SWISS EPHEMERIS. 3

Computer ephemeris for developers of astrological software. 3

Introduction. 4

1. Licensing. 4

Swiss Ephemeris Free Edition. 4

Swiss Ephemeris Professional Edition. 5

2. Descripition of the ephemerides. 5

2.1 Planetary and lunar ephemerides. 5

2.1.1 Three ephemerides. 5

2.1.2.1 Swiss Ephemeris and the Astronomical Almanac. 7

2.1.2.2 Swiss Ephemeris and JPL Horizons System.. 8

2.1.2.3 Differences between Swiss Ephemeris 1.70 and older versions. 8

2.1.3 The details of coordinate transformation. 9

2.1.4 The Swiss Ephemeris compression mechanism.. 10

2.1.5 The extension of the time range to 10'800 years. 11

2.2 Lunar and Planetary Nodes and Apsides. 12

2.2.1 Mean Lunar Node and Mean Lunar Apogee ('Lilith', 'Black Moon') 12

2.2.2 The 'True' Node. 12

2.2.3 The Osculating Apogee (so-called 'True Lilith' or 'True Dark Moon') 13

2.2.4 The Interpolated or Natural Apogee and Perigee (Lilith and Priapus) 14

2.2.5 Planetary Nodes and Apsides. 14

2.3. Asteroids. 17

Asteroid ephemeris files. 17

How the asteroids were computed. 18

Ceres, Pallas, Juno, Vesta. 19

Chiron. 19

Pholus. 19

”Ceres” - an application program for asteroid astrology. 19

2.4 Comets. 19

2.5 Fixed stars and Galactic Center 20

2.6 ‚Hypothetical' bodies. 20

Uranian Planets (Hamburg Planets: Cupido, Hades, Zeus, Kronos, Apollon, Admetos, Vulkanus, Poseidon) 20

Transpluto (Isis) 20

Harrington. 21

Nibiru. 21

Vulcan. 21

Selena/White Moon. 21

Dr. Waldemath’s Black Moon. 21

The Planets X of Leverrier, Adams, Lowell and Pickering. 21

2.7 Sidereal Ephemerides. 22

Sidereal Calculations. 22

The problem of defining the zodiac. 22

The Babylonian tradition and the Fagan/Bradley ayanamsha. 23

The Hipparchan tradition. 23

The Spica/Citra tradition and the Lahiri ayanamsha. 25

The sidereal zodiac and the Galactic Center 25

Other ayanamshas. 26

Conclusions. 26

In search of correct algorithms. 27

More benefits from our new sidereal algorithms: standard equinoxes and precession-corrected transits. 29

3. Apparent versus true planetary positions. 29

4. Geocentric versus topocentric and heliocentric positions. 30

5. Eclipses, occultations, risings, settings, and other planetary phenomena. 30

6. AC, MC, Houses, Vertex. 31

6.1. House Systems. 31

6.2. Vertex, Antivertex, East Point and Equatorial Ascendant, etc. 33

6.3. House cusps beyond the polar circle. 33

6.3.1. Implementation in other calculation modules: 34

6.4. House position of a planet 34

6.5. Gauquelin sector position of a planet 35

7. D T (Delta T) 35

8. Programming Environment 37

9. Swiss Ephemeris Functions. 37

9.1 Swiss Ephemeris API 37

Other functions that may be useful 39

9.2 Placalc API 40

Appendix. 40

A. The gravity deflection for a planet passing behind the Sun. 40

B. The list of asteroids on the software CDROM... 41

#

# SWISS EPHEMERIS

# Computer ephemeris for developers of astrological software

© 1997 - 2003 by

Astrodienst AG

Dammstr. 23

Postfach (Station)

CH-8702 Zollikon / Zürich, Switzerland

Tel. +41-1-392 18 18

Fax +41-1-391 75 74

Email to devlopers swisseph-owner@astro.ch

Email to users mailing list swisseph@astro.ch

Authors: Dieter Koch and Dr. Alois Treindl

Editing history:

14-sep-97 Appendix A by Alois

15-sep-97 split docu, swephprg.doc now separate (programming interface)

16-sep-97 Dieter: absolute precision of JPL, position and speed transformations

24-sep-97 Dieter: main asteroids

27-sep-1997 Alois: restructured for better HTML conversion, added public function list

8-oct-1997 Dieter: chapter 4 (houses) added

28-nov-1997 Dieter: chapter 5 (delta t) added

20-Jan-1998 Dieter: chapter 3 (more than...) added, chapter 4 (houses) enlarged

14-Jul-98: Dieter: more about the precision of our asteroids

21-jul-98: Alois: houses in PLACALC and ASTROLOG

27-Jul-98: Dieter: True node chapter improved

2-Sep-98: Dieter: updated asteroid chapter

29-Nov-1998: Alois: added info on Public License and source code availability

4-dec-1998: Alois: updated asteroid file information

17-Dec-1998: Alois: Section 2.1.5 added: extended time range to 10'800 years

17-Dec-1998: Dieter: paragraphs on Chiron and Pholus ephemerides updated

12-Jan-1999: Dieter: paragraph on eclipses

19-Apr-99: Dieter: paragraph on eclipses and planetary phenomena

21-Jun-99: Dieter: chapter 2.27 on sidereal ephemerides

27-Jul-99: Dieter: chapter 2.27 on sidereal ephemerides completed

15-Feb-00: Dieter: many things for Version 1.52

11-Sep-00: Dieter: a few additions for version 1.61

24-Jul-01: Dieter: a few additions for version 1.62

5-jan-2002: Alois: house calculation added to swetest for version 1.63

26-feb-2002: Dieter: Gauquelin sectors for version 1.64

12-jun-2003: Alois: code revisions for compatibility with 64-bit compilers, version 1.65

10-jul-2003: Dieter: Morinus houses for Version 1.66

12-jul-2004: Dieter: documentation of Delta T algorithms implemented with version 1.64

7-feb-2005: Alois: added note about mean lunar elements, section 2.2.1

22-feb-2006: Dieter: added documentation for version 1.70, see section 2.1.2.1-3

17-jul-2007: Dieter: updated documentation of Krusinski-Pisa house system.

28-nov-2007: Dieter: documentation of new Delta T calculation for version 1.72, see section 7

Swiss Ephemeris Release history:

1.00 30-sept-1997

1.01 9-oct-1997 simplified houses() and sidtime() functions, Vertex added.

1.02 16-oct-1997 houses() changed again

1.03 28-oct-1997 minor fixes

1.04 8-Dec-1997 minor fixes

1.10 9-Jan-1998 bug fix, pushed to all licensees

1.11 12-Jan-98 minor fixes

1.20 21-Jan-98 NEW: topocentric planets and house positions

1.21 28-Jan-98 Delphi declarations and sample for Delphi 1.0

1.22 2-Feb-98 Asteroids moved to subdirectory. Swe_calc() finds them there.

1.23 11-Feb-98 two minor bug fixes.

1.24 7-Mar-1998 Documentation for Borland C++ Builder added

1.25 4-June-1998 sample for Borland Delphi-2 added

1.26 29-Nov-1998 source added, Placalc API added

1.30 17-Dec-1998 NEW:Time range extended to 10'800 years

1.31 12-Jan-1999 NEW: Eclipses

1.40 19-Apr-1999 NEW: planetary phenomena

1.50 27-Jul-1999 NEW: sidereal ephemerides

1.52 15-Feb-2000 Several NEW features, minor bug fixes

1.60 15-Feb-2000 Major release with many new features and some minor bug fixes

1.61 11-Sep-2000 Minor release, additions to se_rise_trans(), swe_houses(), ficitious planets

1.62 23-Jul-2001 Minor release, fictitious earth satellites, asteroid numbers > 55535 possible

1.63 5-Jan-2002 Minor release, house calculation added to swetest.c and swetest.exe

1.64 7-Apr-2002 NEW: occultations of planets, minor bug fixes, new Delta T algorithms

1.65 12-Jun-2003 Minor release, small code renovations for 64-bit compilation

1.66 10-Jul-2003 NEW: Morinus houses

1.67 31-Mar-2005 Minor release: Delta-T updated, minor bug fixes

1.70 2-Mar-2006 IAU resolutions up to 2005 implemented; "interpolated" lunar apsides

1.72 28-nov-2007 Delta T calculation according to Morrison/Stephenson 2007

## Introduction

Swiss Ephemeris is a function package for the computation of planetary positions. It includes the planets, the moon, the lunar nodes, the lunar apogees, the main asteroids, Chiron, Pholus, the fixed stars and several ”hypothetical” bodies. Hundreds of other minor planets are included as well. Ephemeris files all 10000 numbered asteroids are available for download or on CDROM.

The precision of the Swiss Ephemeris is very high. It is at least as accurate as the Astromical Almanac, the standard planetary and lunar tables astronomers refer to. Swiss Ephemeris will, as we hope, be able to keep abreast to the scientific advances in ephemeris computation for the coming decades. The expense will be small. In most cases an update of the data files will do.

The Swiss Ephemeris package consists of a DLL, a collection of ephemeris files and a few sample programs which demonstrate the use of the DLL and the Swiss Ephemeris graphical label. The ephemeris files contain compressed astronomical ephemerides (in equatorial rectangular coordinates referred to the mean equinox 2000 and the solar system barycenter). The DLL is mainly the code that reads these files and converts the raw data to positions as required in astrology (calculation of light-time, transformation to the geocenter and the true equinox of date, etc.).

Full C source code is included with the Swiss Ephemeris, so that not-Windows programmers can create a linkable or shared library in their environment and use it with their application.

# 1. Licensing

The Swiss Ephemeris is not a product for end users. It is a toolset for programmers to build into their astrological software.
The Swiss Ephemeris is available under two different licensing models:

### Swiss Ephemeris Free Edition

Under the Swiss Ephemeris Public License (SEPL) the Swiss Ephemeris is made available with complete source code to programmers free of charge. They can find the Swiss Ephemeris on Astrodienst's website http://www.astro.ch/swisseph. The ftp download area is ftp://www.astro.ch/pub/swisseph.

The Public License is applicable for two kinds developers:
- those who work only privately with the Swiss Ephemeris and make no software or services built upon the Swiss Ephemeris available to others;
- programmers who publish their software with full source code for free under an equivalent open source model.

The Public License is free. Those who want to receive the software and standard set of ephemeris files on a CDROM - instead of downloading it from the Internet - are charged a nominal fee of 39.90 Swiss Francs (approx. 28 USD). Support services are available for a fee if our time schedule and workload allows it.

### Swiss Ephemeris Professional Edition

This version is available to those programmers who do not qualify for the Public License. Examples are
- Programmers who develop commercial software for sale, or build commercial services upon it, for which a fee is charged;
- Programmers who offer free software but do not want to publish their own source code under an equivalent open source license.

The Swiss Ephemeris Professional Edition can be purchased from Astrodienst for a one-time fixed fee for each commercial programming project. The commercial license includes a CDROM with complete source code, pre-built DLLs and libraries, the standard set of ephemeris files and a four hours of support.

Professional license: The license fee for the first license is CHF 750.- (approx. USD 500), and CHF 400.- (approx. USD 270.-) for each additional license by the same licensee.

# 2. Descripition of the ephemerides

## 2.1 Planetary and lunar ephemerides

### 2.1.1 Three ephemerides

The Swiss Ephemeris package allows planetary and lunar computations from any of the following three astronomical ephemerides:

#### 1. The Swiss Ephemeris

The core part of Swiss Ephemeris is a compression of the JPL-Ephemeris DE406. Using a sophisticated mechanism, we succeeded in reducing JPL's 200 MB storage to only 18 MB. The agreement with DE406 is within 1 milli-arcsecond (0.001”). Since the inherent uncertainty of the JPL ephemeris for most of its time range is much greater, Swiss Ephemeris should be completely satisfying even for computations demanding very high accuracy.

The time range of the JPL ephemeris is 3000 BC to 3000 AD or 6000 years. We have extended this time range to 10'800 years, from 2 Jan 5401 BC to 31 Dec 5399. The details of this extension are described below in section 2.1.5.

Each Swiss Ephemeris file covers a period of 600 years; there are 18 planetary files, 18 Moon files and 18 main-asteroid files for the whole time range of 10'800 years.

The file names are as follows:

Planetary file | Moon file | Main asteroid file | Time range
seplm54.se1 | semom54.se1 | seasm54.se1 | 5401 BC – 4802 BC
seplm48.se1 | semom48.se1 | seasm48.se1 | 4801 BC – 4202 BC
seplm42.se1 | semom42.se1 | seasm42.se1 | 4201 BC – 3602 BC
seplm36.se1 | semom36.se1 | seasm36.se1 | 3601 BC – 3002 BC
seplm30.se1 | semom30.se1 | seasm30.se1 | 3001 BC – 2402 BC
seplm24.se1 | semom24.se1 | seasm24.se1 | 2401 BC – 1802 BC
seplm18.se1 | semom18.se1 | seasm18.se1 | 1801 BC – 1202 BC
seplm12.se1 | semom12.se1 | seasm12.se1 | 1201 BC – 602 BC
seplm06.se1 | semom06.se1 | seasm06.se1 | 601 BC – 2 BC
sepl_00.se1 | semo_00.se1 | seas_00.se1 | 1 BC – 599 AD
sepl_06.se1 | semo_06.se1 | seas_06.se1 | 600 AD – 1199 AD
sepl_12.se1 | semo_12.se1 | seas_12.se1 | 1200 AD – 1799 AD
sepl_18.se1 | semo_18.se1 | seas_18.se1 | 1800 AD – 2399 AD
sepl_24.se1 | semo_24.se1 | seas_24.se1 | 2400 AD – 2999 AD
sepl_30.se1 | semo_30.se1 | seas_30.se1 | 3000 AD – 3599 AD
sepl_36.se1 | semo_36.se1 | seas_36.se1 | 3600 AD – 4199 AD
sepl_42.se1 | semo_42.se1 | seas_42.se1 | 4200 AD – 4799 AD
sepl_48.se1 | semo_48.se1 | seas_48.se1 | 4800 AD – 5399 AD

The blue file names in the table indicate that a file is derived directly from the JPL data, whereas the other files are derived from Astrodienst's own numerical integration.

All Swiss Ephemeris files for Version 1 have the file suffix .se1.

A planetary file is about 500 kb, a lunar file 1300 kb.

Swiss Ephemeris files are distributed with the SWISSEPH package. They are also available for download from Astrodienst's web server.

The time range of the Swiss Ephemeris

Start date 2 Jan 5401 BC (jul. calendar) = JD -251291.5

End date 31 Dec 5399 AD (greg. Cal.) = JD 3693368.5

A note on year numbering:

There are two numbering systems for years before the year 1 AD. The historical numbering system (indicated with BC) has no year zero. Year 1 BC is followed directly by year 1 AD.

The astronomical year numbering system does have a year zero; years before the common era are indicated by negative year numbers. The sequence is year -1, year 0, year 1 AD.

The historical year 1 BC corresponds to astronomical year 0,

the historical your 2 BC corresponds to astronomical year -1, etc.

In this document and other documents related to the Swiss Ephemeris we use both systems of year numbering. When we write a negative year number, it is astronomical style; when we write BC, it is historical style.

#### 2. The Moshier Ephemeris

This is a semi-analytical approximation of the JPL planetary and lunar ephemerides, currently based on the DE404 ephemeris, developed by Steve Moshier. Its deviation from JPL is well below 1 arc second with the planets and a few arc seconds with the moon. No data files are required for this ephemeris, as all data are linked into the program code already.

