Gaseous Accretion Disk Research Articles

Coupled Disk-star Evolution in Galactic Nuclei and the Lifetimes of QPE Sources

A modest fraction of the stars in galactic nuclei fed toward the central supermassive black hole (SMBH) approach on low-eccentricity orbits driven by gravitational-wave radiation (extreme mass ratio inspiral (EMRI)). In the likely event that a gaseous accretion disk is created in the nucleus during this slow inspiral (e.g., via an independent tidal disruption event (TDE)), star–disk collisions generate regular short-lived flares consistent with the observed quasiperiodic eruption (QPE) sources. We present a model for the coupled star-disk evolution, which self-consistently accounts for mass and thermal energy injected into the disk from stellar collisions and associated mass ablation. For weak collision/ablation heating, the disk is thermally unstable and undergoes limit-cycle oscillations, which modulate its properties and lead to accretion-powered outbursts on timescales of years to decades, with a time-averaged accretion rate ∼0.1Ṁ Edd. Stronger collision/ablation heating acts to stabilize the disk, enabling roughly steady accretion at the EMRI-stripping rate. In either case, the stellar destruction time through ablation, and hence the maximum QPE lifetime, is ∼102–103 yr, far longer than fallback accretion after a TDE. The quiescent accretion disks in QPE sources may at the present epoch be self-sustaining and fed primarily by EMRI ablation. Indeed, the observed range of secular variability broadly matches those predicted for collision-fed disks. Changes in the QPE recurrence pattern following such outbursts, similar to that observed in GSN 069, could arise from temporary misalignment between the EMRI-fed disk and the SMBH equatorial plane as the former regrows its mass after a state transition.

Ultraviolet Quasiperiodic Eruptions from Star–Disk Collisions in Galactic Nuclei

“Quasiperiodic eruptions” (QPE) are recurrent nuclear transients with periods of several hours to almost a day, which thus far have been detected exclusively in the X-ray band. We have shown that many of the key properties of QPE flares (period, luminosity, duration, emission temperature, alternating long-short recurrence time behavior, and source rates) are naturally reproduced by a scenario involving twice-per-orbit collisions between a solar-type star on a mildly eccentric orbit, likely brought into the nucleus as an extreme mass-ratio inspiral (EMRI), and the gaseous accretion disk of a supermassive black hole (SMBH). The flare is generated by the hot shocked debris expanding outwards from either side of the disk midplane, akin to dual miniature supernovae. Here, we consider the conditions necessary for disk–star collisions to generate lower-temperature flares that peak in the ultraviolet (UV) instead of the X-ray band. We identify a region of parameter space at low SMBH mass M • ∼ 105.5 M ⊙ and QPE periods P ≳ 10 hr for which the predicted flares are sufficiently luminous L UV ∼ 1041 erg s−1 to outshine the quiescent disk emission at these wavelengths. The prospects to discover such “UV QPEs” with future satellite missions such as ULTRASAT and Ultraviolet Explorer depend on the prevalence of very low-mass SMBHs and the occurrence rate of stellar EMRIs onto them. For gaseous disks produced by the tidal disruption of stars, we predict that X-ray QPEs will eventually shut off, only to later reappear as UV QPEs as the accretion rate continues to drop.

