[Paper Review] Periodic orbits of planets in binary systems

G. Voyatzis|arXiv (Cornell University)|Apr 3, 2017

Astro and Planetary Science1 references3 citations

TL;DR

This paper presents a computational framework for computing and continuing periodic orbits in coplanar three-body systems, focusing on planetary orbits in binary star systems. By continuing solutions from two-planet systems around single stars, it identifies stable P-type (circumbinary) and S-type (circumstellar) orbits, with linear stability indicating long-term dynamical stability domains.

ABSTRACT

Periodic solutions of the three body problem are very important for understanding its dynamics either in a theoretical framework or in various applications in celestial mechanics. In this paper we discuss the computation and continuation of periodic orbits for planetary systems. The study is restricted to coplanar motion. Staring from known results of two-planet systems around single stars, we perform continuation of solutions with respect to the mass and approach periodic orbits of single planets in two-star systems. Also, families of periodic solutions can be computed for fixed masses of the primaries. When they are linearly stable, we can conclude about the existence of phase space domains of long-term orbital stability.

Motivation & Objective

• To investigate the existence and stability of periodic orbits for planets in binary star systems using the planar three-body problem.
• To extend known periodic solutions from two-planet systems around single stars to binary systems via mass and phase space continuation.
• To identify stable periodic orbits that indicate long-term orbital stability domains in binary systems.
• To explore the transition from circumstellar (S-type) to circumbinary (P-type) orbits through mass continuation.
• To provide a computational methodology applicable to both circular and elliptic binary systems, with potential extension to triple systems.

Proposed method

• Uses the general planar three-body problem (GTBP) in a rotating frame centered on the binary pair, transforming inertial coordinates to a co-rotating system to simplify dynamics.
• Applies coordinate transformations using rotation and translation operators to express kinetic and potential energy in the rotating frame, enabling numerical integration.
• Performs numerical continuation of periodic orbits with respect to mass (increasing the planet's mass from a small value to that of a primary) and initial conditions in phase space.
• Employs monodromy matrix analysis to determine linear stability of computed periodic orbits.
• Constructs families of periodic solutions by varying initial conditions such as the planet’s starting distance from the binary.
• Validates results by comparing stability boundaries with known limits from Pilat-Lohinger & Dvorak (2002) and Musielak et al. (2005).

Experimental results

Research questions

• RQ1 How can periodic orbits in binary systems be computed and continued from known solutions of two-planet systems around single stars?
• RQ2 What is the transition behavior from S-type to P-type planetary orbits as the planet's mass increases?
• RQ3 Which families of periodic orbits remain linearly stable, and what do they indicate about long-term orbital stability?
• RQ4 How do the shapes and stability of periodic orbits change as the planet approaches the binary's orbit?
• RQ5 Can the mass-continuation method be extended to elliptic binary systems and systems with comparable-mass components?

Key findings

• Families of stable circular periodic orbits exist for circumbinary (P-type) configurations, extending to infinite orbital radii, and remain linearly stable in the initial segment of the family.
• As the planetary orbit shrinks and approaches the outer primary, the orbit becomes increasingly non-circular and loses stability, terminating at the 3:1 resonance with the binary.
• For a Jupiter-mass planet (m_i = 0.001) in a binary with total mass 0.999, stable S-type orbits are found when the planet is a satellite of the primary star.
• Continuation from the outer planet’s mass increase leads to S-type orbits, while increasing the inner planet’s mass yields P-type orbits.
• The method successfully generates stable periodic orbit families in phase space, with stability confirmed via monodromy matrix analysis.
• The approach is extendable to elliptic binaries and systems with comparable-mass components, offering a pathway to study triple systems.