Requiem for Relativity

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Joe Keller</i>
<br />More About the Asteroid Resonance with Barbarossa
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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
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Another way to express the above, from a post of mine June 6, facts (again based on the JPL minor planet database; epoch Aug. 27, 2011 this time) is

6340 Julian yr = Davida period * 1126.090
=Laetitia period * 1337.044
=Monterosa period * 1390.007
=Arlon period * 1948.455

It is seen that these four asteroids, which rotate with period near 5.14535 hr, and as near as I and others have determined, have at least approximately the same rotation axes as well, revolve a whole or half-whole number of times during a 6340.0 Julian yr "Barbarossa period". So, if they lie along a line through the Sun in Dec. 2012, they lay along a line through the Sun 6340.0 yr earlier, neglecting perihelion advance. Perihelion advance, causes the true anomaly of the asteroids, to differ a variable amount, from the mean anomaly. How much is this perihelion advance?

For Laetitia, I found four "osculating" (i.e., ellipse fit to actual orbit) perihelion determinations (omega + Omega = omegatilde). The longitude of perihelion, omegatilde, I also correct to equinox 2000.0, using Newcomb's precession 50.2564" - 0.0222" * T (T in centuries after 1900AD).

AJ 4:159 (1856)
epoch 1856.0; equinox 1856.0
2.010(equinox corr.)+0.662(given by authors as omegatilde)=2.672

PASP 54:24+, Table III (1942)
epoch 1939; equinox 1950.0
0.698(equinox corr.)+207.251+157.437=5.386

Bender, chapter in Gehrels, "Asteroids", 1st ed. (1979)
epoch Oct. 19, 1978; equinox 1950.0?
0.698(equinox corr.)+207.477+156.880=5.055

JPL online minor planet database (2011)
epoch Aug. 27, 2011; equinox J2000.0
208.475+157.137=5.612

The rates of perihelion change implied by these four, vary wildly, and are sometimes even negative. According to Lautman, AJ 68:539-540 (abstract)(1963),

"A simplified analysis based on the theory of secular perturbation shows that the motion of the perihelion of an asteroid is specified by a single parameter alpha dependent on the proper eccentricity and semimajor axis of the asteroid's orbit. Depending on the magnitude of alpha the longitude of perihelion will either oscillate about the position of Jupiter's perihelion or show a continuous positive motion. The vast majority of the asteroids fall in the latter goup."

Mercury's perihelion advance due to Jupiter has been calculated by LeVerrier and later by others, as about 153"/cyr = 1.53"/yr. To lowest order of precision, the time rate of advance is proportional to a^1.5, so, to lowest order of precision, about 29"/yr would be expected for Laetitia. However, the lowest order formula converges poorly for orbits so large relative to Jupiter's; these have considerable positive higher degree terms. It ignores the effect of Saturn and Mars, and Jupiter's and Laetitia's eccentricities and their nearly 5:2 resonance.

The best effort of celestial mechanics experts, to determine the long-term "secular" perihelion advance, free of short-term "periodic" fluctuations, is called the "proper" (as opposed to actual or "osculating") perihelion advance. According to the chapter by JG Williams, in Gehrels, "Asteroids", 2nd ed. (1989), the "proper" perihelion advances are

Davida 58.9"/yr
Laetitia 50.2"/yr
Monterosa 56.0"/yr
Arlon 31.3"/yr

also

Uranus 3.34"/yr observed (R Fitzpatrick, 2011, www.farside.ph.utexas.edu/teaching , Table 2)

(Uranus' rotation axis is roughly the same as these asteroids'; also, harmonics of the 5.145 hr period are common, among Uranus' moons & rings.)

Somewhat reassuringly, Laetitia's observed rate of perihelion advance from 1856 to 2011, according to the above sources, was 68.0"/yr; and from 1978 to 2011, 61.0"/yr.

With the "proper" rate of perihelion advance, I can find asteroid perihelion longitudes some whole number of revolutions earlier, for any time. To a good approximation,

true anomaly = mean anomaly + 2*eccentricity*sin(true anomaly)radians

and also

true anomaly = mean anomaly + 2*eccentricity*sin(mean anomaly)radians

With a computer, it's easy to use Kepler's eqn. (Roy, "Orbital Motion", 1st ed., eqn. 4.60, p. 84) to convert "eccentric anomaly" to "mean anomaly" though the reverse conversion is not in closed form (ten iterations gives about the computer's "single precision"). A calculator is enough for conversion from "true anomaly" to "eccentric anomaly" and the reverse (Roy, eqn. 4.57, p. 83).

