Evolution of Supermassive Black Hole Pairs on Inclined Orbits in Post-Merger Galaxies

TL;DR

The authors address how the initial orbital inclination of SMBH pairs formed in galaxy mergers affects their inspiral times within the inner kiloparsec. They develop a 3D dynamical model with a refined Poisson solver to compute the gravitational potential and dynamical friction from stars and gas, and they run 972 configurations across a grid of galaxy parameters. The results show that higher inclinations generally lengthen pairing times, with a critical angle beyond which pairing may fail within a Hubble time; disk properties such as $v_g$ and the primary mass $M_1$ strongly modulate these times, sometimes allowing pairing at all inclinations. These findings imply that dual-AGN and gravitational wave progenitors are more likely to arise from nearly coplanar mergers, and they establish a framework for predicting SMBH coalescence rates in diverse post-merger environments.

Abstract

Theoretical models of the evolution of supermassive black hole (SMBH) pairs in post-merger remnant galaxies are necessary to motivate observational searches for dual active galactic nuclei (AGN) and gravitational wave sources. Studies have explored the dynamical evolution of SMBH pairs under the influence of dynamical friction to calculate pairing times and predict the expected population of dual-AGNs at various redshifts. We formulate a three-dimensional dynamical model of SMBH pairs in the innermost kiloparsec of a post-merger galaxy to investigate the impact of orbital inclination with respect to the galactic disk on pairing times. The SMBH pairs are evolved in 81 different galaxy configurations initialized using a Gauss-Seidel Poisson solver. The dynamics are calculated for 12 distinct initial inclinations ranging from 0 to 75 degrees in each of the galaxies to gauge the impact of inclination on pairing time. Orbits characterized by initial inclinations greater than 20 degrees frequently require longer pairing times when compared to uninclined orbits. Pairing times for orbits with inclinations $\gtrsim 45$ degrees often exceed 14 Gyr. Galaxies with higher mass SMBH pairs and faster rotating disks generally shorten pairing times relative to galaxies with less massive or slower rotating disks when the inclination is $\lesssim 45$ degrees. The model suggests that SMBH pairs that form from mergers at inclinations $\lesssim 20$ degrees are likely progenitors of dual-AGN and gravitational wave sources.

Figures (12)

• Figure 1: Illustration of a gravitational potential field computed from the specified mass density profiles for the galactic bulge, stellar disk, and gas disk. In this scenario, the mass of the central SMBH is $M_1 = 10^6$$M_{\rm \odot}$, the central gas number density is $n_{\rm \rm gd} = 100$ cm$^3$, and the gas fraction is $f = 0.7$.
• Figure 2: Example orbits in a galaxy with $n_{\rm gd} = 100$ cm$^3$, $f_{\rm g}$ = 0.7, $v_{\rm g} = 0.7v_c$, and $M_1 = 10^6$$M_{\rm \odot}$. Left: An orbit at an initial inclination of 0$^{\circ}$ which merges in 5.75 Gyr staying at the same inclination of 0$^{\circ}$ for the whole orbit. Middle: An orbit at an initial inclination of 25$^{\circ}$ which merges in 7.90 Gyr. The orbital inclination of the secondary decreases, and the SMBH settles into the plane of the disk due to vertical DF. Right: An SMBH with initial inclination of 45$^{\circ}$, which does not pair within a Hubble time. The secondary maintains a large inclination throughout the entire orbit since vertical DF is inefficient far from the plane of the disk when $v_z \gg c_s$.
• Figure 3: Left: Projection of the full 25 degree orbit from the middle panel of Figure \ref{['fig:2']} into the $rz$-plane represented by the solid red curve. The amplitude of vertical oscillations abruptly decays when the vertical Mach number is of order unity, leading to the very thin envelope observed at $r \lesssim 600$ pc. The dashed blue line indicates the shape of the orbit for the first 200 Myr when DF has not substantially altered the dynamics. Right: Orbital inclination as a function of time for the same orbit plotted as the solid orange curve. The vertical Mach number is plotted as the dashed green curve alongside the inclination to demonstrate the strong relationship between vertical speed and vertical DF strength. At a critical time around 4 Gyr, vertical DF becomes extremely efficient and drags the secondary close to the plane where it maintains a low inclination for the rest of the orbit. This coincides with the vertical Mach number decreasing to the range where $\langle \mathcal{M}_z\rangle \sim 1$.
• Figure 4: Left: Projection of the full 45 degree orbit from the right panel of Figure \ref{['fig:2']} into the $rz$-plane. The amplitude of vertical oscillations decays more gradually compared to Figure \ref{['25inc']} since the vertical Mach number is always much greater than 1. The envelope gradually shifts due to DF, but does not collapse into a thin line as in the 25 degree case. Right: Orbit inclination as a function of time for the 45$^{\circ}$ inclined orbit. In contrast to the behavior observed in the right panel of Figure \ref{['25inc']}, the inclination $i$increases on average. The inclination increases due to the azimuthal DF being more efficient than vertical DF, which drains the $z$-component of the angular momentum faster than the other components of the angular momentum.
• Figure 5: Left axis: Pairing time (purple circles) as a function of initial inclination angle $i_0$ for orbits in the same galaxy displayed in Figure \ref{['fig:1']}. Right axis: The average vertical Mach number (green diamonds) as a function of initial inclination, calculated as the time average of $\mathcal{M}_{\rm z}$ over the whole evolution. The SMBH pair fails to merge at an inclination of 45$^{\circ}$ or more. Additionally, the average vertical Mach number very closely follows the trend of the pairing time with both increasing along with initial inclination.