The Rhythmic Movements Of The Earth's Crust
( Originally Published 1915 )
DEVIATIONS OF THE VERTICAL
IN our study of the Earth's movements, we have seen how great is the complexity of the motion of any given point upon its surface; we have also demonstrated and sought to explain the continuous displacement of the Earth's poles. But, in all this account, the Earth has been assumed to be rigid, and to retain always the flattened ellipsoidal form imposed upon it by the combined laws of universal attraction and centrifugal force.
But perhaps the Earth is not really rigid. The crust may not be truly undeformable. It is possible that it suffers deformation under certain influences. We have to inquire whether this is so and under what conditions such non-rigidity becomes manifest, also its extent and importance.
In the course of this chapter, we shall find that the crust is in a state of perpetual movement and incessant vicissitude which still further emphasises the life, figuratively speaking, of the Earth.
The illustrious English physicist Lord Kelvin was the first to suggest the question whether the Earth is an undeformable solid, or if, on the contrary, it is an elastic body whose form is incessantly modified by exterior causes, in particular by the combined periodical and variable attractions of the Moon and the Sun. The problem resolves itself into finding proof of the elasticity of the terrestrial crust and the measurement of such elasticity.
Any body whatever, that is free to move at the Earth's surface, for example, the heavy ball of a plumb line, is always subjected to the attractive forces exercised upon it by the Moon and the Sun. The prolongation of the plumb line should therefore describe some kind of curve on the ground beneath it. If the Earth were rigorously rigid and undeformable, it would not change its form under the action of these attractive forces, the only effect of which is to impress upon the Earth the movements of rotation, revolution, precession, nutation, etc., which have been described in detail in Chapter IV.
Assuming the Earth to be strictly rigid, what would be the value of this luni-solar attraction? At first sight, it would seem a large one. The Sun has a mass about 325,000 times greater than the Earth and is at a distance from the latter equal to 23,400 terrestrial radii. If we evaluate the attractive force strictly according to Newton's law, viz., as proportional to the product of the masses and in inverse ratio to the square of the distances, a result is obtained which is about 1300 times less than that of gravity [at the Earth's surface.—Ed.]. Consequently the luni-solar attraction is sufficient to produce an apparent diminution of the weight of bodies here equal to 1/1300 part of their real weight.
But it must not be forgotten that the Earth, under the action of the solar attraction, executes its orbital movement. Now it is a fundamental principle in mechanics that a force already obeyed does not enter into play except as regards the effect already produced. A heavy body, suspended at the Earth's surface, and which the solar attraction tends to draw aside from the vertical, is already moving with the whole Earth under the influence of that attraction. There would consequently only remain, as an effective deviating force, the difference between the attractive force at the surface and that at the centre of the Earth. The result so obtained is, for the Sun, nearly 20,000 times less than that above given and is equivalent to only the 1/26,000,000 part of the force of gravity [at the Earth's surface.—Ed.].
The small mass of the Moon is largely compensated, from the point of view of the extent of attractive force it exercises on a body placed at the surface of our globe, by its much greater proximity; the Moon's centre is only distant sixty terrestrial radii from that of the Earth. On applying to our satellite the same reasoning and the same calculation that we have already done in the case of the Sun, we obtain the result that the perturbing effect of the lunar attraction produces a diminution of gravity [at the Earth's surface.—Ed.] of about 1/12,000,000 part. As the arc corresponding to an angle of a second is about 1/200,000 it is evident that the deviation from the vertical due to the influence of the Moon attains about 1/60 of a second.
We are indebted to Victor Puiseux for the complete analysis of this perturbing action. At a later date Gaillot put it into a simplified form and Radau made a more elementary calculation for the case where the Moon is in the plane of the equator. The astronomer Gaillot has traced the theoretical curves which the prolongation of a plumb line should describe on a horizontal sheet, under the influence of the lunar attraction, the Earth being assumed absolutely rigid. These curves are shown in the accompanying diagrams (Figs. 14, 15, 16, 17). It will be observed that they differ in accordance with the Moon's declination, or in other words, with its angular distance from the equator. When these results were known, and the minuteness of the quantity to be measured, in order to prove the diurnal variations of the vertical, realised, many physicists gave up hope of achieving it. But others attempted to overcome the difficulties of the experiment. In 1873, Zöllner tried, for the first time, a horizontal pendulum, to which we will return later in fuller detail, and which had extreme sensitiveness; in 1874, Bouquet de la Grye used a pendulum, connected with an amplifying balance, at Campbell Isle where he had gone to observe the transit of Venus; and in 1878, Lord Kelvin made use of a long pendulum the deviations of which were multiplied by means of a small rotating mirror. In 1879, G. and H. Darwin perfected this apparatus by immersing it in a liquid bath to preserve it from disturbing effects; in 1881, d'Abbadie installed in his observatory at Hendaye, an auto-collimating telescope directed perpendicularly downward on a bath of mercury placed at the bottom of a deep shaft ; the variations from coincidence of a reticle at the focus and its reflected image should be double the variations of the vertical. In 1883, Professor C. Wolf, of the Sorbonne, set up an analogous apparatus, but a horizontal one, in the vaults of the Paris Observatory ; and finally, in 1890, in the same place the mining engineer Léon and I attempted to arrange a very sensitive instrument, with communicating liquid baths, the differences of level of which were observed by interference fringes in yellow light. In spite of the extreme precision of the method employed, our apparatus gave no more clearly affirmative results than those of our predecessors.
