Dissipation and acoustic tunnelling about the sonic horizon of Bondi accretion

TL;DR

This paper investigates how viscous dissipation affects acoustic wave propagation near the sonic horizon in Bondi accretion, revealing wave blocking and tunneling phenomena analogous to Hawking radiation.

Contribution

It introduces a perturbative analysis of viscous effects on acoustic waves in Bondi flow, highlighting wave tunneling through the sonic horizon influenced by viscosity and analogue Hawking temperature.

Findings

Viscous dissipation shrinks the sonic sphere.

Acoustic waves are blocked near the horizon but can tunnel outward.

Tunneling amplitude depends on viscosity and analogue Hawking temperature.

Abstract

Viscous dissipation, as a small perturbative effect about the Bondi flow, shrinks its sonic sphere. An Eulerian perturbation on the steady flow gives a wave equation and the corresponding dispersion relation. The perturbation is a high-frequency travelling acoustic wave, in which small dissipation is taken iteratively. The wave, propagating radially outwards against the bulk inflow, is blocked just within the sonic horizon, where the amplitude of the wave diverges because of viscosity. The blocked acoustic wave can still tunnel outward through the horizon with a viscosity-dependent decaying amplitude, scaled by the analogue Hawking temperature. The escape of acoustic waves (analogue Hawking phonons) through the sonic horizon is compatible with the radial contraction of the sonic sphere.

Equations

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \mathbf{v}\right)=0,$$

$$\frac{\partial \mathbf{v}}{\partial t}+\left(\mathbf{v}\cdot \nabla \right)\mathbf{v}+\frac{\nabla P}{\rho }+\frac{GM}{r^2}\hat{\mathbf{r}}=\frac{1}{\rho }\left[{\eta }_1{\nabla }^2\mathbf{v}+\left(\frac{{\eta }_1}{3}+{\zeta }_2\right)\nabla \left(\nabla \cdot \mathbf{v}\right)\right].$$

$$P=K{\rho }^{\gamma },$$

$$\frac{\mathrm{d}K}{K}=\frac{\mathrm{d}T}{T}-(\gamma -1)\frac{\mathrm{d}\rho }{\rho },$$

$$\frac{\partial \rho }{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\rho v{r}^2\right)=0$$

$$\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial r}+\frac{1}{\rho }\frac{\partial P}{\partial r}+\frac{GM}{r^2}=\frac{\eta }{\rho }\frac{\partial }{\partial r}\left[\frac{1}{r^2}\frac{\partial }{\partial r}\left(v{r}^2\right)\right],$$

$$4\pi \rho v{r}^2=-\dot{m},$$

$$\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{v^2}{2}+\frac{a^2}{\gamma -1}-\frac{GM}{r}\right)=\frac{2\eta }{(\gamma -1)\rho }\frac{\mathrm{d}}{\mathrm{d}r}\left(-\frac{v}{a}\frac{\mathrm{d}a}{\mathrm{d}r}\right).$$

$$\left(v-\frac{a^2}{v}-\frac{2\eta }{\rho r}\right)\frac{\mathrm{d}v}{\mathrm{d}r}=\frac{2a^2}{r}-\frac{GM}{r^2}+\frac{\eta }{\rho }\left(\frac{{\mathrm{d}}^{2}v}{\mathrm{d}{r}^2}-\frac{2v}{r^2}\right).$$

$$r_{\mathrm{c}}=\frac{GM}{2a_{\mathrm{c}}^2},v_{\mathrm{c}}^2=a_{\mathrm{c}}^2,$$

$$\frac{v^2}{2}+n{a}^2-\frac{GM}{r}=E,$$

$$\frac{v^2}{2}+\frac{\gamma K}{\gamma -1}{\left(\frac{-\dot{m}}{4\pi v{r}^2}\right)}^{\gamma -1}-\frac{GM}{r}=E.$$

$$\dot{m}=\frac{\pi (GM{)}^2}{(\gamma K{)}^n}{\left(\frac{2E}{2n-3}\right)}^{n-1.5}.$$

$$2a_{\mathrm{c}}^2{r}_{\mathrm{c}\eta }-GM\simeq -\frac{\eta v_{\mathrm{c}}}{{\rho }_{\mathrm{c}}}\left({\xi }^2+\xi -2\right),$$

$$v_{\mathrm{c}\eta }-\frac{a_{\mathrm{c}}^2}{v_{\mathrm{c}\eta }}\simeq \frac{2\eta }{{\rho }_{\mathrm{c}}{r}_{\mathrm{c}}},$$

$$\frac{r_{\mathrm{c}\eta }-r_{\mathrm{c}}}{r_{\mathrm{c}}}\sim -{\eta }_{\star }{\left(\frac{2n-3}{2n}\right)}^{n-0.5}.$$

$$v_{\mathrm{c}\eta }\simeq \pm a_{\mathrm{c}}+\frac{\eta }{{\rho }_{\mathrm{c}}{r}_{\mathrm{c}}}.$$

$$v_{\mathrm{c}\eta }\simeq -a_{\mathrm{c}}+4a_{\mathrm{c}}{\eta }_{\star }{\left(\frac{2n-3}{2n}\right)}^{n-0.5},$$

$$\frac{f^{\prime }}{f_0}=\frac{{\rho }^{\prime }}{{\rho }_0}+\frac{v^{\prime }}{v_0},$$

$$\frac{\partial {\rho }^{\prime }}{\partial t}=-\frac{1}{r^2}\frac{\partial f^{\prime }}{\partial r}.$$

$$\frac{\partial v^{\prime }}{\partial t}=\frac{v_0}{f_0}\left(\frac{\partial f^{\prime }}{\partial t}+v_0\frac{\partial f^{\prime }}{\partial r}\right).$$

$$\begin{gathered} \frac{\partial }{\partial t}\left(h^{tt}\frac{\partial f^{\prime }}{\partial t}\right)+\frac{\partial }{\partial t}\left(h^{tr}\frac{\partial f^{\prime }}{\partial r}\right)+\frac{\partial }{\partial r}\left(h^{rt}\frac{\partial f^{\prime }}{\partial t}\right)+\frac{\partial }{\partial r}\left(h^{rr}\frac{\partial f^{\prime }}{\partial r}\right) \\ =\eta v_0{r}^2\frac{\partial }{\partial r}\left[\frac{1}{r^2}\frac{\partial }{\partial r}\left(\frac{1}{{\rho }_0}\frac{\partial f^{\prime }}{\partial t}+\frac{v_0}{{\rho }_0}\frac{\partial f^{\prime }}{\partial r}\right)\right]+\frac{\eta v_0}{{\rho }_0}\frac{\partial f^{\prime }}{\partial r}\times \\ \frac{\partial }{\partial r}\left[\frac{1}{r^2}\frac{\partial }{\partial r}\left(v_0{r}^2\right)\right], \end{gathered}$$

$$h^{\mu \nu }=v_0\left(\begin{array}{cc}1\text{hfill}& v_0\\ v_0& v_0^2-a_0^2\text{hfill}\end{array}\right).$$

$$\Delta \phi \equiv \frac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\phi \right),$$

$$\frac{{\partial }^2{f}^{\prime }}{\partial t^2}=a_0^2\frac{{\partial }^2{f}^{\prime }}{\partial r^2}+\frac{\eta }{{\rho }_0}\frac{\partial }{\partial t}\left(\frac{{\partial }^2{f}^{\prime }}{\partial r^2}+\frac{2}{r}\frac{\partial f^{\prime }}{\partial r}-\frac{2f^{\prime }}{r^2}\right),$$

$${\left(\omega -k{v}_{\mathrm{B}}\right)}^2-\frac{i\eta }{{\rho }_0}\left(-k^2-\frac{2}{r^2}+\frac{2ik}{r}\right)\left(\omega -k{v}_{\mathrm{B}}\right)-k^2{a}_0^2=0,$$

$$\omega \simeq k{v}_{\mathrm{B}}\pm k{a}_0-\frac{\eta k}{{\rho }_{0}r}-\frac{i\eta }{2{\rho }_0}\left(k^2+\frac{2}{r^2}\right).$$

$$v_{\mathrm{g}}=\frac{\partial \omega }{\partial k}\simeq v_{\mathrm{B}}\pm a_0-\frac{\eta }{{\rho }_{0}r}.$$

