Accurizing the "Crouching Tiger"
My most complete and accurate estimate of the period of the proto-Jupiter, now is 11.871464 Julian yr = 6339.501 tropical yr divided by 534 (this precision is limited by, among other things, the seven-digit precision of Jupiter's semimajor axis). As before, I assume that all semimajor axes were preserved, but that proto-Jupiter's mass was distributed symmetrically.
In this estimate, I account accurately for the influence of all other planets and moons, including Barbarossa (a small influence, due to its great distance). Orbital eccentricities are assumed to be zero, except that, for Barbarossa, I use the eccentricity, 0.6106, that I find from the sky surveys.
The most important change in my calculation, is that instead of assuming that the proto-Jupiter mass was at three equally spaced Lagrange points, I assume that it was spread into a wide ring (an infinitesimally narrow ring gives an infinite effect). I estimate the effect of this ring, by assuming that it is wide enough that the net effect is zero nearby. The effect is proportional to 1/2/sqrt(2) at 90deg, and 1/4 at 180deg. The effect, as a function of theta, is periodic with local minima at 0 and 180. So, the best numerical integration, using values at 0, 90, 180, & 270, is with equal weights. This, and some smaller accurizations, decrease the implied interval from 6339.637 tropical yr (with the Lagrange point distribution) to *6339.501 trop yr* (with my new preferred wide ring estimate).
If I were a year off, and the Egyptian calendar began at the summer solstice 4328BC instead of 4329BC, then the time to the winter solstice 2012AD, is (assuming today's precession rate) *6339.5032 trop yr*, accounting for Earth's longitude of perihelion in 4328BC and in 2012AD (est. advance of Earth's perihelion: 360deg in 112,000 yr; est. effective eccentricity, 0.017). The discrepancy, between this time interval, and the interval resonant with the proto-Jupiter, is only 0.002yr = 0.7day.
Another approach to Barbarossa's period, is through the Bessel functions of the second kind (Neumann functions). The ratio of the Mayan Long Count, or rather, more precisely, the best grand visible solar system cycle, 5124.58 Julian yr, which I discuss above, to Barbarossa's period (assumed to be 6339.5032 modern tropical yr) is 1::1.23705. This is close to the ratio, 1.24113, of the first peaks, of the first and third Neumann functions. It is even closer to the ratio, 1.23422, of the first peaks of (Neumann function / (1/sqrt(x) ), that is, the approximate ordinates where the first and third Neumann functions first touch their common envelope, if one approximates that envelope by a function A/sqrt(x). The precise calculation of these envelope points, gives the ratio, 1.23113.
(Neumann functions can be found from rapidly convergent series in Apostol's Calculus, vol. 2, or in Franklin's Calculus. The series in Apostol omits the factor 2/pi which Franklin includes. The formula in Franklin contains a grossly erroneous definition of the "psi" function, which contradicts both Apostol, and Jahnke & Emde.
Franklin gives the method for finding envelope curves. This requires finding the derivative of the Neumann function with respect to its order. Difficulties arise from the definition, of the integer-order Neumann function, as a limit. I use Hankel's complex-valued integral formula in Jahnke & Emde, for H1, with k=1, to overcome these difficulties. The formula ultimately requires values of the "factorial function" near x = -1/2; Jahnke & Emde give the exact values of the factorial function, and its logarithmic derivative and second derivative, at x = -1/2.)
