Tidal Shock Formation in Molecular Clouds Under Intermediate Black Hole Tidal Fields

Abstract

We develop a comprehensive semi–analytical framework to describe tidal shock formation in a molecular cloud encountering a massive black hole. The cloud is initially modeled as a uniform sphere with radius $R$, density $\rho_0$, and mass $M_{\rm cl}=1000\,M_\odot$. Under the influence of the black hole’s tidal field, the cloud is deformed by stretching along the orbital direction and compression in the vertical direction. In addition to self–gravity (with a restoring frequency $\omega_s$) and pressure corrections, we include radiative cooling/heating in the energy equation. Tidal compression increases the density so that the cooling timescale $t_{\rm cool}$ becomes very short and the gas rapidly cools toward an equilibrium temperature $T_{\rm eq}$. We also incorporate the cloud’s orbital velocity and the internal tidal–induced velocity (described in a Lagrangian frame) into an overall effective velocity vector $\mathbf{v}_{u,\mathrm{eff}}$ that governs shock formation. (When analyzing the shock, one projects $\mathbf{v}_{u,\mathrm{eff}}$ onto the shock normal.) Our final normalized formulas are presented in Section 4.

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1. Introduction

Molecular clouds in the vicinity of galactic nuclei are subject to intense tidal forces from supermassive or intermediate mass black holes. When a molecular cloud encounters such a tidal field, its deformation is anisotropic: the cloud is stretched along the orbital direction (which we designate as the $\hat{\mathbf{x}}$ direction) and compressed in the vertical direction (taken as $\hat{\mathbf{z}}$). A shock forms if the effective upstream velocity—that is, the velocity of the gas impinging on the shock front—exceeds the local sound speed. Because tidal compression increases the gas density, radiative cooling becomes highly efficient and prevents the post–shock temperature from reaching values high enough to dissociate molecules (e.g., CO$_2$).

In our treatment the cloud’s center–of–mass moves with an orbital velocity

$$\mathbf{v}_{\rm orb}=v_0\,\hat{\mathbf{x}},\qquad \text{with}\quad v_0=10\,\mathrm{km/s}\;(1\times10^6\,\mathrm{cm/s})\,,$$

while tidal forces introduce an additional internal velocity $\mathbf{v}_{\rm tidal}$. Consequently, the effective upstream velocity is given by the vector sum

$$\mathbf{v}_{u,\mathrm{eff}}=\mathbf{v}_{\rm orb}+\mathbf{v}_{\rm tidal}\,.$$

When computing shock properties, one projects $\mathbf{v}_{u,\mathrm{eff}}$ onto the shock normal $\hat{\mathbf{n}}$ to obtain the scalar upstream speed. In addition, we now define the Mach number along both the normal direction (typically along $\hat{x}$) and in the vertical ($z$) direction.

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2. Theoretical Framework

In what follows we detail the derivation of the key equations in strict vector notation.

2.1. Homologous Displacement and Mass Conservation

Assume that the cloud is initially a uniform sphere of radius $R$ and density $\rho_0$. A self–similar (homologous) displacement is expressed as a mapping from the initial (Lagrangian) coordinates $\mathbf{r}_0=(x_0,y_0,z_0)$ to the current (Eulerian) coordinates:

$$\mathbf{r}(t)=\bigl(f_x(t)x_0,\; f_y(t)y_0,\; f_z(t)z_0\bigr)\,.$$

For an axisymmetric deformation we set

$$f_x(t)=f_y(t)=f_\parallel(t),\qquad f_z(t)=f_\perp(t),$$

with the initial conditions $f_\parallel(0)=f_\perp(0)=1$ and $\dot{f}_\parallel(0)=\dot{f}_\perp(0)=0$. Mass conservation then requires that

$$\rho(t)=\frac{\rho_0}{J(t)}=\frac{\rho_0}{f_\parallel(t)^2\,f_\perp(t)}\,.$$

A rigorous derivation is provided in Appendix A.1.

