Theory of Jupiter’s Satellites—Effects of the Figure of Jupiter upon his
Satellites—Position of their Orbits—Singular Laws among the Motions of
the first Three Satellites—Eclipses of the Satellites—Velocity of
Light—Aberration—Ethereal Medium—Satellites of Saturn and Uranus.
THE changes which take place in the planetary system are exhibited on a
smaller scale by Jupiter and his satellites; and, as the period
requisite for the development of the inequalities of these moons only
extends to a few centuries, it may be regarded as an epitome of that
grand cycle which will not be accomplished by the planets in myriads of
ages. The revolutions of the satellites about Jupiter are precisely
similar to those of the planets about the sun; it is true they are
disturbed by the sun, but his distance is so great, that their motions
are nearly the same as if they were not under his influence. The
satellites, like the planets, were probably projected in elliptical
orbits: but, as the masses of the satellites are nearly 100,000 times
less than that of Jupiter; and as the compression of Jupiter’s spheroid
is so great, in consequence of his rapid rotation, that his equatorial
diameter exceeds his polar diameter by no less than 6000 miles; the
immense quantity of prominent matter at his equator must soon have given
the circular form observed in the orbits of the first and second
satellites, which its superior attraction will always maintain. The
third and fourth satellites, being farther removed from its influence,
revolve in orbits with a very small excentricity. And, although the
first two sensibly move in circles, their orbits acquire a small
ellipticity, from the disturbances they experience (N. 86).
It has been stated, that the attraction of a sphere on an exterior body
is the same as if its mass were united in one particle in its centre of
gravity, and therefore inversely as the square of the distance. In a
spheroid, however, there is an additional force arising from the bulging
mass at its equator, which, not following the exact law of gravity, acts
as a disturbing force. One effect of this disturbing force in the
spheroid of Jupiter is to occasion a direct motion in the greater axes
of the orbits of all his satellites, which is more rapid the nearer the
satellite is to the planet, and very much greater than that part of
their motion which arises from the disturbing action of the sun. The
same cause occasions the orbits of the satellites to remain nearly in
the plane of Jupiter’s equator (N. 87), on account of which the
satellites are always seen nearly in the same line (N. 88); and the
powerful action of that quantity of prominent matter is the reason why
the motions of the nodes of these small bodies are so much more rapid
than those of the planet. The nodes of the fourth satellite accomplish a
tropical revolution in 531 years, while those of Jupiter’s orbit require
no less than 36,261 years;—a proof of the reciprocal attraction between
each particle of Jupiter’s equator and of the satellites. In fact, if
the satellites moved exactly in the plane of Jupiter’s equator, they
would not be pulled out of that plane, because his attraction would be
equal on both sides of it. But, as their orbits have a small inclination
to the plane of the planet’s equator, there is a want of symmetry, and
the action of the protuberant matter tends to make the nodes regress by
pulling the satellites above or below the planes of their orbits; an
action which is so great on the interior satellites, that the motions of
their nodes are nearly the same as if no other disturbing force existed.
The orbits of the satellites do not retain a permanent inclination,
either to the plane of Jupiter’s equator, or to that of his orbit, but
to certain planes passing between the two, and through their
intersection. These have a greater inclination to his equator the
farther the satellite is removed, owing to the influence of Jupiter’s
compression; and they have a slow motion corresponding to secular
variations in the planes of Jupiter’s orbit and equator.
The satellites are not only subject to periodic and secular inequalities
from their mutual attraction, similar to those which affect the motions
and orbits of the planets, but also to others peculiar to themselves. Of
the periodic inequalities arising from their mutual attraction the most
remarkable take place in the angular motions (N. 89) of the three
nearest to Jupiter, the second of which receives from the first a
perturbation similar to that which it produces in the third; and it
experiences from the third a perturbation similar to that which it
communicates to the first. In the eclipses these two inequalities are
combined into one, whose period is 437·659 days. The variations peculiar
to the satellites arise from the secular inequalities occasioned by the
action of the planets in the form and position of Jupiter’s orbit, and
from the displacement of his equator. It is obvious that whatever alters
the relative positions of the sun, Jupiter, and his satellites, must
occasion a change in the directions and intensities of the forces, which
will affect the motions and orbits of the satellites. For this reason
the secular variations in the excentricity of Jupiter’s orbit occasion
secular inequalities in the mean motions of the satellites, and in the
motions of the nodes and apsides of their orbits. The displacement of
the orbit of Jupiter, and the variation in the position of his equator,
also affect these small bodies (N. 90). The plane of Jupiter’s equator
is inclined to the plane of his orbit at an angle of 3° 5ʹ 30ʺ, so that
the action of the sun and of the satellites themselves produces a
nutation and precession (N. 91) in his equator, precisely similar to
that which takes place in the rotation of the earth, from the action of
the sun and moon. Hence the protuberant matter at Jupiter’s equator is
continually changing its position with regard to the satellites, and
produces corresponding mutations in their motions. And, as the cause
must be proportional to the effect, these inequalities afford the means,
not only of ascertaining the compression of Jupiter’s spheroid, but they
prove that his mass is not homogeneous. Although the apparent diameters
of the satellites are too small to be measured, yet their perturbations
give the values of their masses with considerable accuracy—a striking
proof of the power of analysis.
