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 nemesis</i>
<br />Shouldn't Barbarossa cause observable perturbations of the outer planets?
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Thanks for bringing up this important question again. Scattered through this long thread, is more information about this.

Goldreich & Ward, PASP 84:737, 1972, claimed that even a moderate Planet X, unless almost exactly coplanar with the rest of the solar system, would move Neptune to high inclination, within much less than the age of the solar system. Their article doesn't consider chaos theory. Also, I've counted four questionable simplifying assumptions explicit in their calculation:

1. p. 737, col. 2, par. 2, sentence 2
2. p. 738, col. 2, par. 1, sen. 3
3. p. 739, col. 1, 3rd to last sentence
4. p. 739, col. 2, last par.


Zakamska & Tremaine, AJ 130:1939+, 2005, measured acceleration relative to those millisecond pulsars (MSPs) which are not within globular clusters. In their sample, 35 of 48 MSPs were binary (typically MSPs are close binaries). The contemporary theory of MSPs is that the original supernova sweeps its environs so clean, that except for companion stars which often endure, and rarely planet-size masses (maybe really stellar fragments), there's nothing nearby to produce an acceleration more than about that of Barbarossa on the sun: a distant brown dwarf companion (there is evidence that many or most stars have these; see my posts) survives the supernova, but planets (closer and smaller) don't; stellar companions, if they survive, are detected and the MSP period corrected for their effects. Thus the scatter in Pdot/P for MSPs, corrected for obvious binary effects, should be about what would come from a brown dwarf companion at a few hundred AU, which endured the supernova! Accordingly, Zamaska & Tremaine believe themselves statistically able, based on the scatter of Pdot/P for their sample of 48 MSPs, to rule out as little as 1/10 ~ 1/sqrt(48) the acceleration due to Barbarossa.

Zakamska & Tremaine used nonparametric statistics (rank correlation) to look for the acceleration effect of Barbarossa, then parametric statistics (scatter of Pdot/P) to calculate the significance of their finding. It would have been better to use parametric statistics for both parts, rather than discard statistical significance by using nonparametric statistics, then use parametric statistics to show that statistical significance is lacking.

That articles resorting to such approaches have been published recently, indicates that ephemerides are considered, at least by some peer reviewers, unable, alone, to rule out bodies like Barbarossa. Standish's use of space probe data to remove residuals from Uranus and Neptune, was complicated. Only a partial orbit of Neptune is available.

"It is true that if you put in the new mass of Neptune, some of the key residuals in Uranus do decrease. But in my opinion it is not correct to say that they disappear entirely."

- Robert Harrington of the U.S. Naval Observatory (New Scientist, Jan. 30, 1993, p. 18, cited by Ken Croswell)


Perhaps the simplest approach is to consider the Pioneer acceleration. Barbarossa's tidal force, i.e., the difference between Barbarossa's pull on Pioneer 10 or 11, and the sun's, would have been, in the outer solar system, of the same order of magnitude as the anomalous Pioneer deceleration (and indeed seems to explain much of the apparent variation in the anomalous acceleration; for discussion and references see, Joe Keller's March 29, 2007 post, p. 10 of this thread, and April 24, 2007 post, near the bottom of the post, p. 13 of this thread). I recall from an article by JD Anderson, that with the help of space probes, Mars' orbital period has been measured accurately enough to rule out any steady unexplained sunward force on Mars, more than ~ 1% as big as the Pioneer deceleration.

On the other hand, Barbarossa's tidal force, for any planet, though also radial, would mostly cancel over a period (it's half as much at quadrature as at opposition/conjunction, so averages to 1/4 of the maximum); for outer planets, whole orbital periods are practically unavailable to measurement anyway. (For Earth at this epoch: net toward the sun in December and June, net away from the sun in March and September.) It would be &lt; 1/10 as big in the inner solar system, as in the outer solar system where it becomes comparable to the Pioneer force. That, combined with averaging to zero, would make it undetectable even by the measurement of Mars' period, which Anderson put forth as the most sensitive way to detect such radial forces.