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2.2. Restoring Acceleration: Self–Gravity and Pressure

Within a uniform sphere, Newton’s shell theorem gives the gravitational acceleration at radius $r$ as

$$\mathbf{g}(r)=\frac{4\pi G\rho_0}{3}\,\mathbf{r}\,.$$

A small radial displacement $\delta \mathbf{r}$ results in a change in acceleration,

$$\delta\mathbf{g}\approx\frac{4\pi G\rho_0}{3}\,\delta \mathbf{r}\,,$$

so that the restoring acceleration is

$$\mathbf{a}_{\rm rest}=-\delta \mathbf{g}=-\omega_s^2\,\delta \mathbf{r}\,,$$

with

$$\omega_s^2=\frac{4\pi G\rho_0}{3}\,.$$

Pressure gradients provide a small additional correction, and we define an effective restoring frequency by

$$\omega_{\rm eff}^2=\omega_s^2+\Delta\omega_{\rm press}^2\,.$$

Under nearly isothermal or nearly incompressible conditions, it is appropriate to take $\omega_{\rm eff}\approx\omega_s$. (See Appendix A.2 for further details.)

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2.3. Affine Model and Tidal Forcing

A point mass $M_{BH}$ at a distance $r_p$ produces a tidal field whose principal tidal tensor (in a coordinate system with $\hat{\mathbf{x}}$ radial and $\hat{\mathbf{z}}$ vertical) has eigenvalues

$$\lambda_\parallel=\frac{2GM_{BH}}{r_p^3} \quad \text{and} \quad \lambda_\perp=-\frac{GM_{BH}}{r_p^3}\,.$$

The evolution equations for the scale factors then become

$$\ddot{f}_\parallel(t)=\Bigl(\lambda_\parallel-\omega_{\rm eff}^2\Bigr)f_\parallel(t),\qquad \ddot{f}_\perp(t)=\Bigl(\lambda_\perp-\omega_{\rm eff}^2\Bigr)f_\perp(t)\,.$$

In the tidal limit, the approximate solutions are

$$f_\parallel(t)=\cosh\Bigl(\sqrt{\lambda_\parallel-\omega_{\rm eff}^2}\,t\Bigr),\qquad f_\perp(t)=\cos\Bigl(\sqrt{\omega_{\rm eff}^2+\frac{GM_{BH}}{r_p^3}}\,t\Bigr)\,.$$

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2.4. Radiative Cooling and the Cooling Timescale

The energy equation can be written as

$$\frac{d\epsilon}{dt}=-\frac{P}{\rho}\,\nabla\cdot\mathbf{v}+\Gamma-\Lambda\,,$$

and upon substituting $\epsilon=c_vT$, one obtains

$$\frac{dT}{dt}=-\left(\gamma-1\right)T\,\nabla\cdot\mathbf{v}-\frac{T-T_{\rm eq}}{t_{\rm cool}}\,,$$

where $\Gamma$ and $\Lambda$ are the heating and cooling rates, respectively. For a net cooling rate per unit volume given by

$$\Lambda_{\rm net}=n^2\,\Lambda(T)\,,$$

the cooling time is approximately

$$t_{\rm cool}\sim\frac{n\,k_B\,T}{n^2\,\Lambda(T)}\propto\frac{1}{n}\,.$$

Since tidal compression increases the density by a factor $1/(f_\parallel^2f_\perp)$, the cooling time is reduced accordingly. For instance, if $f_\perp\approx0.1$, then $t_{\rm cool}$ is decreased by roughly an order of magnitude, causing the gas to cool rapidly toward $T_{\rm eq}$.

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2.5. Effective Divergence

In a compressing or collapsing flow the divergence $\nabla\cdot\mathbf{v}$ may be estimated via the free–fall time,

$$t_{\rm ff}\sim\frac{1}{\sqrt{G\rho}},$$

which implies

$$\left|\nabla\cdot\mathbf{v}\right|\sim\frac{1}{t_{\rm ff}}\sim\sqrt{G\rho}\,.$$

For example, if the post–compression number density is $n\sim10^5\,\mathrm{cm}^{-3}$ (corresponding to $\rho\sim 3.84\times10^{-19}\,\mathrm{g/cm^3}$), then

$$\sqrt{G\rho}\sim\sqrt{6.67\times10^{-8}\times3.84\times10^{-19}}\approx1.6\times10^{-13}\,\mathrm{s}^{-1}\,.$$

An effective divergence of approximately $-1\times10^{-13}\,\mathrm{s}^{-1}$ is therefore plausible in a strongly compressed cloud.

