The Effect of Star–Disk Interactions on Highly Eccentric Stellar Orbits in Active Galactic Nuclei: A Disk Loss Cone and Implications for Stellar Tidal Disruption Events

We model the stellar cluster surrounding the supermassive black hole of mass ${M}_{\mathrm{bh}}$ as follows. We adopt the simplification that stars share a single stellar mass, ${M}_{* }$, and stellar radius, ${R}_{* }$, and a spherical, power-law stellar distribution.

The stellar number density is arranged according to a power-law distribution in radius,

$$\begin{eqnarray}&&{\nu }_{* }={\nu }_{{\rm{m}}}{\left(\displaystyle \frac{r}{{r}_{{\rm{m}}}}\right)}^{-\gamma },\end{eqnarray} \tag{ 1 }$$

with γ = 1.5, and where ${r}_{{\rm{m}}}$ is the radius that encloses twice the black hole mass in stars (Merritt 2013, Equation (2.11)). The normalization (Merritt 2013, Equation (3.48)),

$$\begin{eqnarray}&&{\nu }_{{\rm{m}}}=\displaystyle \frac{3-\gamma }{2\pi }\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{* }}{r}_{{\rm{m}}}^{-3},\end{eqnarray} \tag{ 2 }$$

assures that the integrated stellar mass is $2{M}_{\mathrm{bh}}$ within ${r}_{{\rm{m}}}$. To set the length scale of ${r}_{{\rm{m}}}$, we turn to the black hole mass–velocity dispersion relation, which implies that the typical velocity dispersion is

$$\begin{eqnarray}&&{\sigma }_{{\rm{h}}}=2.3\ \mathrm{km}\ {{\rm{s}}}^{-1}{\left({M}_{\mathrm{bh}}/{M}_{\odot }\right)}^{1/4.38},\end{eqnarray} \tag{ 3 }$$

with the numerical values derived from the fit of Kormendy & Ho (2013). This means that the black hole is the dominant influence on stellar motions within an influence radius,

$$\begin{eqnarray}&&{r}_{{\rm{h}}}=\displaystyle \frac{{{GM}}_{\mathrm{bh}}}{{\sigma }_{{\rm{h}}}^{2}},\end{eqnarray} \tag{ 4 }$$

where we follow the notation of Merritt (2013, Equation (2.12)). We will assume that ${r}_{{\rm{m}}}={r}_{{\rm{h}}}$, thus setting the normalization of the stellar density profile.

We adopt the Keplerian limit in which the black hole dominates the gravity in which stars orbit. The orbital period is, therefore, $P=2\pi {\left({r}^{3}/{{GM}}_{\mathrm{bh}}\right)}^{1/2}$ for orbits of semimajor axis a = r. If the stellar distribution function is isotropic in angular momentum space, the distribution function depends on specific energy (which is defined positive),

$$\begin{eqnarray}&&{ \mathcal E }=\displaystyle \frac{{{GM}}_{\mathrm{bh}}}{2a},\end{eqnarray} \tag{ 5 }$$

only, and simplifies to a power law,

$$\begin{eqnarray}&&f({ \mathcal E })={f}_{0}{{ \mathcal E }}^{\gamma -\displaystyle \frac{3}{2}},\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&{f}_{0}={\left(2\pi \right)}^{-\displaystyle \frac{3}{2}}{\nu }_{{\rm{m}}}{{\rm{\Phi }}}_{0}^{-\gamma }\displaystyle \frac{{\rm{\Gamma }}(\gamma +1)}{{\rm{\Gamma }}(\gamma -\tfrac{1}{2})},\end{eqnarray} \tag{ 7 }$$

in which ${{\rm{\Phi }}}_{0}={{GM}}_{\mathrm{bh}}/{r}_{{\rm{m}}}$ (Vasiliev & Merritt 2013).

The three-dimensional stellar velocity dispersion for such a structure is approximately

$$\begin{eqnarray}&&{\sigma }^{2}\approx \displaystyle \frac{{{GM}}_{\mathrm{bh}}}{(1+\gamma )r}.\end{eqnarray} \tag{ 8 }$$

The two-body stellar-relaxation time is

$$\begin{eqnarray}&&{t}_{\mathrm{rel}}\approx \displaystyle \frac{0.34{\sigma }^{3}}{{G}^{2}{M}_{* }^{2}{\nu }_{* }\mathrm{ln}{\rm{\Lambda }}},\end{eqnarray} \tag{ 9 }$$

where $\mathrm{ln}\,{\rm{\Lambda }}\approx \mathrm{ln}\left({M}_{\mathrm{bh}}/{M}_{* }\right)$ is the Coulomb logarithm (Merritt 2013). Over this timescale, the energy and angular momentum of stellar orbits in the cluster are randomized by the cumulative effect of two-body scatterings. Over an orbit, the magnitude of the typical root mean square (rms) change in energy is

$$\begin{eqnarray}&&{\rm{\Delta }}{{ \mathcal E }}_{\mathrm{rel}}\approx { \mathcal E }{\left(\displaystyle \frac{P}{{t}_{\mathrm{rel}}}\right)}^{1/2},\end{eqnarray} \tag{ 10 }$$

where $P\approx 2\pi {\left({r}^{3}/{{GM}}_{\mathrm{bh}}\right)}^{1/2}$ is the orbital period of a stellar orbit with semimajor axis r. Similarly, the magnitude of the rms change in angular momentum is

$$\begin{eqnarray}&&{\rm{\Delta }}{J}_{\mathrm{rel}}\approx {J}_{{\rm{c}}}{\left(\displaystyle \frac{P}{{t}_{\mathrm{rel}}}\right)}^{1/2},\end{eqnarray} \tag{ 11 }$$

where ${J}_{{\rm{c}}}={\left({{GM}}_{\mathrm{bh}}r\right)}^{1/2}$ is the circular angular momentum for semimajor axis equal to r.

An accretion disk coexists with the nuclear stellar cluster in the AGN nucleus. Here we describe our disk model.

We adopt a disk model in which the disk has constant mass flux, $\dot{M}$; constant Toomre Q parameter, describing its susceptibility to gravitational instability; and constant α, parameterizing the efficiency of instability-driven "viscosity" in the disk (for more description of the role of α, see, for example, the review of Papaloizou & Lin 1995).