From the true anomaly in late 2012, I'll find the mean anomaly, also the mean anomaly a whole number of periods (or half-whole, for Arlon with odd multiples of 6340yr ago) earlier, and finally the true anomaly then. A quick test of this method, using the JPL ephemeris for Luna, shows the gain or loss, of the true anomaly vis-a-vis the mean anomaly, due to perihelion advance, to be within 10% of that expected according to the roughest estimate based on Luna's eccentricity.

I'll also find what length of time ago, gives the most equal longitudes mod 180deg at best alignment, assuming that the periods are adjusted so that 6340 sidereal years is a whole or half-whole number of periods for all four asteroids and for Uranus. I find that multiples of 6340 yr ago, give much poorer alignments of these asteroids. However, I find that if I assume no change in asteroid perihelion, the alignment is twice as good (2deg rms longitude difference mod 180, vs. 4deg mod 180) 6340 yr before 2012, as in late 2012 - early 2013AD. This is due to the half-orbit of Arlon. Furthermore if the period resonance is assumed to be with 6340 sidereal Earth years, the best alignment occurs at what was then about two days before the winter solstice, according to Newcomb's first-order precession formula. This harks to the reference to the first day of winter as the beginning of eternity, in the famous inscription of Seti I.

If Uranus (also without perihelion advance, though this hardly matters for Uranus; and Uranus also is half-whole resonant, like Arlon) is included as a fifth asteroid, the best alignment, 6340 years ago, is practically as good (2.4 instead of 2.3deg rms difference), and occurs only half a day before the winter solstice. On the other hand, inclusion of Uranus in Dec. 2012, worsens the best alignment from 4deg to 5.5deg (rms difference) and this best alignment occurs a month later than without Uranus.

So the evidence is, that somehow, perihelion advance, at least of the four asteroids, did not occur over the last 6340 yr. Furthermore their true orbital periods are whole or half-whole divisors of 6340 sidereal Earth yr. Excellent (2deg rms difference mod 180) alignment, in ecliptic longitude modulo 180, of all four of these " ~5.1 hr rotation" asteroids, plus Uranus, occurred 6340 years before the winter of 2012-2013AD, and it occurred at the winter solstice, just as the inscription of Seti I says of "the beginning of eternity".

From the IAU Minor Planet data base, I printed out the 57 pp. (as I happened to paste it into a Word document) of information on asteroid rotation periods. I scanned the pages hastily and imperfectly. I happened to find two more asteroids which are only slightly farther than my four, from the 5.14535h period; these two do not lie near longitude 0 or 180 at 2013.0AD.

I also wrote down asteroid rotation periods near 1.5*5.14535h = "1.5P", and 2.0*5.14535h = "2P". I chose to write down, those about as near to these values, as my four (Davida, Laetitia, Monterosa and Arlon) are to 5.14535h. There was a significant tendency for these, especially the 1.5P asteroids, to lie within about 8 deg of longitude 0 or 180, at 2013.0AD. Four of the five 1.5P asteroids, that had ID numbers &lt; 1000 (these generally would be the most massive ones) lay within 8 deg of longitude 0 or 180, just like my original four asteroids with rotation period "P", at 2013.0AD.

Unlike my original four, their orbital periods did not tend to be whole or half-whole divisors of 6340 yr. Two of my original four asteroids, divide the Mayan Long Count to give a quotient N + 2/3 or N + 1/3 (Davida does this very precisely). I found that all four of the "1.5P" asteroids with ID numbers &lt; 1000 and which lie near 0 or 180 at 2013.0AD, also give quotients N + 2/3 or N + 1/3, sometimes very precisely, when divided into the Mayan Long Count.

If we are nearing the end of a special string of three Mayan Long Counts = 3/5 precession period, and also the end of a 6340 year period = 1/4 precession period, it would be the end of a period equal to the least common multiple of those, namely 3 precession periods = approx. 78,000 yr.

Addendum Oct. 27, 2011: Above, I note that four of the five asteroids, with ID number &lt; 1000 (i.e. relatively massive asteroids) and rotation period near 5.14535*1.5 = 1.5*P (as found by my quick check of the IAU Minor Planet Center list), lie no more than 8 deg from 0 or 180 ecliptic longitude, at 2013.0AD. These asteroids are 118 Peitho, 340 Eduarda, 776 Berbericia and 796 Sarita.

