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

Abstract

We develop a comprehensive semi-analytical framework to describe how molecular clouds form tidal shocks under the influence of massive or intermediate-mass black holes. As a cloud nears the black hole, differential gravitational forces (“tidal forces”) stretch it along the radial direction and compress it in the transverse directions. This anisotropic deformation can induce internal collisions and strong shocks if the velocity differences become supersonic. We examine the conditions for shock formation, the role of radiative cooling, and the post-shock thermal state of the gas. A numerical example illustrates that even moderate bulk orbital velocities and black hole masses (~10^5 M_ $\odot$ ) can yield large Mach numbers in the normal (shock) direction, thereby generating strong shocks. We conclude with a discussion of pancake-like compression in the vertical direction and how repeated tidal encounters can further shape the cloud’s structure and fate.

1. Introduction

Molecular clouds in galactic nuclei can wander perilously close to supermassive or intermediate-mass black holes (BHs). When this happens, the intense gravitational gradient (tidal field) can distort and, at times, disrupt these clouds. This process is astrophysically significant for several reasons:

Shock Formation: Tidal stretching and shearing can force portions of the cloud to collide supersonically, forming shocks that heat the gas, create turbulence, and possibly trigger star formation in certain conditions.
Enhanced Accretion: Repeated shocks can dissipate the cloud’s orbital energy, allowing some fraction of the gas to become bound to the black hole and contribute to accretion flows.
Observational Evidence: High-velocity dispersions (tens of km/s to over a hundred km/s) in nearby molecular clouds have been linked to encounters with massive compact objects.

In this post, we present a theoretical and semi-analytical treatment of tidal shock formation, focusing on the self-consistent interplay of tidal forces, self-gravity, pressure support, and cooling. We then apply this formalism to a numerical example with an intermediate-mass black hole of $10^5\,M_\odot$ .

2. Background: Tidal Forces and Cloud Stability

2.1. Tidal Acceleration

Tidal forces arise because different parts of an extended cloud experience slightly different gravitational pulls from the black hole. In vector form, the gravitational field at position $\mathbf{r}$ from a point mass $M_{\bullet}$ is

\mathbf{g}(\mathbf{r}) = -\frac{G\,M_{\bullet}}{r^3}\,\mathbf{r}.

The tidal force per unit mass is given by the gradient (Jacobian) of $\mathbf{g}$ , which effectively stretches the cloud along the radial direction toward the black hole and compresses it in perpendicular directions. In simpler scalar form, if a cloud of radius $\Delta r$ is at distance $R$ from the black hole, the differential acceleration across the cloud is approximately

\Delta a_{\rm tidal} \,\approx\, \frac{2\,G\,M_{\bullet}}{R^3}\,\Delta r.

As $R$ decreases (the cloud gets closer), $\Delta a_{\rm tidal}$ can become large enough to either disrupt the cloud or produce internal collisions that generate shock fronts.

2.2. Cloud Survival Criterion

Whether a cloud survives an encounter depends on the competition between tidal forces and the cloud’s self-gravity (plus any internal pressure support). One often defines a tidal radius $R_{\rm tidal}$ by equating the black hole’s tidal force to the cloud’s self-gravity:

\frac{2\,G\,M_{\bullet}}{R_{\rm tidal}^3}\,R_c \;\sim\;\frac{G\,M_{\rm cloud}}{R_c^2},

where $R_c$ is the cloud’s radius. Rearranging gives $R_{\rm tidal} \,\sim\, R_c \sqrt[3]{\frac{M_{\bullet}}{M_{\rm cloud}}}.$
If the cloud passes well inside this radius (i.e., $R \ll R_{\rm tidal}$ ), it is typically disrupted. If $R$ is comparable to or larger than $R_{\rm tidal}$ , the cloud can survive long enough to experience partial compression and shock formation.

