[Joe Keller]

(Moderator: for the historical record, please do not delete my previous post. It contains only text and therefore consumes little memory.)

"Dot theory" update: high significance.

With a more accurate calculation of Earth parallax, the above-mentioned alignment of presumed centers of gravity of A (1954), C (1987) & D (1997) objects became less perfect. I've been resorting to the IBM486 computer more, to avoid such inaccuracies.

Also, I've found two "disappearing dots" of interest on the B (1986) plate:

"B3" 11 16 51.55 -7 49 41.1
"B" 11 16 56.07 -7 55 14.3.

The following theory is hindered by my lack of success in finding accurately corresponding objects on the C plate. However A2 & A, B3 & B, and J. Genebriera's (March 25) & S. Riley's (April 1) objects (see above for coordinates and other details) seem to correspond to Barbarossa & Frey, resp.

I found accurate heliocentric coordinates for these six objects, using old ephemerides and the above-mentioned "BASIC" computer program. I assumed that all six objects have the same distance from the sun. This distance, 197.7283 AU, was adjusted, as above, so that the overall angular speed 1954-2007 (equivalent to a circular orbit with period 2811.866 yr) would equal that for an elliptical orbit with period 2688.000 yr.

The mass ratio of the contemporaneous pairs of objects was adjusted so that the two great-circle arcs, from center-of-mass A to center-of-mass B and from c.o.m. B to c.o.m. Genebriera/Riley, had exactly the same direction to within a millionth of a radian. The resulting mass ratio was 0.87710:0.12280. Generally there is a ratio which will cause the directions to be the same, because often an equation will have a real solution. Generally a different equation will not be simultaneously satisfied. Here, I found simultaneously that the c.o.m. angular speeds from 1954 to 1986 and from 1986 to 2007 became nearly equal, only a -0.0508% change (-0.001915% per yr). This is typical of the speed change expected in the lower range of possible solar orbital eccentricities.

I checked this graphically as above. By successive approximations at the computer, I found the geocentric coordinates (for G.'s & R.'s objects, coordinates always were adjusted to the midpoint of Genebriera's & Riley's observation times) which would put Genebriera's object on a perfect constant-speed heliocentric great circle with objects A2 & B3. Then I graphed all six objects on the same sheet, with these three points superimposed as the origins for objects of their epoch. I approximated the short great circles between contemporaneous points, as lines, and neglected the nonparallelism of longitude lines. (Also on my graph I neglected the difference between RA degrees & Decl degrees: this changes 0.7% between 6 & 9deg Decl.)

When I found the centers of gravity according to the above ratio, they were, as expected, collinear and showing constant speed. On the graph paper, the errors of the collinearity and of the constancy of speed both were about 1% of the graphed portion, which was in turn about 1% of the total motion; i.e., about 10^(-4) accuracy overall. Translating the Genebriera & Riley objects equally so that Genebriera's object also lay at the origin, I found by quick but generalized searching at the computer, the approximate unique ellipse centered at that origin and passing through the three presumed Frey objects (A, B, and Riley's). The ellipse was tipped 10deg NE-SW (i.e., 37deg to Barbarossa's orbit). The major and minor semiaxes were 0.25deg (0.86 AU) and 0.10deg, resp., giving i=22deg and a total tip for the Barbarossa-Frey system, of 43deg to Barbarossa's solar orbit.

The positions on the presumed orbital circle then were found. This was consistent with an orbital period of 8.3 yr (almost 4 revolutions, 1954-1986, & almost 2.5, 1986-2007) with only 3% discrepancy between the A-B and B-Riley arcs. The total mass Barbarossa+Frey would be 0.0094 solar mass (Barbarossa alone, 0.00825 solar mass).

The theoretical mass at this distance, needed to produced the CMB dipole, was calculated as above using precise 200- or 2000-step trapezoidal rule integration, and found to be 0.0116 solar mass. (Any additional bodies in the Barbarossa-Frey system, unless closely orbiting Barbarossa or Frey, or very distant, would need to have small mass because of the precise motion of the Barbarossa-Frey center of mass).

An inaccuracy in the tidal Pioneer 10/11 acceleration calculation above, was fixed, and the new parameters applied. The net sunward anomalous Pioneer 10/11 acceleration, after subtracting the tidal forces from the presumed 0.0116 solar mass Barbarossa system, at the four distances tabulated by O. Olsen (A&A, op. cit. 2007 above) becomes:

27 AU: 6.03*10^(-8) cm/s^2
45: 5.38* "
52: 5.68* "
63: 4.25* "

The best figure for the Hubble shift is 72 km/s/Mpc, which is equivalent to 7.0*10^(-8) cm/s^2. Assuming that near the sun, the net anomalous acceleration equals the Hubble-equivalent acceleration (the Galileo and Ulysses measurements of the anomalous acceleration lack the accuracy to confirm or refute this) the anomalous acceleration approximates a normal distribution with peak 7.0 and standard deviation 53 AU:


observed/predicted/difference (relative scale, i.e., 7.0*10^(-8) is 1.00)

0 AU: ~1.0+-0.3/1.00/?
27: 0.86/0.88/-0.02
45: 0.77/0.70/0.07
52: 0.81/0.61/0.20
63: 0.61/0.50/0.11

So, accounting for the tidal force of Barbarossa et al, the magnitude of the anomalous acceleration, as a function of radius, approximates a normal curve with standard deviation roughly 53 AU; with a small hump added near 53 AU.
Object A2 has four dimmer disappearing dots within about an arcminute (6,000,000 mi at 198 AU); two of these are a pair about 20" (2,000,000 mi) apart. Object B3 has probably one dimmer disappearing dot about 45" S. Barbarossa is a cold brown dwarf: these might be Barbarossa's inner planets. Frey is Barbarossa's Jupiter. Barbarossa's next-biggest satellite already has been named Freya.

Five additional more-or-less starlike disappearing dots have been found (on my "C", 1987 Red, and "D" 1997 Optical IR) sky survey images, which seem bright enough to be Barbarossa, Frey or Freya and are close enough to fall on the same sheet of graph paper on which I graphed the Barbarossa-Frey orbit. The largest telescopes used so far to search prospectively for Barbarossa or Frey have been the 16" telescopes of J. Genebriera and S. Riley. Comparison with sky surveys shows that the sky survey dots I seek to relocate are near, if not beyond, the detection limits of these telescopes. Genebriera and Riley have achieved, so far, one detection apiece (Barbarossa and Frey, resp., according to this latest theory).