Hi Jim, I did give a link to an article on Hawking radiation and I mention power several times throughout the posts on this. The Hawking equation for power, where we look at the Planck mass, gives us a power in Joules per Planck second. When we alter the equation and put in the speed of gravity, then we get 2.898E-19 Watts.

What does this mean? Well, a Planck mass micro black hole evaporates in about 1E-41 seconds! yet if we allow for a ftl speed of gravity then it radiates very slowly, it would take about 70 billion years to disappear.

It looks rather like a mini supernova. Interesting questions would be, this thing is a source, does it have a twin that's a sink? Or is it its own antiparticle? Could protons/antiprotons and electrons/positrons be slightly different types of mini nova?

Hi Sloat, I try to follow and have yet to get more than a few thoughts into a post before I'm lost. This latest post has me wondering what a Planck second is. Can you define that term? I am going to study the post while avoiding that term until you define what it is. I'll try to figure out the part about the universal charge and the watt unit you have uncovered.

Otto Struve, Aitken and others long have noted that there might be poorly understood systematic errors in measuring binary orbits. Barbarossa's eccentricity (0.6106, which I calculated from the sky surveys) is, in any case, near the measured mean for wide (>100 yr) binaries (details in my previous posts).

Suppose the early Sun were surrounded by a disk of distant particles. For simplicity suppose these particles are of equal, small mass. Suppose that the mean energy (potential plus kinetic) of the particles is zero, but that every particle has the same angular momentum.

Suppose interactions occur between the particles, involving only forces radial to the Sun, so the particles keep the same angular momentum but their energies acquire a Boltzmann (i.e. exponential) distribution with mean zero.

Let a0 be the radius of a circular orbit which has the given angular momentum. Those particles having negative energy (i.e. bound, i.e. elliptical orbits) will have semimajor axes between a0 and infinity, corresponding to eccentricities between 0 and 1. The Boltzmann (i.e. exponential) distribution of energy levels, contains a number of particles, between energies E and E + deltaE, that is proportional to exp(-k*E) where k is a constant determined so that the mean energy will be zero. Elementary integral calculus shows that the mean energy of the bound particles (not of all the particles, only of the bound particles) corresponds to a semimajor axis 2.71828... - 1 = 1.71828... times as large as a0.

If all the particles then agglomerate into one body, while conserving energy and angular momentum, that body's eccentricity must satisfy the equation

Dr Joe, Binary orbits are determined by assuming the mass center is also the location of mass whereas the mass is located far from the point called the barycenter of a binary. If all of the system's mass was located at the barycenter the Kepler Law exactly predicts the orbits of all other bodies is the system. But, as the mass of the system can never by exactly located on any one point the Kepler Law cannot be expected to explain everything. I would rather like to see the 2/3 relationship pop up(~.6 or so in your terms) but mostly for my own sense of neatness. In the real universe how would the particles you want to exist work-they seem to violate natural laws in some ways in order to merge into an independent body bound to it's partner because it cannot gain enough angular momentum to go off on its own. The rules of of the force we call gravity have not yet been totally revealed or discovered.

Dr Joe, ...how would the particles you want to exist work - they seem to violate natural laws in some ways in order to merge...

Hi Jim,

Thanks for the post. You make several relevant points, but this one hits the nail on the head. I'm hoping that though the behavior of real particles would differ from mine in several important ways, those differences would not affect the final eccentricity much.

Dr Joe, The real final structure would be a galactic disk and lesser mass particles would be captured by greater mass particles-right? The lesser structures within the galactic disk such as our solar system or any other star system would be bound to the disk structure which must have some unknown effect on our Kepler model. Our star has a tiny effect on the galactic structure which might be covered by Newton's 3rd law. If you find action you can assume it will be observed in some other form in some other place.

