Requiem for Relativity

Hi Joe, I was looking in an old book by Carl Sagan, and in it he talks about a strange radio emission lobe about thirty degrees from galactic centre. Actually it looks a lot like a "willy." It curves round, ending up at its tip near Crater.

[Today, March 3, 2011, I formally completed the submission of this paper to the peer-reviewed journal, Celestial Mechanics (Springer). The submitted version is on p. 53 of this thread, originally dated Nov. 15, 2010, but improved considerably until Feb. 28, 2011. Before submitting to Celestial Mechanics, I had submitted it twice to Observatory, but minus the best section, Section IX. In contrast to Celestial Mechanics, whose editor rejected the paper within a day without sending it for peer review, Observatory responded appropriately, peer-reviewing it after each submission: the peer reviews amounted to several pages, including a few quantitative checks of my figures, which found no errors. However, Observatory declined to publish the paper. Here is the most improved version. I'll continue to make improvements by editing here on p. 56, while leaving on p. 53 of this thread, the version submitted to Celestial Mechanics.

(Apr. 9) I have now sent the paper to the editor of Astronomische Nachrichten but have heard nothing, not even an acknowledgment (as of May 5); maybe sending the paper in the body of an email was not acceptable to him and deemed unworthy of response. So, tomorrow I'll arrange to print up nice copies that I'll distribute myself.]


Mars' medieval and ancient orbital period

Abstract. Vedic planetary and lunar parameters, mostly originating c. 3100BC, conform superbly to modern theory, except that the Vedic orbital period for Mars is short. Most of the best ancient and medieval information (Seleucid Babylonian, Arabic/Byzantine/Persian, and most moderns before LeVerrier) shows a swift linear time trend in Mars' orbital period. However, Ptolemy and Kepler, and also the Seleucid Babylonian record of a Mars stellar conjunction at stationarity, conform well to modern theory. The apparent anomaly of Mars' orbital period, shows suggestive relationships not only with Earth's precession period, but also with the deviation of the Jupiter-Saturn Great Inequality from its average value.



Contents

I. Introduction
II. The 1336AD Trebizond & related 1353AD almanac tables
III. The Trebizond and Alfonsine almanac parameters
IV. Refining the almanac parameters
V. The 1590AD Heidelberg Venus-Mars occultation
VI. The 251BC Seleucid conjunction of Mars with Eta Geminorum at stationarity
VII. Seleucid and Ptolemaic parameters
VIII. Han China
IX. Vedic India
X. Discussion
Appendix 1. Plethon's mean motion of Mars
Appendix 2. Kepler's mean motion of Mars
Appendix 3. Ancient sidereal year and mean solar day



I. Introduction.

The well-known 13::8 Venus::Earth conjunction resonance point, regresses with period 1192.8 yr, according to prevailing modern estimates of the planets' periods. Likewise the analogous 19: Mars::Jupiter conjunction resonance point, progresses with period 2*1188.48 yr, twice the period of the Venus-Earth point.

Suppose this -2::1 ratio, of the Venus::Earth regression to the Mars::Jupiter progression rates, is constant. Maybe an unknown force, acting on triads instead of pairs of bodies, causes this. The orbits of Venus and Earth are nearly circular and relatively close to the Sun. The orbital period of Mars might be much more affected by Jupiter or other perturbations. Suppose the orbital periods of Venus and Earth are practically constant, so the Venus::Earth regression rate stays the same. Then, the orbital period of Mars must change by the same proportion as that of Jupiter, if the Mars::Jupiter progression rate is to stay the same too.

If Jupiter and Saturn maintain constant total J+S orbital angular momentum, nearly constant semimajor axes, and small eccentricity, then their orbital frequencies must change by almost exactly equal and opposite amounts. Then the Great Inequality of Jupiter and Saturn, can be restored to its calculated average value, 1092.9 yr [1, p. 287], only by shortening Jupiter's period, by about 1 part in 3000. According to the speculation of the previous paragraph, the orbital period of Mars also would shorten by 1 part in 3000.

The published calculations based on the theory of gravity, say that planetary orbital periods change only very slowly; Mars' orbital period shortens only 1 part in 350,000,000 per 1000 yrs [2, p. 24][3]. At that rate, 10^8 yr might be needed, in a linear extrapolation, to restore the Great Inequality to its average value. If fluctuations toward or away from the average Great Inequality occur much faster, there might be historical evidence. The purpose of this paper is to examine medieval and ancient records for evidence of a change in the orbital period of Mars.

Sometimes upper and lower bounds will be found, often due to rounding error in the information source, restricting the result to an interval of a priori uniform probability. Sometimes several results found by similar methods, will be averaged with equal weight. Always, uncertainty will be given as root-mean-square ("rms") using, when appropriate, Standard Error of the Mean ("SEM").



II. The 1336AD Trebizond & related 1353AD almanac tables.

Trebizond was "the last refuge of Hellenistic civilization" [4, article "Trebizond, empire of"]. The 1336AD Almanac for Trebizond was authored by Manuel, a Christian priest of Trebizond, but apparently referred to parameters from contemporary Persian and Arabic almanacs. Mercier [5] also includes fragments of three 1353AD almanacs: the 1353AD Constantinople almanac by Manuel's student Chrysococces, and two 1353AD Persian almanacs, the Zij-i Ilkhani & Zij-i Alai, on which that of Chrysococces might in part be based [5, p. 161].

The Trebizond almanac table is said [5, p. 60] to give noon planetary, lunar and solar longitudes to the day and arcminute (the other almanacs resemble it, except that the Persian almanacs give longitudes to 0.1'). Because of the mathematical and astrological importance of planetary conjunction, the almanac should be most accurate then. Consideration of longitude differences, circumvents exact determination of the almanac's equinox. Interpolating, I find the geocentric ecliptic longitude difference between Venus and the Sun, at the time when Mars and Venus have equal geocentric longitudes, according to the Trebizond almanac for Muharram 10 & 11, 1336AD, and again according to the JPL ephemeris for 0h Aug. 19 & 20 [6]. In [6] I work from heliocentric planetary positions corrected for light time and for Earth surface parallax at Trebizond. I neglect the small error due to use of the J2000.0 ecliptic for heliocentric longitudes in [6] but use of the approximate ecliptic of date in the Trebizond almanac.

Ideally, the Venus-minus-Sun longitude difference, at Venus-Mars conjunction, would be the same in the medieval almanacs as in [6]. Actually, when Venus-minus-Sun is the same, the discrepancy in the Venus & Mars longitudes is 10.3 arcminutes in the Trebizond almanac; almost three times it, in the Persian almanacs; and almost ten times it, in Chrysococces' almanac. So henceforth I consider only the Trebizond almanac: the most favorable 0.5 arcminute rounding corrections, reduce its discrepancy to 8.0'. This discrepancy can be removed by moving Mars 857 minutes of time, prograde in its orbit in [6].

In my Sec. V, I find that the Moestlin/Kepler description of the 1590AD Venus/Mars occultation, most reasonably implies that Venus' orbital period is slightly longer than in [6]; this amounts to moving Venus 93.6 minutes of time prograde in 1336AD in [6]: with such a corrrection, Mars need be moved only 418 minutes of time prograde in [6]. (Relying on [6] circumvents the need to make elliptical orbit calculations requiring the formulas given in, inter alia, [7, eqns. 4.57 & 4.60, pp. 84, 85].) Mars' orbit is more eccentric and more subject to perturbation by Jupiter, so Mars' JPL ephemeris for a remote time would be less accurate than for Venus or Earth.

Kieffer's modern figure [2] for 2000.0AD is equivalent to a Mars sidereal orbital period of

686.9798529 d

(Newcomb [ 8 ] cites LeVerrier's figure, which is 686.9798027 d, epoch 1850.0AD, corrected for day lengthening, 2ms/century from 2000.0AD.)

Thus the Trebizond almanac likely implies a Mars orbital period of

686.9798529 d * (1 + 418min/(1982-1336yr)) = 686.980698 Julian d

Indeed with these adjustments to the JPL ephemeris, not only are the relative longitudes of the Sun, at Venus-Mars conjunction, the same for the JPL and Trebizond ephemerides; also, the Venus-Mars conjunction occurs at the same time of day in the two ephemerides. To show this, I first show that this portion of the Trebizond almanac gives positions for midnight, rather than noon as stated in [5, p. 60]: the three differences, between Sun, Venus or Mars longitudes in the Trebizond and in the JPL ephemeris, respectively, should be nearly equal when the time of the JPL ephemeris equals the time of the Trebizond almanac entry for, say, Muharram 10. These differences have least variance when the time of the JPL ephemeris is 0.19 day before 0h UT August 19. Correcting for 2ms/cyr day lengthening, this is 0.21 day before Greenwich mean midnight Aug. 19, and due to the longitude of Trebizond (modern Trabzon) 0.10 day before Trebizond mean midnight. Twelve hours earlier or later, the variance of the three differences is 16 times larger. So reference of this portion of the Trebizond almanac, to midnight rather than noon, explains roughly 7/8 of the variance in the longitude differences. The Sun's geocentric longitude in the JPL ephemeris (geocentric longitudes in [6] are for the equinox and ecliptic of date) for Trebizond mean midnight Muharram 10 (i.e. 21:50 UT Aug. 18) is 0.4061 deg less than in the Trebizond almanac; this error could arise from refraction when the Sun is nearly on the horizon.