3. Tidal Shock Formation Mechanism

3.1. Shock Trigger by Differential Velocities

A tidal shock in the cloud occurs when different regions acquire sufficiently different velocities that, upon intersection, they produce supersonic collisions. Broadly:

Leading vs. Trailing Edge: As the cloud approaches the black hole, the leading edge (closest to the BH) accelerates more than the trailing edge.
Shearing & Collisions: If the cloud has nonzero impact parameter, parts of it swing around the BH on slightly different trajectories. This shearing can cause the cloud’s own gas streams to collide internally.
Supersonic Condition: For a genuine shock, the relative speed of colliding gas parcels must exceed the local sound speed $c_s$ , yielding a Mach number $\mathcal{M} > 1$ . The resulting shock waves convert kinetic energy into thermal energy and compressed gas.

3.2. Self-Similar Deformation and Velocity

One can model the cloud via a time-dependent scaling approach (an affine or homologous model). If the cloud is initially a sphere of radius $R$ , we write:

x(t) = f_\parallel(t)\,x_0, \quad y(t) = f_\parallel(t)\,y_0, \quad z(t) = f_\perp(t)\,z_0,

where $f_\parallel(t)$ captures in-plane stretching (or compression) and $f_\perp(t)$ describes the vertical dimension’s evolution. Mass conservation implies

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

Differentiating twice and including gravitational (self-gravity + tidal) and pressure terms yields ordinary differential equations for $f_\parallel$ and $f_\perp$ . In the linear tidal approximation about distance $r_p$ :

\ddot{f}_\parallel \;=\;\bigl(\lambda_\parallel - \omega_{\rm eff}^2\bigr)\,f_\parallel, \quad \ddot{f}_\perp \;=\;\bigl(\lambda_\perp - \omega_{\rm eff}^2\bigr)\,f_\perp,

where $\lambda_\parallel \approx \tfrac{2\,G\,M_{\bullet}}{r_p^3}$ , $\lambda_\perp \approx -\tfrac{G\,M_{\bullet}}{r_p^3}$ , and $\omega_{\rm eff}$ is a restoring frequency incorporating self-gravity (plus possibly pressure). The solutions describe an in-plane stretch (cosh-type growth if $\lambda_\parallel > \omega_{\rm eff}^2$ ) and a vertical compression (cos-type oscillation if $\omega_{\rm eff}^2 > \lambda_\perp$ ).

3.3. Effective Upstream Velocity

A key quantity for shock formation is the effective upstream velocity that impinges on the shock front. Suppose the cloud’s center of mass orbits with velocity $\mathbf{v}_{\rm orb}$ . Meanwhile, internal cloud elements gain additional velocity from the tidal deformation, $\mathbf{v}_{\rm tidal}$ . We combine them:

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

To find the Mach number associated with a potential shock surface of normal $\hat{\mathbf{n}}$ , one projects:

v_{u,\mathrm{eff}}^{(n)} \;=\; \mathbf{v}_{u,\mathrm{eff}}\cdot\hat{\mathbf{n}}, \quad \mathcal{M}^{(n)} \;=\; \frac{v_{u,\mathrm{eff}}^{(n)}}{c_s}.

A shock occurs if $\mathcal{M}^{(n)} > 1$ . Because the cloud’s deformation might also compress it vertically, we define a vertical Mach number $\mathcal{M}^{(z)}$ using the $z$ -component of tidal velocity:

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

3.4. Radiative Cooling and Post-Shock Temperature

In strong shocks without radiative cooling, the post-shock temperature $T_2$ scales with $v_{u,\mathrm{eff}}^{(n)2}$ . For molecular gas, if $v_{u,\mathrm{eff}}^{(n)}$ is tens of km/s, the adiabatic post-shock temperature could exceed several thousand K. However, if the cooling time $t_{\rm cool}$ is short (high density and efficient molecular/atomic line emission), the gas may remain near an equilibrium temperature $T_{\rm eq}$ . Formally,

T_{\rm eff} \;=\;\min\bigl\{T_2,\;T_{\rm eq}\bigr\}.

