Orbital stability in static axisymmetric fields
- 115 Downloads
We investigate the stability of circular orbits in static axisymmetric, but otherwise arbitrary, gravitational and electromagnetic fields. We extend previous studies of this problem to include a toroidal magnetic field. We find that even though the toroidal magnetic field does not alter the location of circular orbits, given by the critical points of the effective potential, it does affect their stability. This is because a circular orbit located at an isolated maximum of the effective potential—which in the absence of a toroidal magnetic field is an unstable configuration—can be rendered stable by a toroidal magnetic field through the phenomenon of gyroscopic stabilization. We find that for any such maximum, gyroscopic stabilization is always possible given a sufficiently strong toroidal magnetic field. We also show that no isolated maxima exist in source-free regions of space. As an example of a force field produced in part by a continuous charge distribution throughout space, we consider a rotating dipolar magnetosphere. We show that in this case a toroidal magnetic field can indeed provide gyroscopic stabilization for positively charged particles in prograde equatorial orbits.
KeywordsGyroscopic stabilization Magnetic fields Axisymmetry Orbital stability
We thank Pablo Benítez-Llambay, Luis García-Naranjo and Jihad Touma for insightful comments. We are grateful for the hospitality of the Institute for Advanced Study where part of this work was carried out. We thank the anonymous referees whose inquiries and comments contributed to improving this manuscript. The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework programme (FP/2007–2013) under ERC Grant Agreement No. 306614.
- Chetaev, N.G.: The Stability of Motion. Pergamon Press, Oxford (1961)Google Scholar
- Kolmogorov, A.: On the conservation of conditionally periodic motions under small perturbation of the hamiltonian. Dokl. Akad. Nauk. SSR 98(527), 2–3 (1954)Google Scholar
- Modesitt, G.E.: Maxwell’s equations in a rotating reference frame. Am. J. Phys. 38(1970), 1487 (1970). http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970AmJPh..38.1487M&link_type=EJOURNAL%5Cnpapers://ad348f84-eb9b-4b43-b3a1-f1af266abee1/Paper/p4499
- Moser, J.: Lectures on Hamiltonian Systems: Rigorous and Formal Stability of Orbits about an Oblate Planet; Memoirs of the American Mathematical Society, 81 (1968)Google Scholar
- Routh, E.J.: The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies. Cambridge University Press, Cambridge (1884)Google Scholar
- Thyagaraja, A., McClements, KG.: Plasma physics in noninertial frames. Phys. Plasm. 16(9), 092506 (2009). http://link.aip.org/link/PHPAEN/v16/i9/p092506/s1&Agg=doi