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Daniel Proga

Publications and source records attributed to Daniel Proga.

At least 19 recordsLinked to original sources
arXiv2026

Wind Acceleration as a Driver of Detached Blueshifted Absorption in Quasar Disk Winds

Active galactic nuclei (AGN) unification models often emphasize the viewing angle, $i$, but $i$ alone does not determine quasar properties. This is crucial for quasar outflows: UV absorption in Extremely High Velocity Outflow (EHVO) quasars can reach blueshifted velocities of $\sim0.2~c$. In disk-wind models, both $i$ and internal wind structure shape the emergent spectrum. We test their interplay using biconical quasar disk-wind models with different acceleration lengths, $R_v$, and generate synthetic spectra over a range of $i$. We use Monte Carlo radiative transfer to account for finite continuum sources, wind attenuation, scattering, reprocessing, and emission. Changing $R_v$ greatly alters the ionization structure, continuum shape, and absorption-line profiles. At intermediate viewing angles, sightlines pass through the fastest wind. Even there, highly detached and blueshifted \CIV\ absorption like that observed in EHVO quasars appears only in models with small $R_v$. In these models, the gas reaches high velocity before attaining the ionization and density conditions favorable for \CIV. Models with larger $R_v$ instead produce broader, less detached troughs, even when the terminal velocity is very high. Thus, highly detached and blueshifted absorption requires both a high terminal velocity and small $R_v$, making such features diagnostics of disk-wind acceleration and structure. EHVO quasars provide a clear example, but the same principle applies more broadly to highly detached and blueshifted absorption in quasar outflows. Our results support an extended disk-wind view of AGN unification: $i$ selects the observed wind region, while $R_v$ shapes the emergent spectrum and absorption morphology.

astro-ph.GA↗
arXiv2026

Flux-Averaged Force Multipliers

We apply novel developments in photoionization modeling and multi-frequency radiation hydrodynamics to the study of line driven AGN disc winds. We use a flux-averaged force multiplier approach to compute the radiation force due to lines for hydrodynamics simulations using 4 frequency bands - infrared (IR), optical (O), ultraviolet (UV) and X-rays. Though line driving is dominated by the UV, contributions from the O and X-ray bands are non-negligible and can lead to enhancements in the wind both in terms of mass flux and outflow velocity. Crucially, these effects are not captured when using a ``grey'' approach to the radiation modeling in the hydrodynamics, where frequency information is averaged over during the photoionization modeling. These results further strengthen the case for frequency dependent radiation dynamics studies for line driven winds.

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arXiv2025

AGN STORM 2. XI. Spectroscopic reverberation mapping of the hot dust in Mrk 817

The AGN Space Telescope and Optical Reverberation Mapping 2 (STORM 2) campaign targeted Mrk 817 with intensive multi-wavelength monitoring and found its soft X-ray emission to be strongly absorbed. We present results from 157 near-IR spectra with an average cadence of a few days. Whereas the hot dust reverberation signal as tracked by the continuum flux does not have a clear response, we recover a dust reverberation radius of $\sim 90$ light-days from the blackbody dust temperature light-curve. This radius is consistent with previous photometric reverberation mapping results when Mrk 817 was in an unobscured state. The heating/cooling process we observe indicates that the inner limit of the dusty torus is set by a process other than sublimation, rendering it a luminosity-invariant `dusty wall' of a carbonaceous composition. Assuming thermal equilibrium for dust optically thick to the incident radiation, we derive a luminosity of $\sim 6 \times 10^{44}$ erg s$^{-1}$ for the source heating it. This luminosity is similar to that of the obscured spectral energy distribution, assuming a disk with an Eddington accretion rate of $\dot{m} \sim 0.2$. Alternatively, the dust is illuminated by an unobscured lower luminosity disk with $\dot{m} \sim 0.1$, which permits the UV/optical continuum lags in the high-obscuration state to be dominated by diffuse emission from the broad-line region. Finally, we find hot dust extended on scales $> 140-350$ pc, associated with the rotating disk of ionised gas we observe in spatially-resolved [SIII] $\lambda 9531$ images. Its likely origin is in the compact bulge of the barred spiral host galaxy, where it is heated by a nuclear starburst.

