The Recession of the Moon
and the Age of the Earth-Moon System
by Tim ThompsonCopyright © 2000
[Last Update: December 1999]
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ne of the common arguments made in
support of young-Earth creationism is that the dynamic age of the Earth-moon system (as
determined by the physics of the Earth-moon tidal interaction) is too young to support a
multi-billion year age for the system. In this article I will (a) review the basic physics of
gravity and tides, (b) review the history of theoretical models for Earth-moon tides, (c) review
the paleontological evidence relevant to the history of the Earth-moon system, and (d)
demonstrate that the combination of theory and observation refute the young-Earth creationist
arguments, with reference to specific young-Earth arguments and their specific failures. This is
intended as a review for readers not versed in physics and math, so the arguments are presented
as non-technically as possible. There are references to more technical work, for those who are
interested in following up any of the arguments presented here as accepted assertions.
While this article is intended as a refutation of yet another ill conceived young-Earth argument, the introductory reviews do not refer to creationism at all. Therefore, the article should work just as well as an introduction to the physics of the evolution of the Earth-moon system, even for those readers not interested in the issue of creation vs. evolution.
Although gravity has been known to exist since people knew they could fall, it was not until Isaac Newton came along that a mathematical description of gravity was forthcoming. It was Newton who showed that the force of gravity obeyed a simple algebraic equation, shown here as equation 1.
| Fg = Gm1m2 / R2 | equation 1 |
In equation 1, Fg is the gravitational force between two objects of mass m1 and m2 and R is the distance that separates the two masses. This equation is important because it is the fundamental equation for describing the force of gravity in Newtonian physics. It is, however, an idealization; it assumes the masses m1 and m2 are point masses, in that they have no physical size. But, of course, all real masses are not point masses, and therefore do not exactly obey Newton's equation. However, as an approximation the equation works very well for masses that are separated by distances that are very large compared to their physical size. For instance, in analyzing Earth's orbit around the sun, one needs to include the gravitational effect of the other planets, as expressed by equation 1, but one need not worry about the fact that they are not point masses, since the differential effect is not measurable.
A tide is what happens when the masses we see in equation 1 are not separated by distances that are large compared to their physical size. A tide is a "differential gravity", the result of the fact that extended bodies do not pull equally on all parts of each other, as equation 1 would imply. In figure 2, below, we see how the tidal force operates between Earth and the moon, where the red arrows show the relative pull of the moon's gravity on Earth.
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As figure 1 shows, the force is not constant over the distance between the moon and the various parts of the Earth. The moon, being rather closer to the near-side of Earth, pulls harder on it (where the red arrows are longer), while it pulls more lightly on the side of Earth that is farther away (where the red arrows are shorter). In physics, we call this kind of effect a "gradient", and it represents the differences in force applied at different points. The strength of that gradient is represented in equation 2 below.
| DF / DR = 2Gm1m2 / R3 | equation 2 |
In equation 2, DF / DR represents a change in the force (DF) with respect to a change in distance (DR). That variation in force, or tidal gradient, is what produces the distortion in the shape of both Earth and the moon, while the force seen in equation 1 is what keeps Earth and the moon in orbit around each other. As the red arrows in figure 1 imply, there is a "inward" pull on the poles of the Earth, towards the equator, which would tend to squeeze the planet. Squeeze a rubber ball that way, and you can see for yourself that the inward squeeze causes an outward squish at the "equator" of the ball. Add to that the effect that the moon pulls harder on those parts of the Earth that are closer to it, and the result is that the Earth is squished, bulging towards the moon, and away from the moon. The effect is illustrated below, in figure 2.
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The illustration in figure 2 above shows the solid earth (green) and the oceans (blue) in schematic form. The "solid" Earth really isn't all that solid, and it does bend under the moon's tidal stress, but the water oceans are clearly far less "solid" than the rest of the Earth, and so they will be much more deformed by the moon's tidal squeeze. Hence, the bulge is mostly ocean, and only a little bit ground. The gaseous atmosphere is tidally squished too, but it does not figure much in the total system, and I will ignore it here (a detailed study of tides should not ignore atmospheric tides, I only do it here because it does not figure prominently in this particular discussion).
In a static system such as in figure 2, the mostly ocean bulge points right at the moon. But the real system is not static; the moon goes around the Earth, but the Earth spins on its daily axis much faster than that. So the spin of the Earth pulls the bulge out in front of the moon. The result of this is illustrated below in figure 3, and we are now ready to understand the greater mysteries of tides and the Earth-moon system.
