Requiem for Relativity

Kimura phenomenon.

Refs.
H Kimura, Astronomical Journal 22(517):107-108, 1902.
SC Chandler, AJ 22(524):164, 1902.

"[Kimura's latitude variation] does not depend upon the longitude...[it] is not due to the wanderings of the pole. ...No plausible explanation of the Kimura term has as yet made its appearance,..."

"...southern hemisphere [observations show variations of the same sign as the northern hemisphere]"

- Frank Schlesinger, Proceedings of the American Philosophical Society 54(220) (available on internet) 1915, p. 357.


Circa 1900 the Astronomical Journal has many papers, often by famous astronomers, about "variation of latitude" and the related problem "aberration of starlight". Usually, the altitude of a star would be measured vs. a plumb line or mercury pool, when the star crossed the meridian. Often, bright stars and smallish refractors were used, day and night, to measure a star throughout the year. The observatories then involved, were in the northern hemisphere, often northern Europe, so to maximize accuracy, almost-circumpolar northern stars were used because these were near the zenith.

Kimura's meta-analysis included many studies. These seem to have been corrected by their authors, for aberration of starlight (neglecting Earth's rotation and orbital eccentricity) and for precession; and by Kimura, for Chandler wobble.

It would seem that the 9" "lunar" term of Earth's nutation also was corrected as part of the authors' "reduction procedure"; its long (18.6 yr) period would be forgiving of small errors, because our interest is in a phenomenon with period one year. The 1.2" "solar" nutation term has period one-half yr, so could contribute negligibly to Kimura's phenomenon, which has period one yr. The 0.1" "fortnightly" nutation has period 15 days, so could be trusted to average out.

Parallax corrections sometimes would be insignificant, and sometimes were significant and made. Their effect is opposite in phase, to the Kimura phenomenon (with my sign presumption; see below).

Chandler said that Kimura's "phenomenon may have an objective existence", perhaps "an annual periodical oscillation" of "earth's center of gravity". Chandler did not propose that nutation was the explanation. Chandler noted two imperfections (both of about the magnitude of Kimura's phenomenon) in the correction for the aberration of light:

1. Chandler indicated that often observations were not timed to the nearest hour, and that the average hour might vary enough, seasonally, to give the Kimura phenomenon. This author calculates an average observation time four hours earlier (!) in winter than summer, to give Kimura's 0.03" amplitude, even assuming usually a sensitively positioned star such as Vega or Zeta Draconis near the ecliptic pole, and assuming observation times seldom were recorded to the hour. This suggestion of Chandler's gives the correct period and phase for Kimura's phenomenon, but cannot give enough amplitude. Also this suggestion reveals that Chandler uses the same phase for the Kimura phenomenon that I do: a reversed phase would imply *later* observations on winter nights than summer, which is absurd.

2. Chandler indicated that stellar aberration wasn't corrected for Earth's orbital eccentricity (correction for Earth's rotation is unneeded for declinations). Chandler doesn't give the period of this term but my own approximations suggest that the period is 1/4 yr; therefore it averages out.

On the other hand, the retardation of stellar aberration, to the time the light crosses the 52.6 AU barrier, agrees with Kimura's phenomenon in period, phase and magnitude. A sensitively positioned star, i.e., +/- 90 ecliptic latitude, e.g. Vega or, better, Zeta Draconis, would move 0.107". A star near the arctic circle and 6h RA, e.g. Capella or, better, Delta Aurigae, would move 0.0765". For a crude estimate, suppose the stars studied are distributed randomly over the northern hemisphere. At ecliptic (lat,long)=(0, 0 or 180), the movement is 0.107 * sin(23.5) = 0.043"; at (0, 90 or 270), it's zero. This averages 0.050", vs. Kimura's amplitude of about 0.03".

