Lunar Theory—Periodic Perturbations of the Moon—Equation of
Centre—Evection—Variation—Annual Equation—Direct and Indirect
Action of Planets—The Moon’s Action on the Earth disturbs her
own Motion—Excentricity and Inclination of Lunar Orbit
invariable—Acceleration—Secular Variation in Nodes and
Perigee—Motion of Nodes and Perigee inseparably connected with
the Acceleration—Nutation of Lunar Orbit—Form and Internal
Structure of the Earth determined from it—Lunar, Solar, and
Planetary Eclipses—Occultations and Lunar Distances—Mean
Distance of the Sun from the Earth obtained from Lunar
Theory—Absolute Distances of the Planets, how found.
OUR constant companion, the moon, next claims our attention. Several
circumstances concur to render her motions the most interesting, and at
the same time the most difficult to investigate, of all the bodies of
our system. In the solar system, planet troubles planet; but, in the
lunar theory, the sun is the great disturbing cause, his vast distance
being compensated by his enormous magnitude, so that the motions of the
moon are more irregular than those of the planets; and, on account of
the great ellipticity of her orbit, and the size of the sun, the
approximations to her motions are tedious and difficult, beyond what
those unaccustomed to such investigations could imagine. The average
distance of the moon from the centre of the earth is only 238,793 miles,
so that her motion among the stars is perceptible in a few hours. She
completes a circuit of the heavens in 27^d 7^h 43^m 11^s·5, moving
in an orbit whose excentricity is about 12,985 miles. The moon is about
four hundred times nearer to the earth than the sun. The proximity of
the moon to the earth keeps them together. For so great is the
attraction of the sun, that, if the moon were farther from the earth,
she would leave it altogether, and would revolve as an independent
planet about the sun.
The disturbing action (N. 101) of the sun on the moon is equivalent to
three forces. The first, acting in the direction of the line joining the
moon and earth, increases or diminishes her gravity to the earth. The
second, acting in the direction of a tangent to her orbit, disturbs her
motion in longitude. And the third, acting perpendicularly to the plane
of her orbit, disturbs her motion in latitude; that is, it brings her
nearer to, or removes her farther from, the plane of the ecliptic than
she would otherwise be. The periodic perturbations in the moon, arising
from these forces, are perfectly similar to the periodic perturbations
of the planets. But they are much greater and more numerous; because the
sun is so large, that many inequalities which are quite insensible in
the motions of the planets, are of great magnitude in those of the moon.
Among the innumerable periodic inequalities to which the moon’s motion
in longitude is liable, the most remarkable are, the Equation of the
Centre, which is the difference between the moon’s mean and true
longitude, the Evection, the Variation, and the Annual Equation. The
disturbing force which acts in the line joining the moon and earth
produces the Evection: it diminishes the excentricity of the lunar orbit
in conjunction and opposition, thereby making it more circular, and
augments it in quadrature, which consequently renders it more
elliptical. The period of this inequality is less than thirty-two days.
Were the increase and diminution always the same, the Evection would
only depend upon the distance of the moon from the sun; but its absolute
value also varies with her distance from the perigee (N. 102) of her
orbit. Ancient astronomers, who observed the moon solely with a view to
the prediction of eclipses, which can only happen in conjunction and
opposition, where the excentricity is diminished by the Evection,
assigned too small a value to the ellipticity of her orbit (N. 103). The
Evection was discovered by Ptolemy from observation, about A.D. 140. The
Variation produced by the tangential disturbing force, which is at its
maximum when the moon is 45° distant from the sun, vanishes when that
distance amounts to a quadrant, and also when the moon is in conjunction
and opposition; consequently, that inequality never could have been
discovered from the eclipses: its period is half a lunar month (N. 104).
The Annual Equation depends upon the sun’s distance from the earth: it
arises from the moon’s motion being accelerated when that of the earth
is retarded, and _vice versâ_—for, when the earth is in its perihelion,
the lunar orbit is enlarged by the action of the sun; therefore, the
moon requires more time to perform her revolution. But, as the earth
approaches its aphelion, the moon’s orbit contracts, and less time is
necessary to accomplish her motion—its period, consequently, depends
upon the time of the year. In the eclipses the Annual Equation combines
with the Equation of the Centre of the terrestrial orbit, so that
ancient astronomers imagined the earth’s orbit to have a greater
excentricity than modern astronomers assign to it.