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2.6. Orbital and Tidal–Induced Velocities in Vector Notation

Let the cloud’s center–of–mass (orbital) velocity be

$$\mathbf{v}_{\rm orb}=v_0\,\hat{x},$$

with $v_0=10\,\mathrm{km/s}\;(1\times10^6\,\mathrm{cm/s})$. In our axisymmetric model the tidal deformation in the plane produces identical velocity contributions in both the $x$ and $y$ directions. Thus, we define the tidal–induced velocities as

$$v_{\rm tidal}^x=f_\parallel\,R\,\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)},$$ $$v_{\rm tidal}^y=f_\parallel\,R\,\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)},$$

and in the vertical ($z$) direction,

$$v_{\rm tidal}^z=f_\perp\,R\,\sqrt{\frac{GM_{BH}}{r_p^3}(1-\eta)}\,.$$

Accordingly, the full effective upstream velocity vector becomes

$$\boxed{ \mathbf{v}_{u,\mathrm{eff}} = \mathbf{v}_{\rm orb} + \mathbf{v}_{\rm tidal} = v_0\,\hat{x} + \Bigl( v_{\rm tidal}^x\,\hat{x} + v_{\rm tidal}^y\,\hat{y} + v_{\rm tidal}^z\,\hat{z} \Bigr)\,. }$$

When analyzing shock formation one projects $\mathbf{v}_{u,\mathrm{eff}}$ onto the shock normal $\hat{n}$:

$$v_{u,\mathrm{eff}}^{(n)} = \mathbf{v}_{u,\mathrm{eff}}\cdot\hat{n}\,.$$

In addition to the usual Mach number based on the normal component,

$$\mathcal{M}^{(n)} = \frac{v_{u,\mathrm{eff}}^{(n)}}{c_s}\,,$$

we also define the Mach number in the vertical direction as

$$\mathcal{M}^{(z)} = \frac{|v_{\rm tidal}^z|}{c_s}\,.$$

The self–gravity parameter is defined as

$$\eta=\frac{M_{\rm cl}}{M_{BH}}\Bigl(\frac{r_p}{R}\Bigr)^3\,.$$

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2.7. CO$_2$ Survival Criterion

In the strong–shock limit (in the absence of cooling), the post–shock temperature based on the normal component of the effective velocity is

$$T_2\sim\frac{3\,\mu\,m_p}{16\,k_B}\,\Bigl(v_{u,\mathrm{eff}}^{(n)}\Bigr)^2\,.$$

However, efficient cooling forces the gas to relax toward the equilibrium temperature $T_{\rm eq}$. We therefore define the effective post–shock temperature as

$$T_{\rm eff}=\min\{T_2,\,T_{\rm eq}\}\,.$$

For CO$_2$ survival it is necessary that

$$T_{\rm eff}<T_{\rm diss}\quad \text{with}\quad T_{\rm diss}\sim3000\,\mathrm{K}\,.$$

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2.8. Cooling Timescale Revisited

The net radiative cooling rate per unit volume is

$$\Lambda_{\rm net}=n^2\,\Lambda(T)\,,$$

and the cooling time (i.e., the time to radiate away the thermal energy density $n\,k_B\,T$) is approximately

$$t_{\rm cool}\sim\frac{n\,k_B\,T}{n^2\,\Lambda(T)}=\frac{k_B\,T}{n\,\Lambda(T)}\,.$$

For example, if after tidal compression $n\sim10^6\,\mathrm{cm}^{-3}$ and $\Lambda(T)\sim10^{-23}\,\mathrm{erg\,cm^3/s}$ (for $T\sim10^4\,\mathrm{K}$), then

$$t_{\rm cool}\sim\frac{1.38\times10^{-16}\,T}{10^6\times10^{-23}}\sim1.38\times10\,T\quad\text{s}\,.$$

For $T\sim10^4\,\mathrm{K}$ this gives $t_{\rm cool}\sim1.38\times10^5\,\text{s}$ (a few days), which is much shorter than the typical dynamical timescale (e.g., $t_{\rm ff}\sim10^{12}\,\text{s}$), confirming the efficiency of cooling at high densities.

