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Constraining AGN Torus Sizes with Optical and Mid-Infrared Ensemble Structure Functions
Astrophysics of Galaxies · Submitted 24 Feb 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Applying this method to the ensemble SFs measured for 587 SDSS quasars, we measure a torus $R-L$ relation of ${\rm log}\,R_{\rm eff}\, {(\rm pc)} = 0.51_{-0.04}^{+0.04} \times {\rm log}\,(L_{\rm bol}/10^{46}\,\rm erg\,s^{-1}) -0.38_{-0.01}^{+0.01}$ in the WISE $W1$ band, and sizes $\sim1.4$ times larger in the $W2$ band, which are in good agreement with dust reverberation mapping measurements.
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The abstract, sentence by sentence
We propose a new method to constrain the size of the dusty torus in broad-line active galactic nuclei (AGNs) using optical and mid-infrared (MIR) ensemble structure functions (SFs). Because of the geometric dilution of the torus, the MIR response to optical continuum variations has suppressed variability with respect to the optical that depends on the geometry (e.g., size, orientation, opening angle) of the torus. More extended tori have steeper MIR SFs with respect to the optical SFs. We demonstrate the feasibility of this SF approach using simulated AGN light curves and a geometric torus model. While it is difficult to use SFs to constrain the orientation and opening angle due to insensitivity of the SF on these parameters, the size of the torus can be well determined. Applying this method to the ensemble SFs measured for 587 SDSS quasars, we measure a torus $R-L$ relation of ${\rm log}\,R_{\rm eff}\, {(\rm pc)} = 0.51_{-0.04}^{+0.04} \times {\rm log}\,(L_{\rm bol}/10^{46}\,\rm erg\,s^{-1}) -0.38_{-0.01}^{+0.01}$ in the WISE $W1$ band, and sizes $\sim1.4$ times larger in the $W2$ band, which are in good agreement with dust reverberation mapping measurements. Compared with the reverberation mapping technique, the SF method is much less demanding in data quality and can be applied to any optical+MIR light curves for which a lag measurement may not be possible, as long as the variability process and torus structure are stationary. While this SF method does not extract all information contained in the light curves (i.e., the transfer function), it provides an intuitive interpretation for the observed trends of AGN MIR SFs compared with optical SFs.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose a new method to constrain the size of the dusty torus in broad-line active galactic nuclei (AGNs) using optical and mid-infrared (MIR) ensemble structure functions (SFs). | contribution 0.99 | 0.75 | 0.12 | asserted 0.73 |
| s1 | Because of the geometric dilution of the torus, the MIR response to optical continuum variations has suppressed variability with respect to the optical that depends on the geometry (e.g., size, orientation, opening angle) of the torus. | background 0.87 | 0.10 | 0.04 | asserted 1.00 |
| s2 | More extended tori have steeper MIR SFs with respect to the optical SFs. | interpretation 0.38 | 0.39 | 0.10 | asserted 1.00 |
| s3 | We demonstrate the feasibility of this SF approach using simulated AGN light curves and a geometric torus model. | method 0.58 | 0.93 | 0.04 | asserted 1.00 |
| s4 | While it is difficult to use SFs to constrain the orientation and opening angle due to insensitivity of the SF on these parameters, the size of the torus can be well determined. | limitation 0.85 | 0.83 | 0.13 | asserted 0.98 |
| s5 | Applying this method to the ensemble SFs measured for 587 SDSS quasars, we measure a torus $R-L$ relation of ${\rm log}\,R_{\rm eff}\, {(\rm pc)} = 0.51_{-0.04}^{+0.04} \times {\rm log}\,(L_{\rm bol}/10^{46}\,\rm erg\,s^{-1}) -0.38_{-0.01}^{+0.01}$ in the WISE $W1$ band, and sizes $\sim1.4$ times larger in the $W2$ band, which are in good agreement with dust reverberation mapping measurements. | result 1.00 | 0.97 | 0.76 | asserted 1.00 |
| s6 | Compared with the reverberation mapping technique, the SF method is much less demanding in data quality and can be applied to any optical+MIR light curves for which a lag measurement may not be possible, as long as the variability process and torus structure are stationary. | contribution 0.49 | 0.43 | 0.11 | asserted 0.99 |
| s7 | While this SF method does not extract all information contained in the light curves (i.e., the transfer function), it provides an intuitive interpretation for the observed trends of AGN MIR SFs compared with optical SFs. | limitation 0.82 | 0.54 | 0.07 | asserted 0.98 |
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The paper
Original abstract
We propose a new method to constrain the size of the dusty torus in broad-line active galactic nuclei (AGNs) using optical and mid-infrared (MIR) ensemble structure functions (SFs). Because of the geometric dilution of the torus, the MIR response to optical continuum variations has suppressed variability with respect to the optical that depends on the geometry (e.g., size, orientation, opening angle) of the torus. More extended tori have steeper MIR SFs with respect to the optical SFs. We demonstrate the feasibility of this SF approach using simulated AGN light curves and a geometric torus model. While it is difficult to use SFs to constrain the orientation and opening angle due to insensitivity of the SF on these parameters, the size of the torus can be well determined. Applying this method to the ensemble SFs measured for 587 SDSS quasars, we measure a torus $R-L$ relation of ${\rm log}\,R_{\rm eff}\, {(\rm pc)} = 0.51_{-0.04}^{+0.04} \times {\rm log}\,(L_{\rm bol}/10^{46}\,\rm erg\,s^{-1}) -0.38_{-0.01}^{+0.01}$ in the WISE $W1$ band, and sizes $\sim1.4$ times larger in the $W2$ band, which are in good agreement with dust reverberation mapping measurements. Compared with the reverberation mapping technique, the SF method is much less demanding in data quality and can be applied to any optical+MIR light curves for which a lag measurement may not be possible, as long as the variability process and torus structure are stationary. While this SF method does not extract all information contained in the light curves (i.e., the transfer function), it provides an intuitive interpretation for the observed trends of AGN MIR SFs compared with optical SFs.
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References
How the literature uses it
1 arXiv paper cite this in 3 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 1 paper
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In contrast, [ref] and [ref] reported $R_dust,W1 ∝ L_bol^0.47$ ($R_dust,W2 ∝ L_bol^0.45$), and $R_dust,W1 ∝ L_bol^0.51$, respectively, in their findings, which differ considerably from our results.
Revisiting the dust torus size-luminosity relation based on a uniform reverberation mapping analysis · The torus size correlation with AGN bolometric luminosity · 0.98
What it cites · 1 sentence without a confident label
Uses the method or tool of 1 paper
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The optical light curve is generated from a DRW model using celerite [ref] assuming a mean flux of 1.0, a $∼10%$ fractional variability amplitude ($=0.1$) and a decorrelation timescale $τ=700$ days, as motivated by observations (Table [ref]).
Fast and scalable Gaussian process modeling with applications to astronomical time series · Idealized Light Curves · 1.00
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