In Petrovich & Antonini 2017 we studied the orbital evolution of stellar binaries in our galactic center and proposed that a sizable fraction of the LIGO sources come from mergers of binary black holes in galactic nuclei through a combination of three-body interactions (the third-body being a central massive black hole) and the perturbations from the dense stellar cluster.
First, some context. The stellar binaries in the inner parsec of our galactic center orbit around the massive black hole at the center (Sag A) in nearly Keplerian orbits, but after many orbits they can undergo eccentricity-inclination oscillations through the so-called Lidov-Kozai mechanism. This mechanism is known to produce extreme eccentricities, bringing the stars very closely together during periastron passages, and has previously been proposed as a channel to produce stellar mergers, low-mass x-ray binaries, gravitational-wave sources, and others interesting outcomes. However, the Lidov-Kozai mechanism requires that the binary orbits in a plane that is nearly orthogonal to its orbit around Sag A ("in" orbit orthogonal to "out" orbit in the cartoon below), allowing for a very small set of orbits that can reach extreme eccentricities. But, how small? For example, for binary black holes mergers is only ~0.1% of the total phase-space if this is populated ergodically, making this channel insignificant to contribute to the LIGO merger rates.
This is where our work comes into play. We propose that the perturbations from the dense stellar cluster, where the binary embedded in, greatly enhances the merger rates (a factor of ~10-100) by allowing for a much wider range of orbits that can undergo extreme eccentricity excitation (see Figure below where red is the no cluster case). The Figure shows that 1-e_max<0.0001 occurs in ~40% of the systems if epsilon=1 and for any values of the initial inclination.
As a first approximation we only considered the potential from the cluster, which in our galactic center it is observed not to be fully spherical. As sketched in the cartoon above, a flattened (axissymetric) cluster makes the outer precess around n_z. We find that if this precession occurs in timescales comparable to the Lidov-Kozai timescale, there can be appreciable changes in the mutual inclination between the inner and outer orbit which would intensify the eccentricity oscillations. Interestingly, we realized that this is the dynamical regime of typical binaries at ~0.1 pc for clusters with ellipticities of ~10%. Since most of the massive stars, progenitors of the compact objects, are observed to lie at ~0.1 pc, our proposed mechanism is a good candidate for gravitational-wave sources as we show below.
Let's show the dynamical behavior with some movies. I consider three extreme cases to illustrate the effect from the cluster potential and show the Keplerian orbits (ellipses) with the white arrow pointing towards the pericenter and the red arrow indicates the orientation of the angular momentum vector j_in. The ellipse is in the frame co-precessing with the outer orbit so as we increase the value of epsilon_sc,z in the movies the precession of j_out around n_z becomes faster and the ellipse will move faster.
1) No cluster perturbations. There are no e-i oscillations because the initial mutual inclination is below the critical angle of 39.2 deg and the orbit only precesses: j_in around j_out (red arrow) and the pericenter (white arrow), as expected from secular three-body interactions.
2) LK timescale/cluster precession=0.2. Some modest inclination oscillations are observed overcoming the critical 39.2 degrees from the LK mechanism, time at which the eccentricity grows as expected. However, the eccentricity reaches only up to ~0.7 and no merger is expected.
3) LK timescale/cluster precession=1. The inclination rapidly crosses the 39.2 deg and increases up to ~90 deg in only a few LK cycles allowing for the eccentricity to reach extreme values (e>0.999). The behavior is chaotic and the orbit flips episodically to retrograde configurations (i>90 deg).
As shown in the surfaces of section below the system can become chaotic as the ratio of timescales in the problem approaches unity ("epsilon=0" means no cluster perturbations). In particular, for epsilon_sc,z=1 (panel d) any point in phase-space will find its way to reach e=1 given enough time for diffusion to occur.
As a final piece of our work we looked at the possible mergers through gravitational-wave radiation between compact objects: black holes (BHs) and Neutron stars (NSs). In order to do so, we fixed the background density of the cluster and evolve tens of thousands observationally-motivated main sequence massive (>8 Solar masses) binaries:
- radial separation from Sag A from observations of young stars in ~0.04-0.5 pc;
- binary separation from binaries in the galatic field;
- IMF from the galactic field and top heavy;
First, we use binary evolution code (BSE) to get a population of compact objects. Second, we use our secular code including GR precession and vary the value of epsilon_z (departure from spherical symmetry ~ ellipticity of the cluster) and assess whether a merger occurs before the binary is disrupted by interactions with other stars in the cluster.
We summarize the results in the table below. We observe that the fraction of mergers (last column) for epsilon_z>0.1 is ~30-100 times larger than for epsilon_z=0.001 (spherically symmetric cluster).
The final piece is to derive a merger rate that can be compared with that from LIGO. Here, we assume the following:
- the star formation rate has been constant and equal to ~0.001 Msun/yr (divide the mass of cluster by 10 Gyr);
- only Milky Way-like galaxies will have rates that are similar to our galactic center and their number density is 0.02/Mpc^3.
By factoring in all these ingredients, we get the following rates:
compared to ~12-210 Gpc^-3 yr^-1(BH-BH) from LIGO. There are many uncertainties in our estimates mostly due to the poor constraints on the star formation history in galactic nuclei. However, our work shows that the orders-of-magnitude merger rate enhancement due to the cluster potential can lead to rates that are high enough that many detections by LIGO are expected to come from galactic centers.
We plan to pursue this research venue further, stay tuned!