Innermost stable circular orbits in the Majumdar-Papapetrou dihole spacetime

We investigate the positions of stable circular massive particle orbits in the Majumdar-Papapetrou dihole spacetime with equal mass. In terms of qualitative differences of their sequences, we classify the dihole separation into five ranges and find four critical values as the boundaries. When the separation is relatively large, the sequence on the symmetric plane bifurcates, and furthermore, they extend to each innermost stable circular orbit in the vicinity of each black hole. In a certain separation range, the sequence on the symmetric plane separates into two parts. On the basis of this phenomenon, we discuss the formation of double accretion disks with a common center. Finally, we clarify the dependence of the radii of marginally stable circular orbits and innermost stable circular orbits on the separation parameter. We find a discontinuous transition of the innermost stable circular orbit radius. We also find the separation range at which the radius of the innermost stable circular orbit can be smaller than that of the stable circular photon orbit.

Figure 1

Positions of stable circular orbits of massive particles in $\rho \text{−}z$ plane of the Majumdar-Papapetrou dihole spacetime with equal mass $M_{+}=M_{-}=1$ for the separation range $0\le a\le 5$. The solid black lines represent the curves satisfying $U_z=0$ (i.e., $z=0$ and $\rho ={\rho }_0$). The shaded regions show the region $D$, where $h_0>0$, $k_0>0$, and $L_0^2>0$ are satisfied. The sequence of stable circular orbits is the solid black curves included in the region $D$. The solid blue lines are the boundary of $D$ where $h_0=0$, $L_0^2>0$, and $k_0>0$. The dashed blue lines are the boundary of $D$ where $h_0>0$, $k_0>0$, and $L_0^2$ diverges. The red dots are the positions of ISCOs. The green dots are the positions of marginally stable circular orbits (MSCOs) except for the ISCO. The orange triangles and dots are the positions of unstable circular photon orbits and stable ones, where $L_0^2$ diverges. (a) $a=5$. When $a$ is large enough, stable circular orbits exist not only on $z=0$ line in the range $\rho \in (\sqrt{2}a,\infty )$ but on $\rho ={\rho }_0$ line. An MSCO exists at $(\rho ,z)=(\sqrt{2}a,0)$, and the ISCOs are located around each black hole. (b) $a=a_0=1.401\cdots$. There exist stable circular orbits on $z=0$ plane in the range $\rho \in (\sqrt{2}{a}_0,\infty )$. The ISCO is located at $(\rho ,z)=(\sqrt{2}{a}_0,0)$, where three MSCOs for $a>a_0$ are degenerate. (c) $a=1$. There exist stable circular orbits on $z=0$ plane in the range $\rho \in (\sqrt{2},\infty )$. The point $(\rho ,z)=(\sqrt{2},0)$ is the ISCO. (d) $a=a_{*}=0.9713\cdots$. The region $D$ is marginally connected at $(\rho ,z)=({\rho }_{*},0)$. There exist stable circular orbits on $z=0$ plane in the range $\rho \in (\sqrt{2}{a}_{*},\infty )$, of which the boundary $(\rho ,z)=(\sqrt{2}{a}_{*},0)$ is the ISCO. (e) $a=0.95$. The sequence of stable circular orbits splits into two parts. As a result, two additional MSCOs appear at the boundary of the outer sequence and the outer boundary of the inner sequence. The point $(\rho ,z)=(\sqrt{2}a,0)$ is the ISCO. (f) $a=a_{\infty }=0.5433\cdots$. There are two sequences of stable circular orbits. The outer boundary of the inner sequence is no longer physical because $L_0^2$ diverges there. An MSCO appears at the boundary of the outer sequence, and the ISCO appears at $(\rho ,z)=(\sqrt{2}{a}_{\infty },0)$. (g) $a=0.53$. There are two sequences of stable circular orbits. At the outer boundary of the inner sequence, $L_0^2$ diverges. An MSCO appears at the boundary of the outer sequence, and the ISCO appears at $(\rho ,z)=(\sqrt{2}a,0)$. (h) $a=a_{\mathrm{c}}=0.3849\cdots$. The divergence of $L_0^2$ occurs at $(\rho ,z)=(\sqrt{2}{a}_{\mathrm{c}},0)$ and the inner sequence of stable circular orbits disappears. There only exists the outer sequence and its inner boundary becomes the ISCO. (i) $a=0$. There exist stable circular orbits in the range $\rho \in (6,\infty )$ on $z=0$. The point $(\rho ,z)=(6,0)$ is the ISCO, which connects to the ISCO of the single extremal Reissner-Nordström black hole with mass 2.

Figure 2

Dependence of the radii of marginally stable circular orbits and circular photon orbits on the separation parameter $a$ in the Majumdar–Papapetrou dihole spacetime with equal unit mass. The red and green solid lines mark the radii of the MSCOs and ISCOs, respectively. The orange dashed lines and the orange solid line are the unstable circular photon orbits and the stable circular photon orbits, respectively. For $a_{\mathrm{c}}<a\le a_{\infty }$, the radius of the ISCO is smaller than that of the stable circular photon orbit. At $a=a_{\mathrm{c}}$, a discontinuous transition of ISCO occurs.