This may be sufficient accuracy for most astrologers, since the moon moves 1 arc second in 2 time seconds and the sun 2.5 arc seconds in one minute.

However, if you work with the 'true' lunar node, which is derived from the lunar ephemeris, you will have to accept an error of about 1 arc minute. To get a position better than an arc second, you will have to spend 1.3 MB for the lunar ephemeris file 'semo_18.se1' of Swiss Ephemeris.

The advantage of the Moshier ephemeris is that it needs no disk storage. Its disadvantage besides the limited precision is reduced speed: it is about 10 times slower than JPL and Swiss Ephemeris.

The Moshier Ephemeris covers the interval from 3000 BC to 3000 AD.

#### 3. The full JPL Ephemeris

This is the full precision state-of-the-art ephemeris. It provides the highest precision and is the basis of the Astronomical Almanac.

JPL is the Jet Propulsion Laboratory of NASA in Pasadena, CA, USA (see http://www.jpl.nasa.gov ). Since many years this institute which is in charge of the planetary missions of NASA has been the source of the highest precision planetary ephemerides. The currently newest version of JPL ephemeris is the DE405/DE406. As most previous ephemerides, it has been created by Dr. Myles Standish.

According to a paper (see below) by Standish and others on DE403 (of which DE406 is only a slight refinement), the accuracy of this ephemeris can be partly estimated from its difference from DE200:

With the inner planets, Standish shows that within the period 1600 – 2160 there is a maximum difference of 0.1 – 0.2” which is mainly due to a mean motion error of DE200. This means that the absolute precision of DE406 is estimated significantly better than 0.1” over that period. However, for the period 1980 – 2000 the deviations between DE200 and DE406 are below 0.01” for all planets, and for this period the JPL integration has been fit to measurements by radar and laser interferometry, which are extremely precise.

With the outer planets, Standish's diagrams show that there are large differences of several ” around 1600, and he says that these deviations are due to the inherent uncertainty of extrapolating the orbits beyond the period of accurate observational data. The uncertainty of Pluto exceeds 1” before 1910 and after 2010, and increases rapidly in more remote past or future.

With the moon, there is an increasing difference of 0.9”/cty2 between 1750 and 2169. It comes mainly from errors in LE200 (Lunar Ephemeris).

The differences between DE200 and DE403 (DE406) can be summarized as follows:

1980 – 2000 all planets < 0.01”,

1600 – 1980 Sun – Jupiter a few 0.1”,

1900 – 1980 Saturn – Neptune a few 0.1”,

1600 – 1900 Saturn – Neptune a few ”,

1750 – 2169 Moon a few ”.

(see: E.M. Standish, X.X. Newhall, J.G. Williams, and W.M. Folkner, JPL Planetary and Lunar Ephemerides, DE403/LE403, JPL Interoffice Memorandum IOM 314.10-127, May 22, 1995, pp. 7f.)

The DE406 is a 200 Megabyte file available for download from the JPL server ftp://navigator.jpl.nasa.gov/ephem/export or on CD-ROM from the astronomical publisher Willman-Bell, see http://www.willbell.com.
Astrodienst has received permission from Dr. Standish to include the file on the Swiss Ephemeris CD-ROM.

There are several versions of the JPL Ephemeris. The version is indicated by the DE-number. A higher number stands for a later update. SWISSEPH should be able to read any JPL file from DE200 upwards.

The time range of this ephemeris (DE406) is:

start date 23 Feb 3001 BC (28 Jan greg.) = JD 625360.5,

end date 3 Mar 3000 AD = JD 2816848.5.

Swiss Ephemeris is based on the latest JPL file, and reproduces the full JPL precision with better than 1/1000 of an arc second, while requiring only 18 Mb instead of 200 Mb. Therefore for most applications it makes little sense to get the full JPL file, except to compare the precision. Precision comparison can also be done at the Astrodienst web server, because we have a test utility online which allows to compute planetary positions for any date with any of the three ephemerides.

For the extension of the JPL time range to 5400 BC - 5400 AD please see section 2.5.1 below.

### 2.1.2.1 Swiss Ephemeris and the Astronomical Almanac

The original JPL ephemeris gives barycentric equatorial Cartesian positions of the equinox 2000. Moshier provides heliocentric positions. The conversions to apparent geocentric ecliptical positions were done with the algorithms and constants of the Astronomical Almanac as described in the ”Explanatory Supplement to the Astronomical Almanac”. Using the DE200 data file, it is possible to reproduce the positions given by the Astronomical Almanac 1995, 1996, and 1997 down to the last digit. Editions of other years have not been checked.

Since 2003, the Astronomical Almanac has been using JPL ephemeris DE405, and since Astronomical Almanac 2006 all relevant resolutions of the International Astronomical Union (IAU) have been implemented. Versions 1.70 and higher of the Swiss Ephemeris also follow these resolutions and reproduce the sample calculation given by AA2006, page B61-B63, to the last digit, i.e. to better than 0.001 arc second. (To avoid confusion when checking this, it may be useful to know that the JD given on page B62 does not have enough digits in order to produce the correct final result.)

### 2.1.2.2 Swiss Ephemeris and JPL Horizons System

The Swiss Ephemeris, from Version 1.70 on, reproduces astrometric planetary positions of the JPL Horizons System precisely. However, there are small differences with the apparent positions. The reason is that the Horizons System still uses the old precession model IAU 1976 (Lieske) and nutation IAU 1980 (Wahr). This was confirmed by Jon Giorgini from JPL in an E-mail of 3 Feb. 2006.

### 2.1.2.3 Differences between Swiss Ephemeris 1.70 and older versions

With version 1.70, the standard algorithms recommended by the IAU resolutions up to 2005 were implemented. The following calculations have been added or changed with Swiss Ephemeris version 1.70:

- "Frame Bias" transformation from ICRS to J2000.

- Nutation IAU 2000B (could be switched to 2000A by the user)

- Precession model P03 (Capitaine/Wallace/Chapront 2003), including improvements in ecliptic obliquity and sidereal time that were achieved by this model

The differences between the old and new planetary positions in ecliptic longitude (arc seconds) are:

year new - old

2000 -0.00108

1995 0.02448

1980 0.05868

1970 0.10224

1950 0.15768

1900 0.30852

1800 0.58428

1799 -0.04644

1700 -0.07524

1500 -0.12636

1000 -0.25344

0 -0.53316

-1000 -0.85824

-2000 -1.40796

-3000 -3.33684

-4000 -10.64808

-5000 -32.68944

-5400 -49.15188

The discontinuity of the curve between 1800 and 1799 is explained by the fact that the old Swiss Ephemeris used different precession models for different time ranges: the model IAU 1976 by Lieske for 1800 - 2200, and the precession model by Williams 1994 outside of that time range.

Note: In the literature there are no indications concerning the long-term use of the precession model P03. It is said to be accurate to 0.00005 arc second for CE 1000-3000. However, there is no reason to trust alternative models (e.g. Bretagnon 2003) more for the whole period of the Swiss Ephemeris.

The differences between version 1.70 and older versions for the future are as follows:

2000 -0.00108

2010 -0.01620

2050 -0.14004

2100 -0.29448

2200 -0.61452

2201 0.05940

3000 0.27252

4000 0.48708

5000 0.47592

5400 0.40032

The discontinuity in 2200 has the same explanation as the one in 1800.

Jyotish / sidereal ephemerides:

The ephemeris changes by a constant value of about +0.3 arc second. This is because all our ayanamsas have the start epoch 1900, for which epoch precession was corrected by the same amount.

Fictitious planets / Bodies from the orbital elements file seorbel.txt:

There are changes of several 0.1 arcsec, depending on the epoch of the orbital elements and the correction of precession as can be seen in the tables above.

The differences for ecliptic obliquity in arc seconds (new - old) are:

5400 -1.71468

5000 -1.25244

4000 -0.63612

3000 -0.31788

2100 -0.06336

2000 -0.04212

1900 -0.02016

1800 0.01296

1700 0.04032

1600 0.06696

1500 0.09432

1000 0.22716

0 0.51444

-1000 1.07064

-2000 2.62908

-3000 6.68016

-4000 15.73272

-5000 33.54480

-5400 44.22924

The differences for sidereal time in seconds (new - old) are:

5400 -2.544

5000 -1.461

4000 -0.122

3000 0.126

2100 0.019

2000 0.001

1900 0.019

1000 0.126

0 -0.122

-500 -0.594

-1000 -1.461

-2000 -5.029

-3000 -12.355

-4000 -25.330

-5000 -46.175

-5400 -57.273

### 2.1.3 The details of coordinate transformation

The following steps are applied to the coordinates between reading from the ephemeris files and output to the user:

Correction for light-time. Since the planet's light needs time to reach the earth, it is never seen where it actually is, but where it was some time before. Light-time is a few minutes with the inner planets and a few hours with distant planets like Uranus, Neptune and Pluto. For the moon, the light-time correction is about one second. With planets, light-time correction may be of the order of 20” in position, with the moon 0.5”

Conversion from the solar system barycenter to the geocenter. Original JPL data are referred to the center of the gravity of the solar system. Apparent planetary positions are referred to an imaginary observer in the center of the earth.

Light deflection by the gravity of the sun. In gravitational fields of the sun and the planets light rays are bent. However, within the solar system only the sun has mass enough to deflect light significantly. Gravity deflection is greatest for distant planets and stars, but never greater than 1.8”. When a planet disappears behind the sun, the Explanatory Supplement recommends to set the deflection = 0. To avoid discontinuities, we chose another procedure. See Appendix A.

”Annual” aberration of light. The velocity of light is finite, and therefore the apparent direction of a moving body from a moving observer is never the same as it would be if both the planet and the observer stood still. For comparison: if you run through the rain, the rain seems to come from ahead even though it actually comes from above. Aberration may reach 20”.

Frame Bias (ICRS to J2000). The JPL ephemeris DE405/DE406 is referred to the International Celestial Reference System, a time-independent, non-rotating reference system which was recommended by the IAU in 1997. The planetary positions and speed vectors are rotated to the J2000 system. This transformation makes a difference of only about 0.0068 arc seconds in right ascension. (Implemented from Swiss Ephemeris 1.70 on)

Precession. The motion of the vernal equinox, which is an effect of the influences of solar, lunar, and planetary gravity on the equatorial bulge of the earth. Original JPL data are referred to the mean equinox of the year 2000. Apparent planetary positions are referred to the equinox of date. (From Swiss Ephemeris 1.70 on, we use the precession model P03. Older versions used precession model IAU 1976 (Lieske) for 2000 +- 200 years and the model Williams 1994 outside that time range.)

Nutation (true equinox of date). A short-period oscillation of the vernal equinox. It results from the moons gravity which acts on the equatorial bulge of the earth. The period of nutation is identical to the period of a cycle of the lunar node, i.e. 18.6 years. The difference between the true vernal point and the mean one is always below 17”. (From Swiss Ephemeris 1.70 on, we use the nutation model IAU 2000. Older versions used the nutation model IAU 1980 (Wahr).)

Transformation from equatorial to ecliptic coordinates.

For precise speed of the planets and the moon, we had to make a special effort, because the Explanatory Supplement does not give algorithms that apply the above-mentioned transformations to speed. Since this is not a trivial job, the easiest way would have been to compute three positions in a small interval and determine the speed from the derivation of the parabola going through them. However, double float calculation does not guarantee a precision better than 0.1”/day. Depending on the time difference between the positions, speed is either good near station or during fast motion. Derivation from more positions and higher order polynomials would not help either.

Therefore we worked out a way to apply directly all the transformations to the barycentric speeds that can be derived from JPL or Swiss Ephemeris. The speed precision is now better than 0.002” for all planets, and the computation is even much faster than it would have been from three positions. A position with speed takes in average only 1.66 times longer than one without speed, if a JPL or a Swiss Ephemeris position is computed. With Moshier, however, a computation with speed takes 2.5 times longer.

### 2.1.4 The Swiss Ephemeris compression mechanism

The idea behind our mechanism of ephemeris compression was developed by Dr. Peter Kammeyer of the U.S. Naval Observatory in 1987.

To make it simple, it works as follows:

The lunar and the inner planets ephemerides require by far the largest part of the storage. A more sophisticated mechanism is needed for them than for the outer planets. Instead of the positions we store the differences between JPL and the mean orbits of the analytical theory VSOP87. These differences are much smaller than the position values, wherefore they require less storage. They are stored in Chebyshew polynomials covering a period of an anomalistic cycle each. (By the way, this is the reason, why Swiss Ephemeris begins only with 4 Nov -3000 (instead of 23 Feb -3000 as JPL). This is the date, when the last of the inner planets Mars has its first perihelion after the start date of DE406.)

With the outer planets from Jupiter through Pluto we use a simpler mechanism. We rotate the positions provided by DE406 to the mean plane of the planet. This has the advantage that only two coordinates have high values, whereas the third one becomes very small. The data are stored in Chebyshew polynomials that cover a period of 4000 days each. (This is the reason, why Swiss Ephemeris stops in the year 2991 AD. 4000 days later is a date beyond 3000 AD)

### 2.1.5 The extension of the time range to 10'800 years

The JPL ephemeris covers the time range from 3000 BC to 3000 AD. While this is an excellent range covering all precisely known historical events, there are some types of astrological and historical research which would welcome a longer time range.

In December 1998 we have made an effort to extend the time range by our own numerical integration. The exact physical model used by Standish et. al. for the numerical integration of the DE406 ephemeris is not fully documented (at least we do not understand some details), so that we cannot use the same integration program as had been used at JPL for the creation of the original ephemeris.

The previous JPL ephemeris, the DE200, however has been reproduced by Steve Moshier over a very long time range with his integration program, which has been available to us. We have used this integration program with start vectors taken at the end points of the DE406 time range. To test our numerical integrator, we ran it upwards from 3000 BC to 600 BC for a period of 2400 years and compared its results with the DE406 ephemeris itself. The agreement is excellent for all planets except the Moon (see table below). The lunar orbit creates a problem because the physical model for the Moon's libration and the effect of the tides on lunar motion is quite different in the DE406 from the model in the DE200. We have varied the tidal coupling parameter (love number) and the longitudinal libration phase at the start epoch until we found the best agreement over the 2400 year test range between our integration and the JPL data. We could reproduce the Moon's motion over a the 2400 time range with a maximum error of 12 arcseconds. For most of this time range the agreement is better than 5 arcsec.

With these modified parameters we ran the integration backward in time from 3000 BC to 5400 BC. It is reasonable to assume that the integration errors in the backward integration are not significantly different from the integration errors in the upward integration.

planet | max. error arcsec | avg. error arcec
Mercury | 1.67 | 0.61
Venus | 0.14 | 0.03
Earth | 1.00 | 0.42
Mars | 0.21 | 0.06
Jupiter | 0.85 | 0.38
Saturn | 0.59 | 0.24
Uranus | 0.20 | 0.09
Neptune | 0.12 | 0.06
Pluto | 0.12 | 0.04
Moon | 12.2 | 2.53
Sun bary. | 6.3 | 0.39

The same procedure was applied at the upper end of the DE406 range, to cover an extension period from 3000 AD to 5400 AD. The maximum integration errors as determined in the test run 3000 AD down to 600 AD are given in the table below.

planet | max. error arcsec | avg. error arcsec
Mercury | 2.01 | 0.69
Venus | 0.06 | 0.02
Earth | 0.33 | 0.14
Mars | 1.69 | 0.82
Jupiter | 0.09 | 0.05
Saturn | 0.05 | 0.02
Uranus | 0.16 | 0.07
Neptune | 0.06 | 0.03
Pluto | 0.11 | 0.04
Moon | 8.89 | 3.43
Sun bary. | 0.61 | 0.05

We expect that in some time a full integration program modeled after the DE406 integrator will become available. At that time we will rerun our integration and report any significant differences.