We parameterize the disk accretion rate in terms of the Eddington mass accretion rate, ${\dot{M}}_{\mathrm{Edd}}={L}_{\mathrm{Edd}}/\eta {c}^{2}$, or

$$\begin{eqnarray}&&\dot{M}=\lambda {\dot{M}}_{\mathrm{Edd}}=0.22\,\lambda \left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{10}^{7}\,{M}_{\odot }}\right)\,{M}_{\odot }\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 12 }$$

where we have adopted η = 0.1 and used ${L}_{\mathrm{Edd}}=4\pi {{GM}}_{\mathrm{bh}}{m}_{{\rm{p}}}c/{\sigma }_{\mathrm{es}}$, in which m_p is the proton mass and σ_es is the Thompson cross section to electron scattering.

The disk surface density is given by the mass flux and the radial velocity of material through the disk,

$$\begin{eqnarray}&&{\rm{\Sigma }}=\displaystyle \frac{\dot{M}}{2\pi {{rv}}_{r}},\end{eqnarray} \tag{ 13 }$$

where the radial velocity, ${v}_{r}=\alpha {h}^{2}{({{GM}}_{\mathrm{bh}}/r)}^{1/2}$, and where h = H/R is the dimensionless disk scale height (Papaloizou & Lin 1995). Under these assumptions, the average volume density within the disk is approximately

$$\begin{eqnarray}&&\rho \approx \displaystyle \frac{{\rm{\Sigma }}}{2H}.\end{eqnarray} \tag{ 14 }$$

The dimensionless scale height, h, is given by the relation

$$\begin{eqnarray}&&{h}^{3}\approx \displaystyle \frac{Q}{2\alpha }\displaystyle \frac{\dot{M}}{{M}_{\mathrm{bh}}\ {\rm{\Omega }}},\end{eqnarray} \tag{ 15 }$$

where ${\rm{\Omega }}={\left({{GM}}_{\mathrm{bh}}{/r}^{3}\right)}^{1/2}$ is the disk angular frequency. Under these conditions, the surface density can also be written in terms of the critical surface density for gravitational instability,

$$\begin{eqnarray}&&{\rm{\Sigma }}=\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{c}}}}{Q}=\displaystyle \frac{h}{Q}\displaystyle \frac{{M}_{\mathrm{bh}}}{\pi {r}^{2}}=\displaystyle \frac{1}{{\left(2\alpha \right)}^{1/3}{Q}^{2/3}}\displaystyle \frac{{M}_{\mathrm{bh}}^{2/3}{\dot{M}}^{1/3}}{\pi {r}^{2}{{\rm{\Omega }}}^{1/3}}.\end{eqnarray} \tag{ 16 }$$

Thus, the scaling of Σ with disk parameters is Σ ∝ α^−1/3 λ^1/3Q^−2/3. Figure 1 shows example disk profiles for the model in which α = λ = Q = 1, which we consider in Section 3.

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We adopt an orientation such that the disk is in the X−Y plane, with angular momentum in the +Z-direction. The disk velocity, ${{\boldsymbol{V}}}_{\mathrm{disk}}$, at position ${\boldsymbol{X}}$ is set by the circular Keplerian velocity at cylindrical radius $R=\sqrt{{X}^{2}+{Y}^{2}}$.

We define stellar orbits on the basis of the orbital elements: semimajor axis, a; eccentricity, e; longitude of the ascending node, Ω; argument of periapsis, ω; inclination I; and true anomaly, f. Together, this set of elements allows complete specification of the orbital position and velocity (Murray & Dermott 1999).

The ranges of the orbital elements are $0\lt a\lt \infty$ and e ≤ 0 < 1, implying bound orbits, −π < Ω ≤ π, −π < ω ≤ π, and 0 ≤ I < π. The magnitude of the specific orbital energy is ${ \mathcal E }$, and the orbital vector angular momentum is

$$\begin{eqnarray}&&{\boldsymbol{J}}={{\boldsymbol{X}}}_{* }\times {{\boldsymbol{V}}}_{* },\end{eqnarray} \tag{ 17 }$$

where ${{\boldsymbol{X}}}_{* }$ and ${{\boldsymbol{V}}}_{* }$ are the stellar position and velocity in the reference frame. Inclinations 0 ≤ I < π/2 are "prograde," implying that orbit and disk angular momentum vectors both have positive Z-components, while orbits π/2 < I < π are "retrograde."

We will work in the approximation that the disk is thin (h ≪ 1), in which case disk crossings occur over a small arc length of the orbit near the nodes. We will, therefore, consider positions and relative velocities to be those evaluated at the disk midplane or node crossings, but we note that this approximation would not be appropriate for a thick disk. Because we are interested in the positions and velocities of node crossings, in which the stellar orbit transects the disk plane, we begin with the true anomaly of node crossings, f = −ω and f = −ω + π. Given these, we compute positions and velocities in the plane of the individual stellar orbit, then rotate these to the reference frame.

Node crossings correspond to Z = 0 in the reference frame. Using the orbital elements to compute the position, ${{\boldsymbol{X}}}_{* }$, and velocity, ${{\boldsymbol{V}}}_{* }$, of the star at node crossings, we then apply the drag force resulting from relative motion between the star and the disk. The relative velocity is

$$\begin{eqnarray}&&{{\boldsymbol{V}}}_{\mathrm{rel}}={{\boldsymbol{V}}}_{* }-{{\boldsymbol{V}}}_{\mathrm{disk}}.\end{eqnarray} \tag{ 18 }$$

Given the magnitude of a drag force, F_drag, the change in orbital velocity, in the impulse approximation, is

$$\begin{eqnarray}&&{\rm{\Delta }}{\boldsymbol{V}}=-\displaystyle \frac{{F}_{\mathrm{drag}}}{{M}_{* }}{\rm{\Delta }}{t}_{\mathrm{cross}}\displaystyle \frac{{{\boldsymbol{V}}}_{\mathrm{rel}}}{| {{\boldsymbol{V}}}_{\mathrm{rel}}| },\end{eqnarray} \tag{ 19 }$$

in which the crossing time through the disk is

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{cross}}=\displaystyle \frac{2H}{{V}_{* ,Z}}.\end{eqnarray} \tag{ 20 }$$

