Let x' G): sin E:: R: r cos l. R sin E cos 7 sin (L G) SP cos PSP =r r cos l, NSP, and L"N+x=VSN + NSP centric orbit longitude of the planet (199). Then DP helio SD tang tang x' tang x or, tang (LN) tang (LN) (G) cos I , cos I Consequently, L" N+ x', becomes known. 274. Longitude of the perihelion, &c. Assuming the orbit of the planet to be an ellipse, if its heliocentric orbit longitude, and its radius vector be found for three different times (273), we may thence determine the longitude of the perihelion, the eccentricity, and the semi-transverse axis, of the orbit. Let PDG, Fig. 48, be the orbit, P the perihelion, and D, E and F, the three positions of the planet in its orbit. Then, SD, SE and SF are known, and from the longitudes, the angles DSE and DSF are also known. Putr SD, SE, SF, angle r'' = r'' = DSE, & = DSF, x = PSD, a = PC = semi-transverse axis, and e = the eccentricity. Then, ae = SC (151). Hence (Conic Sections), In like manner, from (H) and (K), we have, r'' cos (x + ¢) Put ' (M) r = m, and r'' — r = n; then from (L) and (M), we have, m The value of x being subtracted from the orbit longitude of the planet in the first position, the remainder must be the orbit longitude of the perihelion. Then, if L and L" be the ecliptic and orbit longitudes of the perihelion, we have (273 G), tang (L ~ N) = tang (L” Whence, L, the heliocentric ecliptic longitude of the perihelion, becomes known. The value of e, the eccentricity, may be found from either of the expressions (L) and (M), and a, the semi-transverse axis, from (H), (I), or (K). Scholium. In the above investigation it has been assumed, in accordance with Kepler's first law, that the orbits of the planets are ellipses. That they are so, or, at least, nearly so, is established by the fact, that different sets of observations, made on the planet in various parts of its orbit, give very nearly the same results for the longitude of the perihelion, the eccentricity, and the semitransverse axis. The semi-transverse axes of the orbits of the planets, or their mean distances determined as above, and the periodic times determined by a previous article (272), are found to accord with Kepler's third law (154), or nearly so. As the truth of this law has been confirmed by investigations in physical astronomy, and as the periodic times of the planets may be determined with great precision, we may, from these, obtain more accurate values of the semi-transverse axis. Thus, putting P and p for the periodic times of the earth and planet respectively, and A and a for their mean distances, we have P2: p2:: A3: a3. Whence, a becomes known, A3: P2 when P, p, and A are known. It is, however, usual to assume the earth's mean distance from the sun to be a unit, and to express the mean distances of the planets and the radii vectores of the earth and planets, in accordance with this assumption. We then have, P2: p2:: 1: a3; or, P2 275. Motions of the perihelions and changes in the eccentricities. From observations made on each of the different planets, at distant periods, it is found that the perihelions of all their orbits have slow motions. The motion of the perihelion of the orbit of Venus is retrograde. Those of the other planets are direct. The eccentricities of the orbits are also subject to small secular variations. Some of them are at present increasing, and others decreasing. 276. Semi-transverse axes of the orbits. The semi-transverse axes of the orbits, or, which is the same, the mean distances of the planets from the sun, do not change. This fact was first discovered by Lagrange, from investigations in physical astronomy, and it accords with observation. 277. Epoch of a planet's being at its perihelion. From several observations of the planet about the time it has the same longitude as the perihelion, the exact time that it is at the perihelion may be obtained by proportion or interpolation. 278. Scholium. There are various other methods for determining most of the elements of the orbit of a planet, besides those given in the preceding articles. Those which are founded on ob servations of the planet when in conjunction or opposition, and at the nodes, are the most convenient and accurate. The elements of the orbit may, also, be determined with tolerable accuracy, by certain methods of estimation and computation, without extending the observations to the time of the planet's passage through the node. These methods were applied on the discovery of the new planets. In determining the elements, many hundreds, or even thousands of observations have been employed; and, with the exception of those of the new planets, they are now known with a high degree of precision. 279. Tables of the planets. When the elements of a planet's elliptical orbit have been determined, its place in the orbit may be calculated by Kepler's Problem, or by a table of the equation of the centre, computed by means of that problem. But the motion of a planet is subject to small perturbations, produced by the actions of the other planets. Investigations in physical astronomy have furnished the means of computing these perturbations and forming tables by which their values for a given time may be easily obtained. A complete set of tables for any planet includes tables of the mean heliocentric places and motions of the planet, and of the perihelion and ascending node of the orbit, the equation of the centre, the values of the perturbations, the reduction of the planet's place in its orbit to the ecliptic, and the radius vector of the planet or its logarithm.* 280. Geocentric longitude and latitude. From the heliocentric longitude, latitude and radius vector of a planet as obtained from the tables, it is often required to find the geocentric longitude and latitude. Referring to Fig. 47, and designating the quantities as in article (273), we have, by trigonometry, SE + Sp: SE Sp :: tang (SpE + SEp): tang 1 (SpE SEp), or, R+r cos 7: Rr cos :: tang (p + E): tang 1 (p ∞ E), l * The best tables of the planets are those by Lindenau, for Mercury, Venus and Mars, with the explanations in Latin; and those by Bouvard, for Jupiter, Saturn and Uranus, with the explanations in French. Then, as (273), p + EL G+ G- L'L-L', we have, 1 + tang : 1 tang ◊ : : tang 1⁄2 (L – L') : tang 1⁄2 (p ∞ E). Ꮎ ∞ Hence, tang (pE) or, (App. 15), tang 1 tang (45°). tang (L (p2 E) Then, 1⁄2 (p2 E), added to (p + E), or its equal } (L – L'), for a superior planet, or subtracted from it for an inferior planet, gives E, the elongation. And E added to L' gives G, the geocentric longitude. As (L —L'), or its equal (p + E), is the supplement of S, and as the sine of an angle is the same as that of its supplement, we have for 2, the geocentric latitude (273, F), tang a sin E tang l ..(N) When the planet is in conjunction or opposition, this formula for the geocentric latitude is not applicable; for, then, E and (L - L') are either 0° or 180°, and, consequently, their sines are, each, zero. But, as E, S and p are then in a straight line, we Rr cos l. Hence, SE + Sp have Ep The upper sign appertains to the conjunction of a superior planet or superior conjunction of an inferior, and the lower, to opposition or inferior conjunction. 281. Distance of a planet from the earth. For the distance of a planet from the earth, we have, Another expression, more accurate in practice, especially when the latitudes are small, may be easily obtained. For, sin E: sin S: Sp: Ep, or, sin E: sin (L L'): :r cos 7: EP cos 2. |
