Requiem for Relativity

The surprising resonance of the large outer moons

Summary. Titan, Triton and all five large moons of Uranus, resonate with the commonest orbital period of classical Kuiper belt objects. The high numerical values of these resonances suggest that they are due not to Newtonian gravity, but to an undiscovered force.


Using the posted (online Feb. 3, 2012) NASA Fact Sheet periods for solar system moons, the following relations hold (sometimes the last digit is not significant):

exactly 6335 Titan (a Saturn moon) orbital periods (6335=5*7*181)
= 71464.975 Miranda (Uranus moon) periods (71465=5*14293)
= 40078.989 Ariel periods (40079=13*3083)
= 24374.983 Umbriel (24375=3*5^4*13)
= 11603.001 Titania (!) (11603=41*283)
= 7502.967 Oberon (7503=3*41*61)
= 17188.489 (half-whole resonance) Triton (Neptune moon)(17188.5*2=34377=3*7*1637)

To confirm this, I repeated the calculations, using the (online, Feb. 4, 2012) Jet Propulsion Lab values. These usually were given to more significant figures, though I had to convert mean motions to orbital periods. The JPL values are referred to local Laplace planes for Jupiter's, Saturn's, and Neptune's moons, but are simply mean equatorial elements for Uranus' moons. Their Uranus moon values are from a 1987 Astronomy & Astrophysics paper.

6335 Titan periods
= 71465.076 Miranda
= 40079.056 Ariel
= 24375.027 Umbriel
= 11603.027 Titania
= 7502.983 Oberon
= 17188.518 (half-whole) Triton

Whichever Titan period I use, multiplying it by 6335 also gives 147.04 Mars revolution periods.

The Julian day very nearly equals the mean solar day, and according to a USNO graph posted online, exactly equals it in about 2013. If I adopt the JPL value for Titan's period, 15.945448 Julian day, I find that 6335 times this is

15.945448*6335*366.25636/365.25636 = 101,290.97 sidereal Earth day

but on the other hand using the NASA value for Titan, I find

15.945421*6335/1.025956756 = 98,458.58 sidereal Mars day

and 15.945421*6335*(1/1.025956756 - 1/686.9798529)

= 98,311.54 synodic Mars day,

a possible half-whole resonance, using Folkner's 1997 Mars rotation value as cited by Bouquillon & Souchay, A&A, 1999, Table 5 (Folkner's differs inconsequentially from the value of Ashbrook, Astronomical Journal, 1953).

This mysterious resonant period, equal to 6335 Titan orbits, is, using the JPL value, 276.56239 Julian yr. This is the orbital period corresponding to a semimajor axis of 42.5 AU, which distance is, the last time I looked at a chart of it, within a few tenths of an AU of the mode of the distribution (i.e. peak) of the semimajor axes of classical (i.e. non-plutino) Edgeworth-Kuiper belt objects. The tidal effect of these rather symmetrically distributed, low mass, distant objects, hardly could cause these high numerical resonances of the outer planet moons. So, likely there is an undiscovered force at work, causing these resonances.

The surprising resonance of the large outer moons, continued:
statistical significance

Using the NASA values in the preceding post, I consider the sum "S", for the five Uranus moons, of the squares of the differences from the nearest whole cycle. Trying all numbers of Titan cycles from 1 to 10,000, S is minimum for 6335.

For 100 "Monte Carlo" trials, I altered the orbital periods of Titan and the Uranus moons individually, randomly within an interval of +/- 1%. For each trial, S was minimized at some number of Titan cycles in the range [1,10,000]. The mean such minimum was 0.01130; the standard deviation of these 100 minima was 0.00420. If the distribution is normal near the mean, 16% of these S minima should be <0.00710.

Actually 13 of the 100, are <0.00600 (sigma = 1.26), vs. 10 expected for a normal curve. Also 7/100 are <0.00500 (sigma = 1.50) vs. 7 expected. Furthermore 3/100 are <0.00400 (sigma = 1.74) vs. 4 expected. The real value, 0.00210 (sigma = 2.19) should correspond to a tail of 1.4% of Monte Carlo trials, if the normal approximation holds this close to zero, but because S cannot be <0, the significance is better than this. Indeed the smallest of the 100 Monte Carlo trials was 0.00325.

Addendum Feb. 9: In a Monte Carlo trial of 800, 6 had minimum S values (for the 10,000 variations) less than the actual 0.00210. This corresponds to p=6/800=0.75% (one-tailed).

The surprising resonance of the large outer moons, continued:
link to Mayan Long Count

Let's adopt the NASA value for Titan's orbital period, and Uranus' & Saturn's. Recall that the Mayan Long Count is 360*5200 days. Because of its exactness, let's take 6335 times Titan's orbital period, as the Kuiper belt orbital period.

Then f(Kuiper) + 0.25*f(Saturn) - f(Uranus)
= 1/101,014.242d + 0.25/10,759.22d - 1/30,685.4d
= 1/1,829,200d = 1/5008yr = approx. 1/5125 yr = 1/MayanLongCount

Though this relation gives the Mayan Long Count with 2% error, the Mayan frequency term is given as a difference of terms that are as much as 60 times larger. If the formula were rearranged, it would be seen to give Saturn's orbital period to 0.05% error.