According to the Trebizond almanac, Venus-Mars conjunction occurs 0.344 d after the reference time (shown above to be Trebizond mean midnight) Muharram 10. With the above adjustments, conjunction occurs 0.205 d after 0h Aug. 19 according to [6]; due to 2ms/cyr day lengthening, this is 0.186 d after mean Greenwich midnight, and due to Trebizond's longitude, 0.296 d after Trebizond mean midnight, differing only 0.048 d = 69 minutes of time (equivalent to 1.5' Venus-Mars longitude difference) from the Trebizond almanac of 1336AD. Had I not adjusted Venus, but only adjusted Mars prograde 857 minutes of time (see above) the JPL time would have been 0.458 d after Trebizond mean midnight.



III. The Trebizond and Alfonsine almanac parameters.

Mercier [5; pp. 180, 181, 183, 184] identified many medieval Islamic almanacs as possible sources for the parameters of the Trebizond almanac. Four of these likely give especially accurate Mars tropical periods, because they seem to have chosen observation intervals optimized via the approximation, 66*365d = 35 Mars periods. The four are:

Zij al-Sanjari, c. 1135AD, 0;31,26,39,36,34,05,16,50
(astronomer: al-Khazini; reference meridian: Merv)

Taj al-Azyaj, late 13th cent. AD, 0;31,26,38,16,02,26
(astronomer: al-Maghribi; ref. merid. Damascus)

Adwar al-Anwar, late 13th cent. AD, 0;31,26,39,44,40,48
(astronomer: al-Maghribi; ref. merid. Maraghah)

Zij al-Alai, 12th cent., 0;31,26,39,51,20,01
(astronomer: al-Shirwani; ref. merid. Shirwan)

with Mars' tropical orbital daily mean motion expressed in their sexagesimal notation (deg; ', '', etc.). Using either epochs 1135AD or ~1280AD as appropriate, and assuming that these mean motions are simply Mars' sidereal mean motion plus the sidereal motion of Earth's equinox (see Part IV below), and using Newcomb's precession formula P = 50.2564" + 0.0222" * T (in tropical cent. since 1900.0AD)[9], I find the three implied Mars sidereal periods:

686.9780971 d, epoch 1100AD
686.9862748 d, epoch 1252AD
686.9773086 d, epoch 1252AD
686.9765930 d, epoch 1100AD

in days of epoch, where I have subtracted ~66/2 yr from what I might otherwise have estimated historically as the epoch.

The Alfonsine astronomical tables, commissioned by Alfonso the Wise of Toledo [10][11] have

stated epoch 1252AD: 0;31,26,38,40,05

from which I find, as above, the implied sidereal period

686.9838366 d, epoch 1252AD

The Yuan Mongol dynasty ruled China c. 1300AD; Yuan Chinese records give the synodic period of Mars as 779.93d [12, Table 10.2, p. 518]. My conversion from synodic to sidereal period, gives 686.985d, epoch 1300AD, where the last digit is of doubtful significance. The Yuan figure might also indicate borrowing from contemporary Persian sources.



IV. Refining the almanac parameters.

The mean daily motions in Part III, are in days of the epoch, not Julian days. I will correct the period for day lengthening, 2ms/century [13][14]. The correction factor for 1200AD is 1 - 1.85/10^7.

Vimond, a medieval French astronomer, published mean motions of Mars, Jupiter and Saturn, which all are less than those of the Parisian Alfonsine tables by about 68.3"/yr. Chabas & Goldstein [11, p. 271] note that the difference is near 54.5" + 12.9" = 67.4", where 54.5" is the precession given by the medieval authority al-Battani as 1deg/66yr, and 12.9" is the advance of Earth's perihelion given by its 11th century AD discoverer Azarquiel as 1deg/279yr (or rediscoverer: Hindu figures, which, controversially, might or might not originate prior to Azarquiel, give (1/9)"/yr = 11"/century for Earth's perihelion advance [15][16], which I surmise might in the uncorrupted text have been 11"/yr, which is more accurate than Azarquiel's 12.9"). This indicates that Vimond thought, that the Alfonsine tables give the mean sidereal orbital motion plus the motion of Earth's equinox. Vimond converted the Alfonsine mean motions, to the mean sidereal orbital motion minus the motion of Earth's perihelion.

Of these five especially credible and influential medieval published Mars periods (which in Part III, I converted to sidereal periods assuming, like Vimond, that the originals were tropical periods) three fall into a short-period group,

686.9780971 d, 686.9765952, & 686.9773086 d (resp. two early & one late Trebizond precursor)

and two fall into a long-period group,

686.9862748 d (late Trebizond precursor) & 686.9838366 d (Alfonsine)

A likely explanation for the differences among the three epoch ~1252AD periods, is that starting with Mars at some special longitude, such as the vernal equinox or Mars' perihelion, teams of astronomers waited ~66 yr for Mars to return to that longitude (heliocentric longitude, i.e. geocentric longitude at opposition) for the 35th time. At 1252AD, the vernal equinox was at J2000 10.4deg longitude [9], Mars' perihelion at J2000 332.7deg, and Mars' eccentricity 0.0927 [2]; in 35*1.881 yr, the equinox had regressed 0.916deg. If Mars' orbit were circular, Mars' tropical year then would be shorter than Mars' sidereal year by a fraction 0.916/360/35 = 7.27/10^5. Instead, the reciprocal of Mars' angular speed, at the vernal equinox, was about (1-e^2)^1.5 / (1+e*cos(10.4-332.7))^2 = 0.857 times as much as for a circular orbit, i.e. the shortening effect was only a fraction 6.23/10^5. Likewise at Mars' perihelion was 0.827 times, i.e. a fraction 6.01/10^5.

Maybe the shortest of the epoch ~1252AD periods, and the two 1100AD periods, are averages effectively based on Mars' mean sidereal motion (as Vimond supposed for the Alfonsine) but the Alfonsine is effectively based on Mars' heliocentric motion at the vernal equinox, and the longest ~1252AD period is effectively based on Mars' heliocentric motion at perihelion. Then the shortest should differ from the Alfonsine by a fraction (7.27-6.23)/10^5 = 1/96,000, and the shortest should differ from the longest by a fraction (7.27-6.01)/10^5 = 1/79,000. The actual differences are 1/105,000 and 1/77,000, resp.

Summarizing these sidereal period corrections:

686.9773086 d, epoch ~1252AD: no correction
686.9780971 d, epoch ~1100AD: no correction
686.9765952 d, epoch ~1100AD: no correction
686.9838366 d, epoch 1252AD: *(1-1/96,000)--> 686.9766805
686.9862748 d, epoch ~1252AD: *(1-1/79,000)--> 686.9775788

Averaging, then applying the 2ms/century day length correction, gives

686.9771248 Julian d, epoch ~1200AD (ave. of 5; +/- SEM 0.0002813 d)

Two more Arabic or Persian tables are listed among the Trebizond almanac sources [5, pp. 179-184]. One of these listings, Zij-i Ilkhani, involves grossly ambiguous numbers.

The other, Zij al-Adudi (by astronomer Ibn al-Alam, c. 970AD [5, pp. 70, 179]) is optimized via the approximation 70*365d = 37 Mars periods. Using Newcomb's equinox with linear secular trend, and 2ms/cyr day length correction, I find that this table implies a Mars sidereal period

686.9779785 Julian d.



V. The 1590AD Heidelberg Venus-Mars occultation.

Kepler wrote that the Heidelberg astronomer Moestlin (also spelled Maestlin) recorded an occultation of Mars by Venus, which occurred Oct. 3, 1590AD (Julian calendar; Oct. 13 Gregorian) at about 5AM:

"De Venere et Marte experimentum refert idem Moestlinus, anno 1590.3.Octobris mane hora 5. Martem totum a Venere occultatem, colore Veneris candido rursum indicante, quod Venus humilior fuerit."

- [17, lines 11-13, p. 264]

(A copy of this book of Kepler's, of the original 1604 edition, is listed as stolen by the U. S. Federal Bureau of Investigation [18].)