astro-ph.GA↗
arXiv2025

Time-Dependent AGN Disc Winds II -- Effects of Photoionization

We use a combination of radiation hydrodynamics (rad-HD) and photoionization modeling to study line-driven disc winds for a range of black hole masses. We refined previous models by incorporating heating, cooling, and radiation forces from spectral lines calculated using a photoionization code, assuming that composite AGN spectra irradiate the gas. For black holes with masses $3 \times 10^{6} \lesssim {\rm M_{BH}/M_{\odot}} \lesssim 10^{8}$, the mass loss rate, ${\rm \dot{M}_w}$ increases proportionally with the disk Eddington fraction, $\Gamma$. The insensitivity of ${\rm \dot{M}_w}$ to the hardness of the spectral energy distribution (SED) arises because the central region is dominated by radiation in the frequency range with ample spectral lines for the range of $M_{BH}$ considered here. Disc winds are suppressed or fail outside the above mass range because of a dearth of line-driving photons. We find \emph{stronger} winds, both in terms of ${\rm \dot{M}_w}$ and wind velocity compared to previous disc wind models. Our winds are stronger because of an enhanced line force from including many spectral lines in the X-ray band. These lines were unavailable and, hence, unaccounted for in previous photoionization studies and their subsequent application to AGN wind models. For $\Gamma \gtrsim 0.4$, ${\rm \dot{M}_w}$ is higher than the assumed disc accretion rate, implying that the wind feeds back strongly. Our findings indicate the necessity of utilizing comprehensive and current atomic data along with a more thorough approach to radiation transfer - both spatially and temporally - to accurately calculate the line force.

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arXiv2025

Oscillatory Line-Driven Winds: The Role of Atmospheric Stratification

Time-dependent numerical studies of line-driven winds using the Sobolev approximation have a history spanning over three decades. In many of these studies, the wind solutions display notorious oscillations. Two clues suggest the oscillations originate at the wind base: (i) simulations reach a steady state without oscillations when the base density is sufficiently low and (ii) the oscillation dominant frequency is comparable to the Lamb cut-off frequency, {\omega}c, of acoustic waves propagating in a stratified hydrostatic atmosphere. Recently, Dannen et al. observed another clue: when the line force significantly weakens due to ionization, the winds become increasingly sensitive to the self-excited oscillations. Here, we present a set of simulations and perturbation analyses that further elucidate the source and characteristics of oscillations. We found that the line force adds wave energy and amplifies perturbations with frequencies near {\omega}c. This selective amplification results from the coupling between the natural tendency of velocity perturbations to grow in a stratified atmosphere and from the line force dependence on the velocity gradient, per the Castor-Abbott-Klein line-driven wind theory. We also found that the variability stems from self-excitation that occurs in the exponential atmosphere due to the non-linearity introduced by the absolute value of the velocity gradient in the line force prescription. We conclude that self-consistently calculating ionization is insufficient for modeling the dynamics in the subsonic atmosphere. Instead, future wind or unified models should relax the Sobolev approximation, or model the radiative transfer to properly capture the resulting radiation-induced instabilities and dynamics at the wind base.

astro-ph.GA↗
arXiv2024

AGN STORM 2: VIII. Investigating the Narrow Absorption Lines in Mrk 817 Using HST-COS Observations

We observed the Seyfert 1 galaxy Mrk817 during an intensive multi-wavelength reverberation mapping campaign for 16 months. Here, we examine the behavior of narrow UV absorption lines seen in HST/COS spectra, both during the campaign and in other epochs extending over 14 years. We conclude that while the narrow absorption outflow system (at -3750 km/s with FWHM=177 km/s) responds to the variations of the UV continuum as modified by the X-ray obscurer, its total column density (logNH =19.5 cm-2) did not change across all epochs. The adjusted ionization parameter (scaled with respect to the variations in the Hydrogen ionizing continuum flux) is log UH =-1.0. The outflow is located at a distance smaller than 38 parsecs from the central source, which implies a hydrogen density of nH > 3000 cm-3. The absorption outflow system only covers the continuum emission source and not the broad emission line region, which suggests that its transverse size is small (< 1e16 cm), with potential cloud geometries ranging from spherical to elongated along the line of sight.

astro-ph.GA↗
arXiv2024

AGN STORM 2: IX. Studying the Dynamics of the Ionized Obscurer in Mrk 817 with High-resolution X-ray Spectroscopy