The planets disturb the motion of the moon both directly and indirectly;
their action on the earth alters its relative position with regard to
the sun and moon, and occasions inequalities in the moon’s motion, which
are more considerable than those arising from their direct action; for
the same reason the moon, by disturbing the earth, indirectly disturbs
her own motion. Neither the excentricity of the lunar orbit, nor its
mean inclination to the plane of the ecliptic, have experienced any
changes from secular inequalities; for, although the mean action of the
sun on the moon depends upon the inclination of the lunar orbit to the
ecliptic, and the position of the ecliptic is subject to a secular
inequality, yet analysis shows that it does not occasion a secular
variation in the inclination of the lunar orbit, because the action of
the sun constantly brings the moon’s orbit to the same inclination to
the ecliptic. The mean motion, the nodes, and the perigee, however, are
subject to very remarkable variations.
From the eclipse observed at Babylon, on the 19th of March, seven
hundred and twenty-one years before the Christian era, the place of the
moon is known from that of the sun at the instant of opposition (N. 83),
whence her mean longitude may be found. But the comparison of this mean
longitude with another mean longitude, computed back for the instant of
the eclipse from modern observations, shows that the moon performs her
revolution round the earth more rapidly and in a shorter time now than
she did formerly, and that the acceleration in her mean motion has been
increasing from age to age as the square of the time (N. 105). All
ancient and intermediate eclipses confirm this result. As the mean
motions of the planets have no secular inequalities, this seemed to be
an unaccountable anomaly. It was at one time attributed to the
resistance of an ethereal medium pervading space, and at another to the
successive transmission of the gravitating force. But, as La Place
proved that neither of these causes, even if they exist, have any
influence on the motions of the lunar perigee (N. 102) or nodes, they
could not affect the mean motion; a variation in the mean motion from
such causes being inseparably connected with variations in the motions
of the perigee and nodes. That great mathematician, in studying the
theory of Jupiter’s satellites, perceived that the secular variation in
the elements of Jupiter’s orbit, from the action of the planets,
occasions corresponding changes in the motions of the satellites, which
led him to suspect that the acceleration in the mean motion of the moon
might be connected with the secular variation in the excentricity of the
terrestrial orbit. Analysis has shown that he assigned the true cause of
the acceleration.
It is proved that the greater the excentricity of the terrestrial orbit,
the greater is the disturbing action of the sun on the moon. Now, as the
excentricity has been decreasing for ages, the effect of the sun in
disturbing the moon has been diminishing during that time. Consequently
the attraction of the earth has had a more and more powerful effect on
the moon, and has been continually diminishing the size of the lunar
orbit. So that the moon’s velocity has been gradually augmenting for
many centuries to balance the increase of the earth’s attraction. This
secular increase in the moon’s velocity is called the Acceleration, a
name peculiarly appropriate at present, and which will continue to be so
for a vast number of ages; because, as long as the earth’s excentricity
diminishes, the moon’s mean motion will be accelerated; but when the
excentricity has passed its minimum, and begins to increase, the mean
motion will be retarded from age to age. The secular acceleration is now
about 11ʺ·9, but its effect on the moon’s place increases as the square
of the time (N. 106). It is remarkable that the action of the planets,
thus reflected by the sun to the moon, is much more sensible than their
direct action either on the earth or moon. The secular diminution in the
excentricity, which has not altered the equation of the centre of the
sun by eight minutes since the earliest recorded eclipses, has produced
a variation of about 1° 48ʹ in the moon’s longitude, and of 7° 12ʹ in
her mean anomaly (N. 107).
The action of the sun occasions a rapid but variable motion in the nodes
and perigee of the lunar orbit. Though the nodes recede during the
greater part of the moon’s revolution, and advance during the smaller,
they perform their sidereal revolution in 6793^d 9^h 23^m 9^s·3, or
about 18-6/10 years; and the perigee accomplishes a revolution, called
of the moon’s apsides, in 3232^d 13^h 48^m 29^s·6, or a little more
than nine years, notwithstanding its motion is sometimes retrograde and
sometimes direct: but such is the difference between the disturbing
energy of the sun and that of all the planets put together, that it
requires no less than 109,830 years for the greater axis of the
terrestrial orbit to do the same, moving at the rate of 11ʺ·8 annually.