Compact gaseous accretion disk in Keplerian rotation around MWC 147

Context. The disks around some Herbig Be stars have been observed to be more compact than the expected dust sublimation radius for such objects. Highly refractory dust grains and optically thick gas emission have been proposed as possible explanations for this phenomenon. Aims. Previously, the “undersized” Herbig Be star MWC 147 was observed with interferometry, and the results indicated a presence of a compact gaseous disk based on the measured wavelength-dependence of near-infrared or mid-infrared visibilities. Our aim is to search for direct evidence for the presence of hot gas inside of the expected dust sublimation radius of MWC 147. Methods. By combining VLTI/AMBER spectro-interferometry (R = 12 000) with CRIRES spectroscopy (R = 100 000) we can both spectrally and spatially resolve the Brγ line-emitting gas around MWC 147. Additionally, using CHARA/CLIMB enables us to achieve baseline lengths up to 330 m, offering ~2 times higher angular resolution (and a better position angle coverage) than has previously been achieved with interferometry for MWC 147. To model the continuum we fit our AMBER and CLIMB measurements with a geometric model of an inclined Gaussian distribution as well as a ring model. We fit our high-resolution spectra and spectro-interferometric data with a kinematic model of a disk in Keplerian rotation. Results. Our interferometric visibility modelling of MWC 147 indicates the presence of a compact continuum disk with a close to face-on orientation. We model the continuum with an inclined Gaussian and a ring with a radius of 0.60 mas (0.39 au), which is well within the expected dust sublimation radius of 1.52 au. We detect no significant change in the measured visibilities across the Brγ line, indicating that the line-emitting gas is located in the same region as the continuum-emitting disk. Using our differential phase data we construct photocentre displacement vectors across the Brγ line, revealing a velocity profile consistent with a rotating disk. We fit our AMBER spectro-interferometry data with a kinematic model of a disk in Keplerian rotation, where both the line-emitting and continuum-emitting components of the disk originate from the same compact region close to the central star. The presence of line-emitting gas in the same region as the K-band continuum supports the interpretation that the K-band continuum traces an optically thick gas disk. Conclusions. Our spatially and spectrally resolved observations of MWC 147 reveal that the K-band continuum and Brγ emission both originate from a similar region which is 3.9 times more compact than the expected dust sublimation radius for the star; Brγ is emitted from the accretion disk or disk wind region and exhibits a rotational velocity profile. We conclude that we detect the presence of a compact, gaseous accretion disk in Keplerian rotation around MWC 147.

Numerical Studies of Astrophysical Objects at Fesenkov Astrophysical Institute (FAI)

Numerical studies of astrophysical objects are a relatively new direction in Fesenkov Astrophysical Institute (FAI) and is mainly represented by the Laboratory of Cosmology, Stellar Dynamics and Computational Astrophysics. The lab seeks to understand the evolution of gravitating systems at various scales – from star clusters to galaxies to large-scale structure of the universe as a whole, and tackles these problems both through analytical methods and through numerical simulations. The particular focus is on numerical simulations of star clusters, especially those found in active galactic nuclei – this is a topic of oldestablished collaboration with colleagues from Astronomisches Rechen-Institut (Heidelberg) and National Astronomical Observatories of China (Beijing). The prominent example is STARDISK project dedicated to the numerical research of active galactic nuclei as multicomponent systems composed of compact stellar cluster, gaseous accretion disk and a supermassive black hole. It is demonstrated that an accretion disk can noticeably decelerate stars and thus enhance the accretion rate onto the black hole. In 2013 FAI hosted the MODEST-13 International Workshop dedicated to modeling of star clusters. Recently a new project has been approved aimed at construction of triaxial equilibrium N-body systems that can be of great help in various numerical experiments with disk galaxies. There are also long standing plans to perform cosmological simulations of large scale structures to test a new approach to dark matter and energy actively developed at FAI. For numerical calculations, FAI has a small, but growing computer cluster consisting of several high-performance computing servers equipped with computational GPU cards.

Local models of astrophysical discs

Local models of gaseous accretion discs have been successfully employed for decades to describe an assortment of small-scale phenomena, from instabilities and turbulence, to dust dynamics and planet formation. For the most part, they have been derived in a physically motivated but essentially ad hoc fashion, with some of the mathematical assumptions never made explicit nor checked for consistency. This approach is susceptible to error, and it is easy to derive local models that support spurious instabilities or fail to conserve key quantities. In this paper we present rigorous derivations, based on an asympototic ordering, and formulate a hierarchy of local models (incompressible, Boussinesq, and compressible), making clear which is best suited for a particular flow or phenomenon while spelling out explicitly the assumptions and approximations of each. We also discuss the merits of the anelastic approximation, emphasising that anelastic systems struggle to conserve energy unless strong restrictions are imposed on the flow. The problems encountered by the anelastic approximation are exacerbated by the disk's differential rotation, but also attend non-rotating systems such as stellar interiors. We conclude with a defence of local models and their continued utility in astrophysical research.