For molecules like CO or CO $_2$ to survive, one requires $T_{\rm eff} < T_{\rm diss} \sim 3000\text{ K}$ . Rapid cooling helps keep the temperature below dissociation thresholds, preserving molecular species. In dense molecular clouds, line emission can be extremely efficient once the density surpasses $10^5\text{–}10^6\ \mathrm{cm}^{-3}$ , leading to typical cooling times of days to weeks, much shorter than the dynamical timescale of years to centuries.

4. Numerical Example

4.1. Setup

We illustrate the formalism with a fiducial set of parameters:

Black Hole Mass: $M_{\bullet} = 10^5\,M_{\odot}$ .
Cloud Mass: $M_{\rm cl} = 1000\,M_{\odot}$ .
Cloud Radius: $R = 1.5\,\mathrm{pc}$ .
Pericenter Distance: $r_p = 3\,\mathrm{pc}$ .
Orbital Velocity: $\mathbf{v}_{\rm orb} \approx 10\,\mathrm{km/s}\,\hat{x}$ .
Initial Temperature: $T_{\rm cloud}=10\,\mathrm{K}$ (with equilibrium $T_{\rm eq} \approx 20\,\mathrm{K}$ ).

We assume the cloud approaches the black hole on a near-hyperbolic orbit with pericenter $r_p$ . Tidal deformations develop most strongly around pericenter.

4.2. Density and Self-Gravity

The initial uniform density is

\rho_0 \;=\;\frac{3\,M_{\rm cl}}{4\,\pi\,R^3}.

With the given numbers, $\rho_0 \approx 4.8\times10^{-22}\,\mathrm{g\,cm}^{-3}$ . The characteristic self-gravity frequency is

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

which is typically $\sim10^{-14}\,\mathrm{s}^{-1}$ for these parameters.

4.3. Tidal Velocities and Mach Numbers

At a characteristic time near pericenter, one can estimate the tidal velocity components as

v_{\rm tidal}^x \,\sim\, f_\parallel\,R\,\sqrt{\tfrac{2\,G\,M_{\bullet}}{r_p^3}(\,1-\eta\,)}, \quad v_{\rm tidal}^z \,\sim\, f_\perp\,R\,\sqrt{\tfrac{G\,M_{\bullet}}{r_p^3}(\,1-\eta\,)},

where $\eta$ is a “self-gravity correction”

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

For the chosen numbers, $\eta \approx 0.08$ . One finds that $v_{\rm tidal}^x$ and $v_{\rm tidal}^y$ can reach a few $\times10^5\,\mathrm{cm/s}$ (several km/s), with the cloud’s orbital velocity adding $\sim10^6\,\mathrm{cm/s}$ . Hence:

\mathbf{v}_{u,\mathrm{eff}} \;\approx\; (1.0\times10^6 + 4.8\times10^5)\,\hat{x} \;+\; (4.8\times10^5)\,\hat{y} \;+\; (1.2\times10^5)\,\hat{z} \;\mathrm{cm/s}.

Projecting onto $\hat{x}$ can yield normal speeds of $\sim1.5\times10^6\,\mathrm{cm/s}$ .

The sound speed at $T_{\rm eq}\approx 20\,\mathrm{K}$ is a few $\times10^4\,\mathrm{cm/s}$ , so the normal Mach number $\mathcal{M}^{(n)}$ can exceed 40. Even the vertical Mach number $\mathcal{M}^{(z)}$ may be a few, meaning the cloud is easily compressed from above and below.

4.4. Shock Heating and Cooling

In an adiabatic strong-shock limit (ignoring cooling), the post-shock temperature

T_2 \,\sim\, \frac{3\,\mu\,m_p}{16\,k_B}\,\bigl(v_{u,\mathrm{eff}}^{(n)}\bigr)^2

could be tens of thousands of Kelvin for $\sim\,10\,\mathrm{km/s}$ speeds, enough to dissociate molecules. However, at high densities the cooling time $t_{\rm cool}$ is drastically reduced. If the compression increases the density by factors of $10^2$ to $10^3$ , line emission from molecules (CO, H $_2$ , etc.) keeps the gas near a modest $\sim\,20\text{–}100\,\mathrm{K}$ . This effectively clamps the post-shock temperature to $T_{\rm eff} = \min\{T_2,\;T_{\rm eq}\}$ . For typical interstellar molecules, this ensures survival as long as the collision timescale does not drive the gas above 2000–3000 K for too long.

