ON THE OFFSET OF BARRED GALAXIES FROM THE BLACK HOLE M_BH–σ RELATIONSHIP

The disk component is an evolved Kuz'min–Toomre disk with the following surface density distribution:

$$\begin{equation} \Sigma (R) = \frac{M_d}{2\pi R_d^2}\left(1+\frac{R^2}{R_d^2}\right)^{-3/2}, \end{equation} \tag{ 1 }$$

where R is the radial distance from the axis of rotation and R_d is the disk scale length. The disk is spread vertically as an isothermal sheet and truncated at R = 5R_d. Particles are drawn from a distribution function which yields a Toomre Q ≃ 1.5. The resulting structure is unstable to bar formation (Athanassoula & Sellwood 1986). The bar forms and is vertically thickened via the buckling instability, resulting in a stable bar (Toomre 1966; Raha et al. 1991; Sellwood & Wilkinson 1993).

We choose a DM halo with the well-known logarithmic potential

$$\begin{equation} \Phi _{\mathrm{halo}}(r) = \frac{V^{2}_{0}}{2}\ln \left(1+\frac{r^2}{c^2}\right), \end{equation} \tag{ 2 }$$

which yields a flat circular velocity when r ≫ c, where c is the core radius (Binney & Tremaine 2008). We choose c = 30R_d = 90 kpc and V₀ = 0.7(GM_d/R_d)^1/2 = 187 km s⁻¹. Since we use a rigid halo as opposed to a live halo, the halo in our simulations cannot exchange energy or angular momentum with the disk and/or bulge particles. Shen & Sellwood (2004) found that replacing their rigid logarithmic halo with a live one resulted in little change to the evolution of the bar in their simulations. In these simulations, the central region of the halo is shallow, preventing the halo from affecting the evolution of the angular momentum significantly. However, Athanassoula et al. (2005) found in their simulations with live halos that the survival of the bar depended quite strongly on the density profile of the dark matter halo. They found that for a CMC of the same mass, a bar in a DM halo with a shallow central cusp is more easily destroyed than a bar in a DM halo with a steeply rising DM cusp. In this paper we will assume only a rigid logarithmic halo with a core. We address the effect of this assumption on our results in Section 6.

In the disk+bulge simulations, the bulge component has a mass of 0.15M_d and is initially truncated at a radius of 0.9R_d. The two component system is constructed using a method first proposed by Prendergast & Tomer (1970), used in Raha et al. (1991), and described in Jarvis & Freeman (1985) and Appendix A of Debattista & Sellwood (2000). Using the integrals of motion (E, J_z), a distribution function f(E, J_z) is chosen that corresponds to a King model (King 1966) with some net rotation (Jarvis & Freeman 1985). Integrating the distribution function over velocity yields a density ρ(R, z). The density is converted to a mass, which is added to that of a smooth disk component. The potential due to this new mass distribution is computed, yielding a new distribution function. This process is iterated until convergence.

The CMC representing a SMBH is modeled as a Plummer sphere with potential of the form

$$\begin{equation} \Phi _{\mathrm{CMC}}(r) = -\frac{GM_{\mathrm{CMC}}(t)}{\sqrt{r^2+\epsilon _{\mathrm{CMC}}}}, \end{equation} \tag{ 3 }$$

where _CMC is the softening length. The softening length corresponds to the compactness of the CMC. A large value of _CMC is representative of a relatively diffuse CMC (e.g., molecular gas clouds or a nuclear star cluster), whereas a small value represents a relatively compact (hard) CMC. Shen & Sellwood (2004) showed that the effect of a very compact CMC is much greater than that of a softer CMC. Here we set _CMC = 0.001R_d (corresponding to a length scale of a few parsecs) since we wish to assess the stronger effect of its growth on the observable kinematics.

In half of our simulations, we choose a final M_CMC of 0.2% M_d, which for our choice of physical units corresponds to 10⁸ M_☉. Note that this CMC is a factor of 6.5 more massive than the SMBH mass predicted from scaling relation M_BH ≲ 0.002 M_bulge (Häring & Rix 2004). For this reason, we also carry out an investigation of the effect of a CMC with 10 times smaller mass (M_CMC = 10⁷ M_☉) and show that the effects of this smaller black hole on the stellar velocity dispersion are similar to those resulting from the M_CMC = 10⁸ M_☉. More importantly, the fractional difference in σ between the barred and axisymmetric models is nearly independent of M_CMC. To remind readers that the central point mass in some of our simulations are somewhat overmassive, we will henceforth refer to it as a CMC rather than a SMBH.

We adopt the definition for a black hole's "sphere of influence," r_s, as the radius within which the mass of stars is equal to the mass of the black hole. For M_CMC = 10⁸ M_☉, the sphere of influence r_s = 0.17 ± 0.078 kpc. Since r_s is directly proportional to the mass of the CMC, it is about a factor of 10 less for M_CMC = 10⁷ M_☉, which would make r_s much smaller than the particle softening and therefore unresolvable by our current simulations. Nevertheless, we will show that despite the factor of 10 difference in final masses of the two CMC, they both affects on the observed values of σ in qualitatively similar ways and differing quantitatively by at most a few percent—a difference that is unlikely to be observationally detectable.

The CMC is grown adiabatically on a timescale that is much longer than the orbital period of stars near the disk center. M_CMC is a function of time given by

$$\begin{equation} M_{\mathrm{CMC}}(\tau) =\left\lbrace \begin{array}{@{}l@{\quad }l@{}}0 & \tau < 0\\ \ms M_{\mathrm{CMC}}\sin ^2(\pi \tau / 2) & 0 \le \tau \le 1,\\ \ms M_{\mathrm{CMC}}& \tau > 1 \end{array}\right. \end{equation} \tag{ 4 }$$

where τ ≡ (t − t_CMC)/t_grow for a CMC that began growing at t_CMC. We increase M_CMC over t_grow = 50 dynamical times.

