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

Technical Summary: Orbital Stability of Exoplanets Formed Around Double White Dwarfs

Problem Statement
While theoretical models suggest that "second-generation" planets can form from circumbinary discs created by material ejected during the common-envelope phase of double white dwarf (DWD) binaries, the long-term dynamical stability of such systems remains untested. Previous studies have demonstrated the potential for giant planet formation in these environments (Ledda et al. 2023), yet it is unclear whether these systems can survive the chaotic gravitational interactions inherent to multi-planet circumbinary architectures over million-year timescales. Furthermore, the stability of these systems is critical for assessing their detectability by the upcoming Laser Interferometer Space Antenna (LISA) mission, which aims to detect exoplanets orbiting DWDs via gravitational wave (GW) frequency modulations.

Methodology
The authors performed $N$-body simulations of circumbinary planetary systems orbiting three specific DWD configurations (DWD2, DWD3, and DWD4) drawn from the LISA DWD population. The study employed a hybrid symplectic integrator (Chambers et al. 2002) specifically adapted for binary systems to handle both long-term Keplerian evolution and close encounters/collisions.

The simulations covered four distinct populations based on initial conditions:

• Population A: Non-resonant systems with lower-mass planets ($M_p \leq 1.2 M_J$) around lower-mass binaries ($M_{bin} < 0.6 M_\odot$).
• Population B: Non-resonant systems with higher-mass planets ($M_p \geq 2.5 M_J$) around more massive binaries ($M_{bin} \sim 1 M_\odot$).
• Population C: Resonant counterparts to Population B, where planets are initially locked in 2:1 mean-motion resonances.
• Population D: Five-planet systems in a specific "NMS" resonance (a combination of 4:3 and 5:3 resonances) around DWD*4.

Initial conditions included planet masses drawn from distributions based on Ledda et al. (2023), semi-major axes between the critical stability limit ($a_{crit}$) and 1 au (or scaled for resonant cases), and low initial eccentricities ($e < 0.01$) and inclinations ($1^\circ-3^\circ$). Simulations ran for $3 \times 10^6$ to $10^7$ years.

To analyze the outcomes, the authors utilized several metrics:

1. Normalized Angular Momentum Deficit (NAMD): To quantify the dynamical excitation (temperature) of the system.
2. Catastrophic Event Tracking: Recording ejections (scattering beyond 100 au), collisions, and close approaches to the binary.
3. Orbital Spacing and Mass Concentration: Analyzing changes in orbital spacing ($S_s$) and the mass fraction of the most massive planet ($S_m$).
4. Center of Mass (CoM) Shift: Calculating the displacement of the planetary system's barycenter to infer which planets were lost.

Key Results

• Stability and Multiplicity: Planetary systems orbiting DWDs can remain stable over the studied timescales, but stability is strongly dependent on initial multiplicity. Systems starting with two planets are the most likely to survive unaltered. Conversely, systems with three, four, or five planets frequently experience catastrophic instability.
• Population Outcomes:
  • Population A (Low mass, non-resonant): 82.5% retained their initial planet count; 17.5% experienced catastrophic events.
  • Population B (High mass, non-resonant): Only 49% retained their initial multiplicity; 51% experienced planet loss.
  • Population C (High mass, resonant): 65.6% retained initial multiplicity. While resonance enhanced stability for low-multiplicity systems, four-planet resonant systems were more prone to disruption than their non-resonant counterparts.
  • Population D (Five-planet resonant): 100% of systems experienced catastrophic events. The majority (58%) lost exactly two planets, and 17.2% were completely disrupted (all planets lost).
• Dynamical Evolution: Systems that underwent instability typically evolved into "dynamically hot" states (NAMD > $1.3 \times 10^{-3}$), characterized by increased orbital spacing and higher mass concentration in the surviving most massive planet. A strong anti-correlation was observed between final multiplicity and NAMD: systems ending with fewer planets generally exhibited higher dynamical excitation.
• Resonance Effects: While initial 2:1 resonances helped preserve low-multiplicity (2-3 planet) systems, they did not prevent the disruption of higher-multiplicity chains (4-5 planets), which almost universally broke the resonance and lost planets.
• LISA Detectability:
  • Single-Planet Systems: Approximately 7% of the total simulated systems ended up as single-planet systems. Most of these survivors possess masses and periods that fall within LISA's detection window, provided favorable sky location and distance.
  • Multi-Planet Systems: The authors derived a formula (Eq. 15) to estimate the global gravitational wave frequency shift amplitude ($\Delta f$) induced by multiple planets. For the majority of simulated systems, this shift is below LISA's frequency resolution ($\sim 7.92 \times 10^{-9}$ Hz for a 4-year mission). Only a handful of resonant systems (Population C) showed shifts potentially detectable within an 8-year mission.
  • CoM Acceleration: The study noted that planetary ejections can induce measurable accelerations in the binary's center of mass (up to $\sim 0.9$ au/yr$^2$ in specific cases), which could leave an imprint on the GW signal, though a specific detection threshold for LISA has not yet been defined.

Significance and Claims
The paper claims that second-generation planetary systems around DWDs are dynamically viable candidates for observation, but their final architectures are heavily sculpted by early dynamical instabilities. The authors emphasize that:

1. Survival is Common but Modified: While complete disruption is rare (7.7% of total systems), the loss of planets is common, leading to a significant increase (122%) in the population of two-planet systems derived from higher-multiplicity progenitors.
2. NAMD as a Diagnostic: The NAMD metric serves as a crucial tool to distinguish between systems that formed with low multiplicity and those that were dynamically excited and lost planets. High NAMD values in observed two-planet systems may indicate a history of instability.
3. LISA Potential: The study supports the feasibility of detecting "Magrathea" planets (second-generation planets around DWDs) with LISA, particularly single-planet survivors. However, the detection of multi-planet systems via frequency modulation is unlikely for most configurations due to small frequency shifts, necessitating specific waveform studies and potentially relying on CoM acceleration signatures.

The authors conclude that while their simulations cover the early dynamical evolution (up to $\sim 10$ Myr), the results provide a foundational framework for interpreting future LISA data and understanding the fate of planets in evolved binary systems. They explicitly state that their detectability claims are illustrative and that dedicated waveform analyses are required to confirm observational prospects.