Orbital Precession in an Inverse Square Force Field

G.R.Dixon, 11/28/2004

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Data for the orbital plots in this article were generated by the Visual Basic program in Appendix A.

G.R.Dixon, 11/28/2004

Data for the orbital plots in this article were generated by the Visual Basic program in Appendix A.

This article demonstrates the effect of a negatively charged particle’s speed-dependent mass on its path when it orbits under the influence of a fixed, positively charged particle’s electrostatic field. More specifically, it demonstrates that the aphelion of the negatively charged satellite’s nominally elliptic orbit precesses in the orbital plane when even mildly relativistic speeds are attained. The effect was known when Einstein published the General Theory of Relativity, where it was stressed that the precession of Mercury’s aphelion is not quite what would be predicted by the computed precessions in this article. Einstein theorized that the more accurate precession of Mercury is obtained by factoring in the effect of the Sun on the space-time metric. It might be noted in this regard that the precessions computed in this article can also be further affected by endowing the central, positively charged body with a magnetic dipole moment, such that its magnetic field points perpendicular to the orbiting, negatively charged satellite’s velocity at all times. As discussed in another article, interesting parallels between a charge' s B field and an analogous field in the case of mass-mass interactions can be suggested (particularly in light of the fact that the Sun rotates on its axis).

The radiation reaction force that theoretically acts on the negative charge is presumably counteracted non-electromagnetically at all points. Thus the radiation reaction force has no effect on the negative particle’s motion; the net force acting on the negative charge is solely that experienced in the central, positive charge’s electrostatic field.

The orbital plotting software in Appendix A is patterned after logic suggested in The Feynman Lectures on Physics, V1, Sect. 9-7 (Planetary Motions). The added feature herein is that the dependence of the orbiting particle’s mass on its speed is factored in. The formulas for the orbiting charge’s x- and y-components of acceleration, in terms of the components of the electrostatic force acting on it, were derived in a previous article. Multiple swings around the central body are computed and plotted in order to demonstrate the aphelion’s precession when the maximum speed is more or less relativistic.

Fig. 1 depicts the computed orbits when the minimum speed is a non-relativistic .001c. The maximum speed attained in this case is approximately .004c, which is also non-relativistic. Only a single path appears to be plotted since the multiple orbits overlay one another. In brief, there is no significant precession of the aphelion at these speeds.

Figure 1

Path When Maximum Speed is .004c

Fig. 2 depicts the computed orbits when the minimum speed is .05c. The maximum speed attained in this case is approximately .214c, which is not very relativistic. Nevertheless the precession of the aphelion is obvious.

Figure 2

Path When Maximum Speed is .214c