Finally, the drag force may be set by either the gravitational cross section or geometric cross section of the star, depending on the relative velocity of the star through the gas. To order of magnitude, these forces are

$$\begin{eqnarray}&&{F}_{\mathrm{drag},\mathrm{grav}}\approx \displaystyle \frac{4\pi {\left({{GM}}_{* }\right)}^{2}\rho }{| {{\boldsymbol{V}}}_{\mathrm{rel}}{| }^{2}}\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&{F}_{\mathrm{drag},\mathrm{geo}}\approx \pi {R}_{* }^{2}\rho | {{\boldsymbol{V}}}_{\mathrm{rel}}{| }^{2}.\end{eqnarray} \tag{ 22 }$$

The resultant drag force is the maximum of these two,

$$\begin{eqnarray}&&{F}_{\mathrm{drag}}=\max \left({F}_{\mathrm{drag},\mathrm{grav}},{F}_{\mathrm{drag},\mathrm{geo}}\right),\end{eqnarray} \tag{ 23 }$$

where, in general, the geometric cross section dominates when $| {{\boldsymbol{V}}}_{\mathrm{rel}}| \gg {\left({{GM}}_{* }/{R}_{* }\right)}^{1/2}$ and the gravitational cross section is important when $| {{\boldsymbol{V}}}_{\mathrm{rel}}| \ll {\left({{GM}}_{* }/{R}_{* }\right)}^{1/2}$.

To illustrate how eccentric orbits are modified by drag forces as they pass through the disk, we select a representative Sun-like star (${M}_{* }={M}_{\odot }$ and ${R}_{* }={R}_{\odot }$) and place it in an orbit in which its periapse distance is 30 times the tidal disruption radius,

$$\begin{eqnarray}&&{r}_{{\rm{t}}}={\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{* }}\right)}^{1/3}{R}_{* }.\end{eqnarray} \tag{ 24 }$$

If we write ${r}_{{\rm{p}}}={\beta }^{-1}{r}_{{\rm{t}}}$, then β = 1/30 for this orbit. We select an initial semimajor axis inside the black hole's sphere of influence by a factor of 10, such that ${a}_{0}=0.1{r}_{{\rm{h}}}\ \approx 0.52$ pc. Together, properties imply an initial eccentricity of e ≈ 0.9997. Given these fixed orbital dimensions, we randomize the orientation relative to the disk by generating 10⁴ samples of the orbital elements Ω, ω, and I. We note that because the system is axisymmetric, we expect outcomes to be independent of Ω. From the initial random distribution, we remove any orbits that are inclined such that they pass within a scale height of the disk plane. Because h is an increasing function of r as seen in Figure 1, we evaluate this condition at apoapse, removing orbits that satisfy $| \tan I| \lt h({r}_{\mathrm{apo}})$, where r_apo = a(1 + e).

Figure 3 examines the distributions of orbital elements that result from a single orbital cycle given initially isotropic orientations. We imagine the orbit starting at apoapse, passing through the disk once on the way to its periapse passage, and again after periapse (as shown in Figure 2). Figure 3 labels the corresponding distributions of orbital elements: initial, at periapse (after one disk crossing), and final (after the second disk crossing).

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In Figure 3, we observe that the orientation of this eccentric orbit is essentially unmodified by the disk crossings. The angles Ω, ω, and I all remain within 1% of their initial quantities. By contrast, the semimajor axis and eccentricity both decrease significantly. The periapse distance ${r}_{{\rm{p}}}$, is mildly affected, with most orbits showing slightly smaller ${r}_{{\rm{p}}}$ and a few acquiring larger ${r}_{{\rm{p}}}$.

We can qualitatively understand the relative magnitude of these changes in terms of the orbital and disk-crossing geometry. Disk crossings occur very close to periapse for an eccentric orbit. As a result, any drag force applied can significantly modify the orbital energy (this manifests as a decrease in orbital semimajor axis and eccentricity). However, the torques applied to the orbit are relatively small because of the short lever arm of the small periapse distance, leading to comparatively minor changes in the orbit's vectorial angular momentum, Equation (17). Therefore, the angular momentum's magnitude, which determines ${r}_{{\rm{p}}}$, and direction, which determines Ω, ω, and I, both experience little modification.

Rauch (1995) showed that as orbits become less eccentric (e ≲ 0.9), the angular torques become comparable to the orbital energy dissipation. This results in a decrease in orbital inclination relative to the disk as an orbit circularizes over subsequent passages. Thus, while orientation is initially unaffected while the orbit is still eccentric, it eventually decreases and stars are entrained into the disk plane if they fully circularize.

One clear conclusion from Figure 3 is that a distribution of outcomes is realized depending on the orbital orientation. Focusing on the ratio of final-to-initial orbital semimajor axis, a_f/a₀, Figure 4 illustrates some of the dependence of outcome on orientation. We see that orbits that near the disk plane, $I\to 0$ and $I\to \pi$, experience the most dramatic effect from disk interaction. These trajectories have low $\hat{Z}$ velocities, and therefore pass the slowest through the disk material (Equation (20)); as a result, they traverse the most significant column of disk mass each orbit. Among out-of-plane orbits, those that are retrograde generally experience a greater reduction of the semimajor axis. This is attributable to the larger relative velocity between the disk gas and the star that results from retrograde motion and the resulting larger drag force in the geometric limit.

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Figure 3 also highlights the dependence of a_f/a₀ on the argument of periapse, ω. We see that the reduction of the semimajor axis is maximized when $\omega \to \pm \pi /2$. In these orientations, the two disk crossings are roughly equidistant from the black hole, maximizing the amount of dense inner disk material that the star intercepts. As $\omega \to 0$ or ±π, the orbit and disk geometry is such that there is one disk crossing near periapse and one near apoapse. As expected, we do not observe a dependence on Ω, which amounts to azimuthal rotation relative to the axisymmetric disk.

The preceding analysis has shown that highly eccentric orbits can be modified, in some cases dramatically, by their interaction with the accretion disk. Here, we compare the magnitude of these disk-related changes to those that arise from two-body scattering of stellar orbits, which occurs continuously in galactic nuclei as stars trace orbits influenced by both the black hole and the surrounding stellar cluster.

