Swiss ephemeris 4 Computer ephemeris for developers of astrological software 4



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Introduction


Swiss Ephemeris is a function package of astronomical calculations that serves the needs of astrologers, archaeoastronomers, and, depending on purpose, also the needs of astronomers. It includes long-term ephemerides for the Sun, the Moon, the planets, more than 300’000 asteroids, historically relevant fixed stars and some “hypothetical” objects.

The precision of the Swiss Ephemeris is at least as good as that of the Astromical Almanac, which follows current standards of ephemeris calculation. Swiss Ephemeris will, as we hope, be able to keep abreast to the scientific advances in ephemeris computation for the coming decades.

The Swiss Ephemeris package consists of source code in C, 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

Full C source code is included with the Swiss Ephemeris, so that non-Windows programmers can create a linkable or shared library in their environment and use it with their applications.


  1. 1. Licensing


The Swiss Ephemeris is not a product for end users. It is a toolset for programmers to build into their astrological software. 
Swiss Ephemeris is made available by its authors under a dual licensing system. The software developer, who uses any part of Swiss Ephemeris in his or her software, must choose between one of the two license models, which are

a) GNU public license version 2 or later

b) Swiss Ephemeris Professional License

The choice must be made before the software developer distributes software containing parts of Swiss Ephemeris to others, and before any public service using the developed software is activated.

If the developer choses the GNU GPL software license, he or she must fulfill the conditions of that license, which includes the obligation to place his or her whole software project under the GNU GPL or a compatible license. See http://www.gnu.org/licenses/old-licenses/gpl-2.0.html

If the developer choses the Swiss Ephemeris Professional license, he must follow the instructions as found in http://www.astro.com/swisseph/ and purchase the Swiss Ephemeris Professional Edition from Astrodienst and sign the corresponding license contract.

The Swiss Ephemeris Professional Edition can be purchased from Astrodienst for a one-time fixed fee for each commercial programming project. The license is just a legal document. All actual software and data are found in the public download area and are to be downloaded from there. 

Professional license: The license fee for the first license is Swiss Francs (CHF) 750.-, and CHF 400.- for each additional license by the same licensee. An unlimited license is available for CHF 1550.-.

  1. 2. Descripition of the ephemerides

    1. 2.1 Planetary and lunar ephemerides

      1. 2.1.1 Three ephemerides


The Swiss Ephemeris package allows planetary and lunar computations from any of the following three astronomical ephemerides:
      1. 2.1.1.1 The Swiss Ephemeris


The core part of Swiss Ephemeris is a compression of the JPL-Ephemeris DE431, which covers roughly the time range 13’000 BCE to 17’000 CE. Using a sophisticated mechanism, we succeeded in reducing JPL's 2.8 GB storage to only 99 MB. The compressed version agrees with the JPL Ephemeris to 1 milli-arcsecond (0.001”). Since the inherent uncertainty of the JPL ephemeris for most of its time range is a lot greater, the Swiss Ephemeris should be completely satisfying even for computations demanding very high accuracy.

(Before 2014, the Swiss Ephemeris was based on JPL Ephemeris DE406. Its 200 MB were compressed to 18 MB. The time range of the DE406 was 3000 BC to 3000 AD or 6000 years. We had 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. To make sure that you work with current data, please check the date of the ephemeris files. They must be 2014 or later.)

Each Swiss Ephemeris file covers a period of 600 years; there are 50 planetary files, 50 Moon files for the whole time range of almost 30’000 years and 18 main-asteroid files for the time range of 10'800 years.

