Skyscript Astrology Forum

Hello
I'm sure this must have been dealt with somewhere on the forum, but I can't find it.
I've just finished working through Joseph Crane's book "Astrological Roots: The Hellenistic Legacy" (a good introduction and overview, but rather a lot of confusing mistakes). This prompted me to re-read Deborah Houlding's excellent article on the Classical Basis of Antiscia (here on Skyscript).
I chose sidereal quite early, as I really find it work better and more simply - the first (tropical) astrologer I saw told me there must be a mistake in my birth time, but I later found my military hospital birth certificate, 13:35, while my Mum had said 1:30pm. Crane's book confirms this choice for me - many of his example charts work more simply when shifted to sidereal, and of course the Hellenistic examples can pretty well be considered to be sidereal or tropical, as you like. At one point Crane says "As you may remember, Margaret Thatcher's Saturn is exalted in Libra" and uses this to prove his (tropical) point - I did remember that it's exactly at 21? Libra, but sidereally.
But the symmetries of antiscia and contra-antiscia in tropical astrology are very appealing, and while they can of course be worked out sidereally, tropical astrology makes them clearer. Antiscial signs (around the solstice axis, which have equal rising times at the equator, and equal day lengths and equal night lengths, e.g. Taurus days = Leo days) are deemed to have "equivalent" qualities. Contra-antiscia (around the equinox axis, and which have equal rising times and complementary day and night lengths, e.g. Taurus days = Aquarius nights) are deemed to have "complementary" qualities. So far, so elegant! But why not go the whole hog and put the solstice axis at 0? Leo/Aquarius, the equinox at 0? Taurus/Scorpio? Cyril Fagan suggested this back in the 70s, claiming that the zodiac first took shape in Egypt when the VP was in Taurus, and that the fixed signs are the natural "pillars" of the zodiac. Whether or not this is true, if the zodiac is considered to be arbitrary in terms of the constellations, why not go for symmetry? Antiscial "equivalent" signs would then have the same rulers, and contra-antiscial "complementary" ones would have complementary rulers (Mars/Venus, Mercury/Jupiter, Sun-Moon/Saturn).
I'm sure many tropical astrologers would say that the Aries zodiac "works", but we all find this, with the systems we know and have got used to. Or perhaps it's ,simply a tribute to Ptolemy and his epoch?
As things stand, it's true that a sidereal zodiac is, most of the time, out of sync in terms of symmetry of rising signs. But once every 26000 years, it is perfectly symmetrical: rising times, day and night lengths, rulerships. The modern tropical zodiac is permanently symmetrical around the solstice/equinox axis in terms of rising times, but eternally out of sync in terms of the rulership scheme. A Taurus-based tropical zodiac would be SO elegant!
(Perhaps even stranger than starting a tropical zodiac with Aries is saying that there is a "natural" sidereal one from Aries, Aries being the ""natural" first house, etc. This is frequently encountered in Indian sidereal astrology, and probably underlines the debt to the Greeks: it muyst have been introduced when already tropicalised, and was then re-siderealised and combined in a system with the Indian lunar mansions. As Fagan again pointed out, this has also led, for example, to the calculations of the Indian divisional charts from 0? Aries, when the navamsa, at least, could give better results when taken from 0? Taurus (for the harmonic 9th, repeating every 120?, this also lines it up with the other fixed signs: 0? Leo is 0? Leo in navamsa etc).
And my tropical chart actually makes some sense, I get the ascendant that that astrologer thought I was - so I might even convert, to a properly symmetrical tropical zodiac...
I would be interested in references to any documents you might know of about this question.
Graham

Graham Fox wrote:

But the symmetries of antiscia and contra-antiscia in tropical astrology are very appealing, and while they can of course be worked out sidereally, tropical astrology makes them clearer. Antiscial signs (around the solstice axis, which have equal rising times at the equator, and equal day lengths and equal night lengths, e.g. Taurus days = Leo days) are deemed to have "equivalent" qualities. Contra-antiscia (around the equinox axis, and which have equal rising times and complementary day and night lengths, e.g. Taurus days = Aquarius nights) are deemed to have "complementary" qualities. So far, so elegant! But why not go the whole hog and put the solstice axis at 0? Leo/Aquarius, the equinox at 0? Taurus/Scorpio? Cyril Fagan suggested this back in the 70s, claiming that the zodiac first took shape in Egypt when the VP was in Taurus, and that the fixed signs are the natural "pillars" of the zodiac. Whether or not this is true, if the zodiac is considered to be arbitrary in terms of the constellations, why not go for symmetry? Antiscial "equivalent" signs would then have the same rulers, and contra-antiscial "complementary" ones would have complementary rulers (Mars/Venus, Mercury/Jupiter, Sun-Moon/Saturn).
I'm sure many tropical astrologers would say that the Aries zodiac "works", but we all find this, with the systems we know and have got used to. Or perhaps it's ,simply a tribute to Ptolemy and his epoch?

