Reverberation Mapping of Lamp-post and Wind Structures in Accretion Thin Disks

Active Galactic Nuclei (AGNs) are astrophysical sources powered by the accretion of hot gas onto supermassive black holes (SMBHs) at the centers of galaxies. Gas and dust surrounding an SMBH orbit in a plane, forming what is known as an accretion disk. The emission from this disk is a combination of internal heat generated by viscous dissipation and external heat from the reprocessing of radiation by the corona near the SMBH. The corona, consisting of a medium of hot electrons, Compton up-scatters radiation, which is subsequently emitted as X-rays (e.g. Lightman & White, 1988). In the standard theory of AGN, the central accretion disk is typically modeled as optically thick and geometrically thin (Lynden-Bell, 1969; Pringle & Rees, 1972; Shakura & Sunyaev, 1973; Novikov & Thorne, 1973; Friedjung, 1985), emitting a blackbody continuum that exhibits variability. While the physical origin of this variability remains unclear, several studies indicate that ultraviolet (UV) to optical variability may be driven by preceding X-ray variations (e.g. Cackett et al., 2007; McHardy et al., 2014).

The reprocessing of the driving X-ray variability into the UV/optical emitting regions of the accretion disk is known as continuum reverberation mapping. Reverberation mapping, in general, encompasses several types, each probing different regions of AGN (see the review by Cackett et al., 2021). For instance, continuum reverberation mapping focuses on the accretion disk, emission-line reverberation mapping probes the structure and kinematics of the broad-line and narrow-line regions (BLR and NLR), and dust reverberation mapping explores the torus surrounding the AGN. This phenomenon was first observed through the time lag between the continuum emission and the H$\alpha$ emission lines (Cherepashchuk & Lyutyi, 1973). The theoretical framework for this technique was later established by Blandford & McKee (1982), and it has since been applied to observations to estimate the size of BLR (Gaskell & Sparke, 1986). Assuming that broad-line emission is triggered by central emission, the lag represents the light-travel time from the central illuminating source to the BLR. This allows for calculating the BLR’s size as $R_{\rm BLR}=c\ \tau_{\rm lag}$. Similarly, this method can be applied to determine the size of the accretion disk, as the continuum emission across the disk is also driven by the central source. The time lags measured through continuum reverberation mapping of UV/optical light curves provide insights into the relative size scales of the emitting regions. These measurements are closely linked to the properties of the accretion disk and the black hole (e.g. Fausnaugh et al., 2017). In the remainder of this work, we refer to reverberation mapping specifically as continuum reverberation mapping of the accretion disk.

To understand the growth and evolution of SMBHs in AGNs, it is crucial to study their accretion disk structures. Studies on quasar variability has shown that accretion disk sizes are larger than those predicted by the thin-disk model, by a factor of about 2–4 (Fausnaugh et al., 2016; Jiang et al., 2017; Mudd et al., 2018; Yu et al., 2020; Guo et al., 2022; Jha et al., 2022). This observation is also supported by accretion disk size measurements obtained through gravitational microlensing (Morgan et al., 2010; Blackburne et al., 2011; Muñoz et al., 2016; Morgan et al., 2018; Cornachione & Morgan, 2020). Some studies suggest that this size discrepancy may be attributed to factors such as an underestimation of black hole mass, which could be larger by a factor of 3.3 when the geometric scaling factor is 6 times greater (Pozo Nuñez et al., 2019), or the negligence of reddening, which can underestimate the disk size by a factor of 2.4 (Gaskell et al., 2023). In addition, the optical flux measured from the source is significantly lower than that predicted by a standard geometrically thin disk emitting at the observed wavelength. To explain this discrepancy, we explore several extensions of the models for UV/optical reverberation mapping, which have also been applied in microlensing analysis (Zdziarski et al., 2022).