Jupiter/Luna occultation timing anomalies in the 19th century: Occultation of Jupiter, 1889 September 3
adsabs.harvard.edu/full/1889AJ......9...84K

When comparing the reported timings with the values derived from Stellarium.
Considering the time lapse between first contact of Jupiter with the events as published:
- observed 2nd contact is 6 seconds earlier as per Stellarium
- observed reappearance of Callisto, IO + 3th and 4th contact are around 29 seconds earlier as per Stellarium

Explanation:
- The moon appears to move down relative to the position of Jupiter (and stars)
- This is because the observer is positioned on a rotating earth of which the axis has an inclination relative to the path of the Moon
- The observed position of Jupiter is different from the calculated position because of the effect of planetary aberration
- Planetary Aberration = Effect of Light-time delay + Aberration of light (or however we want to call this effect)
- The effect of Ligth-time delay is taken into account through the Stellarium software (parameter selection)
- Jupiter is observed 'to the right' of its calculated position which explains why the observed first contact is earlier then calculated
- The sooner the first contact is made, the shorter the time between first contact and the reappearance.
(because the Moon appears to move upward: the later the first contact, the longer the 'path behind the moon')

By displacing Jupiter and its planets respectively 5, 10, 15, 20, 25, 30 arcsec (on the Stellarium images), I derived that the duration until reappearance is decreasing with around 1 seconds for every increase of planetary aberration with 1 arcsec. The observed timings match with an aberration of around 25 arcsec.

On 1889 September 3, the Earh and Jupiter were moving in a direction almost perpendicular to each other (85 degrees). (The Earth was moving away from Jupiter). Stars showing in a direction of Jupiter were therefore subject to only 2 arcsec of stellar aberration.

On a side note: Following the logic above: the reported reappearance of Europa (satellite II) looks off with 30 seconds.

Correction:

"because the Moon appears to move upward: the later the first contact, the longer the 'path behind the moon'":

because the Moon appears to move downward: the later the first contact, the longer the 'path behind the moon'

Catastrophism: Ice Ages and redomestication of species

Paragraph from the Wikipedia article "Dog":

"The present lineage of dogs was domesticated from gray wolves about 15,000 years ago. Remains of domesticated dogs have been found in Siberia and Belgium from about 33,000 years ago. The earlier specimens not only show shortening of the snout but widening of the muzzle and some crowding of teeth making them clearly domesticated dogs and not wolves. There are more sites of varying ages in and around Europe and Asia younger than 33,000 years ago but significantly older than 15,000 years ago. None of these early domestication lineages seem to have survived the Last Glacial Maximum. Although mDNA suggest a split between dogs and wolves around 100,000 years ago, no specimens predate 33,000 years ago that are clearly morphologically domesticated dog."

Budiansky's book on the cat, says that the subspecies Felis silvestris lybica (the North African wildcat), though wild, is far more tameable than other Felis species or subspecies such as Felis silvestris silvestris (the European wildcat). This tameability has been documented by many objective criteria of its behavior, or the behavior of its hybrids with the domestic cat. One such "tame" behavior is called neotenic behavior, i.e. a preservation of kitten behavior such as play, in the behavior of the adult. This subspecies, lybica, also happens to be the most widely distributed subspecies of F. silvestris.

Budiansky remarks that besides cats, other mammalian genera or families, such as sheep/goats, contain one special wild species that is both: (A) by far the closest to being tame; and ( the most widespread. To me, the situation with cats seems the most clear:

Sometime, perhaps during the last interglacial (130,000-100,000 yr ago according to the "Quaternary glacial cycles" graph in Wikipedia's "Timeline of glaciation" article) Felis silvestris lybica was domesticated; this domestic cat was so successful that it accompanied humans over a large area, greatly exceeding the original wild subspecies in range and numbers. When humankind nearly collapsed during the last Ice Age, Felis sylvestris lybica became essentially a feral, i.e. wild-reverted, subspecies, but still far more tameable than any other cat species or subspecies. Then beginning with the Holocene (i.e. present interglacial) ~10,000 yr ago, some individuals of F. s. lybica were redomesticated, giving rise to what we call F. s. catus.

Likewise these other exceedingly widespread and tameable mammalian species that Budiansky mentions, might be essentially feral subspecies from the last interglacial. In particular, the dog seems to have diverged from the wolf, mitochondrially, at about the end of the last interglacial. Apparently dogs were widespread during Cro-Magnon times ~33,000 years ago (the last relatively mild interlude within the "Wisconsin" Ice Age before the worst of that Ice Age suddenly hit) and again beginning with the sudden end of the Wisconsin Ice Age ~15,000 yr ago. The unrelatedness of the modern dogs and the Cro-Magnon dogs, might be because both were redomesticated separately from a 100,000 year-old feral subspecies of semi-tame wolf analogous to Felis silvestris lybica. Maybe these special wolves survive unidentified somewhere in the world, or maybe all of them have been caught and absorbed into modern dog breeds.

Be that as it may, my theory is that again and again we retame our domestic animals after catastrophes almost wipe out their populations so that only a scant but widespread feral population (such as, perhaps, Felis silvestris lybica) survives. These catastrophes would have had to be something more extreme, than a mere ice sheet advance or reduction of CO2 levels to 2/3 of what they are now.