CHAPTER XIII.
GRAVITATIONAL EFFECTS OF SUN AND MOON UPON THE EARTH.
THE TIDES.—The familiar phenomena of the tides are of such importance to commerce in so many parts of the world that they have been carefully investigated from very early times. The necessities of coast navigation would soon lead to the recognition of a periodic character in the tides, as well as to their association with the age and position of the moon. With the march of science, an explanation of tidal phenomena was therefore sought in the motion of the moon. A great impetus was given to this inquiry by Newton’s generalisation, and the tides were shown to be a necessary consequence of the gravitational attraction of the sun and moon. Regarding the earth merely as a cosmical particle, we have seen that its orbital motion is perfectly explained by the gravitational attraction of the sun, and some of its minor movements by the attractions of other members of the solar system. The law of gravitation, however, compels us, in a closer investigation of these mutual attractions, to regard each globe as an assemblage of particles, each of which individually influences and is influenced by other particles. If such a collection of particles be spherical and perfectly rigid, it will behave precisely as a simple particle in which the whole mass is concentrated.
When we cease to consider the earth as a mere particle, we must regard the waters of the oceans as being free to move over the more rigid crust of the globe. Imagine our globe to be a spherical mass completely surrounded by a liquid envelope. At any moment one half of this is presented towards the moon. The solid earth we may conceive to be attracted by the moon as a simple particle; but the water on the side nearest to the moon is attracted with a greater force than the solid globe, because of its greater proximity to the attracting body, and it has therefore a tendency to heap itself up directly under the moon. Being free to move, the water thus remains heaped up under the moon, notwithstanding the earth’s rotation, and if there were only one such elevation, there would only be one tide a day. Observation shows us that there are two high tides a day, and the water must therefore be heaped up on the side of the earth which is turned away from the moon. This is perfectly true, though seemingly at first sight inconsistent with the moon’s attraction. The fact is that the solid earth is attracted by the moon with greater energy than the water on the side most remote from it, so that the heaping up of the water on the side away from the moon is to be regarded as due to the earth having left it behind.
[Illustration:
FIG. 45.—_The Tides._ ]
There is thus a double tidal wave produced by a spheroid of water which, in the simple case we have considered, has its axis directed towards the moon, as in Fig. 45. The earth, rotating within this liquid shell, successively brings different parts of the solid earth to the points of high and low water. If the moon were fixed, we should then experience two high and two low waters every day, but as it revolves in the same direction that the earth rotates, the average interval between two successive meridian passages is 24 hours 51 minutes. This, then, is the period in which alternate high waters or alternate low waters are experienced.
A similar train of reasoning applies to the attraction of the sun upon different parts of our planet, so that there are solar as well as lunar tides. Nevertheless, the moon is the dominating cause, for although the total attraction of the sun upon the earth is about 200 times that of the moon, its differential attraction upon the opposite sides of the earth, which is alone effective in producing tides, is only about ⅖ths that of the moon.
A simple mathematical investigation shows that the tide-raising force of a body is proportional to its mass, and approximately in inverse proportion to the cube of its distance from the affected body. Thus, it appears that if the moon were removed to 1·36 times its present distance, solar and lunar tides would be equal.
At the times of new and full moon, the sun and moon will produce two tidal spheroids of water upon our imaginary earth, having their axes coincident, and an exceptionally high tide will occur. This is a _spring tide_. When the moon is at its quarters the two ellipsoids tend to neutralise each other, and an exceptionally low or _neap tide_ results. Two spring tides and two neap tides thus occur in each synodic month of 29½ days.
The height of the tide will also be affected by the variations in the distance of the moon. If the moon be at perigee the tide will be greater because of the smaller distance, and if this occur at new or full moon there will be a very high spring tide, while a less notable spring tide will occur when the new or full moon is at apogee.
The combination of the solar and lunar tides gives rise to what is called the _priming_ and _lagging_ of the tides. At new and full moons the combined tides will produce a spheroid of water with its axis directed towards the moon. When the moon is a few days old however, the crest will take up a position intermediate between the direction of the moon and that of the sun, and high water will therefore be accelerated. The same thing will happen during three or four days after full moon. Three days before full or new moon the combination of the two tides will displace the crest towards the sun, and therefore in advance of the moon, so that high water will be retarded. The retardation and acceleration correspond to lagging and priming respectively.
At the quadratures the combined tides simply reduce the height of the crest, since there is no reason why the deviation should be to one side any more than to the other. On account of priming and lagging, the tides on successive days are accelerated or retarded by as much as 13 minutes when the effects are greatest.
