Resolving the relative influence of strong field spacetime dynamics and MHD on circumbinary disk physics

In this paper we evolve magnetized and unmagnetized circumbinary accretion disks around supermassive black hole binaries in the relativistic regime. We use a post-Newtonian expansion to construct an analytical spacetime and determine how the order of the post-Newtonian (PN) expansion affects the dynamics of the gas. We find very small differences in the late-time bulk dynamics of nonmagnetized hydrodynamic evolutions between the two spacetimes down to separations of approximately $40\mathit{GM}/c^2$ where $M$ is the total mass of the binary. For smaller separations, the differences due to PN order become comparable to differences caused by using initial data further from equilibrium. For magnetized gas, magnetohydrodynamic stresses, which drives the accretion dynamics, tends to mask all higher order PN effects even at separations of $20\mathit{GM}/c^2$, leading to essentially the same observed electromagnetic luminosity. This implies that our calculations of the electromagnetic signal may be robust down to small binary separations. Our investigation is the first to demonstrate how the level of PN accuracy affects a circumbinary disk’s evolution and informs us of the range in separation within which to trust the PN approximation for this kind of study. We also address the influence the initial conditions and binary separation have on simulation predictions.

Figure 1

Covariant metric components of the time-averaged metric used for constructing disk initial data, for $a=20M$ (left column) and $a=100M$ (right column) binary separations, and the 1PN metric (top row) and 2.5PN metric (middle row). The relative differences (bottom row) are shown between the 2.5PN metric and the 1PN metric (solid lines), the 2.5PN metric and its “closest” Kerr solution metric (dots), and the 1PN metric and its closest Kerr solution metric (dashes). Note that all off-diagonal components, except for $g_{t\varphi }$, are very small. This is the crucial ingredient for our disk initial data construction procedure. Here, the numerical indices $\{0,1,2,3\}$ refer to the coordinates $\{t,r,\theta ,\varphi \}$, respectively.

Figure 2

Torque density per unit radius due to binary potential for different separations. Quantities were time averaged over the “quasisteady state” period.

Figure 3

Same as Fig. 2, but for separations $a/M=40,30,20$. Recall that the disk was initialized with a different configuration than that of Fig. 2. Note that the $y$-axis range of these figures is different, as the finer details of the right-hand side figure would not be noticeable otherwise.

Figure 4

Color contour of surface density $\mathrm{\Sigma }(t,r)$, Eq. (3.4) (in logarithmic units) for some simulations. $y$ axis is the simulation time in units of total mass $M$ and $x$ axis is the coordinate radial distance in units of binary separation $a$.

Figure 5

Color contour of (relative difference of) surface density $\mathrm{\Sigma }(t,r)$, Eq. (3.4) (in logarithmic units) for simulations hydro1PN_a30, hydro2.5PN_a30, hydro1PN_a20, and hydro2.5PN_a20. $y$ axis is the simulation time in units of total mass $M$ and $x$ axis is the coordinate radial distance in units of binary separation $a$.

Figure 6

Mass (of the gas) enclosed within several radii as functions of time. Full lines show 1PN evolutions while dotted lines show the corresponding 2.5PN versions.

Figure 7

Different contributions to the flux of angular momentum through the accretion flow seen in the MHD1PN_H (left panel) and MHD2.5PN (right panel) runs. Radial derivatives of the angular momentum flux due to shell-integrated Maxwell stress in the coordinate frame (red lines), the angular momentum flux due to shell-integrated Reynolds stress in the coordinate frame (green lines), advected angular momentum (gold lines), and net rate of change of angular momentum ${\partial }_r{\partial }_{t}J$ (solid black lines). Also shown are torque densities per unit radius due to the actual binary potential (blue lines) and radiation losses (cyan lines). All quantities are time averaged over the secularly evolving period. To clarify colors used, note that at $r=4a$ colors are (from bottom to top): red, green, cyan, blue, black, gold.

Figure 8

Color contour of surface density in logarithmic units, ${\log }_{10}\mathrm{\Sigma }(t,r)$. (Left panel) MHD1PN_H evolution. (Right panel) MHD2.5PN evolution.

Figure 9

Surface density $\mathrm{\Sigma }(r/a)$ for fixed time steps, from beginning of the secularly evolving state (blue lines, bottom) to end of the simulation (red lines, top). The dotted curve shows the initial condition, while the dashed curved shows the average of the colored curves. (Left panel) MHD1PN_l. (Middle panel) MHD1PN_H. (Right panel) MHD2.5PN.

Figure 10

Mass enclosed within $r<(1,1.5,2,3,4)a$ (bottom to top, respectively) in logarithmic units for simulations MHD1PN_H (left panel) and MHD2.5PN (right panel).

Figure 11

FFT of luminosity for our MHD simulations MHD1PN_l (left panel), MHD1PN_H (middle panel), MHD2.5PN (right panel). Highlighted frequencies include the orbital frequency at time-averaged radius of $\max (\mathrm{\Sigma })\equiv {\mathrm{\Omega }}_1$ (green line); ${\mathrm{\Omega }}_2\equiv 1.375\mathrm{\Omega }\_\text{bin}$ (blue dashes); ${\mathrm{\Omega }}_1\pm {\mathrm{\Omega }}_2$ (blue dots); and the overtones of ${\mathrm{\Omega }}_1\pm {\mathrm{\Omega }}_2$ (red dots) and ${\mathrm{\Omega }}_2$ (red dashes). Green, blue and red lines appear in the leftmost, central and rightmost part of the figure respectively.

Figure 12

High resolution version of the lower panel of Fig. 5. We repeated runs hydro1PN_a20 and hydro2.5PN_a20 using $480\times 480$ cells, in order to test the accuracy of our evolution.