We first consider mass loss due to winds from the disk surface, which can significantly alter the predicted mass accretion rate between disks with and without wind, thereby reconciling theoretical predictions with observations (Li et al., 2019; Sun et al., 2019). A second effect involves local color corrections to the disk’s blackbody emission, with $f_{\rm col}\geq 1$. Several studies have explored the concept of accretion disks sustained by magnetic pressure, which are substantially hotter than traditional thin disk models (Blaes et al., 2006; Begelman & Pringle, 2007; Salvesen et al., 2016). While some studies have found $f_{\rm col}\approx 1$ in optical wavelengths (Hubeny et al., 2001), this assumes no dissipation in the disk’s surface layers. Relaxing this assumption to allow for moderate dissipation in the surface layers can potentially resolve the observed discrepancy (e.g. Ross et al., 1992; Kammoun et al., 2021a). In this work, we study the effects of these phenomena within the frameworks of the standard accretion disk model and the lamp-post model, with a particular focus on their implications for UV/optical reverberation mapping.

This paper is organized as follows. In Section 2, we derive the physical quantities for the thin-disk, lamp-post, and wind models. The results of the observed time-lag spectra, analyzed using our model, are presented in Section 3. We discuss model parameters in Section 4 and conclude our findings in Section 5. When required, a flat cosmology is used with $H_{0}=70~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$, $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.

The structure of an AGN can be postulated as a geometrically thin, optically thick accretion disk irradiated by external sources. This configuration predicts specific dependencies of observed time lags and fluxes on wavelength $\lambda$ (or frequency $\nu=c/\lambda$), expressed as the time-lag spectrum $\tau_{\lambda}$ and the flux spectrum $f_{\nu}$. By measuring these spectra, we can infer the physical structure of the accretion disk. This section presents a comprehensive calculation of light curves for an accretion disk model irradiated by a “lamp post” geometry (see Section 2.1). We build upon previous works on irradiated disk light curves (e.g. Cackett et al., 2007; Starkey et al., 2016; Kovačević et al., 2022; Pozo Nuñez et al., 2023) by introducing an approach that incorporates a power-law dependence for the accretion rate and explores the influence of the disk structure. Furthermore, we provide analytical formulas derived using a power-law approximation of the temperature profile, allowing for more efficient application to observational data (see Section 2.2).

To evaluate the practical efficacy of the power-law approximation, we fit the time lags of AGNs with well-sampled light curves. Our sample comprises (type I) AGNs with known time-lag spectra obtained from intensive, multi-wavelength campaigns: NGC 4593 (Cackett et al., 2018), NGC 5548 (Fausnaugh et al., 2016), Fairall 9 (Hernández Santisteban et al., 2020), and Mrk 817 (Kara et al., 2021). These four AGNs have also been used to demonstrated the work in Kammoun et al. (2023). Furthermore, we utilize luminosity measurements at $\lambda=5100~{}\textrm{\AA}$ in the rest frame from Netzer (2022) as prior information, aiding in the constraint of the power index $\beta$. Further details of this sample are listed in Table 1. We note that for all time lag measurements, we adopt the centroid values of the ICCF as reported in the corresponding papers.

Fitting the time lag spectra while allowing all parameters, including $M_{\rm bh}$, $a$, ${\dot{M}_{\rm in}}$, $\beta$, $i$, $H$, $\eta_{X}$, $f_{\rm col}$, and ${R_{\rm out}}$, to vary freely is not feasible, as some parameters are inherently degenerate. To address this, we propose an alternative approach for fitting the observed time lags and luminosity at $5100~{}\textrm{\AA}$ for the four AGNs in our sample. First, we keep the spin $a=0$ (i.e. $\alpha=6$) since $\alpha$ is degenerate with ${\dot{M}_{\rm in}}$ (i.e. ${R_{c}}^{3-\beta}\propto{\nu L_{\nu}}^{\frac{3-\beta}{2}}\propto{\dot{M}_{\rm in}}\alpha^{-\beta}$). Second, we fix the inclination to $i=0$ for this analysis, given that the test cases are type I AGNs; further discussion of this parameter is provided in Section 4. Last, we adopt the time lag derived from the power-law approximation in Equation (30), which assumes ${R_{\rm in}},H\ll R\ll{R_{\rm out}}$. In essence, we reduce the parameters to three by relating them to one another, within the fitting ranges of $0\leq\beta<1$, $-3\leq\log\frac{{\dot{M}_{\rm in}}}{M_{\odot}/{\rm yr}}<0$, and $0.5\leq f_{\rm col}<3$. In addition, we provide the fits with fixed $f_{\rm col}=1$ and with fixed $\beta=0$ for comparison, if needed.