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The ocean bulge is pulled in front of the moon by Earth's spin; since the ocean is gravitationally stuck to the Earth, it has to go where the Earth goes. But it can't go too far, because it is pulled back by the moon. The result, illustrated in figure 3, is that the ocean bulge is in equilibrium, remaining essentially fixed with respect to the Earth and moon, while the solid Earth spins under the ocean. The ocean is gravitationally bound to the Earth, but it is still fluid, and not stuck to the Earth the way a rock or a mountain is. There is an interface, namely the ocean bottom, where the water and the Earth are free to move with respect to each other. That interface, like any other real physical interface, is not totally frictionless, and that too is illustrated in figure 3 by the small caption that reads "Friction force". But in this case, "friction" includes all of the ways that the ocean and the Earth impede each other. The ocean runs into the continents and has to wash around them (so how they are distributed around the Earth makes a difference).
Since the Earth is trying to spin forward, but the ocean is held back by the moon, the Earth winds up trying to move through the oceans. Just as you can feel the resistance if you try to walk through water, so the Earth feels the resistance trying to move through the water of the oceans, and that resistance transfers energy from the Earth (causing its spin rate to slow), and to the oceans (sloshing them around and heating them up). But the Earth-ocean system also exerts a torque (a "twisting" force) on the moon, because the line along the arrow labeled "B" in figure 3 is at an angle to the line that connects the center of the Earth to the center of the moon. As a result of that torque, the Earth also transfers energy (causing its spin rate to slow) through the ocean bulge, and gravity, to the moon (causing it to speed forward in its orbit, and therefore move farther away from the Earth).
At this point we are ready to understand two important observations. First, the high and low ocean tides we all know about, are caused by the Earth moving through the high and low parts of the ocean, seen in either figure 2 or figure 3. Since we are on the Earth, it looks to us, from our frame of reference, as if the ocean is doing the moving, but however you want to look at it, the result is the same. The Earth and its oceans move with respect to each other, because of the pull of the moon, and we see that motion as what we call high & low tides. Second, the moon is slowly drifting away from the Earth. That means that the moon is not where it has always been with respect to the Earth; the Earth-moon system clearly must have evolved over time. Can we figure out how the Earth-moon system has evolved? I will review the answer to that question in the next section.
The description I have given so far is necessarily general, and leaves out a lot of details. But there is a lot of physics and mathematics hidden behind that layman's facade, and it has to be dealt with in order to understand the real nature of the tidal relationship between Earth and the moon. I will not develop any of that mathematics here. I will concentrate instead on reviewing the history of the scientific efforts to understand the Earth-moon tidal system. Along the way I will make reference to numerous original sources, books, journal papers and the like. Those sources will provide the reader with all of the mathematical and/or physical details one could wish to see. Readers eager to know more are encouraged to consult those sources.
It was not possible to study tides in any quantitative, physical or mathematical sense, until Isaac Newton essentially invented the science of mechanics, with the publication of his Philosophiae Naturalis Principia Mathematica in 1687. Since then a number of eminent scientists have struggled with the problem of tides, including Edmond Halley, Pierre Laplace, and William Thomson (Lord Kelvin). But it was the celebrated English mathematician and geophysicst George Howard Darwin Who really attacked the problem of Earth's rotation and the Earth-moon system with analytical zeal (G.H.Darwin; 1877, 1879, 1880; with an ironic twist on the creation-evolution issue, he was the son of Charles Darwin, the founding father of biological evolution). Darwin considered ocean tides, and made some significant advances there, but he concentrated mostly on solid body tides in a homogenous Earth. Today we know that ocean tides are much more important than solid body tides. Thomson was the first to show that tides transferred angular momentum from Earth to the moon, and that transfer of momentum is what causes the moon to recede from Earth. But Darwin was the first to cast the problem into analytical detail, setting the stage for explorations in the early 20th century.