$$\left(v_{0}^{2}-a_{0}^{2}\right)\left[i\frac{{\mathrm{d}}^{2}s}{{\mathrm{d}}r^{2}}-\left(\frac{{\mathrm{d}}s}{{\mathrm{d}}r}\right)^{2}\right]+\frac{i}{v_{0}}\frac{{\mathrm{d}}}{{\mathrm{d}}r}\left[v_{0}\left(v_{0}^{2}-a_{0}^{2}\right)\right]\frac{{\mathrm{d}}s}{{\mathrm{d}}r}\\ +2v_{0}\omega\frac{{\mathrm{d}}s}{{\mathrm{d}}r}-2i\omega\frac{{\mathrm{d}}v_{0}}{{\mathrm{d}}r}-\omega^{2}=\eta\Bigg{\{}-\frac{iv_{0}}{\rho_{0}}\left(\frac{{\mathrm{d}}s}{{\mathrm{d}}r}\right)^{3}\\ -\frac{3v_{0}}{\rho_{0}}\frac{{\mathrm{d}}s}{{\mathrm{d}}r}\frac{{\mathrm{d}}^{2}s}{{\mathrm{d}}r^{2}}+\left(\frac{{\mathrm{d}}s}{{\mathrm{d}}r}\right)^{2}\left[\frac{2v_{0}}{r\rho_{0}}-2\frac{\mathrm{d}}{{\mathrm{d}}r}\left(\frac{v_{0}}{\rho_{0}}\right)+\frac{i\omega}{\rho_{0}}\right]+\frac{iv_{0}}{\rho_{0}}\frac{{\mathrm{d}}^{3}s}{{\mathrm{d}}r^{3}}\\ +\frac{{\mathrm{d}}^{2}s}{{\mathrm{d}}r^{2}}\bigg{[}\frac{\omega}{\rho_{0}}-\frac{2iv_{0}}{r\rho_{0}}-2i\frac{\mathrm{d}}{{\mathrm{d}}r}\left(\frac{v_{0}}{\rho_{0}}\right)\bigg{]}+\frac{{\mathrm{d}}s}{{\mathrm{d}}r}\bigg{[}i\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}r^{2}}\left(\frac{v_{0}}{\rho_{0}}\right)-\frac{2i}{r}\frac{\mathrm{d}}{{\mathrm{d}}r}\left(\frac{v_{0}}{\rho_{0}}\right)\\ -\frac{i}{\rho_{0}}\frac{\mathrm{d}}{{\mathrm{d}}r}\left(\frac{v_{0}}{\rho_{0}}\frac{{\mathrm{d}}\rho_{0}}{{\mathrm{d}}r}\right)-\frac{2\omega}{r\rho_{0}}-\frac{2\omega}{\rho_{0}^{2}}\frac{{\mathrm{d}}\rho_{0}}{{\mathrm{d}}r}\bigg{]}-\frac{2i\omega}{r\rho_{0}^{2}}\frac{{\mathrm{d}}\rho_{0}}{{\mathrm{d}}r}-i\omega\frac{{\mathrm{d}}^{2}\rho_{0}^{-1}}{{\mathrm{d}}r^{2}}\Bigg{\}}.$$

$${\alpha }_0=\int \frac{\omega }{v_0\mp a_0}\mathrm{d}r.$$

Full text

Infall, accretion, and accretion discs; Quantum aspects of black holes, evaporation; Perturbation methods

I Introduction

Bondi (1952) accretion is a classic textbook example of a transonic flow in astrophysics (Frank et al., 2002; Clarke and Carswell, 2007). This compressible astrophysical flow is steady and spherically symmetric, with its fluid elements being driven radially inwards by the gravity of a centrally located accretor, which can be an ordinary star or a neutron star or a black hole (Petterson et al., 1980). Far away from the accretor, the conservative velocity field has a very low subsonic value (idealized to vanish at infinity), but it becomes highly supersonic as it approaches the accretor. With the flow thus being subsonic at the outer boundary and supersonic at the inner boundary, it crosses the sonic barrier at an intermediate radius (Bondi, 1952; Ray and Bhattacharjee, 2002). The surface of this sonic barrier, where the flow becomes transonic, is spherical. Travelling acoustic waves in the supersonic region are completely trapped within this spherical sonic surface. As a result the region bounded by the surface can be viewed as a spherically symmetric acoustic black hole (Moncrief, 1980; Unruh, 1981, 1995; Visser, 1998), and the sonic surface itself becomes a sonic horizon – the acoustic analogue of the event horizon of a general-relativistic black hole.

The distinctive features of Bondi (1952) accretion, namely, spherically symmetric, compressible, convergent and irrotational, suit it well as a three-dimensional transonic potential flow. Natural examples of such flows are otherwise uncommon. These reasons make the Bondi (1952) inflow, apart from being a paradigm in studies of astrophysical accretion, a physical model of interest in analogue gravity as well, from multiple perspectives (Moncrief, 1980; Das, 2004; Mach and Malec, 2008; Sen and Ray, 2014). In this work we take up Bondi (1952) accretion as an analogue-gravity problem. Our aim is to investigate how the transonic conditions at the sonic horizon of conservative spherically symmetric accretion are affected by small viscous dissipation. Particularly since we know that the merest presence of viscosity disrupts the acoustic geometry of conserved flows (Visser, 1998). Certain effects of viscosity in spherically symmetric accretion have been taken up in previous studies (Axford and Newman, 1967; Ray, 2003a), but these have not explored the impact of viscous dissipation on the black-hole-like properties of the sonic horizon of transonic accretion. This is our specific quest.

In Sec. II, we set down the governing hydrodynamic equations of a spherically symmetric inflow, incorporating viscous dissipation in the equation of momentum balance. In Sec. III, we study the impact of a perturbatively small order of viscosity on the steady transonic conditions and estimate how viscosity contracts the sonic sphere, which points to the shrinking of acoustic black holes (Unruh, 1981; Jacobson, 1991; Unruh, 1995). In Sec. IV, by using an Eulerian perturbation scheme, we derive a viscous wave equation and establish the metric of an acoustic black hole in the inviscid limit. The wave equation also gives a dispersion relation and the group velocity of the waves, using which we argue that viscosity diffuses the sharp condition of the inviscid sonic horizon. Consistent with the acoustic black hole, in Sec. V we show that acoustic waves of high-frequency, travelling outwards against the radial fluid inflow, are blocked inside the sonic surface, accompanied by an unstable growth in the amplitude of the wave. Cauchy’s residue analysis at the sonic horizon, carried out in Sec. VI, allows the blocked wave to tunnel outward through the impenetrable sonic barrier, with the viscosity-dependent decaying amplitude of the wave being scaled by the analogue Hawking temperature. The weak viscosity-induced tunnelling concurs with the shrinking of the spherical acoustic black hole by viscosity. The acoustic tunnelling demonstrated here is the fluid analogue of Hawking radiation as a tunnelling phenomenon in general-relativistic cases (Parikh and Wilczek, 2000; Mitra, 2007). Hawking radiation as the tunnelling of acoustic waves through a critical point in a fluid flow is known for laboratory fluids (Volovik, 2006; Bhattacharjee, 2017). Here we establish the same for astrophysical accretion.

In summary, the theme of this study is the fluid analogue of gravity. The convergent inflow of viscous spherically symmetric accretion is an astrophysical fluid system where the formulations of analogue gravity have been applied. Viscosity is taken as a perturbative effect about the inviscid Bondi (1952) inflow, which is a fundamental model of astrophysical accretion. The main results of this theoretical study are that viscous dissipation contracts the spherical sonic horizon, destabilizes its neighbourhood under high-frequency perturbations, but lets acoustic waves tunnel through the barrier.

II The hydrodynamic equations

The flow of a viscous compressible fluid (in our study, an astrophysical gas flow, to be precise) is governed by the equation of continuity (Landau and Lifshitz, 1987), given as

[TABLE]

and by the Navier-Stokes equation (Landau and Lifshitz, 1987), which, for a flow driven radially by the gravity of an accretor of mass, $M$, is expressed in full as

[TABLE]

In the foregoing equation, $\eta_{1}$ and $\zeta_{2}$ are the first (shear) and second (bulk) coefficients of viscosity, respectively, both with positive values of the same order (Landau and Lifshitz, 1987). Since the bulk flow here is vorticity-free, the left hand side of the vector identity, ${\mathbf{\nabla}}\times({\mathbf{\nabla}}\times{\mathbf{v}})={\mathbf{\nabla}}({\mathbf{\nabla}}\cdot{\mathbf{v}})-\nabla^{2}{\mathbf{v}}$, vanishes to give ${\mathbf{\nabla}}({\mathbf{\nabla}}\cdot{\mathbf{v}})=\nabla^{2}{\mathbf{v}}$, which we use to simplify the viscosity-dependent terms in Eq. (2). Further, since accretion is a compressible fluid flow, ${\mathbf{\nabla}}\cdot{\mathbf{v}}\neq 0$, as seen in Eq. (1). The pressure, $P$, in Eq. (2) is prescribed in terms of the polytropic process (Chandrasekhar, 1939) as

[TABLE]

in which $\gamma$, the polytropic exponent, is restricted by isothermal and adiabatic limits over the range, $1\leq\gamma\leq c_{\mathrm{P}}/c_{\mathrm{V}}$, with $c_{\mathrm{P}}$ and $c_{\mathrm{V}}$ being the two coefficients of specific heat capacity of a gas (Chandrasekhar, 1939). With $P\equiv P(\rho)$, both the density, $\rho$, and the flow velocity, $\mathbf{v}$, are mathematically closed in Eqs. (1) and (2).