The two relations between Barbarossa's orbital parameters, and fundamental astrophysical constants

I mention both these relations in recent posts but will restate them here:

1. speed / acceleration = A * time

Here the speed is sqrt(k*T/m) = 14996.77 cm/s, where k is Boltzmann's constant, 1.38053/10^16 erg/deg, T is the cosmic microwave background temperature, estimated as 2.72553deg Kelvin; and m is the mass of the hydrogen atom, 1.00797 atomic mass units = 1.67302/10^24 gm. This is the isothermal speed of sound in atomic hydrogen, however rarified, at the CMB temperature. The acceleration is the Hubble acceleration, i.e. the Hubble parameter times the speed of light: H0 * c; in this post, I'll find the value of the Hubble parameter that fits both equations. The time is the Barbarossa period; my best estimate is from the Egyptian & Mayan calendars, 6339.5 tropical yr, adjusted to Newcomb's mean tropical year during the interval 4328BC - 2012AD: 6339.5yr*365.2424d/yr = 2.00055*10^11 sec (the 1.8 day correction for Earth's orbital eccentricity's effect on this solstice-to-solstice time, is insignificant at this precision). "A" is an adjustable parameter (see below).

2. speed * A * time = distance

The speed and time are as above. The distance is Barbarossa's latus rectum, a*(1-e^2) = 343.84AU * (1-0.610596^2) = 3.22619*10^15 cm. The parameters of Barbarossa's orbit are from my own work; the value of the CMB constant is as I recall it from a recently published journal article; and other physical constants are from the CRC Handbook, 44th ed., 1962.

To exactly satisfy both equations simultaneously, requires two adjustable parameters. The first adjustable parameter is the Hubble parameter, which is known observationally only to within ~5%. The second adjustable parameter is the constant, A, by which Barbarossa's period is multiplied.

Eqn. #2 implies that A = 1.075331 = approx. 1 + 1/13.

With this value of A, eqn. #1 implies that the acceleration is 6.97118/10^8 cm/s^2 = 71.7556 km/s/Mpc. This is the 72 km/s/Mpc value for H0 which long has been the most accepted, and still is within the error bar of the latest best Hubble telescope effort.

More reasons a binary's eccentricity should be ~0.6

In my Oct. 28, 2010, post, I explain that if all the original particles have the same angular momentum, and if their mean orbital energy is zero, and if the orbital energies follow a Boltzmann distribution, then, if all the bound (i.e., negative orbital energy) particles agglomerate without loss of orbital energy, the eccentricity will be 0.64655. Particles are likelier to agglomerate when they intersect with similar velocities; at similar velocities, rather little orbital energy is lost. Some of the orbital energy lost when the particles agglomerate, might be recoverable (e.g. orbital energy of one particle about the other). Yet some orbital energy will be lost, and this will cause the final eccentricity to be < 0.64655.

If the apses of the original particles are distributed symmetrically in the plane, then the final eccentricity (the eccentricity after all particles are agglomerated into one) must be zero, because there is no preferred direction. On the other hand, Barbarossa's aphelion aligns with the galactic center, so, plausibly the original particles all aligned with the galactic center for the same reason; let's suppose that the original particles did have the same aphelion direction (and the same angular momentum vector).

The simplest particle interaction, is that of point masses, as discussed in the dynamics chapter of most freshman college physics texts. Particles with the same aphelion direction and angular momentum vector, have the same latus rectum segment, and for different eccentricities, intersect only at the ends of the latus rectum. At the latus rectum, the potential energy is the same for all particles because the position is the same; the energy of transverse velocity is the same because the angular momenta are the same; only the energies of radial velocity differ, proportional to eccentricity^2. The eccentricity is proportional to the radial velocity; so the mean eccentricity, will be the eccentricity of the agglomeration. This mean eccentricity, e, can be found as the ratio of two integrals:

Numerator: integral of d(e^2)*exp(-e^2)*e, from 0 to 1

Denominator: integral of d(e^2)*exp(-e^2), from 0 to 1

(The exponential function gives a Boltzmann distribution, with mean corresponding to eccentricity = 1.) Integrating the numerator numerically (use the substitution u = e^2), I find

eccentricity = 0.59948... . The eccentricity might be higher than this if some particles in hyperbolic orbits agglomerate also, on their one-time pass through the system; this describes Barbarossa (e = 0.6106 according to the orbit I fit to the sky surveys last year).