Through most of the first couple of decades of the 20th century, the chief investigator of this problem was Harold Jeffreys. Jefferies published a number of papers during the early 1900's, and extensively summarized the then current state of affairs in the first edition of his landmark book The Earth (Jefferys, 1924). In that book (chapter XIV, Tidal Friction, pp 205-237 of the 1st edition) Jeffreys uses an estimate of tidal friction to derive a maximum age for the Earth-moon system of 4 billion years. That estimated age remained unchanged in later editions at least through 1952. The main problem that vexed Jeffreys, and later researchers, was their inability to fully describe ocean tides analytically, or even to know the numerical values of oceanic tidal friction. But it is quite clear that by then, about 44 years after Darwin's work, Jeffreys knew that oceanic tides were more important than solid body tides. The search for oceanic tidal response functions was on.
Later researchers came to the conclusion that Jeffreys had rather severely underestimated the true numerical value for oceanic tidal dissipation, and had therefore overestimated the age of the Earth-moon system. Although they do not offer an age, Munk & McDonald (1960) said that Jeffreys had the oceanic dissipation wrong by a factor of 100. It soon became apparent that the pendulum had swung the other way, and that there was a fundamental problem. Slichter (1963) reanalyzed the Earth-moon torque by devising a new way to use the entire ellipsoid of Earth rather than treating it as a series of approximations. He decided that, depending on the specifics of the model, the moon would have started out very close to Earth anywhere from 1.4 billion to 2.3 billion years ago, rather than 4.5 billion years ago. Slichter remarked that if "for some unknown reason" the tidal torque was much less in the past than in the present (where "present" means roughly the last 100 million years), this would solve the problem. But he could not supply the reason, and concluded his paper by saying that the time scale of the Earth-moon system "still presents a major problem"; I call this "Slichter's dilemma".
Despite the effort expended on the problem over the years, a truly complete mathematical method for handling the tidal dissipation had not yet been forthcoming. That problem was redefined by Peter Goldreich. Goldreich (1966) extended the realm of the problem well beyond the limits that Slichter had set, as Goldreich had included solar tides and precessional torques. However, the age of the system being dependent on observed quantities, and arbitrary factors in the model, Goldreich did not approach the question of age.
The years that followed saw the rise of plate tectonics and a major shift in geophysical thinking because of it. The mobility of the drifting continents is a matter of major import, for by this time it was well realized that tidal dissipation in shallow seas dominated the interaction between Earth and the moon. Kurt Lambeck was a major player in the tidal game at that time, authoring several papers. His study of the variable rotation of Earth (Lambeck, 1980) remains the most extensive such study ever done. Lambeck noted that after the struggles of Slichter, Goldreich, and others, the observed and modeled values for tidal dissipation were finally in agreement (Lambeck, 1980, page 286). However, this still left a time scale problem. According to Lambeck, " ... unless the present estimates for the accelerations are vastly in error, only a variable energy sink can solve the time-scale problem and the only energy sink that can vary significantly with time is the ocean." (Lambeck, 1980, page 288). In section 11.4, "Paleorotation and the lunar orbit", Lambeck explicitly points out that paleontological evidence shows a much slower lunar acceleration in the past, and that this is compatible with the models for continental spreading from Pangea (Lambeck, 1980, pages 388-394). It is important to remember that by 1980, Lambeck had pointed out the essential solution to Slichter's dilemma - moving continents have a strong effect on tidal dissipation in shallow seas, which in turn dominate the tidal relationship between Earth and the moon.
While Lambeck pointed the way, Kirk Hansen (1982) got on the right road. Hansen's models assumed an Earth with one single continent, placed at the pole for one set of models, and at the equator for another (the location is chosen to simplify the computations, but the basic idea of a one-continent Earth may not be all that bad; Piper, 1982 suggests that our current multi-continent Earth is actually abnormal, and that one continent is the norm). His continent doesn't move around as a model of plate tectonics would do it, but Hansen was the first to make a fully integrated model for oceanic tidal dissipation directly linked to the evolution of the lunar orbit. As Hansen says, his results are in "sharp contrast" with earlier models, putting the moon at quite a comfortable distance from Earth 4.5 billion years ago.
Hansen had already all but eliminated Slichter's dilemma with his integrated model of continents and tides. Kagan & Maslova (1994) treat the oceanic tidal dissipation with fully mobile and arbitrary continents. Like Hansen, their models show time scales that are not a problem for matching the radiometric age of Earth with the dynamic age of the Earth-moon system. Kagan & Maslova (1994), Kagan (1997), and Ray, Bills & Chao (1999) have continued the study in even more detail, with plate tectonics fully integrated into their models of Earth-moon tidal evolution. Touma & Wisdom (1994) do the calculation in a fully integrated multi-planet chaotically evolving solar system.