Hi Graham,

I dont have much time tonight but I thought you might be interested in the pioneering work of one of our members here called Lunlumo. He is a German astrologer and has proposed looking at zodiacal quarter charts from the middle of the fixed signs rather than the start of the cardinal ones. He claims these times are more descriptive of mundane events. Its not identical to what you suggest but I thought it was close enough to be worth a look.

Here is a thread where Lunlumo introduces the idea. He calls it the Octilogram:

http://skyscript.co.uk/forums/viewtopic ... highlight=

regards

Mark

Mark
Thank you for this link. As you say, it's not quite the same as the symmetry I was thinking of. I probably shouldn't speculate on why the Chinese New Year corresponds to Sun at 15? Aquarius tropical, but it looks rather like the sort of thing that has happened in the West and India, with a sidereal zodiac becoming tropicalised in Ptolemy's time, exported to India where it was re-siderealised and then used to misdate obviously seasonal festivals (solstice/equinox related) which have now slipped to dates way off the phenomena they are supposed to mark. This problem is developed notably by Krishen Kaul in his critiques of the what he sees as the incoherence of the Indian sidereal zodiac. Kaul, and this problem, is mentioned in an interesting article by Dieter Koch:
http://www.astro.com/astrology/in_vedic2_e.htm
A problem with the application of this hypothesis to the "Chinese New Year at tropical 15 Aquarius" as noted by Lunlumo is that it would push back the origins of some sort of Chinese zodiacal astrology to somewhere around 3000BC, when the winter solstice was at around 15 Aquarius, which is may be possible but is not, I think, universally accepted.
Graham

Good morning,

The constellations are of unequal length and really sidereal. There are astrologers and schools of astrology that work directly with them instead of equal-length signs of any kind.

Those suggesting equal-length non-tropical signs would enhance their arguments by proposing objective criteria for selection of an arc of precession. Such criteria might be for example:

- inclusion of alpha stars of all 12 (13?) ecliptical constellations (see prior thread);

- most frequently used in practice for a) Indian b) Western astrologies;

- inclusion of the greatest total number of degrees of the constellations bearing the same names (this would mean subtracting the most degrees from the longest ecliptical constellations Virgo and Pisces).

Criteria like 'it works for me' and 'it apparently works better on such-and-such deliberately selected genitures' are obviously subjective. Proofs proposed by some of the 20th authors of 'Western sidereal astrology' have since been closely examined by Mr John Addey's associates and successors who demonstrated them, to the extent they were statistically valid, to be harmonic effects that do not depend on zodiac selection.

At times certain visible fixed stars are brighter than some wandering stars and thus have been explicitly included in astrological analyses by authors applying the tropical zodiac of the northern hemisphere, ex. gr. Klaudios Ptolomaios.

Best regards,

lihin

Lihin
The fixed stars are markers, "aide-m?moires", if you will. They do not define the sidereal zodiac. We all know that some astrologers work with fixed stars, unequal constellations, 13 constellations etc. This is not what most sidereal astrology is about. I suggest you look up to the sky at night: what do you see? If the sky is clear, you see the Milky Way, i.e. the galactic equator, a great circle, Addey's requirement for a starting point from which to calculate harmonics, and a second best, in his view, to the equinoctial colures.
The arc of precession is not the problem, it has been accurately determined by astronomers. The problem is where to start a sidereal division of the ecliptic.
I notice at least that you are no longer claiming that the "Hipparchus" ayanamsha puts all the alpha stars in the right sectors (it puts alpha Virginis at 3? Libra), so there is some progress! Try looking for an offset about 3? less, perhaps, if this criterion is the one that interests you.
In any case this thread is about the symmetry of the tropical zodiac. If you just wish to criticise the sidereal zodiac with the same tired old arguments, please start a new thread for that purpose.
Best regards
Graham

Good afternoon,

Is the fixed star Spica, brightest of the constellation Virgo, not Alpha Virginis? Hipparchos' zodiac showed it at about 25 ecliptical degrees of Virgo (see Prof. Dr. Auguste Bouche-Leclercq's L'astrologie grecque, page 140).