Hi Joe, with Jim I'd like to see that 2/3 but I think we have to allow for the fact that our orbit round the galactic centre is not going to be a perfect circle. In which case the apses will sometimes lead or lag by a tiny amount. i can't help thinking that this has something to do wuth the neutrino, that temperature of about 2.752 Kelvin looks to e in the mass range of the neutrino. We have basically hydrogen atoms orbiting in a sea of neutrinos with little interaction but won't there be slightly more neutrinos coming from galactic centre? A particle diving in an elliptical orbit through this soup wouldn't see the mass "above" it, it would want to describe a roseate (it would look like the petals of a flower) orbi, all things being equal but that slight excess of neutrinos could tend to lock it.

Well, there must be a way to determine the angular momentum of the sun relative to the galatic center and using that to determine the orbit of the sun around the galaxy. You guys have unused observational details of the Milky Way that can be useful to prove your hypothesis

Dr Joe, I don't understand what the "isothermal speed of sound" means. How can sound exist in the vacuum of space? I suppose the existence of sound needs matter unlike light that exists even without matter.

Dr. Joe, ...what the "isothermal speed of sound" means. ...vacuum of space?...

Hi Jim,

My source for the formula, was Hausmann & Slack's old college physics text. There isn't any definite cutoff at which sound disappears. For example, a rocket can exceed the speed of sound even though the atmosphere is very thin at very high altitude.

There are a few hydrogen atoms and/or protons per cubic centimeter in interplanetary space. It would be difficult to impart much sound energy to such a medium using, say, stereo speakers, but something huge like a nova could impart much sound energy to it. Also, the continuum approximation used for studying sound, would apply (in interplanetary space) only to long wavelengths.

Isothermal means that the temperature of the medium is constant over time. The speed of sound isn't the same, if the sound isn't isothermal (i.e. if the medium is significantly heated and cooled by the compression waves each cycle).

Dr Joe, Sound waves transmit energy but, not much even in dense matter. How much of an effect would sound waves have when the density is a few particles per cubic meter?

Open letter to the Director of the Lowell Observatory

...Assuming a circular orbit and making first order approximations to correct for Earth parallax, Barbarossa has period 2640 yr. and is 191 AU from the sun. Accordingly, the resonances of the orbital periods of the outer planets have discrepancies which advance prograde with periods

Jupiter:Saturn 5:2 2780 yr ...

In March, 2007, I told the Lowell Observatory (in direct emails similar to the above "open letter" on p. 8 of this messageboard thread) that Barbarossa's orbital period is similar to the period of advance of a Jupiter-Saturn conjunction; that is, Barbarossa shepherds the Great Inequality.

In 2009 I fitted an elliptical orbit to the sky surveys. At the latus rectum of an elliptical orbit, the relationship between distance and angular speed, is the same as for a circular orbit. So, near the latus rectum, as Barbarossa has been in recent decades, a circular orbit will fit the (geocentric) observational data with only a small error. In its actual elliptical orbit, Barbarossa's angular speed at its latus rectum, matches the average speed of advance of the Jupiter-Saturn conjunction.

Using the 6339.36 Julian yr orbital period and 0.610596 eccentricity, I find at Barbarossa's latus rectum (reached at 2003.94 AD) an angular speed consistent with a period of 3148.7 Julian yr. At Dec. 21, 2012 (true anomaly = 91.022deg) I find (heliocentric) angular speed consistent with a period of 3218.4 Julian yr.

The mean period of advance of a given Jupiter-Saturn conjunction point (i.e. a given corner of the "trigon") is exactly 3x the "Great Inequality", i.e. 3 * 1/(5/S - 2/J) where J & S are the periods of Jupiter & Saturn, resp. The present-day Great Inequality (as in the time of Laplace) is about +900 yr, but

"...the GI's average period is...1092.9 years;..."

- Ferenc Varadi et al, "Jupiter, Saturn, and the Edge of Chaos", Icarus 139:286-294, 1999, p. 287, col. 2

Three times the average GI given by Varadi, is 3278.7 yr, only 1.9% longer than period corresponding to the heliocentric angular speed of Barbarossa at the end of the Mayan Long Count.

