It is argued that, for motion in a central force field, polar reciprocals of trajectories are an elegant alternative to hodographs. The principal advantage of polar reciprocals is that the transformation from a trajectory to its polar reciprocal is its own inverse. The form of polar reciprocals $k_*$ of Kepler problem orbits is established, and then the orbits $k$ themselves are shown to be conic sections using the fact that $k$ is the polar reciprocal of $k_*$. A geometrical construction is presented for the orbits of the Kepler problem starting from their polar reciprocals. No obscure knowledge of conics is required to demonstrate the validity of the method. Unlike a graphical procedure suggested by Feynman (and amended by Derbes), the algorithm based on polar reciprocals works without alteration for all three kinds of trajectories in the Kepler problem (elliptical, parabolic, and hyperbolic).
Approximately, a hodograph is a plot of velocities along a trajectory; more precisely, it is the locus of the tips of the velocity vectors after they have been parallelly transported until their tails are at the origin (in velocity space). There have been many articles on the pedagogic virtues of hodographs. [1][2][3][4][5] Feynman's Lost Lecture 6 contains, amongst other things, a recipe for drawing the elliptical path of a planet given its hodograph. The procedure, as reproduced in Ref. 6, has some shortcomings, 7,8 but, fortunately, these have been more than satisfactorily rectified by Derbes. 9 Derbes notes that Feynman's scheme also works for hyperbolic orbits and he is able to devise another construction for the exceptional case of parabolic orbits. A completely different way of tracing all three kinds of orbits with the help of hodographs has been successfully developed by Salas-Brito and co-workers in a series of publications culminating in Ref. 10.
The various geometrical methods of the previous paragraph are easily implemented, but, to prove their validity, a student would have to be acquainted with many properties of conics which no longer form part of most school curricula. Indeed, at least one reader of Derbes’ article feels that his exposure to the hodograph left him “disappointed by the trade-off of intricate calculus for obscure geometry”. 11 There is, however, an alternative approach which requires only some elementary vector algebra and calculus for its justification.
The seed for this other construction can be traced back to a result of Newton (Proposition I, Corollary I on page 41 of Ref. 12) arising from the conservation of angular momentum in a central force field. Let ⇀ r be the position vector of a body relative to the center O of the force field and let ⇀ C be the body’s angular momentum per unit mass with respect to O; then, It is the inverse of the mapping P → P * in Fig. 1 which can be used to draw trajectories.
In fact, there is a pleasing symmetry. The mapping P → P * is involutory, i.e. (P * ) * = P (see Appendix A for an elementary proof), so trajectories in a central force field and their hodographs (after rotation and rescaling as in the preceding paragraph) are polar reciprocals of each other. The point-by-point determination of a trajectory from a rotated and rescaled hodograph involves exactly the steps depicted in Fig. 1. 14 Unlike the methods of Refs. 9 and 10, polar reciprocation is valid for any smooth hodograph associated with any central force field. Be that as it may, I will now specialize to orbits of the Kepler problem. I will also use the identity
which takes advantage of the fact that, for the trajectory τ depicted in Fig. 1, ⇀ C points perpendicularly out of the page, so that ⇀ ν × ⇀ C is parallel to ⇀ OP * and has magnitude ν C.
For an inverse-square force per unit mass of -γ ⇀ r/r 3 (γ > 0), the equation of motion for the position vector ⇀ OP * of a typical point P * on the polar reciprocal of an orbit (see Fig. 1)
where ρ ≡ γ/C 2 and plane polar coordinates r and φ have been adopted for the orbital plane; the corresponding unit vectors are e r and e φ with the origin of the coordinate system at the force center O, and the azimuthal angle φ defined as in Fig. 2 (so that ⇀ C = r 2 φ k). Equation (3) implies that ⇀ OP * -ρ e r is a (vectorial) constant of the motion, say ⇀ OO * (drawn in Fig. 2). Setting
which demonstrates that the polar reciprocal k * of an orbit k of the Kepler problem is a circle or an arc of a circle (as suggested by the plot in Fig. 2). 15 The polar reciprocal of k * (which would be the corresponding orbit k) comprises points like P * * in Fig. 2, the image (under polar reciprocation) of P * on k * . In terms of the angles in Fig. 2,
with Eq. ( 4),
Comparison of Eq. ( 5) with the standard equation 16 for a conic in polar coordinates confirms that P * * is on a conic with focus O, eccentricity e = | ⇀ ε * |, latus rectum 2/ρ, and directrix perpendicular to OO * . The angle φ can thus be identified as the true anomaly (of celestial mechanics), and ⇀ ε * can be reinterpreted as a vector of magnitude equal to the eccentricity e of the orbit k, directed from the force center O to the point on k of closest approach (i.e.
the periapse). In fact, to within a constant, ⇀ ε * is the Laplace-Runge-Lenz vector. 17
In the previous two paragraphs, it has been shown that it is easy to establish the character of the polar reciprocal k * of an orbit k of the Kepler problem and even easier to infer from k * that the orbit k must be a conic section. It is now possible to indicate how, as an alternative to the methods of Refs. 9 and 10, polar reciprocation may be used, in principle, to draw orbits with a compass and ruler.
Suppose that the orbiting body’s velocity is given at some point P 0 on the orbit (which need not coincide with an apse of the orbit); let ⇀ r 0 be the position vector of P 0 relative to O, and let the associated velocity (in
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