4.5. Vertical “Pancaking” and Oscillations

Tidal forces along the vertical ( $z$ ) direction cause a “pancaking” effect. In a quasi-static treatment, one balances tidal compression $\sim\frac{G\,M_{\bullet}}{r_p^3}\,z$ with pressure gradient $\sim c_s^2/H$ . This yields an approximate vertical thickness:

H \,\sim\, c_s\,\sqrt{\frac{r_p^3}{G\,M_{\bullet}}},

which can be much smaller than the cloud’s original size if $\tfrac{r_p}{R}$ is only modestly larger than $\sqrt[3]{\tfrac{M_{\bullet}}{M_{\rm cl}}}$ . In a dynamic treatment, the scale height $H(t)$ obeys

\ddot{H}(t) \;=\; -\,\frac{G\,M_{\bullet}}{r_p^3}\,H(t) \;+\;\frac{c_s^2}{H(t)},

so the cloud may oscillate about its equilibrium thickness with a characteristic frequency $\omega \approx \sqrt{\tfrac{2\,G\,M_{\bullet}}{r_p^3}}$ . This can lead to repeated compression/expansion cycles if the orbital time is comparable to the vertical dynamical time.

5. Discussion and Conclusions

We have presented a framework for tidal shock formation in molecular clouds near black holes:

Tidal Field & Cloud Deformation: As the cloud approaches the black hole, it experiences stretching along the radial direction and compression in perpendicular directions, described by simple affine scale factors $f_\parallel(t)$ and $f_\perp(t)$ .
Shock Trigger: Differential velocities within the cloud (the sum of orbital motion plus tidal-induced velocity) can reach supersonic values. Where these flows intersect, a shock forms, particularly at the leading or trailing edges (or along shear layers).
Cooling and Molecular Survival: Tidal compression can raise the density to levels where radiative cooling is extremely efficient. Even if the shock would nominally heat the gas to thousands of Kelvin, cooling can clamp the post-shock temperature to a modest $T_{\rm eq}$ . This allows molecules like CO, CO $_2$ , etc., to remain intact.
Vertical “Pancake” Structure: The tidal field in the vertical direction flattens the cloud into a pancake geometry. A quasi-static approximation predicts a scale height $H$ , while a dynamic model gives vertical oscillations about that equilibrium.
Example Outcome: A numerical scenario with $M_{\bullet} = 10^5\,M_\odot$ , $M_{\rm cl} = 10^3\,M_\odot$ , and an initial orbit of a few pc yields normal Mach numbers as high as 30–40. The post-shock temperature remains low if cooling is rapid, protecting molecules from dissociation.

Altogether, these results demonstrate that tidal shock formation is a plausible mechanism for producing strong, supersonic compression and significant internal heating in molecular clouds near black holes — but with substantial radiative energy losses that keep the gas from fully ionizing or destroying all molecules. Repeated encounters (for instance, on eccentric orbits) may strip more kinetic energy over time, allowing partial capture of the cloud or fragmentation into smaller clumps. Observationally, bright emission lines and large velocity spreads within certain central molecular clouds can be explained by exactly these tidal shock processes.

Acknowledgments

Referenced Observations: Some aspects of tidal shock formation are supported by studies of unusual molecular clouds near galactic centers, such as CO–0.40–0.22, which shows velocity dispersions on the order of 100 km/s that can be explained by an encounter with an unseen ~ $10^5\,M_\odot$ BH.
Numerical Simulations: The theoretical formalism here aligns with hydrodynamic simulations showing how shearing flows in tidally stretched clouds collide, forming shock fronts that dissipate orbital energy and drive turbulence.