A better estimate realizes that the stars used for such studies basically were evenly distributed on the arctic circle. I tallied all variation of latitude studies, and included the (more numerous) aberration of starlight studies, published in the Astronomical Journal 1856-1899. There were 46, excluding one using Polaris. I calculated the exact effect predicted by my theory at 18h, 6h, 0h & 12h; interpolated at 3h, 9h, 15h & 21h using a quarter-wave spline; and weighted by the number of studies using stars nearest those RAs. The resulting predicted amplitude of the Kimura phenomenon, is 0.0304".

Unaccounted stellar parallax (see below) is thought to be up to one-fourth the magnitude of the observed Kimura phenomenon (but opposite in sign, I assume here), which would decrease my prediction by as much as 25%. On the other hand, the calculation above assumes all stars are on the arctic circle; really, few were above the arctic circle; some ranged as far south as Arcturus; usually the latitude of the observatory was preferred, which according to Kimura averaged 42N. Use of the 42nd parallel instead of the arctic circle increases my estimate by 41%. Stars might sometimes have been measured on the far meridian (under the pole); this would subtract from the phenomenon. So, the agreement between theory and observation remains good, when I explain Kimura's phenomenon as a retardation of Earth's effective velocity vector, by 52.6 AU * 8.3 minutes/AU.


Sign confusion in Kimura phenomenon.

Kimura uses "phi" for baseline latitude on p. 108. This suggests that phi minus phi0, essentially Kimura's "xi", is the baseline latitude, minus the latitude measured by the star's altitude on the meridian. That's the sign I used. The correctness of this also is suggested by Kimura's formula in his introduction, which puts his "xi" (sometimes called "Kimura's Z") term on the same side of the equation as what looks like the Chandler pole wobble (hence opposite in sign, to the measured minus baseline latitude).

However, in "Physics of the Earth - II The Figure of the Earth - Bulletin of the National Research Council" (1931)(available on the internet), p. 269:

"...annual parallaxes of the stars...cannot account for greater than one-fourth of the [Kimura phenomenon] effect."

This usage implies that the sign of the Kimura phenomenon is the same as that of stellar parallax, i.e., opposite what I assumed. Then phi would be the latitude measured by the star's altitude, and phi0 the baseline latitude. If so, the phase of Kimura's phenomenon is opposite that expected from retardation of Earth's velocity vector.

Except for a sine function in place of a cosine, an article by SC Chandler, Astronomical Journal 17(400):125-127, 1897, resolves the sign ambiguity of the Kimura phenomenon in favor of my usage and my theory. Furthermore Chandler gives an observed dependence, of catalog declination errors, on right ascension, which might conform to my theory.

The three catalogs considered by Chandler were Russian/French, or German productions. In this era such catalogs typically covered the sky to Decl -30. Catalog declinations typically were measured when a star was on the meridian at midnight.

In 1902 Chandler mentioned that stellar aberration corrections then customarily ignored Earth's orbital eccentricity. (This reflected not so much laziness, as lingering doubt of the true cause of the aberration.) Using Simpson's rule to integrate over the truncated lune from -30 to +90, I find that correction for eccentricity, requires on average that 0.182" be added to declinations of stars at RA 180#, and that as much be subtracted from the declinations of stars at RA 0#.

Let's assume that the correct sign of Kimura's phenomenon, makes it consistent with the theory that Kimura's phenomenon is caused by a seven-hour (i.e., 52.6 AU / c) retardation of Earth's orbital velocity vector. Again using Simpson's rule to integrate over the truncated lune, I find that correction for said retardation, requires on average that 0.0665" be added to declinations of stars at RA 270#, and 0.0755" subtracted from declinations at RA 90#, with declinations 0# & 180# requiring intermediate, small positive, corrections.

The above two corrections together amount to adding approximately:

0.126" * cos( RA + 139 )

to the declinations. Except for the presence of a sine rather than a cosine in Chandler's formula, this agrees perfectly with Chandler's "systematic correction needed by the new Pulkowa declinations":

+0.14 * sin ( RA + 135 )

and with Chandler's correction to the previous 1865 version of the Pulkovo catalog:

+0.09 *sin ( RA + 153 ).