How bright is the Milky Way? To me it looks like a nebulous, uneven and not straight band of varying thickness, scarcely if at all visible against urban light pollution.

I have no basic objections to any astrological systems but prefer those that attempt to be coherent within their own frameworks. My question is entirely within the framework of systems proposing 'sidereal' zodiacs of equal-length signs. What are objective selection criteria, please? How should a student choose amongst the dozens of such zodiacs listed?

Best regards,

lihin

Lihin
Using Morinus, in Options, you can select "ayanamsha Hipparchos" and Spica (and others) in the Fixed Stars option. Then you load any chart, and see where those fixed stars are in the Tables > Fixed Stars. With "Hipparchos" (these are the Swiss Ephemeris definitions of ayanamsha), you will see that it is at about 3?35'30" Libra. With Lahiri, you will find it at 29?58'59" Virgo, with Fagan at 29?18'21 Virgo. You can do the same for any other so-called "fixed" stars (which actually have their own very slow proper movment from our point of view, so over the centuries the values will move by a few minutes, even sidereally).
The 25? Libra value given by Bouch?-Leclercq must refer to the tropical (moving, subject to precession, calculated from so-called "0? Aries") value in Hipparchos day. Cast a tropical chart for 120BC with Spica, you'll see. If you are interested in how Hipparchos determined which point on the ecliptic he would call "0 Aries", etc, I suggest you have a look at the post in French which I sent you by PM some time ago.
The Milky Way does indeed look milky, but then the earth might look muddy to an earthworm, yet it is still deemed to have an equator, which divides it into two distinct halves with different polarities and where water goes down the sink in different directions. It is our galaxy, we are inside it, it has polarity and it is much more symmetrical than it looks to us; although astronomers are divided about how precisely its poles and central plane can be calculated, the US Naval Observatory felt confident enough to state that on 27/10/98 it was at exactly 270? on the ecliptic (i.e. at the winter solstice point, 0? so-called "Capricorn" tropical, with the opposite colure at the summer solstice point, of course, since it is a great circle bisecting the ecliptic, as Addey pointed out). The respected Belgian astronomer Jean Meeus calculated May 1998. I'll spare you the numerous New Age fantasies on the subject, but here's a link to a hard-headed illustrated article which seeks to downplay the whole phenomenon:
http://www.2012hoax.org/galactic-equator-vs-plane
As you may not have time or inclination to read it, here is a passage relevant to this question:

We sometimes hear that the Sun aligns with the Galactic Equator near the Winter Solstice of each year, but that's only because we tend to look at things from our from our provincial geocentric perspective. The ecliptic plane (i.e., the plane that contains the Earth's orbit) passes through the Center of the Sun, just like the "Galactic Equator", but the two are not parallel. Therefore, the ecliptic plane and the "Galactic Equator" intersect along a line that passes through the Sun's center. When the Earth crosses that line in its yearly orbit around the Sun, the Sun appears (from our perspective) to "align with the intersection of the ecliptic and the galactic equator". Actually, it's the Earth that's getting in line. [?] Conclusion: A calculation of the date on which the solstice points cross the galactic equator gives 1998, but the probable errors in the position of the galactic pole as defined by the IAU mean that this date could fall anytime between the years 1986 and 2011.

This galactic/ecliptic colure is currently one of the most "fixed" points we have, on the ecliptic, but a problem for siderealists is of course how to deal with this margin of error (just under 21' arc minutes) and where to locate this crossing point on the ecliptic. In terms of the traditional ecliptical constellations, it's somewhere in early Sagittarius. Most commonly used ayanamshas would place it (if it is defined as by the US Naval Observatory) between about 5?15' (Fagan) and 7?36 (Raman), though "Hipparchos" would give 9?58 (the ayanamsha just being a facility for converting from the conventional tropical values - but which unfortunately gives the impression that the sidereal zodiac is the moving one). I readily admit that these disagreements are a problem, as are house systems and many other things in astrology. We tend to deal with them in the same way: we read, follow the example of others, try things out, adjust them by experience, etc. Many, especially in India, just take what is proposed by their teacher, and stick with it, which may not be such a bad idea.
All this is rather off-topic, so perhaps we can leave it there, unless you wish to start a new thread.
Any ideas about a symmetrical tropical zodiac starting with 0? "Taurus" (or Leo, Scorpio or Aquarius)?
Best regards
Graham

Graham F wrote:... the US Naval Observatory felt confident enough to state that on 27/10/98 it was at exactly 270? on the ecliptic... The respected Belgian astronomer Jean Meeus calculated May 1998.