The viscosity of the accreting gas, being the outcome of internal friction on the molecular scale, is small. Viscosity of this order suits our purpose, as we argue now. In fundamental models of analogue gravity, the invariance of the acoustic geometry requires an inviscid potential flow (Unruh, 1981; Visser, 1998). To see how the conditions of this conservative background are violated, we introduce viscosity as a feeble perturbative presence about the inviscid Bondi (1952) inflow. We also know that the acoustic structure is violated on the molecular scale (Visser, 1998), where the mean free path of the molecules is comparable to the short wavelength of acoustic waves. The known physical means of dissipation about this scale is, of course, molecular viscosity. Therefore, from this physical standpoint alone, a small molecular viscosity suffices in our quest. Besides the physics itself, the mathematical justification for considering a small molecular viscosity lies in our linearized and iterative approach to a second-order nonlinear hydrodynamical system (as we shall demonstrate from Sec. III onwards).

Inasmuch as the molecular viscosity of a gas depends on its pressure and temperature (hence, on its density as well), the spatial variation of the molecular viscosity is less than that of the bulk velocity of a spherically symmetric inflow. This is certainly true for a transonic radial flow in an open astrophysical system, where radiative processes facilitate efficient cooling and maintain the system close to being isothermal. By this argument, we approximate $\eta_{1}$ and $\zeta_{2}$ as constants (Balbus and Hawley, 1998; Ray, 2003a).

Heating of a fluid, undergoing transonic accretion in spherical symmetry, can occur due to a variety of physical causes. These are viscosity (Axford and Newman, 1967; Ray, 2003a), thermal conductivity (Axford and Newman, 1967), turbulence (Turolla and Nobili, 1989; Ray and Bhattacharjee, 2005) and radiative transfer (Nobili et al., 1991) in both Newtonian gravity (Axford and Newman, 1967; Ray, 2003a; Ray and Bhattacharjee, 2005) and Schwarzschild geometry (Turolla and Nobili, 1989; Nobili et al., 1991). A polytropic process, which is more general than the limiting case of an adiabatic process, encompasses the physical reality of heating in the fluid (Chandrasekhar, 1939). For a value of $\gamma$, the polytropic function in Eq. (3) forms a one-parameter family of curves (Chandrasekhar, 1939). This parameter, given by $K$ in Eq. (3), is related to the “polytropic temperature”, which becomes equal to the actual temperature of the fluid in the isothermal limit (Chandrasekhar, 1939). Eliminating $P$ in Eq. (3) by using the equation of state, $P=\rho k_{\mathrm{B}}T/\mu m_{\mathrm{H}}$ (Frank et al., 2002), and then taking the logarithmic derivative, give

[TABLE]

which shows how $K$ varies with the temperature, $T$, and the density (Chandrasekhar, 1939). With small viscous dissipation and efficient radiative cooling of the open astrophysical flow, the thermodynamic condition of the accreting fluid will be nearly isothermal. In that case, $\gamma\longrightarrow 1$ and $T$ will be almost fixed, which, considered together in Eq. (4), will physically justify approximating $K$ as constant.

Tailored according to spherical symmetry, with $v\equiv v(r,t)$ and $\rho\equiv\rho(r,t)$, Eqs. (1) and (2) are now written as

[TABLE]

and

[TABLE]

respectively (Landau and Lifshitz, 1987). The two coefficients of viscosity are added in Eq. (6), to give a total viscosity, $\eta=(4/3)\eta_{1}+\zeta_{2}$. With Eqs. (5) and (6), the coupled dynamics of $\rho$ and $v$ in the viscous spherically symmetric inflow is formulated completely.

III The steady sonic horizon

In the steady state, partial time derivatives vanish, i.e. $\partial/\partial t\equiv 0$, leaving behind only full spatial derivatives. As such, integrating Eq. (5) gives

[TABLE]

in which $\dot{m}$, a constant of the motion, is the matter inflow rate. Its negative sign, due to $v<0$, indicates the inward flux of matter (Frank et al., 2002). Using the constraint in Eq. (7) and substituting the spatial derivatives of $\rho$ with the spatial derivatives of the local speed of sound, $a=\sqrt{\partial P/\partial\rho}=\sqrt{\gamma K\rho^{\gamma-1}}$, we reduce Eq. (6) in its steady limit ($\partial v/\partial t=0$) to

[TABLE]

The viscosity-dependent term on the right hand side of Eq. (8) is positive (Ray, 2003a), whose physical implication is that viscosity opposes the gravity-driven infall in spherical symmetry (Ray, 2003a). In contrast, by the way, viscosity in accretion discs effects the outward transport of angular momentum, and aids the axially symmetric infall (Balbus and Hawley, 1998; Frank et al., 2002). After substituting $\rho$ with $a$ in Eq. (7), we replace the spatial derivatives of $a$ in Eq. (8) by the spatial derivatives of $v$. This gives us

[TABLE]

When $\eta=0$, Bondi (1952) accretion becomes transonic at the sonic horizon (Frank et al., 2002; Ray and Bhattacharjee, 2002). A smooth passage of the inflow through the sonic horizon requires both the right hand and the left hand sides of Eq. (9) to vanish together, while ${\mathrm{d}}v/{\mathrm{d}}r\neq 0$, as in a first-order dynamical system (Ray and Bhattacharjee, 2002). The values of $r$ and $v$ at the sonic horizon are thus obtained from Eq. (9) as

[TABLE]

with the subscript “$\mathrm{c}$” indicating the critical values when $\eta=0$. We now introduce the polytropic index, $n=(\gamma-1)^{-1}$ (Chandrasekhar, 1939). Using it, the integral of Eq. (8), for $\eta=0$, follows as

[TABLE]

with $E$ being the Bernoulli constant. The boundary conditions of the flow are $v\longrightarrow 0$ and $a\longrightarrow a_{\infty}$ (an ambient limit of $a$), for $r\longrightarrow\infty$. Applying both boundary conditions in Eq. (11) gives $E=na_{\infty}^{2}$. Further, putting the critical values given by Eqs. (10) in Eq. (11), gives us $a_{\mathrm{c}}^{2}=2na_{\infty}^{2}/(2n-3)$. This fixes $v_{\mathrm{c}}$ and $r_{\mathrm{c}}$ in terms of the outer boundary conditions. Since $a$ is related to $\rho$, we also get $\rho_{\mathrm{c}}=\rho_{\infty}[2n/(2n-3)]^{n}$, with $\rho\longrightarrow\rho_{\infty}$ (the ambient limit of $\rho$) for $r\longrightarrow\infty$.

Our discussion so far has dwelt on the transonic inflow solution for $\eta=0$. But this solution is just one special solution in the $r$–$v$ phase plane, which also contains multiple classes of other solutions, both inflows and outflows (Frank et al., 2002; Ray and Bhattacharjee, 2002). These solutions are not constrained by the critical conditions of Eqs. (10). For inflows, in which we are interested, a general solution of $v(r)$ results from combining Eqs. (7) and (11). Eliminating $\rho$ and $a$, this solution with only $v$ and $r$ is

[TABLE]

Solving for $v(r)$ in Eq. (12) requires the prescription of two independent boundary conditions to determine the two independent constants of the motion, $\dot{m}$ and $E$. This is generally true for all $v(r)$. However, the transonic inflow comes up with its own set of unique conditions, given by the critical coordinates in Eqs. (10). In such a situation, $\dot{m}$ and $E$ are no longer independent of each other. Through the critical condition they become related to each other by

[TABLE]

It is now clear that we can realize the transonic inflow in two equivalent ways. One way is to provide the values of $\dot{m}$ and $E$ required for transonicity in Eq. (12). The alternative way (which is also the usual way) is through the critical coordinates in Eqs. (10) and specifying either $\dot{m}$ or $E$ (but not both) from a boundary condition. With viscosity included, we cannot obtain one constant of the motion, because Eq. (8), with $\eta\neq 0$, no longer stands for a conserved physical condition. The steady continuity condition, however, continues to be conservative. Hence, we are now left with only one constant of the motion. One might then suggest that at least for the transonic flow the loss of the second constant of the motion can be compensated by the critical conditions. But that will be mathematically self-contradictory, because if we do not have two independent constants of the motion, then its equivalent condition of a set of critical coordinates and one independent constant of the motion does not stand either. Therefore, we conclude that for a viscous inflow our inability to find a constant of the motion from Eq. (8) translates equivalently to the mathematical impossibility of a precise determination of the critical coordinates of the inflow. Deficiency of exact information about the critical coordinates in viscous transonic accretion has indeed been noted in studies on accretion (Axford and Newman, 1967).