Chandler's correction to declinations in Kuestner's 1890 catalog was:

+0.12 *sin ( RA + 120 ).

With standard error of the mean, the average of the three is:

(+0.117+/- 0.024 ) * sin ( RA + 139+/- 8 ).

The remarkable resemblance of this to my theory, suggests that the replacement of cosine with sine is a transcription error, an error by Chandler, or due to an unknown complication.

If alternatively, an advancement rather than a retardation ( by 52.6 AU / c ) is assumed for Earth's velocity (i.e., other sign convention for Kimura phenomenon) 180+34 --> 180-34 for the axis needing positive correction. That is, RA-214 --> RA-146, giving cos(RA-146) = -sin(RA+124) or rather -sin(RA+117) considering that Earth's perihelion (1889 ephemeris) is 11.5# past the solstice and that most of the effect depends on the former, not the latter. This too is acceptably close to Chandler's values, especially if Kuestner's catalog (which differs only 5% in amplitude and 0.05 radian in phase) is emphasized. The problem now is that the sign is reversed from Chandler's. Maybe there is a misunderstanding about what is being corrected to what. For example, the catalogs might be based on declinations measured on the meridian at all seasons and Chandler's supposed correct values based on midnight declinations.

Wagner's 1861-1872 right ascensions of Polaris (quoted in Chandler AJ 19(444):89-92, 1898) are consistent with advancement rather than retardation of Earth's velocity as it affects stellar aberration. The table on p. 90 gives Wagner's "Observed minus Calculated" values for the RA of Polaris. I used the 1887 (oldest ephemeris in the library) coordinates, with yearly changes given, for Polaris, extrapolating them to 1867. Including along with the velocity advancement also the small effect of parallax, I found an expected excess, of "Obs - Calc", with period one year, and maximum 0.289s (i.e., seconds of RA) when the sun is at ecliptic longitude 108. I averaged Wagner's figures for each month, weighted proportionally to the number of observations, then found the first-order Fourier coefficients. This gave for the "O - C" excess a maximum of 0.066s when the sun is at ecliptic longitude 82, a phase discrepancy of 26#. Wagner's twelve-year study averaged out the effect of the 14-month Chandler wobble, unless the wobble's axes varied much. The text indicates that Wagner sometimes used inferior instead of superior culminations of Polaris; if 3/8 of the time he used the inferior culmination, then the amplitude would be reduced, as it is, to 1/4 of predicted.

Now three studies - Kimura's meta-analysis of latitude determinations, Chandler's analysis of catalog declinatons, and Wagner's right ascensions of Polaris - roughly agree with my theory that for stellar aberration, Earth's velocity is advanced by about seven hours (there is a questionable sign disagreement for Chandler's analysis).

Proposed experiment:

Get a big old refracting telescope, with a dome and stone pier. Refractors are best for this experiment because of their stable alignment. Let the refractor have an eyepiece with two very straight lines crossing at some angle.

Each night, put the crosshairs on a known star that is as close as possible to the pole, record the time, and fix the telescope in that position though the star eventually will move away from the crosshairs. Also at this moment rotate the crosshairs so that each of the four branches has a known visible star slightly clockwise of it.

Due to precession, uncorrected J2000.0 coordinates don't accurately tell us which stars now are closest to the pole, but at epoch 2000.0, two stars with USNO-B "Red 1" magnitudes of +13.4, were < 2.4' from the north pole. So with a big refractor we typically expect to see a +13.4 star 2' from the pole, maybe closer. The billion-star printed & bound "Millenium Catalog" shows two pairs of almost opposite stars, none dimmer than magnitude +10 or +11, 20-30' from the N pole, and the pairs making about a 60# angle with each other (so crosshairs 60# apart would be best).

Four crossings will be timed soon after aiming the telescope, and four more crossings (for these stars) four or eight hours later (the latter precluding summer observation). Opposite pairs of angles will be perfectly equal. The inequality of the times required for the stars to move from one crosshair to the next, will tell how much the crosshair is displaced in that direction, from the pole. So, the position of the pole will be found.

