MASSIVE BLACK HOLES IN STELLAR SYSTEMS: "QUIESCENT" ACCRETION AND LUMINOSITY

To model the accretion rate, we must choose three-dimensional stellar distributions for the various stellar systems we consider here. For globular clusters and dwarf spheroidals, we assume the stars to be distributed following a Plummer profile:

$$\begin{equation} \rho (r)=\frac{3}{4\pi }\frac{M_{\rm stellar}}{a^3}\left(1+\frac{r^2}{a^2}\right)^{-5/2}, \end{equation} \tag{ 1 }$$

where a = R_eff is the core radius.

Early-type galaxies and nuclear clusters are modeled as Hernquist spheres:

$$\begin{equation} \rho (r)=\frac{M_{\rm stellar}}{2\pi }\frac{r_h}{r(r+r_h)^3}, \end{equation} \tag{ 2 }$$

where the scale length r_h ≈ R_eff/1.81. To fully define the stellar systems we have only to relate the stellar mass, M_stellar, to the effective radius, R_eff.

For globular clusters, we recall that simulations by Baumgardt et al. (2004, 2005) suggest that globular clusters with MBHs have relatively large cores a ∼ 1–3 pc (see also Trenti et al. 2007). Consistent results were found using Monte Carlo simulations (Umbreit et al. 2009) and in analytical models (Heggie et al. 2007). The core radii (where measured) of globular clusters, hosting intermediate-mass BH candidates, are roughly consistent with the values we considered, ranging from approximately 0.5 pc in M15 (Gerssen et al. 2002; core radius from the catalog presented in Harris et al. 2010³), up to few pc in ω Centauri (Noyola et al. 2008).

For early-type galaxies, we adopt the fits by Shen et al. (2003) for stellar mass versus effective radius in Sloan Digital Sky Survey galaxies:

$$\begin{equation} R_{\rm eff}=2.5\,\left(\frac{M_{\rm stellar}}{4\times 10^{10}\,{M_\odot }}\right)^{0.56}\, {\rm kpc}. \end{equation} \tag{ 3 }$$

The scatter is roughly 0.2 dex for stellar masses between 10⁸ M_☉ and 10¹⁰ M_☉: σ_ln R = 0.34 + 0.13/[1 + (M_stellar/4 × 10¹⁰ M_☉)].

We note that for five galaxies (NGC 4697, NGC 3377, NGC 4564, NGC 5845, NGC 821) where measurements of the effective radius are available (along with stellar masses, BH masses, and gas density—see Soria et al. 2006a and Marconi & Hunt 2003) the fits derived by Shen et al. (2003) provide values of the effective radius roughly 55% times larger than the measured value. This is likely due to Shen et al. (2003) definition of effective radius as the radius enclosing 50% of the Petrosian flux. This definition differs from the standard definition of projected radius enclosing half of the total luminosity. We therefore scale the fit for early-type galaxies by a factor of 0.55 for consistency. As shown below (Figure 3), this small correction does not influence the accretion rate we derive.

For dwarf spheroidals, we fit the data presented in Walker et al. (2009, 2010). We assume a constant mass-to-light ratio of 2 for the visible component and derive stellar masses from the total luminosities:

$$\begin{equation} R_{\rm eff}=0.93\,\left(\frac{M_{\rm stellar}}{10^7 \,{M_\odot }}\right)^{0.36}\, {\rm kpc}, \end{equation} \tag{ 4 }$$

where the uncertainties in the slope and in the normalization are 0.06 and 0.2 dex, respectively. Finally, for nuclear clusters we fit the stellar mass versus effective radius data presented in Seth et al. (2008), leading to

$$\begin{equation} R_{\rm eff}=7.9 \times 10^{-3}\,\left(\frac{M_{\rm stellar}}{10^7 \,{M_\odot }}\right)^{0.17}\, {\rm kpc}, \end{equation} \tag{ 5 }$$

where the uncertainties in the slope and in the normalization are 0.05 and 0.3 dex, respectively. These scalings are shown in Figure 1.

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We develop here a simple geometrical model to estimate the accretion rate onto an MBH in a stellar system, fueled by mass loss from stars (Quataert et al. 1999). If a star is located at a distance r from the MBH, and if it produces an isotropic wind, with velocity v_wind, only the fraction of gas which passes within the accretion radius of the MBH,

$$\begin{equation} R_{\rm acc}=2GM_{\rm BH}\big/\big(v^2_{\rm wind}+\sigma ^2+c^2_s\big), \end{equation} \tag{ 6 }$$

can be accreted (ignoring gravitational focusing). Here, σ² = GM_stellar/(2.66r_h) is the velocity dispersion of the stellar system at the half-mass radius. For a Hernquist profile, where the density in the inner region ρ ∝ r⁻¹, the velocity dispersion decreases toward the center. Estimating σ at the half-mass radius gives a conservative lower limit to the accretion radius, and hence the accretion rate. Following Miller & Hamilton (2002), we assume that in Equation (6) the sound speed c_s = 10 km s^-1, and, v_wind = 50 km s^-1 as reference values, although we study the effect that a different v_wind has on our model (see Figure 2).