Sufficient has been said to indicate that tidal phenomena are very complex even when we suppose the earth to be very simply constituted. When we take into account the actual configuration of the land and the consequent restrictions in the movements of the water, these complications are increased tenfold. Yet, by continued observations, the recurrence of tides at any port can be predicted with tolerable accuracy. It is observed that there is a certain pretty regular interval of time between the moon’s meridian passage and the time of next high water; this is different at different ports, but is so nearly constant at a given place as to be called _the establishment of the port_. Observations being made at a great many places, the peculiar movements of the tidal wave can be investigated. For this purpose, it is convenient to draw on a map what are called _co-tidal lines_ that is, lines passing through places at which high water occurs at the same moment. It then appears that it is only in the Southern Pacific where the water is of sufficient extent to permit the formation of the tide crest. The effect of this wave, which commences twice a day, is gradually spread over different parts of the world, but before it reaches most places other waves have commenced a similar journey. The tide at London, for example, coming round the north of Scotland and down the North Sea, really started in the Southern Pacific 66 hours before, and in the same way the tide at New York is a little over 40 hours old.
The height of a tide is thus regulated by the conditions of the sun and moon with regard to the earth when the primary tide was formed, and not by their relation when a tide is actually observed.
In the Pacific Ocean the tides are very feeble, but near the coast they vary enormously, and sometimes reach great heights. At Bristol the difference between high and low water sometimes amounts to fifty feet, and in the Bay of Fundy, Nova Scotia, it has been as much as a hundred feet.
The peculiarities of the tides at many places are due to interference. The primary tidal wave striking the British Islands travels partly up the English Channel, and partly round to the North Sea by the north of Scotland. At some places on the east coast the two waves almost neutralise each other, while at others there are even four high tides in a day.
The circumstances under which tides occur at a given place can only be determined by actual observations, as theory is at present utterly inadequate to deal with the manifold complications brought about by the configuration of the land, and the varying depth of the water.
TIDAL FRICTION.—The regular influx of the tide supplies us with a source of mechanical energy, which in the future will no doubt become of immense importance to mankind. A great mass of water is raised to a higher level, and by suitable contrivances it can be made to do useful work during its subsequent flow to the ocean from which it came. Ordinarily, however, the water simply rushes back without its energy being utilised, and the potential power is merely transferred to another locality. It is manifest, however, that a certain amount of tidal energy is lost by friction as the water rolls to and from the rocky shores. This energy is converted into heat, and finally radiated into space, or dissipated. Now, the principle of the conservation of energy tells us that energy can neither be created nor destroyed, although its form may change from a useful to a useless one. It follows, therefore, that the energy lost through the tides must be abstracted from one source or another, and it has been shown that this energy is really derived from the earth’s rotation. As the earth steadily ploughs its way through its liquid envelope, the tides act as a break, and its rotational velocity is reduced; it is part of this lost energy of rotation which is dissipated by the tides.
One tendency of tidal friction is accordingly to lengthen the period of the earths rotation, and, therefore, to increase the length of the day. There are, however, counteracting causes, so that there is no certain direct evidence that the day has actually lengthened in historical times.
All the energy of rotation which is lost by the earth is not, however, dissipated by the tides. Some of it is transferred to the moon, with the result that the velocity of our satellite, and consequently the size of its orbit, must be increasing. From this it is inferred that the moon was formerly very much closer than at present, and an elaborate investigation of the conditions of its retreat has led Professor G. H. Darwin to his interesting theory of “tidal evolution.” (See p. 236.)
Professor Darwin has shown that if the term “tide” be extended to include distortions of the earth and moon at an earlier stage of their history, when both were fluid or viscous, a similar grinding down of the energies of rotation of both bodies must have taken place. The axial rotation of the moon, under these circumstances, would be retarded by the attraction of the earth on the tides raised in the moon, while that of the earth would also be slowed down, but in a less degree because of the moon’s smaller mass.
CAUSE OF PRECESSION.—On account of the spheroidal form of the earth, we may regard it as a sphere which is surrounded by a ring of protuberant matter at the Equator. Now the attraction of the sun upon the spherical part will be quite independent of the position of its axis of rotation, and will, therefore, not affect the position of the Equator. It is different, however, with the ring; at the solstices the ring is inclined to the line joining its centre with the sun, and the near side is subject to a greater attraction than the side more remote from the sun. On account of this difference of pull, there is a tendency for the ring to move into the plane of the ecliptic, and this is what would happen if the ring were not in rotation. The practical outcome of this tendency, combined with the rotation, is to produce the twisting of the plane of the ring, and, therefore, of the plane of the Equator. At the equinoxes the plane of the ring passes through the sun, and although there is still a difference of attraction on opposite sides of the ring, the differential force is entirely directed to the sun, and therefore cannot produce any precessional effect.
