Requiem for Relativity

When the editor of Icarus declined to forward my latest submission, for peer review, he remarked that it seemed unlikely that the Maya (or their predecessors) could have known about the 5.14535 +/- 0.00717 hr period found from asteroid rotation (and from simple relations of outer planet moon revolutions, and of Uranus' ring revolutions). The value of the paper doesn't depend on whether the Maya did or didn't know about the 5.145 hr time period, but, just as European science draws on anthropomorphism ("foot" of a perpendicular) or mythology (e.g. "boreal forest", "thorium") for its vocabulary, likewise Mayan texts might be about something altogether different from the literal "hero twins", "macaw", "jaguar", "Bolon", etc. Much can be lost in translation. If Mayans didn't happen to discover some things themselves, they might at least have preserved knowledge from their remote predecessors or even from more or less unrelated civilizations. The fall of classical civilization in the Mediterranean c. 400AD was accompanied by the destruction of ~90% of significant literature, mainly as part of the associated religious revolution; scientific literature and the persons of scientists (e.g. Alexandria, Hypatia) weren't spared. The fall of Mayan civilization c. 900AD might not have involved so much destruction of information.

Indeed the Mayans did know about the "5.14535 hr period". Consider the difference between the mean synodic month, and the sidereal month, as measured in sidereal days:

29.53059 - 27.32166 = 2.20893 Julian (i.e. mean solar) days
= 2.21498 sidereal days = 2 sidereal days + 5.1454 hr

That is, the synodic month exceeds the sidereal month, by two sidereal days, plus "the 5.145 hr period". The thoroughness and numerical precision of Mayan astronomy was such that they hardly could have been unaware of this time period.

In my post of June 25, 2011, I note that the "day" equal to the reciprocal, of Earth's sidereal rotation frequency minus twice Luna's, equals 5 * 5.1639 hr. Akin to my post of July 7, I note that the "day" equal to the reciprocal, of Mars' synodic rotation frequency ("synodic" = mean solar) minus Luna's sidereal, equals 5 * 5.1167 hr.

The simplest manifestation of the "5.145 hr period", is that Luna's sidereal rotation period is 27 Earth synodic days (i.e. mean solar days, Julian days) plus 1.5 * 5.1466 hr.

Dr Joe, When will all these odd facts get connected and thereby creating an "I see" moment? I guess you have a method and direction to all the work you are doing that I can't see yet. Is it related to the 2012 date?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />...When will all these odd facts get connected and thereby creating an "I see" moment? ...related to the 2012 date?
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Thanks for the timely questions. In review, I first discovered this 5.145 hr period, as the rotation period (or very close to it) of four large asteroids which align (to within a few degrees of ecliptic longitude) in Dec., 2012 - a rotation period also shorter than that of almost any other known asteroids; and furthermore these four asteroids have about the same rotation axes.

I found further evidence elsewhere in the solar system (particular resonances, of orbital periods of Uranus' moons & rings; and of orbital periods of the moons of Saturn, Jupiter and Mars) that this 5.145 hr period is special and not accidental. What the physical basis of it is, what kind of tinkertoy model can be made of it, is another question. At least I know that the 5.145 hr period is so ubiquitous that it must be a manifestation of a physical phenomenon. Also, in a way, it marks the month of Dec. 2012.

Earlier I had found some quantitative, but somewhat complicated, inexact and doubtful, involvement of the 5.145 hr period with the Earth-Luna system. When the editor of Icarus gave, as his main reason for not sending my article for peer review, that the Maya couldn't have known about the 5.145 hr period, it motivated me to find an important, simple manifestation of the 5.145 hr period, not needing a telescope, so that the Maya must have known of it. I found such a relation. It is described in my previous post:

mean synodic month minus mean sidereal month

= 2 sidereal days + 5.145 hr

The sidereal month, year and day all change very slowly, so this relation is persistent: at 1.5ms/cyr, the sidereal day changes only 1 part in 1000, in 6,000,000 yr. The relation is much more persistent, even than this. According to present estimates, these quantities change in such a way as to preserve the relation to first order; that is, the first derivative of

synodic month minus sidereal month minus two sidereal days

is close to zero.