Figure 5 compares the rms changes in orbital energy and angular momentum from two-body relaxation (per orbit, Equations (10) and (11), respectively) to the impact of the disk. For disk quantities, we show the median value and the 5%–95% region of results, marginalizing over orbital orientation. We show orbits of three representative eccentricities and a range of semimajor axes relative to the black hole sphere of influence radius, ${r}_{{\rm{h}}}$.

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At fixed eccentricity, the most compact orbits (smallest a₀) have the largest ${\rm{\Delta }}{ \mathcal E }/{ \mathcal E }$ and $| {\rm{\Delta }}J| /{J}_{c}$. These orbits also have the smallest periapse distances and pass through the highest-density inner regions of the disk. As previously discussed, the magnitude of changes relative to orbital energy from the disk is larger than those to orbital angular momentum. The change in slope seen in the various disk lines occurs due to the transition from geometrically dominated drag cross section (small a₀) to gravitationally dominated drag cross section (large a₀, where the orbital velocity is lower).

Figure 5 implies that for most of the phase space of stellar orbits, two-body relaxation is the dominant dynamical mechanism driving the evolution of orbital energy and angular momentum relaxation. For orbits within the black hole sphere of influence, ${a}_{0}\lesssim {r}_{{\rm{h}}}\ \sim 5$ pc, the disk becomes a significant driver of orbital energy change when orbits reach eccentricities e ≳ 0.99. By comparison, the disk rarely dominates orbital angular momentum evolution.

To give a sense of the approximate form of the results reported above and their scalings to other orbital, black hole, and accretion disk properties, we derive order of magnitude estimates here for the disk changes to orbital energy and angular momentum. For related derivations, see Syer et al. (1991) and Section 2.2 of Miralda-Escudé & Kollmeier (2005).

The disk mass intersected by a crossing star is key in estimating the magnitude of changes to orbital energy and angular momentum. For the highly eccentric orbits we consider here, the geometric cross section is typically appropriate, and, therefore the disk mass intersected in a passage is

$$\begin{eqnarray}&&{\rm{\Delta }}m\approx \pi {R}_{* }^{2}{\rm{\Sigma }},\end{eqnarray} \tag{ 25 }$$

where we have adopted the simplification of the star's orbit passing perpendicular to the disk plane. Thus, a key quantity to consider is the stellar mean surface density ${M}_{* }/{R}_{* }^{2}$, as compared to Σ. Main-sequence stars of different masses or stars at differing evolutionary states will have different ${M}_{* }/{R}_{* }^{2}$, as we will consider below. Expanding Equation (25) in terms of the disk surface density, we find

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}\approx \displaystyle \frac{h}{Q}{\left(\displaystyle \frac{{R}_{* }}{{r}_{{\rm{p}}}}\right)}^{2}\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{* }}\right),\end{eqnarray} \tag{ 26 }$$

where ${r}_{{\rm{p}}}$ is the periapse radius where the star crosses the disk. Substituting representative parameters for a thin disk and periapse distance of $1000{R}_{* }$,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}\approx {10}^{-3}\left(\displaystyle \frac{h/Q}{{10}^{-3}}\right){\left(\displaystyle \frac{{R}_{* }/{r}_{{\rm{p}}}}{{10}^{-3}}\right)}^{2}\left(\displaystyle \frac{{M}_{\mathrm{bh}}/{M}_{* }}{{10}^{6}}\right).\end{eqnarray} \tag{ 27 }$$

This expression implies that when the periapse distance is relatively small, the star can intersect a significant fraction of its own mass in a single disk crossing.

We can rewrite the periapse distance in terms of the tidal disruption radius for stars of a given mass and radius,

$$\begin{eqnarray}&&{r}_{{\rm{p}}}={\beta }^{-1}{r}_{{\rm{t}}}={\beta }^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{* }}\right)}^{1/3}{R}_{* },\end{eqnarray} \tag{ 28 }$$

where the dimensionless impact parameter is $\beta ={r}_{{\rm{t}}}/{r}_{{\rm{p}}}$. Substituting this into Equation (26) to parameterize the disk mass intersected by stars with periapse distance a fixed multiple of their tidal disruption radius, we have

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}\approx \displaystyle \frac{h}{Q}{\beta }^{2}{\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{* }}\right)}^{1/3}.\end{eqnarray} \tag{ 29 }$$

Therefore, in this form, we note that the intersected mass scales with the square of the dimensionless impact parameter, β (becoming larger as the periapse distance becomes smaller), and only weakly with the ratio of black hole to stellar mass.

From Equation (19), we can estimate the change in orbital energy and angular momentum that results from a disk crossing. To do so, we make several approximations. We imagine that the star crosses the disk only once, at periapse. We further approximate the relative velocity between the star and the disk to be the periapse velocity (equivalent to disk gas that is at rest rather than orbiting), and assume that the crossing is perpendicular to the disk plane. Given these approximations, the magnitude of the change in specific energy is

$$\begin{eqnarray}&&{\rm{\Delta }}{ \mathcal E }\approx 2H\displaystyle \frac{{F}_{\mathrm{drag}}}{{M}_{* }}=\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}{v}_{{\rm{p}}}^{2},\end{eqnarray} \tag{ 30 }$$

where 2H is the distance traversed as the star crosses the disk and where we have used a periapse velocity, ${v}_{{\rm{p}}}\approx \sqrt{2{{GM}}_{\mathrm{bh}}/{r}_{{\rm{p}}}}$, for a highly eccentric orbit. Similarly, the change in specific angular momentum is

$$\begin{eqnarray}&&{\rm{\Delta }}J\approx -\displaystyle \frac{{F}_{\mathrm{drag}}}{{M}_{* }}\displaystyle \frac{2H}{{v}_{{\rm{p}}}}{r}_{{\rm{p}}}=-\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}{r}_{{\rm{p}}}\ {v}_{{\rm{p}}},\end{eqnarray} \tag{ 31 }$$

where 2H/v_p ≈ Δt_cross. The fractional change in specific energy is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{ \mathcal E }}{{ \mathcal E }}\approx 4\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}\displaystyle \frac{a}{{r}_{{\rm{p}}}}=4\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}{\left(1-e\right)}^{-1}.\end{eqnarray} \tag{ 32 }$$

Here, the positive sign indicates that orbits become more bound following interaction with the disk (higher ${ \mathcal E }$). The fractional change due to loss of specific angular momentum is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}J}{J}\approx -\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}.\end{eqnarray} \tag{ 33 }$$

In Figure 5, we apply Equations (30) and (31) (plotted with dotted lines). We see that these simple expressions provide an excellent description of the median value of the numerical results, provided that the passage is in the geometrically dominated limit (as assumed in Equation (26)). We also note here that comparison to Figure 5 gives a sense of the spread about the median due to differing orbital orientations.