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2.9. Vertical Pancake Structure: Quasi–Static and Dynamic Treatment

When a molecular cloud approaches a black hole, tidal forces compress it preferentially in the vertical ($z$) direction, producing a “pancaked” structure. In the simplest quasi–static picture one balances the tidal compression against the pressure gradient.

2.9.1. Quasi–Static Equilibrium

The tidal acceleration in the $z$ direction for small displacements is

$$a_{\rm tidal}^{(z)}=-\frac{GM_{BH}}{r_p^3}\,z\,.$$

For a cloud with vertical thickness $H$, the pressure gradient produces an acceleration of order

$$a_{\rm press}\sim\frac{c_s^2}{H}\,,$$

where $c_s$ is the sound speed. Equating these in magnitude at $z\sim H$ leads to

$$\frac{c_s^2}{H}\sim\frac{GM_{BH}}{r_p^3}\,H\,.$$

Solving for $H$ yields

$$H^2\sim\frac{c_s^2\,r_p^3}{GM_{BH}},\quad\text{or}\quad \boxed{H\sim c_s\,\sqrt{\frac{r_p^3}{GM_{BH}}}\,.}$$

2.9.2. Dynamic Vertical Evolution

When the sound–crossing time is comparable to the dynamical timescale of tidal forcing, a time–dependent treatment is required. Suppose that the vertical coordinate of a fluid element evolves as

$$z(t)=H(t)\,z_0\,,$$

with $z_0$ the initial (Lagrangian) coordinate and $H(t)$ the vertical scaling factor. Then,

$$\dot{z}(t)=\dot{H}(t)\,z_0,\quad \ddot{z}(t)=\ddot{H}(t)\,z_0\,.$$

The two contributions to the vertical acceleration are:

1. Tidal Compression:

   $$a_{\rm tidal}^{(z)}=-\frac{GM_{BH}}{r_p^3}\,z=-\frac{GM_{BH}}{r_p^3}\,H(t)\,z_0\,.$$

2. Pressure Gradient:
   Assuming the cloud remains nearly isothermal with pressure $P(t)=c_s^2\,\rho(t)$ and a density scaling $\rho(t)=\rho_0/H(t)$ (for self–similar contraction), the pressure gradient is approximated by

   $$\frac{dP}{dz}\sim\frac{P}{H(t)}=\frac{c_s^2\,\rho_0}{H(t)^2}\,.$$

   The corresponding acceleration is then

   $$a_{\rm press}=-\frac{1}{\rho(t)}\frac{dP}{dz}=-\frac{c_s^2}{H(t)}\,.$$

The net vertical acceleration is given by

$$\ddot{z}(t)=\ddot{H}(t)\,z_0=a_{\rm tidal}^{(z)}+a_{\rm press}\,.$$

Dividing by $z_0$ leads to the evolution equation for $H(t)$:

$$\boxed{\ddot{H}(t)=-\frac{GM_{BH}}{r_p^3}\,H(t)+\frac{c_s^2}{H(t)}\,.}$$

The quasi–static equilibrium is recovered by setting $\ddot{H}(t)=0$, which yields

$$H_{\rm eq}=c_s\,\sqrt{\frac{r_p^3}{GM_{BH}}}\,.$$

A linear perturbation analysis about $H_{\rm eq}$ shows oscillatory behavior with frequency

$$\omega=\sqrt{\frac{2GM_{BH}}{r_p^3}},$$

and an oscillation period

$$T=2\pi\sqrt{\frac{r_p^3}{2GM_{BH}}}\,.$$

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3. Numerical Example

We now illustrate the framework with representative parameters:

• Black Hole Mass:

  $$M_{BH}=10^5\,M_\odot\approx2.0\times10^{38}\,\mathrm{g}\,.$$

• Cloud Mass:

  $$M_{\rm cl}=1000\,M_\odot\approx2.0\times10^{36}\,\mathrm{g}\,.$$

• Cloud Radius:

  $$R=1.5\,\mathrm{pc}\approx4.63\times10^{18}\,\mathrm{cm}\,.$$

• Periastron Distance:

  $$r_p=3\,\mathrm{pc}\approx9.26\times10^{18}\,\mathrm{cm}\,.$$

• Initial Cloud Temperature:

  $$T_{\rm cloud}=10\,\mathrm{K}\,,$$

  with the equilibrium temperature $T_{\rm eq}=20\,\mathrm{K}$.

