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their common centre of gravity. But by the property of the centre
Ση . x = 0, Σm. y = 0, Σm. z = 0;
By article 353 the centre of gravity is urged in a direction parallel
to the co-ordinates, by the forces
therefore the perturbations in the radius vector SG are very nearly proportional to, that is, to
The masses of Jupiter's satellites compared with the mass of that planet are so small, and their elongations seen from the sun subtend so small an angle, that the perturbations produced by them in Jupiter's motions are insensible; and there is reason to believe this to be the case also with regard to Saturn and Uranus.
495. But the Earth is sensibly troubled in its motions by the Moon, her action produces the inequalities
in the radius vector, longitude and latitude of the Earth, E and m
being the masses of the Earth and Moon.
DATA FOR COMPUTING THE CELESTIAL MOTIONS.
596. THE data requisite for computing the motions of the planets determined by observation for any instant arbitrarily assumed as the epoch or origin of the time, are
The masses of the planets;
Their mean sidereal motions for a Julian year of 365.25 days;
The mean distances of the planets from the sun;
The ratios of the eccentricities to the mean distances;
The inclinations of the orbits on the plane of the ecliptic;
The longitudes of the perihelia;
The longitudes of the ascending nodes on the ecliptic;
Masses of the Planets.
597. Satellites afford the means of ascertaining the masses of their primaries; the masses of such planets as have no satellites are found from a comparison of their inequalities determined by analysis, with values of the same obtained from numerous observations. The secular inequalities will give the most accurate values of the masses, but till they are perfectly known the periodic variations must be employed. On this account there is still some uncertainty as to the masses of several bodies. It is only necessary to know the ratio of the mass of each planet to that of the sun taken as the unit; the masses are consequently expressed by very small fractions.
598. If T be the time of a sidereal revolution of a planet m, whose mean distance from the sun is a, the ratio of the circumference to the diameter, and μm + S the sum of the masses of the sun and planet, by article 383,
From this expression the masses of such planets as have satellites may be obtained.
Suppose this equation relative to the earth, and that the mass of the earth is omitted when compared with that of the sun, it then
Again, let μm + m' the sum of the masses of a planet and of its satellite m', T' being the time of a sidereal revolution of the planet at the mean distance a' from the sun, then
and dividing the one by the other the result is,
If the values of T, T', a and a', determined from observation, be substituted in this expression, the ratio of the sum of the masses of the planet and of its satellite to the mass of the sun will be obtained; and if the mass of the satellite be neglected when compared with that of its primary, or if the ratio of these masses be known, the preceding equation will give the ratio of the mass of the planet to that of the sun. For example,
599. Let m be the mass of Jupiter, that of his satellite being omitted, and let the mass of the sun be taken as
Jm the mean radius of the orbit of the fourth satellite at the mean distance of the earth from the sun taken as the unit, is seen under the angle JEm 2580".579
The radius of the circle reduced to seconds is 206264".8; hence the mean radii of the orbit of the fourth satellite and of the terrestrial orbit are in the ratio of these two numbers. The time of a sidereal revolu
tion of the fourth satellite is 16.6890 days, and the sidereal year is 365.2564 days, hence
With these data it is easy to find that the mass of Jupiter is
The sixth satellite of Saturn accomplishes a sidereal revolution in 15.9453 days; the mean radius of its orbit, at the mean distance of the planet, is seen from the sun under an angle of 179"; whence the mass of Saturn is
By the observations of Sir William Herschel the sidereal revolutions of the fourth satellite of Uranus are performed in 13.4559 days, and the mean radius of its orbit seen from the sun at the mean distance of the planet is 44".23. With these data the mass of Uranus is found to be
600. This method is not sufficiently accurate for finding the mass of the Earth, on account of the numerous inequalities of the Moon. It has already been observed, that the attraction of the Earth on bodies at its surface in the parallel where the square of the sine of the latitude is, is nearly the same as if its mass were united at its centre of gravity. If R be the radius of the terrestrial spheroid drawn to that parallel, and m its mass, this attraction will be
Then, if a be the mean distance of the Sun from the Earth, T the duration of the sidereal year,
R, g, T, and a, are known by observation, therefore the ratio of the