We now look at how the transonic conditions of Eq. (9), established clearly in the inviscid limit, are affected by viscosity, when it is treated as a small perturbative effect about the inviscid state (Ray, 2003a). Molecular viscosity is physically sufficient for this purpose, and we scale it in terms of quantities that are determined by the microscopic molecular properties of the fluid on the large scale of the accretion radius, $r_{\mathrm{a}}\sim 2GM/a_{\infty}^{2}$, where gravity has no noticeable effect (Frank et al., 2002). We write $\eta=\eta_{\star}\eta_{\mathrm{s}}$, in which $\eta_{\star}$ is dimensionless and $\eta_{\mathrm{s}}=a_{\infty}\rho_{\infty}r_{\mathrm{a}}$. The latter sets the scale of $\eta$, and the former determines its magnitude, with $\eta_{\star}\ll 1$ in our perturbative analysis.

Non-zero viscosity, howsoever small, renders Eq. (9) a differential equation of the second-order for a compressible fluid. Without the analytical solution of such a mathematical problem, we lose the precise sonic conditions, as set down in Eqs. (10), which are derived from a viscosity-free first-order dynamical system (Ray and Bhattacharjee, 2002). Nevertheless, the viscous flow will still be transonic (Axford and Newman, 1967),111 In the thin boundary layer of a liquid flow, the second-order derivative in the viscous terms has a serious physical impact, but in the bulk its effect is not quite as qualitatively significant. Spherically symmetric accretion is an open gas inflow with no boundary layer, and as such, with weak molecular viscosity, the bulk flow passes through a critical zone and becomes transonic (Axford and Newman, 1967), much like the inviscid inflow. and for small $\eta$, the essential transonic features can be captured by a viscosity-dependent perturbative expansion in the neighbourhood of the inviscid equilibrium point (Axford and Newman, 1967). In a similar approach, we expand the viscous effect about the inviscid sonic conditions of Bondi (1952) accretion, by iteratively applying the inviscid velocity profile to the $\eta$-dependent second-order derivative of $v$ in Eq. (9). Near the accretor, for a freely falling inviscid inflow, $v\sim r^{-1/2}$, and far away from the accretor, for a highly subsonic inflow, $v\sim r^{-2}$ (Petterson et al., 1980). Going by these power laws in the two extreme limits of the inflow, we prescribe $v\sim r^{-\xi}$ for the steady velocity, with $0.5<\xi<2$ in the vicinity of the sonic point. Iteratively bringing in the inviscid power law of $v$ to the second-order derivative of $v$ on the right hand side of Eq. (9), reduces the derivative to an algebraic form, $v/r^{2}$.222The second-order spatial derivative of $v$ is similarly approximated in the shallow flow of a hydraulic jump (see (Singha et al., 2005; Ray and Bhattacharjee, 2007) and references therein). The resulting first-order system lets us see how the inviscid critical coordinates, $r_{\mathrm{c}}$ and $v_{\mathrm{c}}$ in Eqs. (10), shift, respectively, to viscosity-dependent values, $r_{{\mathrm{c}}\eta}$ and $v_{{\mathrm{c}}\eta}$. To that end, the right and left hand sides of Eq. (9) give us the approximate conditions,

[TABLE]

and

[TABLE]

respectively. In Eqs. (14) and (15), terms with $\eta$ carry the inviscid critical values of $v_{\mathrm{c}}$, $\rho_{\mathrm{c}}$ and $r_{\mathrm{c}}$, which is consistent with our iterative approach to estimate the viscosity-induced correction on $v_{\mathrm{c}}$ and $r_{\mathrm{c}}$. Further, $a(r)$ is a slowly-varying function compared to the rate at which $v$ varies with $r$. This is very true when the flow becomes transonic and it is also consistent with the dependence of $a$ on the temperature, $T$, which, for a polytropic gas flow (Chandrasekhar, 1939), is $a\propto\sqrt{T}$. Efficient radiative processes in the open astrophysical flow maintain near-isothermal conditions, and as such, $a$ does not vary overmuch. With these arguments, and with viscosity as a small effect, we, therefore, retain the inviscid critical value of $a_{\mathrm{c}}$ in Eqs. (14) and (15). Now we divide Eq. (14) throughout by $2a_{\mathrm{c}}^{2}$. Then in the resulting $\eta$-dependent term, noting that $v_{\mathrm{c}}=-a_{\mathrm{c}}$ in the inviscid background flow, we relate $\eta$, $\rho_{\mathrm{c}}$ and $a_{\mathrm{c}}$ to $\eta_{\star}$, $n$, $a_{\infty}$, $\rho_{\infty}$ and $r_{\mathrm{a}}$, as has been outlined following Eq. (11). Next, we relate $r_{\mathrm{a}}$ to $r_{\mathrm{c}}$. The physical part played by viscosity in the spherically symmetric inflow is to resist gravity and diminish its field of attraction (Ray, 2003a). At or about the critical point, the polytropic pressure term in Eq. (6) is balanced against gravity (Ray, 2003a). With gravity weakened by viscosity, the flow becomes transonic deeper within the gravitational potential well (Ray, 2003a), compared to the inviscid sonic radius. According to this physical reasoning, $r_{{\mathrm{c}}\eta}<r_{\mathrm{c}}$, which is achieved with $\xi\lesssim 1$. The viscous spherically symmetric inflow now becomes transonic through a diffused region in the neighbourhood of $r_{\mathrm{c}}$. For small viscosity this region has a thin radial width. To an order-of-magnitude, the fractional width of the transonic region, with a correction in $\eta_{\star}$, is

[TABLE]

The negative sign on the right hand side of Eq. (16) shows that viscosity shrinks the sonic sphere of the inviscid transonic inflow to a reduced radius, $r_{{\mathrm{c}}\eta}$, when $\eta\neq 0$. The physical significance of this result is that it holds viscous dissipation responsible for the loss of the surface area of the spherical sonic horizon (Unruh, 1981; Jacobson, 1991). In Sec. VI, we connect this viscosity-induced contraction of the acoustic black hole to a viscosity-induced tunnelling of phonons through the acoustic horizon.

When the flow is in the transonic stage, the inflow velocity also undergoes a small correction in proportion to $\eta_{\star}$. We estimate this correction by solving the quadratic equation of $v_{{\mathrm{c}}\eta}$ in Eq. (15). Neglecting the term with $\eta^{2}$ (for small $\eta$) in the discriminant of the quadratic solution, we get

[TABLE]

For inflows, $v_{{\mathrm{c}}\eta}\simeq-[a_{\mathrm{c}}-(\eta/\rho_{\mathrm{c}}r_{\mathrm{c}})]$, and for outflows, $v_{{\mathrm{c}}\eta}\simeq a_{\mathrm{c}}+(\eta/\rho_{\mathrm{c}}r_{\mathrm{c}})$. Evidently, $\eta$ breaks the symmetry between the two critical speeds. Pertaining to inflows, we relate $\eta$ and $\rho_{\mathrm{c}}$ to ambient values at the outer boundary, as done to derive Eq. (16). Then we arrive at the approximate result,

[TABLE]

in which the smallness of $\eta_{\star}$ ensures that the Mach number is almost unity in the transonic region (Axford and Newman, 1967). From Eqs. (16) and (18) we get an approximate idea of the sonic conditions of the compressible viscous spherically symmetric inflow.

The sonic coordinates of the inviscid Bondi (1952) solution are determined with full precision from the single equilibrium point of the first-order dynamical system in Eq. (9), when $\eta=0$ (Ray and Bhattacharjee, 2002). This precision is lost as soon as viscosity is operative, although a transonic solution still exists (Axford and Newman, 1967; Turolla and Nobili, 1989). To go past this difficulty we have prescribed the flow velocity as a power law of the radial distance, and thus obtained Eq. (14). This approach is convenient when viscosity is in effect (Ray, 2003a), and its quantitative results do not deviate significantly from the inviscid case, unless the effective viscosity is high, such as in turbulence (Turolla and Nobili, 1989; Ray and Bhattacharjee, 2005). In fact, even the inviscid Bondi (1952) accretion admits self-similar solutions (Gaite, 2006), established by power laws of the velocity and density profiles. With self-similar profiles, it happens that one power law is valid near the centre of coordinates and another is valid asymptotically at the outer boundary. We have seen this feature while arriving at Eq. (14). The transition from one power law to another takes place about the critical point. In many cases, viscosity is necessary for the simultaneous existence of two distinct self-similar solutions, each separated from the other by a narrow crossover zone. This is known well in a wide range of hydrodynamical problems, for example, gaseous baryonic structures on cosmological scales (Hansen and Stadel, 2003), and the shallow-water hydraulic jump on laboratory scales (Bohr et al., 1993; Singha et al., 2005; Ray and Bhattacharjee, 2007). The inner and outer boundary conditions of these systems often select the physically relevant self-similar solutions, which in turn determine the position of the critical point (Bohr et al., 1993; Singha et al., 2005; Ray and Bhattacharjee, 2007). Inappropriate choice of boundary conditions can, therefore, result in extraneous critical points. In the particular context of spherically symmetric accretion on to a Schwarzschild black hole, more than one critical point do appear (Nobili et al., 1991; Mandal et al., 2007; Chaverra et al., 2016). However, we do not expect such a situation to come about in our present case of a viscous inflow, since it involves only a small viscosity under Newtonian gravity. In fact, our results about the critical point, especially Eq. (17), are independently supported by the group velocity of radially propagating waves that we get by perturbing the steady flow. We show this in Sec. IV.