Brackettγradiation from the inner gaseous accretion disk, magnetosphere, and disk wind region of Herbig AeBe stars

Various disk and outflow components such as the magnetosphere, the disk wind, the gaseous accretion disk, and other regions may contribute to the hydrogen line emission of young Herbig AeBe stars. Non-LTE modeling was performed to show the influence of the model parameters of each emitting region on the intensity and shape of the Br γ line profile, to present the spatial brightness distribution of each component, and to compare the contribution of each component to the total line emission. The modeling shows that the disk wind is the dominant contributor to the Br γ line rather than the magnetosphere and inner gaseous accretion disk. The contribution of the disk wind region to the H α line is also considered.

The evolution of a binary in a retrograde circular orbit embedded in an accretion disk

Supermassive black hole binaries may form as a consequence of galaxy mergers. Both prograde and retrograde orbits have been proposed. We study a binary of a small mass ratio, q, in a retrograde orbit immersed in and interacting with a gaseous accretion disk in order to estimate time scales for inward migration leading to coalescence and the accretion rate to the secondary component. We employ both semi-analytic methods and two dimensional numerical simulations, focusing on the case where the binary mass ratio is small but large enough to significantly perturb the disk. We develop the theory of type I migration for this case and determine conditions for gap formation finding that then inward migration occurs on a time scale equal to the time required for one half of the secondary mass to be accreted through the unperturbed disk, with accretion onto the secondary playing only a minor role. The semi-analytic and fully numerical approaches are in good agreement, the former being applicable over long time scales. Inward migration induced by interaction with the disk alleviates the final parsec problem. Accretion onto the secondary does not significantly affect the orbital evolution, but may have observational consequences for high accretion efficiency. The binary may then appear as two sources of radiation rotating around each other. This study should be extended to consider orbits with significant eccentricity and the effects of gravitational radiation at small length scales. Note too that torques acting between a circumbinary disk and a retrograde binary orbit may cause the mutual inclination to increase on a timescale that can be similar to, or smaller than that for orbital evolution, depending on detailed parameters. This is also an aspect for future study (abridged).

MODELING TRANSITING CIRCUMSTELLAR DISKS: CHARACTERIZING THE NEWLY DISCOVERED ECLIPSING DISK SYSTEM OGLE LMC-ECL-11893

We investigate the nature of the unusual eclipsing star OGLE LMC-ECL-11893 (OGLE J05172127-6900558) in the Large Magellanic Cloud recently reported by Dong et al. 2014. The eclipse period for this star is 468 days, and the eclipses exhibit a minimum of ~1.4 mag, preceded by a plateau of ~0.8 mag. Spectra and optical/IR photometry are consistent with the eclipsed star being a lightly reddened B9III star of inferred age ~150 Myr and mass of ~4 solar masses. The disk appears to have an outer radius of ~0.2 AU with predicted temperatures of ~1100-1400 K. We model the eclipses as being due to either a transiting geometrically thin dust disk or gaseous accretion disk around a secondary object; the debris disk produces a better fit. We speculate on the origin of such a dense circumstellar dust disk structure orbiting a relatively old low-mass companion, and on the similarities of this system to the previously discovered EE Cep.

The effect of gaseous accretion disk on dynamics of the stellar cluster in AGN

AbstractThere is a supermassive black hole, a gaseous accretion disk and compact star cluster in the center of active galactic nuclei, as known today. So the activity of AGN can be represented as the result of interaction of these three subsystems. In this work we investigate the dynamical interaction of a central star cluster surrounding a supermassive black hole and a central accretion disk. The dissipative force acting on stars in the disk leads to an asymmetry in the phase space distribution of the central star cluster due to the rotating accretion disk. In our work we present some results of Stardisk model, where we see some changes in density and phase space of central star cluster due to influence of rotating gaseous accretion disk.

