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Elliptical Motion - Mean and True Motion Equinoctial Ecliptic
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 attraetion 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 carth 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 nude: 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° 42' 29.8" to it; on which account it is more difficult to determine their motions.
Were the planets attracted by the sun only, they would always inove 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 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 thirda 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.