Accretion-ejection in rotating black holes: a model for ‘outliers’ track of radio-X-ray correlation in X-ray binaries

Abstract

We study the global accretion-ejection solutions around a rotating black hole considering three widely accepted pseudo-Kerr potentials that satisfactorily mimic the space-time geometry of rotating black holes. We find that all the pseudo potentials provide standing shock solutions for large range of flow parameters. We identify the effective region of the shock parameter space spanned by energy ($\mathcal{E}_{\text{in}}$) and angular momentum ($\lambda _{\text{in}}$) measured at the inner critical point ($x_{\text{in}}$) and find that the possibility of shock formation becomes feeble when the viscosity parameter ($\alpha$) is increased. In addition, we find that shock parameter space also depends on the adiabatic index ($\gamma$) of the flow and the shock formation continues to take place for a wide range of $\gamma$ as $1.5 \le \gamma \le 4/3$. For all the pseudo potentials, we calculate the critical viscosity parameter ($\alpha _{\text{shock}}^{\text{cri}}$) beyond which standing shock ceases to exist and compare them as function of black hole spin ($a_{k}$). We observe that all the pseudo potentials under consideration are qualitatively similar as far as the standing shocks are concerned, however, they differ both qualitatively and quantitatively from each other for rapidly rotating black holes. Further, we compute the mass loss from the disc using all three pseudo potentials and find that the maximum mass outflow rate ($R^{ \mathrm{max}}_{{\dot{m}}}$) weakly depends on the black hole spin. To validate our model, we calculate the maximum jet kinetic power using the accretion-ejection formalism and compare it with the radio jet power of low-hard state of the black hole X-ray binaries (hereafter XRBs). The outcome of our results indicate that XRBs along the ‘outliers’ track might be rapidly rotating.

Appendix: Calculation of sound speed (\(a_{c}\)) at the critical point (\(x_{c}\))

Putting $N = 0$ in Eq. (17b) and using Eq. (20), we get an algebraic equation of $a_{c}$ which is given by,

$$\begin{aligned}& \frac{A\alpha (g+ \gamma \frac{v_{c}^{2}}{a_{c}^{2}} ) ^{2}a_{c}^{4}}{\gamma x_{c}} + \frac{3 a_{c}^{2} v_{c}^{2}}{(\gamma -1)x _{c}} -\frac{ a_{c}^{2} v_{c}^{2}}{(\gamma -1)} \biggl( \frac{d\ln F _{i}(x)}{dx} \biggr)_{c} \\& \quad{}+ \frac{3 A \alpha g (g {+} \gamma \frac{v_{c} ^{2}}{a_{c}^{2}})a_{c}^{4}}{\gamma x_{c}}- \frac{A \alpha g(g {+} \gamma \frac{v_{c}^{2}}{a_{c}^{2}})a_{c}^{4}}{ \gamma } \biggl(\frac{d\ln F_{i}(x)}{dx} \biggr)_{c} \\& \quad{}- \frac{2 A\lambda (g + \gamma \frac{v_{c}^{2}}{a_{c}^{2}})a_{c}^{2}v_{c}}{x_{c}^{2}} - \biggl[2A\alpha g \biggl(g + \gamma \frac{v_{c}^{2}}{a_{c}^{2}} \biggr) a_{c}^{2} \\& \quad{}+ \frac{(\gamma +1)v_{c}^{2}}{(\gamma -1)} \biggr] \biggl( \frac{d \varPhi _{i}^{\text{eff}}}{dx} \biggr)_{c} = 0. \end{aligned}$$

(29)

Using $M_{c} = v_{c}/a_{c}$, we get

$$\begin{aligned}& \frac{A\alpha (g+ \gamma M_{c}^{2} )^{2}a_{c}^{4}}{\gamma x_{c}} + \frac{3 M_{c}^{2} a_{c}^{4} }{(\gamma -1)x_{c}} -\frac{M_{c} ^{2} a_{c}^{4}}{(\gamma -1)} \biggl( \frac{d\ln F_{i}(x)}{dx} \biggr) _{c} \\& \quad+ \frac{3 A \alpha g (g + \gamma M_{c}^{2})a_{c}^{4}}{\gamma x _{c}} \\& \quad{} - \frac{A \alpha g(g + \gamma M_{c}^{2})a_{c}^{4}}{\gamma } \biggl(\frac{d \ln F_{i}(x)}{dx} \biggr)_{c} \\& \quad{}- \frac{2 A\lambda (g + \gamma M_{c} ^{2})M_{c} a_{c}^{3}}{x_{c}^{2}} - \biggl[2A\alpha g \bigl(g + \gamma M_{c}^{2} \bigr) a_{c}^{2} \\& \quad{}+ \frac{(\gamma +1)M_{c}^{2} a_{c} ^{2}}{(\gamma -1)} \biggr] \biggl( \frac{d\varPhi _{i}^{\text{eff}}}{dx} \biggr) _{c}= 0. \end{aligned}$$

(30)

After some simple algebra, we have

$$ a_{1} a_{c}^{2} + a_{2} a_{c} + a_{3} = 0, $$

(31)

where

$$\begin{aligned} a_{1} = {}& -\frac{A\alpha (g + \gamma M_{c}^{2})^{2}}{\gamma x_{c}} - \frac{3 M_{c}^{2}}{(\gamma -1)x_{c}}\\ & + \frac{M_{c}^{2}}{(\gamma -1)} \biggl(\frac{d\ln F_{i}(x)}{dx} \biggr)_{c} - \frac{3A \alpha g (g + \gamma M_{c}^{2})}{\gamma x_{c}} \\ &+ \frac{A \alpha g(g + \gamma M_{c} ^{2})}{\gamma } \biggl(\frac{d\ln F_{i}(x)}{dx} \biggr)_{c}, \\ a_{2} = {}& \frac{2A\lambda M_{c} (g + \gamma M_{c}^{2})}{x_{c}^{2}}, \\ a_{3} = {}& \biggl[2A\alpha g \bigl(g + \gamma M_{c}^{2} \bigr) + \frac{(\gamma + 1)M _{c}^{2}}{(\gamma - 1)} \biggr] \biggl(\frac{d\varPhi _{i}^{\text{eff}}}{dx} \biggr) _{c}. \end{aligned}$$

In may be noted that the trivial solutions are avoided in Eq. (31). Finally, we solve this equation to obtain $a_{c}$ and consider only positive root as $a_{c}>0$ always.