## 2.2 Lunar and Planetary Nodes and Apsides

### 2.2.1 Mean Lunar Node and Mean Lunar Apogee ('Lilith', 'Black Moon')

Our mean node and mean apogee are computed from Moshier's lunar routine, which adjusts the ELP2000-85 lunar theory of Chapront-Touzé and Chapront to fit the JPL ephemeris on the interval from 3000 BC to 3000 AD. Its deviation from Chapront's mean node is 0 for J2000 and keeps below 20 arc seconds for the whole period. With the apogee, the deviation reaches 3 arc minutes at 3000 BC

Lilith or the Dark Moon is either the apogee (”aphelion”) of the lunar orbital ellipse or, for some people, its empty focal point. As seen from the geocenter, this makes no difference. Both of them are located in exactly the same direction. But the definition makes a difference for topocentric ephemerides.

Because the Earth is located in one of the two focuses of the ellipse, it has also been argued that the second focal point ought to be called ”Dark Earth” rather than ”Dark Moon” (Ernst Ott).

The opposite point, the lunar perigee or orbital point closest to the Earth, is also known as Priapus. However, if Lilith is understood as the second focus, an opposite point makes no sense, of course.

Originally, the term ”Dark Moon” was used for a hypothetical second body that was believed to move around the earth. There are still ephemerides around for such a body, but today’s observational skills and knowledge in celestial mechanics clearly exclude the possibility of such an object. As a result of confusion, the term ”Dark Moon” was later given to the lunar apogee. However, from the astrological symbolism of the lunar apogee, the expression ”Dark Moon” seems to be appropriate.

The Swiss Ephemeris apogee differs from the ephemeris given by Joëlle de Gravelaine in her book ”Lilith, der schwarze Mond” (Astrodata 1990). The difference reaches several arc minutes. The mean apogee (or perigee) moves along the mean lunar orbit which has an inclination of 5 degrees. Therefore it has to be projected on the ecliptic. With de Gravelaine's ephemeris, this has been forgotten and therefore the book contains a false ephemeris. As a result of this projection, we also provide an ecliptic latitude of the apogee, which will be of importance if you work with declinations.

There may be still another problem. The 'first' focal point does not coincide with the geocenter but with the barycenter of the earth-moon-system. The difference is about 4700 km. If one took this into account, it would result in a monthly oscillation of the Black Moon. If one defines it as the apogee, this oscillation would be about +/- 40 arc minutes. If one defines it as the second focus, the effect is much greater: +/- 6 degrees! However, we have neglected this effect.

[added by Alois 7-feb-2005, arising out of a discussion with Juan Revilla] The concept of 'mean lunar orbit' means that short term. e.g. monthly, fluctuations must not be taken into account. In the temporal average, the EMB coincides with the geocenter. Therefore, when mean elements are computed, it is correct only to consider the geocenter, not the Earth-Moon Barycenter.

In addition, computing topocentric positions of mean elements is also meaningless and should not be done.

### 2.2.2 The 'True' Node

The 'true' lunar node is usually considered to be the osculating node element of the momentary lunar orbit. I.e., the axis of the lunar nodes is the intersection line of the momentary orbital plane of the moon and the plane of the ecliptic. Or in other words, the nodes are the intersections of the two great circles representing the momentary apparent orbit of the moon and the ecliptic.

The nodes are considered to be important because they are connected with the eclipses. They are the meeting points of the sun and the moon. From this point of view, a more correct definition might be: The axis of the lunar nodes is the intersection line of the momentary orbital plane of the moon and the momentary orbital plane of the sun.

This makes a difference! Because of the monthly motion of the earth around the earth-moon barycenter, the sun is not exactly on the ecliptic but has a latitude, which, however, is always below an arc second. Therefore the momentary plane of the sun's motion is not identical with the ecliptic. For the true node, this would result in a difference in longitude of several arc seconds! However, Swiss Ephemeris computes the traditional version.

The advantage of the 'true' nodes against the mean ones is that when the moon is in exact conjunction with them, it has indeed a zero latitude. This is not true with the mean nodes.

However, in the strict sense of the word, even the ”true” nodes are true only twice a month, viz. at the times when the moon crosses the ecliptic. Positions given for the times in between those two points are just a hypothesis. They are founded on the idea that celestial orbits can be approximated by elliptical elements. This works well with the planets, but not with the moon, because its orbit is strongly perturbed by the sun. Another procedure, which might be more reasonable, would be to interpolate between the true node passages. The monthly oscillation of the node would be suppressed, and the maximum deviation from the conventional ”true” node would be about 20 arc minutes.

Precision of the true node:

The true node can be computed from all of our three ephemerides. If you want a precision of the order of at least one arc second, you have to choose either the JPL or the Swiss Ephemeris.

Maximum differences:

JPL-derived node – Swiss-Ephemeris-derived node ~ 0.1 arc second

JPL-derived node – Moshier-derived node ~ 70 arc seconds

(PLACALC was not better either. Its error was often > 1 arc minute.)

### 2.2.3 The Osculating Apogee (so-called 'True Lilith' or 'True Dark Moon')

The position of 'True Lilith' is given in the 'New International Ephemerides' (NIE, Editions St. Michel) and in Francis Santoni 'Ephemerides de la lune noire vraie 1910-2010' (Editions St. Michel, 1993). Both Ephemerides coincide precisely.

The relation of this point to the mean apogee is not exactly of the same kind as the relation between the true node and the mean node. Like the 'true' node, it can be considered as an osculating orbital element of the lunar motion. But there is an important difference: The apogee contains the concept of the ellipse, whereas the node can be defined without thinking of an ellipse. As has been shown above, the node can be derived from orbital planes or great circles, which is not possible with the apogee. Now ellipses are good as a description of planetary orbits, but not of the lunar orbit which is strongly perturbed by the gravity of the sun. The lunar orbit is far away from being an ellipse!

However, the osculating apogee is 'true' twice a month: when it is in exact conjunction with the moon, the moon is most distant from the earth; and when it is in exact opposition to the moon, the moon is closest to the earth. In between those two points, the value of the osculating apogee is pure imagination. The amplitude of the oscillation of the osculating apogee around the mean apogee is +/- 25 degrees, while the true apogee's deviation from the mean one never exceeds 5 degrees.

It has also to be mentioned, that there is a small difference between the NIE's 'true Lilith' and our osculating apogee, which results from an inaccuracy in NIE. The error reaches 20 arc minutes. According to Santoni, the point was calculated using 'les 58 premiers termes correctifs au perigée moyen' published by Chapront and Chapront-Touzé. And he adds: ”Nous constatons que même en utilisant ces 58 termes correctifs, l'erreur peut atteindre 0,5d!” (p. 13) We avoid this error, computing the orbital elements from the position and the speed vectors of the moon. (By the way, there is also an error of +/- 1 arc minute in NIE's true node. The reason is probably the same.)

Precision:

The osculating apogee can be computed from any one of the three ephemerides. If you want a precision of the order of at least one arc second, you have to choose either the JPL or the Swiss Ephemeris.

Maximum differences:

JPL-derived apogee – Swiss-Ephemeris-derived apogee ~ 0.9 arc second

JPL-derived apogee – Moshier-derived apogee ~ 360 arc seconds = 6 arc minutes!

There have been several other attempts to solve the problem of a 'true' apogee. They are not included in the SWISSEPH package. All of them work with a correction table.

They are listed in Santoni's 'Ephemerides de la lune noire vraie' mentioned above. With all of them, a value is added to the mean apogee depending on the angular distance of the sun from the mean apogee. There is something to this idea. The actual apogees that take place once a month differ from the mean apogee by never more than 5 degrees and seem to move along a regular curve that is a function of the elongation of the mean apogee.

However, this curve does not have exactly the shape of a sine, as is assumed by all of those correction tables. And most of them have an amplitude of more than 10 degrees, which is much too high. The most realistic solution so far was the one proposed by Henry Gouchon in ”Dictionnaire Astrologique”, Paris 1992, which is based on an amplitude of 5 degrees.

In ”Meridian” 1/95, Dieter Koch has published another table that pays regard to the fact that the motion does not precisely have the shape of a sine. (Unfortunately, ”Meridian” confused the labels of the columns of the apogee and the perigee.)

### 2.2.4 The Interpolated or Natural Apogee and Perigee (Lilith and Priapus)

As has been said above, the osculating lunar apogee (so-called "true Lilith") is a mathematical construct which assumes that the motion of the moon is a two-body problem. This solution is obviously too simplistic. Although Kepler ellipses are a good means to describe planetary orbits, they fail with the orbit of the moon, which is strongly perturbed by the gravitational pull of the sun. This solar perturbation results in gigantic monthly oscillations in the ephemeris of the osculating apsides (the amplitude is 30 degrees). These oscillations have to be considered an artifact of the insufficient model, they do not really show a motion of the apsides.

A more sensible solution seems to be an interpolation between the real passages of the moon through its apogees and perigees. It turns out that the motions of the lunar perigee and apogee form curves of different quality and the two points are usually not in opposition to each other. They are more or less opposite points only at times when the sun is in conjunction with one of them or squares them. The amplitude of their oscillation about the mean position is 5 degrees for the apogee and 25 degrees for the perigee.

This solution has been called the "interpolated" or "realistic" apogee and perigee by Dieter Koch in his publications. Juan Revilla prefers to call them the "natural" apogee and perigee. Today, Dieter Koch would prefer the designation "natural". The designation "interpolated" is a bit misleading, because it associates something that astrologers used to do everyday in old days, when they still used to work with printed ephemerides and house tables.

Note on implementation (from Swiss Ephemeris Version 1.70 on):

Conventional interpolation algorithms do not work well in the case of the lunar apsides. The supporting points are too far away from each other in order to provide a good interpolation, the error estimation is greater than 1 degree for the perigee. Therefore, Dieter chose a different solution. He derived an "interpolation method" from the analytical lunar theory which we have in the form of moshier's lunar ephemeris. This "interpolation method" has not only the advantage that it probably makes more sense, but also that the curve and its derivation are both continuous.

Literature (in German):

- Dieter Koch, "Was ist Lilith und welche Ephemeride ist richtig", in: Meridian 1/95

- Dieter Koch and Bernhard Rindgen, "Lilith und Priapus", Frankfurt/Main, 2000. (http://www.vdhb.de/Lilith_und_Priapus/lilith_und_priapus.html)

- Juan Revilla, "The Astronomical Variants of the Lunar Apogee - Black Moon", http://www.expreso.co.cr/centaurs/blackmoon/barycentric.html

### 2.2.5 Planetary Nodes and Apsides

Note to specialists in planetary nodes and apsides: If important publications or web sites concerning this topic have been forgotten in this summary, your clue will be appreciated.

Methods written in small characters are not supported by the Swiss Ephemeris software.

Differences between the Swiss Ephemeris and other ephemerides of the osculation nodes and apsides are probably due to different planetary ephemerides being used for their calculation. Small differences in the planetary ephemerides lead to much greater differences in nodes and apsides.

Definitions of the nodes

The lunar nodes indicate the intersection axis of the lunar orbital plane with the plane of the ecliptic. At the lunar nodes, the moon crosses the plane of the ecliptic and its ecliptic latitude changes sign. There are similar nodes for the planets, but their definition is more complicated. Planetary nodes can be defined in the following ways:

1) They can be understood as a direction or as an axis defined by the intersection line of two orbital planes. E.g., the nodes of Mars are defined by the intersection line of the orbital plane of Mars with the plane of the ecliptic (or the orbital plane of the Earth).

Note: However, as Michael Erlewine points out in his elaborate web page on this topic (http://thenewage.com/resources/articles/interface.html), planetary nodes could be defined for any couple of planets. E.g. there is also an intersection line for the two orbital planes of Mars and Saturn. Such non-ecliptic nodes have not been implemented in the Swiss Ephemeris.

Because such lines are, in principle, infinite, the heliocentric and the geocentric positions of the planetary nodes will be the same. There are astrologers that use such heliocentric planetary nodes in geocentric charts.

The ascending and the descending node will, in this case, be in precise opposition.

2) There is a second definition that leads to different geocentric ephemerides. The planetary nodes can be understood, not as an infinite axis, but as the two points at which a planetary orbit intersects with the ecliptic plane.

For the lunar nodes and heliocentric planetary nodes, this definition makes no difference from the definition 1). However, it does make a difference for geocentric planetary nodes, where, the nodal points on the planets orbit are transformed to the geocenter. The two points will not be in opposition anymore, or they will roughly be so with the outer planets. The advantage of these nodes is that when a planet is in conjunction with its node, then its ecliptic latitude will be zero. This is not true when a planet is in geocentric conjunction with its heliocentric node. (And neither is it always true for inner the planets, for Mercury and Venus.)

Note: There is another possibility, not implemented in the Swiss ephemeris: E.g., instead of considering the points of the Mars orbit that are located on the ecliptic plane, one might consider the points of the earth’s orbit that are located on the orbital plane of Mars. If one takes these points geocentrically, the ascending and the descending node, will always form an approximate square. This possibility has not been implemented in the Swiss Ephemeris.

3) Third, the planetary nodes could be defined as the intersection points of the plane defined by their momentary geocentric position and motion with the plane of the ecliptic. Here again, the ecliptic latitude would change sign at the moment when the planet were in conjunction with one of its nodes. This possibility has not been implemented in the Swiss Ephemeris.

Possible definitions for apsides and focal points

The lunar apsides - the lunar apogee and lunar perigee - have already been discussed further above. Similar points exist for the planets, as well, and they have been considered by astrologers. Also, as with the lunar apsides, there is a similar disagreement:

One may consider either the planetary apsides, i.e. the two points on a planetary orbit that are closest to the sun and most distant from the sun, resp. The former point is called the ”perihelion” and the latter one the ”aphelion”. For a geocentric chart, these points could be transformed from the heliocenter to the geocenter.

However, Bernard Fitzwalter and Raymond Henry prefer to use the second focal points of the planetary orbits. And they call them the ”black stars” or the ”black suns of the planets”. The heliocentric positions of these points are identical to the heliocentric positions of the aphelia, but geocentric positions are not identical, because the focal points are much closer to the sun than the aphelia. Most of them are even inside the Earth orbit.

The Swiss Ephemeris supports both points of view.

Special case: the Earth

The Earth is a special case. Instead of the motion of the Earth herself, the heliocentric motion of the Earth-Moon-Barycenter (EMB) is used to determine the osculating perihelion.

There is no node of the earth orbit itself.

There is an axis around which the earth's orbital plane slowly rotates due to planetary precession. The position points of this axis are not calculated by the Swiss Ephemeris.

Special case: the Sun

In addition to the Earth (EMB) apsides, our software computes so-to-say "apsides" of the solar orbit around the Earth, i.e. points on the orbit of the Sun where it is closest to and where it is farthest from the Earth. These points form an opposition and are used by some astrologers, e.g. by the Dutch astrologer George Bode or the Swiss astrologer Liduina Schmed. The ”perigee”, located at about 13 Capricorn, is called the "Black Sun", the other one, in Cancer, is called the ”Diamond”.

So, for a complete set of apsides, one might want to calculate them for the Sun and the Earth and all other planets.