The title of the cited subchapter indicates that it deals with occultations. It lists five involving Moestlinus: three were explicitly seen by Moestlinus ("vidit"; for one of these Kepler gives the observation location, namely Tuebingen, near Heidelberg); but the Venus-Mars occultation, and one other, are not explicitly stated to have been seen by him. Yet by giving the hour of observation, Kepler implies that it was observed by someone near Moestlinus' longitude. Moestlin or surely Kepler would have been able to convert sundial to mean time, but likely would have given that result to the minute, because conversion did not become routine until the publication of Flamsteed's tables. So at Tuebingen, 5AM local sundial time would have been

05:00 range +/- 00:30 (rounding to the nearest hour, I hope)
- 9*00:04 (longitude correction)
- 00:16 ("equation of time" conversion from sundial to mean time)
+ 00:10 (correction for 2ms/century day lengthing)
= 04:18 range +/- 00:30 Universal Time (UT)

Here is an authoritative paraphrase of part of Kepler's subchapter on occultations:

"Kepler states...Maestlin and himself witnessed an occultation of Jupiter by Mars. The red colour of the latter on that occasion plainly indicated that it was the inferior planet. He also mentions that on the 3rd of October, 1590, Maestlin witnessed an occultation of Mars by Venus. In this case, on the other hand, the white colour of Venus afforded a clear proof that she was the nearer of the two planets to the Earth."

- [19, p. 433]

At occultation, Mars' visible disk was 97% sunlit. According to [6; I thank the anonymous peer reviewer at the journal "Observatory" for informing me that the Mars system barycenter, though not Mars itself, is given for such old dates]([20] confirms this time) maximum occultation as seen from Tuebingen, occurred at

04:50.3 UT

with Mars' and Venus' centers separated by 8.40". Venus' apparent radius was 6.52" and Mars' 1.95"; 6.52+1.95=8.47", so the greatest overlap was less than 4% of Mars' radius, thus the light blockage negligible. Total occultation would have implied no more than 6.52-1.95 = 4.57" separation of the centers.

Only substantial occultation would be fully consistent with the reported loss of Mars' color. The diffraction-limited resolving power of a 6 mm pupil (Moestlin was forty years old at the time of this presumably unaided eye observation) at 560 nm, is 23", so if the observed color change were due to lack of optical resolution of the sources, the duration of the color change could have been interpreted as a surprisingly large radius for Venus.

My graphical solution based on [6] shows that Venus' center, as seen from the center of Earth, appeared to cross Mars' center's path, as seen from Earth's center, at 06:17 UT. Corroborating this, my computer calculation using the Keplerian Mars orbit with secular trend [2], shows that the (on-center, geometric) Earth-Venus line from [6] crossed Mars' (unperturbed) orbit at 06:06 UT. The aberrations of light from Venus and Mars were about equal, so geometric and apparent occultation times would have been about equal.

Kepler's 5.0 +/- 0.5 AM local sun time for this occultation, argues for an actual time at least two minutes earlier than given by [6], but Kepler's secondhand description of the color change, argues for an actual time (if the Venus ephemeris of [6] is accurate) later than [6]. This author, at age 54, easily saw Venus when near its brightest, with an unaided (except for myopic correction) eye, more than 15 minutes after sunrise when the Sun was blocked by clouds on the horizon. However, the delicate color observations imply that the 1590AD event was seen before sunrise.

Using [6] with extra precision, and correction for light time, at Tuebingen, I find that if both Venus and Mars may be moved elsewhere on their orbits, total occultation can occur at 05:30 Tuebingen sundial time (i.e. 04:48 UT) only if Venus is moved at least 56.8 minutes forward in its orbit. For total occultation with this Venus position, Mars must be moved forward 112.8 minutes in its orbit. Thus during the intervening 1982-1590=392 yr, Mars orbited 112.8 minutes / 392 yr = 1 part in 1,828,000 slower than believed by [2][3][6], corresponding to a period of 686.9798529 * (1 + 1/1,828,000) = 686.980229 d. This is a reasonable sharp lower bound for Mars' period: it results from the smallest perturbation of Venus which preserves Venus' orbit but allows total occultation of Mars (with no change in Mars' orbit except for Mars' position on it) no later than 04:48 UT, consistent with the information provided by Moestlin via Kepler.

Venus would have had to be slowed by 56.8 minutes / 392 yr = 1 part in 3,630,000. This is, proportionally, just half the period increase, vis-a-vis modern ephemerides, that is implied for Mars.



VI. The 251BC Seleucid conjunction of Mars with Eta Geminorum at stationarity.

Modification of my computer program used in Part V, gives, assuming arbitrary mean motion, but otherwise no perturbation except the secular trend of the Keplerian orbit, the position of Mars at its supposed conjunction with Eta Geminorum, Seleucid Baylonian date 61 VII 13 (Oct. 4-5, 251BC, Julian calendar) [13, Table IV.9, p. 144]. For explanation of the Seleucid calendar, see [21, p. 188].

The conjunction was not necessarily observed at Babylon (or nearby Seleucia), because such an abstract event easily could have been interpolated from observations on nearby nights. My program (which corrects for the star's proper motion, and approximately for the barely significant effect of aberration of light) shows conjunction (Mars at the projection of Eta Geminorum, on Mars' orbit as given by [2]) at Oct. 5.0 (i.e. actual mean Greenwich midnight between Oct. 4 and 5) 251BC if Mars' sidereal orbital period, over the nearly 1196 orbits between then and 2000.0AD, is

686.9804164 Julian days

The insensitivity of this result, to observation time, occurs because Mars is near stationarity. Following [13] I assume that the time was between actual (including day lengthening; not UT) mean Greenwich Oct. 4.375 and Oct. 5.375, midpoint Oct. 4.875, giving

686.980424 Julian d +/- 0.000016 d rms

This places Mars, in 251BC, 21 arcminutes of mean motion ahead of Kieffer's mean motion, but this could be due to planetary gravitational perturbation. My small random sample of ten whole numbers ~200, of Mars orbits ending randomly in the interval 2007.0AD +/- 10,000d, and starting some whole number of orbits (of duration estimated according to Kieffer's mean motion with secular trend for 1800AD) randomly 350 to 450 yr earlier, shows that already after ~200 whole orbits, Mars, according to [6], and measuring by projecting the final heliocentric position onto the starting orbit, ranges 19' ahead to 13' behind its mean motion; mean 6' +/- 4'(SEM) ahead. Planetary gravitational perturbation of Mars' Keplerian orbit by several arcminutes, is well known to modern astronomy.

Using the JPL ephemeris for Mars instead of Kieffer's Keplerian orbit, I still must move Mars, 11 arcminutes of mean motion farther ahead to achieve conjunction (i.e. nearest approach on the celestial sphere) at the desired time (it happens also to be 11 arcminutes of true anomaly because of Mars' position). So although the discrepancy between the Keplerian orbit and observation is small enough to be explainable by planetary gravitational perturbation, only half of it is explained by [6]; including the perturbation according to [6], implies that Mars' unperturbed period is

686.980154 Julian d

The "ammat" was a unit of angular measurement thought to be 2.5 degrees [14]. The record says that Eta Geminorum was 2/3 ammat below Mars, i.e. 5/3 degree below Mars. My program finds that the star was instead 2/3 degree below (i.e. ecliptic south of) Mars' unperturbed Keplerian orbit; it can be seen by rough graphing that this is correct. Maybe the transmitted record somehow confused the ammat unit with the degree unit.

According to my program using Kieffer's Keplerian orbit, stationarity, if it occurred exactly when at conjunction with Eta Geminorum, did not occur until Oct. 10.346, in the calendar that uses Julian days of fixed length (Oct. 10.132 in the Greenwich mean calendar of that epoch). This was determined by finding the true anomaly of Mars needed for the conjunction to occur at various dates, then finding the date when the time derivative of this hypothetical needed anomaly, equaled the actual time derivative of the anomaly. If stationarity occurred precisely at conjunction with Eta Geminorum, and planetary gravitational perturbations of Mars' Keplerian orbit are neglected, then the average sidereal orbital period of Mars since 251BC has been

686.9802751 Julian d

amounting to a 15.5 arcmin forward translation of Mars, 21-11=10 arcmin of which are explained by planetary gravitational perturbation according to [6]. In conclusion, [6] might have as little as 5.5 arcminute error for Mars at 251BC, or as much as 11 arcmin.



VII. Seleucid and Ptolemaic parameters.

Surviving tablets from the Seleucid era c. 200BC (late Babylonian astronomy), when Babylonian Systems A and B were used, equate 133 Mars synodic periods to 151 Mars sidereal periods [22, pp. 78, 80] [23, sec. IIA6,1C , eqns. (1)-(4)]. This implies a Mars period of (133+151)/151 Earth sidereal yr

= 686.97223 Julian d

assuming a practically constant length, 365.25636 Julian d, for Earth's sidereal year.

The precision of a fraction involving two three-digit numbers, is about the precision of one six-digit number, but let's suppose the Seleucid Babylonian astronomers chose their 133 & 151 from among integers 1 through 360. The nearest other fractions would be 0.0093 d less or 0.0070 d more, giving an rms uncertainty of about 0.00235 d.

