Influence of geometrical configuration on low angular momentum relativistic accretion around rotating black holes

We illustrate how the formation of energy-preserving shocks for polytropic accretion and temperature-preserving shocks for isothermal accretion are influenced by various geometrical configurations of general relativistic, axisymmetric, low angular momentum flow in the Kerr metric. Relevant pre- and postshock states of the accreting fluid, both dynamical and thermodynamic, are studied comprehensively. Self-gravitational backreaction on the metric is not considered in the present context. An elegant eigenvalue-based analytical method is introduced to provide qualitative descriptions of the phase orbits corresponding to stationary transonic accretion solutions without resorting to involved numerical schemes. Effort is made to understand how the weakly rotating flow behaves in close proximity to the event horizon and how such “quasiterminal” quantities are influenced by the black hole spin for different matter geometries. Our main purpose is thus to mathematically demonstrate that, for non-self-gravitating accretion, separate matter geometries, in addition to the corresponding space-time geometry, control various shock-induced phenomena observed within black hole accretion disks. We expect to reveal how such phenomena observed near the horizon depend on the physical environment of the source harboring a supermassive black hole at its center. We also expect to unfold correspondences between the dependence of accretion-related parameters on flow geometries and on black hole spin. Temperature-preserving shocks in isothermal accretion may appear bright, as a substantial amount of rest-mass energy of the infalling matter gets dissipated at the shock surface, and the prompt removal of such energy to maintain isothermality may power the x-ray/IR flares emitted from our Galactic Center.

Figure 1

$\mathcal{E}\text{−}\lambda$ plot for quasispherical disk geometry ($\gamma =1.35$ and $a=0.1$).

Figure 2

Comparison of the $\mathcal{E}\text{−}\lambda$ plots for three different flow geometries ($\gamma =1.35$ and $a=0.1$). Constant height disk, quasispherical flow, and flow in vertical hydrostatic equilibrium represented by blue dashed lines, green dotted lines, and red solid lines, respectively. The shaded region in the inset depicts $[\mathcal{E},\lambda ]$ space overlap with multicritical solutions for all three models.

Figure 3

Comparison of ${\mathrm{\Omega }}^2$ vs $[\mathcal{E},\lambda ]$ of the inner and outer critical points for constant height flow (CH), quasispherical flow (CF), and flow in vertical hydrostatic equilibrium (VE) ($\gamma =1.35$, $a=0.1$).

Figure 4

Comparison of ${\mathrm{\Omega }}^2$ vs $[\mathcal{E},\lambda ]$ of the middle critical point for constant height flow (CH), quasispherical flow (CF), and flow in vertical hydrostatic equilibrium (VE) ($\gamma =1.35$, $a=0.1$).

Figure 5

Comparison of ${\mathrm{\Omega }}^2$ vs $a$ for constant height flow (blue dashed lines), quasispherical flow (green dotted lines), and flow in vertical hydrostatic equilibrium (red solid lines) ($\gamma =1.35$, $\mathcal{E}=1.003$, $\lambda =3.0$). (Inset) Magnified view of the common monotransonic region for the three flow geometries.

Figure 6

Phase-space portrait (Mach number vs $r$ plot) for quasispherical disk ($\mathcal{E}=1.0003$, $\lambda =3.5$, $\gamma =1.35$, $a=0.1$). $r_{\mathit{sh}}^{\mathit{in}}=4.805$, $r_{\mathit{sh}}^{\mathit{out}}=38.15$, $r^{\mathit{in}}=4.5$ (inner sonic point $I$), $r^{\mathit{mid}}=13.712$ (middle sonic point $M$), $r^{\mathit{out}}=2244.313$ (outer sonic point $O$).

Figure 7

Comparison of $\mathcal{E}\text{−}\lambda$ plots of allowed shocked multitransonic accretion solutions for three different flow geometries ($\gamma =1.35$ and $a=0.57$). Constant height disk, quasispherical flow, and flow in vertical hydrostatic equilibrium represented by blue dashed lines, green dotted lines, and red solid lines, respectively.

