Requiem for Relativity

Uranus' rings also reveal the 5.145 hr period

(Source of data on Uranus' moons & rings: Miner, "Uranus", 2nd ed., 1998; Tables 11.1, 11.2, 12.1, 12.3, pp. 266, 271, 289, 292)

From inside to outside, the eleven rings of Uranus are:

"1986U2R"
6
5
4
alpha
beta
eta
gamma
delta
lambda
epsilon

All were discovered in 1977 except for "1986U2R" & lambda, which were discovered in 1986; these two rings discovered in 1986, are the least substantial of the eleven. The most substantial ring is epsilon, which resonates with Uranus' rotation period and the orbit of Oberon:

1/8.3823h - 1/323.1182h = 2.0034 / 17.24h
i.e. f(epsilon) - f(Oberon) = 2*f(Uranusrot)

Ring 5 is somewhat more substantial than Rings 6 or 4. Ring 5 resonates perfectly, with the orbit of Miranda and 1.5 times the 5.14535h asteroid rotation period estimated in my earlier post:

1/6.2875h - 1/33.924h = 1 / (1.49999 * 5.14535h)

The alpha ring resonates with Ariel:

1/6.8508h - 1/60.4891h = 1 / (1.50151 * 5.14535h)

The beta ring resonates with Umbriel:

1/7.0688h - 1/99.4603h = 1 / (1.47893 * 5.14535h)

The eta ring resonates with Titania:

1/7.4239h - 1/208.9411h = 1 / (1.49599 * 5.14535h)

And the gamma ring resonates with Oberon:

1/7.5307h - 1/323.1182h = 1 / (1.49852 * 5.14535h)

After Uranus' five major moons, the largest of its minor moons are, from inside to outside: Juliet (~42km), Portia (~55km) and Puck (~77km); Rings 6, 4 and delta, resonate (somewhat less accurately) with these moons, resp., though with factor 2.5 instead of 1.5:

1/6.1988 - 1/11.8336 = 1 / (2.53006*5.14535)

1/6.3628 - 1/12.3167 = 1 / (2.55815*5.14535)

1/7.6911 - 1/18.2840 = 1 / (2.58006*5.14535)

Here lies the explanation of the radii of all nine of Uranus' most substantial rings.

I've combined my latest five posts, June 2 - June 14, 2011, into an article which I'll submit today to Icarus. If Icarus rejects it, I'll try Observatory. Due to a browser problem, I don't have time to cut and paste the article here now yet.

I've overcome the browser problem; here's the article:



The Mayan Calendar and the Mysterious 5.145 hr Period
by Joseph C. Keller


Abstract. The Mayan Long Count involves several well-known and not so well-known planetary astronomical and astrophysical phenomena, which must have been known to the Maya or their predecessors. Among these is a previously unreported 5.14535 +/- 0.00717 hr rotation or revolution period, which is prominently involved with the revolution periods of Uranus' moons and ring particles and of Mars' moons, the rotation period of at least four asteroids, the rotation axes of Uranus and of at least some of those asteroids, and the end date of the Mayan Long Count, Dec. 21, 2012.


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

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

So, the Maya, or their predecessors, knew the orbital period of Uranus. People with excellent vision can see Uranus unaided under good conditions. Flamsteed saw Uranus with the unaided eye but recorded it as a star.

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

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

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

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

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

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

6340.5 * 2 = 12681

Studying online sky surveys in Feb. 2007, I discovered "Planet X" near where David Todd, Percival Lowell, Robert Harrington, and Conley Powell thought it would be. Receiving no response from the IAU, I named it myself: "Barbarossa". My search was facilitated by my theory that Planet X (Barbarossa) would be near the so-called (positive) "CMB dipole". In the spring of 2009, before I knew the above historical information, I definitively calculated Barbarossa's orbital period as 6340 +/- 7 yr, and spoke on the subject, to the May 2009 meeting, at Cedar Rapids, Iowa, of the North Central Region of the Astronomy League. However, this planet X (Barbarossa) has a typical orbit for a "brown dwarf companion" (though to be absent from longwave infrared, it must be very cold and/or surprisingly small): it never comes near the known solar system; its semimajor axis is 344 A.U. and its eccentricity 0.61.