• Cooling Timescale:

  $$t_{\rm cool}=1\times10^{12}\,\mathrm{s}\,.$$

• Effective Divergence:

  $$\nabla\cdot\mathbf{v}=-1\times10^{-13}\,\mathrm{s}^{-1}\,.$$

• Orbital Velocity:

  $$\mathbf{v}_{\rm orb}=1\times10^6\,\mathrm{cm/s}\,\hat{x}\,.$$

1. Initial Density:

   The initial density is given by

   $$\rho_0=\frac{3M_{\rm cl}}{4\pi R^3}\approx\frac{3\times2.0\times10^{36}}{4\pi(4.63\times10^{18})^3}\approx4.81\times10^{-22}\,\mathrm{g/cm^3}\,.$$

2. Self–Gravity Frequency:

   Using

   $$\omega_s=\sqrt{\frac{4\pi G\rho_0}{3}},$$

   we obtain $\omega_s\approx1.16\times10^{-14}\,\mathrm{s}^{-1}$.

3. Tidal Eigenvalues:

   The eigenvalues are

   $$\lambda_\parallel\approx\frac{2GM_{BH}}{r_p^3}\quad\text{and}\quad \lambda_\perp\approx-\frac{GM_{BH}}{r_p^3}\,,$$

   which, for these parameters, are on the order of $10^{-26}\,\mathrm{s}^{-2}$.

4. Affine Model Scale–Factors:

   At time $t_0=5\times10^{12}\,\mathrm{s}$ we find

   $$f_\parallel(t_0)\approx\cosh(0.71)\approx1.26,\qquad f_\perp(t_0)\approx\cos(0.87)\approx0.64\,.$$

5. Self–Gravity Parameter:

   The parameter

   $$\eta=\frac{M_{\rm cl}}{M_{BH}}\Bigl(\frac{r_p}{R}\Bigr)^3=0.01\times\Bigl(\frac{3\,\mathrm{pc}}{1.5\,\mathrm{pc}}\Bigr)^3=0.08\,.$$

6. Tidal–Induced Velocity Components:

   • In the $x$–direction:

     $$v_{\rm tidal}^x = f_\parallel(t_0)\,R\,\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)}\,.$$

     With $f_\parallel(t_0)R\approx1.26\times4.63\times10^{18}\approx5.84\times10^{18}\,\mathrm{cm}$ and assuming $\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)}\sim8.2\times10^{-14}\,\mathrm{s}^{-1}$, we have

     $$v_{\rm tidal}^x\approx5.84\times10^{18}\,\mathrm{cm}\times8.2\times10^{-14}\,\mathrm{s}^{-1}\approx4.79\times10^5\,\mathrm{cm/s}\,.$$

   • In the $y$–direction:

     $$v_{\rm tidal}^y = f_\parallel(t_0)\,R\,\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)}\approx4.79\times10^5\,\mathrm{cm/s}\,.$$

   • In the $z$–direction:

     $$v_{\rm tidal}^z = f_\perp(t_0)\,R\,\sqrt{\frac{GM_{BH}}{r_p^3}(1-\eta)}\,.$$

     With $f_\perp(t_0)R\approx0.64\times4.63\times10^{18}\approx2.96\times10^{18}\,\mathrm{cm}$ and assuming $\sqrt{\frac{GM_{BH}}{r_p^3}(1-\eta)}\sim4.1\times10^{-14}\,\mathrm{s}^{-1}$, we obtain

     $$v_{\rm tidal}^z\approx2.96\times10^{18}\,\mathrm{cm}\times4.1\times10^{-14}\,\mathrm{s}^{-1}\approx1.21\times10^5\,\mathrm{cm/s}\,.$$

7. Effective Upstream Velocity Vector:

   The complete effective velocity is then

   $$\mathbf{v}_{u,\mathrm{eff}} = \mathbf{v}_{\rm orb} + \mathbf{v}_{\rm tidal} = v_0\,\hat{x} + \Bigl( v_{\rm tidal}^x\,\hat{x} + v_{\rm tidal}^y\,\hat{y} + v_{\rm tidal}^z\,\hat{z} \Bigr)\,.$$