The file names are as follows:



Planetary file

Moon file

Main asteroid file

Time range

Seplm132.se1

Semom132.se1




11 Aug 13000 BC – 12602 BC

Seplm126.se1

Semom126.se1




12601 BC – 12002 BC

Seplm120.se1

Semom120.se1




12001 BC – 11402 BC

Seplm114.se1

Semom114.se1




11401 BC – 10802 BC

Seplm108.se1

Semom108.se1




10801 BC – 10202 BC

Seplm102.se1

Semom102.se1




10201 BC – 9602 BC

Seplm96.se1

Semom96.se1




9601 BC – 9002 BC

Seplm90.se1

Semom90.se1




9001 BC – 8402 BC

Seplm84.se1

Semom84.se1




8401 BC – 7802 BC

Seplm78.se1

Semom78.se1




7801 BC – 7202 BC

Seplm72.se1

Semom72.se1




7201 BC – 6602 BC

Seplm66.se1

Semom66.se1




6601 BC – 6002 BC

Seplm60.se1

Semom60.se1




6001 BC – 5402 BC

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

sepl_54.se1

semo_54.se1




5400 AD – 5999 AD

sepl_60.se1

semo_60.se1




6000 AD – 6599 AD

sepl_66.se1

semo_66.se1




6600 AD – 7199 AD

sepl_72.se1

semo_72.se1




7200 AD – 7799 AD

sepl_78.se1

semo_78.se1




7800 AD – 8399 AD

sepl_84.se1

semo_84.se1




8400 AD – 8999 AD

sepl_90.se1

semo_90.se1




9000 AD – 9599 AD

sepl_96.se1

semo_96.se1




9600 AD – 10199 AD

sepl_102.se1

semo_102.se1




10200 AD – 10799 AD

sepl_108.se1

semo_108.se1




10800 AD – 11399 AD

sepl_114.se1

semo_114.se1




11400 AD – 11999 AD

sepl_120.se1

semo_120.se1




12000 AD – 12599 AD

sepl_126.se1

semo_126.se1




12600 AD – 13199 AD

sepl_132.se1

semo_132.se1




13200 AD – 13799 AD

sepl_138.se1

semo_138.se1




13800 AD – 14399 AD

sepl_144.se1

semo_144.se1




14400 AD – 14999 AD

sepl_150.se1

semo_150.se1




15000 AD – 15599 AD

sepl_156.se1

semo_156.se1




15600 AD – 16199 AD

sepl_162.se1

semo_162.se1




16200 AD – 7 Jan 16800 AD

All Swiss Ephemeris files have the file suffix .se1.

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

Swiss Ephemeris files are available for download from Astrodienst's web server.

The time range of the Swiss Ephemeris

Versions until 1.80, which were based on JPL Ephemeris DE406 and some extension created by Astrodienst, work for the following time range:

Start date 2 Jan 5401 BC (-5400) jul. = JD -251291.5

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

Versions since 2.00, which are based on JPL Ephemeris DE431, work for the following time range:

Start date 11 Aug 13000 BCE (-12999) jul. = JD -3026604.5

End date 7 Jan 16800 CE greg. = JD 7857139.5

Please note that versions prior to 2.00 are not able to correctly handle the JPL ephemeris DE431.

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.


      1. 2.1.1.2 The Moshier Ephemeris


This is a semi-analytical approximation of the JPL planetary and lunar ephemerides DE404, developed by Steve Moshier. Its deviation from JPL is 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 purposes, since the moon moves 1 arc second in 2 time seconds and the sun 2.5 arc seconds in one minute.

The advantage of the Moshier mode of the Swiss 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 mode and the compressed JPL mode (described above).

The Moshier Ephemeris covers the interval from 3000 BC to 3000 AD. However, Moshier notes that “the adjustment for the inner planets is strictly valid only from 1350 B.C. to 3000 A.D., but may be used to 3000 B.C. with some loss of precision”. And: “The Moon's position is calculated by a modified version of the lunar theory of Chapront-Touze' and Chapront. This has a precision of 0.5 arc second relative to DE404 for all dates between 1369 B.C. and 3000 A.D.” (Moshier, http://www.moshier.net/aadoc.html).


      1. 2.1.1.3 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. Time range:

Start date 9 Dec 13002 BCE (-13001) jul. = JD -3027215.5

End date 11 Jan 17000 CE greg. = JD 7930192.5

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 DE430/DE431.

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

Accuracy of JPL ephemerides DE403/404 (1996) and DE405/406 (1998)

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 is mainly caused by 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.)