Although it may seem to the casual reader that the Earth-moon system is fairly simple (after all, it's just Earth and the moon), this is only an illusion. In fact, it is frightfully complicated, and it has taken over 100 years for physicists to generate the mathematical tools, and physical models, necessary to understand the problem. Slichter's dilemma, as I called it, was a theoretical one. He lacked the mathematical tools, and the observational knowledge, to solve his problem. But those who came after got the job done. Slichter's dilemma is today, essentially a solved problem. Once all of the details are included in the physical models of the Earth-moon system, we can see that there is no fundamental conflict between the basic physics and an evolutionary time scale for the Earth-moon system.
I have thus far illuminated the theory, the construction of the mathematical methods used to understand the details of the Earth-moon tidal interaction. But theory and observation, theory and evidence go hand in hand in the empirical sciences, and this is no exception. Tides, and the Earth's rotation leave behind tell-tale clues about Earth's past. So, when Lambeck (1980) or Stacey (1977) say that tidal dissipation must have been lower in the past, that's neither an idle guess, nor a knee-jerk reaction. It is an attitude consistent the evidence.
The first critical observation is How fast is the moon moving away from Earth now? This linear motion away from Earth had to be estimated from the observed angular acceleration, or it had to be calculated from theory, the former being preferred, since it is an observed quantity. Stacey uses an astronomical estimate of 5.6 cm/year (Stacey, 1977, page 99). Lambeck gives 4.5 cm/year (Lambeck, 1980, page 298). It's an important number, because it reveals the true strength of tidal dissipation. But today the number can be observed directly, as a result of three-corner mirrors left behind by Apollo astronauts. Lunar laser ranging establishes the current rate of retreat of the moon from Earth at 3.82±0.07 cm/year (Dickey et al., 1994).
But what about the past rate of retreat? Paleontological data directly reveals the periodicity of the tides, from which one can derive what the rate of retreat would be to match the frequency. It is also a non-trivial point that it proves the moon was physically there. After all, if your theory implies that the moon was not there at some time in the past, but your observed tidal evidence says that it was there in the past, then it's pretty clear that the theory, and not the observation, needs to be adjusted.
This paleontological evidence comes in the form of tidal rhythmites, also known as tidally laminated sediments. Rhythmites have been subjected to intense scrutiny over the last decade or so, and have returned strong results. Williams (1990) reports that 650 million years ago, the lunar rate of retreat was 1.95±0.29 cm/year, and that over the period from 2.5 billion to 650 million years ago, the mean recession rate was 1.27 cm/year. Williams reanalyzed the same data set later (Williams, 1997), showing a mean recession rate of 2.16 cm/year in the period between now and 650 million years ago. That these kinds of data are reliable is demonstrated by Archer (1996). There is also a very good review of the earlier paleontological evidence by Lambeck (1980, chapter 11, paleorotation)
As you can see, the paleontological evidence indicates that moon today is retreating from Earth anomalously rapidly. This is exactly as expected from the theoretical models that I have already referenced. The combination of consistent results from both theoretical models and paleontological evidence presents a pretty strong picture of the tidal evolution of the Earth-moon system. Bills & Ray (1999) give a good review of the current status of this harmony. Without realizing it, they have also explained well why the creationist arguments are unacceptable.
I don't know who first brought up the age of the Earth-moon system as a pro-creationist argument. But the first example I am aware of is Barnes (1982, 1984). Barnes says, "It has been known for 25 years that the earth-moon system cannot be that old", and assuring us that "Celestial mechanics proves that the moon cannot be as old as 4.5 billion years", goes on to quote the last sentence from Slichter's (1963) paper, "The time scale of the earth-moon system still presents a major problem" (in fact, Barnes should not have capitalized the "T" since this is a sentence fragment, not a full sentence, but in this case the oversight is inconsequential). It is noteworthy that Barnes is happy to quote a paper already 19 years old in 1982, and 21 years old in 1984, yet despite a research physics background, declines to bother researching anything post-Slichter. If he had, he would have found Lambeck (1980), a major work which clearly indicated the real nature of Slichter's dilemma (or even Stacey, 1977, which already showed the conflict between Slichter's theoretical dilemma and the paleontological evidence available at the time). And, of course, Kirk Hansen's 1982 paper predates Barnes' 1984 reiteration by two years, yet is ignored despite being recognized even then as a major step forward. Barnes shows the same kind of sloppy and lazy approach to "research" that permeates young-Earth creationism, although his is a particularly egregious case (as it also was for his arguments concerning Earth's magnetic field).
