compared with astronomical data, which are often referred to a standard equinox.

There are predefined sidereal modes for these standard equinoxes:

18) J2000

19) J1900

20) B1950
Moreover, it is possible to compute precession-corrected transits or synastries with very high precision. An astrological transit is defined as the passage of a planet over the position in your birth chart. Usually, astrologers assume that tropical positions on the ecliptic of the transit time has to be compared with the positions on the tropical ecliptic of the birth date. But it has been argued by some people that a transit would have to be referred to the ecliptic of the birth date. With the new Swiss Ephemeris algorithm (method no. 3) it is possible to compute the positions of the transit planets referred to the ecliptic of the birth date, i.e. the so-called precession-corrected transits. This is more precise than just correcting for the precession in longitude (see details in the programmer's documentation swephprg.doc, ch. 8.1).



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