Diversity and Origin of 2:1 Orbital Resonances in Extrasolar Planetary Systems

(Abridged) A diversity of 2:1 resonance configurations can be expected in extrasolar planetary systems, and their geometry can provide information about the origin of the resonances. Assembly during planet formation by the differential migration of planets due to planet-disk interaction is one scenario for the origin of mean-motion resonances in extrasolar planetary systems. The stable 2:1 resonance configurations that can be reached by differential migration of planets with constant masses and initially coplanar and nearly circular orbits are (1) anti-symmetric configurations with the mean-motion resonance variables theta_1 and theta_2 (in deg.) librating about 0 and 180, respectively (as in the Io-Europa pair), (2) symmetric configurations with both theta_1 and theta_2 librating about 0 (as in the GJ 876 system), and (3) asymmetric configurations with theta_1 and theta_2 librating about angles far from either 0 or 180. There are, however, stable 2:1 resonance configurations with symmetric (theta_1 = theta_2 = 0), asymmetric, and anti-symmetric (theta_1 = 180 and theta_2 = 0) librations that cannot be reached by differential migration of planets with constant masses and initially coplanar and nearly circular orbits. If real systems with these configurations are ever found, their origin would require (1) a change in the planetary mass ratio m_1/m_2 during migration, (2) a migration scenario involving inclination resonances, or (3) multiple-planet scattering in crowded planetary systems. We find that the asymmetric configurations with large e_2 and the theta_1 = 180 and theta_2 = 0 configurations have intersecting orbits and that the theta_1 = theta_2 = 0 configurations with e_1 > 0.714 have prograde periapse precessions.Comment: 24 pages, including 14 figures; uses AASTeX v5.0; accepted for publication in Ap

Secular Evolution of Hierarchical Planetary Systems

(Abridged) We investigate the dynamical evolution of coplanar hierarchical two-planet systems where the ratio of the orbital semimajor axes alpha=a_1/a_2 is small. The orbital parameters obtained from a multiple Kepler fit to the radial velocity variations of a star are best interpreted as Jacobi coordinates and Jacobi coordinates should be used in any analyses of hierarchical planetary systems. An approximate theory that can be applied to coplanar hierarchical two-planet systems with a wide range of masses m_j and orbital eccentricities e_j is the octupole-level secular perturbation theory (OSPT). The OSPT shows that if the ratio of the maximum orbital angular momenta, lambda \approx (m_1/m_2) alpha^{1/2}, for given a_j is approximately equal to a critical value lambda_{crit}, then libration of the difference in the longitudes of periapse, w_1-w_2, about either 0 or 180 deg. is almost certain, with possibly large amplitude variations of both e_j. We establish that the OSPT is highly accurate for systems with alpha<0.1 and reasonably accurate even for systems with alpha as large as 1/3, provided that alpha is not too close to a significant mean-motion commensurability or above the stability boundary. The HD 168443 system is not in a secular resonance and its w_1-w_2 circulates. The HD 12661 system is the first extrasolar planetary system found to have w_1-w_2 librating about 180 deg. The libration of w_1-w_2 and the large-amplitude variations of both e_j in the HD 12661 system are consistent with the analytic results on systems with lambda \approx lambda_{crit}. The HD 12661 system with the best- fit orbital parameters and sin i = 1 is affected by the close proximity to the 11:2 commensurability, but small changes in the outer orbital period can result in configurations that are not affected by mean-motion commensurabilities.Comment: 32 pages, including 8 figures; uses AASTeX v5.0; accepted for publication in Ap

A Primordial Origin of the Laplace Relation Among the Galilean Satellites

Understanding the origin of the orbital resonances of the Galilean satellites of Jupiter will constrain the longevity of the extensive volcanism on Io, may explain a liquid ocean on Europa, and may guide studies of the dissipative properties of stars and Jupiter-like planets. The differential migration of the newly formed Galilean satellites due to interactions with a circumjovian disk can lead to the primordial formation of the Laplace relation n_1 - 3 n_2 + 2 n_3 = 0, where the n_i are the mean orbital angular velocities of Io, Europa, and Ganymede, respectively. This contrasts with the formation of the resonances by differential expansion of the orbits from tidal torques from Jupiter.Comment: 13 pages, including 4 figures; uses scicite.st

