Elliptical Motion—Mean and True
Motion—Equinoctial—Ecliptic—Equinoxes—Mean and True Longitude—Equation
of Centre—Inclination of the Orbits of Planets—Celestial
Latitude—Nodes—Elements of an Orbit—Undisturbed or Elliptical
Orbits—Great Inclination of the Orbits of the New Planets—Universal
Gravitation the Cause of Perturbations in the Motions of the Heavenly
Bodies—Problem of the Three Bodies—Stability of Solar System depends
upon the Primitive Momentum of the Bodies.
A PLANET moves in its elliptical orbit with a velocity varying every
instant, in consequence of two forces, one tending to the centre of the
sun, and the other in the direction of a tangent (N. 38) to its orbit,
arising from the primitive impulse given at the time when it was
launched into space. Should the force in the tangent cease, the planet
would fall to the sun by its gravity. Were the sun not to attract it,
the planet would fly off in the tangent. Thus, when the planet is at the
point of its orbit farthest from the sun, his action overcomes the
planet’s velocity, and brings it towards him with such an accelerated
motion, that at last it overcomes the sun’s attraction, and, shooting
past him, gradually decreases in velocity until it arrives at the most
distant point, where the sun’s attraction again prevails (N. 39). In
this motion the _radii vectores_ (N. 40), or imaginary lines joining the
centres of the sun and the planets, pass over equal areas or spaces in
equal times (N. 41).
The mean distance of a planet from the sun is equal to half the major
axis (N. 42) of its orbit: if, therefore, the planet described a circle
(N. 43) round the sun at its mean distance, the motion would be uniform,
and the periodic time unaltered, because the planet would arrive at the
extremities of the major axis at the same instant, and would have the
same velocity, whether it moved in the circular or elliptical orbit,
since the curves coincide in these points. But in every other part the
elliptical, or true motion (N. 44), would either be faster or slower
than the circular or mean motion (N. 45). As it is necessary to have
some fixed point in the heavens from whence to estimate these motions,
the vernal equinox (N. 46) at a given epoch has been chosen. The
equinoctial, which is a great circle traced in the starry heavens by the
imaginary extension of the plane of the terrestrial equator, is
intersected by the ecliptic, or apparent path of the sun, in two points
diametrically opposite to one another, called the vernal and autumnal
equinoxes. The vernal equinox is the point through which the sun passes
in going from the southern to the northern hemisphere; and the autumnal,
that in which he crosses from the northern to the southern. The mean or
circular motion of a body, estimated from the vernal equinox, is its
mean longitude; and its elliptical, or true motion, reckoned from that
point, is its true longitude (N. 47): both being estimated from west to
east, the direction in which the bodies move. The difference between the
two is called the equation of the centre (N. 48); which consequently
vanishes at the apsides (N. 49), or extremities of the major axis, and
is at its maximum ninety degrees (N. 50) distant from these points, or
in quadratures (N. 51), where it measures the excentricity (N. 52) of
the orbit; so that the place of the planet in its elliptical orbit is
obtained by adding or subtracting the equation of the centre to or from
its mean longitude.
The orbits of the principal planets have a very small obliquity or
inclination (N. 53) to the plane of the ecliptic in which the earth
moves; and, on that account, astronomers refer their motions to this
plane at a given epoch as a known and fixed position. The angular
distance of a planet from the plane of the ecliptic is its latitude
(N. 54), which is south or north according as the planet is south or
north of that plane. When the planet is in the plane of the ecliptic,
its latitude is zero; it is then said to be in its nodes (N. 55). The
ascending node is that point in the ecliptic through which the planet
passes in going from the southern to the northern hemisphere. The
descending node is a corresponding point in the plane of the ecliptic
diametrically opposite to the other, through which the planet descends
in going from the northern to the southern hemisphere. The longitude and
latitude of a planet cannot be obtained by direct observation, but are
deduced from observations made at the surface of the earth by a very
simple computation. These two quantities, however, will not give the
place of a planet in space. Its distance from the sun (N. 56) must also
be known; and, for the complete determination of its elliptical motion,
the nature and position of its orbit must be ascertained by observation.