IV A wave equation

In the steady state, the solutions of Eqs. (5) and (6) are two coupled time-independent fields, that we write as $\rho_{0}(r)$ and $v_{0}(r)$. About these steady flow profiles, we impose small time-dependent radial perturbations and then linearize the perturbed quantities. The prescription for the time-dependent radial perturbation is $v(r,t)=v_{0}(r)+v^{\prime}(r,t)$ and $\rho(r,t)=\rho_{0}(r)+\rho^{\prime}(r,t)$, with the primed quantities being perturbations about a steady background. Following an Eulerian perturbation scheme employed by Petterson et al. (1980), we define a new variable, $f(r,t)=\rho vr^{2}$, which emerges as a constant of the motion from the steady limit of Eq. (5). This constant, $f_{0}=\rho_{0}v_{0}r^{2}$, is the matter flow rate, within a geometrical factor of $4\pi$ (Frank et al., 2002), as seen in Eq. (7). On applying the perturbation scheme for $v$ and $\rho$, the perturbation in $f$ is derived as

[TABLE]

connecting $v^{\prime}$, $\rho^{\prime}$ and $f^{\prime}$. To relate $\rho^{\prime}$ and $f^{\prime}$ to each other, we apply the perturbation scheme to Eq. (5), resulting in

[TABLE]

To obtain a similar relation between $v^{\prime}$ and $f^{\prime}$, we combine the conditions given in Eqs. (19) and (20), to get

[TABLE]

Together, Eqs. (20) and (21) form a closed set, with $\rho^{\prime}$ and $v^{\prime}$ expressed exclusively in terms of $f^{\prime}$. We now need an independent condition, in which we can use Eqs. (20) and (21). Just such a condition is offered by Eq. (6). We take the second-order time derivative of Eq. (6), to which we then apply Eqs. (20), (21) and the second-order time derivative of Eq. (21). This exercise leads us to the wave equation,

[TABLE]

in which $h^{tt}=v_{0}$, $h^{tr}=h^{rt}=v_{0}^{2}$ and $h^{rr}=v_{0}(v_{0}^{2}-a_{0}^{2})$, with $a_{0}$ in $h^{rr}$ being the steady value of $a$.

We find it instructive to examine Eq. (22) for $\eta=0$. In this inviscid limit, going by the symmetry of the left hand side of Eq. (22), we can recast it in a compact form as $\partial_{\mu}\left(h^{\mu\nu}\partial_{\nu}f^{\prime}\right)=0$, with the Greek indices running from [math] to $1$, under the equivalence that [math] stands for $t$ and $1$ stands for $r$. All the $h^{\mu\nu}$ in Eq. (22) are to be seen as elements of the matrix,

[TABLE]

Now, in Lorentzian geometry the d’Alembertian of a scalar field in curved space is obtained from the metric, $g_{\mu\nu}$, as

[TABLE]

where $g^{\mu\nu}$ is the inverse of the matrix, $g_{\mu\nu}$ (Moncrief, 1980; Visser, 1998; Barceló et al., 2011). Comparing Eq. (24) with Eq. (22) (when $\eta=0$), we identify $h^{\mu\nu}=\sqrt{-g}\,g^{\mu\nu}$ (Visser, 1998), and see that the wave equation of $f^{\prime}$ in Eq. (22) is similar to Eq. (24). The metric that is implicit in Eq. (22), is to be read from Eq. (23), and its inverse establishes an acoustic metric and an acoustic horizon, when $v_{0}^{2}=a_{0}^{2}$ (Visser, 1998). In the radial inflow of the Bondi (1952) accretion, this horizon is due to an acoustic black hole. The radius of the horizon is the critical radius, $r_{\mathrm{c}}$, given in Eqs. (10), which cannot be breached by any acoustic wave propagating against the bulk inflow, after having originated in the supercritical region, where $v_{0}^{2}>a_{0}^{2}$ and $r<r_{\mathrm{c}}$. Borne by the wave, the flow of a signal across the acoustic horizon is, therefore, only inwards. With the inclusion of viscosity ($\eta\neq 0$), however, a source term appears in Eq. (22) to compromise the symmetry of the acoustic metric, which we can relate to the violation of the acoustic Lorentz invariance by viscous dissipation (Visser, 1998). Without viscosity the acoustic horizon is determined precisely when $v_{0}=\pm a_{0}$, the positive sign of $a_{0}$ standing for outflows and the negative sign for inflows. Accounting for viscosity, perturbatively small albeit, the acoustic horizon is diffused about the sharp inviscid condition, $v_{0}=\pm a_{0}$. For small viscosity, this loss of precision has been approximately estimated in Eq. (17).

This point of view is also favoured by the dispersion relation that we can extract from Eq. (22), and the consequent group velocity of the acoustic waves. With respect to a homogeneous fluid background, a viscous wave equation is yielded by Eq. (22) as,

[TABLE]

in a familiar form (Visser, 1998). The solution, $f^{\prime}(r,t)\sim\exp[i(kr-\omega t)]$, applied to Eq. (25), gives a quadratic equation,

[TABLE]

in which $v_{\mathrm{B}}$ stands for the bulk motion of the fluid. For small viscosity, we neglect terms with $\eta^{2}$ in the discriminant of the quadratic form in Eq. (26), to get a dispersion relation,

[TABLE]

Dissipation happens because of the viscosity-dependent imaginary term in Eq. (27). The real terms, taken together, contribute to the group velocity of the wave,

[TABLE]

First of all, the wave is carried by the bulk motion of the fluid, with a velocity, $v_{\mathrm{B}}$. With respect to the fluid itself, outgoing waves have the group velocity, $a_{0}-(\eta/\rho_{0}r)$, and incoming waves have the group velocity, $-[a_{0}+(\eta/\rho_{0}r)]$. But for $\eta\neq 0$, the two waves would have had the same speed, and the sonic points of inflows and outflows would have coincided. The breaking of symmetry in the group velocities of outgoing and incoming waves, entirely due to viscosity, agrees with what Eq. (17) suggests – that the acoustic horizons of inflows and outflows of a viscous compressible fluid are diffused about the inviscid sonic condition, $v_{\mathrm{c}}=\pm a_{\mathrm{c}}$, by a narrow velocity width of approximate order, $\eta/(\rho_{\mathrm{c}}r_{\mathrm{c}})$. The viscous inflow becomes transonic through this non-zero velocity width, and not through a sharply defined sonic point, as in Eqs. (10).

V Wave blocking and instability about the sonic horizon

Without viscosity on the right hand side of Eq. (22), linear perturbations do not destabilize the steady background flow (Petterson et al., 1980). We now look at the effect that viscosity has on this condition. We treat the perturbation as a high-frequency travelling wave, whose wavelength, $\lambda$, is much less than the natural length scale in the inflow, the radius of the acoustic horizon, $r_{\mathrm{c}}$. Thus, specifying $\lambda\ll r_{\mathrm{c}}$, we use a separable solution for the travelling wave as $f^{\prime}(r,t)=\exp[is(r)-i\omega t]$, with the understanding that $\omega$ is much greater than any characteristic frequency of the system. Applying the foregoing solution to Eq. (22) and carrying out algebraic simplifications, deliver

[TABLE]

From Eq. (29) it is clear that $s$ has both real and imaginary parts. Therefore, we prescribe $s(r)=\alpha(r)+i\beta(r)$, with both $\alpha$ and $\beta$ being real. Going by the separable form of $f^{\prime}$, we see that while $\alpha$ contributes to the phase of the perturbation, $\beta$ contributes to its amplitude. Solutions of both $\alpha$ and $\beta$ are found by a WKB analysis of Eq. (29), according to which we consider $\alpha\gg\beta$ for travelling waves of high frequency. In Eq. (29), the term of the highest order in $s$ is cubic. Fortunately, this term is dependent on $\eta$, whose presence in our analysis is very feeble anyway. We use this fact to our advantage by first setting $\eta=0$ in Eq. (29), after which we solve a second-order differential equation that is independent of $\eta$. In this special case of $\eta=0$, we also modify our solution as $s_{0}(r)=\alpha_{0}(r)+i\beta_{0}(r)$, with the subscript “[math]” denoting solutions in the absence of viscosity. Using this in the inviscid limit of Eq. (29), we first separate the real and the imaginary parts, which are then individually set equal to zero. The WKB prescription stipulates that $\alpha_{0}\gg\beta_{0}$. Accordingly we collect only real terms without $\beta_{0}$, and solve a resulting quadratic equation in ${\mathrm{d}}\alpha_{0}/{\mathrm{d}}r$ to obtain

[TABLE]

Likewise, from the imaginary part, in which we need to use the solution of $\alpha_{0}$, we obtain

[TABLE]

with $C$ being a constant of integration (Sarkar et al., 2013).