Unless one wants its "mean" precessing position only, the "apparent" position of the galactic pole fluctuates a little due to annual aberration and nutation like any other fixed star. It reached 270 on 3-4 Feb 1998, then went retrograde and crossed it backwards on 18-19 June, crossing the 270 line forward once more on Nov 10-11 1998.

The statistical marging of error (calculated in the late 1950's) is 7.5 arcminutes (30s) in right ascension, which precession-wise is the equivalent of about 12.5 years.

You can download the original IAU technical paper here:

http://adsabs.harvard.edu/abs/1960MNRAS.121..123B

J.

Juan
Thank you. This 1958 paper is included, along with several more recent published work, in the bibliography of the article which I linked to in my previous post:
http://www.2012hoax.org/galactic-equator-vs-plane
In fact this recent article is broadly a simplified, illustrated and updated summary of key aspects of the 1958 paper, dealing notably with a clarification of the reference systems used ("galactic equator"/H1 plane on the one hand, "galactic plane" on the other). The author is writing from an astronomer's point of view, and seems more interested in a heliocentric perpective on these, but does remind us that the earth crosses the intersection of the ecliptic and the galactic equator (twice) each year, and that in 1998 that crossing was at the solstice points:
"Conclusion: A calculation of the date on which the solstice points cross the galactic equator gives 1998, but the probable errors in the position of the galactic pole as defined by the IAU mean that this date could fall anytime between the years 1986 and 2011."
The author's explanation that this alignment during our our epoch is geocentric is rather laboured, and fairly obvious to astrologers:
"We sometimes hear that the Sun aligns with the Galactic Equator near the Winter Solstice of each year, but that's only because we tend to look at things from our from our provincial geocentric perspective. The ecliptic plane (i.e., the plane that contains the Earth's orbit) passes through the Center of the Sun, just like the "Galactic Equator", but the two are not parallel. Therefore, the ecliptic plane and the "Galactic Equator" intersect along a line that passes through the Sun's center. When the Earth crosses that line in its yearly orbit around the Sun, the Sun appears (from our perspective) to "align with the intersection of the ecliptic and the galactic equator". Actually, it's the Earth that's getting in line."

In any case, the world didn't end, and this 1998 solstice axis alignment is only of interest in as much as it gives a round figure to locate the intersection (colure) when expressed tropically. Once located, some astrologers have suggested using it sidereally as an anchor on the ecliptic, considering it to mark, for example, 5? Sag sidereal.

You refer to the the galactic pole as having very small fluctuations in its precessed position on the tropical ecliptic; perhaps part of this can be accounted for by the earth's nutation and polar motion, which also affect anything observed from earth, I think, tropically or sidereally (precession is not quite uniform throughout the year or from one year to the next, and surely it's the earth that "wobbles" more than the "fixed" stars or the galactic poles, which are themselves an extrapolation from an aggregate of fixed stars etc). You say the galactic pole "reached 270 on 3-4 Feb 1998, then went retrograde and crossed it backwards on 18-19 June, crossing the 270 line forward once more on Nov 10-11 1998." Don't you mean the galactic equator, as described above? The galactic poles are perpendicular to the equator, and thus must have been at 0? and 180? tropical in 1998. The precise dates you give (you don't say where they're from) cover quite neatly the dates (presumably the mean ones), for the coincidence of galactic equator (not the pole) and the solstice axis (90?-270?) given in 1997 by Jean Meuss and the US Naval Observatory. Assuming this is what you mean, you give a statistical margin of error considerably less than that given by the author I linked to, but it is fairly clear he was erring on the side of skepticism with regard possible accuracy, as his aim was to debunk the more far-fetched 2012 excitement.
Funny how we got here from a question about the assymmetry of rulerships versus the symmetry of rising times in the tropical zodiac...
Graham

Graham F wrote:... some astrologers have suggested using it sidereally as an anchor on the ecliptic, considering it to mark, for example, 5? Sag sidereal.

This implies a zero point (or tropical/sidereal alignment) in 1998, so the sidereal zodiac would be just like a precession-corrected tropical zodiac for someone born in 1998.

You refer to the the galactic pole as having very small fluctuations in its precessed position on the tropical ecliptic; perhaps part of this can be accounted for by the earth's nutation and polar motion...