Confirming this, are the Seleucid Babylonian "Goal Year Text" tablets, dated c. 250BC, though with earlier Babylonian and Assyrian predecessors [24, pp. 26, 27]. These say that 79 years + 7 days, is a whole multiple of Mars' synodic period. The Babylonian calendar added a 13th month to 7 of the years in each 19-year cycle; months had either 29 or 30 days to match the synodic month, 29.53d. Using 29.53d, rather than 29.500d, as the average month length, gives too different an answer. The Seleucid Babylonian calendar gave a 13th month to years 1 & 18 but to no others among the three at either end of the cycle [21, p. 188]; so, the needed two extra months in the leftover 79 - 4*19 = 3 years, are just possible. Another version of the Babylonian calendar gave a 13th month to years 17 & 19, again making the needed two extra months just possible [25, p. 24]. Mars' synodic period must have been (29.5*(79*12 + 4*7 + 2) + 7)/37 = 779.945946 days of epoch, which implies a Mars sidereal period of

686.972515 Julian d

if the Babylonian months strictly alternated 29 & 30 d, and days lengthen 2ms/century. The denominator, 37, presumably was dictated by the approximation, 37 Mars synodic yr = 42 Mars sidereal yr. A +/- 0.5 day rounding, without changing the denominator, is +/- 0.0105d, corresponding to rms error +/- 0.00606.

The latest observation in Ptolemy's Almagest is from 141AD. Ptolemy's "Planetary Hypotheses" is a later work, maybe Ptolemy's last [26][27]. It has the one place in Ptolemy's writings where precise synodic planetary periods are listed explicitly [26, sec. "VB7,3", Table 15, p. 906]. It gives 473 Mars sidereal periods as equal to 1010*365d + 259;22,50,56,16,27,50d (sexagesimal notation), which is equivalent to giving Mars' sidereal period as

686.980745 Julian d

using a 2ms/century day length correction for 150AD, and an Earth sidereal year of 365.25636 d. Ptolemy's earlier estimate, in his "Almagest", is a mean daily motion with respect to the (continuously changing) equinox of date [26, sec. "VB7,3", Table 17, p. 907][28]. Using Ptolemy's estimate, or rather his only explicit estimate that we still possess, of the equinox motion, i.e. 36" per tropical year (and the usual 2ms/cyr correction for day lengthening, for 141AD) gives

686.9802165 Julian d

for Mars' sidereal period according to the Almagest. Apparently Ptolemy applied a very rough estimate of precession, to a very accurate estimate, perhaps inherited from Hipparchos, of Mars' mean sidereal motion.


VIII. Han China.

The Si Feng almanac of Han China (c. 100AD) has a table listing intervals of days, with corresponding changes in Mars' position, measured in whole "du" = 360/365.25deg [22, pp. 85-86]. The table is symmetrical about opposition, but grossly inconsistent with either opposition at Mars' perihelion or opposition at Mars' aphelion.

The table says that an 11 day interval, beginning 31 days, after opposition (i.e. after the midpoint of retrogression), corresponds to zero change in Mars' geocentric position. If the table refers to circularized Mars and Earth orbits, and is denominated in sidereal days, not synodic days, then this entry of the table, referring to stationarity, is correct with less than 4 arcsec error, neglecting the aberration of light. The error disappears if Mars' period is 686.86966 Julian d, including the 2ms/century correction for day lengthening, but this result is not significantly different from 686.98, because greater precision than 4", hardly can be expected among the choices of whole day intervals. This device, i.e. the substitution of sidereal for synodic days, does not fit Jupiter or Saturn.

The table usually is assumed to be denominated in Earth synodic days, and assumed to sum to one Mars synodic year. The intervals are

184, 92, 11, 62, 11, 92, 184, 143, and the fraction 1872/3516 = (156*12)/(293*12).

If the fraction were intended to be one of the doubled intervals, then Mars' synodic period, assuming denomination in mean synodic Earth days, would be 780.06450 Julian d, including 2ms/century day lengthening. This corresponds to a Mars sidereal period of 686.88025 Julian d. (Nor does this device, i.e. counting the fraction twice, fit Jupiter or Saturn.) If the fraction were instead 158/293 (and doubled), this corresponds to Mars sidereal period 686.86966 Julian d, exactly matching any then extant recorded period based on the 11 sidereal day stationarity interval discussed in my previous paragraph.

The foregoing suggests that the original table was denominated in sidereal days. Someone mistook it for synodic days, and misguidedly repaired it by subtracting 2 whole days (because 2*(184+92+11)+62+143 = 779 = approx. 2*365) and choosing the (doubled) fraction 158/293 to match then extant records of Mars' period.

If there were a rival table, also denominated in synodic days, using a different fraction (also doubled) with numerator 156, confusion between the two tables might have caused the substitution of 156 for 158 in our surviving table. The correct denominator, for numerator 156, would have been relatively prime to 156, perhaps 329, whose digits in base 10, permute the digits of 293. With 2ms/cyr day lengthening, using 156/329 (doubled) gives sidereal period

686.97062 Julian d

An epoch many centuries earlier than 100AD, would have allowed time for these several transcription errors to occur.



IX. Vedic India

"One sidereal period is called a Bhagana. In an Equinoctial Cycle of 4,320,000 yr, called a Maha Yuga...revolutions done by...Sun 4320000...Saturn 146564 Jupiter 364224 Mars 2296824 Moon's Apogee 488219 Mercury's Perihelion 17937020 Venus' Perihelion 7022388 Moon's Ascending Node 232226..."

- Kalidasa, "Uttara Kalamrita" [29]

This quote suggests that the Sun's "revolutions" are probably tropical, but the other revolutions unambiguously sidereal, except for the inferior planets, whose revolutions are explicitly vis a vis their perihelia, i.e. anomalistic. If the epoch is 3102BC, the above implies that Mars' sidereal period is 686.96933 Julian d, assuming Earth's sidereal year is 365.25636 d, and using Newcomb's [9] precession correction. Likewise, epoch c. 1700AD (an historical date for Kalidasa) implies period 686.968765.

The former epoch is the start of the Kali Yuga, 3102BC according to an astronomical multiple conjunction date [15]. This is near the start (c. 3110BC) of the Egyptian pharaonic dynasties according to Manetho, and the start (3114BC) of the Mayan Long Count too. The sidereal apogee advance of Luna given by [29] would have agreed, allowing for rounding the number of revolutions, with modern theory [30, sec. 3.4.a.1, p. 669] sometime between 3145BC and 3092BC, assuming a tropical year Maha Yuga and Newcomb's linear (or quadratic [31, p. 90]) secular precession.

Kalidasa's figures for the other superior planets, Jupiter and Saturn, also support a Maha Yuga denominated in tropical years. With the precession of 3102BC, Kalidasa implies Jupiter would have sidereal mean motion 3035.262deg / Julian cy, and Saturn 1221.392. Including the T^2 but omitting the sinusoidal Great Inequality ("GI") terms, [6; Standish, "Keplerian Elements...", Table 2a] gives, for 3102BC, 3034.9164 & 1222.0885, resp. The discrepancies, for Jupiter & Saturn, are in the ratio -1::2.015, near the ratio -1::2.444 which [6, Table 2b] gives for the (constant) amplitudes of the GI terms, and even nearer the -1::2.080 which [6, Table 2b] gives for the coefficients of T^2.

The Great Inequality period, including GI terms, implied by [6, Tables 2a,b] has a ~ 900-year peak of 968.58 yr, at 3141BC. The GI period implied by Kalidasa is 988 with a range of approx. +/- 1 yr due to his rounding Jupiter's & Saturn's data to whole numbers. Thus Kalidasa's figures are consistent with the modern calculated phase of the GI c. 3100BC but require a 21% greater amplitude than [6], for the GI sinusoidal terms: the mean motion of Jupiter c. 3102BC would have to be greater than [6] predicts, amounting to an additional secular trend of a part in 305,000,000 per year.

Clemence [31, p. 90] gives quadratic polynomials in T, for the coefficients of the GI terms. Though the precision of Clemence's polynomials is dubious for such large T, Clemence's amplitude of Jupiter's GI sinusoid (assuming that Saturn's GI sinusoid has -2.444x Jupiter's amplitude) is large enough, for any time earlier than 2350BC, to cause [6], at its 3141BC peak, to match Kalidasa's GI period (the mean motion's sinusoidal terms alone for Jupiter & Saturn in [6] peak, resp. trough, at 3189BC, resp. 3187BC). As a check, I find that [31, p. 90] matches Kalidasa's Jupiter period for any time earlier than 2440BC, if the phase of the GI sinusoid is near enough maximum. Also [6, Table 2a,b] implies that the ~ 900-yr peak GI period is, due to the slow T^2 term, slightly greater and closer to Kalidasa's value, at c. 3100 BC than at later peaks.