The pole change due to change in Earth's axis (i.e., precession & nutation) can be removed by calculation based on ephemeris data. Chandler wobble doesn't move the pole; it only changes the latitude. Herein lies the superiority of my method, to that of using zenith cameras: daylight prevents 24-hour zenith camera observations, so zenith cameras don't distinguish pole motion (axis change) from Chandler wobble (latitude change). Stellar aberration due to Earth's rotation will not affect the time intervals, so the only effect remaining is stellar aberration due to Earth's orbital motion (and to the sun's motion).

The altitude of observation is constant, but temperature, pressure and humidity also correlate with atmospheric refraction, and these might change in 4-8 hours. Adjustment can be made for these, using the formula in PASP 108(729):1051+, 1996. This formula is said to agree with the empirical Pulkovo tables within 0.01", if the altitude of observation is > 15#.

"Anomalous refraction" limits accuracy. Over several hours, anomalous refraction amounts to 0.05-0.20" (A&A 459(1):283+, 2006). So observations on ~100 nights are needed, not necessarily all from the same observatory.

A book says that one-hour clock-drive exposures sometimes are made unattended. Then a firmly grounded fixed telescope should remain within one arcsecond for eight hours.

Another book says that for typical eyepieces, eyepiece fields of view are 40-50#, and that the observer's actual field of view equals the eyepiece field of view divided by the magnification. So, a one or two degree field of view would correspond to 20-50x magnification, which seems plausible.

Earth's motion may be found by numerical differentiation of the sun's apparent ephemeris position, projected onto Earth's equatorial plane. The difference between the sun's apparent and geometric position might as a first approximation be neglected. Also the sun's motion due to Jupiter might be neglected.

If Kimura's phenomenon is due to a lead or lag in stellar aberration, then the direction of deviation of the pole will lead or lag (the opposite of) the direction of Earth's velocity vector. This will provide another accurate test of Special Relativity over solar system distances.

Dimitroff & Baker, "Telescopes & Accessories", 1948, lists the world's big (15"+) refractors. In the US these include, in order of aperture:

Yerkes (40 inch)
Lick
Pittsburgh
USNO - DC
U. of Va.
Lowell
Swarthmore
Princeton
Wesleyan (Middletown, Conn.)
U. of Denver
Oakland, Cal.
Northwestern
U. of Penn.
Amherst
Carleton Coll.
U. of Cincinnati
Harvard
Mt. Lowe
U. of Wisc.
U. of Miss.
Yale (15")

Special Relativity is disproven by double stars. There should be several arcseconds difference in stellar aberration between the primary and the secondary, because their velocities differed by several km/s when their light departed. Therefore stellar aberration is determined by something besides mere relative motion of source and observer.

Red giant variables give a clue. Over many years, the angular diameter of Antares, the brightest red giant that can be occulted by the moon, was measured many times at lunar occultations (Astronomical Journal 62(2):83+, 1957). One of the most precise such diameter measurements, was 0.0413" +/- 0.0001" (41.3 +/- 0.1 mas)(A&A 230(2):355&, 1990; see also Table 2); this and other references (ApJ 242:646&, 1980, Table 5; this article is missing from "Web of Science")(ApJ 463(1):336&, 1996) list 22 determinations of Antares' diameter since 1920; some used visible light and others infrared. Ten were by occultation, eleven by any of three different kinds of interferometry (Michelson, amplitude, and speckle); one used flux curves. Fourteen of the 22 were in the range 40-43 mas inclusive. Only four (35,28,29 & 28 mas) of the 22 were outside the range 39-45 mas inclusive. Yet the Radial Velocity measurements 1905-1909 (op. cit., 1957) showed about 6 km/s "full amplitude" (max minus min) in roughly a sinusoidal pattern with period 6-7 years. The period is somewhat irregular and uncertain (AJ 98:2233+, 1989). No one seems to think that such periods in RV, typical of 25-30% of pulsating asymptotic giant-branch stars (Astrophysical Journal 604:800+, 2004), are due to engulfed companions. An average 6/2*2/pi*1.35("projection factor" to find velocity radial to Antares; because of "limb darkening", the factor is less than 1.5) km/s radial velocity for 6.5 / 4 years, even using Antares' most generous (Hipparcos) distance estimate of 185 parsec, gives +/- 9.5 mas in diameter. If the RV indicates expansion of Antares, and the period 1905-1909 wasn't unusual, then Antares' diameter should vary roughly sinusoidally +/- 9.5 mas. So, the occultation-measured diameters are too tightly clustered near 42 mas.

