Soft Lags Research Articles

Effects of ultrafast outflows on X-ray time lags in active galactic nuclei

The time lag between soft (e.g., $0.3 1$\,keV) and hard (e.g., $1 4$\,keV) X-ray photons has been observed in many active galactic nuclei (AGNs) and can reveal the accretion process and geometry around supermassive black holes. High-frequency Fe K and soft lags are considered to originate from the light-travel distances between the corona and the accretion disk, while the propagation of the inward mass accretion fluctuation usually explains the low-frequency hard lags. Ultrafast outflows (UFOs) with a velocity range of $ 0.3$c have also been discovered in numerous AGNs and are believed to be launched from the inner accretion disk. However, it remains unclear whether UFOs can affect the X-ray time lags. As a pilot work, we aim to investigate the potential influence of UFOs on X-ray time lags of AGNs in a small sample. By performing the UFO-resolved Fourier spectral timing analysis of archival XMM-Newton observations of three AGNs with transient UFOs -- PG 1448+273, IRAS 13224-3809, and PG 1211+143 -- we compare the X-ray timing products, such as lag-frequency and lag-energy spectra, of observations with and without UFO obscuration. Our results find that in each AGN, low-frequency hard lags become weak or even disappear when they are accompanied by UFOs. This change is confirmed by Monte Carlo simulations at a confidence level of at least $2.7 In the high-frequency domain, soft lags remain unchanged, while the Fe K reverberation lags tentatively disappear. The comparison between timing products of low- and high-flux observations on another three AGNs without UFOs (Ark 564, NGC 7469, and Mrk 335) suggests that the disappearance of low-frequency hard lags is likely related to the emergence of UFOs, not necessarily related to the source flux. The presence of UFOs can affect X-ray time lags of AGNs by suppressing the low-frequency hard lags, which can be explained by an additional time delay introduced by UFOs or disk accretion energy, which should transferred to heat the corona, carried away by UFOs.

Large and complex X-ray time lags from black hole accretion disks with compact inner coronae

Abstract Black hole X-ray binaries in their hard and hard-intermediate states display hard and soft time lags between broadband noise variations (high-energy emission lagging low-energy and vice versa), which could be used to constrain the geometry of the disk and Comptonising corona in these systems. Comptonisation and reverberation lag models, which are based on light-travel delays, can imply coronae which are very large (hundreds to thousands of gravitational radii, Rg) and in conflict with constraints from X-ray spectral modelling and polarimetry. Here we show that the observed large and complex X-ray time lags can be explained by a model where fluctuations are generated in and propagate through the blackbody-emitting disk to a relatively compact (∼10 Rg) inner corona. The model naturally explains why the disk variations lead coronal variations with a Fourier-frequency dependent lag at frequencies <1 Hz, since longer variability time-scales originate from larger disk radii. The propagating fluctuations also modulate successively the coronal seed photons from the disk, heating of the corona via viscous dissipation and the resulting reverberation signal. The interplay of these different effects leads to the observed complex pattern of lag behaviour between disk and power-law emission and different power-law energy bands, the energy-dependence of power-spectral shape, and a strong dependence of spectral-timing properties on coronal geometry. The observed spectral-timing complexity is thus a natural consequence of the response of the disk-corona system to mass-accretion fluctuations propagating through the disk.

Detection of QPO Soft Lag during the Outburst of Swift J1727.8-1613: Estimation of Intrinsic Parameters from Spectral Study

The recently discovered bright transient black hole candidate Swift J1727.8-1613 is studied in a broad energy range (0.5–79 keV) using combined NICER and NuSTAR data taken on 2023 August 29. A prominent type C quasiperiodic oscillation (QPO) at 0.89 ± 0.01 Hz with its harmonic was observed in NICER data of 0.5–10 keV. Interestingly, the harmonic becomes weaker in the lower energy bands (0.5–1 and 1–3 keV). We also report the first detection of a soft time lag of 0.014 ± 0.001 s at the QPO frequency between harder (3–10 keV) and softer (0.5–3 keV) band photons observed with the NICER/X-ray timing instrument. This indicates that the inclination of the accretion disk in the binary system might be high. From the detailed spectral analysis with the relxill reflection model, we found the disk inclination angle of the source to be ∼85°. We discuss how the accretion flow configuration inferred from spectral analysis can help us understand the origin of QPOs and soft lag in this source.