Most of the discrepancy disappears when the motion of the trigon point, is projected onto Barbarossa's orbital plane. Barbarossa is inclined ~14.5deg to the angular momentum-weighted mean orbital plane of Jupiter + Saturn (about the same as the principal plane of the solar system) and at 2013.0AD is ~68deg past its descending node on that plane. So the angular speed of the trigon point, projected onto Barbarossa's orbital plane, corresponds to a period of roughly

3278.7 * cos(14.5*sin(90-2*(90-68))) = 3224.5 Julian yr

only 0.19% longer than the period, P = 3218.4 yr, corresponding to Barbarossa's angular speed at the end of the Mayan Long Count. A more precise calculation of the projection, using a modern value of the Jupiter + Saturn combined angular momentum vector, and spherical trigonometry, gives

3278.7 / 1.02287 = 3205.4 Julian yr

which is 0.40% shorter than P. A 0.4% error in the speed of the trigon point, corresponds to only 1 part in 30,000 error in Jupiter's average orbital period.

According to NASA's Fact Sheets, for sidereal years of the planets, J = 4332.589 days, S = 10,759.22 d, U = 30,685.4 d, N = 60,189 d, P = 90,465 d, M=686.980 d, and V = 224.701 d (I'll use Wikipedia's V = 224.70069 d). Also, E = 365.25636 d. The Neptune-Pluto inequality is 1/(3/P-2/N); this is -41,068 Julian yr (the same length as the Milankovitch cycle), and also equals the period of regression of the Neptune-Pluto conjunction point. The Jupiter-Neptune inequality is 1/(14/N-1/J) = +1528.0 Julian yr. To get the period of progression of the conjunction point, this must be multiplied by 14-1=13, giving +19,864 yr, (negative) half the Neptune-Pluto regression period. Likewise 1/(11/N-2/S) = -874.9 yr; *(11-2) = -7873.8 yr, a fifth of the Pluto-Neptune regression period. We have

S / J = 2.48 and 19864 / 7907.9 = 2.52

that is, the mean conjunction points of Jupiter (13 points, i.e. a 13-gon) and Saturn (9 points, a 9-gon) with Neptune, move in the same 5::2 ratio as do Jupiter and Saturn themselves, though the J-N conjunction is the slower, and the S-N conjunction moves retrograde.

Earth's conjunction points with Jupiter (an 11-gon), move prograde with period equal to the Great Inequality:

11*1/(12/J-1/E) = 944.0 Julian yr

Varadi gives the average value of the Jupiter-Saturn Great Inequality as 1092.9 yr, but the current value is

1/(5/S-2/J) = 883.19 yr

so the Jupiter-Earth mean conjunction progression is between the current and average values of the GI.

The Venus-Earth conjunction point moves retrograde, with period 5*1/(13/E-8/V) = -1192.8 Julian yr. This is also rather near the long term average value of the GI.

The mean conjunction of Saturn and Uranus moves prograde with period 2*1/(3/U-1/S) = 1135.4 yr. Suppose that originally the PI had its average value, and the Saturn-Uranus conjunction moved forward with period equal to the PI. If the semimajor axes of Jupiter and Saturn are conserved, and their eccentricities remain small, then conservation of angular momentum between Jupiter and Saturn requires that the increase in Saturn's angular speed almost exactly equals the decrease in Jupiter's angular speed. So, half the change in the PI is due to Saturn and half to Jupiter. If the Saturn-Uranus interaction is comparatively small, then the decrement in the frequency of the Saturn-Uranus conjunction progession would be due only to Saturn and would be 1/2 * (3/2)/(5/3) = 9/20 as much as the increment for the Jupiter-Saturn progression (proportionally, 9/20*1/3 = 3/20):

1092.9 / (1 - 3/20*(1092.9/883.19-1)) = 1133.3 yr

only 0.19% less than the actual value. Because the (proportional) change in frequency due to the Jupiter-Saturn interaction, is only 3/20 as much, this might afford a better estimate of the average PI:

1092.9*1133.5/1130.9 = 1094.925 yr

then times 3, and correcting for projection of the Jupiter+Saturn plane,

1094.71*3/1.02287 = 3211.33 yr

only 0.22% shorter than P above.

