the orbit is therefore a “reciprocal spiral,” except in the special case of A = 0, when it is a circle. It will be seen that unless the conditions be exactly adjusted for a circular orbit the particle will either recede to infinity or approach the pole asymptotically. This problem was investigated by R. Cotes (1682-1716), and the various curves obtained arc known as Coles’s spirals.
A point on a central orbit where the radial velocity (dr/dt) vanishes is called an apse, and the corresponding radius is called an apse-line. If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed. It follows that the angle between successive apse-lines is constant; it is called the apsidal angle of the orbit.
If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),
| h2 | = 2 ∫∞r P dr; |
| p2 |
(26)
this determines the form of the critical orbit, as it is called. If P = μ/rn, its polar equation is
rm cos mθ = am,
(27)
where m = 1⁄2(3 − n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2 gives the parabola as before.
If we eliminate dθ/dt between (15) and (16) we obtain