Influence of flow thickness on general relativistic low angular momentum accretion around spinning black holes

General relativistic, axisymmetric flow of low angular momentum accretion around a Kerr black hole can assume certain geometric configurations where the flow is maintained in hydrostatic equilibrium along the vertical direction (the direction orthogonal to the equatorial plane of the flow). The flow thickness for such accretion models becomes a function of the local radial distance measured from the black hole horizon. There are three types of functions defined in the literature which resemble the thickness of the flow for such a configuration. We formulate the equations governing the steady state astrophysical accretion characterized by both the polytropic as well as the isothermal equation of state in classical thermodynamics. We solve the equations within the framework of such geometric configuration for three different thickness functions to obtain the multitransonic, shocked, stationary integral accretion solutions. Such solutions enable us to study how flow thickness influences the dependence of the properties of postshock flows on black hole spin angular momentum, i.e., the Kerr parameter. For temperature-preserving standing shocks, we find that the postshock part of the disc can become luminous, and a considerable amount of gravitational energy carried by the accreting fluid gets liberated at the shock. We find which kind of thickness function produces the maximum liberated energy, making the disc most luminous.

Figure 1

$\mathcal{E}$ vs $\lambda$ plot for polytropic NT disc with $a=0.3$ and $\gamma =4/3$.

Figure 2

(a) Disc height vs radial distance, (b) Central region in (a) is magnified, (c) Transonic boundaries–Cyan (lighter shade) region represents subsonic flow and red (darker shade) region represents supersonic flow, (c) Phase space trajectories–Mach number vs radial distance.

Figure 3

Generic flow profiles–(a) Advective velocity ($u$) vs $R_g$, (b) Flow ion temperature ($T$) vs $R_g$, (c) Rest mass density ($\rho$) vs $R_g$, (d) Pressure ($P$) vs $R_g$. $u$ is in units of ${10}^{10}\mathrm{cm}/\sec$, $T$ is in units of ${10}^{10}$ Kelvin, $\rho$ is in units of $\mathrm{gm}/\mathrm{cc}$ and $P$ is in units of $\mathrm{dyn}/{\mathrm{cm}}^2$.

Figure 4

Parameter space overlap for shocked solution–(a) $\mathcal{E}-\lambda$ plot and (b) $\lambda -a$ plot. Insets and shaded areas depict the common regions of shock solutions. ALP shown by red solid lines, RH shown by green dotted lines and NT shown by blue dashed lines.

Figure 5

$\gamma \text{−}a$ plot with different $\mathcal{E}$ values (${\mathcal{E}}_1=1.001$, ${\mathcal{E}}_2=1.0005$, ${\mathcal{E}}_3=1.0001$, ${\mathcal{E}}_4=1.00001$) for (a) $\lambda =3.6$, (b) $\lambda =3.3$, (c) $\lambda =3.0$ and (d) $\lambda =2.7$.

Figure 6

Shock location–(a) $r_{\mathrm{sh}}$ (in terms of $R_g$) vs $a$, Ratios at shock–(b) $M_{-}/M_{+}$ vs $a$, (c) $T_{+}/T_{-}$ vs $a$, (d) ${\rho }_{+}/{\rho }_{-}$ vs $a$, (e) $P_{+}/P_{-}$ vs $a$ and (f) ${\dot{\mathrm{\Xi }}}_{+}/{\dot{\mathrm{\Xi }}}_{-}$ vs $a$. “$-$” and “$+$” refer to values “before” and “after” the shock respectively. ALP shown by red solid lines, RH shown by green dotted lines and NT shown by blue dashed lines.

Figure 7

$T-\lambda$ parameter space plot for accretion and wind in isothermal NT disc at $a=0.2$.

Figure 8

Isothermal NT disc–(a) Disc height vs radial distance (in units of $R_g$) with vertical dotted lines depicting the shock locations ($r_{\mathrm{sh}}=24.15$), (b) Magnified view of the central region depicting the truncation radius ($R_T$) and inner critical point ($r_c^{\mathrm{in}}$), (c) Face-on view of the disc (solid curves represent sonic points and dashed curve represents shock front. Regions shaded in cyan and red depict subsonic and supersonic flows respectively), (d) Mach number vs radial distance profile (red path depicts physical flow in the direction indicated with arrows).

Figure 9

Isothermal NT flow profile–(a) Advective flow velocity ($u$) vs $r$, (b) Rest mass density ($\rho$) vs $r$, (c) Pressure ($P$) vs $r$. $u$ in units of $\mathrm{cm}/\sec$, $\rho$ in units of $\mathrm{gm}/\mathrm{cc}$, $P$ in units of $\mathrm{dyn}/{\mathrm{cm}}^2$ and $r$ in units of $R_g$.

Figure 10

Parameter space for shocked accretion flows–(a) Flow temperature ($T$) vs $\lambda$ ($a=0.2$), (b) $\lambda$ vs $a$ ($T={10}^{10}\mathrm{K}$). Solid red curves, green dotted curves and blue dashed curves represent ALP, RH and NT discs respectively. Shaded regions depict overlap zones for the three disc models.

Figure 11

Shock location–(a) $r_{\mathrm{sh}}$ (in terms of $R_g$) vs $a$, Ratios at shock–(b) $M_{-}/M_{+}$ vs $a$, (c) ${\rho }_{+}/{\rho }_{-}$ vs $a$ and (d) $P_{+}/P_{-}$ vs $a$. “$-$” and “$+$” refer to “before” and “after” shock respectively. ALP shown by red solid lines, RH shown by green dotted lines and NT shown by blue dashed lines.

Figure 12

Energy dissipation at shock–${\xi }_{-}/{\xi }_{+}$ vs $a$ along with the corresponding $r_{\mathrm{sh}}$ (in terms of $R_g$) vs $a$ for $T={10}^{10}\mathrm{K}$ and (a) $\lambda =3.95$, (b) $\lambda =3.75$ and (c) $\lambda =3.45$. “-” and “+” represent quantities “before” and “after” shock respectively. ALP shown by red solid lines, RH shown by green dotted lines and NT shown by blue dashed lines.