For the inferior planets, Kalidasa is accurate if the Maha Yuga is denominated in sidereal yr. Mercury's & Venus' anomalistic periods for 2000AD correspond to 149472.514 & 58517.813 deg/cyr, resp. [6]. Assuming a sidereal year of 365.25636 Julian d for the Maha Yuga, Kalidasa gives periods corresponding to 149472.564 & 58518.881 deg/cyr, resp. Venus' error, 1.068 deg/cyr, suggests that someone corrected it to a tropical period, by adding Ptolemy's precession value, 1.0 deg / cy.

In [32] are similar tables from five more Hindu astronomical books:

1) Aryabhatta, "Arya Siddhanta", 1322AD (it bears a date vis a vis the Kali Yuga) [32, pp. 138,139].
2) Aryabhatta, but attributed by him to Parasara, "Parasara Siddhanta", thus written c. 1322AD (see (1) above) but maybe based on work c. 540BC (an historical date for Parasara) [32, pp. 78,144,145].
3) Varaha, "Vasishtha" + "Surya" + "Soma" Siddhantas, c. 940AD [32, pp. 116,117,126]. These mention an observation of the longitude of Canopus which Bentley dates to 928AD. From the parameters of Varaha's Siddhantas, and their positions for Mercury, Venus, Jupiter and Saturn for 3102BC, Bentley infers four more dates near the Canopus date, ranging from 887AD to 945AD.
4) Anonymous, "system of 538AD" [32, pp. 81,82,92]. Bentley confidently dates this using astronomical and calendrical information.
5) Anonymous, "spurious Arya Siddhanta", bearing the date 522AD, but Bentley thinks it is more recent [32, pp. 179,180]. The parameters given by this book are the same as the Uttara Kalamrita.

The number of Mars periods per Maha Yuga, in the five distinct books (i.e. excluding (5)) ranges from 2296824 to 2296833.037. With a Maha Yuga denominated in tropical years, a 365.25636 Julian day Earth sidereal year, and Newcomb's quadratic precession for 3102BC, this gives Mars sidereal periods ranging from 686.9666207 to 686.96932 Julian d. The sidereal period of Mars, averaged from the five books is

686.967616 +/- 0.000482 (SEM) Julian d, epoch 3102BC ? (no later than 538AD)

The number of Jupiter (resp. Saturn) periods per Maha Yuga, ranges in the five distinct books from 364219.682 to 364226.455 (resp. 146564 to 146571.813). The amplitude of the GI sinusoidal term of Jupiter's (resp. Saturn's) mean motion [6], amounts to +/- 43 (resp. 106) periods per Maha Yuga. So, the five books give Jupiter, and especially Saturn, periods corresponding to nearly the same phase of the ~ 900-year Great Inequality cycle, and therefore likely of the same epoch.

The Uttara Kalamrita plus (1)-(4) above comprise five estimates of Jupiter's & Saturn's mean sidereal motions, averaging 3035.246 deg/cy +/- 0.011 SEM & 1221.425 +/- 0.011 SEM, resp., using Newcomb's quadratic precession formula [31, p. 90] to determine the length of the tropical year in 3102BC. The discrepancies, vs. [6] including its T^2 but omitting its sinusoidal GI terms, are, as for the Uttara Kalamrita, in the ratio -1::2.015. The GI period implied by the Hindu means, is 982.7 +/- 1.6, which would be consistent with the GI at its maximum phase in 3141 BC, with a 15.5% greater amplitude than [6], for the GI sinusoidal terms: here the mean motion of Jupiter c. 3102BC would have to be greater than [6] predicts, amounting to an additional secular trend of a part in 413,000,000 per year.

Sometimes the deviation of a book's Jupiter & Saturn motions, from the mean of all the books, likely is due to a slightly different epoch of observation, not random error:

book / Jupiter motion, deviation from ave. of all books, deg per cy / Saturn, " / ratio
Uttara Kalamrita / +0.016 / -0.033 / -2.05 (vs. -2.444 for the GI sinusoid of [6])
Parasara Siddhanta / -0.17 / +0.032 / -1.82

Book (3) above lists 488203 Lunar apogee advance periods per Maha Yuga. This conforms to 2298BC using the method of the third paragraph, above, of this section. This epoch, like 3100BC, would be near a peak, 2250BC according to [6], of the sinusoidal term of Jupiter's mean motion, so could relatively easily conform to the Jupiter and Saturn periods given by these Hindu books.

Books (1), (2) & (4) list 488108.674, 488104.634 & 488105.858 apogee periods, conforming to 1800AD, 1960AD & 1910AD resp. If the true dates of these books are resp. 1322AD, 540BC & 538AD, this is an increasingly accurate estimate of our own time, as the books get older. Maybe someone intended to state the Lunar apogee periods for the beginning and ending of what we know as the Mayan Long Count.

The Lunar node regression periods of three of the books, also yield plausible epochs. Apparently the Lunar node was measured vis a vis Earth's perihelion (i.e. the Sun's perigee) rather than sidereally. The Lunar apse lies often 5deg from the ecliptic but the Lunar node by definition always lies on the ecliptic. So, Lunar node measurements easily may be made vis a vis something that moves slowly and steadily on the ecliptic. Not only does Earth's perihelion move only 1/5 as fast as does Earth's equinox; it also is changing its speed only 2/3 as fast over the centuries [33, Table VI, p. 294]. Equinox measurements must contend with nutation; sidereal measurements must contend with stellar proper motion.

Books (1), (2) and (4) above, list respectively 232313.354, 232313.235 and 232311.168 Lunar node periods per Maha Yuga. Using Newcomb's quadratic precession formula [31, p. 90] to find the length of the tropical year, interpolation in Dziobek's table [33, Table VI, p. 294] to find the motion of Earth's perihelion, and the polynomial of [30, sec. 3.4.a.1, p. 669] for Lunar node regression rate, I find that when Lunar node regression is measured vis a vis Earth's perihelion, the Arya Siddhanta (1) and Parasara Siddhanta (2) are consistent with 2880BC & 2840BC resp., and the "system of 538AD" (4) is consistent with 2110BC.

Corroborating my interpretation of the books' node periods, I find that the other node periods listed, are explained by an erroneous subtraction, rather than addition, of the magnitude of the motion of Earth's perihelion, from the magnitude of the motion of Luna's node. Book (3) lists 232238 Lunar node periods per Maha Yuga, and the Uttara Kalamrita lists 232226; these would conform to 1900BC & 900AD, resp.

The most difficult sidereal periods for the Hindu astronomers to determine accurately, must have been those of Mars and Luna. For the period they give for Luna, to be roughly consistent with the accepted dates of Homo sapiens, requires a Maha Yuga denominated in sidereal years, not tropical or anomalistic years. Likewise, the periods given by (1)-(5) for the inferior planets, are as for Kalidasa's, most consistent with a sidereal Maha Yuga.

Even with correction for the equation of center, accurately accounting for perigee advance, Luna's sidereal period fluctuates due to "evection" [34, sec. 169, p. 128], "variation" [34, sec. 166, p. 125], "annual equation" [34, sec. 171, p. 129] and "parallactic inequality" [34, sec. 167, p. 126; the minus sign is missing in Brown's text but occurs in the approximate equation above it]. The modern definition of mean period, amounts to the arithmetic mean [34, sec. 165, p. 124]. It seems that the Vedic astronomers considered the harmonic mean period instead. If the arithmetic mean of {1 + dP(i)} is 1, dP(i) << 1, then the harmonic mean is approx. 1 - mean((dP(i))^2).

In Brown's notation [34] phi is Luna's mean longitude and phi' the Sun's mean longitude. Vedic astronomers seem to have chosen the endpoint phi, so that phi-phi' at the endpoints of the sidereal month measured, was symmetrically more or less than 45deg (315 will give the same answer; but 135 or 225 a different answer). Such endpoints nullify the effect of "variation" on the measured sidereal month. Averaging, over phi, the squared sum of effects of "evection", "annual equation", and "parallactic inequality", I find, using Delaunay's values as given by Brown [34], that such a standardized harmonic mean sidereal month is less than the arithmetic mean sidereal month, by a fraction 1/175,072.

If Kalidasa's datum, 57753336 months (books (3) & (5) above, concur), refers to the harmonic mean of such a standardized sidereal month, then the implied arithmetic mean month conforms per [30, sec. 3.4.a.1, p. 669] to 3116BC, given a Maha Yuga in sidereal years. Delaunay's evection value is given by Brown only to the nearest arcsecond; 0.5" corresponds to almost 14yr in the date. In Kalidasa's datum, 0.5 months corresponds to a century in the date; but the first three digits, 432, of the Maha Yuga number, likely were chosen to lessen this error. Modern estimates of Earth's sidereal year with secular trend (see Appendix 3) are near enough the 365.25636 d used throughout this paper, to affect the date by no more than about a century. Books (2), (1), & (4), resp., above, list 57753334.114, 57753334, & 57753300 months; 57753334.114 conforms to 2730BC. The determination of Mars' sidereal period, involves different mathematical problems than Luna's, but such an accurate Vedic determination of Luna's period, argues that their determination of Mars' period is similarly accurate.