   Substituting numbers, we have

   $$\mathbf{v}_{u,\mathrm{eff}} \approx \Bigl(1.0\times10^6 + 4.79\times10^5\Bigr)\,\hat{x} + \Bigl(4.79\times10^5\Bigr)\,\hat{y} + \Bigl(1.21\times10^5\Bigr)\,\hat{z}\quad\mathrm{cm/s}\,.$$

   When the shock normal is chosen along $\hat{x}$, the normal component is

   $$v_{u,\mathrm{eff}}^{(n)}\approx1.48\times10^6\,\mathrm{cm/s}\,,$$

   and the Mach number in that direction is

   $$\mathcal{M}^{(n)}=\frac{1.48\times10^6}{c_s}\,.$$

   Similarly, the Mach number in the vertical ($z$) direction is given by

   $$\mathcal{M}^{(z)}=\frac{|v_{\rm tidal}^z|}{c_s}=\frac{1.21\times10^5}{c_s}\,.$$

8. Sound Speed and Mach Numbers:

   Rapid cooling forces the post–shock temperature to $T_{\rm eq}\approx20\,\mathrm{K}$, which gives a sound speed of

   $$c_s\approx3.46\times10^4\,\mathrm{cm/s}\,.$$

   Thus, the normal Mach number becomes

   $$\mathcal{M}^{(n)}\approx\frac{1.48\times10^6}{3.46\times10^4}\approx42.7\,,$$

   and the vertical Mach number is

   $$\mathcal{M}^{(z)}\approx\frac{1.21\times10^5}{3.46\times10^4}\approx3.5\,.$$

9. Post–Shock Temperature:

   In the absence of cooling, the strong–shock limit yields

   $$T_2\sim\frac{3\,\mu\,m_p}{16\,k_B}\,\Bigl(v_{u,\mathrm{eff}}^{(n)}\Bigr)^2\sim1.14\times10^4\,\mathrm{K}\,.$$

   However, because efficient cooling forces the effective post–shock temperature to be

   $$T_{\rm eff}=\min\{T_2,\,T_{\rm eq}\}\approx20\,\mathrm{K}\,,$$

   CO$_2$ survival is ensured as $T_{\rm eff}\ll T_{\rm diss}\sim3000\,\mathrm{K}$.

10. Vertical Pancake Thickness:

    In the quasi–static approximation, the vertical thickness is given by

    $$H\sim c_s\,\sqrt{\frac{r_p^3}{GM_{BH}}}\,.$$

    Inserting the numerical values yields

    $$H\sim 3.5\times10^4\,\mathrm{cm/s}\times7.72\times10^{12}\,\mathrm{s}\sim2.70\times10^{17}\,\mathrm{cm}\sim0.09\,\mathrm{pc}\,.$$

11. Dynamic Time–Scales:

    The oscillation frequency for vertical motions is

    $$\omega=\sqrt{\frac{2GM_{BH}}{r_p^3}}\approx5.79\times10^{-13}\,\mathrm{s^{-1}},$$

    so that the oscillation period is

    $$T\approx\frac{2\pi}{5.79\times10^{-13}}\approx1.085\times10^{13}\,\mathrm{s}\approx3.44\times10^5\,\mathrm{yr}\,.$$

    The sound–crossing time is approximately

    $$t_{\rm sc}\sim\frac{H}{c_s}\sim\frac{2.70\times10^{17}\,\mathrm{cm}}{3.5\times10^4\,\mathrm{cm/s}}\sim7.7\times10^{12}\,\mathrm{s}\sim2.4\times10^5\,\mathrm{yr}\,.$$

    These comparable timescales indicate that a full time–dependent treatment may be required when external forcing varies on similar time–scales.