All reference star positions including the galactic center, pole, node, etc. are mean heliocentric positions, and the effect when moving to a geocentric perspective is called "annual aberration" which amounts to about 20" twice a year. This results in a small apparent "retrogradation" which affects both sidereal and tropical positions. Tropically the 2nd largest deviation from a mean position is caused by nutation which amounts to approx. 19" every 18 or 19 years.

Don't you mean the galactic equator, as described above?

I think it is the same. When the Sun aligns with the galactic pole (i.e. is in conjunction with the ecliptic position of the pole) it is perpendicular to it and exactly parallel to the plane of the galaxy (as has been conventionally defined), which is the definition of the galactic node on the ecliptic. So the node on the ecliptic and the ecliptic position of the pole are the same point... that's how I understand it at least.

J.

Juan wrote:

Graham F wrote:... some astrologers have suggested using it sidereally as an anchor on the ecliptic, considering it to mark, for example, 5? Sag sidereal.

"This implies a zero point (or tropical/sidereal alignment) in 1998, so the sidereal zodiac would be just like a precession-corrected tropical zodiac for someone born in 1998."

[No - if the galactic node is considered to be at 5? sag sidereally (in 1998 or any other year) someone born in 1998 with e.g. 0? Aries tropical rising would have 5? Pisces rising sidereally. The "zero point" would be in 221 AD. I'm not interested in 1998 or 2012, it's just that the "solstice alignment" prompted astronomers to issue a precise location for the galactic equator/ecliptic intersection. They could just as well have said that it was at 269? in 1926. I think this great circle intersection gives, as John Addey said back in the 70s, a better relatively fixed point than such and such a "fixed" star from which to define a non-precessing division of the ecliptic. We still have the problem of deciding what to call it: 5? Sag? 6?? 7?? So we can still argue about ayanamsha.]

You refer to the the galactic pole as having very small fluctuations in its precessed position on the tropical ecliptic; perhaps part of this can be accounted for by the earth's nutation and polar motion...

"All reference star positions including the galactic center, pole, node, etc. are mean heliocentric positions, and the effect when moving to a geocentric perspective is called "annual aberration" which amounts to about 20" twice a year. This results in a small apparent "retrogradation" which affects both sidereal and tropical positions. Tropically the 2nd largest deviation from a mean position is caused by nutation which amounts to approx. 19" every 18 or 19 years."

[OK, but I think aberration is a phenomenon of light, and many astronomical calculations (including galactic) also involve radio sources ("aberration causes a displacement of the apparent position of an object from its true position" - Wikipedia). Astronomy and astrology programs give the option of "true" versus "apparent" positions (this is not the same as parallax or not), and I'm not sure that astronomers usually calculate stars' positions and galactic coordinates in apparent terms, as you suggest. But if you're right, and it is the apparent position that is used, it is indeed another factor of imprecision for "siderealists" in general, or anyone using "fixed" stars. But I'm rather surprised that Meuss, Smelyakov and the US Naval Observatory didn't think to take account of aberration in giving their rahter precise dates in 1998.]

Don't you mean the galactic equator, as described above?

"I think it is the same. When the Sun aligns with the galactic pole (i.e. is in conjunction with the ecliptic position of the pole) it is perpendicular to it and exactly parallel to the plane of the galaxy (as has been conventionally defined), which is the definition of the galactic node on the ecliptic. So the node on the ecliptic and the ecliptic position of the pole are the same point... that's how I understand it at least."

I'm sure that's not right. The galactic pole is perpendicular to the galactic equator, which from a geocentric point of view is oblique to the ecliptic. It's like the relationship between the ecliptic and the earth's equator: the points on the equator which bisect the ecliptic are 90? of longitude away from the projection of the poles onto the equator, otherwise the equinoxes and solstices would happen at the same time.

Do you have that source reference for your three dates in 1998 when the galactic something (you say pole, I say equator)

"reached 270 on 3-4 Feb 1998, then went retrograde and crossed it backwards on 18-19 June, crossing the 270 line forward once more on Nov 10-11 1998.

?

Here's a brief and recent r?sum? of this business, with some links for those who want to go into more depth. Curiously doesn't mention the US Naval date of October 1998, but to Meuss adds Smelyakov, who "refined" Meuss's May 98 date:
http://2012wiki.com/index.php?title=Galactic_Alignment
Thanks for your interest!
Graham

Thanks for your comments, Graham. It gives me the opportunity to clarify things in my head... hopefully.

No - if the galactic node is considered to be at 5? sag sidereally (in 1998 or any other year) someone born in 1998 with e.g. 0? Aries tropical rising would have 5? Pisces rising sidereally. The "zero point" would be in 221 AD.