The simulations use a three-dimensional (3D), cylindrical, polar grid-based N-body code described in Sellwood & Valluri (1997). The gravitational field at a distance d from a particle is given by a Plummer sphere Φ(d) = −G/(d² + ²)^1/2. We use a constant particle softening length, = 0.02R_d, in all of our simulations. See Table 1 for the full set of numerical parameters.

Table 1. Summary of Model Setup

Parameter	Disk	Disk+Bulge
Numerical Parameters
Number of particles	2.8 × 106	1.15 × 106
Grid size (R, ϕ, z)	55 × 64 × 375	58 × 64 × 375
Vertical plane spacing	0.02	0.01
Grid boundaries (R, z)	(20.0, ± 3.74)	(26.8, ± 3.74)
Particle softening length	0.02	0.01
Time step Δt0 without CMC	0.04	0.04
Time step Δt0 with CMC	0.01	0.01
Number of guard shellsa	9	9
Outermost guard radius rmax	0.127	0.127
Innermost guard radius rmin	0.008	0.008
Smallest time step	tstep/29	tstep/29
Initial Disk
Toomre Q	1.5	1.2
rms vertical thickness	0.3	0.5
Truncation radius	5	5
Fixed Halo
V0	0.7	0.8
Core radius c	30	8
Bulge
Mass	⋅⋅⋅	0.15
Truncation radius	⋅⋅⋅	0.9
CMC
MCMC (1)	0.002	0.002
MCMC (2)	0.0002	0.0002
Softening length CMC	0.001	0.001
Growth time tgrow	50	50

Note. ^aSee the Appendix of Shen & Sellwood (2004) for guard shell details.

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Because of the differing time scales associated with each particle, the simulation is divided into four spherical zones and different time steps are used in each zone, with the minimum time step of 0.01/128 (for details see, Shen & Sellwood 2004). Additionally, the "guard shell" scheme described in detail in the Appendix of Shen & Sellwood (2004; the CMC is enclosed by a number of spherical regions with successively shorter time steps as R decreases) helps ensure accurate orbit integrations in areas where particles are subjected to relatively strong accelerations.

Our analysis begins by "observing" each snapshot at a specific angle of inclination of the disk to the line of sight, i, and the angle formed by the bar (if present) to the line of nodes, Φ_LON.

Because of our focus on the nuclear region of the models, we restrict our field of view of the simulations to ±10.5 kpc (and ±7.5 kpc) in the x and y directions for the disk-only (and disk+bulge) simulations, respectively. We binned all the particles that fall within this projected rectangular region on a 300 × 300 Cartesian grid corresponding to a pixel size of 0.07 × 0.07 kpc in the pure disk models (and pixels of 0.05 × 0.05 kpc in the disk+bulge models). This is roughly equal to the particle softening length. We then adaptively bin the square pixels to maintain a minimum signal-to-noise ratio ${\rm (S/N)} \equiv \sqrt{N} \ge 50$ using the Voronoi binning scheme outlined in Cappellari & Copin (2003).⁵ Our choice of pixel size and S/N was a compromise between maintaining computational economy and attempting to resolve the sphere of influence r_s ∼ 0.17 kpc of the M_CMC = 10⁸ M_☉.⁶ We found that the resulting kinematics were relatively insensitive to our choice of pixel size and S/N threshold, given a S/N ≳ 30. On average, each Voronoi bin is composed of ∼300 pixels, with the smallest and largest Voronoi bins containing 3 and 767 pixels, respectively. Inside R ∼ 2 kpc, individual pixels are comparable to the size of the Voronoi bins; outside of R ∼ 2 kpc, the Voronoi bins are considerably larger than a single pixel.

We construct line of sight velocity distributions (LOSVDs) from all particles that fall within a Voronoi bin. Since the LOSVDs of such systems generally depart from pure Gaussian shapes, following the standard practice we parameterized the LOSVD within each Voronoi bin using a Gauss-Hermite expansion (van der Marel & Franx 1993; Gerhard 1993) and define v_los as the mean line of sight velocity, define σ_los as the line of sight velocity dispersion, and describe the asymmetric and symmetric departures from a Gaussian LOSVD by the Hermite coefficients h₃, h₅ and h₄, h₆, respectively. The parameters characterizing the LOSVD in each Voronoi bin were obtained by using the MPFIT procedure implemented in IDL (Markwardt 2009) to simultaneously fit γ, v_los σ_los, h₃, h₄, h₅, and h₆.

Because of the anisotropic velocity distribution inherent to barred galaxies, both the inclination of the disk i and the angle made by the bar to the line of nodes⁷ Φ_LON are likely to alter the measured nuclear kinematics.

Figure 1 (top) shows the 2D kinematics fields (from left to right: v_los, σ_los, h₃, h₄, and projected surface brightness log₁₀Σ) for i = 45° and Φ_LON = 45° for the disk-only simulation with a bar after the growth of the M_CMC = 10⁸ M_☉. The bottom panels show the kinematics that would be observed along the artificial "slit" oriented along the major axis of the bar (shown as a red line in the top panels). For each rectangular "aperture" along the slit, we average the kinematics of the bins that fall within that aperture. While we do not weight the bins according to the area of the slit they occupy (i.e., bins that fall only partially within a slit aperture are given the same weight as those that fall entirely within the aperture), we find that a more careful treatment of apertures with partial overlap accounted for does not produce noticeable differences in the resulting slit profiles. In these figures we use a slit of length −6 kpc ⩽ r ⩽ 6 kpc and a width of 0.075 kpc (a factor of a few smaller than the sphere-of-influence of 0.17 kpc).