Vedic sidereal year and mean solar day lengths also seem to be very accurate (see Appendix 3). This accuracy is realized only with the above emendation, for the harmonic mean Vedic sidereal month.



X. Discussion.

The sidereal orbital periods determined for Mars, from my collection of the most authoritative, documented, and believable modern, medieval and ancient observations or calculations, fall into three groups. Here is a recapitulation (all corrected to constant Julian days assuming 2ms/century day lengthening) of these three groups.

The modern value:

686.9798529 - Bretagnon 1982AD, accepted by Standish & others

686.9798027 - LeVerrier 1850AD, accepted by Newcomb

Essentially five slightly high estimates:

686.980424 +/- 0.000016 - 251BC Mars / eta Geminorum conjunction near stationarity (Bretagnon/Kieffer's orbit, disregarding planetary gravitational perturbation)(calculation of perturbation, and/or alternative interpretation of the conjunction record, would move this about two or four times closer to the modern value)
686.980154 +/- 0.000016 - above, adjusted for planetary gravitational perturbations of Mars, according to [6]; this is the best empirical value

686.980698 - Trebizond almanac, with help from 1590AD occultation information

686.980229 - sharp lower bound deduced from [6] and Moestlin/Kepler's 1590AD Venus-Mars occultation report

686.980745 - ~150AD parameter given by Ptolemy, "Planetary Hypotheses"
686.980217 - Ptolemy, ~141AD, "Almagest", implicit with his 36"/yr precession

686.980353 - Kepler, 1598AD, from Brahe's observations (see Appendix 2)

If Kepler simply had copied the number of days implied by the Almagest, my assumption of 2ms/cyr day lengthening, would have given 686.980449 for Kepler. Instead, Kepler minus Ptolemy = 0.000136; 2 ms * 0.000136/0.000232 = 1.17 ms/cyr, is the actual rate of day lengthening that would give the number of days implied by Kepler's ephemeris, if Kepler simply copied Ptolemy's number of days. Vedic records imply 1.4917 ms/cyr day lengthening (see Appendix 3).

Eight low estimates showing time trend upward:

(for comparison, 686.9798529 - Bretagnon 1982AD, accepted by Standish & others)

686.9795233 +/- 0.0000029 rms rounding error - Pontecoulant (following La Place) c. 1840AD [35, vol. 3] as cited in [36, Bowditch's note 9159f, Bk. X, ch. ix, sec. 25; vol. IV, p. 681]; from sidereal mean motion per 365 1/4 d.

686.9791639 +/- 0.0002880 rms rounding error - Bailly 1787AD [37, Ch. 7, sec. II, p. 174]; from "moderne" mean motion per 365.0 d, vis a vis equinox of date

686.97937 +/- 0.00011 - likeliest for Gemistus Plethon, 1446AD (see Appendix 1)

686.977125 +/- 0.00028 - ~1200AD corrected ave. of best 5 implicit almanac parameters for Mars' sidereal period

686.9779785 - al-Alam, 970AD

686.97223 +/- 0.00235 - ~250BC Seleucid Babylonian tablet parameter

686.97062 ? - predecessor of 100AD Han value (predecessor epoch probably much earlier) reconstructed by this author

686.96762 +/- 0.00048 - India, 3102BC ? 2200BC ? (surely no later than 538AD)

The time trend within this low group, is linear:

(686.9798529-686.9795233)/(1982-1840) = 2.32/10^6 day/yr
between about 1840AD and about 1982AD

(686.9798529-686.977125)/(1982-1200) = 3.49/10^6 day/yr
between about 1200AD and about 1982AD

(686.977125-686.97223)/(1200+250) = 3.38/10^6 day/yr
between about 250BC and about 1200AD

(686.9798529-686.97062)/(1982-100+ ? ) < 4.91/10^6 day/yr
between sometime earlier than 100AD, and 1982AD

(686.9798529-686.96762)/(1982+3102- ? ) > 2.41/10^6 day/yr
between 3102BC or later, and 1982AD

Suppose Mars' period varies sinusoidally with period 25,785 yr (Newcomb's precession period) and origin at 2013.0 AD (end of Mayan Long Count). We have 1982AD, 1200AD, 250BC & 3102BC equal to phases -0.4, -11.4, -31.6 & -71.4 deg, resp. These data imply almost the same speeds at the node:

3.49*(11.4-0.4deg, in rad)/(sin11.4-sin0.4) = 3.51 / 10^6 d/yr
3.38*(31.6-11.4deg, in rad)/(sin31.6-sin11.4) = 3.65
2.41*(71.4-0.4deg, in rad)/(sin71.4-sin0.4) = 3.17

The Seleucid record of Mars' conjunction with Eta Geminorum at stationarity, indicates that Mars' orbital period either has not changed, or has decreased by at most ~ 1 part in 600,000, during the last 2250 yr. The Seleucid record is, completely independently, corroborated by Ptolemy's "Planetary Hypotheses" and his "Almagest".

On the other hand, most of the accurate ancient, medieval and even early modern efforts to determine Mars' period, find a linear increase of about 1 part in 200,000, per 1000 yr. The Eta Geminorum conjunction was irrelevant to the position of the Sun; on the other hand, most if not all of the medieval and ancient determinations of Mars' sidereal period, are based on Mars' synodic period, which would be defined relative to the position of the Sun.

Epoch 3102BC for the Vedic Mars period, implies a Mars period change of a part in 285,000,000 per yr. This matches the additional part in 305,000,000 for Jupiter implied by the Uttara Kalamrita assuming the same epoch (see Sec. IX), so supports the hypothesis of the Introduction, that the 19: Mars::Jupiter resonance is conserved. If the Vedic Jupiter and Mars epochs are at some other 900-yr peak of Jupiter's mean motion, then the implied additional rates of change of the periods, are larger in the same proportion; for 2200BC and 1300BC they are close to the rates of change implied by the other Mars orbital period values given above.

The rate of change, in the calculated orbital period, is found most confidently from the data of 1200AD and 1982AD. The 1200AD error bar, together with uncertainty in the effective epochs which I approximated as 1200AD and 1982AD, gives an overall error bar, for the slope, of roughly 10%. The time in days, needed for the linear change in orbital period, to displace Mars by one orbit, satisfies

1/2 * t^2 * (3.49/10^6/365.25)/686.98^2 = 1

--> t = 27,211 yr. Newcomb's equinox precession rate with linear secular trend [9] implies that the precession cycle ending in 1982AD, lasted 27,440 yr. This suggests that the explanation for the upward-trending determinations of Mars' orbital period, somehow lies in Earth's precession.

Though in the Introduction, I remark that the Great Inequality ("GI") could be restored to its average value by equal and opposite, i.e. -1::1, changes in the orbital frequencies of Jupiter and Saturn, a more suggestive relation emerges from the -1::2.444 ratio given by [6; Standish, "Keplerian Elements...", Table 2b] for the amplitude ratio of the Jupiter & Saturn 900-yr sinusoidal terms. Assuming an additional term in the trend in Jupiter's mean motion, in proportion to that which I discover for Mars, i.e.

-3.49/10^6 *100/686.98 * 3034.9 = -0.001542deg/cyr^2

& -2.444x as much for Saturn, and disregarding the 900-yr sinusoidal term, [6, "Keplerian Elements...", Table 2a,b] implies that the GI was its theoretical long-term average, 1092.9 yr [1, p. 287], at 29,300BC (31,300BP).

The three upward trendlines from the 1200AD value through each of the most modern (post-Laplace) values, intersect the empirical value, 686.980154d, at

Bretagnon 2088AD weight ((1982-1200)/(2088-1982))^2=54
LeVerrier 1935AD weight ((1850-1200)/(1935-1850))^2=58
Pontecoulant 2008AD weight ((1840-1200)/(2008-1840))^2=15

with a weighted mean of 2009AD.



Appendix 1. Plethon's mean motion of Mars.

"To a certain measure then these revised parameters confirm both Plethon's dependence on al-Battani (Hebrew) and the fact that he must have worked at one or two removes from that Hebrew text as we have it."

- Mercier [38, pp. 248-249]

Gemistus Plethon (c. 1446AD) gives mean motions of Mars, vis a vis Earth's equinox, in original and revised versions. The original version [38, Table 1.3, p. 230] includes a solar mean motion, vis a vis the equinox, implying, with 2ms/cyr day lengthening from 1446AD, an equinox precession rate of 56.24"/yr. The revised version [38, Table 4.5, p. 247] includes a solar motion likewise implying a precession rate 54.69"/yr, even nearer al-Battani's [11, p. 271] 1/66deg = 54.55"/yr.

If Plethon found Mars' sidereal period first, then applied al-Battani's precession, Plethon's original and revised versions give, with 2ms/cyr day lengthening correction and assuming a 365.25636 Julian day Earth sidereal year, Mars' sidereal period as

686.977886 & 686.979197 Julian d, resp.