A singular law obtains among the mean motions and mean longitudes of the
first three satellites. It appears from observation that the mean motion
of the first satellite, plus twice that of the third, is equal to three
times that of the second; and that the mean longitude of the first
satellite, minus three times that of the second, plus twice that of the
third, is always equal to two right angles. It is proved by theory,
that, if these relations had only been approximate when the satellites
were first launched into space, their mutual attractions would have
established and maintained them, notwithstanding the secular
inequalities to which they are liable. They extend to the synodic
motions (N. 92) of the satellites; consequently they affect their
eclipses, and have a very great influence on their whole theory. The
satellites move so nearly in the plane of Jupiter’s equator, which has a
very small inclination to his orbit, that the first three are eclipsed
at each revolution by the shadow of the planet, which is much larger
than the shadow of the moon: the fourth satellite is not eclipsed so
frequently as the others. The eclipses take place close to the disc of
Jupiter when he is near opposition (N. 93); but at times his shadow is
so projected with regard to the earth, that the third and fourth
satellites vanish and reappear on the same side of the disc (N. 94).
These eclipses are in all respects similar to those of the moon: but,
occasionally, the satellites eclipse Jupiter, sometimes passing like
obscure spots across his surface, resembling annular eclipses of the
sun, and sometimes like a bright spot traversing one of his dark belts.
Before opposition, the shadow of the satellite, like a round black spot,
precedes its passage over the disc of the planet; and, after opposition,
the shadow follows the satellite.
In consequence of the relations already mentioned in the mean motions
and mean longitudes of the first three satellites, they never can be all
eclipsed at the same time: for, when the second and third are in one
direction, the first is in the opposite direction; consequently, when
the first is eclipsed, the other two must be between the sun and
Jupiter. The instant of the beginning or end of an eclipse of a
satellite marks the same instant of absolute time to all the inhabitants
of the earth; therefore, the time of these eclipses observed by a
traveller, when compared with the time of the eclipse computed for
Greenwich, or any other fixed meridian (N. 95), gives the difference of
the meridians in time, and, consequently, the longitude of the place of
observation. The longitude is determined with extreme precision whenever
it is possible to convey the time instantaneously by means of
electricity from one place to another, since it obviates the errors of
clocks and chronometers. The eclipses of Jupiter’s satellites have been
the means of a discovery which, though not so immediately applicable to
the wants of man, unfolds one of the properties of light—that medium
without whose cheering influence all the beauties of the creation would
have been to us a blank. It is observed, that those eclipses of the
first satellite which happen when Jupiter is near conjunction (N. 96),
are later by 16ʹ 26ʺ·6 than those which take place when the planet is in
opposition. As Jupiter is nearer to us when in opposition by the whole
breadth of the earth’s orbit than when in conjunction, this circumstance
is to be attributed to the time employed by the rays of light in
crossing the earth’s orbit, a distance of about 190,000,000 of miles;
whence it is estimated that light travels at the rate of 192,000 miles
in one second. Such is its velocity, that the earth, moving at the rate
of nineteen miles in a second, would take two months to pass through a
distance which a ray of light would dart over in eight minutes. The
subsequent discovery of the aberration of light has fully confirmed this
astonishing result.
Objects appear to be situate in the direction of the rays which proceed
from them. Were light propagated instantaneously, every object, whether
at rest or in motion, would appear in the direction of these rays; but,
as light takes some time to travel, we see Jupiter in conjunction, by
means of rays that left him 16^m 26^s·6 before; but, during that time,
we have changed our position, in consequence of the motion of the earth
in its orbit: we therefore refer Jupiter to a place in which he is not.
His true position is in the diagonal (N. 97) of the parallelogram, whose
sides are in the ratio of the velocity of light to the velocity of the
earth in its orbit, which is as 192,000 to 19, or nearly as 10,000 to 1.