A brief web search finds a rotation axis estimate for only one of these, 776 Berbericia:

Durech et al, Astronomy & Astrophysics 465:331+, 2007, find a single solution for the (positive) rotation axis: ecliptic (long, lat) = (347, +12).

Torppa et al, Icarus 198:91+, 2008, find a double solution, two equally valid solutions by their method: (long, lat) = (170, +59) or (347, +11). (I rely on Torppa's citation of the Durech article.)

So it seems reliable, that Berbericia's axis is about (347, +11.5). This is about 60 degrees from the average of my estimates of the axes of Davida, Laetitia, Monterosa & Arlon.

"...at the Corralitos Observatory in New Mexico...On two successive nights just prior to the total lunar eclipse of 24 April 1967, a marked violet excess, centered approximately on the subsolar point but extending generally over the entire visible lunar surface was observed. No explanation - instrumental, astronomical, or meteorological - has been found."

- JA Hynek & JR Dunlap, Astronomical Journal 73:S185 (abstract), 1968

This would have been, the nights of Apr 22-23 and 23-24.

1889 Jupiter occultation report: occultation timing anomaly

J. E. Keeler (Lick Observatory), Astronomical Journal, no. 203, p. 84, reported:

"For observing the occultation of Jupiter on the afternoon of Sept. 3, I used a power of 261 on the 36-inch equatorial, and recorded time on a chronograph by the No. 7 clock, which is regulated by Pacific Standard Time.

"...The time recorded, when I was sure that first contact had occurred, was 5h 32m 14.9s [PM; i.e. Sept. 4, 1889, 01:32:14.9 GMT - JK], and was estimated to be 2s late.

"...The last glimmer of light was noted at 5h 34m 24.5s.

"Clock No. 7 was found by Mr. Hill be be 0.49s slow..."


So, the occultation began at 32:13.4 and was complete at 34:25.0. Let's compare this to the JPL ephemeris. I got the "apparent" celestial coords. and apparent angular diameters for Jupiter and Luna from the JPL Horizons online ephemeris, for 01:32:00 and 01:35:00 UT on Sept. 4, 1889. These are assuming an airless model, but atmospheric refraction should only move the site of interest (the Jupiter-Luna junction) not alter the facts of touching vs. not touching, or covered vs. not covered.

At first, I used the linearly interpolated angular diameters for 01:33; then, I interpolated the diameters (only Luna's changed appreciably) to times of "touching" and "covering", for a more precise result. The difference was only a small fraction of a second, in the times of touching or covering.

JPL gives the equatorial angular diameter of Jupiter. Io's orbital inclination is only 0.04deg. Io was more than 30deg of Io's orbit, away from conjunction with Jupiter. The inclination of Jupiter's equator, to its orbit, is only 3.1deg. So, the angle between the Io-Jupiter line and the Jupiter-Luna line, at 01:33, approximates the latitude on Jupiter which touches Luna's rim both at "touching" and "covering". Due to Jupiter's ellipticity 1 - b/a = 0.06487, its effective diameter was reduced approx. 5.06% vs. equatorial. This correction amounts to 5.06% * 132sec = 6.7sec. Even using the smallest possible (i.e. polar) radius, would give only an extra (6.49% - 5.06%)/5.06% * 6.7sec = 1.9sec shortening of the interval.

For sufficient accuracy (a small fraction of a second) I interpolated Jupiter's and Luna's celestial coordinates to 01:33, and then interpolated the distances between their centers, linearly between 01:32 & 01:33, or between 01:33 and 01:35.

The resulting calculated times of touching & covering are 01:32:21.3 & 01:34:39.9, resp. Thus observation is 8sec & 15sec, resp., earlier than the JPL ephemeris implies, and the interval between touching and covering, is 7sec short. Hardly any of this can be explained by any error in estimating Jupiter's effective radius at the relevant latitude.

In conclusion: Jupiter appeared to touch Luna, 8sec earlier than predicted by the JPL ephemeris. The interval between touching and covering was 7sec/139sec = 5% shorter than expected.

Ok. And what do you think this means?

Dr Joe, How does the data from the 1889 occultation compare with the similar event of 2004?