Spin evolution of supermassive black holes and galactic nuclei

The spin angular momentum $\mathbit{S}$ of a supermassive black hole (SBH) precesses due to torques from orbiting stars, and the stellar orbits precess due to dragging of inertial frames by the spinning hole. We solve the coupled post-Newtonian equations describing the joint evolution of $\mathbit{S}$ and the stellar angular momenta ${\mathbit{L}}_{j}$, $j=1\dots{}N$ in spherical, rotating nuclear star clusters. In the absence of gravitational interactions between the stars, two evolutionary modes are found: (1) nearly uniform precession of $\mathbit{S}$ about the total angular momentum vector of the system and (2) damped precession, leading, in less than one precessional period, to alignment of $\mathbit{S}$ with the angular momentum of the rotating cluster. Beyond a certain distance from the SBH, the time scale for angular momentum changes due to gravitational encounters between the stars is shorter than spin-orbit precession times. We present a model, based on the Ornstein-Uhlenbeck equation, for the stochastic evolution of star clusters due to gravitational encounters and use it to evaluate the evolution of $\mathbit{S}$ in nuclei where changes in the ${\mathbit{L}}_{j}$ are due to frame dragging close to the SBH and to encounters farther out. Long-term evolution in this case is well described as uniform precession of the SBH about the cluster's rotational axis, with an increasingly important stochastic contribution when SBH masses are small. Spin precessional periods are predicted to be strongly dependent on nuclear properties, but typical values are $\ensuremath{\sim}{10}^{7}--{10}^{8}\text{ }\text{ }\mathrm{yr}$ for low-mass SBHs in dense nuclei, $\ensuremath{\sim}{10}^{8}--{10}^{10}\text{ }\text{ }\mathrm{yr}$ for SBH masses $\ensuremath{\sim}{10}^{8}{\mathcal{M}}_{\ensuremath{\bigodot}}$, and $\ensuremath{\sim}{10}^{10}--{10}^{11}\text{ }\text{ }\mathrm{yr}$ for the most massive SBHs. We compare the evolution of SBH spins in stellar nuclei to the case of torquing by an inclined, gaseous accretion disk.

Transient outburst events from tidally disrupted asteroids near white dwarfs

We discuss the possibility of observing the transient formation event of an accretion disk from the tidal destruction process of an asteroid near a white dwarf (WD). This scenario is commonly proposed as the explanation for dusty disks around WDs. We find that the initial formation phase lasts for about a month and material that ends in a close orbit near the WD forms a gaseous disk rather than a dusty disk. The mass and size of this gaseous accretion disk is very similar to that of Dwarf Novae (DNe) in quiescence. The bolometric luminosity of the event at maximum is estimated to be ∼0.001–0.1L⊙. Based on the similarity with DNe we expect that transient outburst events such as discussed here will be observed at wavelengths ranging from visible to the X-ray, and be detected by present and future surveys.

Imprint of Accretion Disk-Induced Migration on Gravitational Waves from Extreme Mass Ratio Inspirals

We study the effects of a thin gaseous accretion disk on the inspiral of a stellar-mass black hole into a supermassive black hole. We construct a phenomenological angular momentum transport equation that reproduces known disk effects. Disk torques modify the gravitational wave phase evolution to detectable levels with LISA for reasonable disk parameters. The Fourier transform of disk-modified waveforms acquires a correction with a different frequency trend than post-Newtonian vacuum terms. Such inspirals could be used to detect accretion disks with LISA and to probe their physical parameters.

The cosmogony of supermassive black holes

We report calculations of the evolution, under Reynolds viscosity, of the massive gaseous accretion discs thought to form in the centres of galaxies as a result of major galactic mergers at early epochs. Starting with the formation of such a disc and the postulated existence of a low-mass 'seed' black hole, we focus on the mass accretion rate, , and hence luminosity, L(GC), of a putative supermassive black hole (SMBH) in the galactic centre. Our computations cover a range of initial disc mass, M(d, 0), and outer radius, s(max, 0), and have been carried out under the assumption that cannot exceed the Eddington limit and that any excess mass supplied from the disc is lost from the system. Our purpose is to understand the extent to which such a model can account for the evolution with redshift of the observed properties of active galactic nuclei (AGNs). We show that within this framework: the initial mass supply rate from the disc increases with M(d, 0) and compactness, adjusts rapidly (on a viscous time-scale) to its peak value and decreases steadily thereafter; peak SMBH mass accretion rate and hence luminosity occurs when the (decreasing) disc flow rate equals the (increasing) black hole (BH) Eddington rate and is proportional to the BH mass, MEdd(BH), at that juncture; the fractional disc mass lost during the Eddington-limited phase increases with M(d, 0) and compactness so strongly that, for given (large) M(d, 0), the peak BH luminosity and corresponding mass occurs for some intermediate value of s(max, 0) so that MEdd(BH) declines for more compact initial discs; SMBHs with mass M(BH) >= 109 M(circle dot) and luminosity L(GC) >= 3 x 1013 L(circle dot) can form in a time t < 109 yr as required by the most luminous high-redshift quasi-stellar objects but the Eddington-induced mass-loss renders unlikely the creation of BHs of comparable or greater mass or luminosity in significantly shorter times; the most luminous quasars form and die at the earliest epochs while those of lower luminosity occur over successively longer periods ranging up to the present time; in general, AGNs that are at an early evolutionary phase accrete at or close to their Eddington rate but evolve into the sub-Eddington regime later on. The proposed scenario appears capable of accounting for many of the features of the observed quasar and AGN luminosity distribution with redshift and seems to be a natural consequence of the formation of massive central discs following major galaxy mergers.