The chi-square value for the time lag is defined as:

where $\Delta\tau_{\lambda}^{\rm meas}$ represents the measured time lag in the rest frame relative to the reference wavelength $\lambda_{0}=\lambda_{\rm ref}/(1+z)$, $\sigma_{\tau}$ is the average of the upper and lower limits of the error bars, and $N_{\lambda}$ is the number of time lag measurements. During the fitting process, we exclude measurements that may be influenced by emission from the much larger Broad Line Region (BLR) relative to the disk (see also Section 4 for further discussion). We further adopt the measured luminosity at $5100~{}\textrm{\AA}$ as our prior, based on Equation (21), defined as

where $\sigma_{\log\nu L_{\nu}}$ is chosen as $\log(0.5)$, indicating that the measurement can be deviated by a factor of 2 due to inclination or contamination of host galaxy. We choose monochramatic luminosity instead of bolometric luminosity as our prior because the bolometric luminosity from literature is estimated considering $\beta=0$, which does not apply to this work. The total chi-square is then defined as $\chi^{2}=\chi^{2}_{\tau}+\chi^{2}_{\nu L_{\nu}}$.

We observe that the fitting process can occasionally become trapped in local minima. Therefore, alongside with the minimization of modified Powell algorithm²²2https://docs.scipy.org/doc/scipy/reference/optimize.minimize-powell.html, we also employ Markov chain Monte Carlo (MCMC)³³3https://emcee.readthedocs.io/en/stable/ to investigate the correlation between fitting parameters. The results are shown in Figure 4, while the MCMC samples are illustrated in Figure 5. Given the current data quality, breaking the degeneracy between $\beta$ and ${\dot{M}_{\rm in}}$ remains challenging. We also observe that fitting based solely on $\chi^{2}_{\tau}$ introduces a degeneracy between ${\dot{M}_{\rm in}}$ and $f_{\rm col}$ (i.e., ${R_{c}}^{3-\beta}\propto{\dot{M}_{\rm in}}{f_{\rm col}}^{4}$). This suggests that incorporating the flux spectrum or the spectral energy distribution (SED) into the fitting process could potentially offer a more robust solution, which may help break the degeneracy and provide a more accurate fit. There are delays that tend to be smaller than the predicted values from the model at longer wavelengths ($\geq 8000~{}{\rm\AA}$), particularly in the cases of NGC 4593 and NGC 5548, although the uncertainties are significant. This discrepancy could be due to the outer edge of the disk, as the power-law approximation assumes ${R_{\rm out}}\to\infty$. One potential solution is to fit the data using a direct integral in Equation (15), which may slow down the fitting process.