If Plethon applied his own precession as I reconstruct it from his solar motions, then his original & revised versions likewise give

686.979575 & 686.979342, resp.

Discarding one of these four possibilities as an outlier and averaging the other three, gives Plethon's Mars sidereal period as

686.97937 +/- 0.00011 d SEM



Appendix 2. Kepler's mean motion of Mars.

The information in Kepler's Rudolphine Tables [39] gives a confident value of the equinox precession that Kepler really used when he gave the mean motion of Mars vis a vis the equinox. Kepler [39, p. 61 in original pagination] gives +1deg51'35" per Julian century aphelion progression for Mars, and +1deg6'15" per Julian century node progression for Mars, both vis a vis the equinox (Earth's equinox regresses faster than Mars' node). By subtracting the modern J2000 values [2] (with secular trend to 1598AD) these give two implicit estimates of Kepler's equinox precession constant: 50.917" & 50.146" per tropical yr, resp. (The modern value of node regression referred to the 1598AD ecliptic [40, p. B18] has been used; it is 0.042" smaller in magnitude than the modern value referred to the 2000AD ecliptic.) Newcomb's value (with secular trend for 1598AD) [9] is 50.1894"/tropical yr. This demonstrates that Kepler's equinox precession was much nearer Newcomb's value, than the 1/66 deg = 54.5"/yr given by al-Battani [11, p. 271] or the 54"/yr of the Brahmins [15, sec. 8]. Kepler [39, p. 43 in original pagination] gives the Sun's mean motion, vis a vis the equinox, between Anni 1 & Anni 97, in Julian yr, as 96*360deg + 1deg29'12" - 45'40"; correction for 2ms/cyr day lengthening, and assumption of a 365.25636 Julian d sidereal Earth year, give Kepler's precession constant c. 1598AD, as 49.89"/tropical yr, again near Newcomb's value, and also near the value, 49.18", implied by the Gregorian calendar's convenient approximate formula.

Kepler's table [39, p. 61] gives Mars' mean motion, vis a vis the equinox, in 100 Julian yr, as X*360 + 2*30 + 1 + 40/60 + 10/3600 deg, where we know X must equal 53. Subtracting Kepler's equinox precession 49.89"/yr, gives sidereal period 686.980353 Julian d, corrected for 2ms/cyr day lengthening.



Appendix 3. Ancient sidereal year and mean solar day.

Newcomb's sidereal year, with linear secular trend, is 365.25636042 d, for 1900AD, and 365.25635492 d, for 3100BC [9, p. 491]. The prevailing more modern value is 365.256363 Julian d [40, p. C1; no trend given]; applying Newcomb's trend to this value gives 365.2563575 Julian d, for 3100BC. A discrepancy of -0.0000025 d, from 365.25636 d, corresponds to +79yr in the epoch of Luna's Vedic sidereal period, i.e. 3195BC instead of 3116BC (see Sec. IX, last par.).

Hindu books (1)-(4) (see Sec. IX) give respectively 1577917542, 1577917570, 1577917828 & 1577916450 "natural" days [32] (perhaps mean solar days; the number of sidereal days, when given, is 4320000 more) per Maha Yuga. For a Maha Yuga of 4320000*365.25636 Julian d, and assuming epoch 3102BC, these imply day lengthening ranging from 9.6 to 11.1 ms/cyr. To emend this, suppose that days per arithmetic mean Lunar sidereal month, were miscopied as days per harmonic mean Lunar sidereal month; the latter are shorter by a fraction 1/175,072 (see Sec. IX, next to last par.), so too, would be the alleged days.

To find the emended day lengthening for book (3) as accurately as possible, assuming epoch 3102BC, I will use a 365.2563575 Julian day year above, and 2013AD as the year when the smoothed trend in mean solar day length equals precisely one Julian day [41, chart]. The result is 1.4917 ms/cyr. Assuming this rate of day lengthening, books (1), (2) & (4) correspond to 2052BC, 2155BC & 2441AD, resp., using year lengths as above.

Though [42, p. 65; Fig. 10, p. 66] finds 2.43+/-0.07ms/cyr day lengthening between the ancient and medieval period, they find 1.40+/-0.04ms/cyr between the medieval and modern. Using mostly modern, but some medieval and ancient data [43, Summary, p. 125; Table X, p. 139] finds 1.35+/-0.38ms/cyr. From ancient eclipses, [44; the abstract gives equivalently -1.7/10^10(day/day)/yr] finds about 1.47ms/cyr (i.e. (ms/day)/cyr).



References.

[1] Varadi et al., Icarus 139:286+ (1999).

[2] Kieffer et al., eds., "Mars" (U. of Arizona, 1992), citing Seidleman & Standish, personal communication, based on [3].

[3] Bretagnon, Astronomy & Astrophysics, 114:278+ (1982).

[4] Columbia Encyclopedia (Columbia, 1975).

[5] Mercier, "Almanac for Trebizond for the Year 1336" (Academia, 1994).

[6] Jet Propulsion Laboratory, "Horizons Ephemeris", internet online (Nov. 2010 & Jan. 2011).

[7] Roy, "Orbital Motion", 3rd ed., (Hilger, 1988).

[ 8 ] Newcomb, "Popular Astronomy" (Harper, 1884).

[9] Astronomical Almanac (U. S. Naval Observatory, 1965), p. 489.

[10] Ratdolt, ed., "Parisian Alfonsine Tables" (1252AD; printed Venice, 1483AD).

[11] Chabas & Goldstein, "The Alfonsine Tables of Toledo" (Kluwer, 2003).

[12] Sivin, "Granting the Seasons" (Springer, 2009).

[13] Newton, Robert, "Ancient Planetary Observations and the Validity of Ephemeris Time" (Johns Hopkins, 1976).

[14] Newton, Robert, "Ancient Astronomical Observations" (Johns Hopkins, 1970).

[15] Playfair, "Remarks on the Astronomy of the Brahmins", Edinburgh Proceedings (1790). Reprinted in [16].

[16] Dharampal, "Indian Science and Technology in the 18th Century" (Delhi; Impex India, 1971).

[17] Kepler, "Ad Vitellionem Paralipomena quibus Astronomiae Pars Optica" (1604), Caput VIII, Sec. 5 "De reliqorum siderum occultationibus mutuis" pp. 304-307 in original pagination or pp. 263-265 in the edition used for this paper: "Johannes Kepler Gesammelte Werke", vol. 2 (C. H. Beck'sche, 1939)(in Latin).

[18] U. S. Federal Bureau of Investigation, "National Stolen Art File" (online, Jan. 2011).

[19] Grant, "History of Physical Astronomy" (Baldwin, 1852; online Google Book).

[20] Albers, Sky & Telescope 57(3):220+ (March 1979).

[21] Evans, James, "History and Practice of Ancient Astronomy" (Oxford, 1998; online Google Book).

[22] Thurston, "Early Astronomy" (Springer, 1994).

[23] Neugebauer, "A History of Ancient Mathematical Astronomy", Part 1 (Springer, 1975).

[24] O'Neill, "Early Astronomy" (Sydney Univ., 1986).

[25] Bickerman, "Chronology of the Ancient World" (Cornell, 1968).

[26] Neugebauer, "A History of Ancient Mathematical Astronomy", Part 2 (Springer, 1975).

[27] Ptolemy, "Planetary Hypotheses" (later than 141AD).

[28] Ptolemy, "Almagest" (c. 141AD).

[29] Kalidasa, "Uttara Kalamrita" (c. 1700AD) as quoted in translation on www.2012-doomsday-predictions.com .

[30] Simon et al, Astronomy & Astrophysics 282:663+ (1994).

[31] Clemence, "On the Elements of Jupiter", Astronomical Journal 52:89+ (1946).

[32] Bentley, "Hindu Astronomy" (Biblio Verlag, 1970; original ed. 1825).

[33] Dziobek, "Mathematical Theories of Planetary Motions" (Dover, 1962; original ed. 1892).

[34] Brown, "Lunar Theory" (Dover, 1960; orginal ed. 1896).

[35] Pontecoulant, "Theorie Analytique" (1829-1846).

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[37] Bailly, "Traite de l'astronomie indienne et orientale" (1787)(in French).

[38] Plethon, George Gemistus, "Manuel D'Astronomie" (1446AD), Tihon & Mercier, transl. (Bruylant-Academia, 1998).

[39] Kepler, "Tabularum Rudolphi Astronomicarum" "Pars Secunda" "Martis" (1627; but title page also refers to Brahe, 1598), pp. 43, 60-61, in original pagination; in "Johannes Kepler Gesammelte Werke", vol. 10 (C. H. Beck'sche, 1939)(in Latin).

[40] Astronomical Almanac (U. S. Naval Observatory, 1990), p. B18.

[41] U.S. Naval Observatory, Time Service Dept., "Leap Seconds", tycho.usno.navy.mil/leapsec.html , as online March, 2011.