Let's consider Mars analogously to Uranus. Most online sources use NASA's value of 686.980 d for Mars' period, though some still quote 686.95 d (the older accepted value, e.g. Inglis, "Planets, Stars & Galaxies", 3rd ed., John Wiley, 1972). Mars' eccentricity and proximity to Earth and Jupiter make exact long-term average orbital period calculation hard. Proceeding anyway, I find that the Mars-Jupiter conjunction point progresses with period

(19-3)*1/(19/J-3/M) = 2376.955 yr = 2*1188.48 yr.

Analogously to Uranus, I get the fraction 1/2*(19/16)/(5/3)*2/3 = 19/80 and

1188.48*(1 - 19/80*(1092.9/883.19-1)) = 1121.46 yr

which is about 2/3 of the correction needed to get the average PI.

From Wikipedia's recent values (citing Chapront, 1991), for the various kinds of month lengths with their linear secular trends, I find P = period of progression of the Lunar apse and N = period of regression of the Lunar node. A mean conjunction of the node and perigee (one of the points of the trigon of three conjunctions), progresses with period

3*1/(1/P-2/N) = 549.49 Julian yr = 1098.98/2 for epoch 2013.0AD = 546.98 J yr = 1093.96/2 for epoch 2013.0AD-6339.36 J yr = ~4328BC

Using instead, the fourth degree polynomials for accumulated advance or regression, in Simon et al, Astronomy & Astrophysics 282:663+, 1994, p. 669, sec. 3.4.a.1, I find

549.48 Julian yr for epoch 2012.97 AD and 547.36 Julian yr for epoch 2012.97AD-6339.36 = ~ 4328BC

(Chapront's linear corrections for the month lengths, amount to second degree truncations of Simon's formulas.) There is a small error from defining the nodes relative to the fixed J2000.0 plane as in Simon's sec. a.1 formulas, but Simon's a.2 formulas refer to the equinox & ecliptic of date; when I use Newcomb's linear formula, for the precession in 4328BC, these a.2 formulas give 547.30 yr, in excellent agreement with a.1.

Doubling the progression period of the Lunar perigee/node conjunction, and using spherical trigonometry to correct precisely for the projection of the present ecliptic (the 6000 yr change in the ecliptic is less than significant, at this precision) on Barbarossa's orbital plane, I find period

547.36*2/1.018001 = 1075.36 yr

and tripling this, 3226.09 yr, only 0.24% longer than P, the period related to the instantaneous angular speed of Barbarossa on Dec. 21, 2012. (Earlier I estimated that my orbit fitting to the sky survey points involved about +/- 0.11% error in the period.)

*********

Digression (Nov. 11, 2010): Force is a signal, not a field.

In late 2000, almost a decade ago, I discovered a new theory of force. I immediately gave a copy of the calculations to Prof. T. at Oxford.

In my freshman physics course at Harvard, Prof. Purcell emphasized the well-known fact that field lines from a relativistic electron in rectilinear motion, point toward its present position, not its retarded position: as if lightspeed were infinite. If the electron is accelerated, then the field lines point as if lightspeed were infinite but the motion were linearly extrapolated from the time the light signal left the electron.

It seems to me that not only must the observer detect a signal from the electron; also the electron must detect a signal from the observer. A lightspeed signal goes from the observer to the electron, and an answering signal immediately returns from the electron to the observer. Each time, the body detecting the signal responds to the linearly extrapolated position of the body emitting the signal.

My theory in 2000, which I quantitatively verified, was that general relativistic perihelion advance, arises simply from force transformed according to special relativity, assuming that there is such a two-way interaction. The general relativistic effect, is the difference between ideal instantaneous transmission, and the actual two-way lightspeed transmission which adjusts itself to emitter motion only to linear order each way.

Gravity between Barbarossa and, say, Jupiter, could be summarized as BJB or JBJ; this is two-trip force, the simplest possible. The next simplest kind of force would be four-trip force, e.g. BJBJB or BJBSB; the latter, rather than any tiny (two-trip) gravitational advantage, would explain Barbarossa's effect on the Jupiter-Saturn resonance (which Barbarossa saves from chaos). Another four-trip force would be JBEBV, which starts and ends nearby, rather than at Barbarossa, therefore must be adjusted by the reciprocal of the projection factor used above. Next, let's use this theory to reconcile the E-J and V-E resonances with the GI.