We present the results of the XMM-Newton and NuSTAR observations taken as part of the ongoing, intensive multi-wavelength monitoring program of the Seyfert 1 galaxy Mrk 817 by the AGN Space Telescope and Optical Reverberation Mapping 2 (AGN STORM 2) Project. The campaign revealed an unexpected and transient obscuring outflow, never before seen in this source. Of our four XMM-Newton/NuSTAR epochs, one fortuitously taken during a bright X-ray state has strong narrow absorption lines in the high-resolution grating spectra. From these absorption features, we determine that the obscurer is in fact a multi-phase ionized wind with an outflow velocity of $\sim$5200 km s$^{-1}$, and for the first time find evidence for a lower ionization component with the same velocity observed in absorption features in the contemporaneous HST spectra. This indicates that the UV absorption troughs may be due to dense clumps embedded in diffuse, higher ionization gas responsible for the X-ray absorption lines of the same velocity. We observe variability in the shape of the absorption lines on timescales of hours, placing the variable component at roughly 1000 $R_g$ if attributed to transverse motion along the line of sight. This estimate aligns with independent UV measurements of the distance to the obscurer suggesting an accretion disk wind at the inner broad line region. We estimate that it takes roughly 200 days for the outflow to travel from the disk to our line of sight, consistent with the timescale of the outflow's column density variations throughout the campaign.

astro-ph.HE↗
arXiv2024

Position dependent radiation fields near accretion disks

In disk wind models for active galactic nuclei (AGN) outflows, high-energy radiation poses a significant problem wherein the gas can become overionized, effectively disabling what is often inferred to be the largest force acting on the gas: the radiation force due to spectral line opacity. Calculations of this radiation force depend on the magnitude of ionizing radiation, which can strongly depend on the position above a disk where the radiation is anisotropic. As our first step to quantify the position and direction dependence of the radiation field, we assumed free streaming of photons and computed energy distributions of the mean intensity and components of flux as well as energy-integrated quantities such as mean photon energy. We find a significant dependence of radiation field properties on position, but this dependence is not necessarily the same for different field quantities. A key example is that the mean intensity is much softer than the radial flux at many points near the disk. Because the mean intensity largely controls ionization, this softening decreases the severity of the overionization problem. The position dependence of mean intensity implies the position dependence of gas opacity, which we illustrate by computing the radiation force a fluid element feels in an accelerating wind. We find that in a vertical accelerating flow, the force due to radiation is not parallel to the radiation flux. This misalignment is due to the force's geometric weighting by both the velocity field's directionality and the position dependence of the mean intensity.

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arXiv2023

Time-Dependent AGN Disc Winds I -- X-ray Irradiation

We study AGN line driven disc winds using time-dependent radiation hydrodynamics. The key criterion for determining wind launching is the coupling strength of the UV radiation field via the spectral lines of the gas. The strength of these lines in turn relies crucially on the gas ionization state, determined by the local X-ray intensity. We consider a suite of models where the central ionizing radiation is affected by scattering, absorption and re-emission by the intervening gas. In a pure attenuation model, the disc launches an episodic wind, as previous studies have shown. Including scattering or re-emission tends to weaken the wind, lowering the mass flux and outflow velocity and if sufficiently dominant, suppressing the outflow entirely. However, the exponential nature of radiative attenuation means only a modest, factor of a few, increase in the absorption cross section can overcome the wind suppression due to scattering and re-emission. We find mass outflow rates of $\sim 20\%$ or more of the assumed inflow rate through the disk, indicating that radiation driven winds may significantly alter the structure of the accretion flow. The winds also supply a large, time-varying column of material above the nominal constant disk scale height, which will determine the geometry of reprocessed emission from the central source. Our results suggest the need for accurate photoionization modeling, radiation transport as well as accretion disc physics, to study their effects on the AGN disc winds

astro-ph.HE↗
arXiv2023

AGN STORM 2. VI. Mapping Temperature Fluctuations in the Accretion Disk of Mrk 817