Antares varies about +/- 0.4 magnitude; possibly as much as +/- 0.15 mag variation (i.e., 15% of power) could occur in a 6 or 7 year cycle (AJ 1989, op. cit.). Maybe the apparent RV change is due to a +/-0.56 AU change in the radius of Antares' "ether iceberg". With about 20x the sun's mass, the radius of this "iceberg" might be 52.6*sqrt(20)=236 AU, so, the change is +/- 0.2%. Antares, like other stars of its type, expels about 0.000002 solar mass per year as wind (ApJ 275:704+, 1983). Even if the boundary or barrier at the edge of Antares' "iceberg" is mainly a gravitational phenomenon, the barrier's position yet might be affected slightly by strong solar wind.

Polaris is the nearest Cepheid variable. How fortunate that the nearest Cepheid variable is placed so prominently! Cepheids are yellow giants, smaller than Antares; only recently have their diameters begun to be measured accurately and directly. So far, only one report (Nature 407(6803):485&, 2000) seems to confirm, even roughly, for a Cepheid (Zeta Geminorum), the diameter periodicity implied by RVs.

There are two more reasons for doubting that there is any real periodic diameter change in Cepheids. The periodic time plot of RV often is grossly nonsinusoidal yet almost identical, except for sign and magnitude, to that for luminosity: e.g., Delta Cephei (Inglis, "Planets, Stars & Galaxies", 3rd ed., Fig. 11-2)(M. Petit, "Variable Stars", 1987, Fig. 4; includes temperature and radius plots). Yet luminosity is supposed to be due to temperature and radius, not RV. The luminosity waveform should resemble the temperature waveform most and the RV waveform least.

If luminosity causes apparent RV by affecting the "ether iceberg" boundary (at distance 52.6 * sqrt(M) A.U.) then the geometric midrange classical (i.e., Population I) Cepheid (Strohmeier, "Variable Stars", 1972, Table 42)(for the period, I used the geometric mean of the luminosity range, and an 0.8-slope log-log period-luminosity relation, with period the abscissa, and through the data point for Delta Cephei) should have (negative) RV lagging luminosity by 7.7% of one period or thereabouts. Indeed the lag is 7% (Carnegie Inst. Yearbook, 36:164, 1937).

Elsewhere on Dr. Van Flandern's messageboard, I've remarked that stability at temperatures proportional to (Z-0.5)^4, involving Special Relativity violations by innermost-shell electrons, might determine, for Hydrogen, the CMB temperature (2.726K); for He, the far CIRB temperature (i.e., about 2.726*1.5^4=13.8K); and for O, the temperature of the main sequence/Cepheid trough intersection. Likewise these temperatures might determine, for Ne, the least-luminous temperature of the beta Cepheids (a.k.a. beta Canis Majoris stars); for O, the least-luminous temperature of the delta Cepheids (aka Population I Cepheids) and/or Pop. II Cepheids; and for C, the least-luminous temperature of the long-period (Mira) variables.

Cepheids range from 5400-6900K according to Strohmeier (op. cit., 1972) but from 6000-8000K according to Dr. Koji Mukai ("Ask an Astrophysicist" website, 1998). The least luminous tend to be hottest (chart in Inglis, 3rd ed., op. cit.). My theoretical oxygen stability temperature is 8625K.

