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By this method it is computed that the discs of the satellites, seen from the centre of Jupiter, and the time they take to penetrate perpendicularly into the shadow, are
Whence the times of immersion and emersion of the satellites and of their shadows on the disc of Jupiter may be found, when they pass between him and the sun.
945. The observations of the eclipses of Jupiter by his satellites, may throw much light on their theory. The beginning and end of their transits may almost always be observed, which with the passage of the shadow afford four observations; whereas the ellipse of a satellite only gives two. La Place thinks these phenomena particularly worthy of the attention of practical astronomers.
946. In the preceding investigations, the densities of the satellites were assumed to be the same with that of Jupiter. By comparing the computed times with the observed times of duration, the densities of the satellites will be found when their masses shall be accurately ascertained.
947. The perturbations of the three first satellites have a great influence on the times of their eclipses. The principal inequality of the first satellite retards, or advances its eclipses 72.41 seconds at its maximum. The principal inequality of the second satellite accelerates or retards its eclipses by 343.2, at its maximum, and the principal inequality of the third satellite advances or retards its eclipses by 261.9 at its maximum.
948. Since the perturbations of the satellites depend only on the differences of their mean longitudes, it makes no alteration in the value of these differences, whether the first point of Aries be assumed as the origin of the angles, or SJ the radius vector of Jupiter sup posed to move uniformly round the sun. If the angles be estimated from SJ, nt, nt, not, become the mean synodic motion of the three first satellites; and in both cases
3nt + 2nt + € − 3€, 26, 180°.
Suppose the longitudes of the epochs of the two first satellites to be
zero or € = 0, €, = 0, so that these two bodies are in conjunction with Jupiter when t = 0, then it follows that e, 90°, and thus when the two first satellites are in conjunction, the third is a right angle in advance, as in fig. 115; and the principal inequalities of the three first satellites become
In the eclipses of the first satellite at the instant of conjunction nt = 0, or it is equal to a multiple of 360°. Let
In the eclipses of the second satellite at the instant of conjunction nt = 0, or it is equal to a multiple of 360°; hence
Sv1 = - 3862".3 sin wt.
Lastly, in the eclipses of the third satellite,
nåt + e, = 0, or it is a multiple of 360° at the instant of conjunction, hence
Sv2 26186 sin wt.
Thus it appears that the periods of these inequalities in the eclipses are the same, since they depend on the same angle. This period is equal to the product of
by the duration of the synodic re
volution of the first satellite, or to 437.659, which is perfectly conformable to observation.
949. On account of the ratio
3nt + 2nt + ε — 3€, + 2€ = 180°,
the three first satellites never can be eclipsed at once, neither can they be seen at once from Jupiter when in opposition or conjunction; for if nt + e, n1t + €1, not + €2,
be the mean synodic
first and second
longitudes, in the simultaneous eclipses of the nt + + = n1t + €1 = 180°;
and from the law existing among the mean longitudes, it appears that
not + eq=270°.
In the simultaneous eclipses of the first and third satellites
nt + e = nåt + €2 = 180°,
and on account of the preceding law, n1t + = 120.
Lastly in the simultaneous eclipses of the second and third satellites
n1t + e1 = n2t += 180°;
hence nt = 0, thus the first satellite in place of being eclipsed, may eclipse Jupiter.
Thus in the simultaneous eclipses of the second and third satel lites, the first will always be in conjunction with Jupiter; it will always be in opposition in the simultaneous transits of the other two.
950. The comparative distances of the sun and Jupiter from the earth may be determined with tolerable accuracy from the eclipses of the satellites. In the middle of an eclipse, the sideral position of the satellite, and the centre of Jupiter is the same when viewed from the centre of the sun, and may easily be computed from the tables of Jupiter. Direct observation, or the known motion of the sun gives the position of the earth as seen from the centre of the sun; hence, in the triangle formed by the sun, the earth, and Jupiter, the angle at the sun will be known; direct observation will give that at the earth, and thus at the instant of the middle of the eclipse, the relative distances of Jupiter from the earth and from the sun, may be computed in parts of the distance of the sun from the earth. By this method, it is found that Jupiter is at least five times as far from us as the sun is when his apparent diameter is 36".742. The diameter of the earth at the same distance, would only appear under an angle of 3".37. The volume of Jupiter is therefore at least a thousand times greater than that of the earth.