Some Political Aspects of Science

According to his online New York Times obituary, Dr. Harrington died in the hospital, of cancer, at age 50, January 23, 1993 ("Saturday"; according to the article "published Jan. 27, 1993") a week before the cover date of the New Scientist article quoting him, as doubting that Standish's work had refuted Planet X. Though Harrington, as recent head of the USNO equatorial division, and well-known Planet X seeker, was an appropriate source for the New Scientist to interview, someone not on his deathbed might have been able to rebut Standish in more detail. Nonetheless, deathbed statements get extra legal weight. This was practically a deathbed statement by Harrington. Surely it was his sincere opinion, free of political influence.

Journalist Peter Jennings waited until he had terminal cancer, to present his expose on UFOs. Another journalist, Bill Moyers, now in his 70s and formally retired since 2004, tried to make a blockbuster statement on the Charlie Rose show last night, but, for the only time in the forty years I've been watching Public Television, the signal was lost for so long, about a minute, that that entire thread of conversation failed to be broadcast. Moyers had been close to Pres. Johnson and was trying to reveal what Johnson really thought about the Vietnam War.

Lowell Knew Barbarossa Fit Uranus Data

From an excerpt of Lowell's writing (see: WG Hoyt, "Planets X & Pluto", 1980, pp. 136-140) I can summarize Lowell's method. Adams had predicted Neptune from discrepancies in Uranus' orbit. Adams, LeVerrier, Galle et al had &gt; 1 orbit of Uranus to guide them since Flamsteed's (accurate, for its era) 1715 position, but Lowell had &lt; 1 orbit of Neptune. Therefore the remaining discrepancies in Uranus' orbit, unexplained by Neptune, were (and are, I gather from statements by Standish and Harrington) a better guide to any Planet X, than are the discrepancies in Neptune.

What follows is my simplified reconstruction of Lowell's method. Lowell assumed zero inclination (the greatest inclination of any outer planet then known was Saturn's, 2.5deg) and, at first, zero eccentricity. He guessed the mass, eventually as 0.00002 solar masses. Then he searched for the radius and phase, that best explained Uranus' discrepancies. This gave the necessary radius as a function of mass. One point, on this mass-radius curve, was 6.5 Earth masses (0.000020 solar masses) and 44 AU. Lowell preferred these values, because they are consistent not only with the geometric sequence ratio 1.5 for the planetary orbital radii (which describes not only the step from Uranus to Neptune, but also the inner planets from Mercury to Mars, though Venus deviates) but also consistent with the geometric sequence ratio 2.64 = (massJupiter/massNeptune)^(1/3) for planetary masses (which describes the outer planets from Jupiter to Neptune, though Uranus deviates).

To investigate the effect of eccentricity, Lowell assumed the largest eccentricity of any known planet (0.20, as for Mercury) and studied the two extreme cases: Planet X at perihelion in 1914, and Planet X at aphelion in 1914. Any phase between perihelion and aphelion, or any lesser eccentricity, would amount to an intermediate case, so knowing these two extremes, Lowell would know what range of ecliptic longitude to search. The two cases (unsurprisingly, because of Planet X's assumed large mass and distance) gave perihelia almost 180deg apart, i.e., almost the same ecliptic longitude for Planet X in 1914. This ecliptic longitude must have been either near 202 for both, or near 202-180=22 for both; it must have been 202, because Pluto, discovered near there, was thought for awhile to be the predicted Planet X. Both solutions had periods near 292yr, so the 199yr of data covered ~240deg of Planet X's orbit. Instantaneous data would have implied a 50% difference in major axis (1+0.2 vs. 1-0.2), if one solution were at perihelion and the other at aphelion; 180deg of orbit equally weighted, would average that to zero, and 240deg of orbit to &lt; 10%. So the two fitted major axes also were almost the same, averaging 44AU.

