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Appendix A: Black hole accretion rate and nuclear star formation rate from disk instabilities: the case of power-law surface density profiles

Following the computation in HQ11, we solve Eqs. (5) and (6) for both Σ_g and Ṁ_infl. To this aim, we consider two distinct regions: an outer region where the potential is dominated by the disk, and an inner region where the BH gravity is dominant. The radius R_BH ~ 10 pc marks the separations between the two regions, while the scale R_acc ~ 0.1 pc corresponds to the inner bound of the innermost regions where we want to compute the BH accretion rate in Eq. (6).

Appendix A.1: The potential

To estimate the potential in the outer regions (from ~10 pc to ~100 pc), HQ11 adopt the WKB approximation to estimate the perturbation amplitude Φ_a ≈ | a | _max2πGΣ_dR | kR | ^-1, taking | kR | ~ 1 since they are considering global modes. On the basis of aimed N-body simulations (Hopkins & Quataert 2010a) the dominant mode is assumed to be m = 2 and the maximal perturbation amplitude | a | _max ≈ 0.3f_d(R) is taken to be proportional to the disk fraction f_d(R) for any radius R. In the inner one (from R_acc~ 0.1 pc to R_BH~ 10 pc), the potential is dominated by the BH and Φ_a ≈ | a | _maxGM_d(R_BH) /R_BH, with | a | _max ≈ 0.2f_d(R_BH). In this region, the dominant mode is m = 1, also motivated by simulations.

Appendix A.2: The gas surface density

In both regions, disk, bulge and gas surface densities are assumed to be described by power-law profiles, that is, Σ_d ∝ R^{− η_d}, Σ_b ∝ R^{− η_d}, Σ_g ∝ R^{− η_g}. In addition, HQ11 assume the disk to be stellar dominated so that the total surface density Σ_t = Σ_b + Σ_d ∝ R^{− η_d}, and the disk fraction is constant with radius.

With this assumptions it is possible to solve Eqs. (5) and (6) for both the gas density Σ_g and the mass inflow rate dM_infl/ dt. In particular, in the inner region R_acc ≤ R ≤ R_BH where the BH dominates the potential Φ in Eq. (5), Eq. (6) yields (A.1)while in the outer region R_BH ≤ R ≤ R₀, where the potential Φ is dominated by the disk, the same equations yield with (A.2)where is the angular velocity of the disk.

Appendix A.3: The boundary conditions

To compute the accretion flow in the inner region R_acc ≤ R ≤ R_BH, we can use Eq. (A.1) to solve Eq. (6). However, the two density profiles in Eqs. (A.1) and (A.2) are not matched to ensure a continuous mass flux at the boundaries R_BH and R₀. In principle, one could impose a continuity condition for such equations at all times. Since the boundaries move as accretion proceeds, this would require a detailed treatment following the dynamical evolution of the boundaries and of the mass flux across them. Following HQ11, we adopt a simplified approach by we keeping the density profiles in Eqs. (A.1) and (A.2), and we impose the mass conservation directly in Eqs. (6). Thus, we write the latter as (A.3)where A(R_BH,R₀) expresses the proper boundary conditions, i.e., the mass conservation across the boundaries R_BH and R₀. To compute such a quantity, we note that the mass accreted into the inner region (R_acc ≤ R ≤ R_BH) must equal the disk mass that has flown across R_BH. We indicate with ΔM_g the mass inside the boundary R_BH computed after integrating Eq. (A.1), and with A(R_BH,R₀)ΔM_g the actual mass accreted in the inner region after matching with it with the mass flown from the disk. In a time interval Δt, we must have (A.4)We must have Ṁ_g(R_BH,outside) = Ṁ_g(R_BH,inside). Estimating exactly such a quantity would require a dynamical description of the variation of the boundary with time and of the mass flux across it. However, a simple – though approximate – estimate of such a quantity can be obtained by computing it as the mass lost from a disk during Δt (divided by Δt). This can be obtained after integrating the disk density profile A.2 extrapolated down to R = 0, so that . The quantity B(R₀) is the normalization corresponding to the boundary condition at the galactic scale R₀, which ensures that the mass flux outside the boundary R₀ matches with that in the region R_BH ≤ R ≤ R₀. We then obtain (A.5)where the last equality follows after performing the integrals over Σ_g,inner and Σ_g,outer given in Eqs. (A.1) and (A.2). The quantity A(R_BH,R₀) acts like a suppression factor; when the mass inflow obtained using the (unmatched) density distributions (A.1) and (A.2) yields an inflow rate that exceeds the gas mass actually available from the disk at the position R_BH, it suppresses the mass flux so as to maintain it lower or equal to this mass. In fact, for high values of ΔM_g(outside,R_BH) we get A(R_BH,R₀) = 1. No suppression is needed since the disk mass available from the disk is high.