Finally, we present the relationship between black hole mass and the fitting parameters of MCMC samples from our four AGNs in Figure 6. Given the small sample size, we recognize the need for caution in making strong claims; however, this work still sheds light on certain aspects and sets the stage for future research with larger samples. In the left panel, our MCMC samples indicate a shallower temperature slope, $\beta\approx 0.35$, corresponding to a temperature slope of $T\propto R^{-0.66}$, which has also been reported in microlensing analyses (Bate et al., 2018; Poindexter et al., 2008; Cornachione & Morgan, 2020). However, the significant uncertainty in $\beta$ complicates the identification of any clear trend. From the Powell best-fit (see Figure 5), only one AGN (NGC 4593) shows a relatively higher $\beta=0.46$, while the remaining AGNs can still be fitted by the traditional thin disk model. From the middle panel, there is modest positive relation between black hole mass and accretion rate (except for Mrk 817), as also reported in Figure 8 in Shen et al. (2019). We note that their result is presented in ${L_{\rm bol}}$; it can be converted to accretion rate, though a more detailed examination of this conversion is necessary. In the right panel, our best-fit yields $f_{\rm col}\approx 1.6$, aligning well with simulation findings from Davis & El-Abd (2019). Based on our results, we do not observe a trend in $\beta$, but we find a positive correlation between the accretion rate and color correction with black hole mass. We restate that given the small sample size and not insignificant uncertainties, it is difficult to assert any strong trends.

In this section, we discuss the impact of alternative disk and wind models, examine the impact of the corona height and color correction, and cover other disk parameters, such as inclination and X-ray luminosity. We also discuss about the continuum diffuse emission on reverberation mapping, and last we compare our simulation with the fitting formulas from the simulation of Kammoun et al. (2021a, 2023).

We investigate the wavelength dependence of flux and time lags in AGN accretion disks, focusing on a geometrically thin and optically thick disk model irradiated by an external “lamp post” source. By introducing a power-law dependence for the accretion rate $\dot{M}={\dot{M}_{\rm in}}\left(R/{R_{\rm in}}\right)^{\beta}$ inspired by disk wind dynamics (Blandford & Begelman, 1999), we extend previous methodologies for reverberation mapping. Our analysis reveals that a higher $\beta$ results in a larger disk size and increased luminosity, offering an alternative approach to addressing the issue of accretion disk sizes. Furthermore, we provide analytical formulas for more efficient application to observational data, particularly in the UV/optical regime. Our fitting analysis of time lags and luminosity for type I AGNs, using a power-law approximation, reveals a shallower temperature slope ($T\propto R^{-0.66}$, or $\beta\approx 0.35$) compared to the traditional thin disk model ($T\propto R^{-3/4}$), and a color correction factor of $f_{\rm col}\approx 1.6$. These results are consistent with previous studies (e.g. Davis & El-Abd, 2019; Cornachione & Morgan, 2020). However, due to high uncertainties, we cannot completely rule out the traditional thin disk model. We observe a positive correlation between the accretion rate and color correction with black hole mass, though the small sample size limits definitive conclusions. We also find a strong degeneracy between $\beta$ and accretion rate ${\dot{M}_{\rm in}}$, which complicates the fitting process. This suggests that incorporating the flux spectrum or spectral energy distribution into the fitting process could help break this degeneracy and provide a more accurate solution. Lastly, we address various aspects including alternative disk and wind models, as well as other disk parameters, demonstrating that our simulation offers a fast and accurate approach for reverberation mapping.

Our work lays the foundation for more realistic simulations, enabling the rapid generation of quasar light curves that are applicable to diverse observational scenarios in training sets, while integrating machine learning modeling through latent stochastic differential equations (latent SDEs; Fagin et al., 2024a, b). This capability is crucial for addressing challenges associated with traditional curve-shifting techniques. Moreover, our efficient approach opens opportunities to combine two dimensional flux variability with microlensing analysis (e.g. Chan et al., 2021; Best et al., 2024a, b), allowing for a deeper investigation into the inner structure of the disk.

We thank D. Sluse, R. Soria, O. Blaes, I. Papadakis, R. Daly, S. Ford and B. McKernan for useful discussion. We also appreciate the constructive feedback provided by the anonymous referee. Support was provided by Schmidt Sciences, LLC. for J. H.-H. Chan, J. Fagin, H. Best, and M. J. O’Dowd.