We fit the UV/optical lightcurves of the Seyfert 1 galaxy Mrk 817 to produce maps of the accretion disk temperature fluctuations $\delta T$ resolved in time and radius. The $\delta T$ maps are dominated by coherent radial structures that move slowly ($v \ll c$) inwards and outwards, which conflicts with the idea that disk variability is driven only by reverberation. Instead, these slow-moving temperature fluctuations are likely due to variability intrinsic to the disk. We test how modifying the input lightcurves by smoothing and subtracting them changes the resulting $\delta T$ maps and find that most of the temperature fluctuations exist over relatively long timescales ($\sim$100s of days). We show how detrending AGN lightcurves can be used to separate the flux variations driven by the slow-moving temperature fluctuations from those driven by reverberation. We also simulate contamination of the continuum emission from the disk by continuum emission from the broad line region (BLR), which is expected to have spectral features localized in wavelength, such as the Balmer break contaminating the $U$ band. We find that a disk with a smooth temperature profile cannot produce a signal localized in wavelength and that any BLR contamination should appear as residuals in our model lightcurves. Given the observed residuals, we estimate that only $\sim$20% of the variable flux in the $U$ and $u$ lightcurves can be due to BLR contamination. Finally, we discus how these maps not only describe the data, but can make predictions about other aspects of AGN variability.

astro-ph.HE↗
arXiv2023

AGN STORM 2: V. Anomalous Behavior of the CIV Light Curve in Mrk 817

An intensive reverberation mapping campaign on the Seyfert 1 galaxy Mrk817 using the Cosmic Origins Spectrograph (COS) on the Hubble Space Telescope (HST) revealed significant variations in the response of the broad UV emission lines to fluctuations in the continuum emission. The response of the prominent UV emission lines changes over a $\sim$60-day duration, resulting in distinctly different time lags in the various segments of the light curve over the 14 months observing campaign. One-dimensional echo-mapping models fit these variations if a slowly varying background is included for each emission line. These variations are more evident in the CIV light curve, which is the line least affected by intrinsic absorption in Mrk817 and least blended with neighboring emission lines. We identify five temporal windows with distinct emission line response, and measure their corresponding time delays, which range from 2 to 13 days. These temporal windows are plausibly linked to changes in the UV and X-ray obscuration occurring during these same intervals. The shortest time lags occur during periods with diminishing obscuration, whereas the longest lags occur during periods with rising obscuration. We propose that the obscuring outflow shields the ultraviolet broad lines from the ionizing continuum. The resulting change in the spectral energy distribution of the ionizing continuum, as seen by clouds at a range of distances from the nucleus, is responsible for the changes in the line response.

astro-ph.GA↗
arXiv2023

AGN STORM 2. IV. Swift X-ray and ultraviolet/optical monitoring of Mrk 817

The AGN STORM 2 campaign is a large, multiwavelength reverberation mapping project designed to trace out the structure of Mrk 817 from the inner accretion disk to the broad emission line region and out to the dusty torus. As part of this campaign, Swift performed daily monitoring of Mrk 817 for approximately 15 months, obtaining observations in X-rays and six UV/optical filters. The X-ray monitoring shows that Mrk 817 was in a significantly fainter state than in previous observations, with only a brief flare where it reached prior flux levels. The X-ray spectrum is heavily obscured. The UV/optical light curves show significant variability throughout the campaign and are well correlated with one another, but uncorrelated with the X-rays. Combining the Swift UV/optical light curves with Hubble UV continuum light curves, we measure interband continuum lags, $\tau(\lambda)$, that increase with increasing wavelength roughly following $\tau(\lambda) \propto \lambda^{4/3}$, the dependence expected for a geometrically thin, optically thick, centrally illuminated disk. Modeling of the light curves reveals a period at the beginning of the campaign where the response of the continuum is suppressed compared to later in the light curve - the light curves are not simple shifted and scaled versions of each other. The interval of suppressed response corresponds to a period of high UV line and X-ray absorption, and reduced emission line variability amplitudes. We suggest that this indicates a significant contribution to the continuum from the broad line region gas that sees an absorbed ionizing continuum.