The quantity B(R₀) in Eq. (A.3) can be obtained by iterating the same mass-matching procedure exactly, but applying it to the outer boundary R₀. In terms of the disk mass M_d(R₀) within R₀ and of the corresponding gas mass M_g(R₀) = f_gasM_d(R₀), the result reads as with (A.6)The physical meaning of the boundary factor B(R₀) is to match the gas available in the galactic disk (given by the semi-analytic model) with the gas corresponding to the profile Σ_g inside R₀. In the original HQ11 model, the gas density profile Σ_g in the external region has a slightly higher normalization to interpolate between the marginal and the strong orbit crossing regimes; however, when such a factor is included, its effects on Eqs. (A.6) and (A.5) balance, leaving our results unchanged.

Appendix A.4: The BH inflow rate

We now proceed to solve Eq. (A.3) to derive the mass inflow rate M_infl close to the inner bound R_acc. HQ11 first consider Eq. (6) in a ring close to R_acc and small enough that the surface density Σ_g can be considered to be constant. This yields (A.7)After substituting Eq. (A.1) computed ad the inner radius R_acc the above equation reads (A.8)Equation (A.8) represents the mass inflow in the inner region corresponding to the BH accretion rate. We can relate R_BH to M_BH in terms of the disk mass within R_BH through the relation M_t(R_BH) ≃ M_BH. The mass M_t is computed as . For a power-law profile Σ_t = A_ΣtR^{− η_d} we can readily express R_BH as a function of M_t = M_BH to obtain (A.9)Substituting in Eq. (A.8) yields (A.10)We are now interested in expressing Eq. (A.8) using physical quantities evaluated far from the very central region, at galactic scales R = R₀ ~ 10²−10³ pc, in order to derive the BH accretion rate from quantities computed from our semi-analytic galaxy formation model. To this aim, we note that, for any disk profile Σ_d, it is possible to combine R₀ and the associated enclosed disk mass to yield convenient constant quantities. In particular for a power-law profile, after some algebra, the last two factors in Eq. (A.10) can be recast as (A.11)After substituting in Eq. (A.10) this reads as (A.12)We can express the above equation in physical units, which yields (A.13)where we have gathered all constant terms into the normalization (A.14)The above expression is the BH accretion rate for generic power-law surface density profiles. After assuming a power law index η_d = 1 / 2 for the gas surface density profile and an index η_k = 7 / 4 for the Kennicut-Schmidt star formation law, the above equation yields Eq. (9) in the main text and a normalization α(η_k,η_d) = 0.18. As noted by HQ11, assuming different values for η_d and η_k has only a minor effect on the exponents in Eq. (A.13), but has a significant impact on α(η_k,η_d). For example, changing the η_k to the value η_k = 3 / 2 yields α(η_k,η_d) = 3.17. In the main text, we assume a reference value for α(η_k,η_d) = 10 to investigate the maximal contribution of the DI mode to the evolution of the AGN population.

Appendix A.5: The nuclear star formation rate

The contribution to the star formation rate can be calculated integrating star formation surface density rate over the region of interest. The latter is related to the gas surface density by the Schmidt-Kennicut law . For the inner region, Σ_g is described by Eq. (A.1). Since all the terms in Eq. (A.1) are constant but the radius, we obtain (A.15)The normalization can be estimated considering the value that the star formation rate assumes in R = R_acc. This can be done easily since the mass inflow in R = R_acc is strictly related to star formation surface density rate by Eq. (6) and gives . Thus, the star formation surface density rate in the inner region is equal to (A.16)The contribution to the star formation rate in the inner region is then obtained by integrating (A.16) from R_acc to R_BH: (A.17)For the fiducial model, assuming η_k = 7 / 4 and R_acc ≃ 10^-2R_BH, we obtain (A.18)The same calculations can be performed for the outer region using the corresponding ; here the scaling and the exponent α_d are given in Eq. (A.2). This yields (A.19)The normalization can be computed imposing continuity between the inner and the outer region. Using Eq. (A.16) (A.20)Integrating Eq. (A.20) from R_BH to R₀, we obtain the star formation rate in the outer region for the fiducial model: (A.21)Assuming η_k = 7 / 4, η_d = 1 / 2, and R_acc ≃ 10^-2R_BH, we obtain (A.22)