Mean and osculating positions

There are serious problems about the ephemerides of planetary nodes and apsides. There are mean ones and osculating ones. Both are well-defined points in astronomy, but this does not necessarily mean that these definitions make sense for astrology. Mean points, on the one hand, are not true, i.e. if a planet is in precise conjunction with its mean node, this does not mean it be crossing the ecliptic plane exactly that moment. Osculating points, on the other hand, are based on the idealization of the planetary motions as two-body problems, where the gravity of the sun and a single planet is considered and all other influences neglected. There are no planetary nodes or apsides, at least today, that really deserve the label ”true”.

Mean positions

Mean nodes and apsides can be computed for the Moon, the Earth and the planets Mercury – Neptune. They are taken from the planetary theory VSOP87. Mean points can not be calculated for Pluto and the asteroids, because there is no planetary theory for them.

Although the Nasa has published mean elements for the planets Mercury – Pluto based on the JPL ephemeris DE200, we do not use them (so far), because their validity is limited to a 250 year period, because only linear rates are given, and because they are not based on a planetary theory. (http://ssd.jpl.nasa.gov/elem_planets.html, ”mean orbit solutions from a 250 yr. least squares fit of the DE 200 planetary ephemeris to a Keplerian orbit where each element is allowed to vary linearly with time”)

The differences between the DE200 and the VSOP87 mean elements are considerable, though:

Node Perihelion

Mercury 3” 4”

Venus 3” 107”

Earth - 35”

Mars 74” 4”

Jupiter 330” 1850”

Saturn 178” 1530”

Uranus 806” 6540”

Neptune 225” 11600” (>3 deg!)

Osculating nodes and apsides

Nodes and apsides can also be derived from the osculating orbital elements of a body, the parameters that define an ideal unperturbed elliptic (two-body) orbit for a given time. Celestial bodies would follow such orbits if perturbations were to cease instantaneously or if there were only two bodies (the sun and the planet) involved in the motion from now on and the motion were an ideal ellipse. This ideal assumption makes it obvious that it would be misleading to call such nodes or apsides "true". It is more appropriate to call them "osculating". Osculating nodes and apsides are "true" only at the precise moments, when the body passes through them, but for the times in between, they are a mere mathematical construct, nothing to do with the nature of an orbit.

I have tried to solve the problem by interpolating between actual passages of the planets through their nodes and apsides. However, this method works only well with Mercury. With all other planets, the supporting points are too far apart as to make an accurate interpolation possible.

There is another problem about heliocentric ellipses. E.g. Neptune's orbit has often two perihelia and two aphelia within one revolution. As a result, there is a wild oscillation of the osculating or "true" perihelion (and aphelion), which is not due to a transformation of the orbital ellipse but rather due to the deviation of the orbit from an elliptic shape. Neptune’s orbit cannot be adequately represented by a heliocentric ellipse. It makes no sense to use such points in astrology.

In actuality, Neptune’s orbit is not heliocentric at all. The double perihelia and aphelia are an effect of the motion of the sun about the solar system barycenter. This motion is much faster than the motion of Neptune, and Neptune cannot react on such fast displacements of the Sun. As a result, Neptune seems to move around the barycenter (or a mean sun) rather than around the real sun. In fact, Neptune's orbit around the barycenter is therefore closer to an ellipse than his orbit around the sun. The same statement is also true, though less obvious, for Saturn, Uranus and Pluto, but not for Jupiter and the inner planets.

This fundamental problem about osculating ellipses of planetary orbits does of course not only affect the apsides but also the nodes.

As a solution, it seems reasonable to compute the osculating elements of slow planets from their barycentric motions rather than from their heliocentric motions. This procedure makes sense especially for Neptune, but also for all planets beyond Jupiter. It comes closer to the mean apsides and nodes for planets that have such points defined. For Pluto and all transsaturnian asteroids, this solution may be used as a substitute for "mean" nodes and apsides. Note, however, that there are considerable differences between barycentric osculating and mean nodes and apsides for Saturn, Uranus, and Neptune. (A few degrees! But heliocentric ones are worse.)

Anyway, neither the heliocentric nor the barycentric ellipse is a perfect representation of the nature of a planetary orbit. So, astrologers, do not expect anything very reliable here either!

The best choice of method will probably be:

For Mercury – Neptune: mean nodes and apsides.

For asteroids that belong to the inner asteroid belt: osculating nodes/apsides from a heliocentric ellipse.

For Pluto and transjovian asteroids: osculating nodes/apsides from a barycentric ellipse.

The modes of the Swiss Ephemeris function swe_nod_aps()

The function swe_nod_aps() can be run in the following modes:

1) Mean positions are given for nodes and apsides of Sun, Moon, Earth, and the planets up to Neptune. Osculating positions are given with Pluto and all asteroids. This is the default mode.

2) Osculating positions are returned for nodes and apsides of all planets.

3) Same as 2), but for planets and asteroids beyond Jupiter, a barycentric ellipse is used.

4) Same as 1), but for Pluto and asteroids beyond Jupiter, a barycentric ellipse is used.

For the reasons given above, Dieter Koch would prefer method 4) as making most sense.

In all of these modes, the second focal point of the ellipse can be computed instead of the aphelion.

## 2.3. Asteroids

### Asteroid ephemeris files

The standard distribution of SWISSEPH includes the main asteroids Ceres, Pallas, Juno, Vesta, as well as Chiron, and Pholus. To compute them, you must have the main-asteroid ephemeris files in your ephemeris directory.

The names of these files are of the following form:

seas_18.se1 main asteroids for 600 years from 1800 - 2400

The size of such a file is about 200 kb.

All other asteroids are available in separate files. The names of additional asteroid files look like:

se00433.se1 the file of asteroid No. 433 (= Eros)

These files cover the period 3000 BC - 3000 AD.
A short version for the years 1500 – 2100 AD has the file name with an 's' imbedded, se00433s.se1.

The numerical integration of the all officiall numbered asteroids is an ongoing effort. In December 1998, 8000 asteroids were numbered, and their orbits computed by the devlopers of Swiss Ephemeris. In January 2001, the list of numbered asteroids has reached 20957, and is growing very fast.

Any asteroid can be called either with the JPL, the Swiss, or the Moshier ephemeris flag, and the results will be slightly different. The reason is that the solar position (which is needed for geocentric positions) will be taken from the ephemeris that has been specified.

Availability of asteroid files:

- all short files (over 20000) are available for free download at our ftp server ftp.astro.ch/pub/swisseph.
The purpose of providing this large number of files for download is that the user can pick those few asteroids he/she is interested in. It is not welcomed that anybody downloads more than 100 such files per day, due to bandwidth problems in our Internet link; the total volume of the short asteroid files is about 500 Mbyte.

- In the standard Swiss Ephemeris CDROM a set of the 200 most interesting asteroids is included, in both the short and long file version. The list of these is found in Appendix B.

- CDROMs with 10'000 short files per CDROM can be purchased from Astrodienst for 49.90 Swiss Francs. The ordering code is SWEAS0 for 1-9999, SWEAS1 for 10000-19999 and so on.

- The long asteroid files are available on a set of CDROMS, with 1000 asteroids per CDROM. The price per CDROM is 49.90 Swiss Francs, the ordering code is
SWEA0 for asteroids with numbers below 1000,
SWEA1 asteroids 1000 – 1999
SWEA2 asteroids 2000 – 2999
SWEA3 asteroids 3000 – 3999
SWEA4 asteroids 4000 – 4999
SWEA5 asteroids 5000 – 5999
SWEA6 asteroids 6000 – 6999
SWEA7 asteroids 7000 – 7999 and so on

Each asteroid CDROM must be individually made when it is ordered, this is the reason for the relatively high price per copy. The asteroid files may be copied and distributed freely under the Swiss Ephemeris Public License.

### How the asteroids were computed

To generate our asteroid ephemerides, we have modified the numerical integrator of Steve Moshier, which was capable to rebuild the DE200 JPL ephemeris.

Orbital elements, with a few exceptions, were taken from the asteroid database computed by E. Bowell, Lowell Observatory, Flagstaff, Arizona (astorb.dat). After the introduction of the JPL database mpcorb.dat, we still keep working with the Lowell data because Lowell elements are given with one more digit, which can be relevant for long-term integrations.

For a few close-Sun-approaching asteroids like 1566 Icarus, we use the elements of JPL’s DASTCOM database. Here, the Bowell elements are not good for long term integration because they do not account for relativity.

Our asteroid ephemerides take into account the gravitational perturbations of all planets, including the major asteroids Ceres, Pallas, and Vesta and also the Moon.

The mutual perturbations of Ceres, Pallas, and Vesta were included by iterative integration. The first run was done without mutual perturbations, the second one with the perturbing forces from the positions computed in the first run.

The precision of our integrator is very high. A test integration of the orbit of Mars with start date 2000 has shown a difference of only 0.0007 arc second from DE200 for the year 1600. We also compared our asteroid ephemerides with data from JPL’s on-line ephemeris system ”Horizons” which provides asteroid positions from 1600 on. Taking into account that Horizons does not consider the mutual perturbations of the major asteroids Ceres, Pallas and Vesta, the difference is never greater than a few 0.1 arcsec.

(However, the Swisseph asteroid ephemerides do consider those perturbations, which makes a difference of 10 arcsec for Ceres and 80 arcsec for Pallas. This means that our asteroid ephemerides are even better than the ones that JPL offers on the web.)

The accuracy limits are therefore not set by the algorithms of our program but by the inherent uncertainties in the orbital elements of the asteroids from which our integrator has to start.

Sources of errors are:

- Only some of the minor planets are known to better than an arc second for recent decades. (See also informations below on Ceres, Chiron, and Pholus.)

- Bowells elements do not consider relativistic effects, which leads to significant errors with long-term integrations of a few close-Sun-approaching orbits (except 1566, 2212, 3200, 5786, and 16960, for which we use JPL elements that do take into account relativity).

The orbits of some asteroids are extremely sensitive to perturbations by major planets. E.g. 1862 Apollo becomes chaotic before the year 1870 AD when he passes Venus within a distance which is only one and a half the distance from the Moon to the Earth. In this moment, the small uncertainty of the initial elements provided by the Bowell database grows, so to speak, ”into infinity”, so that it is impossible to determine the precise orbit prior to that date. Our integrator is able to detect such happenings and end the ephemeris generation to prevent our users working with meaningless data.

### Ceres, Pallas, Juno, Vesta

The orbital elements of the four main asteroids Ceres, Pallas, Juno, and Vesta are known very precisely, because these planets have been discovered almost 200 years ago and observed very often since. On the other hand, their orbits are not as well-determined as the ones of the main planets. We estimate that the precision of the main asteroid ephemerides is better than 1 arc second for the whole 20th century. The deviations from the Astronomical Almanac positions can reach 0.5” (AA 1985 – 1997). But the tables in AA are based on older computations, whereas we used recent orbital elements. (s. AA 1997, page L14)

MPC elements have a precision of five digits with mean anomaly, perihelion, node, and inclination and seven digits with eccentricity and semi-axis. For the four main asteroids, this implies an uncertainty of a few arc seconds in 1600 AD and a few arc minutes in 3000 BC.

### Chiron

Positions of Chiron can be well computed for the time between 700 AD and 4650 AD. As a result of close encounters with Saturn in Sept. 720 AD and in 4606 AD we cannot trace its orbit beyond this time range. Small uncertainties in today's orbital elements have chaotic effects before the year 700.

Do not rely on earlier Chiron ephemerides supplying a Chiron for Cesar's, Jesus', or Buddha's birth chart. They are completely meaningless.

### Pholus

Pholus is a minor planet with orbital characteristics that are similar to Chiron's. It was discovered in 1992. Pholus' orbital elements are not yet as well-established as Chiron's. Our ephemeris is reliable from 1500 AD through now. Outside the 20th century it will probably have to be corrected by several arc minutes during the coming years.

### ”Ceres” - an application program for asteroid astrology

Dieter Koch has written the application program Ceres which allows to compute all kinds of lists for asteroid astrology. E.g. you can generate a list of all your natal asteroids ordered by position in the zodiac. But the program does much more:

- natal positions, synastries/transits, composite charts, progressions, primary directions etc.

- geocentric, heliocentric, topocentric, house horoscopes

- lists sorted by position in zodiac, by asteroid name, by declination etc.

The program is on the asteroid short files CD-ROM and the standard Swiss Ephemeris CD-ROM.

## 2.4 Comets

The Swiss Ephemeris does not provide ephemerides of comets yet.

## 2.5 Fixed stars and Galactic Center

A database of fixed stars is included with Swiss Ephemeris. It contains about 800 stars, which can be computed with the swe_fixstar() function. The precision is about 0.001”.

Our data are based on the star catalogue of Steve Moshier. It can be easily extended if more stars are required.

The database was improved by Valentin Abramov, Tartu, Estonia. He reordered the stars by constellation, added some stars, many names and alternative spellings of names.

In Feb. 2006 (Version 1.70) the fixed stars file was updated with data from the SIMBAD database (http://simbad.u-strasbg.fr/Simbad).

## 2.6 ‚Hypothetical' bodies

We include some astrological factors in the ephemeris which have no astronomical basis – they have never been observed physically. As the purpose of the Swiss Ephemeris is astrology, we decided to drop our scientific view in this area and to be of service to those astrologers who use these ‘hypothetical’ planets and factors. Of course neither of our scientific sources, JPL or Steve Moshier, have anything to do with this part of the Swiss Ephemeris.

### Uranian Planets (Hamburg Planets: Cupido, Hades, Zeus, Kronos, Apollon, Admetos, Vulkanus, Poseidon)

There have been discussions whether these factors are to be called 'planets' or 'Transneptunian points'. However, their inventors, the German astrologers Witte and Sieggrün, considered them to be planets. And moreover they behave like planets in as far as they circle around the sun and obey its gravity.

On the other hand, if one looks at their orbital elements, it is obvious that these orbits are highly unrealistic. Some of them are perfect circles – something that does not exist in physical reality. The inclination of the orbits is zero, which is very improbable as well. The revised elements published by James Neely in Matrix Journal VII (1980) show small eccentricities for the four Witte planets, but they are still smaller than the eccentricity of Venus which has an almost circular orbit. This is again very improbable.

There are even more problems. An ephemeris computed with such elements describes an unperturbed motion, i.e. it takes into account only the Sun's gravity, not the gravitational influences of the other planets. This may result in an error of a degree within the 20th century, and greater errors for earlier centuries.

Also, note that none of the real transneptunian objects that have been discovered since 1992 can be identified with any of the Uranian planets.

SWISSEPH uses James Neely's revised orbital elements, because they agree better with the original position tables of Witte and Sieggrün.

The hypothetical planets can again be called with any of the three ephemeris flags. The solar position needed for geocentric positions will then be taken from the ephemeris specified.

### Transpluto (Isis)

This hypothetical planet was postulated 1946 by the French astronomer M.E. Sevin because of otherwise unexplainable gravitational perturbations in the orbits of Uranus and Neptune.

However, this theory has been superseded by other attempts during the following decades, which proceeded from better observational data. They resulted in bodies and orbits completely different from what astrologers know as 'Isis-Transpluto'. More recent studies have shown that the perturbation residuals in the orbits of Uranus and Neptune are too small to allow postulation of a new planet. They can, to a great extent, be explained by observational errors or by systematic errors in sky maps.

In telescope observations, no hint could be discovered that this planet actually existed. Rumors that claim the opposite are wrong. Moreover, all of the transneptunian bodies that have been discovered since 1992 are very different from Isis-Transpluto.

Even if Sevin's computation were correct, it could only provide a rough position. To rely on arc minutes would be illusory. Neptune was more than a degree away from its theoretical position predicted by Leverrier and Adams.

Moreover, Transpluto's position is computed from a simple Kepler ellipse, disregarding the perturbations by other planets' gravities. Moreover, Sevin gives no orbital inclination.