The ultimate tendency to turn into the plane of the ecliptic thus depends upon the _difference_ of the attractions on opposite sides of the ring, or rather that part of the difference which acts in a direction perpendicular to the Equator.
The terrestrial ring cannot change the position of its plane without taking the whole earth with it, and the rate of precession is thus very slow. The effect of solar precession alone would cause the equatorial plane to twist round with but little change of inclination; or the earth’s axis would travel with a conical movement round a perpendicular to the ecliptic passing through the earth’s centre.
It will be remarked that as the force-producing precession is identical with that which is effective in producing the tides, the moon must have a greater precessional effect than the sun. This is quite true, and on the average the precession-producing force of the moon is 2½ times that of the sun. When the moon is on the celestial equator, as it is twice a month, the differential force acts in the plane of the ring, and no precessional effect results. On the other hand, the greatest effect is produced by the moon when the earth’s Equator is most inclined to the line joining the earth and moon. The amount of this greatest inclination is different in different months according to the position of the moon’s nodes. In consequence of the revolution of the moon’s nodes, the moon’s orbit is inclined to the Equator at all angles from 18° to 28°, and back again to 18° in a period of 19 years. The precessional effect of the moon thus has a principal period of 19 years, while that of the sun has a period of a year during which it has two maxima and two minima. The summation of the effects of the sun and moon gives us the _luni-solar precession_, which is very variable in its actual rate, but averages about 50″·2 per annum.
[Illustration:
FIG. 46.—_Nutation._ ]
NUTATION.—If the precession-producing force were of constant amount, there would be no change in the inclination of the earth’s axis to the ecliptic. When the force is increasing, the equatorial ring is slightly tilted towards the ecliptic, and when it is decreasing the converse takes place. As the moon has the preponderating effect, these changes in the inclination will evidently depend mainly upon the changing value of the moon’s precessional force; that is, they will have a period of 19 years. Thus, if _P_, Fig. 46, represents the pole of the ecliptic, the north celestial pole would travel in a circle of 23½° radius about _P_ if precession were uniform. Suppose, then, the celestial pole to be at _a_ when the moon’s node is on the Equator—that is, when the inclination of the moons orbit to the Equator is greatest—from this time the integrated effects of the moon’s precessional force will be decreasing, and the inclination of the Equator to the ecliptic will be increased; the celestial pole will consequently recede a little more than the average from the pole of the ecliptic, so that after 9½ years it will be at _b_ instead of _c_. During the next 9½ years the inclination of the moon’s orbit to the ecliptic will be gradually getting smaller, the precessional force will be proportionately reduced, and the obliquity of the ecliptic will be increased, so that the north celestial pole will have arrived at _d_ after the lapse of 19 years. The prolongation of the earth’s axis thus describes a wavy curve, each wave extending over 19 years, so that there are about 1,400 waves during the great precessional cycle. This approach and recession of the two poles is called _nutation_, or nodding of the earth’s axis. The most recent investigation of its maximum amount, by Dr. Chandler, gives it as 9″·202. Besides the principal nutation there are others of very much smaller amount, due to the monthly changes of the moon’s declination and to the annual change of the sun’s declination.
The most obvious effect of nutation is that upon the inclination of the earth’s axis to the ecliptic—the “nutation in obliquity.” There is, however, a displacement of the equinoctial point, and corresponding nutations in longitude and right ascension.
As pointed out by Sir John Herschel, we have in nutation a splendid example of a periodical movement in one part of a system giving rise to a motion having the same precise period in another.
EFFECTS OF PRECESSION.—The effects of precession may be conveniently summarised here, although some of them have necessarily been mentioned elsewhere:
(1) The first point of Aries revolves completely round the ecliptic, so that it passes through all the constellations of the zodiac in a period of 25,800 years. The “signs” of the zodiac, accordingly, no longer correspond with the constellations after which they are named.
(2) The Pole Star is constantly changing, since the north celestial pole travels round the pole of the ecliptic at a distance of about 23½° in a period of 25,800 years. About 14,000 years ago the bright star Alpha Lyræ was the Pole Star.
(3) The position of the north celestial pole is in time changed by 47°, and there may accordingly be this change in the north polar distances or declinations of all stars whatsoever. As the position of the ecliptic is almost constant, the celestial latitudes of stars will be but little affected by precession.
(4) The right ascensions and longitudes of stars, being reckoned from the shifting first point of Aries, are themselves changeable, passing through all possible values in the precessional period.
(5) The tropical year is shorter than the sidereal year by the time taken for the earth to travel through 50″·2—that is, 20 minutes 23 seconds.
(6) Celestial globes and maps, as well as star catalogues, can only represent the right ascensions and declinations of stars at a specified epoch.