2017 Outburst of H 1743–322: AstroSat and Swift View

We perform a comprehensive timing and broadband spectral analysis using an AstroSat observation of the low-mass black hole X-ray binary H 1743–322 during its 2017 outburst. Additionally, we use two Swift/XRT observations, one of which is simultaneous with AstroSat and the other taken three days earlier, for timing analysis. The hardness–intensity diagram indicates that the 2017 outburst was a failed one, unlike the previous successful outburst in 2016. We detect type C quasi-periodic oscillation (QPO) in the simultaneous AstroSat and Swift/XRT observations at ∼0.4 Hz, whereas an upper harmonic is noticed at ∼0.9 Hz in the AstroSat data only. Although these features are found to be energy-independent, we notice a shift of ∼0.08 Hz in the QPO frequency over the interval of three days. We also investigate the nature of variability in the two consecutive failed outbursts in 2017 and 2018. We detect soft time lags of 23.2 ± 12.2 ms and 140 ± 80 ms at the type C QPO frequencies in 2017 AstroSat and 2018 XMM-Newton data, respectively. The lag–energy spectra from both the outbursts suggest that the soft lags may be associated with reflection features. The broadband spectral analysis indicates that the source was in the low/hard state during the AstroSat observation. Modeling of the disk and reflection continuum suggests the presence of an accretion disk that is significantly truncated by at least 27.4r g from the innermost stable circular orbit when the source luminosity is ∼1.6% of the Eddington luminosity.

AGN STORM 2. VII. A Frequency-resolved Map of the Accretion Disk in Mrk 817: Simultaneous X-Ray Reverberation and UVOIR Disk Reprocessing Time Lags

X-ray reverberation mapping is a powerful technique for probing the innermost accretion disk, whereas continuum reverberation mapping in the UV, optical, and infrared (UVOIR) reveals reprocessing by the rest of the accretion disk and broad-line region (BLR). We present the time lags of Mrk 817 as a function of temporal frequency measured from 14 months of high-cadence monitoring from Swift and ground-based telescopes, in addition to an XMM-Newton observation, as part of the AGN STORM 2 campaign. The XMM-Newton lags reveal the first detection of a soft lag in this source, consistent with reverberation from the innermost accretion flow. These results mark the first simultaneous measurement of X-ray reverberation and UVOIR disk reprocessing lags—effectively allowing us to map the entire accretion disk surrounding the black hole. Similar to previous continuum reverberation mapping campaigns, the UVOIR time lags arising at low temporal frequencies are longer than those expected from standard disk reprocessing by a factor of 2–3. The lags agree with the anticipated disk reverberation lags when isolating short-timescale variability, namely timescales shorter than the Hβ lag. Modeling the lags requires additional reprocessing constrained at a radius consistent with the BLR size scale inferred from contemporaneous Hβ-lag measurements. When we divide the campaign light curves, the UVOIR lags show substantial variations, with longer lags measured when obscuration from an ionized outflow is greatest. We suggest that, when the obscurer is strongest, reprocessing by the BLR elongates the lags most significantly. As the wind weakens, the lags are dominated by shorter accretion disk lags.

Tracking the X-Ray Polarization of the Black Hole Transient Swift J1727.8–1613 during a State Transition

We report on an observational campaign on the bright black hole (BH) X-ray binary Swift J1727.8–1613 centered around five observations by the Imaging X-ray Polarimetry Explorer. These observations track for the first time the evolution of the X-ray polarization of a BH X-ray binary across a hard to soft state transition. The 2–8 keV polarization degree decreased from ∼4% to ∼3% across the five observations, but the polarization angle remained oriented in the north–south direction throughout. Based on observations with the Australia Telescope Compact Array, we find that the intrinsic 7.25 GHz radio polarization aligns with the X-ray polarization. Assuming the radio polarization aligns with the jet direction (which can be tested in the future with higher-spatial-resolution images of the jet), our results imply that the X-ray corona is extended in the disk plane, rather than along the jet axis, for the entire hard intermediate state. This in turn implies that the long (≳10 ms) soft lags that we measure with the Neutron star Interior Composition ExploreR are dominated by processes other than pure light-crossing delays. Moreover, we find that the evolution of the soft lag amplitude with spectral state does not follow the trend seen for other sources, implying that Swift J1727.8–1613 is a member of a hitherto undersampled subpopulation.