Venus' and Jupiter's tides on Earth, and Luna's rotational angular momentum, are negligible. I'll find the rate of change of the logarithm of the length of the sidereal day, with respect to the logarithm of the angular momentum of the Luna/Earth system; and the rate of change of the logarithm of the time period (synodic month minus sidereal month) also with respect to the log of the Luna/Earth angular momentum.

The vertical (i.e. perpendicular to the ecliptic) component of Earth's spin angular momentum, in convenient units, is

1 Earthmass / 1 sidereal day * (6378km)^2 * 0.3307 * cos(23.44)
= 12.342*10^6

The orbital angular momentum of the Earth/Luna system is

0.012300 / (27.322*366/365) * 384750^2
/ (1+0.0123) * sqrt(1-0.0549^2)
= 65.55*10^6

(The last two factors are the "reduced mass" correction and the eccentricity correction.)

The Sun's tides also rob Earth of spin angular momentum; this is proportional to the square of the tide. The ratio of Luna's to the Sun's tide, is

0.012300/332960/(384750/149598000)^3
*(1-0.0167^2)^3/(1-0.0549^2)^3*(1+1.5*0.0167^2)/(1+1.5*0.0549^2)
= 2.1804

where the factors, near unity, on the second line, comprise the eccentricity correction to the time average 1/r^3.

Also, Earth rotates faster with respect to the Sun than with respect to Luna. The restoring force on the tidal bulge is proportional to (w*t)^2, so the bulge decrement to w^2*t^4, the displacement of the bulge to w/sqrt(w), and the tidal drag to

sqrt(w(Sun)/w(Luna)) = sqrt((1-1/365.25)/(1-1/27.322)) = 1.0174

On average, Earth's spin angular momentum parallel to the ecliptic, is lost to tidal drag only half as fast (because the average value of cos^2 is 1/2). So Earth's total spin is lost only 1-sin(23.44)^2 / 2 = 0.92088 times as fast as the vertical component.

So the rate of change of the logarithm of the length of the sidereal day, with respect to the logarithm of the angular momentum of the Luna/Earth system, is

(*) 65.55/12.342*(1+1.0174/2.1804^2)*0.92088 = 5.938

Let's compare my estimate of the ratio, of the Sun's to Luna's tides, to the estimate of Bursa, Earth, Moon, and Planets 56:57-60 (1992), pp. 58-59, eqs. (5) & (. My ratio is 1.0174/2.1946^2 = 1/4.673. Bursa's is (1.03+/-0.30)/(5.24+/-0.29) = 1/5.09, with a 30% margin of error.

Next I'll find the rate of change of the logarithm of the time period (synodic month minus sidereal month) with respect to the log of the Luna/Earth angular momentum. Let p1 be Luna's sidereal period, and p2 Earth's (constant) orbital sidereal period. Let Luna's (synodic period minus sidereal period) be f = 1/(1/p1 - 1/p2) - p1. We have

df/dp1 = p2^2/(p2-p1)^2 - 1

and d(ln(f))/d(ln(p1)) = df/dp1 * p1/f = 2.08085

From Kepler's laws, the log of Luna's sidereal period, increases exactly three times as fast as the log of Luna's orbital angular momentum. So, log(f) increases 3*2.08085 = 6.243 times as fast as the log of Luna's orbital angular momentum. The lengthening sidereal day cancels 5.938/6.243 = 95.1% of the increase. To maintain f = "2 sidereal days + 5.145 [Julian] hr", the sidereal day should increase (2 sid. day + 5.145 hr)/(2 sid. day) = 1.1075 times as fast as f, not 0.9511 times as fast, as it does. That is, Earth's rotation rate should decrease 6.243*1.1075 = 6.914 times as fast as Luna's angular momentum increases, if the 5.145 hr term is to remain constant to first order in time.