Comparison of JPL ephemerides DE406 (1998) with DE431 (2013)

Differences DE431-DE406 for 3000 BCE to 3000 CE :

Moon < 7" (TT), < 2" (UT)

Sun, Mercury, Venus < 0.4 "

Mars < 2"

Jupiter < 6"

Saturn < 0.1"

Uranus < 28"

Neptune < 53"

Pluto < 129"


Moon, position(DE431) – position(DE406) in TT and UT

(Delta T adjusted to tidal acceleration of lunar ephemeris)

Year dL(TT) dL(UT) dB(TT) dB(UT)

-2999 6.33" -0.30" -0.01" 0.05"

-2500 5.91" -0.62" -0.85" -0.32"

-2000 3.39" -1.21" -0.59" -0.20"

-1500 1.74" -1.49" -0.06" -0.01"

-1000 1.06" -1.50" 0.30" 0.12"

-500 0.63" -1.40" 0.28" 0.09"

0 0.13" -0.99" 0.11" 0.05"

500 -0.08" -0.99" -0.03" 0.05"

1000 -0.12" -0.38" -0.08" -0.06"

1500 -0.08" -0.15" -0.03" -0.02"

2000 0.00" 0.00" 0.00" 0.00"

2500 0.06" 0.06" -0.02" -0.02"

3000 0.10" 0.10" -0.09" -0.09"

Sun, position(DE431) – position(DE406) in TT and UT

Year dL(TT) dL(UT)

-2999 0.21" -0.34"

-2500 0.11" -0.33"

-2000 0.09" -0.26"

-1500 0.04" -0.22"

-1000 0.06" -0.14"

-500 0.02" -0.11"

0 0.02" -0.06"

500 0.00" -0.04"

1000 0.00" -0.01"

1500 -0.00" -0.01"

2000 -0.00" -0.00"

2500 -0.00" -0.00"

3000 -0.01" -0.01"

Pluto, position(DE431) – position(DE406) in TT

Year dL(TT)

-2999 66.31"

-2500 82.93"

-2000 100.17"

-1500 115.19"

-1000 126.50"

-500 127.46"

0 115.31"

500 92.43"

1000 63.06"

1500 31.17"

2000 -0.02"

2500 -28.38"

3000 -53.38"

The 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 a tenth storage. Therefore for most applications it makes little sense to get the full JPL file. Precision comparison can be done at the Astrodienst web server. The Swiss Ephemeris test page http://www.astro.com/swisseph/swetest.htm allows to compute planetary positions for any date using the full JPL ephemerides DE200, DE406, DE421, DE431, or the compressed Swiss Ephemeris or the Moshier ephemeris.

      1. 2.1.2.1 Swiss Ephemeris and the Astronomical Almanac


The original JPL ephemeris provides barycentric equatorial Cartesian positions relative to the equinox 2000/ICRS. Moshier provides heliocentric positions. The conversions to apparent geocentric ecliptical positions were done using 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 1984, 1995, 1996, and 1997 (on p. B37-38 in all editions) to the last digit. Editions of other years have not been checked. DE200 was used by Astronomical Almanac from 1984 to 2002. The sample positions given the mentioned editions of Astronomical Almanac can also be reproduced using a recent version of the Swiss Ephemeris and a recent JPL ephemeris. The number of digits given in AA do not allow to see a difference. The Swiss Ephemeris has used DE405/DE406 since its beginning in 1997.

From 2003 to 2015, 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 (p. B61-B63), AA2011 and AA2013 (both p. B68-B70) to the last digit, i.e. to better than 0.001 arc second. (To avoid confusion when checking AA2006, 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. With later AA2011 and AA2013, there is no such problem.)

The Swiss Ephemeris uses JPL Ephemeris DE431 since version 2.0 (2014). The Astronomical Almanac uses JPL Ephemeris DE430 since 2016. The Swiss Ephemeris and the Astronomical Almanac still perfectly agree.

Detailed instructions how to compare planetary positions as given by the Swiss Ephemeris with those of Astronomical Almanac are given in Appendix D at the end of this documentation.