The preceding analysis indicates that for very eccentric orbits, the fractional change in orbital energy is always larger than the fractional change in orbital angular momentum, by a factor $a/{r}_{{\rm{p}}}={(1-e)}^{-1}$. This implies that orbits will become more circular under the influence of crossing the disk at periapse, as we have previously observed with our numerical approach.

We can observe from Equations (25) and (29) that the magnitude of the disk effect depends linearly on the disk surface density (or, equivalently h/Q) and on the inverse square of the periapse distance relative to the given star's tidal radius. Because disk surface density depends on our model parameters as Σ ∝ α^−1/3 λ^1/3Q^−2/3, the mass intercepted is proportional to $\dot{M}$, or equivalently, the fraction of the Eddington mass accretion rate λ.

More massive black holes will have a larger ${M}_{\mathrm{bh}}$/${M}_{* }$ ratio. Equation (29) shows that this yields mildly larger intercepted masses from disk crossings at fixed β, but the relaxation time (9) is also longer in stellar clusters surrounding more massive black holes. This indicates that disk crossings are more likely to be of significance in systems with high black hole masses than in systems with low black hole mass relative to other stellar dynamical processes.

When the stellar-mass object is a compact object rather than a star, its geometric cross section is reduced accordingly with its compact radius. For illustration, let us take the case of a stellar-mass black hole, for which the gravitational-focus cross section will dominate for all orbital velocities (because the relative star–disk velocity is always less than the speed of light). In this limit, we find that the magnitude of changes to orbital properties is very small, even in relatively extreme configurations of close-in or very eccentric orbits. The simplest way to illustrate this is following the order-of-magnitude scalings of the previous subsection.

In the gravitational-focus limit, the intercepted disk mass is

$$\begin{eqnarray}&&{\rm{\Delta }}m\approx \pi {R}_{{\rm{a}}}^{2}{\rm{\Sigma }},\end{eqnarray} \tag{ 34 }$$

where ${R}_{{\rm{a}}}=2G\,{M}_{* }/| {{\boldsymbol{V}}}_{\mathrm{rel}}{| }^{2}$. If we adopt $| {{\boldsymbol{V}}}_{\mathrm{rel}}| \approx {v}_{{\rm{p}}}$, then

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }}\approx \displaystyle \frac{{M}_{* }}{{M}_{\mathrm{bh}}}\displaystyle \frac{h}{Q},\end{eqnarray} \tag{ 35 }$$

where we note that there is no dependence on r. The implied fractional change in orbital energy can be derived from Equation (32) and the fractional change in angular momentum from Equation (33). Equation (35) implies that Δm/M_* ≪ 1 for compact objects because M_*/${M}_{\mathrm{bh}}$ ≪ 1 and h/Q < 1. As a result, regardless of their orbital configuration, compact objects experience only very minor fractional changes in the orbital configurations due to drag forces during passages through the accretion disk.

The fact that Δm/${M}_{* }$ can be large when orbits become quite eccentric implies that the star is being forced through a column of mass similar to its own. Additionally, for highly eccentric orbits, the periapse velocity of the orbit (and thus the star–disk relative velocity) tends to be much larger than the star's escape velocity. What does this imply for the star itself?

During the disk passage, the outer layers of the star endure hydrodynamic drag and shock heating. Murray et al. (1993) explored hydrodynamic ablation of self-gravitating spheres, while Armitage et al. (1996) and Kieffer & Bogdanović (2016) have performed hydrodynamic simulations of giant stars intersecting columns of material. These numerical results suggest that the momentum imparted in the passage is crucial in the eventual mass removal, as is shock heating of the envelope material. However, whereas the dissipated energy may be radiated away, the total momentum is conserved; for simplicity, we focus on the momentum transfer in what follows.

One simple model for momentum transfer can be derived if we assume that a fraction η < 1 of the dissipated orbital momentum goes into stripping mass from the outer envelope of the star, while the remainder goes into slowing its bulk motion. In this case, the momentum imparted to removing mass from the star at the stellar escape velocity is

$$\begin{eqnarray}&&{\rm{\Delta }}{M}_{* }{v}_{\mathrm{esc}}\sim \eta {\rm{\Delta }}m| {{\boldsymbol{V}}}_{\mathrm{rel}}| \sim \eta {\rm{\Delta }}{{mv}}_{{\rm{p}}},\end{eqnarray} \tag{ 36 }$$

where we have approximated the star–disk relative velocity as the periapse velocity. Thus,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{M}_{* }}{{M}_{* }}\sim \eta \displaystyle \frac{{v}_{{\rm{p}}}}{{v}_{\mathrm{esc}}}\displaystyle \frac{{\rm{\Delta }}m}{{M}_{* }},\end{eqnarray} \tag{ 37 }$$

where Δm/M_* is given in Section 3.3, and where ${v}_{\mathrm{esc}}=\sqrt{2{{GM}}_{* }/{R}_{* }}$, while ${v}_{{\rm{p}}}=\sqrt{2{{GM}}_{\mathrm{bh}}/{r}_{{\rm{p}}}}$.

How does the rate of orbital circularization compare to the rate of mass stripping? To entirely circularize an orbit such that $e\to 0$, the star must pass through a column of mass similar to its own mass Δm/M_* ∼ 1, Equation (30). For the star's envelope to survive this process (ΔM_*/M_* < 1) without being removed requires ηv_p/v_esc < 1. Though the value of η is, a priori, unknown, these scalings indicate a dependence on periapse distance through the ratio of stellar escape velocity to periapse velocity. We can, for example, reexpress this condition in terms of the stellar tidal radius,

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{{\rm{p}}}}{{r}_{{\rm{t}}}}\gt {\eta }^{2}{\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{M}_{* }}\right)}^{2/3}.\end{eqnarray} \tag{ 38 }$$

For example, if η = 0.1, then stars passing with ${r}_{{\rm{p}}}\ \lesssim 460{r}_{{\rm{t}}}$ would have their envelopes ablated rather than fully circularized.