Recovery of High-energy Low-frequency Quasiperiodic Oscillations from Black Hole X-Ray Binary MAXI J1535–571 with a Hilbert–Huang Transform Method

We propose a method based on the Hilbert–Huang transform (HHT) to recover the high-energy waveform of low-frequency quasiperiodic oscillations (QPOs). Based on the method, we successfully obtain the modulation of the phase-folded light curve above 170 keV using the QPO phase reconstructed at lower energies in MAXI J1535–571 with Insight-HXMT observations. A comprehensive simulation study is conducted to demonstrate that such modulation indeed originates from the QPO. Thus, the highest energies turn out to significantly exceed the upper limit of ∼100 keV for QPOs reported previously using the Fourier method, marking the first opportunity to study QPO properties above 100 keV in this source. Detailed analyses of these high-energy QPO profiles reveal different QPO properties between the 30–100 and 100–200 keV energy ranges: the phase lag remains relatively stable, and the amplitude slightly increases below ∼100 keV, whereas above this threshold, soft phase lags and a decrease in amplitude are observed. Given the reports of a hard-tail detection in broad spectroscopy, we propose that the newly discovered QPO properties above 100 keV are dominated by the hard-tail component, possibly stemming from a relativistic jet. Our findings also indicate a strong correlation between the QPOs originating from the jet and corona, supporting the scenario of jet–corona coupling precession. We emphasize that our proposed HHT-based method can serve as an efficient manner in expanding the high-energy band for studying QPOs, thereby enhancing our understanding of their origin.

A theoretical Fourier-transformation model for the formation of X-ray time lags from black hole accretion discs

ABSTRACT X-ray emission from active galactic nuclei (AGNs) often displays complex and rapid variability, which may provide a glimpse into the detailed thermal and dynamical structure of the accreting gas near the event horizon of the central black hole. The observed variability can be analysed using Fourier transforms of the light curves in multiple energy channels, which can be used to generate Fourier phase lags, corresponding to lags in the time domain. The X-ray time lags may be either soft lags or hard lags, depending on whether the variability in the hard energy channel precedes that in the soft channel or vice versa. The physical explanation for the observed X-ray time lags from AGNs has been puzzling, and several scenarios have been proposed. In this paper, we explore the hypothesis that the X-ray time lags are produced as a result of the reprocessing of iron L-line and K-line seed photons generated via fluorescence, which is driven by a variable incident radiation field. The seed photons are reprocessed by a combination of thermal and bulk Comptonization and spatial reverberation. We assume that the inner region of the accretion flow can be approximated as a hot, geometrically thick ADAF disc. The outer radius of the ADAF region is equal to the shock formation radius, which is located just outside the centrifugal barrier. The time-dependent radiative transfer in the disc is analysed using a Fourier-transformed, vertically averaged transport equation in cylindrical coordinates. We demonstrate that the new model can successfully reproduce the complex X-ray variability data for the Seyfert 1 galaxies 1H 0707–495 and Ark 564.

Spectral and timing evolution of GX 340+0 along its Z-track

ABSTRACT We present the results from spectral and timing study of the Z source GX 340+0 using AstroSat’s SXT and LAXPC data. During the observation the source traced out the complete Z-track, allowing for the spectral evolution study of the horizontal, normal, and flaring branches (HB, NB, and FB) as well as the hard and soft apexes (HA and SA). The spectra are better and more physically described by a blackbody component and a hot Comptonizing corona with a varying covering fraction, rather than one having a disc component. Along the track, the Comptonized flux (as well as the covering fraction) monotonically decreases. It is the blackbody component (both the temperature and radius) which varies non-monotonically and hence gives rise to the Z-track behaviour. Rapid timing study reveals a prominent quasi-periodic oscillation (QPO) at ∼50 Hz at the HB, HA, and upper NB, while a QPO at ∼6 Hz is seen for the other branches. The fractional rms of the QPOs increase with energy and exhibit soft lags in all branches except SA and FB.

Detection of an intranight optical hard lag with colour variability in blazar PKS 0735+178

ABSTRACT Blazars are a highly variable subclass of active galactic nuclei that have been observed to vary significantly during a single night. This intranight variability remains a debated phenomenon, with various mechanisms proposed to explain the behaviour including jet energy density evolution or system geometric changes. We present the results of an intranight optical monitoring campaign of four blazars: TXS 0506+056, OJ287, PKS 0735+178, and OJ248 using the Carlos Sánchez Telescope. We detect significant but colourless behaviour in OJ287 and both bluer- and redder-when-brighter colour trends in PKS 0735+178. Additionally, the g band shows a lag of $\sim 10\, \mathrm{min}$ with respect to the r, i, zs bands for PKS 0735+178 on 2023 January 17. This unexpected hard lag in PKS 0735+178 is not in accordance with the standard synchrotron shock cooling model (which would predict a soft lag) and instead suggests the variability may be a result of changes in the jet’s electron energy density distribution, with energy injection from Fermi acceleration processes into a post-shocked medium.