      1. 2.1.2.2 Swiss Ephemeris and JPL Horizons System of NASA


The Swiss Ephemeris, from version 1.70 on, reproduces astrometric planetary positions of the JPL Horizons System precisely. However, there have been small differences of about 52 mas (milli-arcseconds) with apparent positions. The same deviations also occur if Horizons is compared with the example calculations given in the Astronomical Almanac.

Horizons uses an entirely different approach and a different reference system. It follows IERS Conventions 1996 (p. 22), i.e. it uses the old precession models IAU 1976 (Lieske) and nutation IAU 1980 (Wahr) and corrects the resulting positions by adding daily-measured celestial pole offsets (delta_psi and delta_epsilon) to nutation.

On the other hand, the Astronomical Almanac and the Swiss Ephemeris follow IERS Conventions 2003 and 2010, but do not take into account daily celestial pole offsets.

While Horizons’ approach is more accurate in that it takes into account very small and unpredictable motions of the celestial pole (free core nutation), the resulting positions are not relative to the same reference frame as Astronomical Almanac and the Swiss Ephemeris, and they are not in agreement with the recent IERS Conventions 2003 and 2010. Some component of so-called frame bias is lost in Horizons positions. This causes a more or less constant offset of 52 mas in right ascension or 42 mas in ecliptic longitude.

Swiss Ephemeris versions 2.00 and higher contain code to reproduce positions of Horizons with a precision of about 1 mas for 1799 AD – today. Before 1799, the deviations in apparent positions between the Swiss Ephemeris and Horizons slowly increase. This is explained by the fact that Horizons uses the long-term precession model Owen 1990 for the remote past and future, whereas the Swiss Ephemeris uses the long-term precession model Vondrák 2011.

For best agreement with Horizons, current data files with earth orientation parameters (EOP) must be downloaded from the IERS website and put into the ephemeris path. If they are not available, the Swiss Ephemeris uses an approximation which reproduces Horizons still with an accuracy of about 2 mas between 1962 and present.

It must be noted that correct values for delta_psi and delta_epsilon are only available between 1962 and present. For all calculations before that, Horizons uses the first values of the EOP data, and for all calculations in the future, it uses the last values of the existing data are used. The resulting positions are not really correct, but the ephemeris is at least continuous.

More information on this and technical details are found in the programmer’s documentation and in the source code, file swephlib.h.

IERS Conventions 1996, 2003, and 2010 can be read or downloaded from here:

http://www.iers.org/IERS/EN/DataProducts/Conventions/conventions.html

Detailed instructions how to compare planetary positions as given by the Swiss Ephemeris with those of JPL are given in Appendix C at the end of this documentation.

Many thanks to Jon Giorgini, developer of the Horizons System, for explaining us the methods used at JPL.


      1. 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 old versions of the 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 that time range.

Note: Precession model P03 is said to be accurate to 0.00005 arc second for CE 1000-3000.

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

      1. 2.1.2.4 Differences between Swiss Ephemeris 1.78 and 1.77


Former versions of the Swiss Ephemeris had used the precession model by Capitaine, Wallace, and Chapront of 2003 for the time range 1800-2200 and the precession model J. G. Williams in Astron. J. 108, 711-724 (1994) for epochs outside this time range.

Version 1.78 calculates precession and ecliptic obliquity according to Vondrák, Capitaine, and Wallace, “New precession expressions, valid for long time intervals”, A&A 534, A22 (2011), which is good for +- 200 millennia.

This change has almost no ramifications for historical epochs. Planetary positions and the obliquity of the ecliptic change by less than an arc minute in 5400 BC. However, for research concerning the prehistoric cave paintings (Lascaux, Altamira, etc, some of which may represent celestial constellations), fixed star positions are required for 15’000 BC or even earlier (the Chauvet cave was painted in 33’000 BC). Such calculations are now possible using the Swiss Ephemeris version 1.78 or higher. However, the Sun, Moon, and the planets remain restricted to the time range 5400 BC to 5400 AD.

