| d2r | − | h2 | = −P = −ƒ(r), |
| dt2 | r3 |
say. We may apply this to the investigation of the stability of a circular orbit. Assuming that r = a + x, where x is small, we have, approximately,
| d2x | − | h2 | ( 1 − | 3x | ) = −ƒ(a) − xƒ′(a). |
| dt2 | a3 | a |
Hence if h and a be connected by the relation h2 = a3ƒ(a) proper to a circular orbit, we have
| d2x | + { ƒ′(a) + | 3 | ƒ(a) } x = 0 |
| dt2 | a |
(28)
If the coefficient of x be positive the variations of x are simple-harmonic, and x can remain permanently small; the circular orbit is then said to be stable. The condition for this may be written
| d | { a3ƒ(a) } > 0, |
| da |
(29)
i.e. the intensity of the force in the region for which r = a, nearly, must diminish with increasing distance less rapidly than according to the law of the inverse cube. Again, the half-period of x is π / √{ƒ′(a) + 3−1ƒ(a)}, and since the angular velocity in the orbit is h/a2, approximately, the apsidal angle is, ultimately,