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Science Tribune - Article - December 1998

http://www.tribunes.com/tribune/art98/bros.htm

Understanding tidal friction: A history of science in a nutshell



Peter Brosche


Observatorium Hoher List, Institute of Astronomy, University of Bonn, Daun, Germany
E-mail : pbrosche@astro.uni-bonn.de


Summary

For a long time, two separate lines of thought governed our perceptions of tidal friction. Empirical evidence from observations of solar eclipses made as early as in Antiquity pointed to a secular acceleration of the mean angular motion of the Moon amongst the stars which was first noted by Edmund Halley in 1695. However, according to the solitary theoretical speculations of the philosopher Immanuel Kant (1754), oceanic tides retarded the rotation of the earth. When did these two lines of thought converge?


In 1786, Simon Laplace showed that conservative celestial mechanics could explain empirical observations on the acceleration of the lunar orbit. Some 70 years later, Adams (1853) and Delaunay (1859) demonstrated that Laplace's explanation was only half an explanation. The other half was buried in the work of Isaac Newton who thought that the apparent acceleration of the moon could be due to a change in the earth's rotation. Robert Mayer was the first to link the ideas that the moon acts on the earth and the earth on the moon by introducing the concept of angular momentum transfer in the earth-moon-system. He was followed by G.H. Darwin, the son of the famous biologist.

Today, the precise mechanism of energy dissipation is still an open question and, despite a better knowledge of time-scales and sophisticated mathematical modelling techniques, ocean tides are still not known with the desired accuracy.
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Science rarely evolves linearly. The path taken by evolution is far from a straight line and, to learn about science, we should focus on its detours! For instance, if we were to take a look at the genealogical tree describing our perceptions of tidal friction (a) over time, we could build the following series of aphorisms:

Empirism and theory can originate independently, each unaware of the other.

When, in 1695, Edmund Halley (1) (2) noted inconsistencies in how solar eclipses were interpreted (b), the rotation of the earth was paradigmatic for a constant flow of time. It thus seemed obvious to him to ascribe these inconsistencies to an unexplained peculiarity in the motion of the moon. This was also the explanation that Richard Dunthorne gave in 1748 when he first determined from observation the so-called 'secular acceleration' (c) of the lunar orbit: +1011/(century)2 (3).

In 1754, however, the young philosopher Immanuel Kant, who took great interest in scientific issues, reasoned on the basis of pure theory that the action of oceanic tides must slow down the earth's rotation (4) (5). His article appeared in a newspaper devoted in large part to advertisements and warrants amidst a windmill sales-ad and a want-ad for peacocks! Kant apparently did not take his ideas on tides very seriously as they have no mention in his "Allgemeine Naturgeschichte und Theorie des Himmels". In fact, they had no immediate impact. It was only in 1867 that his newspaper article was noticed in England and that his contribution was acknowledged retrospectively (6). No doubt, if it had been anyone but Kant, the article would have been forgotten. The sad truth is therefore that:

A discovery can be annihilated if it does not enter the scientific consciousness of its time.

At the time, observations on lunar motion were well explained by the prevalent scientific theory, celestial mechanics. In 1786, one of its most respected protagonists, Simon Laplace derived a theoretical acceleration value for the lunar orbit that agreed perfectly with the observed value (7).

Nothing is more deadly to an open problem in science than an apparent solution by an acknowledged expert.

Clearly, the matter was settled: "Roma locuta, causa finita" (d) and more than one generation had to elapse before the time was ripe for the accepted view to be revised. Adams in 1853 (8) and Delaunay in 1859 (9), working independently, calculated the lunar orbit to a higher level of approximation and showed that Laplace had literally found only half a truth. Half of the peculiarity in the motion of the moon could be interpreted by pure conservative celestial mechanics (e), the other half remained unexplained.

Progress can arise not only from direct revolutionary attacks but also from old facts interpreted anew.

Given that there were two separate lines of thought - the observations relating solely to the moon and Kant's ideas relating solely to the earth - what was needed was a catalyst - not even a theory - to link them. Somebody, somewhere, had to realise that the lunar inequality (apparent acceleration of the moon's motion) might be due not only to the ordinate (the length of the lunar orbit) but also to the abscissa (time). Time was not the uniform time of celestial mechanics equations but was given by the rotating earth. A constant motion measured with a non-uniform time-scale can appear accelerated or decelerated. This simple idea was a 180um. about-turn in Plato's cave and a move out of the shadows into reality.

Much time is often needed to find an 'obvious' truth. When found, its triviality is so embarrassing that people forget who found it.

