ELLIPTICAL MOTION OF THE PLANETS. 390. If r2 = a2 (1 and ia (1 dv = [Book II, e cos nt)' be put forr = a (1- e cos nt), e cos nt)e sin nt for dr ndt in the development of r in e cos nt) + i.e.sin.nt (1-ecos.nt) + i.e.sin.nt (1 − e cos.nt)—1 2ndt + i.ed.sin.nt (1-ecos.nt)-1 + &c. 2.3.ndt Leti = - 2, then = 1 + 2e. cos If this quantity be substituted in the preceding expression for dv, when the integration is accomplished, and the approximation only carried to the sixth powers of e, the result will be 391. The angles v and nt which are the true and mean anomaly, begin at the perihelion; but if they be estimated from the aphelion, it will only be necessary to make e negative in the values of rand v, or to add 180° to each angle. This expression gives v equation of the centre. nt the True Longitude and Radius Vector in functions of the Mean Longitude. fig. 76. B M 392. Instead of fixing the origin of the time at the instant of the planet's passage at the perihelion, let it be fixed at any point whatever, as E, fig. 76, so that nt = ECB, then by adding the constant angle CE represented by e, the whole angle CB = nt + e is the mean longitude of the planet, being the equinox of m E A P C S Spring; and if the constant angle CP, which is the longitude of the perihelion, be represented by w, the angle nt + e − 5 = PCB must be put for nt, and if v be estimated from ∞, then v for v, and the preceding values of v and r become, nt + € + {2e - + e3} sin (nt + e +5 &c. 11 24 3 8 - ) must be put (97) e*} sin 2(nt + c − a), &c. - ) e3} cos (nt + e {e- &c. (98) cos 2 (nt +€ – 5), 393. v is the true longitude of the planet and nt + ɛ its mean longitude both being estimated on the plane of the orbit. The angle = = CE is the longitude of the point E, from whence the time is estimated, commonly called the longitude of the epoch. 394. In astronomical series, the quantities which multiply the sines and cosines are the coefficients; and the angles are called the arguments: for example in (2e-1e2) sin (nt + - ) the part 2e - 1 e is the coefficient, and nt + ɛ - is the argument. 395. Although the time increases without limit, these series converge: for, as a sine or cosine never can exceed the radius, the values of the sincs and cosines in these series never can be greater than unity, however much the time may increase, and as the powers of e soon become extremely small, they converge rapidly. 396. The values of v and r answer for all the planets and satellites, since they are independent of the masses, for the mass of a planet is so inconsiderable in comparison of that of the sun, that it may be omitted, and as the mass of the sun forms the standard of comparison for the masses of the other bodies of the system, it is assumed to be the unit of measure. The same holds with regard to a planet and its satellites. Determination of the Position of the Orbit in space. 397. The values of vandr give the place of a body in its orbit, but not its position in space; they however afford the means of ascertaining it. For let NpnG, fig. 77, be the plane of the ecliptic, or fixed plane at the epoch, on which the plane of the orbit PnAN has G A fig. 77. m n P N P a very small inclination; then Nn is the line of the nodes; S the sun, and if mp be a perpendicular from the planet on the plane of the ecliptic, it will be the tangent of the latitude mSp. Let SN the longitude of the node be represented by 6 when estimated on the plane of the orbit, and let 0 represent the same angle when projected on the plane of the ecliptic; also let v = Sp be the true longitude Sm or v, when projected on the plane of the ecliptic. Then NSp = 0, 0, NSm = v - 6,. And if be the inclination of the two planes, it appears from the right angled triangle pNm, that 398. This gives v, in terms of v, and the contrary. But these two angles may be obtained in terms of one another in very converging v-0=v-6-tan.sin 2(0-6)+tan.sin 4 (v-6)-&c. (100) v-6=0,-0+tan.sin 2(2-0)+tan.sin 4(-)+&c. (101) 2 Projected Longitude in Functions of Mean Longitude. 399. A value of v -0, or NSp, may be found in terms of the sines and cosines of nt, and its multiple arcs, from the series 5 11 v = nt + e +{2e e3} sin (nt + 6-) + e2 ex sin 2(nt + € – π) + &c. which may be written v = nt + e + eQ. If be subtracted from both sides of this equation, and the sines taken in place of the arcs, it becomes sin (v - 6) = sin (nt + € - 6 + eQ), which may be expanded into a series, ascending, according to the powers of e, by the method already employed for the development of vandr; if = sin (v - 6) = sin (nt + m 6 + eQ). Whence it may be found that, sin i (v - 6) = sin i (nt + 6-6+eQ) = {1-2 × sin i (nt + € - 6) + {ieQ PeQ+ i*e*Q* _ &c.} &c. } 1.2 1.2.3.4 reQ cos i (nt + e-6) + &c. Latitude. &c.} x 400. If mp, the tangent of the latitude, be represented by s, the right-angled triangle mNp gives s = tan & sin (v, - θ). Curtate Distances. 401. Letr, be the curtate distance Sp, then Spm, being a right angle, hence or r = r (1 + s2)− = r {1 - 18+ - &c.} (102) 402. Thus v,, s, and r,, the longitude, latitude, and curtate distance of the planet are determined in convergent series of the sines and cosines of nt and its multiples; if therefore the time be assumed, the place of the body will be known, and the means are thus furnished for computing tables of the motions of the planets and satellites, from which their elliptical places may be ascertained at any instant. 403. A particular period is chosen as an origin from whence the time is estimated, which is called the Epoch of the tables: the elements of the orbits are determined by observation; and the longitude, latitude, and distance of the body from the sun are computed for that period, and for every succeeding day, hour, and minute, if necessary, for any number of years; these are arranged in tables according to the time; so that by inspection alone the corresponding place of the body referred to the fixed plane, or position of the ecliptic at the epoch, may be found. Fortunately for the facility of astronomical calculations, the orbits of the celestial bodies are either very nearly circular, as in the |