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4. Discussion and Final Remarks

Our analysis demonstrates a framework for understanding both the shock formation and the vertical (pancake) structure of molecular clouds as they interact with the tidal fields of black holes. The effective upstream velocity,

$$\boxed{ \mathbf{v}_{u,\mathrm{eff}}=v_0\,\hat{x}+\Bigl(v_{\rm tidal}^x\,\hat{x}+v_{\rm tidal}^y\,\hat{y}+v_{\rm tidal}^z\,\hat{z}\Bigr), }$$

provides the basis for computing shock properties via its normal component,

$$v_{u,\mathrm{eff}}^{(n)}=\mathbf{v}_{u,\mathrm{eff}}\cdot\hat{n}\,.$$

In addition to the Mach number computed along the shock normal,

$$\mathcal{M}^{(n)}=\frac{v_{u,\mathrm{eff}}^{(n)}}{c_s}\,,$$

the vertical Mach number is given by

$$\mathcal{M}^{(z)}=\frac{|v_{\rm tidal}^z|}{c_s}\,.$$

Our numerical example shows that even moderate bulk velocities can lead to highly supersonic shocks in the normal direction ($\mathcal{M}^{(n)}\approx42.7$), while the vertical component corresponds to a moderate Mach number ($\mathcal{M}^{(z)}\approx3.5$). Furthermore, the balance between tidal compression and internal pressure defines a quasi–static vertical thickness

$$H\sim c_s\,\sqrt{\frac{r_p^3}{GM_{BH}}}\,,$$

with typical values of $H\sim0.09\,\mathrm{pc}$. When time–dependent effects are included, the vertical scale factor $H(t)$ obeys the evolution equation

$$\boxed{\ddot{H}(t)=-\frac{GM_{BH}}{r_p^3}\,H(t)+\frac{c_s^2}{H(t)}\,.}$$

Linearization around the equilibrium value reveals oscillations with frequency

$$\omega=\sqrt{\frac{2GM_{BH}}{r_p^3}}\,.$$

Finally, when shock heating, in–plane distortions, and additional forcing (e.g., $F_{\rm shock}(t)$) are considered, the effective sound speed increases and the evolution equation becomes

$$\ddot{H}(t)=-\frac{GM_{BH}}{r_p^3}\,H(t)+\frac{c_{\rm eff}^2(t)}{H(t)}+F_{\rm shock}(t)\,,$$

potentially leading to a vertical thickness significantly larger than the quasi–static prediction.

This framework, complete with numerical estimates and detailed derivations (see Appendices), provides a basis for understanding the shock structure and vertical evolution of molecular clouds interacting with black holes.

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Appendix A: Detailed Derivations

A.1. Homologous Radial Displacement and Mass Conservation

For a uniform sphere with radius $R$ and density $\rho_0$, the mapping

$$x=f_x(t)x_0,\quad y=f_y(t)y_0,\quad z=f_z(t)z_0,$$

with axisymmetry ($f_x=f_y\equiv f_\parallel$ and $f_z\equiv f_\perp$), has Jacobian

$$J=f_\parallel(t)^2\,f_\perp(t)\,.$$

Thus, mass conservation requires

$$\rho(t)=\frac{\rho_0}{J}=\frac{\rho_0}{f_\parallel(t)^2\,f_\perp(t)}\,.$$

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A.2. Restoring Acceleration from Self–Gravity

Within a uniform sphere, Newton’s shell theorem gives

$$\mathbf{g}(r)=\frac{4\pi G\rho_0}{3}\,\mathbf{r}\,.$$

A small radial displacement $\delta \mathbf{r}$ produces

$$\delta\mathbf{g}\approx\frac{4\pi G\rho_0}{3}\,\delta \mathbf{r}\,,$$

and the restoring acceleration is then

$$\mathbf{a}_{\rm rest}=-\delta\mathbf{g}=-\omega_s^2\,\delta \mathbf{r}\,,$$

with

$$\omega_s^2=\frac{4\pi G\rho_0}{3}\,.$$

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A.3. Pressure Correction

For a nearly isothermal fluid, the pressure perturbation is given by

$$\delta P=c_s^2\,\delta\rho\,.$$

Mass conservation implies

$$\frac{\delta\rho}{\rho_0}\approx-3\,\frac{\delta r}{R}\,,$$

so that the pressure gradient force per unit mass is approximately

$$\mathbf{a}_{\rm press}\sim-\frac{1}{\rho_0}\frac{\delta P}{\delta r}\sim\frac{3c_s^2}{R^2}\,\delta \mathbf{r}\,.$$

