
2.2.1 Mean Lunar Node and Mean Lunar Apogee ('Lilith', 'Black Moon' in astrology)
JPL ephemerides do not include a mean lunar node or mean lunar apsis (perigee/apogee). We therefore have to derive them from different sources.
Our mean node and mean apogee are computed from Moshier's lunar routine, which is an adjustment of the ELP200085 lunar theory to the JPL ephemeris on the interval from 3000 BC to 3000 AD. Its deviation from the mean node of ELP200085 is 0 for J2000 and remains below 20 arc seconds for the whole period. With the apogee, the deviation reaches 3 arc minutes at 3000 BC.
In order to cover the whole time range of DE431, we had to add some corrections to Moshier’s mean node and apsis, which we derived from the true node and apsis that result from the DE431 lunar ephemeris. Estimated precision is 1 arcsec, relative to DE431.
Notes for Astrologers:
Astrological Lilith or the Dark Moon is either the apogee (”aphelion”) of the lunar orbital ellipse or, according to some, 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.
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 focal point, an opposite point makes no sense, of course.
Originally, the term ”Dark Moon” stood for a hypothetical second body that was believed to move around the earth. There are still ephemerides circulating for such a body, but modern celestial mechanics clearly exclude the possibility of such an object. Later the term ”Dark Moon” was used for the lunar apogee.
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 was not taken into account. As a result of this projection, we also provide an ecliptic latitude of the apogee, which will be of importance if declinations are used.
There may be still another problem. The 'first' focal point does not coincide with the geocenter but with the barycenter of the earthmoonsystem. 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 the Black Moon as the apogee, this oscillation would be about +/ 40 arc minutes. If one defines it as the second focus, the effect is a lot greater: +/ 6 degrees. However, we have neglected this effect.
[added by Alois 7feb2005, 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 EarthMoon Barycenter.
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 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 important because they are connected with 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, although a small one. Because of the monthly motion of the earth around the earthmoon 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 so with the mean nodes.
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 based on the idea that celestial orbits can be approximated by elliptical elements or great circles. The monthly oscillation of the node is explained by the strong perturbation of the lunar orbit by the sun. A different approach for the “true” node that would make sense, 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:
JPLderived node – SwissEphemerisderived node ~ 0.1 arc second
JPLderived node – Moshierderived node ~ 70 arc seconds
(PLACALC was not better either. Its error was often > 1 arc minute.)
Distance of the true lunar node:
The distance of the true node is calculated on the basis of the osculating ellipse of date.

2.2.3 The Osculating Apogee (astrological '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 19102010' (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 because planetary orbits are close to a twobody problem. But they are not good for the lunar orbit which is strongly perturbed by the gravity of the Sun (threebody problem). The lunar orbit is far from being an ellipse!
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. The motion in between those two points, is an oscillation with the period of a month. This oscillation is largely an artifact caused by the reduction of the Moon’s orbit to a twobody problem. The amplitude of the oscillation of the osculating apogee around the mean apogee is +/ 30 degrees, while the true apogee's deviation from the mean one never exceeds 5 degrees.
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 ChaprontTouzé. 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 a precision of at least one arc second is required, one has to choose either the JPL or the Swiss Ephemeris.
Maximum differences:
JPLderived apogee – SwissEphemerisderived apogee ~ 0.9 arc second
JPLderived apogee – Moshierderived 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 a lot 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 (astrological Lilith and Priapus)
As has been said above, the osculating lunar apogee (socalled "true Lilith") is a mathematical construct which assumes that the motion of the moon is a twobody 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 at an angle of 90° from 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
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 greater differences in nodes and apsides.
Definitions of the nodes
Methods described in small font are not supported by the Swiss Ephemeris software.
The lunar nodes are defined by 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:

They can be understood 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 nonecliptic 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.

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 in the ecliptic plane, one might consider the points of the earth’s orbit that are located in 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.

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 EarthMoonBarycenter (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 sotosay "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 welldefined 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 twobody 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 (twobody) orbit for a given time. Celestial bodies would follow such orbits if perturbations were to cease suddenly or if there were only two bodies (the sun and the planet) involved in the motion 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.
We 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 allow a sensible interpolation.
There is another problem about heliocentric ellipses. E.g. Neptune's orbit has often two perihelia and two aphelia (i. e. minima and maxima in heliocentric distance) 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 heliocentric orbit from an elliptic shape. Neptune’s orbit cannot be adequately represented by a heliocentric ellipse.
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 a lot faster than the motion of Neptune, and Neptune cannot react to 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 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 should 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, method 4) seems to make best sense.
In all of these modes, the second focal point of the ellipse can be computed instead of the aphelion.

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