Lowell's eccentricity 0.2 calculation, seems to have been only exploratory, to find the extreme range of ecliptic longitudes to search. So, Lowell's actual solution was, approximately, a 6.7 Earth mass body in a circular orbit of 44AU radius, phase 202deg ecliptic longitude in 1914.

Uranus' orbit is like a frictionless harmonic oscillator. A perturbation drives it 180deg out of phase. The second harmonic perturbation (see my May 11 post below) predominates. Therefore the lag is 1/2 * 1/2 = 1/4 period. For Lowell's 6.7 Earthmass alternative, to have at present the same effect as Lowell's implicit, not explicitly stated, 3430 Earthmass alternative (which I name Barbarossa), they must have the same longitude, when retarded in their own orbits by a quarter of Uranus' orbital period. The 6.7 Earthmass alternative's retarded ecliptic longitude in 1914 was 202-84/4*360/292=176; likewise Barbarossa's retarded longitude in 1987 was 173-84/4*360/2780=170; in 1914, 170-73*360/2780=160.5. So, Lowell in 1914 essentially predicted Barbarossa's heliocentric ecliptic longitude with only 15 deg error, assuming Lowell's very most recent data got almost all the weight when calculating Planet X's longitude. If the effective mean epoch of Lowell's data is 1900, then the *effective* longitudes of Lowell's Planet X and Barbarossa, are equal. (David Todd in 1877 essentially predicted Barbarossa's longitude with only 11deg error; see subsequent posts.)

The radial component of Barbarossa's tidal force was shown above to be smaller than any measurements hitherto would have detected. The maximum tangential component can be estimated by drawing a tangent from Barbarossa to Uranus' orbit. The ratio of this, for Lowell's 6.7 Earthmass alternative, vs. Barbarossa, is 1:1.075. This should be multiplied by a sine factor to correct for the lesser effectiveness of quickly alternating forces; the corrected result is 1:1.19. So, with only 19% error, Lowell's predicted Planet X, based on 1715-1914 discrepancies in Uranus' orbit (unexplained by Neptune), gives a perturbation of the same strength as Barbarossa.

Planet X: Dynamical Evidence in the Optical Observations

E. Myles Standish's article is entitled, "Planet X: No Dynamical Evidence in the Optical Observations" (Astronomical Journal, 105:2000+, 1993). I'm correcting that title. After Standish carefully removed outliers, adjusted planetary masses to more accurate space probe values, and recomputed orbits, some dynamical evidence for Barbarossa could be seen by inspection of his graphs.

"On 30 Sept 1846, one week after the discovery of Neptune, Le Verrier declared that there may be still another unknown planet out there."

- Paul Schlyter, historian of science (internet)


In 1877 David Todd predicted a planet at ecliptic longitude 170 +/-10 for 1878 (source: Schlyter). Barbarossa's longitude then would have been 159.

By drawing a tangent from the outer to inner orbit, the maximum tangential component of tidal force is found to be approx. (sec^2(theta) - cos(theta))*M/R^2, where theta = arcsin(r/R). The mass of Neptune would need to be increased 7%, to give the magnitude of additional tangential force provided by Barbarossa. So the Voyager/Tyler 0.5% *downward* revision of the 1976 IAU value of Neptune's mass (Standish, Table 1) does little, to explain Lowell's prediction (which was consistent either with Barbarossa or with a smaller, nearer planet). (Lowell explicitly said that he knew only the pull's direction and tidal strength with accuracy; the mass and distance were mutually dependent guesses.) Furthermore, Voyager/Tyler revises Newcomb's 1877 Neptunian mass, likely used by Lowell, only 0.17% downward.