In consequence of the aberration of light, the heavenly bodies seem to
be in places in which they are not. In fact, if the earth were at rest,
rays from a star would pass along the axis of a telescope directed to
it; but, if the earth were to begin to move in its orbit with its usual
velocity, these rays would strike against the side of the tube; it
would, therefore, be necessary to incline the telescope a little, in
order to see the star. The angle contained between the axis of the
telescope and a line drawn to the true place of the star is its
aberration, which varies in quantity and direction in different parts of
the earth’s orbit; but, as it is only 20ʺ·481, it is insensible in
ordinary cases (N. 98).
The velocity of light deduced from the observed aberration of the fixed
stars perfectly corresponds with that given by the eclipses of the first
satellite. The same result, obtained from sources so different, leaves
not a doubt of its truth. Many such beautiful coincidences, derived from
circumstances apparently the most unpromising and dissimilar, occur in
physical astronomy, and prove connections which we might otherwise be
unable to trace. The identity of the velocity of light, at the distance
of Jupiter, and on the earth’s surface, shows that its velocity is
uniform; and as light consists in the vibrations of an elastic medium or
ether filling space, the uniformity of its velocity shows that the
density of the medium throughout the whole extent of the solar system
must be proportional to its elasticity (N. 99). Among the fortunate
conjectures which have been confirmed by subsequent experience, that of
Bacon is not the least remarkable, “It produces in me,” says the
restorer of true philosophy, “a doubt whether the face of the serene and
starry heavens be seen at the instant it really exists, or not till some
time later: and whether there be not, with respect to the heavenly
bodies, a true time and an apparent time, no less than a true place and
an apparent place, as astronomers say, on account of parallax. For it
seems incredible that the species or rays of the celestial bodies can
pass through the immense interval between them and us in an instant, or
that they do not even require some considerable portion of time.”
Great discoveries generally lead to a variety of conclusions: the
aberration of light affords a direct proof of the motion of the earth in
its orbit; and its rotation is proved by the theory of falling bodies,
since the centrifugal force it induces retards the oscillations of the
pendulum (N. 100) in going from the pole to the equator. Thus a high
degree of scientific knowledge has been requisite to dispel the errors
of the senses (N. 237).
The little that is known of the theories of the satellites of Saturn and
Uranus is, in all respects, similar to that of Jupiter. Saturn is
accompanied by eight satellites. The seventh is about the size of Mars,
and the eighth was simultaneously discovered by Mr. Bond in America, and
that distinguished astronomer Mr. Lassell, of Liverpool. The orbits of
the two last have a sensible inclination to the plane of the ring; but
the great compression of Saturn occasions the other satellites to move
nearly in the plane of his equator. So many circumstances must concur to
render the two interior satellites visible, that they have very rarely
been seen. They move exactly at the edge of the ring, and their orbits
never deviate from its plane. In 1789 Sir William Herschel saw them like
beads, threading the slender line of light which the ring is reduced to
when seen edgewise from the earth. And for a short time he perceived
them advancing off it at each end, when turning round in their orbits.
The eclipses of the exterior satellites only take place when the ring is
in this position. Mr. Lassell, with a powerful telescope, made by
himself, has seen Iapetus, the nearest of the two, on several occasions,
even when the opening of the ring was very wide, which made it extremely
difficult to see so minute an object. Of the situation of the equator of
Uranus we know nothing, nor of his compression; but the orbits of his
satellites are nearly perpendicular to the plane of the ecliptic; and,
by analogy, they ought to be in the plane of his equator. Uranus is so
remote that he has more the appearance of a planetary nebula than a
planet, which renders it extremely difficult to distinguish the
satellites at all; and quite hopeless without such a telescope as is
rarely to be met with even in observatories. Sir William Herschel
discovered the two that are farthest from the planet, and ascertained
their approximate periods, which his son afterwards determined to be
13^d 11^h 7^m 12^s·6 and 8^d 16^h 56^m 28^s·6 respectively. The
orbits of both seem to have an inclination of about 101°·2 to the plane
of the ecliptic. The two interior satellites are so faint and small, and
so near the edge of the planet, that they can with difficulty be seen
even under the most favourable circumstances: however, Mr. Lassell has
ascertained that the more distant of the two revolves about Uranus in 4
days, and that nearest to the planet in 2-1/2 days, and from a long and
minute examination he is convinced that the system only consists of four
satellites. Soon after Neptune was seen Mr. Lassell discovered the only
satellite known to belong to that planet. The satellites of Uranus and
Neptune, the two planets on the remotest verge of the solar system,
offer the singular and only instance of a revolution from east to west,
while all the planets and all the other satellites revolve from west to
east. Retrograde motion is occasionally met with in the comets and
double stars.