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It was his favourite cat, and it bothered him constantly, so the master had no other choice. Times came by, the master passed away, and his followers continued the work of their teacher. Before starting a prayer, they looked for a cat, and it became a mandatory part of the praying ritual. They use exactly the same referential frame of the astronomers, except for a small minority of astrologers who study sidereal astrology and use a different ephemeris, based on the constellations.
In order to calculate an ephemeris you need complex mathematical formulas, but these are usually done by specialized software programs. The final result is in the form of a table with the exact position of a planet on the sky at hours every day. The values in the table represent angles, the combination of angles leading us to a financial chart with highs and lows and turning points. Let us see the graphic representation of the ephemerides of a planet. The last two tables showed the longitudinal ephemerides of the strong planets- Jupiter, Saturn, Uranus, Neptune and Pluto, and the latitudinal ephemerides of the weak planets - Mercury and Venus.
This is why the financial chart follows closely the path of these planets. Mercury, Venus and the Moon generate minor fluctuations, representing local Highs an Lows. Anyone interested can do the same and we are opened to dialogues on this topic. We just want to highlight the fact that there are correct ways of predicting the local High and Low and the reversal points. 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. This is the full precision state-of-the-art ephemeris.
It provides the highest precision and is the basis of the Astronomical Almanac. 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. As most previous ephemerides, it has been created by Dr. Myles Standish. According to a paper see below by Standish and others on DE of which DE is only a slight refinement , the accuracy of this ephemeris can be partly estimated from its difference from DE With the inner planets , Standish shows that within the period — there is a maximum difference of 0.
This means that the absolute precision of DE is estimated significantly better than 0. However, for the period — the deviations between DE and DE are below 0. With the moon , there is an increasing difference of 0. It comes mainly from errors in LE L unar E phemeris. Standish, X. Newhall, J. Williams, and W. Astrodienst has received permission from Dr. There are several versions of the JPL Ephemeris. The version is indicated by the DE-number. A higher number stands for a later update. The time range of this ephemeris DE is:.
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. The original JPL ephemeris gives barycentric equatorial Cartesian positions of the equinox Moshier provides heliocentric positions.
Using the DE data file, it is possible to reproduce the positions given by the Astronomical Almanac , , and down to the last digit. As for precession for centuries before or after , we follow Moshier who uses the coefficients of Williams 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.
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. To avoid discontinuities, we chose another procedure. See Appendix A. 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.
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 Apparent planetary positions are referred to the equinox of date. 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.
Oroboros – Astrology Ephemeris Swisseph
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. 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. A position with speed takes in average only 1.
With Moshier, however, a computation with speed takes 2. The idea behind our mechanism of ephemeris compression was developed by Dr. Peter Kammeyer of the U. Naval Observatory in 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.
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. This is the date, when the last of the inner planets Mars has its first perihelion after the start date of DE With the outer planets from Jupiter through Pluto we use a simpler mechanism.
We rotate the positions provided by DE 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 days each. This is the reason, why Swiss Ephemeris stops in the year 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 we have made an effort to extend the time range by our own numerical integration. The exact physical model used by Standish et. The previous JPL ephemeris, the DE, 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 DE time range. To test our numerical integrator, we ran it upwards from BC to BC for a period of years and compared its results with the DE 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 DE from the model in the DE 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 year test range between our integration and the JPL data. We could reproduce the Moon's motion over a the 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 BC to 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. Sun bary. The same procedure was applied at the upper end of the DE range, to cover an extension period from AD to AD. The maximum integration errors as determined in the test run AD down to AD are given in the table below. We expect that in some time a full integration program modeled after the DE integrator will become available.
At that time we will rerun our integration and report any significant differences. Its deviation from Chapront's mean node is 0 for J and keeps below 20 arc seconds for the whole period. With the apogee, the deviation reaches 3 arc minutes at BC. 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.
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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. 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 km. If one took this into account, it would result in a monthly oscillation of the Black Moon. However, we have neglected this effect. The 'true' lunar node is usually considered to be the osculating node element of the momentary lunar orbit. 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. 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. 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:. Michel, 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. 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. The reason is probably the same. The osculating apogee can be computed from any one of the three ephemerides.
There have been several other attempts to solve the problem of a 'true' apogee.
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 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. Note to specialists in planetary nodes and apsides: If important publications or web sites concerning this topic have been forgotten in this summary, your hint 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:. 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.
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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. 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. Here again, the ecliptic latitude would change sign at the moment when the planet were in conjunction with one of its nodes. Possible definitions for apsides and focal points.
The lunar apsides - the lunar apogee and lunar perigee - have already been dealt with 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.
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. 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.
These points form an opposition and are used by some astrologers, e. 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. 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. 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 VSOP 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 DE, we do not use them so far , because their validity is limited to a year period, because only linear rates are given, and because they are not based on a planetary theory.
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. 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. It makes no sense to use such points in astrology. 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. Osculating positions are given with Pluto and all asteroids.
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This is the default mode. 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. The names of these files are of the following form:. The size of such a file is about kb. All other asteroids are available in separate files. The names of additional asteroid files look like:. These files cover the period BC - AD. A short version for the years — AD has the file name with an 's' imbedded, ses.
The numerical integration of the all officiall numbered asteroids is an ongoing effort. In December , asteroids were numbered, and their orbits computed by the devlopers of Swiss Ephemeris. In January , the list of numbered asteroids has reached , 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:. It is not welcomed that anybody downloads more than such files per day, due to bandwidth problems in our Internet link; the total volume of the short asteroid files is about Mbyte. The list of these is found in Appendix B. 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. To generate our asteroid ephemerides, we have modified the numerical integrator of Steve Moshier, which was capable to rebuild the DE JPL ephemeris. Orbital elements, with a few exceptions, were taken from the asteroid database computed by E. Bowell, Lowell Observatory, Flagstaff, Arizona astorb. After the introduction of the JPL database mpcorb. 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 has shown a difference of only 0. 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. 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:. See also informations below on Ceres, Chiron, and Pholus.
The orbits of some asteroids are extremely sensitive to perturbations by major planets. Our integrator is able to detect such happenings and end the ephemeris generation to prevent our users working with meaningless data. The orbital elements of the four main asteroids Ceres, Pallas, Juno, and Vesta are known very precisely, because these planets have been discovered almost 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. But the tables in AA are based on older computations, whereas we used recent orbital elements. AA , page L 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 AD and a few arc minutes in BC.
As a result of close encounters with Saturn in Sept. Small uncertainties in today's orbital elements have chaotic effects before the year Do not rely on earlier Chiron ephemerides supplying a Chiron for Cesar's, Jesus', or Buddha's birth chart. They are completely meaningless. Pholus is a minor planet with orbital characteristics that are similar to Chiron's.
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It was discovered in Pholus' orbital elements are not yet as well-established as Chiron's. Our ephemeris is reliable from AD through now. Outside the 20th century it will probably have to be corrected by several arc minutes during the coming years. Dieter Koch has written the application program Ceres which allows to compute all kinds of lists for asteroid astrology. But the program does much more:.
The Swiss Ephemeris does not provide ephemerides of comets yet. A database of fixed stars is included with Swiss Ephemeris. The precision is about 0. 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. We include some astrological factors in the ephemeris which have no astronomical basis — they have never been observed physically. There have been discussions whether these factors are to be called 'planets' or 'Transneptunian points'.
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 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.
This may result in an error of a degree within the 20 th century, and greater errors for earlier centuries. Also, note that none of the real transneptunian objects that have been discovered since can be identified with any of the Uranian planets. 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.
This hypothetical planet was postulated by the French astronomer M. 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 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.