DeYoung (1992) offers his own model. Actually, he offers an equation. DeYoung asserts that the rate of change of the lunar distance as a function of time must be proportional to the inverse 6th power of the lunar distance (presumably because the lunar tidal amplitude is proportional to the inverse cube of the distance, and the tidal acceleration is proportional to the square of the amplitude, though DeYoung does not say this). He then runs some numbers in the equation, and concludes with remarkable poise that he has demonstrated a maximum possible tidal age for the Earth-moon system of 1.4 billion years. The same calculation can be found in Stacey (1977), with reference to more precise versions. They all get about the same answer as DeYoung, and there is no doubt but that what DeYoung did he did right. However, if you do the "wrong" problem, you may not get the "right" answer! As Stacey pointed out (Stacey, 1977, pages 102-103) it makes more sense to assume that the oceanic tidal dissipation was smaller in the past, which would have the effect of making the calculation that of a minimum age, as opposed to the maximum age proposed by DeYoung. But, of course, we are comparing DeYoung (1992) with Stacey (1977), a gap of 15 years (it's nice to see that DeYoung, like Barnes, is keeping up with the tempo of current research). That gap includes Lambeck (1980) and Hansen (1982) (wherein it was demonstrated that a 4.5 billion years age was compatible). Granted that DeYoung (1992) wrote before the 1994 papers of Kagan & Maslova or Touma & Wisdom, which are directly contradictory to his results. However, Hansen's (1980) results also directly contradict DeYoung, but come 12 years before. This observation does not inspire confidence in the value of DeYoung's one-equation model for the evolution of the lunar orbit. But, as made clear by Bills & Ray (1999), the constant of proportionality, which Stacey suggests is not constant, is in fact a ratio of factors that represent dissipation, and deformation. It is clear that neither of these can be constant, and once that is understood, we can see clearly that DeYoung simply did the wrong thing right, and curiously wound up with a correct form of the wrong answer.
Walter Brown (Brown, 1995) presents essentially the same model as DeYoung. I have seen only the online technical note, but not the printed book. Unfortunate, for the equations do not appear on the webpage, despite being referenced as if they were there. However, Brown does offer the quick-Basic source code for his program that calculates the minimum age of the Earth-moon system. His equations are there, and he seems to be using the inverse 5.5 power of the radius rather than the inverse 6th power used by DeYoung (Brown's usage here is consistent with the equation given by Bills & Ray, 1999; whether one chooses to use the inverse 6 or inverse 5.5 power seems an issue of model dependence). Otherwise, Brown's approach appears to be quite the same as DeYoung's, and subject to exactly the same criticism. He ignores the time variability of dissipation and deformation. It is perhaps humorously ironic that both DeYoung and Brown fail, because they are implicitly making an improper uniformitarian assumption (the constancy of dissipation and deformation), which evolutionists have learned to avoid.
I don't know if there are other, "authoritative" creationist sources for the "speedy moon" argument. But if there are, it is unlikely that their arguments presented differ much from those seen here. I spent quite a bit more time reviewing the actual science of the Earth-moon tidal interaction because once it is well developed, the flaw in the creationist arguments becomes so obvious that it hardly seems necessary to refute them. The most remarkable aspect of this, I think, is the somebody like DeYoung, who certainly has legitimate qualifications (a PhD in physics from Iowa State University), would offer up such a one-equation model as if it was actually definitive. That kind of thing works as a "back-of-the-envelope" calculation, to get the order of magnitude, or a first approximation for the right answer, but it should have been clear to an unbiased observer that it could never be a legitimate realistic model. It is also of considerable interest that both DeYoung and Brown published their refutations of evolution only after evolution had already refuted their refutations! Barnes didn't do all that much better, having overlooked Hansen (1982) for two years. My own conclusion is that my intuitive expectations have been fulfilled, and creation "science" has lived up to its reputation of being either pre-falsified, or easy to falsify once the argument is evident.
As for the real science, remember that science is not a static pursuit, and the Earth-moon tidal evolution is not an entirely solved system. There is a lot that we know, and we do know a lot more than we did even 20 years ago. But even if we don't know everything, there are still some arguments which we can definitely rule out. A 10,000 year age (or anything like it) definitely falls in that category, and can be ruled out both by theory and practice.