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Similarly, Figure 2 shows 2D kinematics (top) and slit kinematics (bottom) (for i = 45° and Φ_LON = 45°) for the snapshot of the disk+bulge simulation with a bar after the growth of the a CMC with M_CMC = 10⁸ M_☉. We note that in both the disk-only and disk+bulge simulations the v_los fields show a slight kinematic twist that is characteristic of triaxial systems and the rotational axis of symmetry is misaligned with the minor axis of the bar. In axisymmetric systems, h₃ is generally anticorrelated with v_los; however, in the region where the bar dominates, h₃ tends to be correlated with v_los (Bureau & Athanassoula 2005). This is indeed what we observe in both the disk-only and disk+bulge barred simulations, even in the presence of a CMC. Finally we observe the regions of negative h₄ that are characteristic of bars that have buckled and are then viewed face-on (Debattista et al. 2005). In the model with the classical bulge (Figure 2), the bar is weaker than in Figure 1; however, the kinematic twist in v_los, the correlation between h₃ and v_los, and the misalignment of the short axis of the central oval and the rotation axis are telltale signs of the presence of a bar.

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The method and aperture used to define σ is a historically contentious issue (e.g., Merritt & Ferrarese 2001; Tremaine et al. 2002). Here we closely follow the observational definition of σ as the luminosity weighted rms velocity within the projected half-light radius R_e:

$$\begin{equation} \sigma ^2 = \frac{\int ^{R_{e}}_{0} I(R) \left(\sigma _{\mathrm{los}}^2+\overline{v}_{\mathrm{los}}^2\right) dR}{\int ^{R_{e}}_{0} I(R) dR}, \end{equation} \tag{ 5 }$$

where I(R) is the luminosity distribution of the bulge as a function of projected radius R, and σ_los and $\overline{v}_{\mathrm{los}}$ are the line of sight velocity dispersion and mean line of sight velocity, respectively. For our simulations, we assume that all particles are stars of the same type, that there is no dust, and that the stars have a constant mass-to-light ratio (i.e., M/L = 1). We then define a circular aperture of radius R_e that we project onto the field of view. We then convert the integral into a sum and compute σ as

$$\begin{equation} \sigma ^2 = \frac{\sum _{R_i \leqslant R_{e}} m_i\left(\sigma _{i,{\rm los}}^2+\overline{v}_{i,{\rm los}}^2\right)}{\sum _{R_i \leqslant R_{e}} m_i}, \end{equation} \tag{ 6 }$$

where the sum is over the cells on the 300 × 300 grid that fall within R_e, and R_i, and m_i, are the projected distance from the center, mass, mean velocity, and velocity dispersion of the ith cell, respectively. Note that this approach allows us to mimic what is done in integral field unit (IFU) observations with a fixed pixel scale.

Since the orientation of the bar to the line of nodes as well as the inclination of the disk to the line of sight can alter σ, we measured this quantity using Equation (6) for nine different orientations, as follows. With i fixed at 45°, we varied the orientation of the bar so that Φ_LON = 0°, 30°, 45°, 60°, and 75°. We obtain four additional measurements with Φ_LON fixed at 45° and inclination of the disk varied so that i = 0°, 30°, 60°, and 75°.

Since a classical bulge is only present in half of the simulations, R_e cannot be defined in a uniform way for all our simulations. Noting that when a bulge is present, its truncation radius is 0.90R_d (2.7 kpc), we computed the mass within this radius (including the mass of disk particles interior to the truncation radius) and then (assuming that mass follows light with constant M/L) we compute the half-mass radius r_1/2 = 0.367R_d = 1.1 kpc.⁸ We note that Hartmann et al. (2013) show that, for their sample of simulations, the values of σ obtained using R_e/8 are consistent with those obtained using R_e. We also tried four other possible values for R_e: 0.04, 0.08, 0.16, and 0.30R_d, which correspond to values of 0.12, 0.24, 0.48, and 0.9 kpc, respectively.

Figure 3 shows how $\langle \sigma \rangle _{\Phi _{\mathrm{LON}},i}$ (the value of rms velocity averaged over all orientations) varies with R_e for M_CMC = 10⁸ M_☉ (top) and M_CMC = 10⁷ M_☉ (bottom). $\langle \sigma \rangle _{\Phi _{\mathrm{LON}},i}$ depends slightly on R_e in the disk-only simulation but is almost independent of R_e for the disk+bulge model. Since $\langle \sigma \rangle _{\Phi _{\mathrm{LON}},i}$ is not strongly dependent on R_e, hereafter we selected R_e = 0.9 kpc unless otherwise noted. In the disk-only simulations, this slightly overestimates the effective R_e, but the difference between the barred and unbarred systems is unlikely to be affected. The error bars represent the standard deviation obtained averaging over nine different orientations. We emphasize that the error bars do not represent the error on the mean σ but are meant to show the scatter introduced by orientation effects. While the error bars for the barred models with CMCs (blue/cyan solid curves) slightly overlap the error bars for the unbarred models (red/pink solid curves), it is clear that the mean values of $\langle \sigma \rangle _{\Phi _{\mathrm{LON}},i}$ for the barred models with CMCs are almost always larger by at least one standard deviation. This figure also shows that in the absence of the CMCs (dashed lines) there is little or no difference between the barred and unbarred galaxies, demonstrating that the orientation of the bar alone cannot be responsible for the observed differences.

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We will discuss the vertical offsets between the different curves (corresponding to models with/without a bar, bulge, CMC in future sections).

In this section we examine various factors that affect the observed σ in our simulations. These include the angle ofl the bar to the line of nodes (Section 4.1.1), the inclination of the disk to the line of sight (Section 4.1.2), and the growth and final mass of a CMC (Section 4.1.3).

