Issue 
A&A
Volume 569, September 2014



Article Number  A37  
Number of page(s)  20  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201424217  
Published online  17 September 2014 
Online material
Appendix A: Black hole accretion rate and nuclear star formation rate from disk instabilities: the case of powerlaw 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  _{max}2πGΣ_{d}R  kR  ^{1}, taking  kR  ~ 1 since they are considering global modes. On the basis of aimed Nbody 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  _{max}GM_{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 powerlaw profiles, that is, Σ_{d} ∝ R^{− ηd}, Σ_{b} ∝ R^{− ηd}, Σ_{g} ∝ R^{− ηg}. In addition, HQ11 assume the disk to be stellar dominated so that the total surface density Σ_{t} = Σ_{b} + Σ_{d} ∝ R^{− ηd}, 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 ≤ R_{0}, 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 R_{0}. 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},R_{0}) expresses the proper boundary conditions, i.e., the mass conservation across the boundaries R_{BH} and R_{0}. 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},R_{0})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(R_{0}) is the normalization corresponding to the boundary condition at the galactic scale R_{0}, which ensures that the mass flux outside the boundary R_{0} matches with that in the region R_{BH} ≤ R ≤ R_{0}. 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},R_{0}) 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},R_{0}) = 1. No suppression is needed since the disk mass available from the disk is high.
The quantity B(R_{0}) in Eq. (A.3) can be obtained by iterating the same massmatching procedure exactly, but applying it to the outer boundary R_{0}. In terms of the disk mass M_{d}(R_{0}) within R_{0} and of the corresponding gas mass M_{g}(R_{0}) = f_{gas}M_{d}(R_{0}), the result reads as with (A.6)The physical meaning of the boundary factor B(R_{0}) is to match the gas available in the galactic disk (given by the semianalytic model) with the gas corresponding to the profile Σ_{g} inside R_{0}. 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 powerlaw profile Σ_{t} = A_{Σt}R^{− ηd} 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 = R_{0} ~ 10^{2}−10^{3} pc, in order to derive the BH accretion rate from quantities computed from our semianalytic galaxy formation model. To this aim, we note that, for any disk profile Σ_{d}, it is possible to combine R_{0} and the associated enclosed disk mass to yield convenient constant quantities. In particular for a powerlaw 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 powerlaw 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 KennicutSchmidt 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 SchmidtKennicut 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^{2}R_{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 R_{0}, 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^{2}R_{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 exponentiallaw: (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 selfconsistently, at variance with the previous case where we assumed a WKB form for the potential and a powerlaw 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},R_{0}) 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 a_{0} ~ 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 R_{0}) (B.6)with (B.7)where we have used the relation Ω^{2}R = G^{∫}2 π Σ_{t}R 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 ≤ R_{0} 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},R_{0}) and B(R_{0}). 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 righthand term in the parenthesis is extremely small, owing to the choice of η_{k}, η_{d} ( η_{d} ~ 0 since Σ_{0} 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},R_{0}) ≈ 1, independently of the outer boundary condition, which anyway reads (in the case η_{k} = 7 / 4) (B.10)where f_{0} and f_{gas} are those given in Eq. (A.6). Thus, in the main text, we adopt the excellent approximation A(R_{BH},R_{0}) = 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 R_{0}, we obtain (B.12)Taking η_{k} = 7 / 4, η_{d} = 0, and R_{0} = 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 R_{0}, due to the exponential decay of the disk surface density.
© ESO, 2014
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