astro-ph.HE↗
arXiv2023

On the transition from efficient to inefficient line-driving in irradiated flows

Observations of ionized AGN outflows have provided compelling evidence that the radiation field transfers both momentum and energy to the plasma. At parsec scale distances in AGN, energy transfer can dominate, in which case the only force needed to launch an outflow is that from gas pressure. Much closer to the black hole, gravity dominates thermal energy due to insufficient heating by the radiation and the gas is in the so-called `cold' regime. Only magnetic or radiation forces can then lead to outflow, but it is unclear at what temperature and ionization state the radiation force weakens, as these properties depend on the spectral energy distribution (SED). In this work, we survey the parameter space of radiation forces due to spectral lines resulting from blackbody SEDs with varying temperatures in the range $\sim 10^4 - 10^6$~K to identify the radiation temperature at which line-driving begins to lose efficiency. We find that the temperature $\lesssim4\times10^5$~K marks the transition to inefficient line driving. We also self-consistently compute the heating and cooling balance to estimate the gas temperature, so that our parameter survey covers the transition where thermal driving goes from negligible to comparable to line driving. We summarize a large set of hydrodynamical simulations of radial flows to illustrate how the wind properties change during the transition and the dependence of these properties on the assumed SED and governing flow parameters.

astro-ph.GA↗
arXiv2023

The saturation mechanism of thermal instability

The literature on thermal instability (TI) reveals that even for a simple homogeneous plasma, the nonlinear outcome ranges from a gentle reconfiguration of the initial state to an explosive one, depending on whether the condensations that form evolve in an isobaric or nonisobaric manner. After summarizing recent developments on the linear and nonlinear theory of TI, here we derive several general identities from the evolution equation for entropy that reveal the mechanism by which TI saturates: whenever the boundary of the instability region (the Balbus contour) is crossed, a dynamical change is triggered that causes the comoving time derivative of the pressure to change sign. This event implies that the gas pressure force reverses direction, slowing the continued growth of the condensation. For isobaric evolution, this 'pressure reversal' occurs nearly simultaneously for every fluid element in the condensation and a steady state is quickly reached. For nonisobaric evolution, the condensation is no longer in mechanical equilibrium and the contracting gas rebounds with greater force during the expansion phase that accompanies gas reaching the equilibrium curve. The cloud then pulsates because the return to mechanical equilibrium becomes wave-mediated. We show that both the contraction rebound event and the subsequent pulsation behavior follow analytically from an analysis of the new identities. Our analysis also leads to the identification of an isochoric TI zone and makes it clear that unless this zone intersects the equilibrium curve, isochoric modes can only become unstable if the plasma is in a state of thermal nonequilibrium.

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arXiv2023

The Black Hole Candidate Swift J1728.9$-$3613 and the Supernova Remnant G351.9$-$0.9

A number of neutron stars have been observed within the remnants of the core-collapse supernova explosions that created them. In contrast, black holes are not yet clearly associated with supernova remnants. Indeed, some observations suggest that black holes are ``born in the dark'', i.e. without a supernova explosion. Herein, we present a multi-wavelength analysis of the X-ray transient Swift J1728.9$-$3613, based on observations made with Chandra, ESO-VISTA, MeerKAT, NICER, NuSTAR, Swift, and XMM-Newton. Three independent diagnostics indicate that the system likely harbors a black hole primary. Infrared imaging signals a massive companion star that is broadly consistent with an A or B spectral type. Most importantly, the X-ray binary lies within the central region of the catalogued supernova remnant G351.9$-$0.9. Our deep MeerKAT image at 1.28~GHz signals that the remnant is in the Sedov phase; this fact and the non-detection of the soft X-ray emission expected from such a remnant argue that it lies at a distance that could coincide with the black hole. Utilizing a formal measurement of the distance to Swift J1728.9$-$3613 ($d = 8.4\pm 0.8$ kpc), a lower limit on the distance to G351.9$-$0.9 ($d \geq 7.5$ kpc), and the number and distribution of black holes and supernova remnants within the Milky Way, extensive simulations suggest that the probability of a chance superposition is $<1.7\%$ ($99.7\%$ credible interval). The discovery of a black hole within a supernova remnant would support numerical simulations that produce black holes and remnants, and thus provide clear observational evidence of distinct black hole formation channels. We discuss the robustness of our analysis and some challenges to this interpretation.