4.1.1. Dependence of σ on Φ_LON

Figure 4 (left) shows the dependence of σ on the choice of Φ_LON, where σ is measured within R_e = 0.9 kpc. The angle of inclination of the disk is fixed at 45°. In the barred cases, the positive correlation between σ and Φ_LON is to be expected from a simple geometrical argument. Bar-supporting x₁ orbits are elongated along the bar, and their primary motion is oscillation back and forth along its major axis (e.g., Sellwood & Wilkinson 1993; Athanassoula 1992; Bureau & Athanassoula 1999; Shen & Sellwood 2004). In the disk-only cases (blue/cyan squares), we see that as the orientation of the bar approaches end-on (i.e., as Φ_LON → 90° and the major axis of the bar aligns with the line of sight) σ increases. This is because a given circular aperture of radius R_e encloses a greater fraction of x₁ orbits for end-on bar orientations. The alignment of these radial orbits with the line of sight results in a wider distribution of line of sight velocities, increasing our measurement of σ. Shen & Sellwood (2004) showed for similar disk-only simulations that the x₁ family that supports the bar is slowly destroyed by a growing CMC. However, they found that the mass of the CMC necessary to completely destroy this family (and the bar) was about 25 times larger than the most massive CMC used in our simulations.

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In the unbarred counterpart (red/pink squares), all the disk particles have been scrambled in azimuth as described in Section 2, erasing the bar but preserving the radially averaged mass and kinematic profiles. For the unbarred models, Φ_LON is not defined (since there is no bar with respect to which the angle of the line of nodes can be measured); however, to make it clear that the velocity dispersion is constant for all line of sights with the same inclination, we mark the measured σ by red/pink squares or solid dots connected by horizontal lines. At time t₂ following the growth of the CMC, the initially unbarred models develop weak spiral patterns that cause small dependence on Φ_LON, which we show connected by solid red/pink lines.

Before the CMC is grown, the barred simulation with the classical bulge (solid blue/cyan dots connected by dashed curves) shows a dependence on Φ_LON similar to the disk-only case (open blue/cyan squares connected with dashed curves). The vertical offset of the former results because of the added mass of the bulge. However, after the growth of the CMC (solid blue/cyan dots and lines), the dependence on Φ_LON is significantly weaker in the presence of the bulge than in the absence of the bulge. This implies that when the CMC grows inside a bulge+bar it results in a more significant reduction in the fraction of x₁ orbits, compared to when the identical CMC grows in a pure bar. We investigate the cause of this in Section 4.3.

Figure 4 (right) shows the fractional difference Δσ/σ_ax = (σ_bar − σ_ax)/σ_ax between σ for a barred model and its unbarred counterpart, relative to the unbarred case. Δσ/σ_ax is plotted as a function of Φ_LON (while keeping the inclination fixed at i = 45°). For the models without a CMC (dashed lines), the orientation of the bar can result in either negative Δσ/σ_ax as the bar becomes parallel to the line of nodes or positive Δσ/σ_ax as the bar is viewed end-on. In contrast, after the growth of a CMC (solid curves) σ is always larger for the barred case than for the unbarred case regardless of the value of Φ_LON, but once again the fractional difference becomes larger as the bar is seen end-on (i.e., Φ_LON → 90°). In the presence of a classical bulge (solid dots), the maximum difference in σ is about 5%. Interestingly, the fractional difference in σ is larger in the absence of a classical bulge, but even so it is ≲ 10%. A comparison of the top panel (M_CMC = 10⁸ M_☉) and bottom panel (M_CMC = 10⁷ M_☉) shows that the overall trends are similar for the two CMCs. In fact the right-hand panels in this figure show that there is almost no difference in Δσ/σ_ax despite the fact that the CMC in the bottom right panel is a factor of 10 smaller than that in the top right panel.

4.1.2. Dependence of σ on Inclination

Figure 5 shows the dependence of σ on the angle of inclination of the disk to the line of sight (Φ_LON = 45°). Once again the dependence of σ on inclination can, in part, be explained with a geometrical argument. At low inclination (i.e., nearly face-on), the contribution of the rotational velocity component of the disk to σ is relatively insignificant. However, the number of disk particles contained within a given aperture of radius R_e increases with inclination. As the inclination increases, a larger number of disk particles on both the near and far side of the nuclear region fall within R_e, causing σ to increase. Note that if the disk orbits were perfectly circular, the orbits falling within R_e would have velocities that are nearly perpendicular to the line of sight and would have little effect on σ. However, since the orbits in the inner region of both the barred and scrambled disks are quite radial, there is a fairly strong dependence on inclination, for both the barred (blue/cyan) and scrambled (red/pink) models (see Figure 5, left).

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There is also a more subtle contribution to the correlation between σ and inclination. As inclination increases and the orientation of the disk becomes more edge-on, the intrinsic (3D) velocity dispersion becomes dominated by the radial and tangential dispersions, σ_R and σ_ϕ respectively, rather than the vertical dispersion σ_z. As we will show in Figure 7, σ_R and σ_ϕ are greater than σ_z, contributing to a positive correlation between σ and inclination.

Figure 5 (right) shows the fractional difference Δσ/σ_ax as a function of inclination (with Φ_LON = 45°). For the models without a CMC (dashed lines), Δσ/σ_ax is almost independent of inclination. After the growth of a CMC (solid curves), σ is larger for the barred cases than for the unbarred cases (i.e., both solid curves are above zero for all values of i) and depends weakly on inclination. In the presence of a classical bulge (solid dots connected by solid lines), the maximum increase in σ is about 3% for a nearly edge-on orientation and about 4% for the pure disk (squares connected by solid lines).

It is important to note that Φ_LON is fixed at 45°, hence the orientation of the bar can essentially be thought of as intermediate between the side-on and end-on orientations. It was evident in Figure 4 that a side-on view of the bar produces values of σ that are less than the unbarred case, while an end-on view of the bar does the opposite. Thus, when Φ_LON is fixed at 45°, the barred and unbarred observations at t₁ produce nearly identical values of σ. This allows for a direct comparison between the t₂ values of σ in the barred and unbarred cases. The growth of a CMC in the presence of a bar clearly produces a greater change in σ than the growth of the same CMC in an unbarred galaxy.