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)}\,.

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}\,.

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\,.

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.

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}\,.

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\,.

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)}\,.

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.

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.

Below is a strict integral derivation of the gravitational field inside a uniform spherical cloud via first principles (rather than simply quoting Newton’s shell theorem). We then apply it to derive the cloud’s self-gravitational acceleration at its surface and show how it leads to the same expression for (\eta).

Appendix: 3D Integral for Newton’s Shell Theorem and Derivation of $\eta$

A.1. Gravitational Field Inside a Uniform Sphere (Integral Approach)

Consider a uniform sphere of total mass $M_{\mathrm{cl}}$ and radius $R$ . Let its (constant) density be

\rho \;=\;\frac{M_{\mathrm{cl}}}{\tfrac{4\pi}{3}R^3}.

We wish to compute the gravitational field $\mathbf{g}$ at a point located a distance $r \le R$ from the center of the sphere.

Geometry Setup:
- Place the center of the sphere at the origin $O$ .
- Let the field point $P$ lie on the $z$ -axis at coordinates $(0,0,r)$ .
- We will integrate over the sphere in spherical coordinates $(\rho,\theta,\phi)$ , where $\rho$ is the radial distance from $O$ , $\theta$ the polar angle measured from the $z$ -axis, and $\phi$ the azimuthal angle about the $z$ -axis. The volume element is $dV \;=\;\rho^2 \sin\theta\,d\rho\,d\theta\,d\phi.$
- Each volume element contributes a gravitational acceleration $d\mathbf{g}$ at $P$ .
Expression for $d\mathbf{g}$ :
A small mass element $dm = \rho\,dV$ at spherical coordinates $(\rho,\theta,\phi)$ has a gravitational effect on $P$ given by
$d\mathbf{g} \;=\; -\,G\,\frac{dm}{r'}\,\frac{\mathbf{r}'}{|\mathbf{r}'|},$
where $\mathbf{r}'$ is the displacement from the mass element to $P$ , and $r' = |\mathbf{r}'|$ is its magnitude. Here, the minus sign indicates attraction toward the mass element.
- Key simplification: by symmetry, the horizontal (x,y) components of $d\mathbf{g}$ from opposite sides of the $z$ -axis will cancel out. Only the $z$ -component survives.
- Let $z'$ be the coordinate of the volume element along the $z$ -axis (with $\theta$ the angle from the axis). One can show that the net gravitational field at $P$ points along the $z$ -axis.
Splitting Into Two Regions:
Traditional derivations separate the sphere into:
- An inner sphere of radius $\rho \le r$ .
- A spherical shell of radius $\rho>r$ .
One shows that the net contribution from any shell of radius $\rho>r$ cancels out inside that shell (the classic shell theorem). Only the mass with $\rho \le r$ contributes net acceleration. We can replicate that logic by direct integration, but it is more illuminating to do the integral piecewise and see that shells outside $\rho>r$ sum to zero.
Inside Contribution ( $\rho \le r$ ):
The mass enclosed within radius $r$ acts as if it were a point mass at the origin with mass
$M_{\mathrm{enc}}(r) \;=\; \frac{4\pi}{3}\,\rho\,r^3.$
Hence, inside that region, the net acceleration at $P$ is
$g(r) \;=\; G\,\frac{M_{\mathrm{enc}}(r)}{r^2} \;=\; G\,\frac{\tfrac{4\pi}{3}\rho\,r^3}{r^2} \;=\; \frac{4\pi}{3}\,G\,\rho\,r.$
This is the linear dependence on $r$ familiar from Newton’s shell theorem. In vector form, for $\mathbf{r}=r\,\hat{z}$ ,
$\mathbf{g}(r) \;=\; -\,\frac{4\pi}{3}\,G\,\rho\,r\,\hat{z}.$
Outer Shells ( $\rho>r$ ):
A spherical shell of uniform density outside radius $r$ contributes no net force inside it (Newton’s shell theorem). The integral of the shell’s gravitational pull in all directions exactly cancels at the point $P$ . This result can also be shown directly by integrating the shell’s mass distribution, but it is standard that the net gravitational field is zero everywhere inside a uniform spherical shell.

