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.