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

velocity / acceleration = time

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

Also, N.B.:

(6340 yr)^2 = 5.14535 hr * 10.90*10^9 yr*2*pi

i.e., Barbarossa's orbital period (which also seems to be the historical interval between catastrophes) is the geometric mean of (a somewhat short estimate of) the Hubble Time (multiplied by 2*pi to convert from one radian to one cycle), and the mysterious 5.145 hr period.

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

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


The asteroid alignment of Dec. 2012. Four large asteroids which, in Dec. 2012, will align with the Sun and Barbarossa, are the same four which cluster near the minimum observed rotation period (about 5.145 hr). At least two of them, and maybe all four of them, have, at least approximately, the same rotation axis. When I tried to get the lightcurve data on the other two, so that I could calculate their rotation axes myself, I met with uncooperation from the possessors of those lightcurves; such uncooperation, which amounts to theft, often occurs when public funds are entrusted to quasi-private organizations. I found the lightcurves published online to be too ambiguously and spottily presented, but I might find time to study them again. I did complain to my Congressman about the unwillingness of these grant-funded lightcurve researchers to really, fully share their data with the public (they ignored my substantive Freedom of Information Act request) but apparently he wasn't able to do enough.

The most accurate determination of any of those asteroid rotation axes, might be the Keck telescope determination for 511 Davida: RA 295, Decl 0 (Conrad et al, Icarus, 2007). The error bars of Conrad's figure were relatively small; the rotation axis was found directly from the projected shape of the asteroid rather than the usual way from lightcurves. This is suspiciously close to the rotation axis of Uranus, whose "North" pole (i.e. North of the ecliptic, but clockwise like Earth's South pole) has RA 257.3, Decl -15.2 (from online reference source www.cseligman.com ). The difference is 40.3 degrees which might at first glance not seem impressive, but which is significant at p=0.24, allowing that either pole of Uranus could as well have been near Davida's.

Furthermore Uranus also aligns fairly well with Barbarossa and the Sun, in Dec. 2012. By my estimate, Barbarossa's barycentric (barycenter of known solar system per JPL) J2000 astrometric celestial coordinates at 12.0h UT Dec. 21, 2012, are RA 11:27:46.95, Decl -9:22:53.1. The JPL Horizons ephemeris offers only heliocentric coordinates for Uranus, but for Barbarossa, heliocentric coords. differ only about an arcsecond from barycentric coords. The JPL heliocentric J2000 astrometric coords. for Uranus at the same time are: RA 6.94508, Decl 2.22893. This is 16.0 deg away from diametric opposition to Barbarossa; out of a 360 arc, this separation from 0 or 180, is significant at p = 16*4/360 = 0.18.

On the "JPL small-body database browser" I find the rotation periods

511 Davida 5.131h
39 Laetitia 5.138h
947 Monterosa 5.164h
1717 Arlon 5.1484h

The difference between the shortest & longest of these periods is only 0.6% = 119 sec. There are no other asteroids whose rotation periods are known to lie in this interval, and there are almost no asteroids whose rotation periods are known to be less than this. So it is fair to say that this is the entire known set of asteroids whose rotation period clusters near the minimum.

According to the JPL Horizons ephemeris, the heliocentric J2000 ecliptic longitudes of these asteroids at 12.0h UT, Dec. 21, 2012, are

Davida 350.79 deg
Laetitia 174.48 deg
Monterosa 179.47 deg
Arlon 174.32 deg
(Uranus 7.26 deg)

By my 2009 calculation, Barbarossa's astrometric heliocentric J2000 ecliptic longitude then is 176.369. So, three of the four asteroids lie within 3.1 deg of Barbarossa's longitude as seen from the Sun (p=0.0020%); Davida is not near conjunction with Barbarossa, but is only 5.6 deg of longitude, away from opposition, as seen from the Sun. For all four to be within 5.6 deg, of either conjunction or opposition, gives p=0.0015%.

Another important coincidence, is that the "Barbarossa period", ~6340 yr, is near a whole or half-whole multiple of the orbital periods of all five of these bodies:

Uranus sidereal orbital period (NASA fact sheet) = 30685.4d = 84.0120 Julian yr; Uranus period * 75.5 = 6342.91 Julian yr

(The orbital periods of the asteroids are from the JPL small-body database, which claims that the five digits after the decimal are significant, except maybe for Monterosa.)

