Requiem for Relativity

Thanks Jim, I passed the link on to her but I think she's asking her university.

Joe, at some point it might be a good idea to post all your Barbarossa data in one place - period, distance from the Sun, location in the sky, etc.

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 Earths 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



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 on which that of Chrysococces might in part be based.

The Persian and Byzantine almanacs often were more accurate than mere extrapolations from Ptolemy's ancient observations. Allowing for various choices of equinox, errors usually were less than a degree.

The Trebizond almanac table gives noon planetary longitudes to the day and arcminute. Interpolating in the table, I find the geocentric ecliptic longitude of the Sun, at the time when Mars and Venus have equal geocentric longitudes. 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.

Next I find the time, according to the JPL ephemeris, when the geocentric longitude difference between Venus and the Sun, is the same as in the Trebizond almanac when Mars and Venus have equal geocentric longitude. I take only longitude differences from the Trebizond Almanac, so I need not know its equinox. I then get the longitudes of Venus and the Sun, from the JPL ephemeris.

The small inclination of Earth's 1336AD ecliptic to the 2000.0AD ecliptic (approx. 0.8' per century = 5') is neglected, but Mars' orbital inclination (approx. 2 degrees) is considered. Mars' orbital elements with secular terms according to [2] are used. My BASIC language computer program solves this problem by iteration on four variables: Mars' orbital radius = r, Mars' true anomaly = f, Mars' orbital radius projected onto the ecliptic = r0, and Mars' heliocentric ecliptic longitude = g.

I begin with rough graphical estimates of r and f. First, Mars' orbital elements imply, if f is known, a sinusoidal approximation for Mars' ecliptic latitude, the cosine of which gives r0 from r. Second, the Law of Sines applied to the ecliptic plane polygon Sun-Earth-Venus-Mars (the projections of Venus and Mars on the ecliptic are used; the polygon is a triangle because E, V and M are collinear) gives g from r0. Third, another sinusoidal approximation involving Mars' inclination, and the definitions of the orbital angles, gives f from g. Fourth, the orbital equation gives r from f. Nine iterations reach double precision.

The Kepler equation and another textbook formula [7, eqns. 4.57 & 4.60, pp. 84, 85] give the eccentric and mean anomalies, which can be compared to Kieffer's mean longitude [2] for 2000.0AD. Correction for actual day length is unneeded, because the JPL ephemeris uses Julian Days. The calculated average Martian sidereal year between the Venus-Mars conjunction in August (then the Islamic month of Muharram) 1336AD (JD 2209262.924) and 2000.0AD (JD 2451545.0) is thus

686.9813730 Julian d

based on the Trebizond almanac table together with the JPL ephemeris for Venus and Earth only. Kieffer's modern figure [2] for 2000.0AD is equivalent to

686.9798529 d

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

If the Trebizond longitudes are accurate except for rounding to the nearest arcminute, then the rms error for a given longitude, is about 2/7 arcminute; interpolation reduces this by ~sqrt(2), though use of three longitudes increases it again by ~sqrt(3). Both the Trebizond and JPL ephemerides give apparent position at the observer; the greatest aberration of light involved is, because of the position of Venus, less than the combined aberrations of Earth and Mars, which for Mars near aphelion, is about 20+15 = 35 arcsec.

These two biggest errors thus are much smaller than the discrepancies between at least some of these almanacs: in 1353AD, my same procedure finds in Chrysococces' almanac a 23.66deg Venus-Sun longitude difference, but in the Zij-Ilkhani and Zij-Alai almanacs, Venus-Sun differences of 24.21 & 24.17deg, resp. Using the average value for the Persian almanacs, I find

686.9764003 Julian d

and from Chrysococces

686.9750673 Julian d

Averaging these results, with the Trebizond result, gives

686.9776135 +/- SEM 0.0019187



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; although the effect of Mars' moons is always < 1/10^4 times the last digit of the JPL positions, not Mars itself but only the Mars system barycenter 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 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.