Figure 8

Shaded region depicts the common domain of $[\mathcal{E},\lambda ]$ ($\gamma =1.35$, $a=0.57$), which allows shock formation in a constant height disk (blue circles), quasispherical disk (green circles), and flow in hydrostatic equilibrium (red dots).

Figure 9

Overlap region of $[a,\lambda ]$ ($\gamma =1.35$, $\mathcal{E}=1.00024$), which allows shock formation in a constant height disk (blue circles), quasispherical disk (green circles), and flow in hydrostatic equilibrium (red dots).

Figure 10

Shock location ($r_{sh}$) vs $a$ plot ($\gamma =1.35$, $\mathcal{E}=1.00024$, $\lambda =2.9$) for a constant height disk (dashed blue line), quasispherical disk (dotted green line), and flow in hydrostatic equilibrium (solid red line).

Figure 11

Variation of shock strength ($M_{+}/M_{-}$), compression ratio (${\rho }_{-}/{\rho }_{+}$), pressure ratio ($P_{-}/P_{+}$), and temperature ratio ($T_{-}/T_{+}$) with black hole spin parameter $a$ ($\gamma =1.35$, $\mathcal{E}=1.00024$, $\lambda =2.9$) for a constant height flow (dashed blue lines), quasispherical flow (dotted green lines), and flow in hydrostatic equilibrium (solid red lines). Subscripts “$+$” and “$-$” represent pre- and postshock quantities, respectively.

Figure 12

Variation of quasiterminal values of Mach number ($M_{\delta }$), density (${\rho }_{\delta }$), pressure ($P_{\delta }$), and temperature ($T_{\delta }$) with $a$ ($\gamma =1.35$, $\mathcal{E}=1.00024$, $\lambda =2.9$) for constant height flow (dashed blue lines), quasispherical flow (dotted green lines), and flow in hydrostatic equilibrium (solid red lines). Density and pressure are in centimeter-gram-second (cgs) units of $\mathrm{g}{\mathrm{cm}}^{-3}$ and $\mathrm{dyn}{\mathrm{cm}}^{-2}$, respectively, and temperature is in absolute units of kelvin.

Figure 13

Variation of quasiterminal values of Mach number ($M_{\delta }$), density (${\rho }_{\delta }$), pressure ($P_{\delta }$), and temperature ($T_{\delta }$) with $a$ ($\gamma =1.35$, $\mathcal{E}=1.2$, $\lambda =2.0$) for monotransonic accretion in constant height flow (dashed blue lines), quasispherical flow (dotted green lines), and flow in hydrostatic equilibrium (solid red lines). Density and pressure are in cgs units of $\mathrm{g}{\mathrm{cm}}^{-3}$ and $\mathrm{dyn}{\mathrm{cm}}^{-2}$, respectively, and temperature is in absolute units of kelvin.

Figure 14

Comparison of $T-\lambda$ plots for three different flow geometries ($a=0.1$, $T$ in kelvins). Constant height disk, quasispherical flow, and flow in vertical hydrostatic equilibrium represented by blue dashed lines, green dotted lines, and red solid lines, respectively. The shaded region allows for multicritical solutions in all flow configurations.

Figure 15

Comparison of ${\mathrm{\Omega }}^2$ vs $[T,\lambda ]$ of the inner and outer critical points for constant height flow (CH), quasispherical flow (CF), and flow in vertical hydrostatic equilibrium (VE) ($a=0.1$). The left and right panels depict ${\mathrm{\Omega }}^2$ for the inner and outer critical points, respectively, for CH, CF, and VE from top to bottom in the respective order.

Figure 16

Comparison of ${\mathrm{\Omega }}^2$ vs $[T,\lambda ]$ of the middle critical point for (top left) constant height flow (CH), (top right) quasispherical flow (CF), and (bottom left) flow in vertical hydrostatic equilibrium (VE) ($a=0.1$).