Therefore, for a uniform sphere of radius $R$ , the gravitational field at $r\le R$ is only due to the inner sphere of radius $r$ . At the surface $r=R$ ,

g(R) \;=\; \frac{4\pi}{3}\,G\,\rho\,R.

Since $\rho = \tfrac{3\,M_{\mathrm{cl}}}{4\pi\,R^3}$ , we get

g(R) \;=\; \frac{4\pi}{3}\,G\,\Bigl(\tfrac{3\,M_{\mathrm{cl}}}{4\pi\,R^3}\Bigr)\,R \;=\; \frac{G\,M_{\mathrm{cl}}}{R^2}.

This matches the simpler result that the entire mass of the sphere can be thought of as concentrated at the center for evaluating the field at the boundary $r=R$ .

A.2. Self-Gravity Versus Tidal Field

Having established that the inward self-gravitational acceleration at the sphere’s surface is

a_{\mathrm{sg}}(R) \;=\; g(R) \;=\; \frac{G\,M_{\mathrm{cl}}}{R^2},

we now compare this to the tidal acceleration from a black hole of mass $M_{\mathrm{BH}}$ at distance $r_p$ .

A.2.1. Black Hole Tidal Expansion

Expanding the BH potential $\Phi_{\mathrm{BH}}$ about the cloud’s center (i.e., for small displacements $\le R$ relative to $r_p$ ) yields a differential acceleration across the cloud of order

a_{\mathrm{tid}} \;\sim\; \frac{G\,M_{\mathrm{BH}}}{r_p^3}\,R.

(Linearizing $\Phi_{\mathrm{BH}}\propto -M_{\mathrm{BH}}/|\mathbf{r}|$ about $|\mathbf{r}|=r_p$ , the first nontrivial term in the Taylor series is $\propto x\,\frac{d^2}{dx^2}(1/r)$ , giving $\sim \tfrac{G\,M_{\mathrm{BH}}\,R}{r_p^3}$ .)

A.2.2. The Ratio and Definition of $\eta$

Thus the ratio of the cloud’s inward self-gravity at its surface to the BH’s tidal pull across that radius is:

\frac{a_{\mathrm{sg}}(R)}{a_{\mathrm{tid}}(R)} \;=\; \frac{\tfrac{G\,M_{\mathrm{cl}}}{R^2}} {\tfrac{G\,M_{\mathrm{BH}}}{r_p^3}\,R} \;=\; \frac{M_{\mathrm{cl}}}{M_{\mathrm{BH}}} \;\Bigl(\frac{r_p}{R}\Bigr)^3.

We define

\boxed{ \eta \;=\; \frac{M_{\mathrm{cl}}}{M_{\mathrm{BH}}} \;\Bigl(\frac{r_p}{R}\Bigr)^3 \;=\; \frac{a_{\mathrm{sg}}(R)}{a_{\mathrm{tid}}(R)}. }

A.3. Physical Interpretation

$\eta\ll1$ : Tidal forces exceed the cloud’s self-gravity, making the cloud susceptible to extreme tidal deformation or disruption.
$\eta\gg1$ : The cloud’s own gravity dominates; the black hole’s tidal effects at radius $R$ are relatively mild.
$\eta \sim 1$ : Comparable influences from self-gravity and tidal pull, fostering strong tidal shocks without outright disruption.

Hence, the uniform-sphere integral approach confirms that Newton’s shell theorem yields $g(R)=\tfrac{G\,M_{\mathrm{cl}}}{R^2}$ at the surface, and combining this with the linearized BH tidal acceleration leads rigorously to

\eta \;=\; \frac{M_{\mathrm{cl}}}{M_{\mathrm{BH}}} \bigl(\tfrac{r_p}{R}\bigr)^3.