Moment of momentum and energy of system. The kinetic energy of the planet's rotation is Cn3. The potential energy of the system is -μMm/r. Since the moon's present radius vector is 11.454, it follows that the orbital momentum of the moon is 3.384. Adding to this the rotational momentum of the earth which is 704, we obtain 4.088 for the total moment of momentum of the moon and earth. The ratio of the orbital to the rotational momentum is 4.80, so that the total moment of momentum of the system would, but for the obliquity of the ecliptic, be 5.80 times that of the earth's rotation. In § 276, where the obliquity is taken into consideration, the number is given as 5.38. It will be noticed that x, the moment of momentum of orbital motion, is equal to the square root of the satellite's distance from the planet. Then the equations (1) and (2) become Two con- (3) is the equation of conservation of moment of momentum, or shortly, the equation of momentum; (4) is the equation of energy. Now, consider a system started with given positive (or say clockwise*) moment of momentum h; we have all sorts of ways in which it may be started. If the two rotations be of opposite kinds, it is clear that we may start the system with any amount of energy however great, but the true maxima and minima of energy compatible with the given moment of momentum are given by dy/dx = 0, This is contrary to the ordinary convention, but I leave this passage as it stood originally. We shall presently see that this quartic has either two real roots and two imaginary, or all imaginary roots*. This quartic may be derived from quite a different consideration, viz., by finding the condition under which the satellite may move round the planet, so that the planet shall always show the same face to the satellite, in fact, so that they move as parts of one rigid body. therefore y = 1/x3. configura The condition is simply that the satellite's orbital angular In these velocity = n the planet's angular velocity of rotation; or since tions the satellite n = y and r12 = 2-3 moves as though rigidly By substituting this value of y in the equation of momentum connected with the (3), we get as before planet. = = x, In my paper on the "Precession of a Viscous Spheroid †," I obtained the quartic equation from this last point of view only, and considered analytically and numerically its bearings on the history of the earth. Sir William Thomson, having read the paper, told me that he thought that much light might be thrown on the general physical meaning of the equation, by a comparison of the equation of conservation of moment of momentum with the energy of the system for various configurations, and he suggested the appropriateness of geometrical illustration for the purpose of this comparison. The method which is worked out below is the result of the suggestions given me by him in conversation, The simplicity with which complicated mechanical interactions may be thus traced out geometrically to their results appears truly remarkable. At present we have only obtained one result, viz.: that if with given moment of momentum it is possible to set the satellite and planet moving as a rigid body, then it is possible to do so in two ways, and one of these ways requires a maximum amount of energy and the other a minimum; from which it is clear that one must be a rapid rotation with the satellite near the planet, and the other a slow one with the satellite remote from the planet. * I have elsewhere shown that when it has real roots, one is greater and the other less than 2 h. Proc. Roy. Soc. No. 202, 1880. + Trans. Roy. Soc. Part 1. 1879. Graphical solution. (6) is the equation of momentum; (7) that of energy; and (8) we may call the equation of rigidity, since it indicates that the two bodies move as though parts of one rigid body. Now, if we wish to illustrate these equations geometrically, we may take as abscissa x, which is the moment of momentum of orbital motion; so that the axis of x may be called the axis of orbital momentum. Also, for equations (6) and (8) we may take as ordinate y, which is the moment of momentum of the planet's rotation; so that the axis of y may be called the axis of rotational momentum. For (7) we may take as ordinate Y, which is twice the energy of the system; so that the axis of Y may be called the axis of energy. Then, as it will be convenient to exhibit all three curves in the same figure, with a parallel axis of x, we must have the axis of energy identical with that of rotational momentum. It will not be necessary to consider the case where the resultant moment of momentum h is negative, because this would only be equivalent to reversing all the rotations; thus h is to be taken as essentially positive. Then the line of momentum, whose equation is (6), is a straight line inclined at 45° to either axis, having positive intercepts on both axes. The curve of rigidity, whose equation is (8), is clearly of the same nature as a rectangular hyperbola, but having a much more rapid rate of approach to the axis of orbital momentum than to that of rotational momentum. The intersections (if any) of the curve of rigidity with the line of momentum have abscissæ which are the two roots of the quartic x-hx3+1=0. The quartic has, therefore, two real roots or all imaginary roots. Then, since x = √r, the intersection which is more remote from the origin, indicates a configuration where the satellite is remote from the planet; the other gives the configuration where the satellite is closer solution. to the planet. We have already learnt that these two cor- Graphical respond respectively to minimum and maximum energy. When x is very large, the equation to the curve of energy is Y= (h-x), which is the equation to a parabola, with a vertical axis parallel to Y and distant h from the origin, so that the axis of the parabola passes through the intersection of the line of momentum with the axis of orbital momentum. small the equation becomes Y= -1/x2. When x is very Hence, the axis of Y is asymptotic on both sides to the curve of energy. Then, if the line of momentum intersects the curve of rigidity, the curve of energy has a maximum vertically underneath the point of intersection nearer the origin, and a minimum underneath the point more remote. But if there are no intersections, it has no maximum or minimum. It is not easy to exhibit these curves well if they are drawn to scale, without making a figure larger than it would be |