Investigating scaling relations in X-ray reverberating AGN using symbolic regression

ABSTRACT Symbolic regression (SR) is a regression analysis based on genetic algorithms to search for mathematical expressions that best fit a given data set, by allowing the expressions themselves to mutate. We use the SR to analyse the parameter relations of the X-ray reverberating active galactic nuclei where the soft Fe-L lags were observed by the X-ray Multi-Mirror Mission (XMM–Newton). First, we revisit the lag–mass scaling relations by using the SR to derive all possible mathematical expressions and test them in terms of accuracy, simplicity, and robustness. We find that the correlation between the lags, τ, and the black hole mass, MBH, is certain, but the relation should be written in the form of log(τ) = α + β(log(MBH/M⊙))γ, where 1 ≲ γ ≲ 2. Moreover, incorporating more parameters such as the reflection fraction (RF) and the Eddington ratio (λEdd) to the lag–mass scaling relation is made possible by the SR. It reveals that α, rather than being a constant, can be −2.15 + 0.02RF or 0.03(RF + λEdd), with the fine-tuned different β and γ. These further support the relativistic disc–reflection framework in which such functional dependences can be straightforwardly explained. Furthermore, we derive their host-galaxy mass, M*, by fitting the spectral energy distribution. We find that the SR model supports a non-linear MBH–M* relationship, while log(MBH/M*) varies between −5.4 and −1.5, with an average value of ∼−3.7. No significant correlation between M* and λEdd is confirmed in these samples.

X-Ray/UVOIR Frequency-resolved Time Lag Analysis of Mrk 335 Reveals Accretion Disk Reprocessing

UV and optical continuum reverberation mapping is a powerful tool for probing the accretion disk and inner broad-line region. However, recent reverberation mapping campaigns in the X-ray, UV, and optical have found lags consistently longer than those expected from the standard disk reprocessing picture. The largest discrepancy to date was recently reported in Mrk 335, where UV/optical lags are up to 12 times longer than expected. Here, we perform a frequency-resolved time lag analysis of Mrk 335, using Gaussian processes to account for irregular sampling. For the first time, we compare the Fourier frequency-resolved lags directly to those computed using the popular interpolated cross-correlation function method applied to both the original and detrended light curves. We show that the anticipated disk reverberation lags are recovered by the Fourier lags when zeroing in on the short-timescale variability. This suggests that a separate variability component is present on long timescales. If this separate component is modeled as reverberation from another region beyond the accretion disk, we constrain a size scale of roughly 15 lt-days from the central black hole. This is consistent with the size of the broad-line region inferred from Hβ reverberation lags. We also find tentative evidence for a soft X-ray lag, which we propose may be due to light travel time delays between the hard X-ray corona and distant photoionized gas that dominates the soft X-ray spectrum below 2 keV.

The origin of long soft lags and the nature of the hard-intermediate state in black hole binaries

ABSTRACT Fast variability of the X-ray corona in black hole binaries can produce a soft lag by reverberation, where the reprocessed thermalized disc photons lag behind the illuminating hard X-rays. This lag is small, and systematically decreases with increasing mass accretion rate towards the hard–soft transition, consistent with a decreasing truncation radius between the thin disc and X-ray hot inner flow. However, the soft lag suddenly increases dramatically just before the spectrum becomes disc dominated (hard-intermediate state). Interpreting this as reverberation requires that the X-ray source distance from the disc increases dramatically, potentially consistent with switching to X-rays produced in the radio jet. However, this change in lag behaviour occurs without any clear change in hard X-ray spectrum, and before the plasmoid ejection event that might produce such a source (soft-intermediate state). Instead, we show how the soft lag can be interpreted in the context of propagation lags from mass accretion rate fluctuations. These normally produce hard lags, as the model has radial stratification, with fluctuations from larger radii modulating the harder spectra produced at smaller radii. However, all that is required to switch the sign is that the hottest Comptonized emission has seed photons that allow it to extend down in energy below the softer emission from the slower variable turbulent region from the inner edge of the disc. Our model connects the timing change to the spectral change, and gives a smooth transition of the X-ray source properties from the bright hard state to the disc-dominated states.

