(164.) Now let us suppose that a planet is rather carelessly launched into space, with a velocity neither at right angles to the sun, nor of sufficient amount to maintain it in a circle, at that distance from the sun. Let P (Fig. 32) represent the position of the planet, a the amount and direction of its haphazard velocity during the first unit of time. The direction of the motion being within a right angle to SP, the action of gravity increases the velocity of the planet, on the same во principle that a falling body increases in velocity; and the planet goes on in a curve,describing equal areas in equal times round the point S; and it will find a point, p, where its increased velocity will be just P a Fig. 32. S equal to the velocity in a circle whose radius is the diminished distance Sp. From the point p, and at right angles to a, draw p C, &c., forming the right angled triangle p C S. is the eccentricity, Sa the mean distance, and p C half the conjugate axis of the orbit. CHAP. II. Planets will find their orbits, whatever be the direction and force of their" original mo tion. The orbits will be sym. metrical on perigee. If the planet is launched into space in the other direction, the action of gravity will diminish its motion, and will bring it at right angles to the line joining the sun; it is then at its each side of apogee, with a motion too feeble to maintain a circle at that apogee and distance; and it will, of course, approach nearer and nearer to the sun by the same laws of motion and force that it receded from the sun; hence the curve on each side of the apogee will be symmetrical; and the same reasoning will apply to the curve on each side of the perigee; and, in short, we shall have an ellipse. An impor. tant conclu. To sum up the whole matter, it is found by a strict examination of the laws of gravity, motion, and inertia, that whatever sion. CHAP. II. may be the primary force and direction given to a planetary body (if not directly to or from the sun), the planet will find a corresponding orbit, of a greater or less eccentricity, and of a greater or less mean distance; and whatever be the eccentricity of the orbit, the real velocity, at the extremity of the shorter axis, will be just sufficient to maintain the planet in a circular orbit, at that mean distance from the sun.* Theory of Dr. Olbers concerning *Let S be the sun, and P the position of a planet as represented in the annexed figure, and we may now suppose it to the asteroids burst into fragments, the figure representing three fragments only; the velocity and direction of one represented by a; of another by b, and of a third by c, &c. Fig. 33. As action is just equal to reaction, under all circumstances, therefore the bursting of a planet can give the whole mass no additional velocity; a small mass may be blown off at a great velocity, but there will be an equal reaction on other masses, On the in the opposite direction. bursting of a planet, the would take The whole might simply burst into about equal parts, and fragments then they would but separate, and all the parts move along in the same general direction, and with the same aggregate sponding to velocity as the original planet. The bursting of a rocket is their veloci- a very minute, but a very faithful representation of such an orbits corre ties and posi tions. explosion. (165.) To see whether Kepler's third law applies to ellipses, CHAP. II. we represent half the greater axis of any ellipse by A, and Kepler's half the shorter axis by B, and then (3.1416)AB is the area third law rigorously true of the ellipse. Also, let a represent the velocity or distance in relation to ellipses, circles. If the velocities of the several fragments were equal, the well times of their revolutions would be equal; but the eccentricities of the several orbits would depend on the angles of a, b, c, &c., with SP. If a is at right angles to SP, and just sufficient to maintain the planet in a circle at that distance, then its orbit would have no eccentricity. If still at right angles, but not sufficient to maintain a circle at that distance, then SP would be the greatest radius of the orbit. Hence, we perceive, there is an abundance of room to have a multitude of orbits passing through the same point, during the first one or two revolutions; and the times of such revolutions may be equal, or very unequal. In short, there is no physical impossibility to be urged against the theory of Dr. Olbers, that the asteroids are but fragments of a planet. The objection is (if an objection it can be called) that these planets have not, in fact, a common node, nor have an approximation to one; nor have they an approximation tɔ a common radius vector, as SP. But the objection vanishes when we consider that the elements of the different orbits must be variable; and time, a sufficient length of time, would separate the nodes and change the positions of the orbits so as to hide the common origin, as is now the case. But if it be true that these planets once had a common origin in one large planet, it is possible to find the variable nature of the elements of their orbits to such a degree of exactness as to trace them back to that origin - define the place where, and the time when, the separation must have occurred. If, however, a planet should burst at one time, and afterward one or more of the fragments burst, there could be no tracing to a common origin; hence it is possible that the asteroids in question may have a common origin, and it be wholly beyond the power of man to show it. as as to CHAP. II. that the planet will move in a unit of time, when at the extremity of its shorter axis; then a B will express the area described in that unit of time. But as equal areas are described in equal times, as often as this area is contained in the whole ellipse will be the number of such units in a revolution. Putt that number, or the time of revolution; then Let A' and B' be the semiaxes of any other ellipse; a' the velocity at the extremity of B', and t the time of revolution; M mass of sun); and putting the values of a and a', in the planetary or By squaring 12: Kepler's third law. : A'3; which is Eccentrici.. (166.) We have seen, in articles 163 and 164, that the ties of the eccentricity of an orbit depends on the direction of the motion bits change to the radius vector, when the planet is at mean distance. If by their mu- that direction is at right angles to the radius vector at that time, then the eccentricity is nothing. If its direction is very acute, then the eccentricity is very great, &c. tual attrac tions. Now suppose another planet to be situated at B (Fig. 32); its attraction on the planet, passing along in the orbit pa, is to give the velocity, a, a direction more at right angles to 1 The mean ver vary. Sp, and thus to diminish the eccentricity of the orbit. If CHAP. II. the disturbing body, B, were anywhere near the line CS, its tendency would be to increase the eccentricity; and thus, in distances ne. general, A disturbing body near a line of the shorter axis of an orbit, has a tendency to diminish the eccentricity of the orbit of the disturbed body; and, anywhere near a line of the greater axis, has a tendency to increase the eccentricity. Hence the eccentricities of the planets change in consequence of their mutual attractions; but their mean distances never change. (167.) As the time of revolution is always the same for the same mean distance, whatever be the eccentricity of the orbit, therefore if we conceive a planet to turn into an infinitely eccentric orbit, and fall directly to the sun, the time of such fall would be half a revolution, in an orbit of half its present mean distance, as we perceive, by inspecting Fig. 34. Hence, by Kepler's third law, we can compute the time that would be required for any planet to fall to the sun. Let x represent the time a planet would revolve in this new and infinitely eccentric orbit; then, by Kepler's law, Therefore half of the revolution, or simply the time of the fall, must be expressed by t t or, 2/8 4/2 Fig. 34. S that is, to find the time in which any planet would By applying this rule, we find that The prin ciples and the computation of the time required for the plan. ets to fall to the sun. The moon would fall to the earth in 4 d. 19 h. 54 m. 36 s. |