*The* unperturbed orbit, is not the same as *an* unperturbed orbit. A large perturbation can cause a perturbed orbit that looks very like some other unperturbed orbit. This does not imply that there is no perturbation. Likewise, a "poorly conditioned", i.e. very acutely angled, 2x2 system of linear equations, isn't solved accurately by projecting a normal from one line to the other; it's solved by moving along one line to the other. In terms of higher mathematics, one must follow the curve of perturbation in Hilbert space, to find the unperturbed orbit; it is not sufficient simply to move to the nearest unperturbed orbit and say, "Look how close this is."

I express the tidal tangential force, as a Fourier series in the longitude of Uranus (assuming a circular orbit), finding the coefficients numerically by convolution. The 1st harmonic is small and its effect on longitude, can be and is, neutralized by assuming a slight alteration in the magnitude and apse of that relatively gigantic orbital parameter, the eccentricity. Relative to the gravitational force of the sun, the 2nd harmonic coefficient is 0.1461 * 0.0103 * (19.18/197.7)^2 = 14.16 * 10^(-6).

To find displacement in RA, the 2nd harmonic amplitude must be divided by 2^2, to account for two integrations; and it's opposite in sign to the input. The effect on orbital radius can be neglected for a second harmonic perturbation, because the radius is a first harmonic response to that input, and cancels. So the 2nd harmonic amplitude of Uranus' longitude displacement is

-0.25 * 14.16*10^(-6) radians = -0.7302"

where the sign signifies that the displacement is opposite in sign to the input tidal tangential perturbation there.

(The 3rd harmonic amplitude of longitude displacement is, analogously, of the order of 0.044".)

The wave seen on, e.g., the rightward portion of Standish's Fig. 5a, is roughly 0.5" peak-to-peak, which might correspond to 0.125" harmonic (semi)amplitude. Uranus' eccentricity strongly affects both the 1st & 2nd harmonic of longitude displacement directly, and also the 2nd harmonic of tidal tangential force (and thereby also the 2nd harmonic of longitude displacement). Optimum choice of eccentricity and apse might partly nullify both 1st and 2nd harmonics. (Likewise, something with both dipole and quadrupole, whose strengths can be multiplied by the same factor, and which can be rotated by the same angle, has two degrees of freedom with which it might partly nullify a dipole and quadrupole simultaneously.) So, the observed amplitude of longitude displacement vis a vis the best fitted orbit, should be &lt; 0.73".

In 1975.0, Uranus' longitude was 211, Barbarossa's 172, & Barbarossa's retarded longitude 172 - 84/4*360/2780 = 169. At this relative phase, 211-169=42deg (~45deg), the second harmonic longitude displacement should peak. Standish's Figs. 5a & 6a show his most highly corrected data for RA. Because of scatter, the harmonic (semi)amplitude might be estimated as 1/4 the raw peak-to-peak amplitude, i.e. 0.125" from c. 1930 to 1990. The peak is c. 1960 (60deg away from the pred. 1975 peak; this amounts to +0.5 correlation with predicted phase) but the seeming period, c. 1940-1980, is close to the 84/2=42yr predicted. The seeming amplitude (considering that due to cancellation through orbital fitting, my 0.73" is only an upper bound) and period do suggest that the effect of Barbarossa is apparent on Standish's graphs.

Standish (sec. 2, p. 2001) estimates 1.2" error for individual time point observations before 1911. Lowell would have needed ~ (1.2/0.73)^2 / mean(sin^2(theta)) * 2^2 = 22 randomly distributed time point observations total, to detect the second-harmonic discrepany in Uranus longitude, due to Barbarossa, to 2-sigma significance. (With the help of the internet browser's magnification feature, I can see that there are &gt; 1000 dots before 1915 on Standish's Fig. 1a.) Lowell would have known that he sought a second harmonic phenomenon. He might have adjusted Uranus' eccentricity and apse by neutralizing the 1st harmonic precisely, without regard to the 2nd harmonic discrepancy in longitude, which then would not be masked.