*********

The difference between the E-J and V-E conjunction progression (or regression) rates corresponds to the period

1/(1/944.0 + 1/1192.8) = 526.96 yr

times 6 and *multiplied* (because it is a four-trip four-body interaction; see "Digression" above) by the projection factor for Earth's orbit,

526.96*6*1.01800 = 3218.7 yr

only 0.01% longer than P, a perfect result to the significant digits available.

In Bretagnon et al, A&A 400:785+, 2003, Table 2, the 7th degree polynomial for accumulated precession (equivalent to a 6th degreee polynomial for precession rate) gives a precession period of 26,391.5 yr at 2012.97AD - 6339.36 = ~4328 BC. This is about eight times the GI, and with correction for projection of the ecliptic onto Barbarossa, gives

26,391.5 / 8 / 1.018001 = 3240.6 yr

only 0.69% longer than P.

Collecting these results, which are corrected for projections *onto* Barbarossa's orbit (or for the Venus relation, *of* Barbarossa's orbit):

Varadi's average Great Inequality x3 underestimates P by 0.40% presumed average Saturn-Uranus resonance x3 underestimates by 0.22% Lunar perigee/node resonance x6 @ 4328BC overestimates by 0.24% modern E-J minus V-E resonance x6 overestimates by 0.01% 4328BC Earth precession /8 overestimates by 0.69%

The five estimates averaged together exceed P by 0.06% with Standard Error of the Mean +/- 0.19%. Omitting the Earth precession result, the average is 0.09% less than P, with SEM +/- 0.16%.

Summary. Barbarossa's angular speed at the critical point in its orbit (at Dec. 21, 2012AD) equals the *average* mean rate of change (*according to Varadi*) of the longitude, in Barbarossa ecliptic coordinates, of the Jupiter-Saturn conjunction trigon at Barbarossa's longitude. The Saturn-Uranus conjunction, the Earth-Jupiter conjunction, and the Venus-Earth conjunction, all progress or regress at ~3x the rate of the Jupiter-Saturn conjunction; for the Saturn-Uranus conjunction, the period of progression equals exactly the average Great Inequality, after correcting for Saturn's contribution to the change in the Great Inequality from its average value. The difference between the E-J and V-E rates, and the Lunar perigee / Lunar ascending node conjunction rate, both progess at 6x the speed rather than 3x. Earth's precession is at 1/8 the speed.

With projection onto Barbarossa's orbital plane at Barbarossa's longitude, the synchrony with Barbarossa is very exact, using 3x the theoretical average Great Inequality, either as given by Varadi or as accurized via the Saturn-Uranus conjunction. It is also very exact, for 6x the mean Lunar perigee / Lunar node conjunction period.

Hi Joe, something that may be of interest. I was taking on a webpage to a woman who is writing her second book on myth and christianity. She asked a question about the planet Mercury, as she was looking at astrological charts. I explained a it about elongation and told her to get a copy of the free Celestia star program.

Then she came back on and cried for help. Wikipedia had given a list of solar eclipses in historic times and there was one for Jerusalem in 303 bce. She's looked at it and nothing! Then she'd found a really good eclipse for the year 301 bce with the aid of the program. She also said that it lasted all day!! I explained that thee date was to with the changes made in the calendar and there being no year zero but with a year zero for astronomy. Two years and ten days error. I also said that a total eclipse can't last all day.

So I looked at the event in celestia. Highly unusual, the moon does partially eclipse the sun for the whole day. Now I've said to her that the drop in light intensity would be difficult to notice, as the human eye is very good at adjustment but the temperature drop over the day would be noticed, and animals would be going ballistic all day. Then we've got Venus in the sky ahead of the sun and Jupiter and Sirius appearing at sun set.

Have you got anything on what the temperature drop would be in Jerusalem?

(Edited) I think I should stress here, that the partial eclipse seems to be a particularly good fit for this eclipse in Jerusalem. Saros cycle?

Sloat, You can use the NASA JPL horizons generator to determine when events like this occurred. I'm confident the NASA generator is the best available but not exact because basically its just a model.

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::3 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)

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:

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

686.9795233 +/- 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

686.97062 ? - 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].

[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).

[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.