We now perform a self-consistency check that $\alpha_{0}\gg\beta_{0}$, as a basic requirement of our WKB analysis. First, we note $\alpha_{0}$ contains $\omega$ (the high frequency of the travelling wave), and in this respect is of a leading order over $\beta_{0}$, which contains $\omega^{0}$. Next, on very large scales of length, i.e. $r\longrightarrow\infty$, the asymptotic behaviour of the background velocity is $v_{0}\longrightarrow 0$, and the corresponding speed of acoustic propagation, $a_{0}$, approaches a constant asymptotic value. In that case, $\alpha_{0}\sim\omega r$ in Eq. (30). Moreover, on similar scales of length, going by $v_{0}\sim f_{0}r^{-2}$, we see that $\beta_{0}\sim\ln r$. Further, near the acoustic horizon, where $|v_{0}|\simeq a_{0}$, for the wave that goes against the bulk inflow with the speed, $a_{0}-|v_{0}|$, there is a singularity in $\alpha_{0}$. All of these facts taken together, we see that our solution scheme is well in conformity with the WKB prescription.

Thus far we have worked with $\eta=0$ (absence of viscosity). To know how viscous dissipation affects the travelling wave, we have to find a solution of $s$ from Eq. (29), with $\eta\neq 0$. To this end, we adopt an iterative approach, exploiting the condition that $\eta=\eta_{\star}\eta_{\mathrm{s}}$ has a very small value, the smallness being set by $\eta_{\star}\ll 1$. Then taking up Eq. (29) in full, we propose a solution for it as $s=s_{0}+\delta s_{\eta}$, with $\delta$ being another dimensionless parameter that, like $\eta_{\star}$, obeys the requirement, $\delta\ll 1$. Therefore, on the right hand side of Eq. (29) all terms that carry the product, $\eta_{\star}\delta$, can be safely neglected as being very small. This, in keeping with the principle of our iterative treatment, effectively means that all the surviving viscosity-related terms on the right hand side of Eq. (29) will go as $\eta s_{0}$. Further, by the WKB analysis, we have also assured ourselves that $\alpha_{0}\gg\beta_{0}$, by which we ignore all dependence on $\beta_{0}$ on the right hand side of Eq. (29), when we compare them with all the terms containing $\alpha_{0}$. With these arguments, we approximate $s\simeq\alpha_{0}$ on the right hand side of Eq. (29), and see here that the most dominant $\alpha_{0}$-dependent real terms are of the quadratic degree. Preserving only these terms on the right hand side of Eq. (29) and extracting only the $\beta$-independent real terms from the left hand side, we arrive at a quadratic equation in ${\mathrm{d}}\alpha/{\mathrm{d}}r$,

[TABLE]

We solve Eq. (32) under the provision of $\eta_{\star}\ll(\lambda/r_{\mathrm{c}})^{2}$, which accords well with our requirement that $\eta_{\star}$ may be arbitrarily small. A binomial approximation of terms with $\eta$ in the discriminant gives us $\alpha=\alpha_{0}+\alpha_{\eta}$, with the viscosity-dependent correction to $\alpha$ being

[TABLE]

Next, to take up $\beta$, we extract all the imaginary terms from the left hand side of Eq. (29), and noting that the most dominant contribution to the imaginary terms on the right hand side is of the cubic degree in $\alpha_{0}$, we need to solve the equation,

[TABLE]

We observe that $\alpha_{0}$ and $\alpha_{\eta}$ are both linear in $\omega$, whereas the right hand side of Eq. (34), with $({\mathrm{d}}\alpha_{0}/{\mathrm{d}}r)^{3}$, is cubic in $\omega$. Therefore, the dominant correction in $\beta$ due to viscosity can be found with the approximation $\alpha\simeq\alpha_{0}$ on the left hand side of Eq. (34). This reasoning gives us $\beta=\beta_{0}+\beta_{\eta}$, in which

[TABLE]

A noteworthy aspect of both Eqs. (33) and (35) is that in the former the correction it provides to $\alpha_{0}$ is of the order of $\omega$ (an odd order contributing to the phase), and in the latter the correction to $\beta_{0}$ is of the order of $\omega^{2}$ (an even order contributing to the amplitude). Since $\alpha_{0}$ is of the order of $\omega$, the correction in $\alpha_{\eta}$ appears comparable to $\alpha_{0}$. More crucially, since $\beta_{0}$ is of the order of $\omega^{0}$, the correction in $\beta_{\eta}$ appears to be dominant over $\beta_{0}$ for large $\omega$. This, however, is not really the case. We have obtained the results given by Eqs. (33) and (35) under the restriction that $\eta_{\star}\ll(\lambda/r_{\mathrm{c}})^{2}$. Considering the wavelength as $\lambda(r)=2\pi(v_{0}\mp a_{0})/\omega$, we immediately see that $\eta_{\star}\omega$ in Eq. (33) and $\eta_{\star}\omega^{2}$ in Eq. (35), reduce both $\alpha_{\eta}$ and $\beta_{\eta}$ to be sub-leading to $\alpha_{0}$ and $\beta_{0}$, respectively. This is true over most of the spatial range of the flow, except in the close neighbourhood of the acoustic horizon, where $|v_{0}|=a_{0}$. In this region, for outgoing waves against the inflow, $\beta_{\eta}$ will diverge, as Eq. (35) shows, while $\beta_{0}$ itself remains finite.

For the travelling wave, $f^{\prime}(r,t)=e^{-\beta}\exp(i\alpha-i\omega t)$, from which, by extracting only the amplitude part, we get $|f^{\prime}(r,t)|\sim e^{-\beta}$. Our primary concern is the stability of waves propagating outwards against the steady Bondi (1952) transonic inflow, for which $v_{0}<0$. We write $v_{0}=-|v_{0}|$ for the Bondi (1952) inflow, and choose the lower sign in Eq. (35) for outgoing waves. These specifications give

[TABLE]

In the supersonic region, bounded within the spherical sonic horizon, $|v_{0}|>a_{0}$. For a wave approaching the sonic horizon from the supercritical side, with $|v_{0}|\longrightarrow a_{0}$, the integrand in Eq. (36) suffers a divergence, and $|f^{\prime}(r,t)|\longrightarrow\infty$. The instability of the wave amplitude stands out very clearly in this case. In contrast, in the subsonic region just outside the sonic horizon, where $|v_{0}|<a_{0}$, the same integrand acquires a negative sign overall, and $|f^{\prime}(r,t)|\longrightarrow 0$. With a small viscosity, the sonic horizon of Bondi (1952) accretion forces a discontinuity in the outward propagation of the wave, and acts like an impenetrable barrier to block acoustic waves within itself.333 A viscosity-induced divergence of the wave amplitude near an acoustic horizon, as Eq. (36) indicates, is similarly seen in the two-dimensional outflow of the shallow-water hydraulic jump, where a perturbation in the sub-critical region, propagating against the steady outflow, accumulates a wall of water at the horizon (Ray and Bhattacharjee, 2007; Kate et al., 2007). However, the propagating perturbation does not breach the horizon, thus characterizing it as a white hole (Ray and Bhattacharjee, 2007; Bhattacharjee, 2017). What is more, specific to astrophysical accretion itself, a small viscosity can cause a perturbation to diverge, leading to a secular instability (Bhattacharjee and Ray, 2007).

The energy flux of the perturbation also behaves in a manner similar to its amplitude. The kinetic energy per unit volume is $E_{\mathrm{kin}}=(1/2)(\rho_{0}+\rho^{\prime})(v_{0}+v^{\prime})^{2}$ (Petterson et al., 1980). The potential energy per unit volume, with contributions from both the gravitational energy and the internal energy (Petterson et al., 1980), is $E_{\mathrm{pot}}=(\rho_{0}+\rho^{\prime})(GM/r)+\rho_{0}\epsilon+[\partial(\rho_{0}\epsilon)/\partial\rho_{0}]\rho^{\prime}+(1/2)[\partial^{2}(\rho_{0}\epsilon)/\partial\rho_{0}^{2}]{\rho^{\prime}}^{2}$, with $\epsilon$ being the internal energy per unit mass (Landau and Lifshitz, 1987). In both of the foregoing expressions of energy, the zeroth-order terms refer to the steady flow, and the first-order terms disappear upon time-averaging. Thereafter, the time-averaged total energy in the perturbation, per unit volume of fluid, is to be obtained by summing the second-order terms in $E_{\mathrm{kin}}$ and $E_{\mathrm{pot}}$. All of these terms go either as ${\rho^{\prime}}^{2}$ or ${v^{\prime}}^{2}$, or as a product of $\rho^{\prime}$ and $v^{\prime}$. Making use of Eqs. (19), (20) and (30), we get $(\rho^{\prime}/\rho_{0})\simeq v_{0}(v_{0}\mp a_{0})^{-1}(f^{\prime}/f_{0})$ and $(v^{\prime}/v_{0})\simeq\mp a_{0}(v_{0}\mp a_{0})^{-1}(f^{\prime}/f_{0})$. Once we have two relations explicitly connecting $\rho^{\prime}$ and $v^{\prime}$ with $f^{\prime}$, the time-averaged total energy per unit volume is $E_{\mathrm{tot}}\sim\langle{|f^{\prime}(r,t)|}^{2}\rangle$. The energy flux of the spherical wavefront, moving with the speed, $(v_{0}\mp a_{0})$, is $F=4\pi r^{2}E_{\mathrm{tot}}(v_{0}\mp a_{0})$. Clearly, for the wave travelling outwards against the Bondi (1952) inflow, very close to the sonic horizon, both $E_{\mathrm{tot}}$ and $F$ will exhibit the same instability implied by Eq. (36).

