Figure 1

Configuration of the compound EMRI/MBHB system. The $XYZ$-frame is defined around the central body $m_0$, and its orientation is spatially fixed. The massive outer body $m_1$ virtually remains on the $XY$-plane (its initial orbital plane and used as our reference plane) with its angular position ${\lambda }_1$. The inclined inner particle $m_2$ intersects with the $XY$-plane at the ascending node specified by ${\Omega }_2$. Its pericenter is described by the angle ${\varpi }_2\equiv {\Omega }_2+{\omega }_2$ (${\omega }_2$: the angle between the ascending node and the pericenter). ${\lambda }_2$, ${\Omega }_1$ and ${\varpi }_1$ can be defined similarly, but are less important in our study.Reuse & Permissions

Figure 2

Evolution of the inner EMRI, as a function of the decaying semimajor axis $a_1$ of the outer MBHB (leftward from $a_1=350$ down to 160). The panel (a) is for the inner semimajor axis $a_2$. The inner eccentricity $e_2$ (panel b) and the inclination $I_2$ (panel c) show sudden changes around $a_2\sim 290$ where the inner EMRI is resonantly captured by the outer MBHB, as shown in panel (d).Reuse & Permissions

Figure 3

The positions (crosses) and pericenters (filled squares) of the inner particle $m_2$ seen in the corotating $X_R{Y}_R{Z}_R$-frame normalized by the outer distance $a_1$. In this frame, the central body $m_0$ is at (0, 0, 0) and the outer body $m_1$ is at (1, 0, 0). The left panel is the projections of the points onto the $X_R{Y}_R$-plane, and the right one onto the $Z_R{Y}_R$-plane. These results are obtained from the same run as Fig. 2. The points are sampled 200 times between $a_1=272$ and 256, corresponding to $\sim 3400$ orbital cycles of the outer MBHB. The pericenter of the EMRI is resonantly trapped by the MBHB, and stays nearly on the analytical curve (8) with $|\mathit{X}|=a_2(1-e_2)R$.Reuse & Permissions

Figure 4

Evolution of the triple systems in the $a_1{a}_2$-plane. The open squares are the initial conditions with $a_{1s}=350$ and $a_{2s}=20–30$. The circles represent the points when $e_2$ exceeded 0.1 (indicating resonant capture), and the triangles are the termination points of the runs (all by the condition C2). The two solid curves are the analytical predictions (12) for the onsets of the resonant capture for $\nu =1$ and 0.Reuse & Permissions

Figure 5

Resonant capture of the inner EMRI by the outer MBHB. We show a schematic illustration for the $\nu =0$ mode with a small eccentricity $e_2\simeq 0$. The thick (solid and dashed) curve is the critical semimajor axis $a_2={\gamma }_{;0}$ of the EMRI for the resonant condition (12). The solid line $a_2={\eta }_c{a}_1$ is the catch-up line where we have $a_1^{-1}d{a}_1/dt=a_2^{-1}d{a}_2/dt$. The dotted line $a_2=\kappa {\eta }_c{a}_1$ with $\kappa =(5/3{)}^{1/4}$ divides the signature of the resonance approach $\mathrm{sign}(\dot{\nu })$, as shown by the three arrows. We put the intersection of the two curves by $(x_{1;0},x_{2;0})$. To realize a capture, the resonant curve $a_2={\gamma }_{;0}$ should be crossed in the direction $\dot{\nu }<0$, corresponding to the upper region of the dotted line (the solid part of the curve $a_2={\gamma }_{;0}$). Therefore, at the critical outer distance $a_1=x_{1;0}$, the capturable inner EMRI must have a semimajor axis ${\kappa }^{-1}{x}_{2;0}<a_2<x_{2;0}$ shown by the vertical double line.Reuse & Permissions