tokens: geometrically thin optically thick, radiatively efficient, and steady-state accretion disk (Shakura Sunyaev 1973 ), which is expected in AGNs. We assume the disk has viscosity parameter α = 0.1, and radiation efficiency , where and are the Eddington luminosity and Eddington accretion rate of an SMBH with mass M • . We further assume the accretion rate is constant during the lifetime τ agn of an AGN. These parameters determine the disk surface density Σ( r ), scale height H ( r ), isothermal sound speed c s ( r ), and midplane temperature T ( r ), which are dependent on the distance r from the SMBH. To derive these quantities we follow the model of Sirko Goodman ( 2003 ), assuming that viscosity in the disk is proportional to the total pressure: where T eff ( r ) is the blackbody temperature of the disk at r . , where r min is the inner radius of the disk; we set here, where . Ω is the Keplerian angular velocity of the disk, which is . τ is the optical depth at midplane. We assume the mean molecular mass . σ is the Stefan–Boltzmann constant, and k is Boltzmann constant. In order to solve these equations, we still need to provide the opacity κ . We approximate the opacity with (see Figure 1 in Thompson et al. 2005 ):where g −1 cm 2 K −2 and κ 1 = κ 0 (100 K) 2 = 2.4 g −1 cm 2 . κ m is the molecular opacity, which is about 0.1 Z , where Z indicates the mass fraction of heavy elements. We adopt Z = 0.01 in our simulations (Iglesias Rogers 1996 ). is the opacity due to the negative hydrogen ion:where and . The electron scattering opacity κ e can be written as X is the mass fraction of hydrogen. The Kramers opacity κ K is due to free–free, bound–free, and bound–bound electronic transitions:The equations above are valid where the Toomre Q -parameter Q = C s Ω/( πG Σ) ≳ 1. Beyond that region, out to 0.1 pc, we assume that the disk self-regulates, which we take into account by fixing the Q -parameter at 1 and replace Equation ( 1 ) with (see Equation (15) of Sirko Goodman 2003 ): 2.2. Orbital Variation Due to AGN Disk Crossing Consider a Cartesian coordinate system with the SMBH in its center, in which the AGN disk lies in the XY plane, and the major axis of the orbit of the stellar-mass BH orbiting the SMBH is in the YZ plane (see Figure 1 ). Let be the unit vector pointing in the direction of the major axis of the BH’s orbit in Cartesian coordinates, with θ being the angle is the orbit’s inclination angle compared to the AGN disk plane. Let a be the semimajor axis, e the eccentricity, and is the normal vector of the BH’s orbital plane. Let be the angle between and the YZ plane, where is the unit vector along the x axis. 5 5 Here we adopt the coordinates ( θ , ψ , φ ) that are the most suitable for the derivation of the results, even though these are not standard Keplerian orbital elements. They can be converted to Keplerian coordinates as follows. Longitude of the ascending mode: . Argument of periapsis: φ . Inclination: . We haveLet φ be the angle
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