Though Sevin gives no inclination for his Transpluto, you will realize that there is a small ecliptic latitude in positions computed by SWISSEPH. This mainly results from the fact that its orbital elements are referred to epoch 5.10.1772 whereas the ecliptic changes position with time.

The elements used by SWISSEPH are taken from ”Die Sterne” 3/1952, p. 70. The article does not say which equinox they are referred to. Therefore, we fitted it to the Astron ephemeris which apparently uses the equinox of 1945 (which, however, is rather unusual!).

### Harrington

This is another attempt to predict Planet X's orbit and position from perturbations in the orbits of Uranus and Neptune. It was published in The Astronomical Journal 96(4), October 1988, p. 1476ff. Its precision is meant to be of the order of +/- 30 degrees. According to Harrington there is also the possibility that it is actually located in the opposite constellation, i.e. Taurus instead of Scorpio. The planet has a mean solar distance of about 100 AU and a period of about 1000 years.

### Nibiru

A highly speculative planet derived from the theory of Zecharia Sitchin, who is an expert in ancient Mesopotamian history and a ”paleoastronomer”. The elements have been supplied by Christian Woeltge, Hannover. This planet is interesting because of its bizarre orbit. It moves in clockwise direction and has a period of 3600 years. Its orbit is extremely eccentric. It has its perihelion within the asteroid belt, whereas its aphelion lies at about 12 times the mean distance of Pluto. In spite of its retrograde motion, it seems to move counterclockwise in recent centuries. The reason is that it is so slow that it does not even compensate the precession of the equinoxes.

### Vulcan

This is a ‘hypothetical’ planet inside the orbit of Mercury (not identical to the “Uranian” planet Vulkanus). Orbital elements according to L.H. Weston. Note that the speed of this “planet” does not agree with the Kepler laws. It is too fast by 10 degrees per year.

### Selena/White Moon

This is a ‘hypothetical’ second moon of the earth (or a third one, after the “Black Moon”) of obscure provenance. Many Russian astrologers use it. Its distance from the earth is more than 20 times the distance of the moon and it moves about the earth in 7 years. Its orbit is a perfect, unperturbed circle. Of course, the physical existence of such a body is not possible. The gravities of Sun, Earth, and Moon would strongly influence its orbit.

### Dr. Waldemath’s Black Moon

This is another hypothetical second moon of the earth, postulated by a Dr. Waldemath in the Monthly Wheather Review 1/1898. Its distance from the earth is 2.67 times the distance of the moon, its daily motion about 3 degrees. The orbital elements have been derived from Waldemath’s original data. There are significant differences from elements used in earlier versions of Solar Fire, due to different interpretations of the values given by Waldemath. After a discussion between Graham Dawson and Dieter Koch it has been agreed that the new solution is more likely to be correct. The new ephemeris does not agree with Delphine Jay’s ephemeris either, which is obviously inconsistent with Waldemath’s data.

This body has never been confirmed. With its 700-km diameter and an apparent diameter of 2.5 arc min, this should have been possible very soon after Waldemath’s publication.

### The Planets X of Leverrier, Adams, Lowell and Pickering

These are the hypothetical planets that have lead to the discovery of Neptune and Pluto or at least have been brought into connection with them. Their enormous deviations from true Neptune and Pluto may be interesting for astrologers who work with hypothetical bodies. E.g. Leverrier and Adams are good only around the 1840ies, the discovery epoch of Neptune. To check this, call the program swetest as follows:

$ swetest -p8 -dU -b1.1.1770 -n8 -s7305 -hel -fPTLBR -head

(i.e.: compute planet 8 (Neptune) - planet 'U' (Leverrier), from 1.1.1770, 8 times, in 7305-day-steps, heliocentrically. You can do this from the Internet web page swetest.htm. The output will be:)

Nep-Lev 01.01.1770 -18° 0'52.3811 0°55' 0.0332 -6.610753489

Nep-Lev 01.01.1790 -8°42' 9.1113 1°42'55.7192 -4.257690148

Nep-Lev 02.01.1810 -3°49'45.2014 1°35'12.0858 -2.488363869

Nep-Lev 02.01.1830 -1°38' 2.8076 0°35'57.0580 -2.112570665

Nep-Lev 02.01.1850 1°44'23.0943 -0°43'38.5357 -3.340858070

Nep-Lev 02.01.1870 9°17'34.4981 -1°39'24.1004 -5.513270186

Nep-Lev 02.01.1890 21°20'56.6250 -1°38'43.1479 -7.720578177

Nep-Lev 03.01.1910 36°27'56.1314 -0°41'59.4866 -9.265417529

(difference in (difference in (difference in

longitude) latitude) solar distance)

One can see that the error is in the range of 2 degrees between 1830 and 1850 and grows very fast beyond that period.

## 2.7 Sidereal Ephemerides

### 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!

For deeper studies of the problem, read:

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)

Thanks to Juan Ant. Revilla, San Jose, Costa Rica, who gave us this precious bibliographic hint.

### 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 which defines 0 Aries as the vernal point (the celestial point where the sun stands at the beginning of spring). The tropical zodiac has actually nothing to do with the star constellations of the same names. Based on these star constellations is the so-called sidereal zodiac, which is used in eastern astrology. Because the vernal point slowly moves through these constellations and completes its cycle once in 26000 years, tropical Aries moves through all sidereal signs, staying in each one for roughly 2160 years. Currently, the vernal point, and the beginning of tropical Aries, is located in sidereal 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.

While the definition of the tropical zodiac is clear and never questioned, sidereal astrology has quite some problems in defining its zodiac. There are many different definitions of the sidereal zodiac, and they differ by several degrees. At a 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 all of them to stem from only one sidereal zodiac. On the other hand, this does not mean that it be simple to give a precise definition of it.

Sidereal planetary positions are usually computed from an equation similar to:

sidereal_position = tropical_position – ayanamsha,

where ayanamsha is the difference between the two zodiacs and changes with time. (Sanskrit ayanâmsha means ”part of a path”; the Hindi form of the word is ayanamsa with an s instead of sh.) ”

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

The zero point of the sidereal zodiac is therefore traditionally defined by the equation

sidereal Aries = tropical Aries – ayanamsha

and by a date for which this equation is true.

The Swiss Ephemeris allows for about twenty different ayanamshas, but the user can also define his or her own ayanamsha.

### 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 from 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:

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) ayanamsha = -4°34´ +/- 20´, t0 = –100 Spica at 29 vi 14

The standard deviation was 1°08’

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) 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) 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”.)

Fagan and Bradley said that the difference between P. Huber’s zodiac and theirs was only 1’. But actually (if Mercier’s value for the Huber ayanamsha is correct) it was 7’.

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. Today, we know that the fixed stars have a proper motion, wherefore such definitions are not a good idea, if an absolute coordinate system independent on moving bodies is intended. But the Babylonians considered them to be fixed.

For this possibility, Swiss Ephemeris gives an Aldebaran ayanamsha:

14) 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”.

### 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. This definition goes back to the great Greek astronomer Hipparchus. How did he choose that point? Hipparchus said that the beginning of Aries rises when Spica sets. This statement was meant for a geographical latitude of 36°, the latitude of the island of Rhodos, which Hipparchus’ descriptions of rises and settings are referred to.

However, there seems to be more behind it. Mercier points out that according to Hipparchus’ star catalogue the stars alpha Arietis, beta Arietis, zeta Piscium, and Spica are located in 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 at the same time 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) ayanamsha = -9°20’ 27 June –128 (jd 1674484) zePsc 29pi33’49” Hipparchos

(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 otherwise correct.)

If we used the explicit statement of Hipparchus 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 Sûrya Siddânta, 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 Citra) is given the longitude 30° Virgo. This is a contradiction, either Spica or Revatî must be considered wrong.

Moreover, if the precession model of the Sûrya Siddânta 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 Ârya Siddânta, 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 Ushâshashî ayanamsha which is still in use. It differs from the Hipparchan one by only 11 arc minutes.

4) ayanamsha = 18°39’39.46 1 Jan. 1900 Ushâshashî

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. Did astronomers or astrologers react on that event? They did! Under the Sassanian ruler Khusrau Anûshirwân, in the year 556, 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 one 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 model used in that epoch 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 today’s 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, and a very precise one (this is not explicitly stated by Mercier, however):

16) 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 Ushâshashî ayanamsha.

(Besides the synchronicity of the Sun-Jupiter conjunction and the coincidence of the two zodiacs, it is funny to note that the cosmos helped the inaccuracy of ancient astronomy by ”rounding” the position of the star zePsc to precisely 10 arc minutes east of the zero point! All Ptolemean star positions were rounded to 10 arc minutes.)

### The Spica/Citra tradition and the Lahiri ayanamsha

After the Babylonian and the Greek definitions of the zero point, there is a third one which fixes the star Spica (in Sanskrit Citra) at 0 Libra. This definition is today the most common one in Hindu astrology. It is named after N.C. Lahiri:

1) ayanamsha = 22°27’37.7 1 Jan. 1900 Lahiri, Spica at 0 Libra

However, and this is very confusing, the same definition seems to have used in Babylon and Greece as well. While the information given in the chapters about the Babylonian and the Hipparchan traditions are based on analyses of old star catalogues and planetary theories, the consideration of 22 ancient Greek and 5 Babylonian birth charts leads to different conclusions: they fit better with Spica at 0 Libra, than with Aldebaran at 15 Taurus and Spica at 29 Virgo (Fagan/Bradley)! See Nick Kollerstrom, in Culture and Cosmos in 1997 (Vol. 1, n.2).

Were there a different zodiacs for astronomical and astrological purposes? May be, Spica was chosen as an anchor star for reasons of more convenience, but it was not originally meant to be located precisely at 0 Libra.

The definition by Spica at 0 Libra would be so simple, clear, and convincing that, had it really been intended, it would probably never have been given up and the other definitions would never have been taken into consideration.

### The sidereal zodiac and the Galactic Center

As said before, there is a very precise definition for the tropical ecliptic. It starts at one of the two intersection points of the ecliptic and the celestial equator. Similarly, we have a very precise definition for the house circle which is said to be an analogy of the zodiac. It starts at one of the two intersection points of the ecliptic and the local horizon. Unfortunately there is no such definition for the sidereal zodiac. Or can a fixed star like Spica be important enough to play the role of an anchor star?

One could try to make the sidereal zero point agree with the Galactic Center. The Swiss astrologer Bruno Huber has pointed out that the Galactic Center enters a new tropical sign always around the same time when the vernal point enters the next sidereal sign. Around the time, when the vernal point will go into Aquarius, the Galactic Center will change from Sagittarius to 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 add an ayanamsha fixed at the Galactic Center:

17) Galactic Center at 0 Sagittarius

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.

### Other ayanamshas

There are a few more ayanamshas, whose provenance is not known to us. They were given to us by Graham Dawson (”Solar Fire”), who took them over from Robert Hand’s Program ”Nova”:

2) De Luce

3) Raman

5) Krishnamurti

David Cochrane adds

7) Yukteshvar

8) JN Bhasin

Graham Dawson adds the following one:

6) Djwhal Khul

He comments it as follows: ”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.”

### Conclusions

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

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

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

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

a) Hipparchus, b) Ushâshashî, c) Sassanian

3. the Greek-Hindu astrological zodiac with Spica at 0 Libra

a) Lahiri

The differences are:

between 1) and 3) is about 1 degree

between 1) and 2) is about 5 degrees

between 2) and 3) is about 4 degrees

It is obvious that all of them stem from the same origin, but it is difficult to say which one should be preferred for sidereal astrology.

1) is historically the oldest one, but we are not sure about its precise astronomical definition. Aldebaran at 15 Tau might be one.

3) has the most striking reference point, the bright star Spica at 0 Libra. But this definition is so clear and simple that, had it really been intended by the inventors of the sidereal ecliptic, it would certainly not have been forgotten or given up by the Greek and Arabic tradition.

2) is the only definition independent on a star – especially, if we take the Sassanian version. This is an advantage, because all stars have a proper motion and cannot really define a fixed coordinate system. Also, it is the only ayanamsha for which there is historical evidence that it was observed and recalibrated at the time when it was 0.

On the other hand, the point 10’ East of zePsc has no astronomical significance at all, and the great difference between this zero point and the Babylonian one raises the question: Did Hipparchus’ definition result from a misunderstanding of the Babylonian definition, or was it an attempt to improve the Babylonian zodiac?

### 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 an equation mentioned before:

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 as an approximation. The precession of the equinoxes 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”. This cycle lasts 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 movement has an influence on the position of the vernal point, as well. (This has nothing to do with the precessional motion of the earth rotation axis; the equator holds an almost stable angle against the ecliptic of a fixed date, e.g. 1900, with a change of only a couple of 0.06” cty-2).

Because the ecliptic is not fixed, it cannot be correct just 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’... 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. This value is also measured on the ecliptic of 1950. But the whole ayanamsha is subtracted from a planetary position which is referred to the ecliptic of the epoch t. This does not make sense.

As an effect of this procedure, objects that do not move sidereally, e.g. the Galactic Center, 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 slightly different:

Date Longitude Latitude

01.01.-5000 2 sag 07'57.7237 -4°41'34.7123 (without aberration)

01.01.-4000 2 sag 07'32.9817 -4°49' 4.8880

01.01.-3000 2 sag 07'14.2044 -4°56'47.7013

01.01.-2000 2 sag 07' 0.4590 -5° 4'39.5863

01.01.-1000 2 sag 06'50.7229 -5°12'36.9917

01.01.0 2 sag 06'44.2492 -5°20'36.4081

01.01.1000 2 sag 06'40.7813 -5°28'34.3906

01.01.2000 2 sag 06'40.5661 -5°36'27.5619

01.01.3000 2 sag 06'44.1743 -5°44'12.6886

01.01.4000 2 sag 06'52.1927 -5°51'46.6231

01.01.5000 2 sag 07' 4.8942 -5°59' 6.3665

The effect can be much greater for bodies with greater ecliptical latitude.

Exactly the same kind of thing happens to sidereal planetary positions, if one calculates them in the traditional way. It is only because planets move that we are not aware of it.

The traditional method of computing sidereal positions is geometrically not sound and can never achieve the same degree of accuracy as tropical astrology is used to.

2) fixed-star-bound ecliptic (not implemented in Swiss Ephemeris)

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.

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 position of the galactic center will always be the same (2 sag 06'40.4915 -5°36' 4.0652 (without aberration))

This method is geometrically sounder than the traditional one, but still it has a problem. For, if we want everything referred to the ecliptic of a fixed 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. And how about that 18 March 564, 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 the Sassanian ayanamsha or 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.

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, we might watch out for a kind of ”average ecliptic”. As a matter of fact, there are some possibilities in this direction. The mechanism of the planetary precession mentioned above works in a similar way as the mechanism of the luni-solar precession. The movement 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 at the moment about 1°33’, but it changes 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.

This algorithm must not be applied to any of the predefined sidereal modes, except the Sassanian one. You can use this algorithm, if you want to research on a better-founded sidereal astrology, experiment with your own sidereal mode, and calibrate it as you like.

### 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).

# 3. Apparent versus true planetary positions

The Swiss ephemeris provides the calculation of apparent or true planetary positions. Traditional astrology works with apparent positions. ”Apparent” means that the position where we see the planet is used, not the one where it actually is. Because the light's speed is finite, a planet is never seen exactly where it is. (see above, 2.1.3 ”The details of coordinate transformation”, light-time and aberration) Astronomers therefore make a difference between apparent and true positions. However, this effect is below 1 arc minute.