The phenomenon to be measured is extremely small. A pendulum loo metres [330 ft.] long would be very difficult to make, and especially to set up and maintain in the necessary conditions of stability and freedom from disturbing effects. Yet even with such a pendulum the deviation in question would be only about 1/100 of a millimetre [3/10,000 in.]!
There is, however, another cause for the lack of success, and this is to be sought for in the elasticity of the Earth.
The mathematical considerations which serve as the base of the preceding experiments all depend upon the hypothesis that the earth is rigid and undeformable; if the Earth has sufficient elasticity to be susceptible to deformation under the influence of luni-solar section, the whole is changed. The entire Earth will then behave similarly to what we already know occurs in the case of the free surface of the oceans, in other words the litho-sphere or solid part will exhibit the phenomenon of tides just as the seas do under the influence of the same forces.
These deformations, which the solid part of the globe suffer, are of two quite distinct kinds; one only affects the superficial layers of the crust while the other acts on the whole body of the Earth. The first is characterised by an apparent deviation of the vertical with respect to the ground; in reality, as the deformations affect the superficial layers of the Earth, it is the ground which suffers displacement relatively to the vertical, which remains fixed. Consequently the deviations are only apparent. The principal cause of these apparent deviations is to be sought in the heating of the surface layers of the Earth by the solar rays. These rays warm the terrestrial globe just as the spirit lamp heats the copper ball in the classical experiment of Gravesand's ring. Since the surface rocks and layers have but slight conductivity for heat, only that part of the Earth turned towards the Sun is affected by the heating action and so only this part is expanded and hence deformed; the antipodes of these regions are not reached by the solar warmth until twelve hours later. For the same reason, viz., the feeble thermal conductivity of the soil, the movements of deformation thus produced are transmitted with difficulty in a downward direction and their amplitudes decrease very rapidly as we penetrate below the surface of the ground. In the Astrophysical Institute of Potsdam, under the direction of Professor Helmert, the apparent oscillation of the vertical has been found to have, at the bottom of a shaft 25 metres [83 ft.] in depth, only 1/8 of its extent at the surface level of the ground.
The heating caused by the solar rays being the principal cause of these superficial deformations, which produce an apparent oscillation of the direction of the vertical, it follows that such oscillations should have an essentially diurnal period; furthermore there should be another period, an annual one, due to the greater or less obliquity of the solar rays, caused by the variation in the Sun's declination according as it is above or below the celestial equator. This period is superposed on the first one and the actual resulting period is a combination of the two.
The second kind of deviation of the vertical is a true one and not only an apparent one; its cause is to be sought in the attractive forces exerted by the Sun and the Moon on the matter which constitutes our Earth.
If the Earth were perfectly rigid, absolutely undeformable, and totally devoid of elasticity, the luni-solar attraction could not produce any possible deformation of it, and in this case the oscillations of the vertical under the influence of these forces could be calculated as previously explained. If the Earth, in its entirety, were perfectly fluid, that is to say, if it behaved as a perfect, and not a viscous, liquid, the exterior surface would have a regular form, which would change continually under the influence of the luni-solar attraction. In these circumstances it would be impossible to prove the slightest change in the vertical, since by definition the terrestrial surface, the fundamental surface, of elevation, would always be normal to the direction of the plumb line. Consequently the terrestrial tides, the deformations, while attaining the greatest amplitude theoretically possible, would not be demonstrable, for lack of reference points, just in the same way as the oceanic tide cannot be appreciated by a navigator in the open sea, out of sight of land, the sea being assumed to be too deep for precise soundings, which. would otherwise prove differences in the depth of the water, to be taken.