Figure 17

Comparison of ${\mathrm{\Omega }}^2$ vs $a$ for constant height flow (blue dashed lines), quasispherical flow (green dotted lines), and flow in vertical hydrostatic equilibrium (red solid lines) ($T={10}^{10}\mathrm{K}$, $\lambda =3.6$). (Inset) Magnified view of the common monotransonic region for the three flow geometries.

Figure 18

Comparison of $T\text{−}\lambda$ plots of allowed shocked multitransonic accretion solutions for three different flow geometries ($a=0.1$, $T$ in kelvins). Constant height disk, quasispherical flow, and flow in vertical hydrostatic equilibrium represented by blue dashed lines, green dotted lines, and red solid lines, respectively. Shaded region depicts the overlapping domain of shock formation in all three geometries.

Figure 19

Shock location ($r_{sh}$) vs $a$ plot ($T={10}^{10}\mathrm{K}$, $\lambda =3.75$) for a constant height disk (dashed blue line), quasispherical disk (dotted green line), and flow in hydrostatic equilibrium (solid red line).

Figure 20

Variation of shock strength ($M_{+}/M_{-}$), compression ratio (${\rho }_{-}/{\rho }_{+}$), and pressure ratio ($P_{-}/P_{+}$) with black hole spin parameter $a$ ($E={10}^{10}\mathrm{K}$, $\lambda =3.75$) for a constant height flow (dashed blue lines), quasispherical flow (dotted green lines), and flow in hydrostatic equilibrium (solid red lines). The subscripts “$+$” and “$-$” represent pre- and postshock quantities, respectively.

Figure 21

Variation of quasispecific energy ratio (${\xi }_{+}/{\xi }_{-}$) with black hole spin parameter $a$ ($T={10}^{10}\mathrm{K}$). In order to obtain multicritical domains with shock over four different ranges of $a$, four different values of $\lambda$ have been set at the given $T$, i.e., $\lambda =3.75$ (upper left panel), $\lambda =3.25$ (upper right panel), $\lambda =3.0$ (lower left panel), $\lambda =2.7$ (lower right panel). Constant height flow, quasispherical flow, and flow in hydrostatic equilibrium are represented by dashed blue lines, dotted green lines, and solid red lines. The subscripts “$+$” and “$-$” represent pre- and postshock quantities, respectively.

Figure 22

Variation of quasispecific energy ratio (${\xi }_{+}/{\xi }_{-}$) with black hole spin parameter $a$ ($T={10}^{10}\mathrm{K}$, $\lambda =4.0$) for retrograde flow. Constant height flow, quasispherical flow, and flow in hydrostatic equilibrium are represented by dashed blue lines, dotted green lines, and solid red lines. The subscripts “$+$” and “$-$” represent pre- and postshock quantities, respectively.

Figure 23

Variation of quasiterminal values of Mach number ($M_{\delta }$), density (${\rho }_{\delta }$), and pressure ($P_{\delta }$) with $a$ ($T={10}^{10}\mathrm{K}$, $\lambda =3.75$) for constant height flow (dashed blue lines), quasispherical flow (dotted green lines), and flow in hydrostatic equilibrium (solid red lines). Density and pressure are in cgs units of $\mathrm{g}{\mathrm{cm}}^{-3}$ and $\mathrm{dyn}{\mathrm{cm}}^{-2}$, respectively, and temperature is in absolute units of kelvin.

Figure 24

Variation of quasiterminal values of Mach number ($M_{\delta }$), density (${\rho }_{\delta }$), and pressure ($P_{\delta }$) with $a$ ($T=2\times {10}^{11}\mathrm{K}$, $\lambda =2.0$) for monotransonic accretion in constant height flow (dashed blue lines), quasispherical flow (dotted green lines), and flow in hydrostatic equilibrium (solid red lines). Density and pressure are in cgs units of $\mathrm{g}{\mathrm{cm}}^{-3}$ and $\mathrm{dyn}{\mathrm{cm}}^{-2}$, respectively, and temperature is in absolute units of kelvin.