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If σ ≫ v_wind, R_acc depends only on the properties of the potential well of the stellar distribution, not on the wind properties. In particular, R_acc ≃ M_BHR_eff/M_stellar ≃ 10⁻³R_eff if M_stellar = 10³M_BH. Note that, at fixed BH mass, the more massive the galaxy, the smaller R_acc is, as the scaling of R_eff with M_stellar is a power law with exponent less than one (see, e.g., Equation (3)). On the other hand, if σ ≪ v_wind, R_acc depends only on the wind velocity. These two limits are apparent in Figures 2 and 3, and they will be discussed in the next section.

Geometrical considerations suggest that for r>R_acc

$$\begin{equation} \dot{M}_{{\rm acc},*}=\frac{1}{2}\dot{M}_*\left[1-\left(1-\frac{R^2_{\rm acc}}{r^2}\right)^{1/2}\right], \end{equation} \tag{ 7 }$$

where $\dot{M}_*$ is the mass-loss rate from the star. If the star lies within R_acc, we consider $\dot{M}_{{\rm acc},*}=\dot{M}_*$. Equation (7) implicitly assumes that the stars have a spherically symmetric distribution and that their velocity field (and, as a consequence, the velocity field of the wind) is isotropic. In a rotating stellar system, the presence of net angular momentum of the gas can change the accretion rate onto the BH (e.g., Cuadra et al. 2008). A study of the dependence of the accretion rate on the degree of rotational support of the stellar distribution is beyond the scope of this paper.

The total contribution from all stars is found by integrating over the density profile of the stellar system:

$$\begin{equation} \dot{M}_{\rm acc}=\int _{R_{\rm acc}}^{\infty } {4\pi r^2\frac{\rho (r)}{\langle m_*\rangle }\dot{M}_{{\rm acc},*} dr}, \end{equation} \tag{ 8 }$$

where 〈m_*〉 is the mean stellar mass and ρ is given by Equations (1) and (2). The normalization in Equation (8) is given by the cumulative mass-loss rate of all the stars in the stellar structure that we estimate following Ciotti et al. (1991):

$$\begin{equation} \dot{M}_{\rm gal}=1.5\times 10^{-11}\,{M_\odot }\,{\rm yr^{-1}}\frac{L_B}{L_{B,\odot }} \left(\frac{t_*}{15\, {\rm Gyr}}\right)^{-1.3}, \end{equation} \tag{ 9 }$$

where t_* is the age of the stellar population and L_B is the total luminosity of the stellar system. We set t_* = 5 Gyr for dSphs and nuclear star clusters, and t_* = 12 Gyr for early-type galaxies and globular clusters. We derive B-band luminosities from stellar masses assuming a mass-to-light ratio of 5 in the B band.

We obtain an upper limit of the luminosity of the MBH by assuming that the whole $\dot{M}_{\rm acc}$ is indeed accreted by the MBH.

Figure 2 shows the resulting accretion rate for a central MBH in different stellar systems, where we assume that the MBH mass scales with the mass of stellar component, M_BH = 10⁻³M_stellar (Marconi & Hunt 2003; Häring & Rix 2004), and we have considered v_wind a free parameter. We have assumed that R_eff scales exactly with M_stellar following the relationships discussed above. Note that for high values of the stellar masses in early-type galaxies and nuclear star clusters, the accretion rate and R_acc do not depend on the wind velocities. In these cases σ ≫ v_wind, and the accretion rate depends only on the properties of the host stellar structure and on the BH mass (see the discussion of Equation (6) above).

In Figure 3 we instead fix v_wind and allow for a scatter in the mass-size relationship. For globular clusters we assume R_eff = 1 pc, R_eff = 2 pc, and R_eff = 4 pc. For galaxies, the middle curve shows the best-fit R_eff for a given stellar mass value (Equations (1), (2), and (3)), the top curves assume that R_eff is half the best-fit value, and the bottom curves assume that R_eff is twice the best-fit value. We have assumed M_stellar = 10⁵–10⁷M_☉ for globular clusters, M_stellar = 10⁵–10⁸M_☉ for dwarf spheroidals and nuclear star clusters, and M_stellar = 10⁸–10¹¹M_☉ for early-type galaxies, limiting our investigation to the mass ranges probed by Shen et al. (2003), Walker et al. (2009), and Seth et al. (2008). In this plot the v_wind ≫ σ limit of Equation (6) becomes evident: at low stellar masses, for every type of stellar distribution but for the early-type galaxies, R_acc does not depend on R_eff, and it is determined only by the BH mass and the assumed v_wind. The early-type galaxies generate deeper potential wells, never reaching the v_wind ≫ σ limit.