But in reality the terrestrial globe is very far from being a perfect fluid. Without being absolutely rigid, it has a considerable degree of hardness. The molten material constituting the internal magma is subjected to such pressures that the state it exists in is hardly conceivable to the mind, which in the attempt to realise it is obliged to picture conditions of which it has had no practical experience. Nevertheless by a rigorous analysis of the question based on the known values of the precession of the equinoxes and of nutation, Lord Kelvin has found that the Earth, taken as a whole, has a rigidity sensibly equal. to that of steel. This result is by no means incompatible with the state of fusion of the metals constituting the centrai nucleus, since this state is largely counteracted by the formidable pressures to which they are subjected. We may therefore admit that the terrestrial globe, taken in its entirety, possesses a certain elasticity.
Owing to the facts of this elasticity and the luni-solar attraction, the form of the globe will be modified. At the same time, the action of the igneous matter will deform the superficial layers, and the deviation which may be shown relatively to the direction of a plumb line will therefore be only that due to the difference of these two effects.
This explains the lack of success of the experiments described above; all were made on the assumption of the absolute rigidity of the Earth with the object of verifying the extremely slight variations of the vertical, which were of theoretical interest. The non-rigidity of the ground diminishes these minute variations still further, hence the failure of methods and apparatus to show it that were hardly sensitive enough even if the Earth had been quite rigid.
The credit of having demonstrated these deviations, not only qualitatively but even quantitatively, falls to Dr. Hecker, of the Geodetic Institute of Potsdam. For this purpose he utilised the wonderful sensitiveness of the horizontal pendulum, an instrument constructed several years before by von Rebeur-Paschwitz, which reached a high degree of perfection, but was still further / improved by Dr. Hecker.
The horizontal pendulum is an instrument of extreme sensitiveness. It is composed essentially of a horizontal rod fixed by two vertical threads F and F' to a strong support S; the points of attachment A and A' are not exactly one above the other, but they may be made as nearly so as is desired. A mass M is fixed at the end of the lever T. In these circumstances the pendulum takes a position of equilibrium for a given direction of the vertical, but, if this latter should change, the pendulum begins to oscillate with a period the same as that which a simple pendulum would have if of length equal to the distance between the mass M and the point V where the vertical M V intersects the straight line joining the points of attachment A A' of the two threads. It may be seen by an inspection of the figure that we are able to make the length M V as great as is desired; all that is necessary is to place the points A and A' more nearly above one another. We thus have a horizontal pendulum H M which oscillates with the same period as a vertical pendulum of very great length V M N and we may make the length of the equivalent simple pendulum so great that its oscillation may show the little displacements from the vertical which we have previously described.
Dr. Hecker took two of these pendulums, the shafts of which were perpendicular to one another; their lengths and the relative positions of the points of attachment had been regulated so that they corresponded respectively to simple pendulums of 175 and 117 metres [574 and 384 ft.] in length. The shafts were orientated symmetrically with regard to the meridian of the place. Two mirrors fixed on the shafts enabled the period of the oscillations to be registered photographically on films, the distance of which further increased the amplitude of the deviations, and this was already doubled by the reflection from the mirror.
By taking the photograms thus obtained and constructing graphs from them by points, having for abscissa and for ordinates the results deduced from the movements of the two pendulums, a curve results which illustrates the displacement of the point of a plumb-line, that is to say of the deviations of the vertical; such a curve has been constructed for every day and the results have been collected in groups of ninety days to furnish three-monthly averages.
A diurnal oscillation of the vertical to the extent of two-thousandths of a second of arc in the direction of the meridian has thus been distinguished; furthermore the three-monthly averages have shown that the amplitude of the oscillation is only half in winter what it is in summer. As Lallemand has justly observed, this is a clear indication of a thermal effect produced by the heating of the peripheral layers of the Earth's surface under the action of the solar rays and "these effects overlie those of the attraction of the Sun on the pendulums and almost mask them entirely."
Dr. Hecker has, however, been able to demonstrate the latter because of the fact that the period of the thermal effect is twenty-four hours, that is to say, is diurnal, while for the purely attractive solar action the period is twelve hours, that is to say, semi-diurnal. The attraction is exercised similarly whether the point in question is directly opposite the Sun on the near or far side of the Earth and therefore makes itself evident twice in every twenty-four hours. By combining the values of the deviations for corresponding hours of the two unsymmetrical periods of twelve hours each, and taking the semi-sum and the semi-difference of the deviations for each pair, the result sought is obtained, for in the semi-sum the thermal effect is naturally eliminated, since it is equal and of contrary sign in the two terms, while the attractive effect is not so counterbalanced. It is the contrary as regards the semi-difference, which isolates the thermal effect by eliminating the attractive one.