Differences in precession (v. 1.78 – v. 1.77, test star was Aldebaran):

Year Difference in arc sec

-20000 -26715"

-15000 -2690"

-10000 -256"

-5000 -3.95388"

-4000 -9.77904"

-3000 -7.00524"

-2000 -3.40560"

-1000 -1.23732"

0 -0.33948"

1000 -0.05436"

1800 -0.00144"

1900 -0.00036"

2000 0.00000"

2100 -0.00036"

2200 -0.00072"

3000 0.03528"

4000 0.59904"

5000 2.90160"

10000 76"

15001 227"

19000 2839"

20000 5218"
Differences in ecliptic obliquity
Year Difference in arc sec

-20000 11074.43664"

-15000 3321.50652"

-10000 632.60532"

-5000 -33.42636"

0 0.01008"

1000 0.00972"

2000 0.00000"

3000 -0.01008"

4000 -0.05868"

10000 -72.91980"

15000 -772.91712"

20000 -3521.23488”

      1. 2.1.2.5 Differences between Swiss Ephemeris 2.00 and 1.80


These differences are explained by the fact that the Swiss Ephemeris is now based on JPL Ephemeris DE431, whereas before release 2.00 it was based on DE406. The differences are listed above in ch. 2.1.1.3, see paragraph on “Comparison of JPL ephemerides DE406 (1998) with DE431 (2013)”.
      1. 2.1.2.6 Differences between Swiss Ephemeris 2.05.01 and 2.06


Swiss Ephemeris 2.06 has a new Delta T algorithm based on:

Stephenson, F.R., Morrison, L.V., and Hohenkerk, C.Y., "Measurement of the Earth's Rotation: 720 BC to AD 2015", Royal Society Proceedings A, 7 Dec 2016,



http://rspa.royalsocietypublishing.org/lookup/doi/10.1098/rspa.2016.0404

The Swiss Ephemeris uses it for calculations before 1948.

Differences resulting from this update are shown in chapter 7 on Delta T.

      1. 2.1.3 The details of coordinate transformation


The following conversions are applied to the coordinates after reading the raw positions from the ephemeris files:

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 amounts to 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 the gravitational fields of the sun and the planets light rays are bent. However, within the solar system only the sun has enough mass 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 a different 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). JPL ephemeredes since DE403/DE404 are referred to the International Celestial Reference System, a time-independent, non-rotating reference system which was introduced 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. Precession is the motion of the vernal equinox on the ecliptic. It results from the gravitational pull of the Sun, the Moon, and the planets 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 1.78 on, we use the precession model Vondrák/Capitaine/Wallace 2011.)

Nutation (true equinox of date). A short-period oscillation of the vernal equinox. It results from the moon’s 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 2.00, we use the nutation model IAU 2006. Since 1.70, we used 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 precision of daily motion is now better than 0.002” for all planets, and the computation is even a lot 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.

      1. 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.

This is how it works: The ephemerides of the Moon and the inner planets require by far the greatest part of the storage. A more sophisticated mechanism is required for these 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 a lot 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 the Swiss Ephemeris does not cover the time range of the full JPL ephemeris. The first ephemeris file begins on the date on which the last of the inner planets (including Mars) passes its first perihelion after the start date of the JPL ephemeris.)

With the outer planets from Jupiter through Pluto we use a simpler mechanism. We rotate the positions provided by the JPL ephemeris 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 before the end date of the JPL ephemeris.)

      1. 2.1.5 The extension of the time range to 10'800 years


This chapter is only relevant for those who use pre-2014, DE406-based ephemeris files of the Swiss Ephemeris.

The JPL ephemeris DE406 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 ancient astrology and archaeoastronomical research which would require a longer time range.

In December 1998 we have made an effort to extend the time range using 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 DE200, however, has been reproduced by Steve Moshier over a very long time range with his numerical integrator, which was available to us. We used this software 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 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

Deviations in heliocentric longitude from new JPL ephemeris DE431 (2013), time range 5400 BC to 3000 BC:

Moon (geocentric) < 40”

Earth, Mercury, Venus < 1.4”

Mars < 4”

Jupiter < 9”

Saturn < 1.2”

Uranus < 36”

Neptune < 76”

Pluto < 120”



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