Beta Cepheids (a.k.a. beta Canis Majoris stars) are said on various websites to range B0 - B2 or B3 in spectral type, i.e. 18,700 or 22,000 - 30,000K. Beta Can Maj itself is said to be 22,000 or 25,000K. For these, the least luminous tend to be coolest (Inglis, op. cit.). My neon stability temperature is 22,200K.

Dyck & van Belle, AJ 112(1):294&, p. 299, quote an article then submitted to the AJ by van Belle, stating that Mira variables average 2700K. My carbon stability temperature is 2490K. Dyck & van Belle also find that "carbon stars" (kin to the Mira variables) average 3000K. The coolest "carbon star" was TW Oph at 2150K. The least luminous tend to be coolest (Inglis, op. cit.).

A delta Cepheid might have a beta Canis Majoris star inside it; and a long-period variable, a Cepheid inside it. The law relating Cepheid period to stellar density, resembles Kepler's for orbital periods. Strohmeier (op. cit., Table 42) gives P(days) = Q /sqrt(density/solar density), where 0.037 < Q < 0.066 days. That is, P = Q / sqrt(M in solar masses) * sqrt(4*pi/3)*(R in solar radii)^1.5. If almost all the star's mass is in a tiny central core, and something orbits between this core and the outer surface, its period is found when Q = 0.041 days. If something orbits between the oxygen-temperature (2.726*7.5^4=8625K) surface and a fluorine-temperature (14,230K) surface beneath that, the ratio of the radii is (7.5/8.5)^(4*2) = 0.367; the period is found when Q = 0.066 d. The periods of variable stars might be orbital periods between one stable surface and another, in the near-vacuum of a giant star's interior. The Q values given by Strohmeier for classic (i.e., delta) Cepheids are the same as the theoretical Q values given for beta Cepheids by Lesch & Aizenman A&A 34:203&, if the 0.066 comes from tripling their 0.022 given for the "second harmonic" (also there are "first harmonic" & "fundamental" modes); their Table 4 shows observational Q values ranging from 0.018 to 0.033 for 12 beta Cepheids.

The Blazhko effect "...occurs among all types of pulsating variables, but...mainly those of short period." [i.e., the RR Lyrae stars, which merge into the low-luminosity side of the Population II Cepheid distribution] (M Petit, "Variable Stars"). The Blazhko effect is, that the amplitude and waveform of a variable star's lightcurve changes grossly, while its period remains nearly constant. The explanation might be that the amount and phase distribution of matter orbiting within the star might vary though the Keplerian period doesn't. More luminous variables, being more fully developed structurally, would tend to approach a mathematical limit whereby the period is longer, the lightcurve more sinusoidal, and the Blazhko effect less.

The three (3/22) anomalously small values for Antares' diameter (0.029", 0.028", 0.028"; see previous post) might be glimpses of the next inner, perhaps N-based, surface. One of these small values was obtained at lunar occultation, observing at 1.04 microns. At this wavelength, the inner surface might be especially visible, with less scattering (long wavelength) yet denser emission from the hotter inner surface than from the outer surface (wavelength not too long).

Two of the small Antares diam values were from the longer (50 ft) Michelson interferometer; as we would expect, Pease noted difficulty from ambiguous readings. If the N-surface were detected on one side only (radius = (5.5/6.5)^8=0.263x the carbon surface), then the angular diameter found should have been 0.0413" * 1.263/2 = 0.026".

The small 1986 occultation value for Antares' diam, was followed by two normal 1987 occultation values (41 & 45 mas). Likewise Pease's small 1934 and 1936 Michelson interferometer values were preceded by a normal (41 mas) 1932 value by Pease with the same interferometer. Such a diameter dip in one year would require (0.041-0.028)"/2*1.35(cosine correction with limb darkening) * 185parsec = 1.6 AU/yr = 7.7 km/s average net positive RV, twice the extreme RV observed during the extensive early 20th century observations. The anomalous small diameters were partial detections of an inner surface. Nor do the RVs indicate real diameter changes.