Davida 2056.39360d = 1126 * 6339.491 Julian yr
Laetitia 1681.63423d = 1377 * 6339.796 Julian yr
Monterosa 1665.95170d = 1390 * 6339.967 Julian yr
Arlon 1188.47218d = 1948.5 * 6340.145 Julian yr

Half an orbital period for these asteroids, is only about two years, but the phase relationship is significant. At 6339.491 Julian yr, when the phase of Davida is zero, I find that the phase of Laetitia is -24deg, the phase of Monterosa -38 deg, the phase of Arlon -72+180 = +108 deg, and the phase of Uranus -15+180 = +165 deg. Plotted on the unit circle, these phases give five lines all through Quadrants II & IV, none through Quadrants I & III. So, not only do the four asteroids and one major planet align with Barbarossa in Dec. 2012; this alignment tends to recur about every 6340 yr.

In Feb. 2007, I found Barbarossa quickly on sky surveys, because I assumed it would be near the positive CMB dipole. However, these four asteroids plus Uranus, on Dec. 21, 2012, match Barbarossa's heliocentric ecliptic longitude better than they do the heliocentric ecliptic longitude of the (+) CMB dipole (171.82 deg, from the galactic coords. given by Lineweaver in 1996). Four of the five bodies are nearer in longitude to Barbarossa or Barbarossa + 180, than to the (+) dipole or (+) dipole + 180; only Davida is nearer to the dipole+180 than to Barbarossa+180. The sum of squared differences from Barbarossa's longitude is 167; from the dipole's, 311. Excluding Uranus, and including only the four asteroids, those figures become 49 & 73, resp. The mean of the longitudes of the five bodies (modulo 180) is 177.26; the sum of squared differences from the mean is 163.


The 5.145 hr Davida-Laetitia-Monterosa-Arlon rotation period resonates with Uranus' major moons, large minor moons, and rings. If the rotation periods of the four asteroids are taken as independent, equally weighted estimates of a fundamental rotation period "P1", then that period is 5.14535 hr +/- 0.00717 hr (Standard Error of the Mean). Another important rotation period, would be that of Uranus; the NASA fact sheet gives 17.24 hr using "magnetic coordinates" (one of several reasonable ways to define the period precisely).

Both these ~5 hr and ~17 hr periods, which I'll call "P1" & "P2", resp., resonate with the differences in the anomalies (i.e. the separation angles) of the major moons of Uranus:

Miranda laps Ariel with period equal to 15.012 * P1
Miranda laps Umbriel, 10.006 * P1
Ariel laps Umbriel, 30.003 * P1

The wholeness of these coefficients is independent: when (f1-f2)*P1 and (f2-f3)*P1 are reciprocals of integers, (f1-f3)*P1 generally is not. There are 5*4/2=10 such differences for the five major moons, so approximately p = 0.012*2*0.006*2*0.003*2*(combinations of 10 things 3 at a time) = 0.02%.

The resonance is less strong, with Uranus' rotation period ("P2"):

Miranda laps Umbriel, 2.986 * P2
Ariel laps Umbriel, 8.955 * P2
Umbriel laps Titania, 11.010 * P2

giving p = 0.014*2*0.045*2*0.010*2*(combinations of 10 things 3 at a time) = 0.6%.

The major moons of Saturn show only a slight tendency for the periods of the differences in their anomalies, to be multiples of P1. The NASA fact sheet lists eight "major" moons for Saturn. Two of the 8*7/2=28 periods of differences, are about 0.01 away from a whole multiple of P1, and one is about 0.02 away. So p = 0.01^2*0.02*2^3*(combs. of 28 things 3 at a time) = 5%.

The Galilean satellites of Jupiter also show a slight tendency. Two of the 4*3/2=6 periods of differences are about 0.04 away from whole multiples of P1. So, p = 0.04^2*2^2*(combs. of 6 things 2 at a time) = 10%.

The two moons of Mars have period, for their difference, equal to 1.990 * P1. This gives simply p = 2%.