A Spectral-timing Study of the Inner Flow Geometry in MAXI J1535-571 with Insight-HXMT and NICER

We have performed a spectral-timing analysis of the black hole X-ray binary MAXI J1535-571 during its 2017 outburst, with the aim of exploring the evolution of the inner accretion flow geometry. X-ray reverberation lags are observed in the hard-intermediate state (HIMS) and soft-intermediate state of the outburst. During the HIMS, the characteristic frequency of the reverberation lags ν 0 (the frequency at which the soft lag turns to zero in lag–frequency spectra) increases when the spectrum softens. This reflects a reduction of the spatial distance between the corona and accretion disk, when assuming the measured time lags are associated with the light travel time. We also find a strong correlation between ν 0 and the type-C quasi-periodic oscillation (QPO) centroid frequency ν QPO, which can be well explained by the Lense–Thirring precession model under a truncated disk geometry. Despite the degeneracy in the spectral modeling, our results suggest that the accretion disk is largely truncated in the low hard state, and moves inward as the spectrum softens. Combine the spectral modeling results with the ν 0 – ν QPO evolution, we are inclined to believe that this source probably has a truncated disk geometry in the hard state.

Correlated short time-scale hard-soft X-ray variability of the blazars Mrk 421 and 1ES 1959+650 using AstroSat

ABSTRACT We study simultaneous soft (0.7–7 keV) and hard (7–20 keV) X-ray light curves at a total of eight epochs during 2016–2019 of two TeV blazars Mrk 421 and 1ES 1959+650 observed by the SXT and LAXPC instruments on-board AstroSat. The light curves are 45–450 ks long and may be sampled with time bins as short as 600–800 s with high signal-to-noise ratio. The blazars show a harder when brighter trend at all epochs. Discrete cross-correlation functions indicate that the hard and soft X-ray variability are strongly correlated. The time lag is consistent with zero in some epochs, and indicates hard or soft lag of a few hours in the rest. In the leptonic model of blazar emission, soft lag may be due to slower radiative cooling of lower energy electrons while hard lag may be caused by gradual acceleration of the high energy electrons emitting at the hard X-ray band. Assuming the above scenario and the value of the Doppler factor (δ) to be 10–20, the hard and soft lags may be used to estimate the magnetic field to be ∼0.1 Gauss and the acceleration parameter to be ∼104 in the emission region. Due to the availability of the high time resolution (∼ minutes to hours) light curves from AstroSat, the value of the illusive acceleration parameter could be estimated, which provides a stringent constraint on the theories of particle acceleration in blazar jets.

Contrasting X-ray/UV time-lags in Seyfert 1 galaxies NGC 4593 and NGC 7469 usingAstroSatobservations

ABSTRACTWe study accretion disc–corona connection in Seyfert 1 galaxies using simultaneous UV/X-ray observations of NGC 4593 (2016 July 14–18) and NGC 7469 (2017 October 15–19) performed with AstroSat. We use the X-ray (0.5–7.0 keV) data acquired with the Soft X-ray Telescope (SXT) and the UV (FUV: 130–180 nm, NUV: 200–300 nm) data obtained with the Ultra-Violet Imaging Telescope (UVIT). We also use the contemporaneous Swift observations of NGC 4593 and demonstrate AstroSat’s capability for X-ray/UV correlation studies. We performed UV/X-ray cross-correlation analysis using the interpolated and the discrete cross-correlation functions and found similar results. In the case of NGC 4593, we found that the variations in the X-rays lead to those in the FUV and NUV bands by ∼38 ks and ∼44 ks, respectively. These UV lags favour the disc-reprocessing model; they are consistent with the previous results within uncertainties. In contrast, we found an opposite trend in NGC 7469 where the soft X-ray variations lag those in the FUV and NUV bands by ∼41 ks and ∼49 ks, respectively. The hard lags in NGC 7469 favour the thermal Comptonization model. Our results may provide direct observational evidence for the variable intrinsic UV emission from the accretion disk, which acts as the seed for thermal Comptonization in a hot corona in a lamp-post-like geometry. The non-detection of disk reverberation photons in NGC 7469, using AstroSat data, is most likely due to a high accretion rate resulting in a hot accretion disc and large intrinsic emission.

