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1st Satellite, 6th day, at 9 m. after 2 morning.
2d Satellite, 20th,
Mercury will be stationary on the 4th, and attain his greatest elongation on the 17th; and Jupiter will be in opposition at 50 m. after 12 on the 30th. Saturn will be in quadrature at 45 m. after 10 in the morning of the 9th. Georgium Sidus will be in opposition at 30 m. after 5 in the morning of the 10th; and the Moon will be in conjunction with the star marked in Libra, at 36 m. after 6 in the evening of the 15th.
On the ELEMENTS of the PLANETARY ORBITS.
The orbits of the planets are curves, whose planes pass through the centre of the Sun; and hence each of these orbits intersect the ecliptic in two opposite points, which are called its nodes; these are situated in the same right line, passing through the centre of the Sun, and called the line of the nodes. The first element of the planetary orbits to be determined by the astronomer, is the situation of these nodes on the ecliptic, and consequently the position of this line. The most favourable times for determining this by observation, are when the planet has no latitude, and is in opposition to or conjunction with the Sun; for then an observer, situated at the centre of the Sun, would see the planet and the earth on the same right line.
The situations of the nodes being thus determined by observations of the kind above indicated, and the
necessary calculations founded upon them, this element is made the foundation of the method for ascertaining the inclination of the plane of the orbit to that of the ecliptic. When the Sun arrives at the node of the planet, or the longitude of the Sun becomes the same as that of the node, the geocentric latitude of the planet, at that moment, is computed from observations, and then a simple trigonometrical calculation will give the inclination of the planetary orbit.
This method, therefore, supposes the node of the planet to be previously and accurately known, and also that the astronomer is able to observe the planet in the whole of its course. It is, however, perhaps impossible to seize the exact moment when the Sun is in the node of the planet; but this difficulty is avoided by observing the two bodies for several days in succession, before and after the epoch of the Sun's passage through the node, and then determining the exact instant in which the phenomenon happened by interpolating the results. Besides, the error in the inclination which would result from a small error in the situation of the node would be so minute, as to be nearly, if not altogether, insensible.
The supposition of the astronomer being able to observe the planet in any part of its orbit, is only applicable to the seven old planets; for the new planets and comets this position and the nature of the curve must be determined from a small part of the orbit. This renders the problem of very difficult solution, and which is accomplished by applying to it the same laws, founded upon the principle of universal gravitation, which exist with respect to the other planetary bodies. The methods of accomplishing this have been explained, in all their generality, with great learning and ingenuity, by Laplace, in his Mécanique Céleste.
Neither the nodes of the planets, nor the inclination of their orbits, are absolutely fixed; for when
the positions of the nodes are determined at distant epochs of time, and referred to the ecliptic, they are found to have experienced variations, and to have a very slow retrograde motion. These variations, as well as those which take place in the inclination of the orbit, are denominated secular inequalities, and are necessary consequences of universal gravitation; which modern analysis not only establishes, but also furnishes the means of calculating their effects on the planetary motions.
When the position of the orbit is thus determined, the law of the planet's motion, and the nature of the curve it describes, are then the objects of research; and these would be known if the length of the radius vector and the angle it makes with a fixed right line situated in the plane of its orbit, and passing through the centre of the Sun, were assigned at every instant. The first object is therefore to ascertain the duration of one complete sidereal revolution of the planet; the most simple means of accomplishing this is, to observe two consecutive passages through the same node. This being found, the mean angular motions of the planet about the Sun are easily deduced, and the variations in its distance from the Sun determined. For these purposes, observations at the time of conjunctions and oppositions are favourable, as these take place in different points of their orbits. Thus a series of similar observations gives different angles and different radii vectores; and as these radii are known in terms of the radius of the solar orbit, and their true directions from the centre of the system ascertained, the figure of the planetary orbit becomes known.
The eccentricities of the planetary orbits experience very slow variations, both the law and extent of which have been determined by theory. The eccentricities of Mercury, Mars, and Jupiter increase, but those of all the other planets diminish.
The Perihelia, or the points of the planet's greatest
distance from the Sun, are not fixed; but have a slow motion in the plane of their orbits, in the same manner as the perigeon of the solar orbit moves along the plane of the ecliptic. For all the planets, except Venus, these movements are direct, or in the same direction as that of the Sun; but for Venus this motion is retrograde. The observations relative to those small variations that can be depended upon, are still of too recent a date to give them with certainty: the theory of attraction is therefore the most accurate.
From a consideration of these principles, it may be perceived that the knowledge of the elliptical movement of each planet depends upon seven elements; and as there are eleven planets, seventy-seven elements must be determined in order to have a complete knowledge of our planetary system, in the present state of astronomy.
It would much exceed the limits, as well as be foreign to the nature, of the present work, to enter into a particular explanation of the methods, and give the formula which the improvements of modern analysis have established for ascertaining these several elements. We must therefore rest satisfied with giving the results that have been obtained by the most eminent astronomers, and refer the reade to their works for the methods by which they have been fonnd.
We have already given the sidereal revolutions and mean distances of the planets from the Sun in our Occurrences for March, inserted in a preceding part of this volume; to which we must therefore refer for these elements. It is also necessary to remark, that the recent discovery of the four new telescopic planets, Ceres, Pallas, Vesta, and Juno, and the small number of observations that astronomers have yet been able to make upon them for the purpose, are not sufficient to determine their secular inequalities with the desirable accuracy: those which are given in the following tables must therefore be regarded as only approximations to the truth. The subsequent
results are extracted from Laplace's Exposition du Systême du Monde.
Eccentricities of the Planetary Orbits, at the Commencement of 1801, expressed in Parts of their semitransverse Axes.
Secular Variations to which the preceding Eccentricities are subject. The Sign-denotes Diminution.
Not yet ascertained
Longitudes of the Planets at the Commencement of 1801; reckoned from the Vernal Equinox.