The five major moons of Uranus, show a rare alignment on Dec. 21, 2012. I'll give these times according to the JPL Horizons online ephemeris: they are times as seen from the Sun's center; i.e., the astrometric times are 2.23h earlier. Ariel's maximum distance from the Sun, during its course around Uranus, occurs at about 19h UT on that date. The minimum distances of Miranda, Umbriel and Titania, occur at 18h, 17h and 15h, resp. (their orbital periods range from 1.4 to 8.7 days).

Equivalently, as seen from the Sun's center, the apparent ecliptic latitude of Ariel equals that of Uranus, at 19.3h UT. For Miranda, Umbriel & Titania, the latitude equals that of Uranus, at 17.9, 17.5 & 15.3h resp. Let's calculate, how rare it is, to have the situation at 17.5h UT (15.3h astrometric time), when Miranda, Ariel, Umbriel & Titania are resp. 0.4, 1.8, 0.0 & 2.2h away from achievement of the heliocentric latitude of Uranus:

0.4*1.8*2.2*4^3 / (1.41*2.52*8.71*24^3) = 1/4220.

This occurs on an exact half orbit of Umbriel, only once in 4.144d / 2 * 4220 = 24 yr. So, only about 4 hr after the solstice on Dec. 21, 2012, the four inner major moons of Uranus achieve a perfection of alignment, in Uranus' orbital plane (the effect of the tilt here of Uranus' orbital plane vis-a-vis the ecliptic, is barely significant at this precision) that occurs on average only once in 24 yr.

Summarizing the analogy between the asteroids Davida, Laetitia, Monterosa & Arlon, and the innermost major Uranian moons Miranda, Ariel & Umbriel:

1a. The asteroids rotate with very nearly the 5.145h period.
1b. The moons lap each other with very nearly whole multiples of the 5.145h period (the moons' rotation and revolution periods are equal).
2a. The asteroids align with Barbarossa (alternatively, the positive CMB dipole) and the Sun, in Dec. 2012.
2b. The moons, together with the next innermost major moon, Titania, lie near Uranus' orbital plane at about 15h UT astrometric time on Dec. 21, 2012; this happens with such perfection, only about every 24 years on average. Since their orbits lie in Uranus' equatorial plane, the moons then lie on the intersection of two planes, i.e., a line. This line passes through Uranus.

Uranus' rings also reveal the 5.145 hr period (source of data on Uranus' moons & rings: Miner, "Uranus", 2nd ed., 1998; Tables 11.1, 11.2, 12.1, 12.3, pp. 266, 271, 289, 292). From inside to outside, the eleven rings of Uranus are:

"1986U2R"
6
5
4
alpha
beta
eta
gamma
delta
lambda
epsilon

All were discovered in 1977 except for "1986U2R" & lambda, which were discovered in 1986; these two rings discovered in 1986, are the least substantial of the eleven. The most substantial ring is epsilon, which resonates with Uranus' rotation period and the orbit of Oberon:

1/8.3823h - 1/323.1182h = 2.0034 / 17.24h
i.e. f(epsilon) - f(Oberon) = 2*f(Uranus rotation)

Ring 5 is somewhat more substantial than Rings 6 or 4. Ring 5 resonates perfectly, with the orbit of Miranda and 1.5 times the 5.14535h asteroid rotation period:

1/6.2875h - 1/33.924h = 1 / (1.49999 * 5.14535h)

Likewise the alpha ring resonates with Ariel:

1/6.8508h - 1/60.4891h = 1 / (1.50151 * 5.14535h)

The beta ring resonates with Umbriel:

1/7.0688h - 1/99.4603h = 1 / (1.47893 * 5.14535h)

The eta ring resonates with Titania:

1/7.4239h - 1/208.9411h = 1 / (1.49599 * 5.14535h)

And the gamma ring resonates with Oberon:

1/7.5307h - 1/323.1182h = 1 / (1.49852 * 5.14535h)

After Uranus' five major moons, the largest of its minor moons are, from inside to outside: Juliet (~42km), Portia (~55km) and Puck (~77km); Rings 6, 4 and delta, resonate (somewhat less accurately) with these moons, resp., though with factor 2.5 instead of 1.5:

1/6.1988 - 1/11.8336 = 1 / (2.53006*5.14535)
1/6.3628 - 1/12.3167 = 1 / (2.55815*5.14535)
1/7.6911 - 1/18.2840 = 1 / (2.58006*5.14535)

Here lies the explanation of the radii of all nine of Uranus' most substantial rings, the rings discovered in 1977.