Accretion disks around binary black holes: A quasistationary model

Tidal torques acting on a gaseous accretion disk around a binary black hole can create a gap in the disk near the orbital radius. At late times, when the binary inspiral time scale due to gravitational wave emission becomes shorter than the viscous time scale in the disk, the binary decouples from the disk and eventually merges. Prior to decoupling the balance between tidal and viscous torques drives the disk to a quasistationary equilibrium state, perturbed slightly by small amplitude, spiral density waves emanating from the edges of the gap. We consider a black hole binary with a companion of smaller mass and construct a simple Newtonian model for a geometrically thin, Keplerian disk in the orbital plane of the binary. We solve the disk evolution equations in the steady state to determine the quasistationary, (orbit-averaged) surface density profile prior to decoupling. We use our solution, which is analytic up to simple quadratures, to compute the electromagnetic flux and approximate radiation spectrum during this epoch. A single nondimensional parameter $\stackrel{\texttildelow{}}{g}$, equal to the ratio of the tidal to viscous torque at the orbital radius, determines the disk structure, including the surface density profile, the extent of the gap, the existence of an inner disk, and the accretion rate. The solution reduces to the Shakura-Sunyaev profile for a stationary accretion disk around a single black hole in the limit of small $\stackrel{\texttildelow{}}{g}$. Our solution may be useful for choosing physical parameters and setting up quasistationary disk initial data for detailed numerical simulations that begin prior to decoupling and track the subsequent evolution of a black hole binary-disk system.

Simulations of direct collisions of gas clouds with the central black hole

We perform numerical simulations of clouds in the Galactic Centre (GC) engulfing the nuclear super-massive black hole and show that this mechanism leads to the formation of gaseous accretion discs with properties that are similar to the expected gaseous progenitor discs that fragmented into the observed stellar disc in the GC. As soon as the cloud hits the black hole, gas with opposite angular momentum relative to the black hole collides downstream. This process leads to redistribution of angular momentum and dissipation of kinetic energy, resulting in a compact gaseous accretion disc. A parameter study using thirteen high resolution simulations of homogeneous clouds falling onto the black hole and engulfing it in parts demonstrates that this mechanism is able to produce gaseous accretion discs that could potentially be the progenitor of the observed stellar disc in the GC. A comparison of simulations with different equations of state (adiabatic, isothermal and full cooling) demonstrates the importance of including a detailed thermodynamical description. However the simple isothermal approach already yields good results on the radial mass transfer and accretion rates, as well as disc eccentricities and sizes. We find that the cloud impact parameter strongly influences the accretion rate whereas the impact velocity has a small affect on the accretion rate.

WANDERING BLACK HOLES IN BRIGHT DISK GALAXY HALOS

We perform SPH+N-body cosmological simulations of massive disk galaxies, including a formalism for black hole seed formation and growth, and find that satellite galaxies containing supermassive black hole seeds are often stripped as they merge with the primary galaxy. These events naturally create a population of "wandering" black holes that are the remnants of stripped satellite cores; galaxies like the Milky Way may host 5 -- 15 of these objects within their halos. The satellites that harbor black hole seeds are comparable to Local Group dwarf galaxies such as the Small and Large Magellanic Clouds; these galaxies are promising candidates to host nearby intermediate mass black holes. Provided that these wandering black holes retain a gaseous accretion disk from their host dwarf galaxy, they give a physical explanation for the origin and observed properties of some recently discovered off-nuclear ultraluminous X-ray sources such as HLX-1.

