EVIDENCE FOR A RECEDING DUST SUBLIMATION REGION AROUND A SUPERMASSIVE BLACK HOLE

According to the accumulated interferometric observations of NGC 4151 and the compiled flux measurements, the radius of the innermost dust distribution has apparently receded in response to the heating flux increase with a significant time delay. The observed delay implies that the overall timescale for the destruction of the innermost dust distribution is at least a few years.

In Figure 3, we also plotted near-IR reverberation radii. These radii have overall increased from the 1970s (time lag of ∼18 days) to the 1990s (∼35 days; Oknyanskij et al. 1999) or the 2001–2006 period (∼50 days; Minezaki et al. 2004; Koshida et al. 2009). The radius also seems to have decreased within the latter 2001–2006 period, which was found by Koshida et al. (2009).

In Hönig & Kishimoto (2011), we re-analyzed the data of Koshida et al. (2009) and used the peak in the cross-correlation function to measure lags, rather than the centroid (which is adopted by Koshida et al.). This ensures the consistency with the other reverberation measurements by Oknyanskij et al. (1999) and those implemented and compiled by Suganuma et al. (2006). The results are plotted in Figure 3. We found slightly shorter lags with larger uncertainties (see Section 4 of Hönig & Kishimoto 2011), but the tendency of the decreasing radius is still roughly consistent with that found by Koshida et al. (2009).

The long-timescale increase of the radius from the 1970s to the 1990s and its subsequent decrease possibly seen in the 2001–2006 period could well be due to the historical flux increase peaking around 1996 (Figure 3) and the subsequent significant decrease in the flux. The former probably destroyed grains in the innermost region, and the latter lead to the reformation of the dust grains in the same region. If this is the case, there must have been quite a long delay of several years between the decrease in flux and that in the reverberation radii measured in the 2001–2006 period. This could, in principle, constrain the dust reformation timescale, implying that it might have taken several years for the innermost dust grains within a ∼0.05 pc radius region to re-form. This reformation timescale was indeed already inferred by Oknyanskij et al. (2006).¹⁰

Note that here we discuss the destruction of the dust distribution, but not necessarily the microscopic destruction of each grain. This distribution radius does not seem to be changing sensitively to the instantaneous change in the incident heating radiation (Pott et al. 2010; Kishimoto et al. 2011; Hönig & Kishimoto 2011). This is also shown in Figure 4(a), where we plot the measured radii against the instantaneous incident flux (in practice, we calculated the flux averaged over the past half a year from the radius measurement). The radius does not show the expected L^1/2 dependence (as already found by Koshida et al. 2009), which is given by the relationship between the reverberation radius and luminosity over a sample of objects (Suganuma et al. 2006) and shown as the dashed line in Figure 4(a). This probably means that the innermost dust grains are in the optically thick accumulations or clouds so that only the dust grains at the surface region sublimate first (Hönig & Kishimoto 2011). This would slowly be followed by the increase of the overall radius of the distribution when the incident flux level is persistently kept high.

Zoom In Zoom Out Reset image size

One simple model to quantify the possible destruction and reformation of the dust distribution is to postulate that the distribution radius is approximately determined by the average of the incident flux over some past period, or an average "retro" flux. The retro flux at a given observing time of a radius measurement is set by averaging the optical flux over the past t_retro years back from the radius observing time. The reformation time and destruction time of the dust distribution are not necessarily the same, with the former likely being longer than the latter. Here, we take the simplest first approximation to use a single averaging timescale t_retro, which can be regarded as an overall mean of the two timescales.

Calculating the deviation from the square-root proportionality and also the Spearman's rank correlation coefficients between the radius and retro flux, we searched for the best value of t_retro for the compiled flux and radius measurements, finding it to be ∼6 yr (Figures 4(c) and (d)). Here we calculated the deviation from the square-root proportionality for reverberation radii using the L^1/2 fit by Suganuma et al. (2006), and the deviation for interferometric radii by fitting a L^1/2 relation separately. This is because we anticipate the latter radii to be slightly but systematically larger than the former. More specifically, the reverberation measurement using the peak of the cross-correlation function tends to probe the fastest responding material (Koratkar & Gaskell 1991), i.e., give a radius close to the inner boundary radius, while the interferometric radius represents an overall average radius of the brightness distribution, thus slightly larger than the reverberation radius. This difference between the interferometric and reverberation radius has been seen in other different objects (Kishimoto et al. 2011). Similarly, we calculated the rank correlation coefficient separately for the two different radii.

Figure 4(b) shows the radii as a function of the retro flux for the case of t_retro = 6 yr. The reduced $\chi ^2_{\nu }$ for the deviation from the square-root proportionality has a minimum of 1.8 at this t_retro (Figure 4(c)). The confidence level of rejecting a null hypothesis for the correlation between the interferometric radii and the retro flux has a peak of 93% (or 1.8σ) at this t_retro, while the rank correlation coefficient for the reverberation radii becomes positive only at t_retro ≳ 6 yr though not with a high confidence level (Figure 4(d)). All these indicate that the radius is relatively well correlated with the long-term average of the incident flux rather than the instantaneous flux, and consistent with being proportional to the square-root of the long-term average luminosity. Finally, Figure 4(b) also suggests that the interferometric radius is larger than the reverberation radius by a factor of ∼1.8 for a given retro luminosity in this AGN.