Hence, the effective restoring acceleration is

$$\mathbf{a}_{\rm net}=-\omega_{\rm eff}^2\,\delta \mathbf{r}\quad\text{with}\quad \omega_{\rm eff}^2=\omega_s^2+\Delta\omega_{\rm press}^2\,.$$

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A.4. Rankine–Hugoniot Jump Conditions in Vector Form

For a shock with upstream velocity $\mathbf{v}_{u,\mathrm{eff}}$, the relevant quantity is its projection onto the shock normal $\hat{\mathbf{n}}$:

$$v_{u,\mathrm{eff}}^{(n)}=\mathbf{v}_{u,\mathrm{eff}}\cdot\hat{\mathbf{n}}\,.$$

This scalar velocity is then used to determine the Mach number and apply the jump conditions.

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A.5. Orbital Dynamics and Affine Deformation

The cloud’s center–of–mass moves with

$$\mathbf{v}_{\rm orb}=v_0\,\hat{x},$$

and its deformation is modeled by the affine transformation

$$\mathbf{r}(t)=\mathbf{A}(t)\,\mathbf{r}_0\quad \text{with}\quad \mathbf{A}(t)= \begin{pmatrix} f_\parallel(t)&0&0\\[1mm] 0&f_\parallel(t)&0\\[1mm] 0&0&f_\perp(t) \end{pmatrix}\,.$$

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A.6. Tidal Acceleration in the Linear Approximation

Expanding the gravitational potential $\Phi(r)=-GM_{BH}/r$ about the periastron $r_p$, one obtains the tidal accelerations:

$$a_{\rm tidal}^\parallel\approx\frac{2GM_{BH}}{r_p^3}\,x,\quad a_{\rm tidal}^\perp\approx-\frac{GM_{BH}}{r_p^3}\,z\,.$$

Including a correction factor $(1-\eta)$ to account for the cloud’s self–gravity leads to

$$a_{\rm eff}^\parallel=\frac{2GM_{BH}}{r_p^3}(1-\eta)\,x,\quad a_{\rm eff}^\perp=-\frac{GM_{BH}}{r_p^3}(1-\eta)\,z\,.$$

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A.7. Induced Velocity Components

Because the cloud deforms affinely, differentiating

$$x(t)=f_\parallel(t)x_0$$

twice gives

$$\ddot{f}_\parallel(t)x_0=\frac{2GM_{BH}}{r_p^3}(1-\eta)f_\parallel(t)x_0,$$

or equivalently

$$\ddot{f}_\parallel(t)=\omega_\parallel^2\,f_\parallel(t),\quad \text{with}\quad \omega_\parallel=\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)}\,.$$

Similarly, in the $z$–direction,

$$\ddot{f}_\perp(t)=-\omega_\perp^2\,f_\perp(t),\quad \text{with}\quad \omega_\perp=\sqrt{\frac{GM_{BH}}{r_p^3}(1-\eta)}\,.$$

Thus, the tidal–induced velocities are

$$v_{\rm tidal}^x\sim f_\parallel(t)R\,\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)},\quad v_{\rm tidal}^y\sim f_\parallel(t)R\,\sqrt{\frac{2GM_{BH}}{r_p^3}(1-\eta)},$$ $$v_{\rm tidal}^z\sim f_\perp(t)R\,\sqrt{\frac{GM_{BH}}{r_p^3}(1-\eta)}\,.$$

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A.8. Self–Gravity Parameter

The self–gravity parameter is defined by

$$\eta=\frac{M_{\rm cl}}{M_{BH}}\Bigl(\frac{r_p}{R}\Bigr)^3\,,$$

which compares the cloud’s binding to the tidal force exerted by the black hole.

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A.9. The Vis–Viva Equation

The Vis–Viva equation,

$$v^2=GM_{BH}\Bigl(\frac{2}{r}-\frac{1}{a}\Bigr)\,,$$

is derived from equating the orbital energy expressions and is used to estimate the orbital velocity of the cloud.

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This revised version now includes the $y$–direction in the effective velocity vector and provides the vertical ($z$–direction) Mach number.