It seems that this is indeed the case. Above, I consider only tidal exchange of angular momentum between Earth's rotation, and Luna's orbit or the Sun's orbit. Artificial satellites reveal a nontidal source of acceleration of Earth's rotation: a steadily decreasing "J2" constant (i.e., some layers of Earth are becoming less oblate). Bursa, in his eqs. (5) & (7), pp. 58-59, estimates this as (1.147+/-0.084)/(5.36+/-0.20) = 1/4.673 as much as Earth's rotation deceleration due to Luna's tide. (This happens to exactly equal my estimate of the ratio of the Sun's to Luna's tidal drag, though Bursa implies an 8% margin of error for his ratio.) My eq. (*) above becomes

(**) 65.55/12.342*(1+1.0174/2.1804^2-1/4.673)*0.92088 = 4.891

Also, Laplace (see Danby, "Fundamentals of Celestial Mechanics", 2nd ed. (1988), Sec. 12.9, pp. 382-384) realized that as Earth's orbital eccentricity slowly changes due to planetary perturbations, the Sun's average tidal effect, on Luna's orbit, changes, and this slowly accelerates Luna. The most precise available calculation is that of John Couch Adams, amounting to +11.44"/cyr/cyr.

According to Huaguan et al, Earth, Moon and Planets 73:101-106 (1996), the total secular acceleration of Luna (from Lunar Laser Ranging to four different reflectors, the first and most important of which was placed by the Armstrong/Aldrin/Collins Apollo 11 Mission) is -25.4" +/- 0.1" /cyr/cyr. Huaguan et al say this agrees within a few percent, with all the many previous researches, including historical researches (see their Table II, p. 105). This implies that the tidal acceleration of Luna is -25.4-11.44 = -36.84, which the Laplace-Adams secular acceleration shrinks by a factor of 25.4/36.84 = 0.6895, so the overall ratio of the log change in Earth's sidereal day, to the log change in Luna's angular momentum, is 4.891 (see my eq. (**)) / 0.6895 = 7.093. This is near the value, 6.914, needed.

Dr Joe, You are using a tidal model I find defective. The sun has little effect on Earth's tides and the moon does not only orbit the Earth. The Earth/moon system is orbiting the sun and Earth has a very real effect on the moon that is ignored in the model you are using here. If the orbit of the moon is viewed without the Earth's orbit it is clear the moon has huge forces effecting it's motion on a 28 day cycle changing it's orbital speed from ~29,000m/s to ~31,000m/s during the 28 day cycle. I am quite sure this force is the real cause of the "spring/neap" tidal effect we observe on Earth. The model says the sun causes the "spring/neap" tides and that is false.

On Jan. 8.75 UT, 2013, the four asteroids 511 Davida, 39 Laetitia, 947 Monterosa & 1717 Arlon are coplanar. On Jan. 14.28 the circumcircles (i.e. circumscribed circles) of Davida-Earth-Laetitia and of Earth-Laetitia-Monterosa have equal radii; that is, Davida, Earth, Laetitia & Monterosa lie on a curve of approximately constant curvature (such as a circle or helix).

This is determined from the online JPL ephemeris of positions as seen from the Sun. I added the small correction for light time, so that the positions are in ephemeris time, not the time when observed at the Sun. I interpolated ecliptic longitude & latitude and distance as seen from the Sun, quadratically on 60 day intervals.

The current (Nov. 2011) issue of Sky & Telescope, has a "News Note" on p. 12, "A Planet Blacker than the Blackest Coal". It says:

"...the planet reflects less than 1% of the light hitting it - making it blacker than any ordinary substance on Earth.

"Astronomers had already found that some, but not all, 'hot Jupiters' are quite dark..."

On this messageboard, I've cited journal articles many years old making this statement about cool brown dwarfs a.k.a. hot Jupiters. As I've been saying since 2007, this might be part of the explanation for the dim visual magnitude of my discovery Barbarossa.

Dr Joe, All the theory that has been developed to explain the moon is nothing more than theory. The real moon is in orbit around the sun as is the Earth. The moon however changes its orbital speed by a lot over and over. The tides on Earth are a result of the 3rd law of Newton with the speed of the moon having some effect as well as the mass of the moon. The idea that the sun is involved in tidal force because the tides are higher when the moon/Earth and sun are radially lined up was invented by Newton to explain why spring tides are higher than neap tides. It is not real, but, it does work. You are mixing real forces with stuff that has been invented to account for mysteries much like witches have been invented to explain something scary.