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(493.) It will first, however, be proper to explain by what process of calculation the expression of a planet's elliptic orbit by its elements can be compared with observation, and how we can satisfy ourselves that the numerical data contained in a table of such elements for the whole system does really exhibit a true picture of it, and afford the means of determining its state at every instant of time, by the mere application of Kepler's laws. Now, for each planet, it is necessary for this purpose to know, 1st, the magnitude and form of its ellipse; 2dly, the situation of this ellipse in space, with respect to the ecliptic, and to a fixed line drawn therein; 3dly, the local situation of the planet in its ellipse at some known epoch, and its periodic time or mean angular velocity, or, as it is called, its mean motion.
(494.) The magnitude and form of an ellipse are determined by its greatest length and least breadth, or its two principal axes; but for astronomical uses it is preferable to use the semi-axis major (or half the greatest length), and the excentricity or distance of the focus from the center, which last is usually estimated in parts of the former. Thus, an ellipse, whose length is 10 and breadth 8 parts of any scale, has for its major semi-axis 5, and for its excentricity 3 such parts ; but when estimated in parts of the semi-axis, regarded as a unit, the excentricity is expressed by the fraction
(495.) The ecliptic is the plane to which an inhabitant of the earth most naturally refers the rest of the solar system, as a sort of ground-plane; and the axis of its orbit might be taken for a line of departure in that plane or origin of angular reckoning. Were the axis fixed, this would be the best possible origin of longitudes; but as it has a motion (though an excessively slow one), there is, in fact, no advantage in reckoning from the axis more than from the line of the equinoxes, and astronomers therefore prefer the latter, taking account of its variation by the effect of precession, and restoring it, by calculation at every instant, to a fixed position. Now, to determine the situation of the ellipse described by a planet with respect to this plane, three elements require to
be known:- 1st, the inclination of the plane of the planet's orbit to the plane of the ecliptic; 2dly, the line in which these two planes intersect each other, which of necessity passes through the sun, and whose position with respect to the line of the equinoxes is therefore given by stating its longitude. This line is called the line of the nodes. When the planet is in this line, in the act of passing from the south to the north side of the ecliptic, it is in its ascending node, and its longitude at that moment is the element called the longitude of the node. These two data determine the situation of the plane of the orbit; and there only remains, for the complete determination of the situation of the planet's ellipse, to know how it is placed in that plane, which (since its focus is necessarily in the sun) is ascertained by stating the longitude of its perihelion, or the place which the extremity of the axis nearest the sun occupies, when orthographically projected on the ecliptic.
(496.) The dimensions and situation of the planet's orbit thus determined, it only remains, for a complete acquaintance with its history, to determine the circumstances of its motion in the orbit so precisely fixed. Now, for this purpose, all that is needed is to know the moment of time when it is either at the perihelion, or at any other precisely determined point of its orbit, and its whole period; for these being known, the law of the areas determines the place at every other instant. This moment is called (when the perihelion is the point chosen) the perihelion passage, or, when some point of the orbit is fixed upon, without special reference to the perihelion, the epoch.
(497.) Thus, then, we have seven particulars or elements, which must be numerically stated, before we can reduce to calculation the state of the system at any given moment. But, these known, it is easy to ascertain the apparent positions of each planet, as it would be seen from the sun, or is seen from the earth at any time. The former is called the heliocentric, the latter the geocentric, place of the planet.
(498.) To commence with the heliocentric places. Let S represent the sun; PAN the orbit of the planet, being an
ellipse, having the S in its focus, and A for its perihelion ; and let pa N r represent the projection of the orbit on the
plane of the ecliptic, intersecting the line of equinoxes Sr in , which, therefore, is the origin of longitudes. Then will S N be the line of nodes ;
and if we suppose B to lie on the south, and A on the north side of the ecliptic, and the direction of the planet's motion to be from B to A, N will be the ascending node, and the angle T S N the longitude of the node. In like manner, if P be the place of the planet at any time, and if it and the perihelion A be projected on the ecliptic, upon the points p, a, the angles r Sp, r S a, will be the respective heliocentric longitudes of the planet and of the perihelion, the former of which is to be determined, and the latter is one of the given elements. Lastly, the angle PSP is the heliocentric latitude of the planet, which is also required to be known.