Most astrological ephemerides provide apparent positions. However, this may be wrong. The use of apparent positions presupposes that astrological effects can be derived from one of the four fundamental forces of physics, which is impossible. Also, many astrologers think that astrological ”effects” are a synchronistic phenomenon (the ones familiar with physics may refer to the Bell theorem). For such reasons, it might be more convincing to work with true positions.

Moreover, the Swiss Ephemeris supports so-called astrometric positions, which are used by astronomers when they measure positions of celestial bodies with respect to fixed stars. These calculations are of no use for astrology, though.

# 4. Geocentric versus topocentric and heliocentric positions

More precisely speaking, common ephemerides tell us the position where we would see a planet if we stood in the center of the earth and could see the sky. But it has often been argued that a planet’s position ought to be referred to the geographic position of the observer (or the birth place). This can make a difference of several arc seconds with the planets and even more than a degree with the moon! Such a position referred to the birth place is called the topocentric planetary position. The observation of transits over the moon might help to find out whether or not this method works better.

For very precise topocentric calculations, the Swiss Ephemeris not only requires the geographic position, but also its altitude above sea. An altitude of 3000 m (e.g. Mexico City) may make a difference of more than 1 arc second with the moon. With other bodies, this effect is of the amount of a 0.01”. The altitudes are referred to the approximate earth ellipsoid. Local irregularities of the geoid have been neglected.

Our topocentric lunar positions differ from the NASA positions (s. the Horizons Online Ephemeris System http://ssd.jpl.nasa.gov) by 0.2 - 0.3 arc sec. This corresponds to a geographic displacement by a few 100 m and is about the best accuracy possible. In the documentation of the Horizons System, it is written that: "The Earth is assumed to be a rigid body. ... These Earth-model approximations result in topocentric station location errors, with respect to the reference ellipsoid, of less than 500 meters."

The Swiss ephemeris also allows the computation of apparent or true topocentric positions.

With the lunar nodes and apogees, Swiss Ephemeris does not make a difference between topocentric and geocentric positions. There are manyfold ways to define these points topocentrically. The simplest one is to understand them as axes rather than points somewhere in space. In this case, the geocentric and the topocentric positions are identical, because an axis is an infinite line that always points to the same direction, not depending on the observer's position.

Moreover, the Swiss Ephemeris supports the calculation of heliocentric and barycentric planetary positions. Heliocentric positions are positions as seen from the center of the sun rather than from the center of the earth. Barycentric ones are positions as seen from the center of the solar system, which is always close to but not identical to the center of the sun.

# 5. Eclipses, occultations, risings, settings, and other planetary phenomena

The Swiss Ephemeris also includes functions for many calculations concerning solar and lunar eclipses. You can:

- search for eclipses or occultations, globally or for a given geographical position

- compute global or local circumstances of eclipses or occultations

- compute the geographical position where an eclipse is central

Moreover, you can compute for all planets and asteroids:

- risings and settings (also for stars)

- midheaven and lower heaven transits (also for stars)

- height of a body above the horizon (refracted and true, also for stars)

- phase angle

- phase (illumined fraction of disc)

- elongation (angular distance between a planet and the sun)

- apparent diameter of a planetary disc

- apparent magnitude.

# 6. AC, MC, Houses, Vertex

The Swiss Ephemeris package also includes a function that computes the Ascendant, the MC, the houses, the Vertex, and the Equatorial Ascendant (sometimes called "East Point").

## 6.1. House Systems

The following house methods have been implemented so far:

#### 6.1.1. Placidus

This system is named after the Italian monk Placidus de Titis (1590-1668). The cusps are defined by divisions of semidiurnal and seminocturnal arcs. The 11 th cusp is the point on the ecliptic that has completed 2/3 of its semidiurnal arc, the 12 th cusp the point that has completed 1/3 of it. The 2 nd cusp has completed 2/3 of its seminocturnal arc, and the 3 rd cusp 1/3.

#### 6.1.2. Koch/GOH

This system is called after the German astrologer Walter Koch (1895-1970). Actually it was invented by Fiedrich Zanzinger and Heinz Specht, but it was made known by Walter Koch. In German-speaking countries, it is also called the "Geburtsorthäusersystem" (GOHS), i.e. the "Birth place house system". Walter Koch thought that this system was more related to the birth place than other systems, because all house cusps are computed with the "polar height of the birth place", which has the same value as the geographic latitude.

This argumentation shows actually a poor understanding of celestial geometry. With the Koch system, the house cusps are actually defined by horizon lines at different times. To calculate the cusps 11 and 12, one can take the time it took the MC degree to move from the horizon to the culmination, divide this time into three and see what ecliptic degree was on the horizon at the thirds. There is no reason why this procedure should be more related to the birth place than other house methods.

#### 6.1.3. Regiomontanus

Named after the Johannes Müller (called "Regiomontanus", because he stemmed from Königsberg) (1436-1476).

The equator is divided into 12 equal parts and great circles are drawn through these divisions and the north and south points on the horizon. The intersection points of these circles with the ecliptic are the house cusps.

#### 6.1.4. Campanus

Named after Giovanni di Campani (1233-1296).

The vertical great circle from east to west is divided into 12 equal parts and great circles are drawn through these divisions and the north and south points on the horizon. The intersection points of these circles with the ecliptic are the house cusps.

#### 6.1.5. Equal System

The zodiac is divided into 12 houses of 30 degrees each starting from the Ascendant.

#### 6.1.6 Vehlow-equal System

Equal houses with the Ascendant positioned in the middle of the first house.

#### 6.1.7. Axial Rotation System

Also called the "Meridian house system". The equator is partitioned into 12 equal parts starting from the ARMC. Then the ecliptic points are computed that have these divisions as their right ascension. Note: The ascendant is different from the 1st house cusp.

#### 6.1.8. The Morinus System

The equator is divided into 12 equal parts starting from the ARMC. The resulting 12 points on the equator are transformed into ecliptic coordinates. Note: The Ascendant is different from the 1st cusp, and the MC is different from the 10th cusp.

#### 6.1.9. Horizontal system

The house cusps are defined by division of the horizon into 12 directions. The first house cusp is not identical with the Ascendant but is located precisely in the east.

#### 6.1.10. The Polich-Page (“topocentric”) system

This system was introduced in 1961 by Wendel Polich and A.P. Nelson Page. Its construction is rather abstract: The tangens of the polar height of the 11th house is the tangens of the geo. latitude divided by 3. (2/3 of it are taken for the 12th house cusp.) The philosophical reasons for this algorithm are obscure. Nor is this house system more “topocentric” (i.e. birth-place-related) than any other house system. (c.f. the misunderstanding with the “birth place system”.) The “topocentric” house cusps are close to Placidus house cusps except for high geographical latitudes. It also works for latitudes beyond the polar circles, wherefore some consider it to be an improvement of the Placidus system. However, the striking philosophical idea behind Placidus, i.e. the division of diurnal and nocturnal arcs of points of the zodiac, is completely abandoned.

#### 6.1.11. Alcabitus system

A method of house division named for Alcabitius, an Arab, who is supposed to have lived in the 1st century A.D. Others connect it with an Arabic system that dates from the 10th century at the earliest, and the name of the astrologer-astronomer with the 12th century Alchabitus. This system is the one used in the few remaining examples of ancient Greek horoscopes.

The MC and ASC are respectively the 10th- and 1st- house cusps. The remaining cusps are determined by the trisection of the semidiurnal and seminocturnal arcs of the ascendant, measured on the celestial equator. The houses are formed by the great circles that pass through these trisection points on the equator and the North and South points of the Horizon.

#### 6.1.12. Gauquelin sectors

This is the “house” system used by the Gauquelins and their epigones and critics in statistical investigations in Astrology. Basically, it is identical with the Placidus house system, i.e. diurnal and nocturnal arcs of ecliptic points or planets are subdivided. There are a couple of differences, though.

- Up to 36 “sectors” (or house cusps) are used instead of 12 houses.

- The sectors are counted in clockwise direction.

- There are so-called plus (+) and minus (–) zones. The plus zones are the sectors that turned out to be significant in statistical investigations, e.g. many top sportsmen turned out to have their Mars in a plus zone. The plus sectors are the sectors 36 – 3, 9 – 12, 19 – 21, 28 – 30.

- More sophisticated algorithms are used to calculate the exact house position of a planet (see chapters 6.4 and 6.5 on house positions and Gauquelin sector positions of planets).

#### 6.1.13. Krusinski system, also known as Amphora/Pisa system

This house system was “discovered” in 1994/1995 by two persons independently, by the Pole Bogdan Krusinski and the Czech Milan Pisa. Pisa gave it the name “Amphora house system”.

Krusinski defines the house system as “… based on the great circle passing through ascendant and zenith. This circle is divided into 12 equal parts (1st cusp is ascendant, 10th cusp is zenith), then the resulting points are projected onto the ecliptic through meridian circles. The house cusps in space are half-circles perpendicular to the equator and running from the north to the south celestial pole through the resulting cusp points on the house circle. The points where they cross the ecliptic mark the ecliptic house cusps.” (Krusinski, e-mail to Dieter Koch)

It may seem hard to believe that a Pole and a Czech could have discovered the same house system at almost the same time. However, this is what happened, and it is not really a miracle, considering the fact that the number of possible house constructions is quite limited. Some more details are given here, in order to refute wrong accusations from the Czech side against Krusinski of having “stolen” the house system.

Out of the documents that Milan Pisa sent to Dieter Koch, the following are to be mentioned: Private correspondence from 1994 and 1995 on the house system between Pisa and German astrologers Böer and Schubert-Weller, two type-written (apparently unpublished) treatises in German on the house system dated from 1994. A printed booklet of 16 pages in Czech from 1997 on the theory of the house system (“Algoritmy noveho systemu astrologickych domu”). House tables computed by Michael Cifka for the geographical latitude of Prague, copyrighted from 1996. The house system was included in the Czech astrology software Astrolog v. 3.2 (APAS) in 1998. Pisa’s first publication on the house system happened in spring 1997 in “Konstelace“ No. 22, the periodical of the Czech Astrological Society.

Bogdan Krusinski first published the house system at an astrological congress in Poland, the “"XIV Szkola Astrologii Humanistycznej" held by Dr Leszek Weres, which took place between 15.08.1995 and 28.08.1995 in Pogorzelica at cost of the Baltic Sea.” Since then Krusinski has distributed printed house tables for the Polish geographical latitudes 49-55° and floppy disks with house tables for latitudes 0-90°. In 1996, he offered his program code to Astrodienst for implementation of this house system into Astrodienst’s then astronomical software Placalc. (At that time, however, Astrodienst was not interested in it.) In May 1997, Krusinski put the data on the web at http://www.ci.uw.edu.pl/~bogdan/astrol.htm (now at http://www.astrologia.pl/main/domy.html) In February 2006 he sent Swiss-Ephemeris-compatible program code in C for this house system to Astrodienst. This code was included into Swiss Ephemeris Version 1.70 and released on 8 March 2006.

Conclusion: While Pisa seems to have discovered the house system a couple of months before Krusinski, Krusinski discovered it independently, at a time when he could not have known of Pisa’s work. Krusinski published the house system more than a year before Pisa. As for the naming of the house system, there is also another important point of view: Krusinski distributed house tables and software for all geographical latitudes years before Pisa and did more in order to introduce them to working astrologers. By March 2006, when the house system was implemented into the Swiss Ephemeris, neither Krusinski nor Astrodienst knew about Pisa. For this reason, we published it as the “Krusinski” house system. Even though we acknowledge Pisa’s independent discovery of the house system, we keep Krusinski’s name in the first place because of his merits and in order to avoid confusion.

### 6.2. Vertex, Antivertex, East Point and Equatorial Ascendant, etc.

The Vertex is the point of the ecliptic that is located precisely in western direction. The Antivertex is the opposition point and indicates the precise east in the horoscope. It is identical to the first house cusp in the horizon house system.

There is a lot of confusion about this, because there is also another point which is called the "East Point" but is usually not located in the east. In celestial geometry, the expression "East Point" means the point on the horizon which is in precise eastern direction. The equator goes through this point as well, at a right ascension which is equal to ARMC + 90 degrees. On the other hand, what some astrologers call the "East Point" is the point on the ecliptic whose right ascension is equal to ARMC + 90 (i.e. the right ascension of the horizontal East Point). This point can deviate from eastern direction by 23.45 degrees, the amount of the ecliptic obliquity. For this reason, the term "East Point" is not very well-chosen for this ecliptic point, and some astrologers (M. Munkasey) prefer to call it the Equatorial Ascendant. This, because it is identical to the Ascendant at a geographical latitude 0.

The Equatorial Ascendant is identical to the first house cusp of the axial rotation system.

Note: If a projection of the horizontal East Point on the ecliptic is wanted, it might seem more reasonable to use a projection rectangular to the ecliptic, not rectangular to the equator as is done by the users of the "East Point". The planets, as well, are not projected on the ecliptic in a right angle to the ecliptic.

The Swiss Ephemeris supports three more points connected with the house and angle calculation. They are part of Michael Munkasey's system of the 8 Personal Sensitive Points (PSP). The PSP include the Ascendant, the MC, the Vertex, the Equatorial Ascendant, the Aries Point, the Lunar Node, and the "Co-Ascendant" and the "Polar Ascendant".

The term "Co-Ascendant" seems to have been invented twice by two different people, and it can mean two different things. The one "Co-Ascendant" was invented by Walter Koch (?). To calculate it, one has to take the ARIC as an ARMC and compute the corresponding Ascendant for the birth place. The "Co-Ascendant" is then the opposition to this point.

The second "Co-Ascendant" stems from Michael Munkasey. It is the Ascendant computed for the natal ARMC and a latitude which has the value 90° - birth_latitude.

The "Polar Ascendant" finally was introduced by Michael Munkasey. It is the opposition point of Walter Koch's version of the "Co-Ascendant". However, the "Polar Ascendant" is not the same as an Ascendant computed for the birth time and one of the geographic poles of the earth. At the geographic poles, the Ascendant is always 0 Aries or 0 Libra. This is not the case for Munkasey's "Polar Ascendant".

### 6.3. House cusps beyond the polar circle

Beyond the polar circle, we proceed as follows:

1) We make sure that the ascendant is always in the eastern hemisphere.

2) Placidus and Koch house cusps sometimes can, sometimes cannot be computed beyond the polar circles. Even the MC doesn't exist always, if one defines it in the Placidus manner. Our function therefore automatically switches to Porphyry houses (each quadrant is divided into three equal parts) and returns a warning.

3) Beyond the polar circles, the MC is sometimes below the horizon. The geometrical definition of the Campanus and Regiomontanus systems requires in such cases that the MC and the IC are swapped. The whole house system is then oriented in clockwise direction.

There are similar problems with the Vertex and the horizon house system for birth places in the tropics. The Vertex is defined as the point on the ecliptic that is located in precise western direction. The ecliptic east point is the opposition point and is called the Antivertex. Our program code makes sure that the Vertex (and the cusps 11, 12, 1, 2, 3 of the horizon house system) is always located in the western hemisphere. Note that for birthplaces on the equator the Vertex is always 0 Aries or 0 Libra.

Of course, there are no problems in the calculation of the Equatorial Ascendant for any place on earth.