The form of the earth has no sensible effect either on the lunar nodes
or apsides. It is evident that the same secular variation which changes
the sun’s distance from the earth, and occasions the acceleration in the
moon’s mean motion, must affect the nodes and perigee. It consequently
appears, from theory as well as observation, that both these elements
are subject to a secular inequality, arising from the variation in the
excentricity of the earth’s orbit, which connects them with the
Acceleration, so that both are retarded when the mean motion is
anticipated. The secular variations in these three elements are in the
ratio of the numbers 3, 0·735, and 1; whence the three motions of the
moon, with regard to the sun, to her perigee, and to her nodes, are
continually accelerated, and their secular equations are as the numbers
1, 4·702, and 0·612. A comparison of ancient eclipses observed by the
Arabs, Greeks, and Chaldeans, imperfect as they are, with modern
observations, confirms these results of analysis. Future ages will
develop these great inequalities, which at some most distant period will
amount to many circumferences (N. 108). They are, indeed, periodic; but
who shall tell their period? Millions of years must elapse before that
great cycle is accomplished.
The moon is so near, that the excess of matter at the earth’s equator
occasions periodic variations in her longitude, and also that remarkable
inequality in her latitude, already mentioned as a nutation in the lunar
orbit, which diminishes its inclination to the ecliptic when the moon’s
ascending node coincides with the equinox of spring, and augments it
when that node coincides with the equinox of autumn. As the cause must
be proportional to the effect, a comparison of these inequalities,
computed from theory, with the same given by observation, shows that the
compression of the terrestrial spheroid, or the ratio of the difference
between the polar and the equatorial diameters, to the diameter of the
equator, is 1/305·05. It is proved analytically, that, if a fluid mass
of homogeneous matter, whose particles attract each other inversely as
the squares of the distance, were to revolve about an axis as the earth
does, it would assume the form of a spheroid whose compression is 1/230.
Since that is not the case, the earth cannot be homogeneous, but must
decrease in density from its centre to its circumference. Thus the
moon’s eclipses show the earth to be round; and her inequalities not
only determine the form, but even the internal structure of our planet;
results of analysis which could not have been anticipated. Similar
inequalities in the motions of Jupiter’s satellites prove that his mass
is not homogeneous, and that his compression is 1/13·8. His equatorial
diameter exceeds his polar diameter by about 6000 miles.
The phases (N. 109) of the moon, which vary from a slender silvery
crescent soon after conjunction, to a complete circular disc of light in
opposition, decrease by the same degrees till the moon is again
enveloped in the morning beams of the sun. These changes regulate the
returns of the eclipses. Those of the sun can only happen in
conjunction, when the moon, coming between the earth and the sun,
intercepts his light. Those of the moon are occasioned by the earth
intervening between the sun and moon when in opposition. As the earth is
opaque and nearly spherical, it throws a conical shadow on the side of
the moon opposite to the sun, the axis of which passes through the
centres of the sun and earth (N. 110). The length of the shadow
terminates at the point where the apparent diameters (N. 111) of the sun
and earth would be the same. When the moon is in opposition, and at her
mean distance, the diameter of the sun would be seen from her centre
under an angle of 1918ʺ·1. That of the earth would appear under an angle
of 6908ʺ·3. So that the length of the shadow is at least three times and
a half greater than the distance of the moon from the earth, and the
breadth of the shadow, where it is traversed by the moon, is about
eight-thirds of the lunar diameter. Hence the moon would be eclipsed
every time she is in opposition, were it not for the inclination of her
orbit to the plane of the ecliptic, in consequence of which the moon,
when in opposition, is either above or below the cone of the earth’s
shadow, except when in or near her nodes. Her position with regard to
them occasions all the varieties in the lunar eclipses. Every point of
the moon’s surface successively loses the light of different parts of
the sun’s disc before being eclipsed. Her brightness therefore gradually
diminishes before she plunges into the earth’s shadow. The breadth of
the space occupied by the penumbra (N. 112) is equal to the apparent
diameter of the sun, as seen from the centre of the moon. The mean
duration of a revolution of the sun, with regard to the node of the
lunar orbit, is to the duration of a synodic revolution (N. 113) of the
moon as 223 to 19. So that, after a period of 223 lunar months, the sun
and moon would return to the same relative position with regard to the
node of the moon’s orbit, and therefore the eclipses would recur in the
same order were not the periods altered by irregularities in the motions
of the sun and moon. In lunar eclipses, our atmosphere bends the sun’s
rays which pass through it all round into the cone of the earth’s
shadow. And as the horizontal refraction (N. 114) or bending of the rays
surpasses half the sum of the semidiameters of the sun and moon, divided
by their mutual distance, the centre of the lunar disc, supposed to be
in the axis of the shadow, would receive the rays from the same point of
the sun, round all sides of the earth; so that it would be more
illuminated than in full moon, if the greater portion of the light were
not stopped or absorbed by the atmosphere. Instances are recorded where
this feeble light has been entirely absorbed, so that the moon has
altogether disappeared in her eclipses.