In this instance, the oblivion is quite surprising as it was Newton's idea that an apparent acceleration of the moon could be due to a change in the rotation of the earth (f). The explanation may be that he based his idea on the contested notion of interplanetary vapors which he probably did not fully accept himself as it appears only in the second edition of his 'Principia' (10).

Julius Robert Mayer (11) was the first to describe both aspects of the earth-moon interaction in his book "Beitrage zur Dynamik des Himmels" (12) at a time when celestial mechanics had not yet been revised. While Kant had only seen the actio on the earth, Mayer also saw the reactio on the moon. Mayer's estimates for the rotational energy of the earth had a sound physical basis and were of the same orders of magnitude as those found today. He thought that, for reasons of energy, there had to be observational effects (a shortening of the length of the day) that would decrease the moment of inertia of the Earth and increase its angular velocity. Just think of a skater. This rotating "object" with shrinking extension (moment of inertia) increases its angular velocity for a constant angular momentum. To his mind, the effects passed unnoticed because they were too small to be measured or hidden.

Mayer was an outsider who was finally accepted by the scientific community not for his work on tidal friction but for the theorem of energy conservation. His theorem's application to the earth-moon-system was neglected because, after Adam's revision, in 1879/80, the astronomer G.H. Darwin produced a full detailed analytical account in the approved insider style (13) (14). Darwin was not fully aware of Mayer's view and, in his popular book on tides (15), mistakenly states that Mayer did not consider the reaction on the moon.

Unfortunately, Darwin's analytical treatment does not really further our understanding. The tidal force field of an external body such as the moon on the Earth diminishes the Earth's gravity along the line connecting the centres of these planets and strengthens it at the girdle perpendicular to the first. Darwin painted a picture of a deformable earth with two tidal bulges in the sea surface, one immediately underneath the moon and the other on the opposite side of the globe. This nice phenomenological description can be woven into a consistent theory but its core, i.e. the strength of the interaction between the earth's oceans and the moon (given by the amplitude and phase lag of the tidal bulges), remains a free parameter!

The richness of a specialist's book can camouflage the crux of the matter.

Not only are the oceans deformed (about 1 cm to 10 metres over 12 hours and 25 minutes) but also the earth (some decimetres). Darwin's description fitted precisely solid earth tides only and this might be why they were temporarily studied in preference to ocean tides. However, the tide phase lag - here of the amplitude of tidal height - had to be measured even for earth tides (g)! As long as observations of the moon's position were affected by both the changes in the lunar orbit and the apparent, more dominant, changes due to a non-uniform time-scale, one could not feel comfortable. This discomfort was dissipated when Cowell (16), Spencer Jones (17) and others obtained "unadulterated" pure values for the changes in the earth's rotation from observations of Mercury and the Sun. (The Sun here is just a 180 mirror for the movement of the earth). Because these values for changes in the earth's motion are unaffected by tidal friction, they provide a clean dynamic time-scale (h).

The 'buildings' of science defy the laws of architecture as often built from the roof down.

Recent history has witnessed two major advances that have furthered our knowledge of tidal friction. The first belongs to the realm of 'pure', empirical science. For John Wells (18) and others, growth rhythms in corals and mussels are produced by the solar day, the synodical (i) month and the tropical year. Corals from different periods are a means of measuring frequency and provide the ratios of astronomical periods for time-scales of several 100 million years. George Williams (19), in particular, detected tidal variations in Precambrian sediments, thus stretching the time-span of observations into the past. This means of measurement has indicated that the length of the day has changed over millions of years at approximately the same rate as in recent history. The second advance is a technological advance: the advent of large electronic computers.

It is only with current scientific and technological means that oceanic tides, which are geometricallv much more complex than solid earth tides and which are the effects of choice to explain tidal interaction since Kant's day, can be treated in any detail. The average torques between the oceans and the moon for the main partial tides (j) calculated with available mathematical models agree with observed values. It thus seems that we have now moved one step up the gradus ad parnassum (k).


Notes

This is a substantially edited and enlarged version of an article published in "Ocean Sciences : Their History and Relation to Man". Proceedings of the 4th Intl. Congress on the History of Oceanography (Edited by W. Lenz and M. Deacon). Hamburg, Sept 1987. Deutsche Hydrographische Zeitschrift (Erg.-H. Series B), Num. 22, 1990, p. 442.

(a) Tidal friction is the time-averaged global transfer of rotational energy and angular momentum by tides. It is caused by tidal forces and the instantaneous local effects of these forces are currents and heights.