In conclusion, Kepler's stated 5AM time for this occultation, argues for an actual time some minutes earlier than given by [6], but Kepler's description of the color, argues for an actual time some minutes later than [6]. This author, at age 54, easily saw Venus when near its brightest, with an utterly 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 1590 event was seen before sunrise.



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][34]. 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.

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/cy, 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 / cy, resp. Venus' error, 1.068 deg/cy, 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]. These periods of planets, Sun, Moon, Lunar apse and node, are identical with Varaha's, (3) above.

The number of Mars periods per Maha Yuga, in these five books, ranges from 2296828.522 to 2296833.037. With a Maha Yuga denominated in tropical years, a 365.25636 Julian day Earth sidereal year, and Newcomb's precession for 3102BC, this gives Mars sidereal periods ranging from 686.96663 to 686.96798 Julian d. The sidereal period of Mars, averaged from the five distinct books (Uttara Kalamrita & (1)-(4) above) 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 these five, from 364219.682 to 364226.455 (resp. 146567.298 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, all six of the Hindu 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.

Not counting (5) above, whose table is identical with (3), 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.

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

Books (3) and (5) above list 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.

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 all 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. Books (3) & (5) list 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. The eight digit precision they give for Luna's period, suggests also a Maha Yuga of sidereal years, because other types of year hardly could be known so precisely.

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. 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.



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

(Also, the 1590AD Heidelberg Venus/Mars occultation supports a value perhaps slightly higher or slightly lower than the modern value.)


Essentially three 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]

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

686.98035 - Kepler, 1598AD (see Appendix 2)


Nine 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)

686.9776 +/- 0.0019 - 1336AD & 1353AD, ave. of essentially 3 Byzantine/Persian almanac tabulations of Venus/Mars conjunctions, vs. Mars position now

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

Medieval almanacs might be expected to give positions accurate within a few arcminutes, based on then recent observations, extrapolated according to Ptolemaic formulas. Their larger errors, though often much less than would be expected from mere extrapolations from Ptolemy, suggest a basic policy of adopting Ptolemy's observations while modifying his formulas.

Basically, they seem to have used Ptolemy's value of "y" and their own value of "dy/dt". If the function y(t) is, say, convex downward, the medieval astronomers are giving us not a tangent at their own point, nor a tangent at Ptolemy's point, but rather a chord from Ptolemy to us, with a slope equal to the tangent at the medieval midpoint. So, the medieval almanac tables differenced from our modern observations (i.e., slope of the chord), roughly confirm the medieval almanac parameters (i.e., slope of the midpoint tangent).

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 (1 part in 100,000, per 2000 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.

The rate of change, in the calculated orbital period, is found most precisely 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 effect of the Great Inequality itself, on the orbital frequencies of Jupiter and Saturn. The GI of 2000AD (from mean motions given by [6, Table 1, which already includes sinusoidal GI terms, per his eqn. 8-30]) is 997.578 yr. Given the -1::2.444 ratio, this changes to the theoretical long-term average GI, 1092.9 yr [1, p. 287] if Jupiter's (and hence, hypothetically, Mars') orbital frequency increases by 1 part in 6671, a change which at the above rate (for Mars, 3.49/10^6 d/yr) would require 29,509 yr. Again this equals Earth's precession period, within my ~10% margin of error.



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.



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).

[36] Bowditch, transl., La Place, "Celestial Mechanics" (Chelsea, 1966; original ed. 1829).

[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.

Hi Joe, according to the Stellarium program, there was an occultation of mars and venus seen from Baghdad on the 22nd of October 864 at 3.50 a.m. Jupiter and mercury were very close together on the horizon as well. They must have seen that sky as a bit of an omen.

Dr Joe, Why not report this Baghdad event to JPL and see how they explain Venus being observed one degree from the location their generator shows it to be? I would go with the observation assuming it was writen down at the time of the event because as good as the the JPL generator is it has its faults.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Stoat</i>
<br />Hi Joe, according to the Stellarium program, there was an occultation of mars and venus seen from Baghdad on the 22nd of October 864 at 3.50 a.m. ... <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Hi Bob!

Fantastic! Thanks for the information. I'll check to see if my own estimate agrees with Stellarium.

- Joe