The fate of a star under circumstances where envelope ablation occurs likely depends on the star's internal structure and its response to mass loss. Stars with highly differentiated core versus envelope densities may have their envelopes removed while a denser core remains. These denser cores would pass through smaller columns of disk mass and, depending on the precise conditions, might not circularize or ablate significantly. By contrast, a more homogeneous star, like a lower-mass main-sequence star, might be fully disrupted if ablative processes are important. The object's response to mass loss is significant here, too, in the context of repeated disk passages. If, upon losing mass, the star contracts such that its surface density, ${M}_{* }/{R}_{* }^{2}$, decreases, it will intercept a smaller column of disk mass. However if the star, or its outer envelope, has lower surface density after mass loss, a runaway of stripping (or orbital circularization) would be expected over the course of subsequent orbits.

Armitage et al. (1996) and Kieffer & Bogdanović (2016) have specifically focused on the case of red giants and the implications of these mass-stripping episodes. Because of their extended radii, giants interact with mass similar to their own in orbits with less extreme periapse radii (or lower eccentricities). Equivalently, we can note that their lower surface density is comparable to the disk surface density at larger radii. Finally, the composite core–envelope structure of these stars implies that even if the hydrogen envelope is removed, the white dwarf core will remain. Armitage et al. (1996) estimate a critical radius for the entire giant envelope to be removed over the giant-branch lifetime and find a semimajor axis of approximately 0.05 pc under typical assumptions for quasar disks, and argue that for giants, the stripping process is much more efficient than trapping within the disk (η ∼ 0.05 if Equation (38) is applied). Following up on Armitage et al.'s (1996) work, Kieffer & Bogdanović (2016) consider the case of our own galactic center and focus on more compact giant-branch stars. They find that collisions with overdense clumps contribute significantly to stripping these giants if the number of clumps in (or equivalently total mass of) a fragmenting gas disk is high enough—approximately several hundred times the mass of the current young stellar disk observed in the galactic center, or several 10⁴ M_⊙. The paucity of observed giants in the galactic center is indicative that such a process may have been at play.

In the previous sections, we have shown that the primary effects on stars in highly eccentric orbits are the circularization of their orbits and hydrodynamic ablation due to accumulated passages through the disk near periapse. Either of these actions removes stars from the phase space of highly eccentric orbits within the nuclear stellar cluster. Here we examine this depletion of stellar orbits in more detail.

Figure 6 defines a "loss cone" of low angular momentum phase space in which orbits are effectively depleted by disk interaction. The upper panel of Figure 6 shows the approximate disk mass intercepted by a Sun-like star on orbits of varying semimajor axis and eccentricity, Equation (26). Here we have again assumed our fiducial disk model of Section 3, a 10⁷ M_⊙ black hole and an α = Q = λ = 1 accretion disk. In the center panel, we compare the per-orbit dissipation of specific orbital energy to the orbital energy, ${\rm{\Delta }}{ \mathcal E }/{ \mathcal E }$, estimated from Equation (32). Finally, the lower panel defines a "loss cone" of orbits that are circularized over the disk lifetime. The critical condition is ${\rm{\Delta }}{{ \mathcal E }}_{\mathrm{total}}/{ \mathcal E }({r}_{{\rm{p}}})=1$, where ${\rm{\Delta }}{{ \mathcal E }}_{\mathrm{total}}={N}_{\mathrm{orb}}{\rm{\Delta }}{ \mathcal E }$ = ${\rm{\Delta }}{ \mathcal E }{\tau }_{\mathrm{disk}}/P$ is the net accumulated orbital energy dissipated over the disk lifetime, assumed to be τ_disk = 1 Myr in Figure 6. From Equation (32), ${\rm{\Delta }}{{ \mathcal E }}_{\mathrm{total}}/{ \mathcal E }({r}_{{\rm{p}}})$ is related to the disk mass intercepted by ${\rm{\Delta }}{{ \mathcal E }}_{\mathrm{total}}/{ \mathcal E }({r}_{{\rm{p}}})\approx 4{\rm{\Delta }}m/{M}_{* }$.

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It is useful to compare the disk loss cone to the tidal disruption loss cone defined by ${r}_{{\rm{p}}}\ \lt {r}_{{\rm{t}}}$. What we observe from Figure 6 is that, for black holes in extremely high accretion states, the disk loss cone is larger (extends to lower eccentricity) than the tidal disruption loss cone, especially for orbits with semimajor axis less than a parsec. The disk-interaction loss cone is especially extended for tight orbits (and has a different slope in phase space than the tidal loss cone) because tight orbits have shorter orbital periods that imply more cumulative passages through the disk over the disk lifetime.

In Figure 7, we compare the relative phase space occupied by the tidal disruption and disk loss cones under varying stellar and black hole parameters. The upper panels assume a 10⁷ M_⊙ accreting black hole and show stars of solar mass and 1, 10, and 100 R_⊙. With increasing stellar radius, both the tidal disruption and disk-interaction loss cones increase in the portion of phase space they occupy. For giant stars of 100 R_⊙, nearly all orbits with a ≲ 0.3 pc are within the disk loss cone. This suggests that disk interaction can efficiently remove giant stars from the innermost regions of a nuclear cluster, as argued by Armitage et al. (1996). Because, as shown in Equation (29), ${\rm{\Delta }}m/{M}_{* }$ is constant with impact parameter in units of the tidal disruption radius, we find that differing stellar properties do not affect the relative configurations of the star and disk loss cones.

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The lower panels of Figure 7 show varying black hole masses for an assumed Sun-like star. As the black hole mass increases, the size of the tidal disruption and disk-interaction loss cones increase in tandem, but do not change their relative configuration significantly. Taken together, this implies that during the active accretion phase, there is an enlarged loss cone on the stellar distribution function due to star–disk interaction.

