Scaling Relations of the Dusty Torus with Luminosity and the Broad-Line Region

TL;DR

This work jointly analyzes dust-reverberation lags from WISE W1/W2 and Hβ BLR lags to map the dusty torus size as a function of AGN luminosity in 182 low-redshift AGNs. Employing accretion-disk contamination corrections and rigorous lag measurements via ICCF and MICA, the study finds torus sizes scale sub-linearly with optical and MIR luminosities (R_tor ∝ L^{~0.34}), with intrinsic scatters ~0.15–0.16 dex, and a torus radius typically 10–14 times larger than the BLR. The BLR–torus size relation is near-linear, with R_BLR ≈ R_tor^{1.1–1.28}, consistent with interferometric constraints, and the torus size–luminosity trend shows hints of dependence on Eddington ratio, possibly due to anisotropic disk emission and self-shadowing in slim disks. These results imply that MIR dust lags can serve as independent tracers for AGN structure and SMBH mass estimation, and they motivate further high-cadence MIR monitoring to refine the torus scaling, particularly at lower luminosities.

Abstract

We measure and compare the size of the dusty torus with active galactic nucleus (AGN) luminosity and the size of the broad-line region (BLR), using a sample of 182 AGNs with the best H$β$ lag measurements. After correcting for accretion-disk contamination, torus sizes are determined from the time lags of the Wide-field Infrared Survey Explorer W1 and W2 band light curves relative to the optical band variability based on the interpolated cross-correlation function (ICCF) analysis and the Multiple and Inhomogeneous Component Analysis. We find that the torus size from the W1-band (W2-band) tightly correlates with the 5100~Å continuum luminosity with an intrinsic scatter of 0.15-0.16 dex and the best-fit slope of $0.35 \pm 0.03$ ($0.33 \pm 0.03$), which is clearly shallower than the expected 0.5 slope from the sublimation radius-luminosity relation. We find a moderate negative trend that higher Eddington AGNs tend to have smaller torus sizes than expected from the best-fit, suggesting the Eddington ratio plays a role in flattening the torus size-luminosity relation. By comparing the torus size with the H$β$ reverberation time lag for a subsample of 67 AGNs, we find that the torus size is a factor of $\sim 10$ and $\sim 14$ larger than the BLR size, respectively for W1 and W2 bands. The torus size based on the W1 (W2) band correlates with the BLR size with the best-fit slope of $1.28 \pm 0.16$ ($1.10 \pm 0.15$), which is comparable but slightly steeper than a linear correlation.

Figures (12)

• Figure 1: Top: Lag analysis result for Mrk 335, shown as an example of a detrended target. The top panel shows intercalibrated optical light curves, with the gray-shaded regions representing the DRW model from PyCALI. The middle panel presents accretion-disk-corrected MIR light curves in the WISE W1 and W2 bands. The blue and red points represent the scaled optical light curves shifted by the $\tau_{cent}$ from ICCF relative to the W1 and W2 light curves, respectively. The dashed lines show the first-order polynomial fit on light curves. The bottom-left panel shows the ICCF (black curve) and CCCD (blue histogram) for W1. The hatched region below the ICCF curve indicates where $r \geq 0.8$ r$_{\rm max}$. The best-fit transfer function derived from the MICA analysis is shown in blue, with the shaded region indicating uncertainties. The bottom-right panel shows W2 data, mirroring the left panel.
• Figure 2: The same as Figure \ref{['fig:example1']} showing lag analysis for PG 1322+659, an example from the SAMP sample.
• Figure 3: Distribution of $p(r_{\rm max})$ versus $r_{\rm max}$ for W1 (left) and W2 (right). Blue circles represent the H$\beta$ RM literature sample, and red circles represent SAMP AGNs. The black dashed lines indicate the selection criteria of $p(r_{\rm max}) = 0.1$ versus $r_{\rm max} = 0.6$. Targets that do not meet these criteria are presented as open circles.
• Figure 4: Comparison of lag measurements obtained using ICCF and MICA in the observed frame for WISE W1 (left) and W2 (right). Colors follow the same convention as in Figure \ref{['fig:rmax_pval']}. The shaded areas show $\pm1\sigma_{\rm int}$, $\pm2\sigma_{\rm int}$, and $\pm3\sigma_{\rm int}$ intrinsic scatter around one-to-one line, illustrating the agreement between the two methods.
• Figure 5: Size of torus ($R_{\rm tor}$) as a function of 5100 Å luminosity for W1 (left) and W2 (right). The green solid line represents our best-fit regression with a free slope. Shaded regions represent the $1\sigma$, $2\sigma$, and $3\sigma$ confidence intervals around the best-fit regression line, derived from the percentiles of model predictions sampled from the posterior distributions of slope and intercept (hereafter this method is applied in all related figures). The black dashed lines represent the best-fit model with a fixed slope of 0.5, while the purple lines show the regression results from Mandal2024.