On the Early In Situ Formation of Pluto's Small Satellites

The formation of Pluto's small satellites - Styx, Nix, Keberos and Hydra - remains a mystery. Their orbits are nearly circular and are near mean-motion resonances and nearly coplanar with Charon's orbit. One scenario suggests that they all formed close to their current locations from a disk of debris that was ejected from the Charon-forming impact before the tidal evolution of Charon. The validity of this scenario is tested by performing $N$-body simulations with the small satellites treated as test particles and Pluto-Charon evolving tidally from an initial orbit at a few Pluto radii with initial eccentricity $e_{\rm C} = 0$ or 0.2. After tidal evolution, the free eccentricities $e_{\rm free}$ of the test particles are extracted by applying fast Fourier transformation to the distance between the test particles and the center of mass of the system and compared with the current eccentricities of the four small satellites. The only surviving test particles with $e_{\rm free}$ matching the eccentricities of the current satellites are those not affected by mean-motion resonances during the tidal evolution in a model with Pluto's effective tidal dissipation function $Q = 100$ and an initial $e_{\rm C}$ = 0.2 that is damped down rapidly. However, these test particles do not have any preference to be in or near 4:1, 5:1 and 6:1 resonances with Charon. An alternative scenario may be needed to explain the formation of Pluto's small satellites.Comment: 27 pages, including 10 figures, accepted for publication in A

Complementarity in Wormhole Chromodynamics

The electric charge of a wormhole mouth and the magnetic flux ``linked'' by the wormhole are non-commuting observables, and so cannot be simultaneously diagonalized. We use this observation to resolve some puzzles in wormhole electrodynamics and chromodynamics. Specifically, we analyze the color electric field that results when a colored object traverses a wormhole, and we discuss the measurement of the wormhole charge and flux using Aharonov-Bohm interference effects. We suggest that wormhole mouths may obey conventional quantum statistics, contrary to a recent proposal by Strominger.Comment: 12 pages and 2 figures, phyzzx, CALT-68-188

On the Origin of Pluto's Small Satellites by Resonant Transport

The orbits of Pluto's four small satellites (Styx, Nix, Kerberos, and Hydra) are nearly circular and coplanar with the orbit of the large satellite Charon, with orbital periods nearly in the ratios 3:1, 4:1, 5:1, and 6:1 with Charon's orbital period. These properties suggest that the small satellites were created during the same impact event that placed Charon in orbit and had been pushed to their current positions by being locked in mean-motion resonances with Charon as Charon's orbit was expanded by tidal interactions with Pluto. Using the Pluto-Charon tidal evolution models developed by Cheng et al. (2014), we show that stable capture and transport of a test particle in multiple resonances at the same mean-motion commensurability is possible at the 5:1, 6:1, and 7:1 commensurabilities, if Pluto's zonal harmonic $J_{2P} = 0$. However, the test particle has significant orbital eccentricity at the end of the tidal evolution of Pluto-Charon in almost all cases, and there are no stable captures and transports at the 3:1 and 4:1 commensurabilities. Furthermore, a non-zero hydrostatic value of $J_{2P}$ destroys the conditions necessary for multiple resonance migration. Simulations with finite but minimal masses of Nix and Hydra also fail to yield any survivors. We conclude that the placing of the small satellites at their current orbital positions by resonant transport is extremely unlikely.Comment: 22 pages, including 7 figures; accepted for publication in Icaru

Complete Tidal Evolution of Pluto-Charon

Both Pluto and its satellite Charon have rotation rates synchronous with their orbital mean motion. This is the theoretical end point of tidal evolution where transfer of angular momentum has ceased. Here we follow Pluto's tidal evolution from an initial state having the current total angular momentum of the system but with Charon in an eccentric orbit with semimajor axis $a \approx 4R_P$ (where $R_P$ is the radius of Pluto), consistent with its impact origin. Two tidal models are used, where the tidal dissipation function $Q \propto$ 1/frequency and $Q=$ constant, where details of the evolution are strongly model dependent. The inclusion of the gravitational harmonic coefficient $C_{22}$ of both bodies in the analysis allows smooth, self consistent evolution to the dual synchronous state, whereas its omission frustrates successful evolution in some cases. The zonal harmonic $J_2$ can also be included, but does not cause a significant effect on the overall evolution. The ratio of dissipation in Charon to that in Pluto controls the behavior of the orbital eccentricity, where a judicious choice leads to a nearly constant eccentricity until the final approach to dual synchronous rotation. The tidal models are complete in the sense that every nuance of tidal evolution is realized while conserving total angular momentum - including temporary capture into spin-orbit resonances as Charon's spin decreases and damped librations about the same.Comment: 36 pages, including 18 figures; accepted for publication in Icaru