This depends upon seven quantities, called the elements of the orbit
(N. 57). These are, the length of the major axis, and the excentricity,
which determine the form of the orbit; the longitude of the planet when
at its least distance from the sun, called the longitude of the
perihelion; the inclination of the orbit to the plane of the ecliptic,
and the longitude of its ascending node: these give the position of the
orbit in space; but the periodic time, and the longitude of the planet
at a given instant, called the longitude of the epoch, are necessary for
finding the place of the body in its orbit at all times. A perfect
knowledge of these seven elements is requisite for ascertaining all the
circumstances of undisturbed elliptical motion. By such means it is
found that the paths of the planets, when their mutual disturbances are
omitted, are ellipses nearly approaching to circles, whose planes,
slightly inclined to the ecliptic, cut it in straight lines, passing
through the centre of the sun (N. 58). The orbits of the
recently-discovered planets deviate more from the ecliptic than those of
the ancient planets: that of Pallas, for instance, has an inclination of
34° 42ʹ 29·8ʺ to it; on which account it is more difficult to determine
their motions.
Were the planets attracted by the sun only, they would always move in
ellipses, invariable in form and position; and because his action is
proportional to his mass, which is much larger than that of all the
planets put together, the elliptical is the nearest approximation to
their true motions. The true motions of the planets are extremely
complicated, in consequence of their mutual attraction, so that they do
not move in any known or symmetrical curve, but in paths now approaching
to, now receding from, the elliptical form; and their radii vectores do
not describe areas or spaces exactly proportional to the time, so that
the areas become a test of disturbing forces.
To determine the motion of each body, when disturbed by all the rest, is
beyond the power of analysis. It is therefore necessary to estimate the
disturbing action of one planet at a time, whence the celebrated problem
of the three bodies, originally applied to the moon, the earth, and the
sun—namely, the masses being given of three bodies projected from three
given points, with velocities given both in quantity and direction; and
supposing the bodies to gravitate to one another with forces that are
directly as their masses, and inversely as the squares of the distances,
to find the lines described by these bodies, and their positions at any
given instant; or, in other words, to determine the path of a celestial
body when attracted by a second body, and disturbed in its motion round
the second body by a third—a problem equally applicable to planets,
satellites, and comets.
By this problem the motions of translation of the celestial bodies are
determined. It is an extremely difficult one, and would be infinitely
more so if the disturbing action were not very small when compared with
the central force; that is, if the action of the planets on one another
were not very small when compared with that of the sun. As the
disturbing influence of each body may be found separately, it is assumed
that the action of the whole system, in disturbing any one planet, is
equal to the sum of all the particular disturbances it experiences, on
the general mechanical principle, that the sum of any number of small
oscillations is nearly equal to their simultaneous and joint effect.
On account of the reciprocal action of matter, the stability of the
system depends upon the intensity of the primitive momentum (N. 59) of
the planets, and the ratio of their masses to that of the sun; for the
nature of the conic sections in which the celestial bodies move depends
upon the velocity with which they were first propelled in space. Had
that velocity been such as to make the planets move in orbits of
unstable equilibrium (N. 60), their mutual attractions might have
changed them into parabolas, or even hyperbolas (N. 22); so that the
earth and planets might, ages ago, have been sweeping far from our sun
through the abyss of space. But as the orbits differ very little from
circles, the momentum of the planets, when projected, must have been
exactly sufficient to ensure the permanency and stability of the system.
Besides, the mass of the sun is vastly greater than that of any planet;
and as their inequalities bear the same ratio to their elliptical
motions that their masses do to that of the sun, their mutual
disturbances only increase or diminish the excentricities of their
orbits by very minute quantities; consequently the magnitude of the
sun’s mass is the principal cause of the stability of the system. There
is not in the physical world a more splendid example of the adaptation
of means to the accomplishment of an end than is exhibited in the nice
adjustment of these forces, at once the cause of the variety and of the
order of Nature.