4.1.3. Dependence of σ on CMC Growth

In Sections 4.1.1 and 4.1.2 we saw that for both the unbarred and barred models σ is more sensitive to the presence/absence of a CMC than to changes in the orientation of the disk to the line of sight. This is surprising since the sphere of influence of the CMC (estimated to be ∼0.17 kpc for models with M_CMC = 10⁸ M_☉) is a factor of six smaller than R_e = 0.9 kpc! This implies that the gravitational potential of the CMC is not directly responsible for this increase, and rather it is the effect that the changing potential has on the evolution of the bar. In this subsection we quantify the effect of the growth of the CMC on σ, and in the following two sections we examine the causes of this increase.

Figure 6 shows the fractional change Δσ/σ_init = (σ(t₂) − σ(t₁))/σ(t₁)) for the unbarred (red/pink) and barred (blue/cyan) models without (squares) and with (solid dots) a classical bulge. Δσ/σ_init is plotted as a function of Φ_LON for models with i = 45° (Figure 6, left) and as a function of inclination for models with Φ_LON = 45° (Figure 6, right). In the left panels we see that in the unbarred models the growth of the CMC produced an increase in σ (∼3%–5% when M_CMC = 10⁸ M_☉) that is essentially independent of Φ_LON (the very small fluctuations with Φ_LON arise from the weak spiral features in the unbarred models at t₂).

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The barred models (blue) display a larger relative increase in σ (∼5%–10% when M_CMC = 10⁸ M_☉). In Figure 4 (left) we saw that the growth of the CMC in a bulge+bar model (solid blue dots) results in no dependence on Φ_LON. This implies that the velocity distribution of stars within R_e is essentially isotropic. It appears that the growth of a CMC scatters and therefore axisymmetrized a significant portion of bar supporting orbits in the innermost regions of the system. This results in a reduced dependence of σ on Φ_LON in the barred disk+bulge simulations at t₂. Thus, Δσ/σ_init decreases with increasing Φ_LON. The weakening of the bar is less significant in the disk-only simulation, resulting in a flatter relationship between Δσ/σ_init and Φ_LON.

In the right-hand panels of Figure 6, all models tend to show a similar dependence on inclination (with Φ_LON = 45°). The increase in Δσ/σ_init following the growth of a CMC is inversely proportional to the inclination. This trend is evident in the unbarred simulations and, to a lesser extent, in the barred disk+bulge simulation. As inclination is increased, the fractional change in σ between t₁ and t₂ decreases. A comparison of top and bottom panels shows that the larger CMC (top) produces a 2%–3% larger increase in Δσ/σ_init only for Φ_LON ∼ i ∼ 30°. For other orientations, we see almost no dependence on the mass of the CMC.

We note that the axisymmetric disk-only simulation with the 10⁷ M_☉ CMC shows a slight decrease in σ between t₁ and t₂ at high inclination. In Figure 3 this simulation also showed a decrease in σ between t₁ and t₂ at small R_e. Both of these trends can be attributed to a significant decrease in radial velocity dispersion after the CMC is grown. This decrease radial dispersion is most prominent at small radii and causes the decrease in σ in this simulation for small values of R_e. We therefore conclude that a less massive CMC mass will produce a slightly smaller increase in σ than a more massive CMC, but it will nonetheless produce an increase that is larger in a barred disk than in an unbarred disk.

We defer a discussion of the cause of the decrease in Δσ/σ_init with increasing inclination to the next section (see Figure 7), where we show that this is because the intrinsic velocity dispersions in the radial, azimuthal, and vertical directions (σ_R, σ_ϕ, and σ_z) all increase by roughly the same amount.

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To analyze the distributions of intrinsic velocity anisotropy, we compute the standard deviation of the radial, tangential, and vertical particle velocity distributions σ_R, σ_ϕ, and σ_z of particles enclosed within cylindrical annular bins in R. The bins have a width of 0.06 kpc and contain ≳ 10⁴ particles on average. We use these quantities to compute the tangential anisotropy parameter $\beta _{\phi }= 1 - \sigma _{\phi }^2/\sigma _{R}^2$ and vertical anisotropy parameter $\beta _{z}= 1 - \sigma _{z}^2/\sigma _{R}^2$ as a function of radius. Figure 7 shows (from top to bottom) σ_R, σ_ϕ, σ_z, β_ϕ, and β_z as a function of cylindrical radius R. These quantities are shown for the disk-only models (left) and disk+bulge models (right). Recall that anisotropy values β_ϕ = 0 and β_z = 0 signify that σ_ϕ = σ_R and σ_z = σ_R, respectively. A positive value of β signifies a larger radial velocity dispersion.

In all cases at time t₁, the barred and unbarred models overlap because of the fact that their cylindrically averaged velocity ellipsoids are identical, and they are therefore represented by the dotted black lines. While we recognize that cylindrically averaging the barred models erases physically important nonaxisymmetric features in the shapes of the velocity ellipsoids, we are justified in doing this because the main purpose of these figures is to understand the differences in the measured values of σ that themselves are obtained by averaging over a circular region of projected radius R_e.

The bottom two rows of Figure 7 show that in both the disk-only (left) and disk+bulge (right) simulations, the growth of a CMC in an unbarred potential (red curves) definitively reduces both β_ϕ and β_z relative to the models at t₁ (black curves) over most of the radial range plotted. From examining the top three rows, it is clear that the decreases in β_ϕ and β_z are because σ_ϕ and σ_z increase slightly between t₁ and t₂, but σ_R (top row) remains essentially unchanged, or even decreases slightly between t₁ and t₂. This comes as no surprise given previous studies (e.g., Goodman & Binney 1984; Quinlan et al. 1995; Sigurdsson 2004) that show that the adiabatic growth of a CMC in an axisymmetric system preferentially increases σ_ϕ over σ_R, thus reducing radial anisotropy. We see here that σ_z also increases quite significantly relative to σ_R, resulting in a decrease in β_z. Notice that the increase in σ_R and σ_ϕ due to the growth of the a CMC with M_CMC = 10⁸ M_☉ in the barred galaxies (blue) are slightly larger than the increase due to the smaller CMC (cyan). However, in the unbarred simulations (red/pink), the difference resulting from the two CMCs is negligible, and both are similar to the initial values of σ_R and σ_ϕ (black curves). For the unbarred galaxies only σ_z differs from the initial models.