[42] Stephenson & Morrison, Philosophical Transactions of the Royal Society of London, Ser. A 313:47-70 (1984).

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[44] Curott, Astronomical Journal 71:264-269 (1966).

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Joe Keller</i>
<br />Bright Stars over the Pyramids: Atlantean Knowledge (Part 4)
by Joseph C. Keller, M. D., September 2, 2009


Luna's vs. Khafre's heights. ...

"...secondary cycle...of perigee...lunar distance varies between 356,375 and 406,720km, the minimum being the 'proxigee' [cites FJ Wood, "Tidal Dynamics", 1986]."

- JH Duke, 2009, johnduke.com


Measuring from the base of Khufu (the biggest pyramid, and the one with the lowest base) the height of Khafre's apex (the highest pyramid) is 5664 +/- 13 inches (per Petrie) above Khafre's base; Khafre's base is 1011 cm (per Vyse) above Khufu's base, giving Khafre's total height as 6062.0 inches.

The ratio of Luna's "proxigee" (nearest perigee, as center-to-center distance) to Khafre's total height, equals the number of days in 6337 +/- 14 yr. My best estimate of Barbarossa's period (either from my four sky survey detections, or from my calculation of Year One of the Egyptian calendar from Sothic dates) is 6340yr.

...
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

The number of Julian days in 6339.5 tropical yr, assuming a 365.25636 Julian d sidereal yr and Newcomb's precession (linear formula) for 6339.5/2 - 113 yr before 1900AD (i.e., midpoint of latest Barbarossa period) is 2315454.1. Assuming 2.0 ms/century day lengthening, this becomes 2315455.8 actual mean solar days. Solstice-to-solstice time also varies a few days because of Earth's eccentricity, but this is negligible anyway. The latter figure, times the height of Khafre's pyramid (6062 +/- 13 inches, referred to Khufu's base) is 356522 km (the last two or three digits are not significant).

The JPL Horizons ephemeris gives the closest approach (center to center) of Luna to Earth on March 19, 2011, as 356574 km. Though of questionable numerical significance, this fits the above, better than the 356375 km theoretical proxigee.

Dr Joe, Can you post an explaination of why you have been logging in and not posting anything for several months? I am puzzled and wonder if I'm missing something important. thanks

So Dr. Joe ....

do you remember the earth quake that happened around 2007 that threw
the moon off its axis ... how long would that normally take for the moon to
fix it self with out a nuclear space ship with a gravitational
tractor beam moving it back in to proper alinment ?

Thanks

MARX

Hi *** & ***,

*** asked me to tell about my Mayan Calendar theories. Everything I know about it, as soon as I know it, I've been posting to the messageboard of the website of the late Dr. Tom Van Flandern, www.metaresearch.org . I always post under my own name, Joe Keller. My posts about the Mayan calendar are all on the subthread, "Requiem for Relativity". Occasionally I post a summary on the messageboard, but here is a special summary just for you, with some never-before-published "new information"!

There is a stone inscription, "Tortuguero Monument 6". Part is missing but what we have, says basically, that on Dec. 21, 2012 "Bolon descends" (some think Dec. 23, but it's one or the other; that's been known since the 1950s, though with a few dissenting experts). "Bolon" sounds like "bola" or "ball" and I think is cognate to the Indo-European word for ball, sphere, or heavenly orb. Dec. 21 (or 23), 2012 is the end of the "Mayan Long Count", an approx. 5125 yr. calendar interval analogous to our millennium. If the Maya or their predecessors deliberately ended the Long Count on the winter solstice, Dec. 21, they must have known the precession of the equinoxes more accurately than anyone else we know of, before the Renaissance.

Unlike our millennium, the Mayan Long Count is not an arbitrary interval of time. The factors going into the time interval were chosen cleverly so that the Long Count is a whole multiple of the orbital period of Mars (new information), a whole multiple of the orbital period of Jupiter, a whole multiple of the orbital period of Saturn, and also a whole multiple of the orbital period of Uranus.

So, the Maya, or their predecessors, knew the orbital period of Uranus. People with excellent vision can see Uranus unaided under good conditions. Flamsteed saw Uranus with the unaided eye but recorded it as a star; so Herschel, who saw it with a telescope, is credited as the modern discoverer.

For Mars, using the modern estimate of the orbital period, I find 360*400*13 days in the Mayan Long Count / 686.97985 Julian days Mars orbital period = 2724.97 orbits. Because of tiny but uncertain variations in day length (I will assume 2ms/day/century and correct all time intervals), it's impossible to be perfectly accurate, but if I use the Seleucid Babylonian value for Mars' sidereal orbital period, which amounts to about 686.9722 Julian days, the result is 2724.9997 orbits. This suggests that the Seleucid Babylonian value for Mars' period was, originally, derived as exactly 1/2725 of the Mayan Long Count (new information).

The Long Count starts at 3114BC. This is important because the Egyptian chronology of Manetho starts with the "1st Dynasty" at about 3110BC. Also, the Hindu "Kali Yuga" starts at 3102BC: Bailly & Playfair proved the truth of this date from the planetary conjunction that happened then, not an especially close or rare one, mentioned in old Hindu almanacs. Using modern ephemerides, it's easy to verify that many of the astronomical parameters given by old Hindu almanacs (e.g., the rate of progression of Luna's perigee, or even the precise period of Earth's rotation) correspond to about 3100BC too. So the Mayan Long Count began at the time that was the main reference point for Manetho's Egyptian chronology, and also the time that was the main reference epoch for Hindu astronomy.

Because the Mayan Long Count is not an arbitrary interval, but rather a time interval really based on some unknown resonance affecting planetary orbits, it is likely that the year of importance is not the beginning of the Long Count, but rather, the end. It is the "Mayan Long Countdown".

The mass extinction c. 12000 yr ago ("black layer" in N. America, all animals larger than bison become extinct in N. America within 100 yrs or less) is well known. Less well known, is that c. 6000 yr ago, according to a study recently published in a mainstream peer-reviewed archaeology journal, there was a drastic reduction in cranial diversity in N. America and maybe also in Europe. Before roughly 6000 yr ago, the cranial diversity in N. America exceeded that of the entire world today; afterward, only the Amerindian type remained, the genetic diversity of which is narrow. Not only bottlenecks in the diversity of cranial anatomy, occurred roughly 6000 yrs ago: the origin of the Indo-European language, a linguistic bottleneck, occurred c. 6500 yrs ago too. Also, the only known simultaneous bicoastal megatsunami occurred c. 6000-6500 yrs ago.

St. Hippolytus and other major early Christian writers say that the created world only lasts "6000 yr.", but there is no really logical derivation of this figure from the Bible. So maybe, as they often did, Christians recorded some "pagan" knowledge here. The Anglican Bishop Ussher did derive from the Bible, that "Creation" was c. 4004BC (+/- 28).

My own interpretation of "Sothic dates" carrying on the work of Eduard Meyer with the help of modern ephemerides and calculators, indicates that the Egyptian calendar began at the summer solstice 4329BC, i.e. 6340.5 yr before the end of the Mayan Long Count. Recently, German scientists studying lake sediments, concluded that the sudden climate change of c. 12000 yr ago (i.e. the onset of the "Younger Dryas") occurred 12683 yr, to the year, before the end of the Mayan Long Count. N.B.:

6340.5 * 2 = 12681

Studying online sky surveys in Feb. 2007, I discovered "Planet X" near where David Todd, Percival Lowell, and Robert Harrington thought it would be. My search was facilitated by my theory that Planet X would be near the so-called (positive) "CMB dipole". In the spring of 2009, before I knew any of the above historical information, I definitively calculated its orbital period as 6340 +/- 7 yr. However, this planet X has a typical orbit for a "brown dwarf companion" (though to be absent from longwave infrared, it must be very cold and/or surprisingly small): it never comes near the solar system; its semimajor axis is 344 A.U. and its eccentricity 0.61.

It is not a "near miss with Planet X" phenomenon. Planet X merely participates, or has participated, in some unknown resonant phenomenon. N.B.:

velocity / acceleration = time

speed of sound in hydrogen / Hubble acceleration (i.e. H0 * c) = approx. 6500 yr

Among other evidence of resonance, I discovered that the four known asteroids whose rotation periods cluster at the minimum observed value (approx. 5 hr) all will be in line with each other, and with the Sun and with my Planet X (which I named Barbarossa), in Dec. 2012. Furthermore at least two of these asteroids, have the same rotation axis.

Conclusion. There is a yet-undiscovered physical process, apparently involving the interplanetary medium (i.e. hydrogen gas or protons) and the so-called Hubble acceleration (which might be a misinterpreted local physical phenomenon). This process has a period of about 6500 yr.; ancient astronomers thought it was due again in 2012AD, hence the Mayan Long Count(down). Every 6000-6500 yr., Earth gets hit by something like a big flyswatter from space. It's not the end of Creation, but it seems like it. Our species gets thinned out, but picks up the pieces and continues, though with some historical amnesia.