Requiem for Relativity

Additional Solar System Features Suggesting the Cold Brown Dwarf Companion, Barbarossa

Earlier on this messageboard I mentioned that if my mass and distance estimates for Barbarossa are accurate, then the classical Kuiper Belt (a.k.a. "cubewanos") occurs at that distance, at which the torque on an orbit, per degree of tilt, due to Barbarossa, equals the torque per degree of tilt due to the entire known solar system combined. That is, for cubewanos, precessions about Barbarossa's orbital plane, and about the principal plane of the known solar system, occur at equal rates.

Yesterday I confirmed that calculation and went further. Including only the known giant planets, JSUN, I found that if Barbarossa is in a circular orbit at 197 AU, then 0.0102 solar masses for Barbarossa and its satellites, gives a ratio for Neptune, torque per unit tilt due to Barbarossa & satellites : torque per unit tilt due to JSU = 1:3. Then the 1:2 ratio (Neptune now included) occurs at 40.2 AU, for a body in a circular orbit; the mode of the semimajor-axis histogram for plutinos is about 39.4 (the semimajor axis of Pluto). The 1:1 ratio occurs at 43.8 AU, vs. the mode of the cubewano histogram, 44.2 AU (the mean and median appear to be slightly less). The fundamental trans-Uranian spacing process might be the torque due to Barbarossa and the resonance of orbital precession rates. The 3:2 Pluto:Neptune orbital period resonance might be a bonus possible because both processes, orbital period and Barbarossa-caused orbital precession rate, go as r^1.5, while the orbital precession rate of Neptune due to JSU equals that of plutinos due to JSUN.

Following Brauer & Nohel, Qualitative Theory of Ordinary Differential Eqns (Dover, 1989), pp. 51-53, I solved the nonhomogeneous system of linear differential equations describing Neptune precessing about the orbital plane of Barbarossa and simultaneously about the presumed fixed principal plane of the known solar system, with exactly 3x the torque, per degree of tilt, as Barbarossa. The result is precession along a small closed curve amounting to a circle superposed on a larger cardiod, or equivalently to a slender ellipse superposed on a hypocycloid of three cusps (see, inter alia, CRC Standard Mathematical Tables, 26th ed., ch. VII - Analytic Geometry, pp. 264,267,269; ES Smith et al, Analytic Geometry, sec. 100, prob. 15; and the Wikipedia article on Lissajous figures). The predicted rms deviation for Neptune resembles that currently observed, and Neptune lies near a point on the predicted curve.

The same equations also predict that Uranus would have an rms deviation resembling that observed. The deviation would be due to the sum of comparable contributions from Neptune and from Barbarossa.

The deviation of Saturn is not explained by the above, but might be related to the Jupiter:Saturn:Barbarossa resonance discussed by me earlier. Suppose that the energies related to this resonance, are proportional to the gravitational forces, and minimized when the resonance is perfectly aligned. Suppose further that there is equipartition of energy between the two degrees of freedom, which correspond to the imperfect alignment, due to Barbarossa's inclination and to Saturn's. I calculate that this equipartition occurs when Saturn's orbit is inclined 1.01 degrees to Jupiter's, vs. 1.25 deg observed.

Assuming, as above, circular orbits, with Barbarossa at 197 AU with 0.0102 solar masses, Saturn's rate (i.e., frequency) of orbital precession, due to J,U & N, and Saturn's hypothetical rate of orbital precession due to Barbarossa, are in the ratio 1:0.00206. Jupiter's rate of orbital precession due to S, U & N, and Jupiter's hypothetical rate due to Barbarossa, are in almost exactly the same ratio, 1:0.00205. Here U and N are almost negligible, so basically, Jupiter and Saturn happen to have the mass ratio needed for them to have the same ratio, of orbital influence by Barbarossa to orbital influence by each other.

The orbital influence (i.e., precession rate) hypothetically caused by Barbarossa goes as r^1.5, like the orbital period. But why should Saturn's 2.5x longer orbital period imply that its orbit precess 2.5x faster than Jupiter's? A distant massive solar companion, such as Barbarossa, seems a likelier reason why the masses of Jupiter and Saturn happen to be just right to conform to this r^1.5 law for their orbital precession rate. If Barbarossa precesses Saturn at 2.5x the frequency it does Jupiter, but Saturn's orbital angular momentum vector's precession cycles around Jupiter's are 2.5x quicker than Jupiter's around Saturn's, then the displacement of Saturn's vector, by Barbarossa, during a quarter cycle of Saturn's precession around Jupiter, equals the displacement of Jupiter's, by Barbarossa, during Jupiter's quarter cycle around Saturn. Regarding the absolute value of the difference between the J & S orbital angular momentum vectors, the discrepancy induced by Barbarossa, thereby amounts to the sum of two sinusoids of different frequency but equal amplitude.

Hi Joe, time maybe for another try at a telescope shot.Here's a screen shot of the ones on the Bradford robotic, Pick out the best position one and I'll put it up again as a job. Though veiwing is not too good at the moment.

Originally posted by Stoat:
"Hi Joe, time maybe for another try at a telescope shot.Here's a screen shot of the ones on the Bradford robotic, Pick out the best position one and I'll put it up again as a job. Though viewing is not too good at the moment. ..."

Great! My notes aren't with me now but I can do it tomorrow.
- Joe Keller

Bode's Law developed into Dermott's Law (MNRAS 141:363+). Dermott found that the satellites of the sun, or of Jupiter, Saturn, or Uranus, orbit at radii that are in geometric sequence. The ratio of that sequence is the cube root of a small integer that is different for different parents: 6 for the sun (gives 1.817121), smaller integers for J, S & U.

As satellites of the sun, J, S & U conform very well to Dermott's Law. Venus & Earth conform very well only when averaged, supporting the double-planet concept of Venus & Earth. Neptune and Pluto (or the plutinos) also conform very well only when averaged, supporting an escaped-satellite or other complicated concept.

Dermott's Law implies a planet at 187.5 AU, i.e., 36 = (cube root of 6)^6 times farther from the sun than Jupiter. Consideration of Saturn implies a planet at 189.0 AU; Uranus implies 209.1 AU. These three estimates together imply a planet at 195.2 +/- 7.0 (standard error of the mean) AU, vs. a little more than 197 AU from my predictions and from Genebriera's, Riley's, Turner's, and the sky surveys', possible detections of Barbarossa.

According to Dermott's Law, Mercury implies a semimajor axis of 151.9 AU for Barbarossa, Mars implies 180.7 AU, Ceres implies 180.9 AU. Ter Haar et al had favored a slightly larger ratio of 1.89, for the planets' orbital radii: this would give 221.6 based on Mercury, 244.1 based on Mars, and 234.5 AU based on Ceres.

Hi Joe, I was going to ask TVF what he made of the anomalous findings of helium in the atmosphere of Jupiter. Which suggests that the planets formed very early. I forgot all about it but it would be of possible importance when considering our solar system as something of a "failed" binary system.