The growth of both CMCs in the barred simulations results in a significantly larger increase in the radial velocity dispersion than in the corresponding unbarred cases. This is seen in the top row of Figure 7, which shows the blue/cyan curves in both the disk-only (left) and disk+bulge (right) models to be significantly higher than for the initial models at t₁ (black curves) and the unbarred models after the growth of the CMC (red/pink curves). The increase in σ_ϕ and σ_z (second and third rows) in the barred simulations are also significantly larger than the unbarred simulations—especially within R = 0.5 kpc. In general, the barred models at t₂ are more radially anisotropic than the unbarred models. This can be attributed to the dramatic increase in radial dispersion accompanied by only moderate increases in tangential and vertical dispersions.

Evidently the presence of the bar facilitates an increase in radial anisotropy during the growth of the CMC. This supports the idea that the elongated bar orbits are scattered by the CMC allowing the system as a whole to become rounder, without individual orbits becoming more tangential. In fact, Shen & Sellwood (2004) showed that low energy bar supporting orbits are converted to rounder, chaotic orbits by the growth of a CMC. In contrast in the unbarred systems, the adiabatic growth of a CMC induces a more tangentially biased velocity ellipsoid (Quinlan et al. 1995), but angular momentum conservation limits the degree to which matter can flow inward.

We now see that the inverse correlation between Δσ/σ_init and inclination seen in Figure 6 (right) can also be explained by considering Figure 7. In both types of models, σ_z undergoes a significant increase due to the growth of the CMC. At low inclinations, σ_z is the primary contributor to σ, because the system is viewed more or less face-on. Thus, the growth of a CMC produces a noticeable increase in σ. However, at high inclinations, σ is dominated by σ_R and σ_ϕ, which, in the unbarred cases, are hardly affected by the growth of a CMC. As a result, the unbarred models (red/pink) show an inverse relationship between inclination and the change in σ between t₁ and t₂. This is also why in Figure 6 (right) the barred disk (open blue squares) simulation showed a weaker dependence between Δσ/σ_init and inclination.

Interestingly, between t₁ and t₂, β_z decreases at small radii, even in the barred case. This can be attributed to the fact that the black hole scatters the low energy (radial) orbits, producing a more isotropic velocity ellipsoid (Shen & Sellwood 2004). Therefore, a consequence of growing a CMC in a barred or unbarred galaxy is an overall decrease in β_z at small radii.

Lynden-Bell & Kalnajs (1972) first showed that angular momentum in collisionless disks can be transferred outward via emission and absorption at the inner and outer Lindblad resonances. Several subsequent studies (Weinberg 1985; Debattista & Sellwood 2000; Athanassoula 2003) showed that resonant material can exchange angular momentum between the bar and halo of a galaxy. Other recent studies (e.g., Saha et al. 2012) have investigated the transfer of angular momentum between the bar and bulge components. The exchange of angular momentum between morphological components of a galaxy has important implications for that galaxy's dynamical evolution.

When a live halo is present, dynamical friction exerted by the halo on the bar can slow it down by allowing angular momentum exchange with the halo. It is important to note that in our simulations, which incorporate a static halo potential, a time-independent (nearly steady state) bar is not expected to transfer significant amounts of angular momentum, since the torque exerted by such a bar on a star during one half of its orbit is of the same magnitude but opposite sign to the torque exerted on the second half of the orbit (Binney & Tremaine 2008). However, when the potential of the bar is changing with time, as is the case when a central SMBH is growing, or if the bar strength or pattern speed are changing because of dynamical friction with the disk and bulge, a net transfer of angular momentum can result.

While an exhaustive discussion of the transfer of angular momentum from the inner to outer regions of our simulations is beyond the scope of this paper, we briefly consider how the presence of a bar influences such angular momentum exchange in our simulations, and how this is related to the changes we saw in the measured σ and σ_R profiles of barred galaxies, and the differences between barred and unbarred galaxies. In the preceding sections, we showed that changing the mass of the CMC by a factor of 10 alters the observed velocity dispersion by a mere 2%. Therefore, for the remainder of this paper we consider only the 10⁸ M_☉ simulations, while examining the cause of the differences in the evolution of the barred and unbarred galaxies.

Figure 8 shows the cylindrically averaged mass density profiles as a function of radius for the initial models and for the barred and unbarred models after the growth of the CMC. The right-hand panels plot the fractional difference in the surface mass density between the barred and unbarred models: ΔΣ_ax = (Σ_barred − Σ_ax)/Σ_ax as a function of cylindrical radius. The increase in central mass surface density is between 5% and 18% higher in the barred galaxy than in unbarred galaxy (although the mass of the CMC is the same).

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Figure 9 shows the cylindrically averaged velocity dispersion profiles as a function of radius for the initial models and for the barred and unbarred models after the growth of the CMC. The right-hand panels plot the fractional difference in the intrinsic velocity dispersion between the barred and unbarred models: Δσ_ax = (σ_barred − σ_ax)/σ_ax as a function of cylindrical radius. The increase in velocity dispersion is systematically higher by 5% in the barred galaxy in the disk-only case (top right panel) and between 2% and 5% higher in the barred disk+bulge model (bottom right panel).

