The quest for Magrathea planets. II. Orbital stability of exoplanets formed around double white dwarfs

Planetary formation might occur at different stages of the stellar evolution. In particular, theoretical studies have been focusing on addressing whether formation can occur around compact binaries that evolved beyond the main sequence. Formation of second-generation planets has been tested in circumbinary disks formed by the ejection of stellar material from binaries composed of either a main-sequence star and a white dwarf or a double white dwarf (DWD). In the latter case, formation appears to be common and to create sub-Neptunian, Neptunian, and giant planets that can migrate within 1 au of the central binary. Nevertheless, the orbital stability of these systems has yet to be studied. We investigate whether planetary systems that formed around compact DWDs in nonresonant and resonant configurations can be dynamically stable over a timescale of a few million years. We performed N-body simulations of circumbinary multiplanetary systems that initially hosted two, three, four or five planets by employing a hybrid symplectic integrator made specifically for circumbinary systems. We recorded the catastrophic events that planetary systems experience and employed a variety of metrics, such as orbital spacing, variation in the center of mass, and normalized angular momentum deficit, to explore the outcomes of their long-term evolution. Furthermore, we evaluated the potential for detecting these systems in their final configurations with the Laser Interferometer Space Antenna mission by measuring the overall amplitude shift in the gravitational-wave frequency induced by their planets. Our results show that planets orbiting DWDs can be stable over the studied timescales. While planetary systems starting with two planets are more likely to survive unaltered, planetary systems with three, four, or five planets experience catastrophic events that cause them to lose some of their original planets. At the end of their phases of dynamical instability, the five-planet population is completely disrupted, and most of the systems host only two surviving planets. This increases the number of two-planet systems by 122% with respect to their initial abundance and creates a single-planet population of 7% of all systems. Additionally, the four-planet population decreases by 56.1% and the three-planet population by 22.5%. Finally, 7.7% of the systems are disrupted; they initially hosted more than two planets. Most of the systems that in the end only host a single planet are potential candidates for the Laser Interferometer Space Antenna mission. A handful of multiplanet systems might be detected. Finally, we provide a formula for estimating the amplitude shift in the gravitational-wave frequency for multiplanet systems orbiting DWDs. Throughout our analysis, we highlight the importance of characterizing the system orbits and estimating their normalized angular momentum deficit in order to distinguish between the different dynamical scenarios presented above. Ultimately, second-generation systems might represent crucial targets for the Laser Interferometer Space Antenna mission because they reside within its observability range.