The sun is eclipsed when the moon intercepts his rays (N. 115). The
moon, though incomparably smaller than the sun, is so much nearer the
earth, that her apparent diameter differs but little from his, but both
are liable to such variations that they alternately surpass one another.
Were the eye of a spectator in the same straight line with the centres
of the sun and moon, he would see the sun eclipsed. If the apparent
diameter of the moon surpassed that of the sun, the eclipse would be
total. If it were less, the observer would see a ring of light round the
disc of the moon, and the eclipse would be annular, as it was on the
17th of May, 1836, and on the 15th of March, 1858. If the centre of the
moon should not be in the straight line joining the centres of the sun
and the eye of the observer, the moon might only eclipse a part of the
sun. The variation, therefore, in the distances of the sun and moon from
the centre of the earth, and of the moon from her node at the instant of
conjunction, occasions great varieties in the solar eclipses. Besides,
the height of the moon above the horizon changes her apparent diameter,
and may augment or diminish the apparent distances of the centres of the
sun and moon, so that an eclipse of the sun may occur to the inhabitants
of one country, and not to those of another. In this respect the solar
eclipses differ from the lunar, which are the same for every part of the
earth where the moon is above the horizon. In solar eclipses, the light
reflected by the atmosphere diminishes the obscurity they produce. Even
in total eclipses the higher part of the atmosphere is enlightened by a
part of the sun’s disc, and reflects its rays to the earth. The whole
disc of the new moon is frequently visible from atmospheric reflection.
During the eclipse of the 19th of March, 1849, the spots on the lunar
disc were distinctly visible, and during that of 1856 the moon was like
a beautiful rose-coloured ball floating in the ether: the colour is
owing to the refraction of the sun’s light passing through the earth’s
atmosphere.
In total solar eclipses the slender luminous arc that is visible for a
few seconds before the sun vanishes and also before he reappears,
resembles a string of pearls surrounding the dark edge of the moon; it
is occasioned by the sun’s rays passing between the tops of the lunar
mountains: it occurs likewise in annular eclipses.