The presence of the extended disk loss cone in the orbital phase space leads to the question of whether stars are able to penetrate to sufficiently eccentric orbits to be directly disrupted by the black hole's tides. The answer to this question depends on the relative magnitudes of the dissipation by the disk crossings and scatterings by two-body relaxation. Imagine a star undergoing a random walk in angular momentum space. For this star to undergo a tidal disruption event, it must reach a very eccentric configuration (with ${r}_{{\rm{p}}}\ \lt {r}_{{\rm{t}}}$) without dissipation acting to circularize its orbit instead. In practice, this implies that the per-orbit scatterings ΔJ are sufficiently large that the star can "jump" across the disk loss cone: from outside (higher angular momentum) the disk loss cone to inside the tidal disruption loss cone. An analogous dynamical process is at play for stellar-mass compact objects in galactic nuclei, where gravitational radiation's dissipation interacts with two-body scattering to form a Schwarzschild barrier to extreme mass-ratio inspirals (Merritt et al. 2011).

To understand the possible role that two-body relaxation plays in scattering stars into and out of this loss cone phase space, we compare the orbital angular momenta within the disk-interaction loss cone to the random walk in angular momentum accrued over the disk lifetime in Figure 8. The accrued random walk over the disk lifetime is proportional to the square root of the number of orbits completed times the per-orbit rms scatter, ${\rm{\Delta }}J({\tau }_{\mathrm{disk}})\approx {J}_{{\rm{c}}}{\left({\tau }_{\mathrm{disk}}/{t}_{{\rm{r}}}\right)}^{1/2}\approx \sqrt{{N}_{\mathrm{orb}}}{\rm{\Delta }}J$. For ν_* ∝ r^−1.5, the relaxation time is constant with radius, and ΔJ (τ_disk) ∝ (1−e²). The comparison of this typical scatter to the orbital angular momentum reveals that all but the most eccentric stars (originating from larger semimajor axes) do not, on average, random walk in or out of the disk-interaction loss cone, due to two-body relaxation. This finding implies that the phase space of the disk-interaction loss cone in which ΔJ (τ_disk) < J is largely emptied over the course of the disk lifetime and is not efficiently repopulated by two-body relaxation.

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4.3.1. Flux into the Tidal Disruption Loss Cone: Default Theory

To assess the flux of stars into the tidal disruption loss cone with and without the presence of the disk, we need to estimate the extent to which two-body relaxation replenishes disk (or tidal disruption) loss cone orbits. In the simplified case of an isotropic, spherically symmetric stellar distribution function that we have adopted, we can follow the well-developed loss cone theory of the flux of stars into disruptive orbits (Lightman & Shapiro 1977; Frank 1978; Magorrian & Tremaine 1999; Alexander 2005; Merritt 2013).

For an entirely isotropic distribution, the number of stars in the (full) loss cone per unit energy is

$$\begin{eqnarray}&&{N}_{\mathrm{flc}}({ \mathcal E })=4{\pi }^{2}{{PJ}}_{{\rm{c}}}^{2}{R}_{\mathrm{lc}}f({ \mathcal E }),\end{eqnarray} \tag{ 39 }$$

where $R\equiv {J}^{2}/{J}_{{\rm{c}}}^{2}$, and ${R}_{\mathrm{lc}}={J}_{\mathrm{lc}}^{2}/{J}_{{\rm{c}}}^{2}$ is the fraction of the total angular momentum phase space occupied by the loss cone. The angular momentum of the loss cone is

$$\begin{eqnarray}&&{J}_{\mathrm{lc}}={\left[2{r}_{\mathrm{lc}}^{2}\left(\displaystyle \frac{{{GM}}_{\mathrm{bh}}}{{r}_{\mathrm{lc}}}-{ \mathcal E }\right)\right]}^{1/2}\approx {\left(2{{GM}}_{\mathrm{bh}}{r}_{\mathrm{lc}}\right)}^{1/2},\end{eqnarray} \tag{ 40 }$$

where r_lc is the periapse distance of the loss cone. In the case of stars being disrupted by tides, ${r}_{\mathrm{lc}}={r}_{{\rm{t}}}$. The flux of stars into the loss cone is then ${F}_{\mathrm{flc}}({ \mathcal E })={N}_{\mathrm{flc}}({ \mathcal E })/P$.

To account for the fact that the loss cone is only partially repopulated by the diffusion process of two-body relaxation, we need to consider the fact that the distribution function depends on angular momentum near the loss cone. A full solution was first derived by Cohn & Kulsrud (1978) and is detailed in Chapter 6 of Merritt (2013), which we follow here. The two-dimensional distribution function is

$$\begin{eqnarray}f(R,{ \mathcal E })=\left\{\begin{array}{ll}0, & R\leqslant {R}_{0}\\ f({ \mathcal E }) & \tfrac{\mathrm{ln}(R/{R}_{0})}{\mathrm{ln}(1/R0)-1+{R}_{0}},\ R\gt {R}_{0}\end{array}\right.,\end{eqnarray} \tag{ 41 }$$

where $f({ \mathcal E })$ is given by Equation (6). R₀ is the dimensionless angular momentum at which the distribution function drops to zero. It is approximately

$$\begin{eqnarray}&&{R}_{0}\approx {R}_{\mathrm{lc}}\exp (-\alpha ),\end{eqnarray} \tag{ 42 }$$

where α = (q⁴ + q²)^1/4 and $q={\rm{\Delta }}{J}_{\mathrm{rel}}^{2}/{J}_{\mathrm{lc}}^{2}$ (Merritt 2013, Equation (6.66)). The periapse distance that corresponds to R₀ is r₀ ≡ R₀a/2.

The number of stars in the loss cone per unit energy can be derived by integration over angular momentum,

$$\begin{eqnarray}&&{N}_{\mathrm{lc}}({ \mathcal E })=4{\pi }^{2}{{PJ}}_{{\rm{c}}}^{2}{\int }_{0}^{{R}_{\mathrm{lc}}}f(R,{ \mathcal E }){dR},\end{eqnarray} \tag{ 43 }$$

from which the flux of stars into the loss cone is ${F}_{\mathrm{lc}}({ \mathcal E })={N}_{\mathrm{lc}}({ \mathcal E })/P$. We note that the loss cone flux is often written as

$$\begin{eqnarray}&&{F}_{\mathrm{lc}}({ \mathcal E })\approx q\displaystyle \frac{{F}_{\mathrm{flc}}({ \mathcal E })}{\mathrm{ln}(1/{R}_{0})},\end{eqnarray} \tag{ 44 }$$

which is an approximation of the integral over the angular momentum distribution in Equation (41). Finally, the integrated loss cone flux (the tidal disruption rate of stars) is

$$\begin{eqnarray}&&{F}_{\mathrm{lc}}=\int {F}_{\mathrm{lc}}({ \mathcal E })d{ \mathcal E }\end{eqnarray} \tag{ 45 }$$

or, explicitly,

$$\begin{eqnarray}&&{F}_{\mathrm{lc}}=4{\pi }^{2}{J}_{{\rm{c}}}^{2}{\int }_{{{ \mathcal E }}_{\min }}^{{{ \mathcal E }}_{\max }}{\int }_{0}^{{R}_{\mathrm{lc}}}f(R,{ \mathcal E }){dRd}{ \mathcal E }.\end{eqnarray} \tag{ 46 }$$