I get it now... thanks for the clarification.

it's just that the "solstice alignment" prompted astronomers to issue a precise location for the galactic equator/ecliptic intersection.

It is exactly the ecliptic position of the galactic pole plus 90 degrees (see below for more on this). The position of the pole is a statistical approximation that was arbitrarily defined as "exact" for the purposes of coordinate conversions, so it would be interesting to know if there has been a new/fresh recalculation of these points recently.

but I think aberration is a phenomenon of light, and many astronomical calculations (including galactic) also involve radio sources. Astronomy and astrology programs give the option of "true" versus "apparent" positions

Yes. But I can't imagine not using apparent for fixed stars, they are so far away that all of them would become invisible, or their "true" position would become meaningless.

and I'm not sure that astronomers usually calculate stars' positions and galactic coordinates in apparent terms, as you suggest

If one were to calculate when the radio waves or light is emitted by a stellar object instead of when it hits the earth, one would have to wait centuries or millenia, not just a few minutes as with objects inside the solar system. So geometric/true positions makes no sense for objects outside the solar system.

But if... it is the apparent position that is used, it is indeed another factor of imprecision for "siderealists" in general, or anyone using "fixed" stars.

You don't have to use a fixed star or radio source to define a sidereal fiducial, all you need is a point in time (an "epoch") and a sidereal reference frame. The sidereal reference frame has already been established with great precision by modern astronomy, so there is no room for imprecision as long as one uses that reference frame.

But I'm rather surprised that Meuss, Smelyakov and the US Naval Observatory didn't think to take account of aberration in giving their rahter precise dates in 1998

If they don't use nutation or aberration the positions will not be very precise. It would be interesting to know how they did it and why they obtained different results from each other, but for the moment I only know how I did it (see the end of this post for more).

So the node on the ecliptic and the ecliptic position of the pole are the same point... that's how I understand it at least."

I'm sure that's not right. The galactic pole is perpendicular to the galactic equator, which from a geocentric point of view is oblique to the ecliptic.

Tanks for the explanation. I went back to how I was making the calculation and realized that the 2 points were exactly 90-degrees apart, they were not the same point (I was using an ephemerides wrongly labelled as "galactic pole" instead of "galactic node" and that had me confused).

A simple ecliptic conversion of the reference right ascension and declination of the galactic pole (192.25 and +27.4 in 1950) shows this relationship: the ecliptic longitude of the pole is right now (as I write this) 0,12'50" Libra, latitude 29n49, and it was 0 Libra in 1998, so the node of the galaxy on the ecliptic is 90 degrees from this minus the latitude. This relationship is evident from the explanation given in the "2012 hoax" link you provided:

We sometimes hear that the Sun aligns with the Galactic Equator near the Winter Solstice of each year... the ecliptic plane and the "Galactic Equator" intersect along a line that passes through the Sun's center. When the Earth crosses that line in its yearly orbit around the Sun, the Sun appears (from our perspective) to "align with the intersection of the ecliptic and the galactic equator".

when does the Solstice occur closest to the time when the Sun is aligned with the intersection of the ecliptic and the galactic equator?

His hand calculation gives the same result as calculating when the ecliptic longitude of the galactic pole is 180 degrees (0 Libra).

Do you have that source reference for your three dates in 1998?

I use my own program Riyal.

Also, I'm not sure what you mean by "reached" and then "backwards"

You are not familiar with calculating the apparent position of stars. The reason is (mostly) the annual aberration as explained.

In the link you provided there is a reference to Meeus, and as it turns out, I have a copy of "Mathematical Astronomy Morsels" which is given as the source. His exposition is in pages 301-303, and basically, he is doing the same thing I did, but with mean positions only: no nutation, no aberration, just precession in longitude. After calculating the B1950 ecliptic coodinates of the pole (lambda) he writes: "Because the galactic equator interects the ecliptic at the 2 points with longitudes lambda-90 and lambda+90, it will pass exactly through the solticial points when lambda is equal to 180 (or 360)"

He proceeds to calculate 5 dates separated by 50 years, and then uses interpolation to find when the tabular value is 180. He says May 1998 without giving any specific day. Using a mean position only (no nutation and no aberration), I get 3-4 May 1998. Nutation at the time was -8 arcseconds, which added to aberration will produce the results I gave.

NOTE: a calculation with the nutation added but with no aberration gives 25-26 June 1998... this covers Meeus. Does anyone know the source for the "Usno date" later that year?

J.