By A. Frank D'Souza

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3 G(ml ,j- mz) [1 = + e cos ((J Case 1: e > I, E > 0. The orbit is a hyperbola that is an open orbit. The particle comes from infinity as shown in Fig. 2I, reaches the minimum distance at the apsis where the potential energy has a minimum and, hence, the kinetic energy a maximum, and escapes to infinity. 77) (Jo)] The constant (Jo can be eliminated by choosing the polar axis so that (Jo = 0 and the constant e can be evaluated from the total mechanical energy of the body. 77) is the equation of a conic section in polar coordinates rand (J and may represent a hyperbola, parabola, ellipse, or circle, depending on the value of e, which is known as the eccentricity of the conic section.

Energy, and Momentum Methods 66 Chap. 3 Sec. 8 m 1; 1 + m2;2 (m1 + mz);c Let; be the radius vector from m 2 to m 1 • From Fig. 20, it is seen that = -il. :.. 8 ORBITS OF PLANETS AND SATELLITES Let ] 11 be the internal force on m 1 due to m 1, there being no external forces. Force ] 0 may be an attractive or repulsive force. :. :. I. a, m2r2 67 where m is the equivalent mass of m1 and f /12 is assumed to be a repulsive central force. 68). 68) are valid for any central force and have applications in atomic and nuclear physics.

Coyly+ kco,l, mis the angular velocity of the body, the velocity of dm is given by v vc +co X ;. Hence, = 2 It is often convenient to deal with rigid-body dynamic problems using the coordinate system fixed in the body for which all products ofti1Iertias are zero simultaneously; that is, the inertia matrix [/] is diagonal. The three mutually orthogonal coordinate axes are known as principal axes and the corresponding moments of inertia are referred to as the principal moments of inertia. The three planes formed by the principal axes are called principal planes.