In the case of the lunar action, the separation of the two effects is more easily made on account of the difference of the periods of the solar day and the lunar day. Dr. Hecker has been able to construct the experimental curve which the point of a plumb-line describes under the Moon's influence, by a graphical interpretation of his observational results.
This curve is shown by a dotted line (Fig. 19). Fig. 20 represents the same curve in the case where the declination of the Moon is a high northern one.
The exact resemblance of these and experimental curves with the scribed by the Bob theoretical ones given by Gaillot, and reproduced in Figs. 14 and 16, is very striking. The only difference is in the lesser amplitude of the real curves. The diminution of amplitude is almost twice as great in the direction of the meridian as in that of the east-west direction at right angles to it; it reaches nearly half in the direction of the meridian. The little closed loop seen on the two curves corresponding to high declinations of the Moon arises from the fact that there are two daily maxima; these maxima are equal if the Moon is in the plane of the equator; inequal if it is to the north or south of the equator, the more so as it is farther away from the equator.
The difference between the calculated and observed curves, assuming that the latter give similar results when the enquiry is pursued over a longer space of time, shows that the Earth, taken in its entirety, possesses a certain degree of elasticity which is of the same order of magnitude as that of steel. In other words the Earth behaves almost as if it were made of solid steel, and of its present dimensions. It is especially remarkable that the consideration of the oceanic tides, the astronomical movements of the Earth, and the displacement of the terrestrial poles all lead us to assign an elasticity of the same order of magnitude to the Earth taken as a whole ; it is an admirable confirmation of the original idea of Lord Kelvin. There is only one point that remains obscure in the interpretation of Dr. Hecker's results, viz., the reduction of the amplitude of the deviations in the direction of the meridian, a reduction of the extent of the phenomenon to almost half its value, while there is scarcely any such reduction in the east-west direction. In the masterly analysis of this question that he has made, Lallemand has sought the cause of this anomaly. It may be due to the instrument itself, or to the relative proximity of the sea, or to a peculiarity of structure of the Earth's crust in the Potsdam region, or again to the tetrahedral form of our globe, of which the Eurasian aręte, oriented in the east-west direction, passes not far from Potsdam. Lallemand favours this last suggestion. In spite of this in-completeness of our knowledge much has been achieved; we know that the globe taken as a whole has an elasticity of the same order of magnitude as that of steel, and we shall see later on that the study of the seismic phenomena brings further confirmation of this fact.
As a result of our study of the combined action of gravity and the luni-solar attraction on a plumb-line we can deduce another consequence; the thread which supports the mass cannot be rectilinear, but has the form of a curve, the equation of which has been given by Puiseux. This curvature has not been detected by any of our methods of measurement so far, but we know that it exists. Since a stretched horizontal thread, however fine it may be and however well it may be stretched, is never rectilinear, because of gravity which imposes upon it the form of a catenary, it will be seen that a straight line is not realisable, at any rate mechanically. It is also the same optically; light, because of the movements of the Earth, and because of refraction and diffraction, is not propagated in a straight line. The idea of a luminous ray has given place to that of a wave. The edge of a crystal is not a right line, for during the time that a second molecule has taken to align itself with the first, the Sun and the Moon have changed position relatively to the Earth and have deviated the molecule from the position it would otherwise have taken.
Is the straight line then entirely a creation of Man's brain? If so, he might be justly proud of it.
However this may be, everything about our Earth is in continual movement, in spite of the deceptive appearance of stability presented to us. The crust expands and contracts under the daily action of the solar heat ; the nucleus, rendered dense and compact by the pressures acting upon it, is subjected to veritable tides under the influence of the luni-solar attraction and we may be certain that the fluid layer, interposed between the nucleus and the crust which covers it, is agitated by perpetual movements, both of tidal and convective origin. Where then may we find real stability? Where is the invariability which the rocks seemed to symbolise so well? In the imagination of poets perhaps, but not in the reality of Nature, where everything moves perpetually. The different forms of movement we have hitherto dealt with, whether affecting the entire Earth or only its crust, are of astronomical origin. We now come to movements of a different nature, viz., the sudden movements which sometimes disturb a large extent of the Earth's crust, known as earthquakes, and also the slow continued movements which produce the raising and lowering of the crust.
The Earth Its Life And Death:
The Birth Of The Earth
The Age Of The Earth
The Form, Magnitude, And Mass Of The Earth
The Movements Of The Earth
The Rhythmic Movements Of The Earth's Crust
The Sudden Movements Of The Earth's Crust
The Magnetism, Electricity, And Radioactivity Of The Earth
The Rhythmic Movements Of The Ocean, Tides, Swell, And Waves
The Circulation Of The Earth, Marine And Atmospheric
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