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The bolometric luminosity of the MBH can be written as

$$\begin{equation} L_{\rm bol}=\epsilon \dot{M} c^2, \end{equation} \tag{ 10 }$$

where represents the fraction of the accreted mass that is radiated away. The nature of the accretion process, and the consequent value of , is rather uncertain. AGNs accrete through accretion disks with a high efficiency ( ∼ 0.1). Supermassive BHs at the centers of quiescent galaxies, including the Milky Way, can have luminosities as low as ∼10⁻⁹ to 10⁻⁸ of their Eddington values (e.g., Loewenstein et al. 2001), and well below the luminosity one would estimate assuming ∼ 0.1.

Following Merloni & Heinz (2008), we define λ ≡ L_bol/L_Edd, where L_bol is the bolometric luminosity and L_Edd = 4πGM_BHm_pc/σ_T ≃ 1.3 × 10³⁸(M_BH/M_☉) erg s⁻¹ is the Eddington luminosity. We write the radiative efficiency, , as a combination of the accretion efficiency, η, that depends only on the location of the innermost stable circular orbit,⁴ here assumed to be η = 0.1, and of a term, η_acc, that depends on the properties of the accretion flow itself: = ηη_acc. We also define $\dot{m}=\eta \dot{M} c^2/L_{\rm Edd}$.

For "radiatively efficient" accretion, η_acc = 1. To estimate the X-ray luminosity, we apply a simple bolometric correction and assume that the X-ray luminosity is a fraction η_X of the bolometric luminosity. Ho et al. (1999) suggest that for low-luminosity AGN, with Eddington rates between 10⁻⁶ and 10⁻³ the luminosity on the [0.5–10] keV band represents a fraction 0.06–0.33 of the bolometric luminosity. We assume here η_X = 0.1, so that $L_{\rm X}=\eta _{\rm X}\, \epsilon \, \dot{M} c^2$, where = η = 0.1. We refer to this model as "radiatively efficient."

Since the accretion rates we find are very sub-Eddington, we assume, in a second model, that the accretion flow is optically thin and geometrically thick. In this state the radiative power is strongly suppressed (e.g., Narayan & Yi 1994; Abramowicz et al. 1988). Merloni & Heinz (2008) suggest that this transition occurs at $\dot{m}<\dot{m}_{\rm cr}=3\times 10^{-2}$, and that $\eta _{\rm acc}=(\dot{m}/\dot{m}_{\rm cr})$, so that $\epsilon =\eta (\dot{m}/\dot{m}_{\rm cr})$. The X-ray luminosity is therefore: $L_{\rm X}=\eta _{\rm X}\, \epsilon \, \dot{M} c^2$, where again η_X = 0.1. We refer to this model as "radiatively inefficient."

In Figure 4 we show the accretion rate, in Eddington units, when we assume η = 0.1. Hereafter, we vary the mass of the MBH from 100 M_☉ to 10⁴ M_☉ for globular clusters, since there is no firm conclusion that MBHs' masses scale with the mass of the stellar component as M_BH = 10⁻³M_stellar. For galaxies we assume instead an upper limit to the MBH mass corresponding to M_BH = 2 × 10⁻²M_stellar, a lower limit of 100 M_☉ for dSph and nuclear clusters and a lower limit of 10⁴ M_☉ for early-type galaxies.

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We complete the exercise by adding observational results for a sample of 29 early-type galaxies where both dynamical BH mass and X-ray luminosity (Pellegrini 2010)⁵ are available (see also Soria et al. 2006a; Gültekin et al. 2009a). Twenty-four of these galaxies also report the stellar mass of the bulge (Marconi & Hunt 2003). For those galaxies where the bulge mass is unavailable we derive stellar masses from B-band magnitudes. For these galaxies we also derive B-band luminosities directly from L_V (Gültekin et al. 2009b), assuming B − V = 1 (Coleman et al. 1980), and we check that our choice of a mass-to-light ratio of 5 agrees well with this complementary technique to derive L_B.

Figure 5 compares the luminosities we predict for these galaxies to the measured X-ray luminosity of the galaxies (or upper limits). In agreement with the conclusions of Pellegrini (2005) and Soria et al. (2006b) the radiatively inefficient case best fits the luminosity of most systems, except the most luminous ones. Overall, even the radiatively inefficient case slightly overestimates the luminosity, at least at the high-mass end, and we find that, for instance, η_X = 0.03 provides a much better fit. As discussed by Pellegrini (2010), there seems to be a smooth transition between radiatively inefficient and radiatively efficient accretion.

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We also estimate the X-ray luminosities for Milky Way dSphs with stellar mass >10⁵ M_☉, where we use directly R_half and M_stellar from Walker et al. (2010). We assume in one case that M_BH = 10⁻³M_stellar, and in another case that BHs have a fixed MBH mass of 10⁵ M_☉, based on models presented in van Wassenhove et al. (2010). We note that in all these cases the X-ray luminosities for MBHs in dwarf galaxies are below 10³⁵erg s^-1. Figure 6 summarizes our primary results: predicted X-ray luminosities for different stellar systems.

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