Probing dust torus radius--luminosity relation: An WISE view

TL;DR

This study tests the dusty torus radius–luminosity relation in AGNs by applying reverberation mapping to WISE W1 and W2 data, using 51 objects with z < 0.8 and long-baseline optical/IR monitoring. Lags are derived with two independent methods, ICCF and MICA, and corrected for redshift and accretion-disk contamination, enabling rest-frame estimates of the torus size. The authors find R_dust ∝ L_BOL^{0.413±0.047} (W1) and R_dust ∝ L_BOL^{0.397±0.058} (W2) when the slope is free (and shallower slopes when incorporating literature), with the torus radii exceeding those of the BLR by factors of roughly 9–12. They also identify a mild anti-correlation between torus size deviations and the Eddington ratio, suggesting accretion-rate–driven self-shadowing in slim-disk regimes as a key factor shaping the dust structure and its luminosity scaling.

Abstract

We present measurements of the dusty torus sizes of 51 active galactic nuclei (AGNs) with a redshift of $z<$ 0.8. Our analysis utilizes about 16 years of optical photometric data of 146 AGNs from various time-domain surveys, including ASAS-SN, CRTS, and ZTF, along with 14 years of infrared data in the $W$1 ($\sim$ 3.4 $μ$m) and $W$2 ($\sim$ 4.6 $μ$m) bands obtained from the Wide-Field Infrared Survey Explorer (WISE). The estimated dust torus size ranges from 1000 to 3000 days, using both the cross-correlation analysis and lightcurve modeling through `MICA'. The measured lag has been corrected by $(1+z)^{-0.37}$, to account for cosmological time dilation and the torus temperature-gradient scaling. We conduct a linear regression analysis for both the $W$1 and $W$2 bands to examine the radius--luminosity ($R$--$L_{BOL}$) relationship under two conditions: one where the slope is fixed at 0.5 and one where it is allowed to vary. For the fixed slope of 0.5, we find the ratio of R$_{\mathrm{BLR}}$: R$_{W1}$: R$_{W2}$ to be 1: 9: 12, indicating that the torus lies outside the BLR and that its size increases with wavelength. Furthermore, we determine the relationship between torus size and L$_{BOL}$, yielding best-fit slopes of $0.413\pm0.047$ for the $W$1 band and $0.397\pm0.058$ for the $W$2 band. Both slopes are shallower than predicted by the dust radiation equilibrium model. Furthermore, our findings indicate that the torus size systematically decreases as the Eddington ratio increases, a trend that can be explained by the self-shadowing effects of slim disks.

Figures (11)

• Figure 1: Comparison of the redshift, z (left), bolometric luminosity (middle), and the Eddington ratio (right) distribution of the various samples with dust lag measurements along with our initial sample are shown in an empty magenta colored histogram, while the final sample of sources with reliable lag measurement in this work is shown in a filled histogram (see details in section \ref{['ss:lag']}).
• Figure 2: Luminosity--redshift density map of the total number of objects selected from rakshit2020spectral within redshift $z$$<$ 0.8, SNR $>$ 10, and g-magnitude $<$ 17. The final sample of 146 objects selected for lag analysis is shown as white dots.
• Figure 3: The light curves of two objects of our sample viz, 105007.75+113228.6 (top) and 141700.82+445606.3 (bottom), where the left panel shows the calibrated unbinned (with lower contrast) and binned optical light curves (upper) and IR light curves for the W1 (middle) and W2 (bottom) bands, respectively. In the lower two panels of the light curves, the optical light curve has been scaled and shifted by the lags found by ICCF. The Right panel illustrates the transfer function and the lag distribution found by employing MICA and the CCF and CCCD found by ICCF.
• Figure 4: Quality assessment through p-valve and maximum cross-correlation coefficient (r$_{max}$) color coded by $\mathcal{R}_{Fe II}$ values. The selection criteria are p-valve $<$ 0.2 and r$_{max}$$>$ 0.5, as shown by the horizontal and vertical dashed lines, respectively.
• Figure 5: Comparison of the lags estimated by ICCF and MICA. The dashed black line shows 1:1 relation. The measured lags show good agreement with the 1:1 relation with an intrinsic scatter of $\sim$ 0.158 dex.