Koshida et al. (2009) have already pointed out that, in their light curves between 2001 and 2006, the reverberation radius in the early part (∼2001) seems to be affected by the past bright flux. Indeed, we see the historical peak around 1996 (see Figure 3). Therefore, it is better to put the data into a perspective covering a much wider time range. With the analysis done here, we indeed find a likely timescale of ∼6 yr, rather than the 1 yr time scale suggested by Koshida et al. (2009). This is depicted in Figures 4(c) and (d).

We can relate this dust timescale t_retro to the radial UV optical thickness of the sublimation region, or more specifically, τ_UV over the inner radial length scale of ∼0.05 pc in this AGN. This is coarsely given as τ_UV ≃ t_retro/t_dest, where we denote the (microscopic) dust destruction timescale as t_dest, within which the region of UV optical thickness unity will be destroyed when flux is increased. The dust evaporation occurs when the dust vapor pressure dominates the ambient dust gas pressure, and the destruction timescale t_dest can roughly be estimated as a function of the dust vapor pressure P_vap and the dust temperature T. This is given as (see, e.g., Phinney 1989; Guhathakurta & Draine 1989; Kimura et al. 2002; Kobayashi et al. 2011)

$$\begin{equation} t_{\rm dest} \simeq \frac{4}{3} \pi a^3 \rho \left[ 4 \pi a^2 \sqrt{ \frac{m}{2 \pi k T} } P_{\rm vap} \right]^{-1} \end{equation} \tag{ 1 }$$

$$\begin{equation} \simeq 4.7 \times 10^{1} \left(\frac{a}{0.1 \ \mu {\rm m}} \right) \left(\frac{T}{T_0} \right)^{1/2} \left(\frac{P_{\rm vap}}{P_0} \right)^{-1} \ {\rm (days)}, \end{equation} \tag{ 2 }$$

where a is the radius of dust grains assumed to be spherical, m the dust molecular mass, k the Boltzmann constant, and ρ the bulk density of a dust particle.

Here we normalized T by T₀ = 1400 K, which is the typically observed color temperature of the dust SED in the near-IR and adopted here as a proxy for the actual temperature of sublimating grains (see below for further information on the temperature). We used the values of m and ρ for silicate grains, but adopting the m and ρ values for other grains does not change the order of magnitude of the estimated t_dest.

The estimation is more sensitive to vapor pressure P_vap(T). We normalized this by the constant P₀ ≡ n₀kT₀, with n₀ = 10⁶ cm⁻³, i.e., P₀ ≡ 2 × 10⁻⁷ dyn cm⁻² (see below for further information on the density), because silicate grains give P_vap(1400 K) ∼ P₀, adopting constants from the literature (Guhathakurta & Draine 1989; Kimura et al. 2002). The vapor pressure as a function of T is given as P_vap∝e^−A/T, where the constant A depends on the dust species and A ∼ 6.8 × 10⁴ K for silicate grains, and is thus very sensitive to T. However, we can still use a fixed temperature for this estimation as long as t_dest is shorter than the flux variation timescale, which seems to be the case here (see Equation (2)). Note that the observed near-IR color temperature is probably a lower limit for the actual temperature of sublimating grains, and thus t_dest can even be shorter.

Thus, we have

$$\begin{equation} \tau _{\rm UV} \simeq 4.7 \times 10^{1} \left(\frac{t_{\rm retro}}{\rm 6 \ {\rm yr}} \right) \left(\frac{a}{0.1 \ \mu {\rm m}} \right)^{-1} \left(\frac{T}{T_0} \right)^{-1/2} \left(\frac{P_{\rm vap}}{P_0} \right). \end{equation} \tag{ 3 }$$

Therefore, the region is estimated to be very optically thick over the radial length scale of ∼0.05 pc. This is consistent with the general expectation from the optically thick equatorial obscuration. Conversely, in order to have such a high optical thickness, the destruction timescale t_dest must be tens of days (recall τ_UV ≃ t_retro/t_dest), and this requires the vapor pressure P_vap to be on the order of P₀. This corresponds to the sublimation at ∼1400 K in the case of silicate grains. In the case of graphite grains, adopting constants again from the literature (Phinney 1989; Guhathakurta & Draine 1989), they should be sublimating at a higher temperature of ∼1800 K, at which they have P_vap ∼ P₀.

This vapor pressure in the sublimation region also implies that the ambient density of gas-phase dust constituents is on the order of ∼10⁶ cm⁻³ (denoted as n₀ above), corresponding to the hydrogen number density of ∼10⁹ cm⁻³ for the normal interstellar medium abundance. It is worth noting that this density is comparable to that estimated for the broad-line region.

It is certainly true that the inference and discussions here are based on a limited number of radius measurements. Further near-IR interferometric observations of this well-monitored object will certainly be yielding. Based on the available optical light curve, we can actually predict that such measurements in the coming few years will show a large radius similar to what we measured in 2012. These and further measurements will better constrain the dust-related timescales and will lead to new physical insights and advance our understanding of the physics and structure of the innermost dusty region.