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We define $\Delta \tilde{\sigma }/\tilde{\sigma }_{\mathrm{init}}$ as the fractional change (between t₁ and t₂) of the 3D intrinsic velocity dispersion $\tilde{\sigma }= \sqrt{\vphantom{A^A}\smash{{(\sigma _{R}^2+\sigma _{\phi }^2+\sigma _{z}^2)}}}$, for all particles within the same cylindrical volume of radius R_e. We define ΔM_enc/M_init as the fractional change (between t₁ and t₂) in the mass enclosed by a specified cylindrical radius (note that ΔM excludes the mass contribution due to the CMC). In Figure 10 we plot $\Delta \tilde{\sigma }/\tilde{\sigma }_{\mathrm{init}}$ versus ΔM_enc/M_init, for four different values of R_e. Adding in the contribution of M_BH would shift all the points toward the right quite significantly for the smaller values of R_e (M_init ∼ 10⁸ M_☉ ∼ M_BH for R_e = 0.12) but cause only a small rightward shift for the larger values of R_e (M_init ∼ 3 × 10⁹ M_☉ ≫ M_BH for R_e = 0.9).

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A clear dichotomy exists between the barred (blue) and unbarred cases (red). At every value of R_e, the barred models shows both a larger fractional increase in the enclosed mass and a larger fractional increase in the velocity dispersion of that. Thus, the presence of a bar during the growth of a CMC facilitates both a higher mass increase within a specified radius and a higher 3D stellar velocity dispersion. This is clear evidence that angular momentum transport in the barred simulations has facilitated the increase in both the enclosed mass and the velocity dispersion.

It is also interesting to note that especially in the two smaller radial bins (R_e = 0.12, 0.24), although the increase in mass in the disk+bulge models (solid dots) is significantly larger than it is in the disk-only models (squares), the 3D velocity dispersion is larger in the disk-only models. Again this is due to the fact that in the disk-only case a larger fraction of the x₁ bar orbits survive the growth of the CMC, while in the disk+bulge case these orbits are more readily destroyed (most probably by the enhanced central density arising from the inflowing disk+bulge material).

As final evidence for our claim that angular momentum transport by the bar plays a significant role in the velocity dispersion increase, in Figure 11 we examine the fractional change in the average specific angular momentum of stars in the disk-only simulations (left) and disk+bulge simulations (right) for the barred (blue) and unbarred models at time t₂ relative to the value at t₁. In the disk-only models, it is clear that the change in the average specific angular momentum of stars in the barred systems is negative over most of the radial range plotted—indicating that on average, stars have lost angular momentum. In contrast, the corresponding unbarred system stars in the inner region have gained angular momentum at time t₂ relative to t₁, because of the adiabatic infall that gives rise to the growth of the central cusp that follows the growth of the CMC (Quinlan et al. 1995). Since this system has only weak spiral features incapable of transporting significant angular momentum, the specific angular momentum of stars has increased as the cusp formed.

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Recall from Figure 10 that at each radius the fractional increase in enclosed mass ΔM_enc/M_init within each cylindrical radial bin is always larger in the barred system than for the corresponding unbarred system. In a collisionless simulation, the net angular momentum of the system is conserved. If significant angular momentum transport does not occur, then an increase in specific angular momentum is expected as matter is drawn inward. The fact that the angular momentum per particle in most of the inner 0.8 kpc of the barred galaxy has decreased shows that some of the angular momentum must have been transported outward by the bar. In the disk+bulge models (right), we see that change in the specific angular momentum of the barred galaxy (blue) is always smaller than for the unbarred galaxy also pointing to outward transport.

Thus, Figures 8–11 clearly demonstrate that the time-dependent bar potential resulting from the growing SMBH results in angular momentum transport that is responsible for increasing the central mass of stars and the radial anisotropy of orbits.

The measured values of M_BH in barred galaxies compiled by Graham et al. (2011) come from a variety of dynamical measurement techniques: gas kinematics, stellar dynamics, and reverberation mapping. In the most recent compilation of galaxies with dynamically measured SMBH masses (McConnell & Ma 2013) consisting of 72 galaxies, nearly 50% of the SMBH mass measurements are derived via stellar dynamical methods. Stellar dynamical methods entail modeling the nuclear stellar kinematics either via the technique referred to as the Schwarzschild orbit superposition method (e.g., Schwarzschild 1979; van der Marel et al. 1998; Cretton et al. 1999; Gebhardt et al. 2003; Valluri et al. 2004; van den Bosch et al. 2008) or by solving the axisymmetric Jeans equations (Binney & Tremaine 2008; Cappellari 2008). Both methods simultaneously optimize the fit to the 3D mass distribution (including the mass of the unknown SMBH), the surface brightness distribution, and the observed LOSVDs to constrain the best fit values of M_BH and the M/L ratio of the stars. A small number of elliptical galaxies in this sample have been modeled with a (nonrotating) triaxial orbit superposition code (e.g., van den Bosch & de Zeeuw 2010), but most stellar dynamical M_BH measurements have been made with axisymmetric modeling codes.

We examined the table of 72 galaxies with dynamically measured SMBH presented by McConnell & Ma (2013) and find that of the sample of ∼35 galaxies in which M_BH was measured via stellar dynamical methods, 17 are S0 or spiral galaxies. Of these, five galaxies (29%) are classified as barred in the NASA Extragalactic Database,⁹ and another six (35%) are edge-on galaxies in which a bar would be difficult to detect should it exist. We note that the total fraction of barred + edge-on galaxies (64%) is comparable to the fraction of local disk galaxies that contain bars (e.g., Knapen 1999; Eskridge et al. 2000; Menéndez-Delmestre et al. 2007; Marinova & Jogee 2007; Sheth et al. 2008). The black hole masses for all these galaxies have been obtained using axisymmetric stellar dynamical codes. In this section we examine qualitatively the possible systematic biases that the assumption of axisymmetry might have on the measured mass of the SMBH in a barred galaxy. We defer a more quantitative study to a future paper.