Appendix B: Black hole accretion rate and nuclear star formation rate from disk instabilities: the case of exponential surface density profiles

Here we follow the same procedure as adopted in Appendix A, changing only the surface density profile (and the corresponding potential) of the galactic disk. In particular, we consider a thin disk whose density profile is described by the following exponential-law: (B.1)where R_d is the exponential length of the disk and its order a few kpc. The potential associated to the disk is (Binney & Tremaine 1987) (B.2)where y ≡ R/ 2R_d and I_n, K_n are the modified Bessel function of first and second kinds. Since both the disk and the bulge exponential length are much greater than scales of our interest, we can use the asymptotic form for the modified Bessel function; as a result, we obtain We note that in this case the disk surface density and the potential are computed self-consistently, at variance with the previous case where we assumed a WKB form for the potential and a power-law form for the surface density.

Following the same procedure as adopted in Appendix A, we consider two regions: an outer region where the potential is dominated by the disk, and an inner region where the BH gravity is dominant. The radius R_BH ~ 10 pc marks the separations between the two regions, while the scale R_acc ~ 0.1 pc corresponds to the inner bound of the innermost regions where we want to compute the BH accretion rate. The potential of the disk is only needed to calculate the analytic expression for Σ_g in the outer region, where the BH does not dominate the potential.

In the inner region, the potential is dominated by the BH, and the computation of the BH accretion rate follows the lines shown in Appendix A exactly, changing only the scaling of the disk radial profile by adopting η_d = 0, to achieve a constant Σ_d as given by Eq. (B.4). Thus, the BH accretion rate is given by the same Eq. (A.13) with η_d = 0. However, the boundary condition A(R_BH,R₀) is different from Appendix A, since it depends on the gas surface density Σ_g,outer in the region R>R_BH where the potential and the surface densities need to be recomputed (see Eqs. (A.5) and (A.6)). Also, the contribution to the nuclear star formation from such a region R>R_BH has to be recomputed, since it also depends on Σ_g,outer.

To computeΣ_g,outer in the outer region we start from the perturbed potential in the outer region (B.5)with a₀ ~ 0.3 and m = 2, motivated by simulations. Substituting the above expression into Eq. (5) we obtain for the outer region (from R_BH to R₀) (B.6)with (B.7)where we have used the relation Ω²R = G^∫2 π Σ_tR dR, and Σ_t is the total (disk + bulge) surface density profiles (see Appendix A). Substituting such an expression in Eq. (6) yields (B.8)That is the equation for Σ_g in the outer region R_BH ≤ R ≤ R₀ and the counterpart of Eq. (A.2) in the case of an exponential disk surface density profile; in the inner region R_acc ≤ R ≤ R_BH, the analytic expression for Σ_g is the same as Eq. (A.1), since this region is dominated by BH potential. We can now recompute the boundary conditions A(R_BH,R₀) and B(R₀). To this aim, we use Eqs. (A.5) and (A.6), but adopting the expression in Eq. (B.8) for the gas surface density in the outer region Σ_g,outer. This yields (B.9)In practice, the right-hand term in the parenthesis is extremely small, owing to the choice of η_k, η_d ( η_d ~ 0 since Σ₀ is almost constant) and to the factor R_d/R_BH, so that the above expression for the case of an exponential disk always yields A(R_BH,R₀) ≈ 1, independently of the outer boundary condition, which anyway reads (in the case η_k = 7 / 4) (B.10)where f₀ and f_gas are those given in Eq. (A.6). Thus, in the main text, we adopt the excellent approximation A(R_BH,R₀) = 1.

As for the nuclear star formation, the contribution from the inner region R_acc ≤ R ≤ R_BH is the same as given in Eq. (A.17), since in this region the potential is dominated

by the BH. In the outer region, we follow the procedure presented in Sect. A5, based on the relation , with the latter quantity taken from Eq. (B.8). This yields (B.11)Integrating Eq. (B.11) from R_BH to R₀, we obtain (B.12)Taking η_k = 7 / 4, η_d = 0, and R₀ = 100 pc, (B.13)At variance with the case of power law profile given in Eq. (A.21), the star formation rate does not diverge with increasing R₀, due to the exponential decay of the disk surface density.

© ESO, 2014