X-ray Time Lag Evaluation of MAXI J1820+070 with a Differential Cross-correlation Analysis

MAXI J1820+070 is a transient black hole binary discovered on 2018 March 11. The unprecedented rich statistics brought by the NICER X-ray telescope allow detailed timing analyses up to ∼1 kHz uncompromised by photon shot noise. To estimate the time lags, a Fourier analysis was applied, which led to two different conclusions for the system configuration: one supporting a lamp-post configuration with a stable accretion disk extending close to the innermost stable circular orbit and the other supporting a truncated accretion disk contracting with time. Using the same data set, we present the results based on the cross-correlation function (CCF). The CCF is calculated between two different X-ray bands where one side is subtracted from the other side, which we call the differential CCF (dCCF). Soft and hard lags of ∼0.03 and 3 s, respectively, are clearly identified without being diluted by the spectral mixture, demonstrating the effectiveness of the dCCF analysis. The evolution of these lags is tracked, along with spectral changes for the first 120 days since discovery. Both the dCCF and spectral fitting results are interpreted as the soft lag being a reverberation lag between the Comptonized emission and the soft excess emission, and that the hard lag is between the disk blackbody emission and the Comptonized emission. The evolutions of these lags are in line with the picture of a truncated disk contracting with time.

Probing the Variation of Reverberation Lags along with X-Ray Flux in the AGN Mrk 704

Understanding the variation of lags with respect to the X-ray flux is important to explore the geometry of the inner region of the accretion disk in AGNs. We performed frequency-lag, energy–lag and spectral studies for two sets of observations, in order to investigate the variations in lags with respect to X-ray flux in the AGN source Mrk 704 using the XMM-Newton observatory. We divided one of the light curves into two sections which were noticed to exhibit a flux variation. The frequency-lag spectra in different energy domains revealed that reverberation (soft) lags varied along with the flux. For the first time, we show that the blurred reflection model can consistently explain the soft excess observed in the X-ray spectra for this source. The fluxes of soft (i.e., reflection) and hard components were noted to vary by ∼18% and ∼9% respectively, across the sections. The soft lag amplitude was found to be larger at the high flux state than the amplitude at the low flux state. Most importantly, we found that both frequency-lag and energy–lag spectra do not display significant variation between two observational data sets despite a flux variation of 43%. This phenomenon cannot be explained by the reflection model because the soft lag amplitudes must be larger in the high flux state. The probable scenario is that, in the low flux state, the obscuring cloud delays the reflected soft photons which increases the soft lag amplitude.

AstroSat observation of the accreting millisecond X-ray pulsar SAX J1808.4–3658 during its 2019 outburst

ABSTRACT We report on the analysis of the AstroSat data set of the accreting millisecond X-ray pulsar SAX J1808.4–3658, obtained during its 2019 outburst. We found coherent pulsations at ∼401 Hz and an orbital solution consistent with previous studies. The 3–20 keV pulse profile can be well fitted with three harmonically related sinusoidal components with background-corrected fractional amplitudes of $\sim 3.5 {{\ \rm per\ cent}}$, $\sim 1.2 {{\ \rm per\ cent}}$ and $\sim 0.37 {{\ \rm per\ cent}}$ for the fundamental, second and third harmonics, respectively. Our energy-resolved pulse profile evolution study indicates a strong energy dependence. We also observed a soft lag in the fundamental and hard lags during its harmonic. The broad-band spectrum of SAX J1808.4–3658 can be described well using a combination of the thermal emission component with kT ∼ 1 keV, a thermal Comptonization (Γ ∼ 1.67) from the hot corona and broad emission lines due to Fe.

An Electron-scattering Time Delay in Black Hole Accretion Disks

Universal to black hole X-ray binaries, the high-frequency soft lag gets longer during the hard-to-intermediate state transition, evolving from ≲1 to ∼10 ms. The soft lag production mechanism is thermal disk reprocessing of nonthermal coronal irradiation. X-ray reverberation models account for the light-travel time delay external to the disk, but assume instantaneous reprocessing of the irradiation inside the electron-scattering-dominated disk atmosphere. We model this neglected scattering time delay as a random walk within an α-disk atmosphere, with approximate opacities. To explain soft lag trends, we consider a limiting case of the scattering time delay that we dub the thermalization time delay, t th; this is the time for irradiation to scatter its way down to the effective photosphere, where it gets thermalized, and then scatter its way back out. We demonstrate that t th plausibly evolves from being inconsequential for low mass accretion rates characteristic of the hard state, to rivaling or exceeding the light-travel time delay for characteristic of the intermediate state. However, our crude model confines t th to a narrow annulus near peak accretion power dissipation, so cannot yet explain in detail the anomalously long-duration soft lags associated with larger disk radii. We call for time-dependent models with accurate opacities to assess the potential relevance of a scattering delay.