The process of measuring the dynamical mass of an SMBH from the kinematics of stars in the nucleus suffers from the well-known mass-anisotropy degeneracy (Binney & Mamon 1982). In this classic paper the authors showed that the degeneracy arises because orbits of stars in elliptical galaxies and the bulges of disk galaxies can have a wide range of possible velocity anisotropy distributions. A large line of sight central stellar velocity dispersion in the nucleus could be the result of a large central SMBH about which stars move on primarily tangential orbits or could equally well be the result of stars on primarily radial orbits moving around a much smaller (or no) central SMBH. In axisymmetric and spherical models, the degeneracy between mass and velocity anisotropy can be lifted by the use of information contained in the shapes of the stellar LOSVDs. It is customary to use Gauss-Hermite coefficients to represent the deviations of an LOSVD from a Gaussian shape (van der Marel & Franx 1993; Gerhard 1993). In axisymmetric or spherical systems, stars on predominantly radial orbits will give rise to LOSVDs with positive h₄ parameters, while stars on predominantly tangential orbits produce LOSVDs with negative h₄ parameters. An isotropic velocity distribution will produce an LOSVD with h₄ ∼ 0. Degeneracy between mass and anisotropy is lifted by ensuring that the orbit superposition method simultaneously fits at least σ_los and h₄.

In the immediate vicinity of a SMBH, the presence of a large fraction of stars at high velocities causes an increase in the amplitudes of the high velocity wings of the LOSVD, resulting in large positive values of h₄ (van der Marel 1994). These high velocity tails provide strong constraints on kinematics of stars in the vicinity of the SMBH and on its mass.

Currently there are no stellar dynamical modeling codes that can measure the masses of SMBHs in barred galaxies. However, recently Lablanche et al. (2012) used an axisymmetric stellar dynamical modeling code (Jeans Anisotropic MGE (JAM) method, Cappellari 2008) to assess the accuracy with which the stellar M/L ratio and intrinsic velocity anisotropy could be recovered. They applied the method to a sample of N-body simulations of barred S0 galaxies and showed that biases in the determination of M/L primarily arise because of the application of an axisymmetric modeling code to a barred galaxy. They find that for Φ_LON = 45° and i > 30, the measured stellar M/L ratio is essentially unbiased, but errors of up to 15% can arise because of varying orientation of the bar and inclination of the disk. Furthermore, they find that when a bar is present, the inferred velocity anisotropy can be significantly in error.

While it is beyond the scope of this paper to carry out a similar exercise to assess the systematic biases that would be introduced into the measured masses of SMBHs by using axisymmetric stellar dynamical codes, we will qualitatively examine the nature and the direction of the bias.

Figure 12 shows the stellar kinematic difference maps in the inner ±1 kpc region for the pure disk simulations (top row) and the disk+bulge simulations (bottom row). The maps show the differences in the kinematic quantities v_los, σ_los, h₃, and h₄ and the projected mass density Σ. In each panel we plot the difference in a specific quantity between barred and unbarred models after the growth of the SMBH in each pixel in the field of view. The angle of inclination of the disk and Φ_LON are both set to 45°. Pixels that are colored green indicate that there is no difference between the barred and unbarred models; yellow and red pixels imply that the quantity in the barred galaxy (v_los, σ_los, h₃, h₄, Σ) is higher, and blue pixels indicate that the quantity in the barred galaxy is lower than it is in its unbarred counterpart. The values in parenthesis above each panel indicate the range of the difference in the quantities. Notice that the difference maps show that σ_los in the inner regions of the map is always red/yellow, indicating that it is systematically higher in the barred models than in the unbarred models (see also Figure 9), while h₄ is green/blue, signifying that it is generally lower than in the unbarred models.

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Therefore, if a barred galaxy is modeled with the assumption of axisymmetry, the dynamical model will attempt to fit the negative h₄ by putting a large fraction of orbits on tangential orbits, while the requirement to simultaneously fit a large σ_los would require a larger enclosed mass than one would infer from the same mass distribution in an unbarred model. The standard approach in stellar dynamical modeling is to hold the M/L ratio of the galaxy fixed (however, see McConnell et al. 2013). When the M/L ratio is held fixed, it is largely determined by kinematic constraints outside the sphere-of-influence of the SMBH, and the M/L ratio of stars in inner part of the bulge is likely to be underestimated. Since bar-induced evolution can significantly increase mass inflow from large radii to small radii, the standard practice of holding M/L fixed will also result in M_BH being overestimated. A striking example of this is seen in NGC 4151, which has recently been modeled by Onken et al. (2013).

We also examined the LOSVDs of stars within the sphere-of-influence of the CMC (r_s) in both the barred and unbarred models (for i = 45, Φ_LON = 45). We find that although h₄ is negative on average within R_e, within r_s—the region where the CMC dominates the dynamics of stars—LOSVDs of the barred galaxies in our simulations have 30%–50% larger values of h₄ than the corresponding unbarred galaxies (for the same mass of CMC). This is due to a combination of the increased radial velocity anisotropy of stars resulting from the growth of the CMC (seen in Figure 7) and streaming motions along the bar. Since the σ_los values are also about 5% larger in the barred models than in the unbarred models, we predict that even if the sphere-of-influence of the SMBH is resolved, the anisotropic velocity distribution will also result in an overestimate of M_BH. The idea that the high central velocity dispersions of nearly end-on bars can be mistaken for central black holes is not new (Gerhard 1988) and has recently been invoked as an alternative explanation (Emsellem 2013) for the claimed overmassive black hole in NGC 1277 (van den Bosch et al. 2012).

Using unbarred dynamical modeling codes to measure the masses of SMBHs in barred galaxies is therefore likely to result in a systematic overestimate of M_BH, regardless of whether the sphere-of-influence of the SMBH is resolved or not. Since the fraction of barred galaxies with stellar dynamical determinations of M_BH is currently quite a small fraction of all the SMBH measurements used in the most recent M_BH–σ relation, this is unlikely to significantly alter this relation or offset of barred galaxies from it. However, the effect of using unbarred models to measure the mass of SMBHs should be examined quantitatively in the future.