A Short History of Astronomy

by Arthur Berry

Stated briefly, the results established by the two astronomers were that the changes in axis, eccentricity, and inclination of any planetary orbit are all permanently restricted within certain definite limits. The perturbations caused by the planets make all these quantities undergo fluctuations of limited extent, some of which, caused by the periodic disturbing forces, go through their changes in comparatively short periods, while others, due to secular forces, require vast intervals of time for their completion.

It may thus be said that the stability of the solar system was established, as far as regards the particular astronomical causes taken into account.

Moreover, if we take the case of the earth, as an inhabited planet, any large alteration in the axis, that is in the average distance from the sun, would produce a more than proportional change in the amount of heat and light received from the sun; any great increase in the eccentricity would increase largely that part (at present very small) of our seasonal variations of heat and cold which are due to varying distance from the sun; while any change in position of the ecliptic, which was unaccompanied by a corresponding change of the equator, and had the effect of increasing the angle between the two, would largely increase the variations of temperature in the course of the year. The stability shewn to exist is therefore a guarantee against certain kinds of great climatic alterations which might seriously affect the habitability of the earth.

It is perhaps just worth while to point out that the results established by Lagrange and Laplace were mathematical consequences, obtained by processes involving the neglect of certain small quantities and therefore not perfectly rigorous, of certain definite hypotheses to which the actual conditions of the solar system bear a tolerably close resemblance. Apart from causes at present unforeseen, it is therefore not unreasonable to expect that for a very considerable period of time the motions of the actual bodies forming the solar system may be very nearly in accordance with these results; but there is no valid reason why certain disturbing causes, ignored or rejected by Laplace and Lagrange on account of their insignificance, should not sooner or later produce quite appreciable effects (cf. chapter XIII., [§ 293]).

246. A few of Laplace’s numerical results as to the secular variations of the elements may serve to give an idea of the magnitudes dealt with.

The line of apses of each planet moves in the same direction; the most rapid motion, occurring in the case of Saturn, amounted to about 15″ per annum, or rather less than half a degree in a century. If this motion were to continue uniformly, the line of apses would require no less than 80,000 years to perform a complete circuit and return to its original position. The motion of the line of nodes (or line in which the plane of the planet’s orbit meets that of the ecliptic) was in general found to be rather more rapid. The annual alteration in the inclination of any orbit to the ecliptic in no case exceeded a fraction of a second; while the change of eccentricity of Saturn’s orbit, which was considerably the largest, would, if continued for four centuries, have only amounted to 1∕1000.

247. The theory of the secular inequalities has been treated at some length on account of the general nature of the results obtained. For the purpose of predicting the places of the planets at moderate distances of time the periodical inequalities are, however, of greater importance. These were also discussed very fully both by Lagrange and Laplace, the detailed working out in a form suitable for numerical calculation being largely due to the latter. From the formulæ given by Laplace and collected in the Mécanique Céleste several sets of solar and planetary tables were calculated, which were in general found to represent closely the observed motions, and which superseded the earlier tables based on less developed theories.[152]

248. In addition to the lunar and planetary theories nearly all the minor problems of gravitational astronomy were rediscussed by Laplace, in many cases with the aid of methods due to Lagrange, and their solution was in all cases advanced.

The theory of Jupiter’s satellites, which with Jupiter form a sort of miniature solar system but with several characteristic peculiarities, was fully dealt with; the other satellites received a less complete discussion. Some progress was also made with the theory of Saturn’s ring by shewing that it could not be a uniform solid body.

Precession and nutation were treated much more completely than by D’Alembert; and the allied problems of the irregularities in the rotation of the moon and of Saturn’s ring were also dealt with.