Stellar discs and intermediate-mass black holes in galactic nuclei I. Fragmenting the disc in an isotropic stellar potential

Technical Summary: Stellar Discs and Intermediate-Mass Black Holes in Galactic Nuclei – I. Fragmenting the Disc in an Isotropic Stellar Potential

Problem Statement
The origin of the complex orbital structure of young massive stars at the Galactic Centre remains an open question. While these stars may have formed from a single, coherently rotating gaseous accretion disc, observations reveal a distribution of angular momentum vectors suggesting multiple orbital components, warped geometry, and a broad eccentricity distribution. This paper investigates whether perturbations from an unseen intermediate-mass black hole (IMBH, $100\text{--}10^5 M_\odot$) could fragment and/or disrupt such a primordial stellar disc into the observed multiple orbital planes. The study focuses on the dynamical interaction between an IMBH and a stellar disc within an isotropic stellar potential, specifically examining the role of vector resonant relaxation (VRR) and resonant dynamical friction (RDF).

Methodology
The authors employ a two-pronged approach combining semi-analytical modeling and direct $N$-body simulations:

1. Semi-Analytical Framework:

   • The system is modeled using a VRR Hamiltonian representing the gravitational interaction energy between rings of stars and the IMBH, averaged over orbital and apsidal precession periods.
   • The authors derive a theoretical criterion for disc disruption by comparing the disc's self-torque (which maintains coherence) against the differential precession torque induced by the IMBH.
   • They analyze three mass regimes: $m_\bullet \ll M_d$ (test particle limit for the IMBH), $m_\bullet \gg M_d$ (test particle limit for stars), and the intermediate case $m_\bullet \simeq M_d$.
   • Analytical expressions for nodal precession rates and disruption conditions are derived for various orbital parameters (inclination, eccentricity, semi-major axis).

2. Direct $N$-Body Simulations:

   • Simulations are conducted using a modified version of the phi-GRAPE code with 4th-order Hermite integration.
   • The system comprises a central supermassive black hole (SMBH, $4 \times 10^6 M_\odot$), a fixed isotropic Plummer potential representing the nuclear star cluster, a stellar disc, and an IMBH.
   • Two initial disc configurations are tested: a "stardisc" model (formed via gas drag, initially circular orbits) and a "thermal" model (thermal eccentricity distribution).
   • A broad parameter space is explored, varying the IMBH mass ($250\text{--}2000 M_\odot$), semi-major axis ($0.05\text{--}0.15$ pc), eccentricity ($0.1\text{--}0.9$), and inclination (prograde, retrograde, and orthogonal).
   • The simulations track the evolution of angular momentum vectors, eccentricities, and disc morphology over timescales of 10–20 Myr.

Key Contributions and Results

• Alignment vs. Anti-Alignment:

  • Prograde IMBHs: An IMBH on a prograde orbit rapidly aligns with the stellar disc via resonant dynamical friction. The overlapping region of the disc aligns with the IMBH, leading to a coherent, albeit potentially thicker, disc structure.
  • Retrograde IMBHs: A massive retrograde IMBH ($m_\bullet \simeq 0.67 M_d$) drives an anti-alignment of the overlapping stellar region relative to the IMBH. The angular momentum vectors of stars in the overlapping region precess around the negative angular momentum vector of the IMBH.

• Disc Fragmentation Mechanism:

  • The study identifies a specific regime where a massive retrograde IMBH fragments the original disc into three distinct components in angular-momentum space:
    1. An inner disc that remains largely intact but may thicken.
    2. A misaligned overlapping region that becomes anti-aligned with the IMBH and fragments.
    3. An unperturbed outer disc that retains the initial conditions.
  • This fragmentation occurs when the IMBH-induced differential precession rate exceeds the disc's self-torque rate. The condition is most easily met for IMBH masses $\sim 2000 M_\odot$ and disc masses $\sim 3000 M_\odot$ within 10–20 Myr.

• Eccentricity Excitation:

  • The IMBH interaction excites eccentricities in the overlapping region, driving stars away from initially circular orbits. This provides a mechanism to explain the observed broad eccentricity distribution of young stars in the Galactic Centre.

• Parameter Dependence:

  • Mass Ratio: Fragmentation is most efficient when the IMBH mass is comparable to the disc mass ($m_\bullet/M_d \sim 0.67$). Lower mass IMBHs primarily cause alignment; higher mass IMBHs can lead to full dispersion depending on the differential torque.
  • Orbital Geometry: The semi-major axis and eccentricity of the IMBH determine the radial location and width of the fragmented region. A retrograde IMBH with $a_\bullet \approx 0.15$ pc and $e_\bullet \approx 0.3\text{--}0.7$ produces fragmentation consistent with the observed substructures.
  • Timescales: The fragmentation and anti-alignment processes occur on timescales of 10–20 Myr, which is comparable to the estimated age of the young Galactic Centre stellar population (6–10 Myr).

Significance
The paper claims that the presence of a massive retrograde IMBH offers a plausible dynamical explanation for the observed complex orbital structure of young massive stars at the Galactic Centre. Specifically, the model reproduces:

1. Multiple Orbital Planes: The fragmentation into inner, misaligned, and outer components matches the observation of multiple counter-rotating or misaligned discs.
2. Warped Geometry: The differential precession creates a warped structure in the overlapping region.
3. Eccentricity Distribution: The excitation of eccentricities accounts for the non-thermal distribution of stellar orbits.

The authors emphasize that while the S-star cluster constraints limit the presence of IMBHs in the innermost regions, the adjacent young stellar disc (0.04–0.5 pc) remains a viable region for such an IMBH to exist and dynamically shape the stellar population without violating current observational limits. The study validates the VRR-based semi-analytical predictions through direct $N$-body simulations, confirming that resonant dynamical friction and differential precession are sufficient to fragment a coherent stellar disc under specific mass and orbital configurations.