Given the power-law distribution function $f({ \mathcal E })$, the limits of integration ${{ \mathcal E }}_{\min }$ and ${{ \mathcal E }}_{\max }$ are not particularly well defined. In practice, this is mitigated by the fact that ${F}_{\mathrm{lc}}({ \mathcal E })$ is highly peaked.

Again adopting our fiducial ${M}_{\mathrm{bh}}={10}^{7}{M}_{\odot }$, α = Q = λ = 1 AGN and cluster models, the upper panel of Figure 9 compares the loss cone periapse radius for tides ${r}_{{\rm{t}}}$ to the periapse distance where the distribution function drops to zero, r₀, as a function of semimajor axis. Below r₀, the phase space is empty. Above ${r}_{{\rm{t}}}$, it is roughly isotropic. At larger semimajor axes, the per-orbit diffusion in angular momentum is larger, and ${r}_{0}\to 0$ (q ≫ 1, the "full loss cone" limit), while at smaller semimajor axes the per-orbit diffusion is weak and ${r}_{0}\to {r}_{{\rm{t}}}$ (q ≪ 1, the "empty loss cone" limit). The lower panel of Figure 9 shows the flux into the loss cone as a function of semimajor axis. In this particular example, the loss cone flux peaks at a ∼ 5 pc.

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4.3.2. Modification by Disk Loss Cone

To consider the role of the disk loss cone, we define equivalent quantities to those of the previous subsection. In particular, we define the disk loss cone periapse distance, r_lc,d, based on the criterion of circularization over the disk lifetime of Section 4.2, $({\tau }_{\mathrm{disk}}/P){\rm{\Delta }}{ \mathcal E }={ \mathcal E }({r}_{{\rm{p}}})$. This periapse radius therefore corresponds to the contours plotted in Figures 6–8. The corresponding zero-point periapse distance r₀ of the resulting distribution function emerges from the definition r₀ = R₀a/2, where we now evaluate R₀ on the basis of the disk loss cone in Equations (41) through (46).

The upper panel of Figure 9 shows r_lc,d and the disk-based r₀. We observe that the disk loss cone periapse distance depends on orbital semimajor axis, as expected from Figures 6–8. The behavior of the disk r₀ relative to r_lc,d is qualitatively similar to that described in the tidal disruption case. For small semimajor axes, two-body relaxation does not refill the phase space carved out by disk interaction. For large semimajor axes, ${r}_{0}\to 0$ and two-body relaxation leads new stars to diffuse into the disk loss cone every orbit. When r₀ < ${r}_{{\rm{t}}}$, then stars diffuse to sufficiently small periapse distances to be disrupted by the black hole. From Figure 9, we see that this primarily occurs for a ≳ 1 pc. By contrast, when r₀ > ${r}_{{\rm{t}}}$ no tidal disruption events can occur, because stars are lost to disk interaction in their angular momentum diffusion prior to reaching sufficiently low angular momentum to tidally disrupt.

We can incorporate the disk-modified distribution function, f_d (R, ${ \mathcal E }$) into our integration of the stellar tidal disruption flux. In ${f}_{{\rm{d}}}(R,{ \mathcal E })$, the disk loss cone is used to define ${J}_{\mathrm{lc}}$, R_lc, and R₀. The number of stars in the tidal disruption loss cone is given by Equation (43), while the total tidal disruption rate is given by Equation (46); in these expressions, by contrast, the upper integration limit R_lc now represents the tidal disruption loss cone.

Pictorially, comparing to Figure 9, we are integrating the stars in the disk-modified distribution function (orange) with periapse distance less than the tidal radius, ${r}_{{\rm{t}}}$. The lower panel of Figure 9 shows the disk-modified tidal disruption flux. We see that disk interaction sharply truncates the flux of stars in orbits of a ≲ 1 pc, which diffuse to the tidal disruption loss cone over many orbits; these stars tend to be captured into disk interactions rather than continuing their random walk in angular momentum. Interestingly, the integrated tidal disruption rate is modified very little, because this cutoff occurs at orbits more tightly bound than the peak of the tidal disruption flux.

Figure 10 shows the integrated tidal disruption event rate with and without a surrounding AGN disk. Here we assume a 10⁷ M_⊙ black hole, surrounded by a stellar cluster as described in Section 2.1. The presence of the disk loss cone depletes some stellar orbits that would have otherwise diffused to the black hole in the empty loss cone limit. We plot the steady-state event rates as a function of the ratio of Σ to a maximal Σ_max, which is derived by adopting parameters of α = 10⁻³, λ = 1, and Q = 1. As ${\rm{\Sigma }}/{{\rm{\Sigma }}}_{\max }\to 1$, there is at most a 30% decrease in tidal disruption event rate for the model parameters we have selected. The diffusion of orbits into the disk loss cone in steady state is also plotted as the "disk capture rate." This is the rate at which new orbits diffuse into the phase space of the disk loss cone to refill it after its initial depletion due to disk interaction and is approximately equal to the decrement in the tidal disruption event rate.

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We conclude by noting that the steady-state approximation that we have made in adopting the Cohn & Kulsrud (1978) solution to the phase-space distribution may not be fully warranted. When stars enter the disk-interaction loss cone, they are brought into more circular orbits, not deleted. Thus, further interaction with the stellar cluster is still possible. A time-dependent solution that follows the combined effects of disk-interaction, two-body, and secular relaxation processes is highly worthwhile to pursue to understand the subsequent evolution of captured stars (e.g., Kennedy et al. 2016 and Panamarev et al. 2018 find that many, but not all, stars captured by the disk subsequently interact with the black hole by migration or subsequent scatterings).