The residuals of Uranus' RA since 1950, seem less grossly sinusoidal in Standish's (1993) Figs. 4a and 6a, than in Fig. 5a. I measured the last 39 data points from the 1600% magnified computer screen (corresponding to the interval 1950-1989), for Figs. 4a, 5a & 6a, and made periodograms. (There was a place, where I found one less point in Figs. 4a & 5a, so interpolated what seemed, by comparison with the abscissas of Fig. 6a, likeliest to be the missing datum.)

Before making the periodograms, I detrended each data set, i.e., adjusted slope and mean to zero, using the average of the first three and average of the last three ordinates, to define the trendline. For Fig. 6a (resp. 4a, 5a), the periodogram peaks at 62 (resp. 68, 60) yr. Fig. 6a's periodogram amplitudes are smaller than Fig. 5a, whose in turn are smaller than Fig. 4a. Well away from the peaks, Figs. 5a & 6a conform well to white (1/f) noise. I used the periodogram amplitude of Fig. 6a at 100 yr, which seemed to be well away from the 62 yr peak and consistent with the amplitude at 10yr, to define the 1/f amplitude curve which I subtracted from Figs. 4a, 5a, & 6a.

Thereby filtering out the white noise, I found the filtered periodogram peaks for Fig. 6a (resp. 4a, 5a) to be 53.9 (resp. 61.5, 55.2) yr. For Figs. 5a & 6a, the signal was almost totally removed by this filtration, leaving only smooth, fairly narrow peaks of heights 0.0983" & 0.0265", resp.

The perturbation signal is a second harmonic; 53.9 yr, the detrended, de-noised periodogram peak for Standish's most highly corrected post-1950 data, corresponds to the net half-period of Uranus vis a vis a Planet X in circular orbit at 52.5 AU. Elsewhere on this messageboard I have discoursed at length on the theoretical and empirical importance of the distance, 52.6 AU.

The amplitude of the RA (essentially, ecliptic longitude) displacement found in Standish's Fig. 6a (resp. 5a), 1950-1989, is only 0.0265"(resp. 0.0983")/0.73" = 4% (resp. 13%) that theorized for Barbarossa (or Lowell's Planet X), but correcting for distance^3 = (52.5(resp. 49.7)/197.7)^3, still should have 2.4 (resp. 7.45) Earth masses.

For all the periodograms, the phase is incoherent. In Fig. 5a, for example, near the de-noised periodogram peak at 55yr, the phase steadily varies ~ 135deg, per year of change in the period. If the discrepancy were due to the tidal gravity of a Planet X, a small change in the period of the fitting sinusoid, would cause only a small change in the best-fitting phase. The periodogram result is consistent with a collection of objects or structures orbiting at ~ 52AU, each with its own phase, affecting Uranus likely nongravitationally. The squares of the interaction amplitudes are additive.

The white noise correction tailored to remove the 100yr & 10yr periodogram amplitudes of Fig. 6a, also removes the 200yr & ~ 10yr periodogram amplitudes of Fig. 5a (the 100yr amplitude of Fig. 5a seems not quite distinct from the 60yr peak); thus the amount of white noise in Figs. 5a & 6a is about the same. The difference between Figs. 5a & 6a, is Earth-based vs. space probe planetary mass determinations. With use of Earth-based determinations, the amplitude of the effect on Uranus' orbit, equals the amplitude (though not the phase, or any phase) of the effect that would be caused by another half of a Uranus (7.45*2 = 14.9 ~ Uranus' 14.6 Earthmasses) orbiting at the special distance, 52.6AU. Use of space probe determinations somehow eliminates most (perhaps, ideally, all) of the effect.

Joe, if I'm following you correctly you're saying a planet of ~3.7 Earth masses at 197.7 AU would fit the Uranus perturbation data. A planet of that mass at that distance would be cold and very dim and would require no masking nebula. It could be the "green dot". Wouldn't it be more parsimonious to make that assumption?