### 6.3.1. Implementation in other calculation modules:

a) PLACALC

Placalc is the predecessor of Swiss Ephemeris; it is a calculation module created by Astrodienst in 1988 and distributed as C source code. Beyond the polar circles, Placalc‘s house calculation did switch to Porphyry houses for all unequal house systems. Swiss Ephemeris still does so with the Placidus and Koch method, which are not defined in such cases. However, the computation of the MC and Ascendant was replaced by a different model in some cases: Swiss Ephemeris gives priority to the Ascendant, choosing it always as the eastern rising point of the ecliptic and accepting an MC below the horizon, whereas Placalc gave priority to the MC. The MC was always chosen as the intersection of the meridian with the ecliptic above the horizon. To keep the quadrants in the correct order, i.e. have an Ascendant in the left side of the chart, the Ascendant was switched by 180 degrees if necessary.

In the discussions between Alois Treindl and Dieter Koch during the development of the Swiss Ephemeris it was recognized that this model is more unnatural than the new model implemented in Swiss Ephemeris.

Placalc also made no difference between Placidus/Koch on one hand and Regiomontanus/Campanus on the other as Swiss Ephemeris does. In Swiss Ephemeris, the geometrical definition of Regiomontanus/Campanus is strictly followed, even for the price of getting the houses in ”wrong” order. (see above, chapter 4.1.)

b) ASTROLOG program as written by Walter Pullen

While the freeware program Astrolog contains the planetary routines of Placalc, it uses its own house calculation module by Walter Pullen. Various releases of Astrolog contain different approaches to this problem.

c) ASTROLOG on our website

ASTROLOG is also used on Astrodienst’s website for displaying free charts. This version of Astrolog used on our website however is different from the Astrolog program as distributed on the Internet. Our webserver version of Astrolog contains calls to Swiss Ephemeris for planetary positions. For Ascendant, MC and houses it still uses Walter Pullen's code. This will change in due time because we intend to replace ASTROLOG on the website with our own charting software.

d) other astrology programs

Because most astrology programs still use the Placalc module, they follow the Placalc method for houses inside the polar circles. They give priority to keep the MC above the horizon and switch the Ascendant by 180 degrees if necessary to keep the quadrants in order.

## 6.4. House position of a planet

The Swiss Ephemeris DLL also provides a function to compute the house position of a given body, i.e. in which house it is. This function can be used either to determine the house number of a planet or to compute its position in a house horoscope. (A house horoscope is a chart in which all houses are stretched or shortened to a size of 30 degrees. For unequal house systems, the zodiac is distorted so that one sign of the zodiac does not measure 30 house degrees)

Note that the actual house position of a planet is not always the one that it seems to be in an ordinary chart drawing. Because the planets are not always exactly located on the ecliptic but have a latitude, they can seemingly be located in the first house, but are actually visible above the horizon. In such a case, our program function will place the body in the 12th (or 11 th or 10 th) house, whatever celestial geometry requires. However, it is possible to get a house position in the ”traditional” way, if one sets the ecliptic latitude to zero.

Although it is not possible to compute Placidus house cusps beyond the polar circle, this function will also provide Placidus house positions for polar regions. The situation is as follows:

The Placidus method works with the semidiurnal and seminocturnal arcs of the planets. Because in higher geographic latitudes some celestial bodies (the ones within the circumpolar circle) never rise or set, such arcs do not exist. To avoid this problem it has been proposed in such cases to start the diurnal motion of a circumpolar body at its ”midnight” culmination and its nocturnal motion at its midday culmination. This procedure seems to have been proposed by Otto Ludwig in 1930. It allows to define a planet's house position even if it is within the circumpolar region, and even if you are born in the northernmost settlement of Greenland. However, this does not mean that it be possible to compute ecliptical house cusps for such locations. If one tried that, it would turn out that e.g. an 11 th house cusp did not exist, but there were two 12th house cusps.

Note however, that circumpolar bodies may jump from the 7th house directly into the 12th one or from the 1st one directly into the 6th one.

The Koch method, on the other hand, cannot be helped even with this method. For some bodies it may work even beyond the polar circle, but for some it may fail even for latitudes beyond 60 degrees. With fixed stars, it may even fail in central Europe or USA. (Dieter Koch regrets the connection of his name with such a badly defined house system)

Note that Koch planets do strange jumps when the cross the meridian. This is not a computation error but an effect of the awkward definition of this house system. A planet can be east of the meridian but be located in the

9th house, or west of the meridian and in the 10th house. It is possible to avoid this problem or to make Koch house positions agree better with the Huber ”hand calculation” method, if one sets the ecliptic latitude of the planets to zero. But this is not more correct from a geometrical point of view.

## 6.5. Gauquelin sector position of a planet

The calculation of the Gauquelin sector position of a planet is based on the same idea as the Placidus house system, i.e. diurnal and nocturnal arcs of ecliptic points or planets are subdivided.

Three different algorithms have been used by Gauquelin and others to determine the sector position of a planet.

1. We can take the ecliptic point of the planet (ecliptical latitude ignored) and calculate the fraction of its diurnal or nocturnal arc it has completed

2. We can take the true planetary position (taking into account ecliptical latitude) for the same calculation.

3. We can use the exact times for rise and set of the planet to determine the ratio between the time the planet has already spent above (or below) the horizon and its diurnal (or nocturnal) arc. Times of rise and set are defined by the appearance or disappearance of the center of the planet’s disks.

All three methods are supported by the Swiss Ephemeris.

Methods 1 and 2 also work for polar regions. The Placidus algorithm is used, and the Otto Ludwig method applied with circumpolar bodies. I.e. if a planet does not have a rise and set, the “midnight” and “midday” culminations are used to define its semidiurnal and seminocturnal arcs.

With method 3, we don’t try to do similar. Because planets do not culminate exactly in the north or south, a planet can actually rise on the western part of the horizon in high geographic latitudes. Therefore, it does not seem appropriate to use meridian transits as culmination times. On the other hand, true culmination times are not always available. E.g. close to the geographic poles, the sun culminates only twice a year.

# 7. D T (Delta T)

The computation of planets uses the so called Ephemeris Time (ET) which is a completely regular time measure. Computations of sidereal time and houses, on the other hand, depend on the rotation of the earth, which is not regular at all. The time used for such purposes is called Universal Time (UT) or Terrestrial Dynamic Time (TDT). It is an irregular time measure, and is roughly identical to the time indicated by our clocks (if time zones are neglected). The difference between ET and UT is called D T (”Delta T”), and is defined as D T = ET – UT.

The earth's rotation decreases slowly, currently at the rate of about 0.5 – 1 second per year. Even worse, this decrease is irregular itself. It cannot precisely predicted but only derived from star observations. The values of D T achieved like this must be tabulated. However, this table, which is published yearly by the Astronomical Almanac, starts only at 1620, about the time when the telescope was invented. For more remote centuries, D T must be estimated from old eclipse records. The uncertainty is in the range of hours for the year 3000 B.C. For future times, D T is estimated from the current and the general changing rate, depending on whether a short-term or a long-term extrapolation is intended.

NOTE: The D T algorithms have been improved with the Swiss Ephemeris release 1.64, mostly according to Stephenson 1997 (s. further below), and further improved with release 1.72, according to Morrison/Stephenson 2004. These changes result in significant changes of the ephemeris for remote historical dates, if Universal Time is used.

The Swiss Ephemeris computes D T as follows.

1620 - today + a couple of years:

The tabulated values of deltaT, in hundredths of a second, were taken from the Astronomical Almanac 1997, page K8. The program adjusts for a value of secular tidal acceleration ndot = -25.7376 arcsec per century squared, the value used in JPL's DE403 ephemeris. ELP2000 (and DE200) used the value -23.8946.

To change ndot, one can either redefine SE_TIDAL_DEFAULT in swephexp.h or use the routine swe_set_tid_acc() before calling the Swiss Ephemeris.

Bessel's interpolation formula was implemented to obtain fourth order interpolated values at intermediate times.

-1000 - 1620:

For dates between -1000 and 1600, the table given by Morrison/Stephenson (2004; p. 332) is used, with linear interpolation. This table is based on an assumed value of ndot = -26. The program adjusts for ndot = -25.7376.

For 1600 - 1620, a linear interpolation between the last value of the latter and the first value of the former table is made.

before -1000:

For times before -600, a formula of Stephenson & Morrison (1995) (Morrison/Stephenson 2004; p. 332) is used: dt=32*t*t-20 sec, where t is centuries from 1820 AD.

For -1100 to -1000, a transition from this formula to the Stephenson table has been implemented in order to avoid a jump.

future:

For the time after the last tabulated value, we use the formula of Stephenson (1997; p. 507), with a modification that avoids a jump at the end of the tabulated period. A linear term is added that makes a slow transition from the table to the formula over a period of 100 years. (Need not be updated, when table will be enlarged.)

Differences between the old and new algorithms (before and after release 1.72):

year difference in seconds (new - old)

-3000 -4127

-2000 -2130

-1000 -760

0 -20

1000 -30

1600 10

1619 0.5

1620 0

Differences between the old and new algorithms (before and after release 1.64):

year difference in seconds (new - old)

-3000 2900

0 1200

1600 29

1619 60

1620 -0.6

1700 -0.4

1800 -0.1

1900 -0.02

1940 -0.001

1950 0

2000 0

2020 2

2100 23

3000 -400

In 1620, where the D T table of the Astronomical Almanac starts, there was a jump of a whole minute in the old algorithms. The new algorithms has no jumps anymore.

The smaller differences for the period 1620-1955, where we still use the same data as before, is due to a correction in the tidal acceleration of the moon, which now has the same value as is also used by JPL for their D T calculations.

References:

- Stephenson, F. R., and L. V. Morrison, "Long-term changes in the rotation of the Earth: 700 BC to AD 1980", Philosophical Transactions of the Royal Society of London, Series A 313, 47-70 (1984)

- Morrison, L. V., and F.R. Stephenson, “Historical Values of the Earth’s Clock Error D T and the Calculation of Eclipses”, JHA xxxv (2004), pp.327-336

- Borkowski, K. M., "ELP2000-85 and the Dynamical Time - Universal Time relation," Astronomy and Astrophysics 205, L8-L10 (1988)

- Chapront-Touze, Michelle, and Jean Chapront, Lunar Tables and Programs from 4000 B.C. to A.D. 8000, Willmann-Bell 1991

- Stephenson, F. R., and M. A. Houlden, Atlas of Historical Eclipse Maps, Cambridge U. Press (1986)

- Morrison, L. V. and F. R. Stephenson, Sun and Planetary System, vol 96,73 eds. W. Fricke, G. Teleki, Reidel, Dordrecht (1982)

- Stephenson, F.R. & Morrison, L.V., "Long-Term Fluctuations in the Earth's Rotation: 700 BC to AD 1990", in: Philosophical Transactions of the Royal Society of London, Ser. A, 351 (1995), 165-202.

- Stephenson, F. Richard, Historical Eclipses and Earth's Rotation, Cambridge U. Press (1997)

- Explanatory Supplement of the Astronomical Almanach, University Science Books, 1992, Mill Valley, CA, p. 265ff.

- For a comprehensive collection of publications and formulae, see: http://www.phys.uu.nl/~vgent/astro/deltatime.htm

# 8. Programming Environment

Swiss Ephemeris is written in portable C and the same code is used for creation of the 32-bit Windows DLL and the link library. All data files are fully portable between different hardware architectures.

To build the DLLs, we use Microsoft Visual C++ version 5.0 (for 32-bit).

The DLL has been successfully used in the following programming environments:

Visual C++ 5.0 (sample code included in the distribution)

Visual Basic 5.0 (sample code and VB declaration file included)

Delphi 2 and Delphi 3 (32-bit, declaration file included)

As the number of users grows, our knowledge base about the interface details between programming environments and the DLL grows. All such information is added to the distributed Swiss Ephemeris and registered users are informed via an email mailing list.

Earlier version up to version 1.61 supported 16-bit Windows programming. Since then, 16-bit support has been dropped.

# 9. Swiss Ephemeris Functions

## 9.1 Swiss Ephemeris API

We give a short overview of the most important functions contained in the Swiss Ephemeris DLL. The detailed description of the programming interface is contained in the document swephprg.doc which is distributed together with the file you are reading.

#### Calculation of planets and stars

/* planets, moon, asteroids, lunar nodes, apogees, fictitious bodies */

swe_calc();

/* fixed stars */

swe_fixstar();

#### Date and time conversion

/* delta t from Julian day number

* Ephemeris time (ET) = Universal time (UT) + swe_deltat(UT)*/

swe_deltat();

/* Julian day number from year, month, day, hour, */

swe_date_conversion ();

/* Julian day number from year, month, day, hour */

swe_julday();

/* year, month, day, hour from Julian day number */

swe_revjul ();

/* get tidal acceleration used in swe_deltat() */

swe_get_tid_acc();

/* set tidal acceleration to be used in swe_deltat() */

swe_set_tid_acc();

#### Initialization, setup, and closing functions

/* set directory path of ephemeris files */

swe_set_ephe_path();

/* set name of JPL ephemeris file */

swe_set_jpl_file();

/* close Swiss Ephemeris */

swe_close();

#### House calculation

/* sidereal time */

swe_sidtime();

/* house cusps, ascendant, MC, armc, vertex */

swe_houses();

#### Auxiliary functions

/* coordinate transformation, from ecliptic to equator or vice-versa. */

swe_cotrans();

/* coordinate transformation of position and speed,

* from ecliptic to equator or vice-versa*/

swe_cotrans_sp();

/* get the name of a planet */

swe_get_planet_name();

/* normalization of any degree number to the range 0 ... 360 */

swe_degnorm();

### Other functions that may be useful

PLACALC, the predecessor of SWISSEPH, included several functions that we do not need for SWISSEPH anymore. Nevertheless we include them again in our DLL, because some users of our software may have taken them over and use them in their applications. However, we gave them new names that were more consistent with SWISSEPH.

PLACALC used angular measurements in centiseconds a lot; a centisecond is 1/100 of an arc second. The C type CSEC or centisec is a 32-bit integer. CSEC was used because calculation with integer variables was considerably faster than floating point calculation on most CPUs in 1988, when PLACALC was written.

In the Swiss Ephemeris we have dropped the use of centiseconds and use double (64-bit floating point) for all angular measurements.

/* normalize argument into interval [0..DEG360]

* former function name: csnorm() */

swe_csnorm();

/* distance in centisecs p1 - p2 normalized to [0..360[

* former function name: difcsn() */

swe_difcsn ();

/* distance in degrees * former function name: difdegn() */

swe_difdegn ();

/* distance in centisecs p1 - p2 normalized to [-180..180[

* former function name: difcs2n() */

swe_difcs2n();

/* distance in degrees

* former function name: difdeg2n() */

swe_difdeg2n();

/* round second, but at 29.5959 always down

* former function name: roundsec() */

swe_csroundsec();

/* double to long with rounding, no overflow check

* former function name: d2l() */

swe_d2l();

/* Monday = 0, ... Sunday = 6

* former function name: day_of_week() */

swe_day_of_week();

/* centiseconds -> time string

* former function name: TimeString() */

swe_cs2timestr();

/* centiseconds -> longitude or latitude string

* former function name: LonLatString() */

swe_cs2lonlatstr();

/* centiseconds -> degrees string

* former function name: DegreeString() */

swe_cs2degstr();

## 9.2 Placalc API

Placalc is a planetary calculation module which was made available by Astrodienst since 1988 to other programmers under a source code license. Placalc is less well designed, less complete and not as precise as the Swiss Ephemeris module. However, many developers of astrological software have used it over many years and like it. Astrodienst has used it internally since 1989 for a large set of application programs.

To simplify the introduction of Swiss Ephemeris in 1997 in Astrodienst's internal operation, we wrote an interface module which translates all calls to Placalc functions into Swiss Ephemeris functions, and translates the results back into the format expected in the Placalc Application Interface (API).

