Some notes on tides, contributed by aoab314@emx.utexas.edu (Srinivas Bettadpur). Please send comments to him. --- The references I have are far too technical for all this. Part A is from my class notes, Part B can be found in "Geophysical Geodesy" by K. Lambeck "Tides of the planet Earth" by P. Melchior People in Astronomy could give you much better references. Let me know if this is too long/short (oh, yeah !). Welcome any comments. --------------------------------------- The answers are in three parts. In the first part, a simple 2-D case is considered as an example of mechanism of tidal deformation. In the second, a brief mention of treatment of tides in practice is given. In the third, an explanation of evolution of the Earth-Moon system is given. PART A : 2-D Example of Tidal Deformation Since gravitational attraction is a function of the distance between two masses, Lunar attraction on the Earth is not uniform. Some parts of the Earth are more strongly attracted to the moon than the others. This *differential* attraction gives rise to tides. The reference attraction is chosen as that acting at the center of the Earth, and the resulting *variation* from this reference is called the tidal attraction. Note that while the gross, orbital motion of an object is governed by the sum of forces acting at the center of mass (CM) of an object, its deformation would be governed by difference in forces between a reference point and the CM of the body. To see the nature of these tidal forces and the resulting deformation, consider a circular sheet of mass, with the moon in the same plane. If the points on the circle at the intersection of the line joining the center C of this circle and the moon are marked as N (for NEAR) and F (for FAR), the forces at these three points can be drawn as F C N ------- to Moon :------> :--------> :----------> (6) (8) (10) Since the deformation of the sheet is proportional to the *difference* in the forces at a point from those at the center, a picture of the relative accelerations of points F, C and N can be drawn as F C N <--: : :--> (-2) (0) (+2) This should show why, in the general case, we have tidal bulges at both the near and far sides from the moon. This same principle can be used to write the tidal attraction at different points along the circumference of this circular sheet. Y | P | / | / | / /________________ * O X M Ang(POX) = A , Ang(OMP) = e , OM = R , Re = Radius of circle Then, approximately (M = Mass of Moon) Fx = GM * Re / (R)^3 * 2 * cos (A) Fy = GM * Re / (R)^3 * sin (A) Draw this function from A=0 to A=360 (set rest of the multipliers equal to one) and you will see why a circular cross section deforms into an ellipse. PART B : Treatment of Tidal Fields in Practice The Earth being the messily complicated object that it is, the picture in Part A is nowhere near adequate from practical applications. First of all, note that a closer picture would be one where the point M goes around O in 27.? days, whereas the axes XY themselves spin around the the point O in 24 hours. Thus the tidal deformation of the circle is changing in both space and time, such that the tidal force acting on you is not the same as that on a person in Tibet, and further, both of you will be subject to different tidal accelerations at different times. This spatial and temporal aspect of the variations are captured in a position dependent function called the Tide Raising Potential (TRP), whose spatial derivatives give the tidal accelerations at a given point. As might be expected, this depends in a complicated way upon the relative Earth-Moon-Sun geometry. In the more precise work, it is usually assumed that the Earth does not instantaneously respond to the temporal variability of the tides. Further, the deformation in the solid earth is assumed to show the same spatial variability as the imposed tides. That is a bad assumption for oceans, which being much more fluid, respond in a spatially much more intricate way than the solid Earth. PART C : Long term evolution of the Earth-Moon system under tides The question on this topic generally refers to the gradual evolution of a two elastic bodies system into a state of tidal lock. Or, in other words, these debates start with "Why does the moon present the same face to the Earth all the time ?" There are many such systems in the solar system, the most obvious of which is the Pluto-Charon system. For the purposes of this discussion, I will define tidal lock to be a situation where *BOTH* the bodies present the same face to each other (as Earth does not, or half the people in the world would never have seen the moon). As mentioned in Part B, there is delay between the imposed tidal acceleration and the Earth deformation response. If the orbital period of the moon is different from the rotation period of the Earth, this means that the bulge due to the deformation does not lie under the line joining the Earth and the moon. In this case, since the rate of rotation is larger than the rate of revolution, the bulge gets ahead of the sub-lunar point on the Earth due to the delayed response. If the case were reversed, the bulge would trail behind. In either case, the phenomena is called a Tidal Lag, only the algebraic sign on the angle is shifted depending on whether it leads or lags. This has the net effect of causing a continuous transverse accleration on the moon, causing it to gain velocity and raise its orbital distance from the Earth. In reaction to delivering the kick to the moon and rasing its orbital angular momentum, the Earth experiences a torque that tends to slow down its rotation. In another example, the orbital periods of Phobos and Deimos are such in relation to Mars rotation period, that while one leads, the other lags. Thus one experiences "drag" while the other experiences "thrust". Thus Phobos and Deimos are said to "exchange orbital angular momentum through the medium of Mars". Of course, as the moon gets farther and farther away, the tidal bulge on the Earth and consequently, the kick to the Moon will weaken. Moreover, the deformation of the Earth in response to Lunar tides is not without dissipation of energy (which is what causes the tidal lag in the first place). Tidal Friction (as it is called) causes the Earth to continually lose kinetic energy of rotation as heat, and as a result, its rotation rate is slowing down. This is case where, if considered in isolation from all else, the system conserves angular momentum while losing energy. Since the Moon is receding due to the tidal kick, its orbital period is also slowing down. It is expected that the system will reach equilibrium when the Moon is just far enough and the Earth just slow enough that the tidal bulge always lies along the line joining the centers of the Earth and Moon. This situation is called Tidal Lock, and in this case, the terrestrial day, the lunar month and the Lunar day would all be equal. At present, only the Lunar month and the Lunar day are equal to each other, which is why the Moon presents the same face to the Earth always. In this picture, Lunar deformations are not commensurate in importance to that of the Earth, because the former is much more of a rigid body than the Earth.