951. On account of Jupiter's distance, some minutes elapse from the instant at which an eclipse of a satellite begins or ends, before it is visible at the earth.
Roëmer observed, that the eclipses of the first satellite happened sooner, than they ought by computation when Jupiter was in opposition, and therefore nearer the earth; and later when Jupiter was in conjunction, and therefore farther from the earth. In 1675, he shewed that this circumstance was owing to the time the light of the satellite employed in coming to the observer at the different distances of Jupiter. It was objected to this explanation, that the circumstance was not indicated by the eclipses of the other satellites, in which it was difficult to detect so small a quantity among their
numerous inequalities then little known; but it was afterwards proved by Bradley's discovery of the aberration of light in the year 1725; when he was endeavouring to determine the parallax of γ He observed that the stars had a small Draconis. annual motion. A star near the pole of the ecliptic appears to describe a small circle about it parallel to the ecliptic, whose diameter is 40", the pole being the true place of the star. Stars situate in the ecliptic appear to describe arcs of the ecliptic of 40" in length, and all stars between these two positions seem to describe ellipses whose greater axes are 40" in length, and are parallel to the ecliptic. The lesser axes vary as the sine of the star's latitude. This apparent motion of the stars arises from the velocity of light combined with the motion of the earth in its orbit. The sun is so very distant, that his rays are deemed parallel; therefore let S'A. SB, fig. 116, be two
rays of light coming from the sun to the earth moving in its orbit in the direction AB. If a telescope be held in the direction AC, the ray S'A in place of going down the tube CA will impinge on its side, and be lost in consequence of the telescope being carried with the earth in the directions AB; but if the tube be in a position SEA, so that BA: BS as the velocity of the earth to the velocity of light; the ray will pass in the diagonal SA, which is the component
of these two velocities, that is, it will pass through the axis of the telescope while carried parallel to itself with the earth. The star appears in the direction AS, when it really is in the direction AS'; hence S'AS ASB is the quantity or angle of aberration, which is always in the direction towards which the earth is moving.
Delambre computed from 1000 eclipses of the first satellite, that light comes from the sun at his mean distance of about 95 millions of miles in 8'.13"; therefore the velocity of light is more than ten thousand times greater than the velocity of the earth, which is nineteen miles in a second; hence BS is about 10000 times greater than AB, consequently the angle ASB is very small. When EAB is a right angle, ASB is a maximum, and then
sin ASB 1: AB: BS: velocity of earth: velocity of light;
the aberration; hence the sine of the greatest aber
ration is equal to
rad. velocity of earth
velocity of light
by the observations of Bradley which perfectly correspond with the maximum of aberration computed by Delambre from the mean of 6000 eclipses of the first satellite.
This coincidence shews the velocity of light to be uniform within the terrestrial orbit, since the one is derived from the velocity of light in the earth's orbit, and the other from the time it employs to traverse its diameter. Its velocity is also uniform in the space included in the orbit of Jupiter, for the variations of his radius vector are very sensible in the times of the eclipses of his satellites, and are found to correspond exactly with the uniform motion of light.
If light be propagated in space by the vibrations of an elastic fluid, its velocity being uniform, the density of the fluid must be proportional to its elasticity.
952. The concurrent exertions of the most eminent practical and scientific astronomers have brought the theory of the satellites to such perfection, that calculation furnishes more accurate results than observation. Galileo obtained approximate values of the mean distances and periodic times of the satellites from their configurations, and Kepler was able to deduce from these imperfect data, proofs that the squares of their periodic times are proportional to the cubes of their mean distances, establishing an analogy between these bodies and the planetary systems, subsequently confirmed.
Bradley found that the two first satellites return to the same relative positions in 437 days. Wangentin discovered a similar inequality in the third of the same period, which was concluded to be the cycle of their disturbances.
In the year 1766, the Academy of Sciences at Paris proposed the theory of the satellites of Jupiter as a prize question, which produced a masterly solution of the problem by La Grange. In the first approximation he obtained the inequalities depending on the elongations previously discovered by Bradley; in the second, he obtained four equations of the centre for each satellite, and by the same analysis shewed that each satellite has four principal equations in latitude, which he represented by four planes moving on each other at different