astro-ph.HE↗
arXiv2023

AGN STORM 2. III. A NICER view of the variable X-ray obscurer in Mrk 817

The AGN STORM 2 collaboration targeted the Seyfert 1 galaxy Mrk 817 for a year-long multiwavelength, coordinated reverberation mapping campaign including HST, Swift, XMM-Newton, NICER, and ground-based observatories. Early observations with NICER and XMM revealed an X-ray state ten times fainter than historical observations, consistent with the presence of a new dust-free, ionized obscurer. The following analysis of NICER spectra attributes variability in the observed X-ray flux to changes in both the column density of the obscurer by at least one order of magnitude ($N_\mathrm{H}$ ranges from $2.85\substack{+0.48\\ -0.33} \times 10^{22}\text{ cm}^{-2}$ to $25.6\substack{+3.0\\ -3.5} \times 10^{22} \text{ cm}^{-2}$) and the intrinsic continuum brightness (the unobscured flux ranges from $10^{-11.8}$ to $10^{-10.5}$ erg s$^{-1}$ cm$^{-2}$ ). While the X-ray flux generally remains in a faint state, there is one large flare during which Mrk 817 returns to its historical mean flux. The obscuring gas is still present at lower column density during the flare but it also becomes highly ionized, increasing its transparency. Correlation between the column density of the X-ray obscurer and the strength of UV broad absorption lines suggests that the X-ray and UV continua are both affected by the same obscuration, consistent with a clumpy disk wind launched from the inner broad line region.

astro-ph.HE↗
arXiv2023

AGN STORM 2: II. Ultraviolet Observations of Mrk817 with the Cosmic Origins Spectrograph on the Hubble Space Telescope

We present reverberation mapping measurements for the prominent ultraviolet broad emission lines of the active galactic nucleus Mrk817 using 165 spectra obtained with the Cosmic Origins Spectrograph on the Hubble Space Telescope. Our ultraviolet observations are accompanied by X-ray, optical, and near-infrared observations as part of the AGN Space Telescope and Optical Reverberation Mapping Program 2 (AGN STORM 2). Using the cross-correlation lag analysis method, we find significant correlated variations in the continuum and emission-line light curves. We measure rest-frame delayed responses between the far-ultraviolet continuum at 1180 A and Ly$\alpha$ $\lambda1215$ A ($10.4_{-1.4}^{+1.6}$ days), N V $\lambda1240$ A ($15.5_{-4.8}^{+1.0}$days), SiIV + OIV] $\lambda1397$ A ($8.2_{-1.4}^{+1.4}$ days), CIV $\lambda1549$ A ($11.8_{-2.8}^{+3.0}$ days), and HeII $\lambda1640$ A ($9.0_{-1.9}^{+4.5}$ days) using segments of the emission-line profile that are unaffected by absorption and blending, which results in sampling different velocity ranges for each line. However, we find that the emission-line responses to continuum variations are more complex than a simple smoothed, shifted, and scaled version of the continuum light curve. We also measure velocity-resolved lags for the Ly$\alpha$, and CIV emission lines. The lag profile in the blue wing of Ly$\alpha$ is consistent with virial motion, with longer lags dominating at lower velocities, and shorter lags at higher velocities. The CIV lag profile shows the signature of a thick rotating disk, with the shortest lags in the wings, local peaks at $\pm$ 1500 $\rm km\,s^{-1}$, and a local minimum at line center. The other emission lines are dominated by broad absorption lines and blending with adjacent emission lines. These require detailed models, and will be presented in future work.

astro-ph.GA↗
arXiv2022

Thermal instability in radiation hydrodynamics: instability mechanisms, position-dependent S-curves, and attenuation curves

Local thermal instability can plausibly explain the formation of multiphase gas in many different astrophysical environments, but the theory is only well understood in the optically thin limit of the equations of radiation hydrodynamics (RHD). Here we lay groundwork for transitioning from this limit to a full RHD treatment assuming a gray opacity formalism. We consider a situation where the gas becomes thermally unstable due to the hardening of the radiation field when the main radiative processes are free-free cooling and Compton heating. We identify two ways in which this can happen: (i) when the Compton temperature increases with time, through a rise in either the intensity or energy of a hard X-ray component; and (ii) when attenuation reduces the flux of the thermal component so that the Compton temperature increases with depth through the slab. Both ways likely occur in the broad line region of active galactic nuclei where columns of gas can be ionization bounded. In such instances where attenuation is significant, thermal equilibrium solution curves become position-dependent and it no longer suffices to assess the stability of an irradiated column of gas at all depths using a single equilibrium curve. We demonstrate how to analyze a new equilibrium curve -- the attenuation curve -- for this purpose, and we show that by Field's instability criterion, a negative slope along this curve indicates that constant density slabs are thermally unstable whenever the gas temperature increases with depth.

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