VI Residues and acoustic tunnelling through the horizon

Waves propagating outwards against the steady Bondi (1952) inflow (for which $v_{0}=-|v_{0}|$), encounter a singularity at the sonic horizon, where $|v_{0}|=a_{0}$. This is obvious from the integrands in Eqs. (30), (33) and (35). In each case, circumvention of the singularity requires rendering it as a simple pole on the path of the integration, and then applying Cauchy’s residue theorem on the path. We first demonstrate this procedure in full for the simplest of the cases, which is in Eq. (30), by considering its lower sign, as only this pertains to an outward wave against the inflow. The main contribution to the integral comes from the immediate neighbourhood of $|v_{0}|=a_{0}$, which is also where $r=r_{\mathrm{c}}$, as Eqs. (10) show. A Taylor expansion about the horizon, up to the first order, gives $a_{0}-|v_{0}|\simeq(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}+[{\mathrm{d}}(a_{0}-|v_{0}|)/{\mathrm{d}}r]_{r_{\mathrm{c}}}(r-r_{\mathrm{c}})$. The Taylor expansion in the neighbourhood of the horizon transforms the singularity at $|v_{0}|=a_{0}$ to a simple pole at $r=r_{\mathrm{c}}$. Going by what Eqs. (10) suggest, the zero-order term in the Taylor expansion vanishes, in consequence of which, we approximate Eq. (30) as

[TABLE]

The analogue surface gravity, $g_{\mathrm{s}}=a_{\mathrm{c}}[{\mathrm{d}}(a_{0}-|v_{0}|)/{\mathrm{d}}r]_{r_{\mathrm{c}}}$, at the sonic horizon (Visser, 1998), and the analogue Hawking temperature, $T_{\mathrm{H}}=(\hbar g_{\mathrm{s}})/(2\pi k_{\mathrm{B}}a_{\mathrm{c}})$ (Visser, 1998). In terms of $g_{\mathrm{s}}$ and $T_{\mathrm{H}}$, the integral in Eq. (37), taking the residue at the pole, becomes

[TABLE]

where ${\mathcal{P}}[\alpha_{0}]$ is the principal value of the integral. Furthermore, the negative sign in $\pm i$ is due to a clockwise detour of the pole, and the positive sign is due to an anti-clockwise detour. Both are valid mathematically, but in a real physical sense, the ultimate choice of the sign is determined by the boundary condition at the pole (Dennery and Krzywicki, 1996). Since $\alpha=\alpha_{0}+\alpha_{\eta}$, the imaginary part of $\alpha_{0}$ in Eq. (38) contributes to the amplitude of $f^{\prime}(r,t)=e^{-\beta}\exp(i\alpha-i\omega t)$. Now, the horizon acts like an unyielding barrier to outgoing waves. This boundary condition at the horizon necessitates the choice of the positive sign of $i$ in Eq. (38), and as such, the wave, which can only be very weak with a decaying amplitude, tunnels through the barrier.

The tunnelling amplitude receives an additional contribution from $\alpha_{\eta}$, as it is given in Eq. (33), with the integral being a sum of two terms. For simplicity of notation, we write $\gamma=2+(4r/v_{0}){\mathrm{d}}v_{0}/{\mathrm{d}}r$. When the inflow is in free fall close to the accretor, $v_{0}\sim r^{-1/2}$, and when the inflow is highly subsonic far away from the accretor, $v_{0}\sim r^{-2}$ (Petterson et al., 1980). In the former case, $\gamma=0$, and in the latter, $\gamma=-6$. We expect $\gamma(r_{\mathrm{c}})$ to have an intermediate value between these two limits. Following what we did to arrive at Eq. (37), a Taylor expansion up to the first order gives $(a_{0}-|v_{0}|)^{2}\simeq(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}^{2}+2(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}[{\mathrm{d}}(a_{0}-|v_{0}|)/{\mathrm{d}}r]_{r_{\mathrm{c}}}(r-r_{\mathrm{c}})$. Now that we explicitly account for a small order of viscosity, we see from Eq. (18) that $(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}^{2}$ is a small non-vanishing quantity of the order of $\eta_{\star}^{2}$. Set against this, the first-order term in the Taylor expansion is of the order of $\eta_{\star}$. This argument allows us to neglect the zero-order term in the Taylor series, and write $(a_{0}-|v_{0}|)^{2}\simeq-2(|v_{0}|-a_{0})_{r_{\mathrm{c}}}[{\mathrm{d}}(a_{0}-|v_{0}|)/{\mathrm{d}}r]_{r_{\mathrm{c}}}(r-r_{\mathrm{c}})$. By the same token, we also approximate $(a_{0}-|v_{0}|)^{3}\simeq 3(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}^{2}[{\mathrm{d}}(a_{0}-|v_{0}|)/{\mathrm{d}}r]_{r_{\mathrm{c}}}(r-r_{\mathrm{c}})$. These conditions, imposed about the sonic horizon (where the singularity contributes the most to the integral), approximates Eq. (33), with its lower sign, to

[TABLE]

About the sonic horizon, the kinematic viscosity, $\nu_{\mathrm{c}}=\eta/\rho_{\mathrm{c}}$. The wave number, $k(r)=\omega/(|v_{0}|-a_{0})$, which is blue-shifted near the horizon (Parikh and Wilczek, 2000). We define a frequency, $\Omega=\nu_{\mathrm{c}}|k(r_{\mathrm{c}})|/r_{\mathrm{c}}$, and a temperature, $T_{\mathrm{s}}=[\hbar(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}]/(2\pi k_{\mathrm{B}}r_{\mathrm{c}})$. With these definitions, and by taking the residue at the pole, the integral in Eq. (39) gives

[TABLE]

with ${\mathcal{P}}[\alpha_{\eta}]$ being the principal value of the integral. The singularity, as noted earlier, stands as a barrier against an outgoing acoustic wave that has to pass from the supersonic region to the subsonic region. This being the physical boundary condition at the singularity, a wave can at best tunnel through it with a decaying amplitude. Since $\alpha_{\eta}$ appears in the phase of $f^{\prime}(r,t)\sim\exp(i\alpha-i\omega t)$ through $\alpha=\alpha_{0}+\alpha_{\eta}$, to make for a decaying amplitude, we select the positive sign of $\pm i$ in Eq. (40). Likewise, not to forget what comes with the positive sign of $\pm i$ in Eq. (38), the total contribution to the amplitude of the tunnelling wave, from both Eqs. (38) and (40), is written as $|f_{\mathrm{T}}^{\prime}|\sim e^{-\Gamma}$, in which

[TABLE]

With $|f_{\mathrm{T}}^{\prime}|$ determined thus, the tunnelling probability is found from $|f_{\mathrm{T}}^{\prime}|^{2}$. The second term in Eq. (41) is of particular interest to us. The frequency, $\Omega$, is dependent on viscosity, and it is the dissipative influence of viscosity that also shrinks the sonic sphere, as we deduce from Eq. (16). What is more, in the tunnelling amplitude, $\hbar\Omega$ is scaled by the fluid analogue of the Hawking temperature. The combined effect of these facts is that the second term in Eq. (41) is responsible for the phenomenon of black hole evaporation by phonon radiation in fluid analogues (Unruh, 1981; Jacobson, 1991; Unruh, 1995). Especially when, with a large blue-shift near the horizon, the corresponding increase in momentum can cause a strong dissipation (Robertson and Parentani, 2015). However, we also note that $T_{\mathrm{s}}\ll T_{\mathrm{H}}$, from which we realize that the third term in Eq. (41) will overwhelm both the terms scaled by $T_{\mathrm{H}}$. We consider the physical meaning of $T_{\mathrm{s}}$ vis-à-vis $T_{\mathrm{H}}$. Phonons are physically valid only as long as the acoustic structure of the fluid flow is preserved. The thermal flux of these phonons through the acoustic horizon has the temperature, $T_{\mathrm{H}}$ (Unruh, 1981; Jacobson, 1991). Close to the horizon, the outgoing waves experience a blue-shift (Parikh and Wilczek, 2000), with $(a_{0}-|v_{0}|)_{r_{\mathrm{c}}}\longrightarrow 0$. The high momentum (and the short wavelength) of the blue-shift disrupts the acoustic structure (Visser, 1998). Physically this happens because the wavelength of the acoustic wave becomes small enough to be comparable to the mean free path of the molecules in the fluid (Visser, 1998). If we go by the definition of $T_{\mathrm{s}}$, we see that with the blue-shift near the acoustic horizon, $T_{\mathrm{s}}\longrightarrow 0$. Since the same blue-shift violates the acoustic structure, $T_{\mathrm{s}}$ should not be associated with the thermal phonon flux. It is named “temperature” because of its dimensional compatibility with $T_{\mathrm{H}}$. One might conjecture that $T_{\mathrm{s}}$ represents the thermal state of the phonon flux when the acoustic geometry is on the verge of breaking down.

