Tidal torques. A critical review of some techniques

[Anon E. Mouse]

In response to a moderators suggestion that I read up on tidal torque
as it relates to the moon's recession, I found several useful
resources. Due to a rather severe visual impairment I can not read
text on white background and I was thus unable to find a readable copy
of Murray and Dermott, "Solar System Dynamics"
(Cambridge U.P., 1999), sections 4.9 "Tidal Torques" and section 4.13
"Tidal Evolution". The explicit statement that the Moon is slowly
*receding* from the Earth can be found in section 4.9, just after
equation (4.152).

However, I did find the above titled paper by Michael Efroimsky
US Naval Observatory, Washington DC 20392 USA e-mail: me @
usno.navy.mil and James G. Williams Jet Propulsion Laboratory,
California Institute of Technology, Pasadena CA 91109 USA
e-mail: james.g.williams @ jpl.nasa.gov

This seems quite comprehensive as it compares and contrasts two of the
most refined methods of tidal torque computation.

Both use methods of Fourier composition of harmonic series for a given
phase lag angle E and assume low inclination angles.

Neither this paper, nor any other source I found report any actual
results. I find this surprising and disturbing. Unless a method of
computation is applied to at least two actual cases how is a person to
estimate the value of the method?

My understanding is that Murray and Dermott likewise do not compare
calculated results to actual data, a matter of 0.38 cm of recession
per year, for Earth's Moon. What this amount of recession amounts to
in terms of force per year I can calculate myself, It will be a pretty
large number, reduced to force per day (Earth revolution) it will
still be a significant figure. The kinematics of the decay of the
Earths rotation is similarly an easy computation, if these match
reasonably well, I will be content. Otherwise, I my present disquiet
will get worse.

Given that no source I found reported any results whatsoever the
question then arises do the methods seem sound, or suspect?

Several sources I consulted including those from reputable
organizations had the Moon's tidal bulges IN LINE with the line
through the Barycenter.

Gravitational force is a strain, balanced by centrifugal force, also a
strain but in an opposite vector orientation. Within the primary,
these strains are opposed by stresses. In plainer English
compressions. Every orbiting body is squeezed along the orbital radius
and bulges nominally orthogonal to those forces.

Because there is no illustration or definition in the above paper I am
uncertain if the above have this right or wrong. They use both sine
and cosine in their preferred computation and say this works better
than other methods using only sine.

Because they refer to r sub i and r sub j in association with a
"single double bulge" it seems possible to tell that they use both
bulges in their computation. Well, their method is a Fourier
composition so perhaps this all works out in the washing, but
gravitation is uni-polar, only the leading bulge can significantly
contribute to the increased delta v and thus, recession of the Moon.

Based on a simple consideration of work in and work out, it is
possible to say that the Earth's rotation must be slowed by the Moon's
gravitation and according to the second law, the Earth's tides must do
a substantially reciprocal amount of work on the Moon and this should,
all other things being equal increase the orbital radius.

I found chat from someone who sounded authoritative stating that tidal
torques were inverse 3rd and 5th order effects. This sounds about
right to me, but at this point I am inclined to think that everyone
involved is simply making well educated guesses.

Meanwhile, in spite of searching hard for several hours I found no
source for data on long term trends in the orbital periods of the
planets of the Solar system. I would very much appreciate it if any
knows of such a data source they please send me a reference.

The Solar tides are approximately 50# of the lunar tides and based on
tidal torque theory the Earth should be recessing from the Sun at
about 0.04 cm per year. I would like to know if this is the case and
how this relates to the recession of the other planets orbital radii
in the Solar system. (I do realize there should be other causes of
planetary orbital period increase or decrease.)

Sincerely,

AAG

[Jonathan Thornburg [remove -animal to reply]]

Anon E. Mouse <agal...@gmail.com> wrote:
> In response to a moderators suggestion that I read up on tidal torque
> as it relates to the moon's recession, I found several useful
> resources. Due to a rather severe visual impairment I can not read
> text on white background

Have you tried contacting Cambridge University Press to see if there's
a large-print or Braill edition?

> and I was thus unable to find a readable copy
> of Murray and Dermott, "Solar System Dynamics"
> (Cambridge U.P., 1999), sections 4.9 "Tidal Torques" and section 4.13
> "Tidal Evolution". The explicit statement that the Moon is slowly
> *receding* from the Earth can be found in section 4.9, just after
> equation (4.152).

[[...]]

>
> Neither this paper, nor any other source I found report any actual
> results. I find this surprising and disturbing. Unless a method of
> computation is applied to at least two actual cases how is a person to
> estimate the value of the method?

For the Earth-Moon system, there are a lot of references cited in

Stephen Merkowitz
"Tests of Gravity Using Lunar Laser Ranging"
Living Reviews in Relativity 13 (2010), 7
http://www.livingreviews.org/lrr-2010-7
(note this is open-access online!)

Some other general reference on lunar laser ranging (which cite more
detailed studies of the effects you ask about) are

S. M. Kopeikin et al
"Prospects in the orbital and rotational dynamics of the Moon
with the advent of sub-centimeter lunar laser ranging"
arXiv:0710.1450

Kenneth Nordtvedt
"Lunar Laser Ranging - A Comprehensive Probe of Post-Newtonian Gravity"
arXiv:gr-qc/0301024

Kenneth Nordtvedt
"30 years of lunar laser ranging and the gravitational interaction"
Classical and Quantum Gravity 16 (1999), A101-A112

James G. Williams, Slava G. Turyshev, and Dale H. Boggs
"Progress in Lunar Laser Ranging Tests of Relativistic Gravity"
Physical Review Letters 96 (2004), 261101
arXiv:gr-qc/0411113

The last of these cites its references 3 & 23 for details of the lunar
dynamical model.

Murray & Dermott's book reports a Moon-receding-from-the-Earth rate
of 3.74 cm/year (text just after equation (4.224)), and for further
details cites (among others)

P. Goldreich and S. Soter
"Q in the solar system"
Icarus 5 (1966), 379-389

D. J. Webb
"Tides and the evolution of the Earth-Moon system"
Geophys. Journal of the Royal Astronomical Society 70, 261-271

There are of course a lot of other moons in our solar system, and
Murray & Dermott's book has extensive discussions of tidal evolution
in various planets' moon systems. In most cases we don't have good
independent knowledge of the tidal dissipation within the planet &
moon (i.e., we don't know their internal structure very well), so
it's hard to directly test the theory, but you could try

R. R. Allan
"Evolution of the Mimas-Tethys commensurability"
Astronomical Journal 74 (1969), 497-506

for an indirect test (whose results are shown in figure 4.18 of
Murray & Dermott's book).

> Several sources I consulted including those from reputable
> organizations had the Moon's tidal bulges IN LINE with the line
> through the Barycenter.

See the paper by Koepekin et al I cited above.

> Meanwhile, in spite of searching hard for several hours I found no
> source for data on long term trends in the orbital periods of the
> planets of the Solar system. I would very much appreciate it if any
> knows of such a data source they please send me a reference.

Table A.3 in Murray & Dermott's book gives the "rates of change of
planetary orbital elements", citing the 1992 Explanatory Supplement
to the Astronomical Almanac, but any recent edition will have this
data (more accurately in newer editions). See
http://en.wikipedia.org/wiki/Astronomical_Almanac
if you're not familiar with this reference.

--
-- "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam