477 



GRAVITATION. 



GRAVITATION. 



478 



is its true anomaly. (The longitude of the planet is, therefore, equal 

 to the sum of the longitude of the perihelion, and the true anomaly of 

 the planet.) The line s p is called the radius rector. 



In all our diagrams it, is to be understood, that the planet, or 

 satellite, moves through its orbit in the direction opposite to the 

 motion of the hands of a watch. This is the direction in which all the 

 planets and satellites would appear to move, if viewed from any place 

 on the north side of the planes of their orbits. 



The time in which the planet moves from any one point of the orbit 

 through the whole orbit, till it cornea to the same point again, is called 

 the planet's periodic time. 



(28.) If we know the mass of the central body, and if we suppose 

 the revolving body to be projected at a certain place in a known 

 direction with a given velocity, the length of the axis major, the 

 excenrricity, the position of the line of apses, and the periodic time, 

 may all be calculated. We cannot point out the methods and formula! 

 used for these, but we may mention one very remarkable result. The 

 length of the axis major depends only upon the velocity of projection, 

 and upon the place of projection, and not at all upon the direction of 

 projection. 



(29.) We shall proceed to notice the principle on which the motion 

 of a planet, or satellite, in its orbit, is calculated. 



It in plain that this is not a very easy business. We have already 

 explained, that the velocity of the planet in its orbit is not uniform 

 (being greatest when the planet's distance from the gun is least, or 

 when the planet is at perihelion) ; and it ia obvious, that the longitude 

 of the planet increases very irregularly; since, when the planet is 

 near to the sun, its actual motion ia very rapid, and, therefore the in- 

 crease of longitude is extremely rapid ; and when the planet is far 

 from the sun, its actual motion is slow, and, therefore, the increase of 

 longitude is extremely slow. The rule which is demonstrated by 

 theory, and which in found to apply precisely in observation, is this : 

 The areas described by the radius vector are equal in equal times. 

 This is true, whether the force be inversely as the square of the 

 distance from the central body, or be in any other proportion, pro- 

 viilril thnt it is directed to the central body. 



(80.) Thus, if in one day a planet, or a satellite, moves from A to 

 (i, iiij. 5 ; in the next day it will mov from a to b, making the area a 



Fig. S. 



8 5 equal to A s n ; in the third day it will move from 6 to e, making 

 the area 6 8 c equal to A 8 a or a s 6, and so on. 



(31.) Upon this principle mathematicians have invented methods of 

 calculating the place of a planet or satellite, at any time for which it 

 may be required. These method* are too troublesome for us to 

 explain here ; but we may point out the meaning of two term* which 

 are frequently used in these computations. Suppose, for instance, as 

 in the figure, that the planet, or satellite, occupies ten days in de- 

 scribing the half of its orbit, A abcdefghiv, or twenty) days in 

 describing the whole orbit ; and suppose that we wished to find its 

 place at the end of three days after leaving the perihelion. If the 

 orbit were a circle, the planet would in three days have moved through 

 an angle of 54 degrees. If the excentricity of the orbit were small 

 (that is, if the orbit did not differ much from a circle), the angle 

 through which the planet would have moved would not differ much 

 from 54 degrees. The eccentricities of all the orbits of the planets 

 are small ; and it is convenient, therefore, to begin with the angle 54 

 as one which is not very erroneous, but which will require some 

 correction. This angle (as 54), which is proportional to the time, is 

 called the mean anomaly ; and the correction which it requires, in 

 ordur to produce the true anomaly, is called the equation of tke centre. 

 If we examine the nature of the motion, while the planet moves from 

 A to B, it will be readily seen, that during the whole of that time, the 

 angle really described by the planet is greater than the angle which is 

 Erroportiana] to the time, or the equation of the centre is to be added 

 to the mean anomaly, in order to produce the true anomaly; but 

 while the planet moves in the other half of the orbit, from B to A, the 

 angle really described by the planet is les than the angle which is 

 proportional to the time, or the equation of the centre is to be tub- 

 traded frirm the mean anomaly, in order to produce the true anomaly. 



i The sum of the mean anomaly and the longitude of perihelion 

 is called the mean longitude of the planet. It is evident, that if we 

 arid the equation of the centre to the mean longitude, while the planet 

 is moving from A to B, or subtract it from the mean longitude, while 

 the planet is moving from A to B, as in (31), we shall form the true 

 longitude. 



i The reader will see, that when the planet's true anomaly is 

 calculated, the length of the radius vector can be computed from a 

 knowledge of the properties of the ellipse. Thus the place of the 



planet, for any time, is perfectly known. This problem has acquired 

 considerable celebrity under the name of Kepler's problem. 



(34.) There remains only one point to be explained regarding the 

 undisturbed motion of planets and satellites ; namely, the relation 

 between a planet's periodic time and the dimensions of the orbit in 

 which it moves. 



Now, on the law of gravitation it has been demonstrated from 

 theory, and it is fully confirmed by observation, that the periodic time 

 does not depend on the excentricity, or on the perihelion distance, or 

 on the aphelion distance, or on any element except the mean distance 

 or semi-major axis. So that if two planets moved round the sun, one 

 in a circle, or in an orbit nearly circular, and the other in a very flat 

 ellipse; provided their mean distances were equal, their periodic times 

 would be equal. It is demonstrated also, that for planets at different 

 distances, the relation between the periodic times and the mean dis- 

 tances is the following : The squares of the number of days (or hours, 

 or minutes, &c.) in the periodic times have the same proportion as the 

 cubes of the numbers of miles (or feet, &c.) in the mean distances. 



(35.) Thus the periodic time of Jupiter round the sun is 4332*7 

 days, and that of Saturn is 10759*2 days ; the squares of these num- 

 bers are 18772289 and 115760385. The mean distance of Jupiter 

 from the sun is about 487491000 miles, and that of Saturn ia about 

 893955000 miles ; the cubes of these numbers are 1158496 (20 ciphers),, 

 and 7144088 (20 ciphers). On trial it will ba found, that 18772289 

 and 115760385 are in almost exactly the same proportion aa 1158496 

 and 7144088. 



(36.) In like manner, the periodic tunes of Jupiter's third and fourth 

 satellites round Jupiter are 7*15455 and 16'68877 days; the squares of 

 these numbers are 51*1876 and 278*515. Their mean distances from 

 Jupiter are 670080 and 1178580 miles ; the cubes of these numbers 

 are 300866 (12 ciphers), and 1687029 (12 ciphers), and the proportion 

 of 51*1876 to 278*515 ia almost exactly the same as the proportion of 

 300866 to 1637029. 



(37.) It must however be observed that this rule applies in com- 

 paring the periodic tunes and mean distances, only of bodies which 

 revolve round the same central body. Thus the rule applies in com- 

 paring the periodic times and mean distances of Jupiter and Saturn, 

 because they both revolve round the sun ; it applies in comparing the 

 periodic times and mean distances of Jupiter's third and fourth satel- 

 lites, because they both revolve round Jupiter ; but it would not apply 

 in comparing the periodic tune and mean distance of Saturn revolving 

 round the sun with that of Jupiter's third satellite revolving round 

 Jupiter. 



(38.) In comparing the orbit* described by different planets, or 

 satellites, round different centres of force, theory gives us the following 

 law : The cubes of the mean distances are hi the same proportion aa 

 the products of the mast by the square of the periodic time. Thua, 

 for instance, the mean distance of Jupiter's fourth satellite from 

 Jupiter is 1178560 miles; its periodic time round Jupiter is 16*68877 

 days; the mean distance of the earth from the sun is 93726900 miles; 

 its periodic time round the sun is 365*2564 days ; also the mass of 

 Jupiter is ^th the sun's mass. The cubes of the mean distances are 

 respectively 1637029 (12 ciphers), and 823365 (18 ciphers) ; the pro- 

 ducts of the squares of the times by the masses are respectively 

 0*265252 and 133412; and these numbers are in the same proportion 

 aa 1687029 (12 ciphers), and 828365 (18 ciphers). 



(39.) The three rules that planets move in ellipses, that the radius 

 vector in each orbit passes over areas proportional to the times, and 

 that the squares of the periodic times are proportional to the cubes of 

 the mean distances, are commonly called Kepler's laws. They were 

 discovered by Kepler from observation, before the theory of gravita- 

 tion was invented; they were first explained from the theory by 

 Newton, about the year 1680. 



(40.) The last of these is not strictly true, unless we suppose that 

 the central body is absolutely immoveable. This however is evidently 

 inconsistent with the principles which we have laid down in Section I. 

 In considering the motion, for instance, of Jupiter round the sun, it is 

 necessary to consider that, while the sun attracts Jupiter, Jupiter 

 is also attracting the sun. But the planets are so small in comparison 

 with the sun (the largest of them, Jupiter, having less than one- 

 thousandth part of the matter contained in the sun), that in common 

 illustrations there ia no need to take this consideration into account. 

 For nice astronomical purposes it is taken into account in the following 

 manner : The motion which the attraction of Jupiter produces in the 

 sun ia less than the motion which the attraction of the sun produces 

 in Jupiter, in the same proportion in which Jupiter is smaller than 

 the sun. If the sun and Jupiter were allowed to approach one another, 

 their rate of approach would be the sum of the motions of the sun and 

 Jupiter, and would therefore be greater than their rate of approach, if 

 the sun were not moveable, in the same proportion in which the sum 

 of the masses of the sun and Jupiter is greater than the sun's mass ; 

 that is, the rate of approach of the sun and Jupiter, both being free, 

 is the game ag the rate of approach would be if the sun were fixed, 

 provided the sun's mass were increased by adding Jupiter's mass to it. 

 Consequently, in comparing the orbits described by different planets 

 round the sun, we must use the rule just laid down, supposing the 

 central force to be the attraction of a mass equal to the sum of the sun 

 and thefplanet; and thus we get a proportion which is rigorously 