A phenomenon altogether unprecedented was seen during the total eclipse
of the sun which happened on the 8th of July, 1842. The moon was like a
black patch on the sky surrounded by a faint whitish light or corona
about the eighth of the moon’s diameter in breadth, which is supposed to
be the solar atmosphere rendered visible by the intervention of the
moon. In this whitish corona there appeared three rose-coloured flames
like the teeth of a saw. Similar flames were also seen in the white
corona of the total eclipse which took place in 1851, and a long
rose-coloured chain of what appeared to be jagged mountains or sierras
united at the base by a red band seemed to be raised into the corona by
mirage; but there is no doubt that the corona and red phenomena belong
to the sun. This red chain was so bright that Mr. Airy saw it illuminate
the northern horizon through an azimuth of 90° with red light. M. Faye
attributes the rose-coloured protuberances to the constitution of the
sun, which, like Sir William Herschel, he conceives to be an
incandescent globe, consisting of two concentric parts of very unequal
density, the internal part being a dark spherical mass, the external a
very extensive atmosphere, at a certain height in which there is a
stratum of luminous clouds which constitutes the photosphere of the sun;
above this rises his real atmosphere, so rare as to be only visible as a
white aureola or corona during total and annular eclipses. M. Faye
conceives that from the central mass gaseous eruptions issue, which form
the spots by dissipating and partly extinguishing the luminous clouds,
and then rising into the rare atmosphere above that they appear as
rose-coloured protuberances during annular eclipses. He estimates that
the volume of these vapours sometimes surpasses that of the earth a
thousand or even two thousand times. Sir William Herschel attributed the
spots to occasional openings in the luminous coating, which seems to be
always in motion; but whatever the cause of the spots may be, it is
certainly periodical. The white corona and beads were seen during the
eclipse of the 15th March, 1858, but there were no rose-coloured
appearances, in England at least; but the sky was clouded, so that the
eclipse was only visible at intervals.
Planets sometimes eclipse one another. On the 17th of May, 1737, Mercury
was eclipsed by Venus near their inferior conjunction; Mars passed over
Jupiter on the 9th of January, 1591; and on the 30th of October, 1825,
the moon eclipsed Saturn. These phenomena, however, happen very seldom,
because all the planets, or even a part of them, are very rarely seen in
conjunction at once; that is, in the same part of the heavens at the
same time. More than 2500 years before our era the five great planets
were in conjunction. On the 15th of September, 1186, a similar
assemblage took place between the constellations of Virgo and Libra; and
in 1801 the Moon, Jupiter, Saturn, and Venus were united in the heart of
the Lion. These conjunctions are so rare, that Lalande has computed that
more than seventeen millions of millions of years separate the epochs of
the contemporaneous conjunctions of the six great planets.
The motions of the moon have now become of more importance to the
navigator and geographer than those of any other heavenly body, from the
precision with which terrestrial longitude is determined by occultations
of stars, and by lunar distances. In consequence of the retrograde
motion of the nodes of the lunar orbit, at the rate of 3ʹ 10ʺ·64 daily,
these points make a tour of the heavens in a little more than eighteen
years and a half. This causes the moon to move round the earth in a kind
of spiral, so that her disc at different times passes over every point
in a zone of the heavens extending rather more than 5° 9ʹ on each side
of the ecliptic. It is therefore evident that at one time or other she
must eclipse every star and planet she meets with in this space.
Therefore the occultation of a star by the moon is a phenomenon of
frequent occurrence. The moon seems to pass over the star, which almost
instantaneously vanishes at one side of her disc, and after a short time
as suddenly reappears on the other. A lunar distance is the observed
distance of the moon from the sun, or from a particular star or planet,
at any instant. The lunar theory is brought to such perfection, that the
times of these phenomena, observed under any meridian, when compared
with those computed for that of Greenwich, and given in the Nautical
Almanac, furnish the longitude of the observer within a few miles
(N. 95.)
From the lunar theory, the mean distance of the sun from the earth, and
thence the whole dimensions of the solar system, are known; for the
forces which retain the earth and moon in their orbits are respectively
proportional to the radii vectores of the earth and moon, each being
divided by the square of its periodic time. And, as the lunar theory
gives the ratio of the forces, the ratio of the distances of the sun and
moon from the earth is obtained. Hence it appears that the sun’s mean
distance from the earth is 399·7 or nearly 400 times greater than that
of the moon. The method of finding the absolute distances of the
celestial bodies, in miles, is in fact the same with that employed in
measuring the distances of terrestrial objects. From the extremities of
a known base (N. 116), the angles which the visual rays from the object
form with it are measured; their sum subtracted from two right angles
gives the angle opposite the base; therefore, by trigonometry, all the
angles and sides of the triangle may be computed—consequently the
distance of the object is found. The angle under which the base of the
triangle is seen from the object is the parallax of that object. It
evidently increases and decreases with the distance. Therefore the base
must be very great indeed to be visible from the celestial bodies. The
globe itself, whose dimensions are obtained by actual admeasurement,
furnishes a standard of measures with which we compare the distances,
masses, densities, and volumes of the sun and planets.