Jupiter and Mars, as well as Venus, interfere in disturbing the elliptical quantity of the Earth's longitude, and the place of its nearest distance from the Sun. If we consider the masses of these bodies as three indeterminate quantities, we must, in order to determine them, use three observations at the least. We may use more: indeed, it is plain, that, the greater the number of observations, (supposing them to be equally accurate) the more exact will be the determination of the masses.. It is of no consequence, in the method which has been described, whether the planet, the mass of which is to be determined, be with or without a satellite. But the mass of a planet of the first kind may be determined most simply (see p. 29.) from the greatest elongation and period of its satellite. There are then, at the least, two methods for determining the mass of Jupiter we may, therefore, use the two methods, the one to serve as a check on the other; or, in determining the mass of Venus from some inequality either in the Earth's motion, or in an element of its orbit, we may contract the investigation by assuming the mass of Jupiter to be that which is determined by means of the period and the greatest elongation of one of his satellites. The mass of a planet that has no satellite must be determined from the effect of its disturbing force; the mass of a planet accompanied by a satellite may be determined by the effect either of its disturbing, or of its attracting force. But, in each case, the principle of the determination is precisely the same. That by which we measure the mass of a planet or the number of its particles, is some effect of their attraction. In one case the effect is the deflection of a satellite from its rectilinear course, and that effect is denominated attraction: in the other case, the effect is the deflection of another planet from its elliptical course; or, from that course which it would pursue did it obey solely the laws of projection and of its centripetal force: and this effect is * Its mass may be determined by comparing, with the best observations, the great inequalitics (see Chap. XIX.) which its action produces in the motion of Saturn. denominated perturbation. The particles, in the two cases, exert their attraction under different circumstances, and the respective effects of their attractions are conveniently distinguished by different denominations. In the following Chapter we will enter more into the details of the methods by which the masses of the Earth, the Moon, the Planets and the Satellites are determined. CHAP. XXIV. On the Method of determining the Masses of Planets that are accompanied by Satellites. Numerical values of the Masses of Jupiter, Saturn, and the Georgium Sidus. The Earth's Mass determined. The Methods for determining the Masses of Venus, Mars, &c. and, generally, of Planets that are without Satellites. The Masses of Satellites and of the Moon determined. THE principle of determining the mass of an heavenly body, whatever it be, Sun, Moon, or Planet, is, as it has been already stated in the close of the preceding Chapter, precisely the same. Under different denominations, because under different circumstances, it is, in every case, some effect of the attracting particles of matter which serves to expound their number, or the mass of the body which they are supposed to constitute. The effect, however, as it has been already stated, in one class of instances, is centripetal force: in another, a force that disturbs: in the former, the effect, according to certain preconceived notions, is regularity, or, the equable description of areas and the observance of Kepler's Laws: in the latter, irregularity, or the perturbation of areas, the progression of the apsides, &c. And such a distinction in the effects of gravitation naturally suggests a convenient distribution of the methods of finding the masses of the heavenly bodies into two classes; one appropriated to the Sun and those Planets that have satellites: the other to Mercury, Venus, Mars, the Moon, and the satellites of Jupiter and Saturn. To begin with the methods of the first class. These methods are contained in the formula of p. 29; according to which μ denoting the attraction residing in, or transferred (see pp. 42, 43, &c.) to the central body, P the period, and a the mean distance of the revolving body. μ the attraction of the central body, (if the revolving body be supposed a material point, or if its mass, relatively to that of the central body, be supposed insignificant) is proportional to its mass. It is, in fact, (see pp. 42, 43.) proportional to the sum of the masses of the central and revolving Bodies. Let 1 denote the Sun's mass, M, Jupiter's, m the mass of Jupiter's fourth satellite; and, moreover, let A, a, P, p, denote, respectively, the mean distances and periods of Jupiter and his satellite: then, by the preceding formula, from which formula M the mass of Jupiter, or the relative quantity of his matter compared with that of the Sun's, may be computed. And, as it is plain, the same formula will serve for determining the masses of Saturn and of the Georgium Sidus. The preceding formula expresses the mathematical dependence of the mass of the attracting on the period of the revolving body. But the process which establishes that formula does not render im mediately obvious their necessary dependence. That, however, may easily be thus shewn. Let C be the centre of the attracting body, P the revolving body; PQ a portion of its orbit, and PR a tangent to the orbit at the point P. Now, according to theory, PQ is described by virtue of the projectile motion PR and RQ the centripetal force : which latter arises from the attraction of the mass at C, and, at a given distance, is proportional to that mass. Suppose the mass to be increased, then RQ would be increased: it might become, in the same time, Rn (= rg). The orbit, therefore, could not remain circular (supposing for simplicity of illustration that to be its form) except the arc PQ became Pq. All portions of the orbit similar to PQ, would, in like manner, be increased by the increase of the mass at C: consequently, the number of portions of the arc described in the same number of portions of time would be diminished: the period, therefore, which is formed of such portions of time, would itself be less. In the same way it would follow that the orbit, supposing it to retain its form, would necessarily be described in an increased period by a diminution of the mass of the attracting body. |