From the perspective of the acoustic continuum, the blue-shift itself calls for careful consideration. For the radially convergent flow of an inviscid compressible fluid, the sonic condition at the horizon is $|v_{0}|=a_{0}$, because of which, for a wave propagating outwards against the flow, the group velocity, $v_{\mathrm{g}}\longrightarrow 0$. This much is clear from Eq. (28) when $\eta=0$. The wavenumber, $k=\omega/(|v_{0}|-a_{0})$, is then blue-shifted without limit near the horizon, and the corresponding wavelength is shortened arbitrarily. As long as the acoustic continuum holds, the blue-shifting can continue, but physically it is limited by the molecular nature of matter. On the molecular scale, arbitrarily short wavelengths will eventually become comparable to the intermolecular spacing, and the fluid continuum needed for the acoustic propagation will no longer hold. Waves travelling outwards from the acoustic horizon can be traced back in time towards the horizon itself. This will entail the blue-shifting of the wave number. But even before the horizon can be reached by the wave travelling backward, arbitrary blue-shifting will ensure the breakdown of the acoustic continuum. Therefore, with the loss of the acoustic continuum at very short wavelengths, the acoustic wave may never seem to have originated at the horizon or arbitrarily close to it. This difficulty, known for long, has been taken up and addressed variously (Jacobson, 1991; Unruh, 1995; Brout et al., 1995; Corley and Jacobson, 1996). With viscosity included in the fluid, however, the physical conditions become qualitatively different. From Eqs. (17) and (28), we realize that viscosity creates a thin layer of uncertainty about the exact transonic condition of the inviscid state. With $|v_{0}|-a_{0}$ restricted to a small non-zero value by viscosity (instead of just vanishing at the horizon), it is possible for an outgoing wave packet to avoid an infinite blue-shift and emerge from the uncertain layer about the horizon. Another consequence of viscosity is that time reversal, $t\longrightarrow-t$, will no longer be symmetric, as shown by the first-order time derivative in Eq. (25). Hence, a wave packet, made to propagate backward in time, will no longer trace back the same physical states of its forward propagation in time.

Change of entropy also prevents time reversal. In a polytropic process, the entropy, $\mathrm{S}$, changes according to (Chandrasekhar, 1939)

[TABLE]

While Eq. (42) gives a general thermodynamic perspective, the specific mechanical means of entropy change in the astrophysical fluid flow of our present interest is viscosity. It causes heating of the fluid, and this heat is then dissipated to the surroundings by radiative processes. This sequence of events, taken together, increases the universal entropy and is fundamentally irreversible. As it happens, the same viscous dissipation breaks the time-reversal symmetry of the propagation of an acoustic wave group, as Eq. (25) indicates. Thus, the breaking of the time-reversal symmetry due to viscous dissipation is consistent with the change of entropy.

The residues in the singularities of $\alpha_{0}$ and $\alpha_{\eta}$ have contributed to the amplitude of the tunnelling wave. In reciprocation, the residue in the singularity of $\beta_{\eta}$, as shown by Eq. (35), will contribute to the phase of the tunnelling wave. This is found to be $\beta_{\eta}\simeq[(\pm i\hbar\Omega_{\mathrm{T}})/(12k_{\mathrm{B}}T_{\mathrm{H}})]+{\mathcal{P}}[\beta_{\eta}]$, in which the tunnelling frequency, $\Omega_{\mathrm{T}}=\nu_{\mathrm{c}}|k(r_{\mathrm{c}})|^{2}\gg\Omega$, and ${\mathcal{P}}[\beta_{\eta}]$ is the principal value of the integral in Eq. (35). We observe that through the singularity, the mutual exchange of amplitude and phase, between $\alpha$ and $\beta$, respectively, occurs only when $\eta\neq 0$.

VII Concluding remarks

We have treated viscous spherically symmetric transonic accretion as an astrophysical model to demonstrate the analogue Hawking radiation of phonons through the sonic horizon of an acoustic black hole. Viscosity plays a part in this process, as well as in shrinking the acoustic black hole. Both phenomena are consistent with each other, when we recall the possibility of the evaporation of black holes (Unruh, 1981; Jacobson, 1991; Unruh, 1995). Since viscosity appears to be instrumental in the Hawking radiation, which is fundamentally a quantum effect, we refer to a study (Bhattacharjee et al., 2009) that shows an equivalence between viscosity in the Navier-Stokes equation and Planck’s constant in Schrödinger’s equation.

In the physical flow that we have studied here, viscosity has a weak perturbative presence about an inviscid background. In astrophysical accretion, however, viscosity is enhanced because of turbulence in the flow, without which, for instance, the outward transport of angular momentum in accretion discs is not feasible (Balbus and Hawley, 1998; Frank et al., 2002). In spherically symmetric accretion, the coupling of the mean flow and the turbulent fluctuations scales viscosity up significantly, and shifts the sonic horizon inwards (Ray and Bhattacharjee, 2005), qualitatively in the same way that we have seen here. Hence, we believe that an effective “turbulent viscosity” (Ray and Bhattacharjee, 2005) in transonic inflows, can markedly alter the tunnelling amplitude of acoustic waves. That said, we should remember that turbulence is a nonlinear phenomenon, while the results of our present study are based on linearization.

Astrophysical flows often possess angular momentum and undergo an axially symmetric differential rotation, as happens in accretion discs (Balbus and Hawley, 1998; Frank et al., 2002). Sub-Keplerian accretion discs with low angular momentum can be treated as conservative flows (Ray, 2003b), and as such they are models of analogue gravity as well (Chaudhury et al., 2006; Das et al., 2007; Nag et al., 2012; Bose et al., 2014). Complications arise for axially symmetric flows with high angular momentum, because in such cases the infall requires a physical means for the outward transport of the angular momentum (Frank et al., 2002). For this purpose, turbulence, with an enhanced effective viscosity, serves well (Balbus and Hawley, 1998; Frank et al., 2002), but with turbulence we lose the basic condition of analogue-gravity modelling, namely, an inviscid and vorticity-free potential flow (Visser, 1998), which, otherwise, is an easy condition to fulfil in the spherically symmetric geometry. Hence, in axially symmetric geometry, the modelling of analogue gravity is limited to flows of low angular momentum, especially when such a flow is driven by the powerful gravity of a black hole (Das et al., 2007).

The physical framework of analogue gravity, the basic subject of our study here, is an inviscid, irrotational and vorticity-free fluid flow in the Newtonian construct of flat space and time (Unruh, 1981; Jacobson, 1991; Unruh, 1995; Visser, 1998). On a closed surface in this fluid, where the flow becomes supersonic, the acoustic analogue of a black-hole horizon is formed. This is the fluid-flow model of a black hole, and the propagation of acoustic waves in the supersonic fluid flow mimics the propagation of scalar waves in the spacetime of a black hole (Unruh, 1981, 1995). As long as an equation of state provides a means for acoustic propagation, transonic hydrodynamic flows can generally produce an analogue metric and an acoustic horizon. While this appears to be a universal feature of flows that pass through a well-defined critical point (Naskar et al., 2007; Ray and Bhattacharjee, 2007; Sarkar et al., 2013; Sen and Ray, 2014), the symmetric form of the acoustic metric can be variously disrupted because of the physical circumstances of a given fluid flow. The coupling of the flow and the geometry of Schwarzschild spacetime is one example of how the acoustic mertic can be affected adversely (Naskar et al., 2007). The same behaviour is also exhibited due to viscous dissipation in the shallow-water circular hydraulic jump (Ray and Bhattacharjee, 2007). Dispersion, arising out of interactions between baryons and vector mesons in a nuclear outflow, similarly breaks the symmetry of the acoustic metric (Sarkar et al., 2013). Perturbations of nonlinear order in spherically symmetric accretion shift the acoustic horizon about its static position, although the symmetry of the acoustic metric remains intact (Sen and Ray, 2014). We surmise that some of the aforementioned fluid systems can help to detect the analogue Hawking radiation.

Acknowledgements.

This research has made use of NASA’s Astrophysics Data System. The author thanks J. K. Bhattacharjee, T. K. Das and T. Jacobson for their comments. He is also grateful for the library support of Jaypee University of Engineering and Technology, Raghogarh, Guna, India.