Abstract
These notes, from a course I gave at a CNRS school in Cargèse in March 2009, have the aim of quickly letting non-experts pick up a physical intuition and a sense of orders of magnitude in the theory of tides. ‘Tides’ include ocean tides as well as tidal effects in astronomy. The theory is illustrated by a variety of back-of-the-envelope problems, some of them surprising, all of them simple.
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Notes
- 1.
Except here.
- 2.
Mathematically we have rediscovered the Green’s function for the Laplacian in ℝn.
- 3.
The zero-gravity conclusion is equally valid for the inside of a uniform ellipsoidal shell; by ‘shell’ is meant a region bounded between similar concentric (not confocal) ellipsoids.
- 4.
Outside an ellipsoidal shell the result is more complicated.
- 5.
A weak hypothesis, only about the line of attraction passing through O, nothing about the size of attraction.
- 6.
Principia book I, proposition LXXIV.
- 7.
Traditionally they are defined as solutions to a certain ODE that crops up when we try to separate ∇2 u=0 in spherical polar coordinates. The definition chosen here is equivalent to the traditional one, but it is better motivated and easier to use for us.
- 8.
It goes without saying that the observed tides of the real ocean are enormously complicated and do not resemble either of these pictures. But we are saying that, if we take the simplest model, of a spherical Earth covered by a sheet of ideal fluid, subjected to the dynamics of the Earth and the Moon, then the picture is (PIC22 ▶) rather than (PIC21 ▶).
- 9.
Undoing an earlier trigonometric transformation of Section 1.4.2.
- 10.
Etymology: syzygy < Greek \(\upsigma\acute{\upsilon}\upzeta\upsilon\upgamma\mbox{o}\upvarsigma\) (spouse) < \(\upzeta\upsilon\upgamma\acute{\mbox{o}}\upvarsigma\) (yoke), cf. conjugate < Latin jugum.
- 11.
- 12.
As does the human society, which insists that the Moon and the Sun rise in the east and set in the west. Actually most of this motion is caused by us spinning from west to east.
- 13.
In order to have the in-phase picture (PIC48 ▶), we would require a deeper ocean H>20 km.
- 14.
A hump spreads and propagates as a ring and meets as a new hump on the antipodes.
- 15.
Rayleigh wave (1885).
- 16.
More precisely, the revolution of the primary and the secondary around each other.
- 17.
\(\ddot{\psi} < 0\) because the friction on the sea floor by the ‘tidally held’ ocean slows down the secondary’s rotation.
- 18.
Lest the astronomers complain: anomalistic, draconic, sidereal, synodic…. For our approximate purposes here it does not matter which, they are all a little under 30 days.
- 19.
Julius Caesar IV. iii. 218–219 may come to some people’s mind.
- 20.
And since simultaneously the Earth is spinning slower, we terrestrials have the impression that the Moon is orbiting all the faster. Halley was the first to notice this (1695).
- 21.
He wrote this paper while serving on the Russian front in WWI, a year before he died. The other paper he wrote in the same year supplied a quantum explanation of the Stark effect.
- 22.
A rigorous calculation using general relativity yields the same expression 2GM/c 2 for D Sch.
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Tokieda, T. (2013). Tides: A Tutorial. In: Souchay, J., Mathis, S., Tokieda, T. (eds) Tides in Astronomy and Astrophysics. Lecture Notes in Physics, vol 861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32961-6_1
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