Subject: Re: On Our Escaping Moon
Date: Sat, 5 Aug 2000 17:09:38 -0700 (PDT)
From: Tim Thompson
To: do_while@ridgecrest.ca.us
My aplogies for the delayed response, I have been on a great deal of
work related travel in the last couple of months.
> Date: Wed, 26 Jul 2000 21:42:01 -0700
> To: Tim Thompson
> Subject: Re: On Our Escaping Moon
> So, by your calculation, how far away was the moon 10,000 years ago, 1 million
> years ago, 10 million years ago, 1 billion years ago, and 4 billion years ago?
There is no possible way to uniquely compute the Earth-Moon distance as a
function of time, because that distance is very sensitive to the oceanic
dissipation, which in turn depends not only on the location of the continents,
but also on the area and depth of the continental shelves. There is no way
to know these parameters, so one has to try alternative methods.
I cited several papers in my web article that address this issue. The analysis
in Kagan & Maslova (1994) shows how that a considerable physically reasonable
parameter space is consistent with a age for the system as old as that of the
Earth (about 4.5 billion years). However, their model is explicitly stochastic.
They do not compute one set of results from one formula. Rather, they use one
formula with a number of free parameters, and then show the range of simultaneous
values required for the Earth-Moon system to have an age of at least 4 billion
years. Such a method does not show how things were, only how they would have been
if the fitted parameters occurred in nature. So the physical plausibility of their
scenarios is important, and they spend some time showing that this is the case.
But the study of Kagan & Maslova is only one of several, along similar lines, that
are best understood by looking at Bills & Ray (1999), where all of the studies
are examined in concert. Here is how Bills & Ray explain it, on page 3046, in the
first paragraph under "Inferences from Ocean Models":
"To our mind, the dynamical solution of the "time scale" problem
for the lunar orbit evolution has been solved by the ocean models
presented by Hansen (1982), Webb (1982), Ooe et al. (1990) and
Kagan and Maslova (1994). These authors all find that ocean tide
models, satisfying various forms of Laplace's hydrodynamic
equations, generate significantly smaller torques in the distant
past than implied by the present f value (9). All of them find
ages for the lunar orbit greater than 3 Gyr. A heuristic explanation,
couched in terms of oceanic normal modes, is as follows: From work
by Platzman et al. (1981) and others, we know that the present day
ocean has a rich spectrum of normnal modes, some having frequencies
close to the main tidal spectral lines. These modes grow increasingly
complex (i.e., higher wavenumber) with increasing frequency. In the
distant past when the Earth's faster rotation forced tidal frequencies
higher, those normal modes with near-tidal frequencies would be spatially
less well matched to the large-scale tidal forcing (a degree 2 spherical
harmonic), and hence they would be less easily excited. Note that this
does not imply smaller ocean tidal heights - in fact they were likely
comparable or larger than present heights (Webb, 1982, figure 5) -
because the tidal potential was larger; rather the tidal admittance to
that potential was reduced. In particular, the degree 2 part of the
admittance, which completely quantifies the dissipation, was reduced.
The torques were therefore correspondingly smaller than they would
otherwise have been if the admittances had maintained their present
day values."
The point to this argument is that the fast rotation of the early Earth weakens
the tidal acceleration of the Moon, whereas most creationist arguments assume
the opposite, that it would strengthen tidal acceleration. So we know from
fundamental considerations that the Moon could not have been accelerated from
the Earth rapidly in the distant past, but we can't compute the specific
acceleration (and therefore the specific distance) without specific knowledge
of the ocean basins and the true dissipation.
However, there is an answer to you question in the form of observational data.
My web article also cites several papers on tidal rhythmites, and gives the
rates of lunar retreat as a function of time, derived from the fossilized
signal of the ocean tides. Since I wrote that article, a much more detailed
analysis is forthcoming from G.E. Williams (2000), from whose Table 1 I borrow
the following (Re = Earth radii):
2.45 billion 900 million 602 mill. Now
(1) (2) (1) (2)
Lunar semi-major axis (Re) 51.9±3.3 54.6±1.8 54.7±0.7 57.1 58.16±0.30 60.27
Lunar recession rate (cm/yr) 2.18±0.86 1.47±0.46 3.95±0.5 2.25 2.17±0.31 3.82±0.07
The time scales on top are years ago. The two columns labeled 2.45 billion are from
the same data set, the cyclic banded iron formation in Weeli Wolli, Australia, but
different analysis. The errors are all 1-sigma, and are clearly compatible. The two
columns labeled 900 million are both based on the Big Cottonwood formation in Utah,
but represent two different studies by Sonnett & Chan (cited in my web article, where
(1) is the 1996 paper, and (2) is 1998 (errors not available). The lone column labled
620 million is based on G.E. Williams' own study of the rhythmites in the Elatina
Formation and Reynella Siltstone, in South Australia
For now, 2.45 billion years ago is as far back as we can reach. As you can see, over that
period of time the Earth-Moon distance appears to have grown from roughly 52 to 62 Earth
radii. This is in keeping with the above cited theoretical considerations, where the moon
should recede very fast from the Earth after formation (because the Earth is still fluid),
but then slow quickly as the Earth hardens and solid tides dominate.
I strongly recomment Williams' new paper, which includes a long discussion of the physical
details which justify the assumption that the rhythmite markings are in fact tidal.
Hopefully this answers what you had in mind. I will be off again for a few days, but will
be back leter in the week if you have more questions.
Bills, Bruce G. & Ray, Richard D.
"Lunar Orbital Evolution: A Synthesis of Recent Results"
Geophysical Research Letters 26(19): 3045-3048 (October 1, 1999)
Kagan, B.A. & Maslova, N.B.
"A stochastic model of the Earth-Moon tidal evolution accounting for the cyclic
variations of resonant properties of the ocean: An asymptotic solution"
Earth, Moon and Planets 66: 173-188 (1994)
Williams, G.E.
"Geological constraints on the Precambrian history of the Earth's rotation and
the Moon's orbit"
Reviews of Geophysics 38(1): 37-59 (February, 2000)
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Timothy J. Thompson
http://www.geocities.com/Tim_J_Thompson/
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