(499.) Now, the time being given, and also the moment of the planet's passing the perihelion, the interval, or the time of describing the portion A P of the orbit, is given, and the periodical time, and the whole area of the ellipse being known, the law of proportionality of areas to the times of their description gives the magnitude of the area A SP. From this it is a problem of pure geometry to determine the corresponding angle A SP, which is called the planet's true anomaly. This problem is of the kind called transcendental, and has been resolved by a great variety of processes, some more, some less intricate. It offers, however, no peculiar difficulty, and is practically resolved with great facility by the help of tables constructed for the purpose, adapted to the case of each particular planet. *
• It will readily be understood, that, except in the case of uniform circular motion, an equable description of areas about any center is incompatible with an equable description of angles. The object of the problem in the text is to pass from the area, supposed known, to the angle, supposed unknown : in other words, to derive the true amount of angular motion from the perihelion, or the true anomaly from what is technically called the mean anomaly, that is, the mean angular motion which would have been performed had the motion in angle been uniform instead of the motion in area. It happens fortunately, that this is the
(500.) The true anomaly thus obtained, the planet's angular distance from the node, or the angle N SP, is to be found. Now, the longitudes of the perihelion and node being respectively r a and Y N, which are given, their difference a N is also given, and the angle N of the spherical rightangled triangle A N a, being the inclination of the plane of the orbit to the ecliptic, is known. Hence we calculate the arc N A, or the angle N S A, which, added to A SP, gives the angle N S P required. And from this, regarded as the measure of the arc N P, forming the hypothenuse of the right-angled spherical triangle P N p, whose angle N, as before, is known, it is easy to obtain the other two sides, N p and P p. The latter, being the measure of the angle PSP, expresses the planet's heliocentric latitude; the former measures the angle N S p, or the planet's distance in longitude from its node, which, added to the known angle TSN, the longitude of the node, gives the heliocentric longitude. This process, however circuitous it may appear, when once well understood may be gone through numerically by the aid of the usual logarithmic and trigonometrical tables, in little more time than it will have taken the reader to peruse its description.
(501.) The geocentric differs from the heliocentric place of a planet by reason of that parallactic change of apparent situation which arises from the earth's motion in its orbit. Were the planets' distances as vast as those of the stars, the earth's orbitual motion would be insensible when viewed from them, and they would always appear to us to hold the same relative situations among the fixed stars as if viewed from the sun, i. e. they would then be seen in their heliocentric places. The difference, then, between the heliocentric and geocentric places of a planet is, in fact, the same thing with its parallax, arising from the earth's removal from the centre
simplest of all problems of the transcendental kind, and can be resolved, in the most difficult case, by the rule of “ false position," or trial and error, in a very few minutes. Nay, it may even be resolved instantly on inspection by a simple and easily constructed piece of mechanism, of which the reader may see a description in the Cambridge Philosophical Transactions, vol. iv. p. 425., by the author of this work.
of the system and its annual motion. It follows from this, that the first step towards a knowledge of its amount, and the consequent determination of the apparent place of each planet, as referred from the earth to the sphere of the fixed stars, must be to ascertain the proportion of its linear distances from the earth and from the sun, as compared with the earth's distance from the sun, and the angular positions of all three with respect to each other.
(502.) Suppose, therefore, S to represent the sun, E the earth, and P the planet; S T the line of equinoxes, r E the earth's orbit, and Pp a perpendicular let fall from the planet on the ecliptic. Then will the angle SPE (according to the general notion of parallax conveyed in art. 69) re
P present the parallax of the planet
arising from the change of station * web p from S to E;EP will be the appa
rent direction of the planet seen from
E; and if SQ be drawn parallel to E p, the angle r S Q will be the geocentric longitude of the planet, while r S E represents the heliocentric longitude of the earth, r Sp that of the planet. The former of these, SEY, is given by the solar tables; the latter, r s p, is found by the process above described (art. 500). Moreover, S P is the radius vector of the planet's orbit, and S E that of the earth's, both of which are determined from the known dimensions of their respective ellipses, and the places of the bodies in them at the assigned time. Lastly, the angle PS p is the planet's heliocentric latitude.
(503.) Our object, then, is, from all these data, to determine the angle T S Q, and PE p, which is the geocentric latitude. The process, then, will stand as follows:- 1st, In the triangle SP p, right-angled at p, given SP, and the angle P S p (the planet's radius vector and heliocentric latitude), find Sp and P p; 2dly, In the triangle S E p, given Sp (just found), S E (the earth’s radius vector), and the angle ESP (the difference of heliocentric longitudes of the earth and planet), find the angle S p E, and the side Ep. The former being equal to the alternate angle pSQ,