Conclusion. Evidence is presented that the time period, 5.14535 +/- 0.00717 hr, has physical significance: the resonances involving this period, are numerous, accurate, and systematic. Furthermore, the main bodies involved in these resonances, show rare alignments in Dec. 2012 (the asteroids Davida, Laetitia, Monterosa & Arlon), or even at c. 15.3h UT astrometric time, Dec. 21, 2012 (the four inner major moons of Uranus). So, the Mayan Long Count(down) really concerns a physical process within our own solar system, a process not yet understood.

Earth, Luna and the 5.145hr period

Let's consider not Earth's surface lapping Luna, but rather Earth's surface lapping an imaginary object with exactly half Luna's orbital period; i.e., Earth's sidereal rotation frequency minus twice Luna's sidereal revolution ( = rotation) frequency:

1/(24/(1+1/365.25636)) - 1/(27.32166*24/2) = 5.163868 * 5.000000

The "5.145 hr period" is estimated by averaging the four asteroid rotation periods, as 5.14535 +/- 0.00717, but the longest of the asteroid rotation periods is that of Monterosa, 5.164h, not significantly different from the 5.163868 found above. Monterosa's orbital period resonates somewhat with Earth's solar and sidereal day, and with Luna's synodic, and especially sidereal months:

1665.95170 Julian day = exactly 1665.95170 mean solar days in approx. 2013AD = approx. 1666 = 2*7*7*17 solar days
= 1670.51275 sidereal day = approx. 3341/2 = 13*257/2 sidereal day
= 56.41444 synodic month = approx. 282/5 = 2*3*47/5 synodic month
= 60.97549 sidereal month = approx. 61 sidereal month (the Mayan Long Count also happens to be 61 Uranian years)

Dr Joe, Now with the moon in the story can you look at it's(the moon) orbit without the Earth's influence on it? The moon orbits the sun and is very erratic in that it speeds up to ~31,000 meters per second and slows to ~29,000mps every 14 days. During this cycle the moon wobbles in what is supposed to be a Kepler modelled orbit. This motion is important to your calculations because it(the motion of the moon) has an effect on the motion of Earth's orbit and position within the solar system. The Earth's orbital speed slows and speeds up on the same time cycle as the moon although at a much lower rate. This action is really not well modelled at this time even though everyone is very comfortable with the models currently in use. They(existing models) could be causing some of the effects you are finding. I hope you will give this some thought.

The rotation period estimates given on the IAU Minor Planet Center website are slightly different, for Davida and Monterosa, from what I found a few weeks ago on the JPL small body database browser, but the differences cancel and don't significantly affect the mean:

511 Davida IAU: 5.1294 (JPL: 5.131)
39 Laetitia IAU: 5.138 (JPL same)
947 Monterosa IAU: 5.1655 (JPL: 5.164)
1717 Arlon IAU: 5.1484 (JPL same)

Searching the web again (Bob Turner helped me find the lightcurves on the web when I investigated this years ago) I'm finding the same two lightcurves: that by Warner, Jan. 2007, which appears in text form on the IAU website and in graphical form on Warner's website, and that by Poncy, Dec. 2006, which appears in graphical form on Prof. Behrend's website which is linked to the IAU website. The former states 5.164h +/- 0.001h and the latter states 0.2138d = 5.1312h, +/- 0.001 days. The textbook way of combining these would weight the former 24^2 times as heavily; this gives, for what it's worth, 5.16394h +/- 0.001h.

Also on the IAU website are many magnitude observations by the USNO at Flagstaff, of Monterosa magnitudes to two-digit precision. These are usually in groups of three or four spread over several nights within a month of opposition. The groups are scattered over many years. I'm working now on a mathematical method of combining the Flagstaff observations, which are coarse but well spread, with the Warner & Poncy observations, which are fine but almost unidirectional. Something easier, which I might do first, is to use only the Flagstaff observations, despite their calibration problems (i.e. dubious photometric standardization between one year and another), to estimate a rotation axis.