Archer, A.W.
Reliability of lunar orbital periods extracted from ancient cyclic tidal rhythmites
Earth and Planetary Science
Letters 141(1-4): 1-10, June 1996
Barnes, Thomas G.
Young Age for the Moon and Earth
Institute for Creation Research,
Impact 110, August 1982
Barnes, Thomas G.
Earth's young magnetic age: An answer to Dalrymple
Creation Research Society Quarterly 21:
109-113, December 1984
Bills, B.G. & R.D. Ray Lunar Orbital Evolution: A Synthesis of Recent Results
Geophysical Research Letters 26(19): 3045-3048, October
1, 1999
Brown, Walter
In the Beginning: Compelling Evidence for Creation and the Flood
Center for Scientific Creation, 1995
Darwin, G.H.
On the influence of geological changes on the Earth's axis of rotation
Philosophical Transactions of the Royal Society of
London, 167, 271, 1877
Darwin, G.H.
On the precession of a viscous spheroid and on the remote history of the earth
Philosophical Transactions of the Royal Society of
London, 170, 447-530, 1879
Darwin, G.H.
On the secular change of the orbit of a satellite revolving about a tidally distorted
planet
Philosophical Transactions of the Royal Society of
London, 171, 713-891, 1880
DeYoung, Don
Is the Moon Really Old?
Creation Ex Nihilo 14(4): 43,
September-November, 1992
Dickey, J.O. et al.
Lunar laser Ranging: A Continuing Legacy of the Apollo Program
Science 265: 482-490, July 22, 1994
Goldreich, Peter
History of the Lunar orbit
Reviews of Geophysics 4(4): 411-439, November 1966
Hansen, Kirk S.
Secular Effects of Oceanic Tidal Dissipation on the Moon's Orbit and the Earth's
Rotation
Reviews of Geophysics and Space Physics 20(3): 457-480, August 1982
(journal title has since then changed to Reviews of
Geophysics)
Jeffreys, Harold
The Earth
Cambridge University Press, 1st edition, 1924 (multiple
expanded editions since then; 4th edition 1959)
Kagan, B.A. & Maslova, N.B.
A stochastic model of the Earth-moon tidal evolution accounting for
cyclic variations of resonant properties of the ocean: An asymptotic solution
Earth, Moon and Planets 66: 173-188,
1994
Kagan, B.A.
Earth-Moon tidal evolution: model results and observational evidence
Progress in Oceanography
40(1-4): 109-124, 1997
Lambeck, Kurt
The Earth's Variable Rotation - Geophysical causes and consequences
Cambridge University Press, 1980
Munk, W.H. & McDonald, G.J.F.
The Rotation of the Earth - A Geophysical Discussion
Cambridge University Press, 1960 (reprinted with
corrections 1975)
Ray R.D., Bills B.G., Chao B.F.
Lunar and solar torques on the oceanic tides
Journal of Geophysical Research - Solid
Earth 104(B8): 17653-17659, August 10, 1999
Slichter, Louis B.
Secular Effects of Tidal Friction upon the Earth's Rotation
Journal of Geophysical Research 68(14), July 15, 1963
(JGR has since broken into 5 separate journals
published by the American Geophysical Union)
Stacey, Frank D.
Physics of the Earth
John Wiley & Sons, 1977 (2nd edition)
Touma, Jihad & Wisdom, Jack
Evolution of the Earth-moon system
Astronomical Journal 108(5):
1943-1961, November 1994
Williams, G.E.
Tidal Rhythmites - Key to the History of the Earth's Rotation and the Moon's Orbit
Journal of the Physics of the Earth 38(6): 475-491, 1990
Williams, G.E.
Precambrian Length of Day and the Validity of Tidal Rhythmite paleotidal Values
Geophysical Research Letters 24(4): 421-424, February 15,
1997
I am no graphic artist, and readily admit lifting the diagrams used from the following sources.
Figures 1 & 2 are both borrowed from Lunar Tides, a chapter in the Astronomy 161 web syllabus, from the Department of Physics & Astronomy, at the University of Tennessee, Knoxville. They are used with permission of the Artist, Mike Guidry.
Figure 3 comes from the 1989 edition of "Introduction to the World's Oceans" by Alyn & Allison Duxbury (the book is now in its 6th edition, as of July 1999).
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