This interface (swepcalc.c and swepcalc.h) is part of the source code distribution of Swiss Ephemeris; it is not contained in the DLL. All new software should be written directly for the SwissEph API, but porting old Placalc software is convenient and very simple with the Placalc API.

# Appendix

## A. The gravity deflection for a planet passing behind the Sun

The calculation of the apparent position of a planet involves a relativistic effect, which is the curvature of space by the gravity field of the Sun. This can also be described by a semi-classical algorithm, where the photon travelling from the planet to the observer is deflected in the Newtonian gravity field of the Sun, where the photon has a non-zero mass arising from its energy. To get the correct relativistic result, a correction factor 2.0 must be included in the calculation.

A problem arises when a planet disappears behind the solar disk, as seen from the Earth. Over the whole 6000 year time span of the Swiss Ephemeris, it happens often.

Planet | number of passes behind the Sun
Mercury | 1723
Venus | 456
Mars | 412
Jupiter | 793
Saturn | 428
Uranus | 1376
Neptune | 543
Pluto | 57

A typical occultation of a planet by the Solar disk, which has a diameter of approx. _ degree, has a duration of about 12 hours. For the outer planets it is mostly the speed of the Earth's movement which determines this duration.

Strictly speaking, there is no apparent position of a planet when it is eclipsed by the Sun. No photon from the planet reaches the observer's eye on Earth. Should one drop gravitational deflection, but keep aberration and light-time correction, or should one switch completely from apparent positions to true positions for occulted planets? In both cases, one would come up with an ephemeris which contains discontinuities, when at the moment of occultation at the Solar limb suddenly an effect is switched off.

Discontinuities in the ephemeris need to be avoided for several reasons. On the level of physics, there cannot be a discontinuity. The planet cannot jump from one position to another. On the level of mathematics, a non-steady function is a nightmare for computing any derived phenomena from this function, e.g. the time and duration of an astrological transit over a natal body, or an aspect of the planet.

Nobody seems to have handled this problem before in astronomical literature. To solve this problem, we have used the following approach: We replace the Sun, which is totally opaque for electromagnetic waves and not transparent for the photons coming from a planet behind it, by a transparent gravity field. This gravity field has the same strength and spatial distribution as the gravity field of the Sun. For photons from occulted planets, we compute their path and deflection in this gravity field, and from this calculation we get reasonable apparent positions also for occulted planets.

The calculation has been carried out with a semi-classical Newtonian model, which can be expected to give the correct relativistic result when it is multiplied with a correction factor 2. The mass of the Sun is mostly concentrated near its center; the outer regions of the Solar sphere have a low mass density. We used the a mass density distribution from the Solar standard model, assuming it to have spherical symmetry (our Sun mass distribution m® is from Michael Stix, The Sun, p. 47). The path of photons through this gravity field was computed by numerical integration. The application of this model in the actual ephemeris could then be greatly simplified by deriving an effective Solar mass which a photon ”sees” when it passes close by or ”through” the Sun. This effective mass depends only from the closest distance to the Solar center which a photon reaches when it travels from the occulted planet to the observer. The dependence of the effective mass from the occulted planet's distance is so small that it can be neglected for our target precision of 0.001 arc seconds.

For a remote planet just at the edge of the Solar disk the gravity deflection is about 1.8”, always pointing away from the center of the Sun. This means that the planet is already slightly behind the Solar disk (with a diameter of 1800”) when it appears to be at the limb, because the light bends around the Sun. When the planet now passes on a central path behind the Solar disk, the virtual gravity deflection we compute increases to 2.57 times the deflection at the limb, and this maximum is reached at _ of the Solar radius. Closer to the Solar center, the deflection drops and reaches zero for photons passing centrally through the Sun's gravity field.

We have discussed our approach with Dr. Myles Standish from JPL and here is his comment (private email to Alois Treindl, 12-Sep-1997):

.. it seems that your approach is

entirely reasonable and can be easily justified as long

as you choose a reasonable model for the density of

the sun. The solution may become more difficult if an

ellipsoidal sun is considered, but certainly that is

an additional refinement which can not be crucial.

## B. The list of asteroids on the software CDROM

# List of asteroids on SwissEph CD-ROM

# ====================================

# At the same time a brief introduction into asteroids

# ====================================================

#

# Ephemerides of all of the asteroids mentioned below

# can be found on the SwissEph CD-ROM.

# For complete Ephemerides of ALL asteroids, order our

# special asteroid CD-ROMS.

#

# Literature:

# Lutz D. Schmadel, Dictionary of Minor Planet Names,

# Springer, Berlin, Heidelberg, New York

# Charles T. Kowal, Asteroids. Their Nature and Utilization,

# Whiley & Sons, 1996, Chichester, England

#

#

# What is an asteroid?

# --------------------

#

# Asteroids are small planets. Because there are too many

# of them and because most of them are quite small,

# astronomers did not like to call them "planets", but

# invented names like "asteroid" (Greek "star-like",

# because through telescopes they did not appear as planetary

# discs but as star like points) or "planetoid" (Greek

# "something like a planet"). However they are also often

# called minor planets.

# The minor planets can roughly be divided into two groups.

# There are the inner asteroids, the majority of which

# circles in the space between Mars and Jupiter, and

# there are the outer asteroids, which have their realm

# beyond Neptune. The first group consists of rather

# dense, earth-like material, whereas the Transneptunians

# mainly consist of water ice and frozen gases. Many comets

# are descendants of the "asteroids" (or should one say

# "comets"?) belt beyond Neptune. The first Transneptunian

# objects (except Pluto) were discovered only after 1992

# and none of them has been given a name as yet.

#

#

# The largest asteroids

# ---------------------

# Most asteroids are actually only debris of collisions

# of small planets that formed in the beginning of the

# solar system. Only the largest ones are still more

# or less complete and round planets.

1 Ceres # 913 km goddess of corn and harvest

2 Pallas # 523 km goddess of wisdom, war and liberal arts

4 Vesta # 501 km goddess of the hearth fire

10 Hygiea # 429 km goddess of health

511 Davida # 324 km after an astronomer David P. Todd

704 Interamnia # 338 km "between rivers", ancient name of

# its discovery place Teramo

65 Cybele # 308 km Phrygian Goddess, = Rhea, wife of Kronos-Saturn

52 Europa # 292 km beautiful mortal woman, mother of Minos by Zeus

87 Sylvia # 282 km

451 Patientia # 280 km patience

31 Euphrosyne # 270 km one of the three Graces, benevolence

15 Eunomia # 260 km one of the Hours, order and law

324 Bamberga # 252 km after a city in Bavaria

3 Juno # 248 km wife of Zeus

16 Psyche # 248 km "soul", name of a nymph

# Asteroid families

# -----------------

# Most asteroids live in families. There are several kinds

# of families.

# - There are families that are separated from each other

# by orbital resonances with Jupiter or other major planets.

# - Other families, the so-called Hirayama families, are the

# relics of asteroids that broke apart long ago when they

# collided with other asteroids.

# - Third, there are the Trojan asteroids that are caught

# in regions 60 degrees ahead or behind a major planet

# (Jupiter or Mars) by the combined gravitational forces

# of this planet and the Sun.

# Near Earth groups:

# ------------------

#

# Aten family: they cross Earth; mean distance from Sun is less than Earth

2062 Aten # an Egyptian Sun god

2100 Ra-Shalom # Ra is an Egyptian Sun god, Shalom is Hebrew "peace"

# was discovered during Camp David mid-east peace conference

# Apollo family: they cross Earth; mean distance is greater than Earth

1862 Apollo # Greek Sun god

1566 Icarus # wanted to fly to the sky, fell into the ocean

# Icarus crosses Mercury, Venus, Earth, and Mars

# and has his perihelion very close to the Sun

3200 Phaethon # wanted to drive the solar chariot, crashed in flames

# Phaethon crosses Mercury, Venus, Earth, and Mars

# and has his perihelion very close to the Sun

# Amor family: they cross Mars, approach Earth

1221 Amor # Roman love god

433 Eros # Greek love god

# Mars Trojans:

# -------------

5261 Eureka a mars Trojan

# Main belt families:

# -------------------

# Hungarias: asteroid group at 1.95 AU

434 Hungaria # after Hungary

# Floras: Hirayama family at 2.2 AU

8 Flora # goddess of flowers

# Phocaeas: asteroid group at 2.36 AU

25 Phocaea # maritime town in Ionia

# Koronis family: Hirayama family at 2.88 AU

158 Koronis # mother of Asklepios by Apollo

# Eos family: Hirayama family at 3.02 AU

221 Eos # goddess of dawn

# Themis family: Hirayama family at 3.13 AU

24 Themis # goddess of justice

# Hildas: asteroid belt at 4.0 AU, in 3:2 resonance with Jupiter

# --------------------------------------------------------------

# The Hildas have fairly eccentric orbits and, at their

# aphelion, are very close to the orbit of Jupiter. However,

# at those times, Jupiter is ALWAYS somewhere else. As

# Jupiter approaches, the Hilda asteroids move towards

# their perihelion points.

153 Hilda # female first name, means "heroine"

# a single asteroid at 4.26 AU, in 4:3 resonance with Jupiter

279 Thule # mythical center of Magic in the uttermost north

# Jupiter Trojans:

# ----------------

# Only the Trojans behind Jupiter are actually named after Trojan heroes,

# whereas the "Trojans" ahead of Jupiter are named after Greek heroes that

# participated in the Trojan war. However there have been made some mistakes,

# i.e. there are some Trojan "spies" in the Greek army and some Greek "spies"

# in the Trojan army.

# Greeks ahead of Jupiter:

624 Hector # Trojan "spy" in the Greek army, by far the greatest

# Trojan hero and the greatest Trojan asteroid

588 Achilles # slayer of Hector

1143 Odysseus

# Trojans behind Jupiter:

1172 Äneas

3317 Paris

884 Priamus

# Jupiter-crossing asteroids:

# ---------------------------

3552 Don Quixote # perihelion near Mars, aphelion beyond Jupiter;

# you know Don Quixote, don't you?

944 Hidalgo # perihelion near Mars, aphelion near Saturn;

# after a Mexican national hero

5335 Damocles # perihelion near Mars, aphelion near Uranus;

# the man sitting below a sword suspended by a thread

# Centaurs:

# ---------

2060 Chiron # perihelion near Saturn, aphelion near Uranus

# educator of heros, specialist in healing and war arts

5145 Pholus # perihelion near Saturn, aphelion near Neptune

# seer of the gods, keeper of the wine of the Centaurs

7066 Nessus # perihelion near Saturn, aphelion in Pluto's mean distance

# ferryman, killed by Hercules, kills Hercules

# Plutinos:

# ---------

# These are objects with periods similar to Pluto, i.e. objects

# that resonate with the Neptune period in a 3:2 ratio.

# There are no Plutinos included in Swiss Ephemeris so far, but

# PLUTO himself is considered to be a Plutino type asteroid!

# Cubewanos:

# ----------

# These are non-Plutiono objects with periods greater than Pluto.

# The word "Cubewano" is derived from the preliminary designation

# of the first-discovered Cubewano: 1992 QB1

20001 1992 QB1 # will be given the name of a creation deity

# (fictitious catalogue number 20001!)

# other Transplutonians:

20001 1996 TL66 # mean solar distance 85 AU, period 780 years

# Asteroids that challenge hypothetical planets astrology

# -------------------------------------------------------

42 Isis # not identical with "Isis-Transpluto"

# Egyptian lunar goddess

763 Cupido # different from Witte's Cupido

# Roman god of sexual desire

4341 Poseidon # not identical with Witte's Poseidon

# Greek name of Neptune

4464 Vulcano # compare Witte's Vulkanus

# and intramercurian hypothetical Vulcanus

# Roman fire god

5731 Zeus # different from Witte's Zeus

# Greek name of Jupiter

1862 Apollo # different from Witte's Apollon

# Greek god of the Sun

398 Admete # compare Witte's Admetos

# "the untamed one", daughter of Eurystheus

# Asteroids that challenge Dark Moon astrology

# --------------------------------------------

1181 Lilith # not identical with Dark Moon 'Lilith'

# first evil wife of Adam

3753 Cruithne # often called the "second moon" of earth;

# actually not a moon, but an asteroid that

# orbits around the sun in a certain resonance

# with the earth.

# After the first Celtic group to come to the British Isles.

# Also try the two points 60 degrees in front of and behind the

# Moon, the so called Lagrange points, where the combined

# gravitational forces of the earth and the moon might imprison

# rocks and stones. There have been some photographic hints

# that there are clouds of such material around these points.

# They are called the Kordylewski clouds.

# other asteroids

# ---------------

5 Astraea # a goddess of justice

6 Hebe # goddess of youth

7 Iris # rainbow goddess, messenger of the gods

8 Flora # goddess of flowers and gardens

9 Metis # goddess of prudence

10 Hygiea # goddess of health

14 Irene # goddess of peace

16 Psyche # "soul", a nymph

19 Fortuna # goddess of fortune

# Some frequent names:

# --------------------

# There are thousands of female first names in the asteroids list.

# Very interesting for relationship charts!

78 Diana

170 Maria

234 Barbara

375 Ursula

412 Elisabetha

542 Susanna

# Wisdom asteroids:

# -----------------

134 Sophrosyne # equanimity, healthy mind and impartiality

197 Arete # virtue

227 Philosophia

251 Sophia # wisdom (Greek)

259 Aletheia # truth

275 Sapientia # wisdom (Latin)

# Love asteroids:

# ---------------

344 Desiderata

433 Eros

499 Venusia

763 Cupido

1221 Amor

1387 Kama # Indian god of sexual desire

1388 Aphrodite # Greek love Goddess

1389 Onnie # what is this, after 1387 and 1388 ?

1390 Abastumani # and this?

# The Nine Muses

# --------------

18 Melpomene Muse of tragedy

22 Kalliope Muse of heroic poetry

23 Thalia Muse of comedy

27 Euterpe Muse of music and lyric poetry

30 Urania Muse of astronomy and astrology

33 Polyhymnia Muse of singing and rhetoric

62 Erato Muse of song and dance

81 Terpsichore Muse of choral dance and song

84 Klio Muse of history

# Money and big busyness asteroids

# --------------------------------

19 Fortuna # goddess of fortune

904 Rockefellia

1338 Duponta

3652 Soros

# Beatles asteroids:

# ------------------

4147 Lennon

4148 McCartney

4149 Harrison

4150 Starr

# Composer Asteroids:

# -------------------

2055 Dvorak

1814 Bach

1815 Beethoven

1034 Mozartia

3941 Haydn

And there are many more...

# Astrodienst asteroids:

# ----------------------

# programmers group:

3045 Alois

2396 Kochi

2968 Iliya # Alois' dog

# artists group:

412 Elisabetha

# production family:

612 Veronika

1376 Michelle

1343 Nicole

1716 Peter

# children group

105 Artemis

1181 Lilith

# special interest group

564 Dudu

349 Dembowska

484 Pittsburghia

# By the year 1997, the statistics of asteroid names looked as follows:

# Men (mostly family names) 2551

# Astronomers 1147

# Women (mostly first names) 684

# Mythological terms 542

# Cities, harbours buildings 497

# Scientists (no astronomers) 493

# Relatives of asteroid discoverers 277

# Writers 249

# Countries, provinces, islands 246

# Amateur astronomers 209

# Historical, political figures 176

# Composers, musicians, dancers 157

# Figures from literature, operas 145

# Rivers, seas, mountains 135

# Institutes, observatories 116

# Painters, sculptors 101

# Plants, trees, animals 63