(b) The place where solar eclipses are observed contains information on the rotational position of the earth.

(c) Secular: Lasting or going on for ages, occurring over an indefinitely long time.

(d) Rome has spoken, the issue is settled.

(e) Conservative celestial mechanics means that forces are derived from a potential, in this instance, the potential of mutual gravitational attraction. As soon as friction is implied, this assumption no longer holds and energy is no longer conserved.

(f) I am grateful to François Mignard (CERGA, Grasse) for this information.

(g) The observed phase lag is small and theorists today consider that the real value is even smaller, in other words, that earth tides are not important.

(h) A dynamic time-scale is a physical time-scale based on the laws of celestial mechanics (e). A time t enters into these laws, and motions and positions x(t) are predicted in dependence of t. It is thus possible to reverse the process and to deduce the corresponding t from an observed x.

(i) Synodical (or synodial) : pertaining to the conjunction of the planets.

(j) Total tides are decomposed mathematically into a sum of simple sine waves with special periods. The main partial tide is the lunar semi-diurnal one with a period of 12 hours and 25 minutes.

(k) Step to Parnassus: Title of a dictionary of Latin prosody used in schools to help in writing Latin verse. Parnassus is a mountain in Greece anciently sacred to the Muses.


References

1. Halley E. Emendationes ac Notae in vetustas Albatenii Observationes Astronomicas cum restitutione Tabularum Lunisolarium ejusdem Authoris. Philos Trans Roy Soc 17, 913, 1693.

2. Halley E. Some account of the ancient state of the city of Palmyra, with short remarks upon the inscriptions found there. Philos Trans Roy Soc 19, 160, 1695.

3. Dunthorne J. XIV. A Letter from the Rev. Mr. Richard Dunthorne to the Reverend Mr. Richard Mason F. R. S. and Keeper of the Woodwardian Museum at Cambridge concerning the Acceleration of the Moon. Philos Trans Roy Soc 46. 162, 1749.

4. Kant I. Untersuchung der Frage, ob die Erde in ihrer Umdrehung um die Achse, wodurch sie die Abwechselung des Tages und der Nacht hervorbringt, einige Veränderung seit den ersten Zeiten ihres Ursprungs erlitten habe und woraus man sich ihrer versichern könne, welche von der Konigl. Akademie der Wissenschaften zu Berlin zum Preise fur das jetzt laufende Jahr aufgegeben worden. Kants Werke, Bd. I, Vorkritische Schriften I, 1747-1756, S. 185, 1754.

5. Brosche P. Kant und die Gezeitenreibung. Die Sterne 53, 114, 1977.

6. Wackerbarth A D. On an astronomical presentiment of Immanuel Kant relative to the constancy of the earth's sidereal period of rotation on its axis. Monthly Notices Roy Astr Soc 27, 199, 1867.

7. Laplace PS de. Sur l'equation séculaire de la Lune. Mem Acad Roy Sci, p. 235, 1786.

8. Adams JC. On the secular variation of the Moon's mean motion. Monthly Notices Roy Astron Soc 14. 59, 1854.

9. Delaunay CE. Comptes Rendus Acad Sci Paris (Séance du 25/4/1859) 48, 817, 1859.

10. Newton J. Philosophia Naturalis Principia Mathematica. 2nd ed. Cambridge. 481, 1713.

11. Friedländer S. Julius Robert Mayer. Leipzig: Thomas, 1905.

12. Mayer R. Beiträge zur Dynamik des Himmels. Chap. 8. Heilbronn: Landherr, 1848.

13. Darwin GH. On the precession of a viscous spheroid and the remote history of the earth. Philos Trans Roy Soc 170 Part 2, 447, 1879.

14. Darwin GH. On the secular changes in the elements of the orbit of a satellite revolving about a tidally distorted planet. Philos Trans Roy Soc 171. 713, 1880.

15. Darwin GH. Ebbe und Flut (translated by A. Pockels, authorized 2nd German ed. from the 3rd English ed.) Leipzig [u. a.], p. 304, 1911.

16. Cowell PH. On the secular acceleration of the earth's orbital motion. Monthly Notices Roy Astr Soc 66. 3, 1905.

17. Spencer Jones H. The rotation of the earth, and the secular accelerations of the sun, moon and p!anets. Monthly Notices Roy Astr Soc 99. 541, 1939.

18. Wells JW. Coral growth and geochronometry. Nature 197, 948, 1963.

19. Williams GE. Precambrian length of day and the validity of tidal rhythmite paleotidal values. Geophys Res Lett 24, 421, 1997.

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