The origin of mid-infrared emission in massive young stellar objects: multi-baseline VLTI observations of W33A

Aims. In this paper we aim to determine the structure on 100 AU scales of the massive young stellar object W33A, using interferometric observations in the mid-infrared. This emission could be caused by a variety of elements, for example, the inner protostellar envelope, outflow cavity walls, or a dusty or gaseous accretion disk. Methods. We used the Unit Telescopes of the VLT Interferometer in conjunction with the MIDI instrument to obtain spectrally dispersed visibilities in the N-band on 4 baselines with an angular resolution between 25 and 60 milli-arcsec (equivalent to 95 and 228 AU at 3.8 kpc). The visibility spectra and spectral energy distribution were compared to 2D-axi-symmetric dust radiative transfer models with a geometry that includes a rotationally flattened envelope and outflow cavities. We assumed an O7.5 ZAMS star as the central source, consistent with the observed bolometric luminosity. The observations were compared to models with and without (dusty and gaseous) accretion disks. Results. The visibilities are between 5% and 15%, and the non-spherically symmetric emitting structure has a typical size of 100 AU. A satisfactory model is constructed to reproduce the visibility spectra for each (u ,v ) point. It fits the N-band flux spectrum, the midinfrared slope, the far-infrared peak, and the (sub)mm regime of the SED. It produces a 350 μm morphology consistent with the observations. Conclusions. The mid-infrared emission of W33A on 100 AU scales is dominated by the irradiated walls of the cavity sculpted by the

Filling the disk hollow following binary black hole merger: The transient accretion afterglow

Tidal torques from a binary black hole empty out the central regions in any circumbinary gaseous accretion disk. The balance between tidal torques and viscosity maintain the inner edge of the disk at a radius $r\ensuremath{\sim}1.5a--2a$, where $a$ is the binary semimajor axis. Eventually, the inspiraling binary decouples from the disk and merges, leaving behind a central hollow (``donut hole'') in the disk orbiting the remnant black hole. We present a simple, time-dependent, Newtonian calculation that follows the secular (viscous) evolution of the disk as it fills up the hollow down to the black hole innermost stable circular orbit and then relaxes to stationary equilibrium. We use our model to calculate the electromagnetic radiation (``afterglow'') spectrum emitted during this transient accretion epoch. Observing the temporal increase in the total electromagnetic flux and the hardening of the spectrum as the donut hole fills may help confirm a binary black hole merger detected by a gravitational wave interferometer. We show how the very existence of the initial hollow can lead to super-Eddington accretion during this secular phase if the rate is not very far below Eddington prior to decoupling. Our model, though highly idealized, may be useful in establishing some of the key parameters, thermal emission features and scalings that characterize this transient. It can serve as a guide in the design and calibration of future radiation-magnetohydrodynamic simulations in general relativity.

EVIDENCE FOR WARPED DISKS OF YOUNG STARS IN THE GALACTIC CENTER

The central parsec around the super-massive black hole in the Galactic Center hosts more than 100 young and massive stars. Outside the central cusp (R~1") the majority of these O and Wolf-Rayet (WR) stars reside in a main clockwise system, plus a second, less prominent disk or streamer system at large angles with respect to the main system. Here we present the results from new observations of the Galactic Center with the AO-assisted near-infrared imager NACO and the integral field spectrograph SINFONI on the ESO/VLT. These include the detection of 27 new reliably measured WR/O stars in the central 12" and improved measurements of 63 previously detected stars, with proper motion uncertainties reduced by a factor of four compared to our earlier work. We develop a detailed statistical analysis of their orbital properties and orientations. Half of the WR/O stars are compatible with being members of a clockwise rotating system. The rotation axis of this system shows a strong transition as a function of the projected distance from SgrA*. The main clockwise system either is either a strongly warped single disk with a thickness of about 10 degrees, or consists of a series of streamers with significant radial variation in their orbital planes. 11 out of 61 clockwise moving stars have an angular separation of more than 30 degrees from the clockwise system. The mean eccentricity of the clockwise system is 0.36+/-0.06. The distribution of the counter-clockwise WR/O star is not isotropic at the 98% confidence level. It is compatible with a coherent structure such as stellar filaments, streams, small clusters or possibly a disk in a dissolving state. The observed disk warp and the steep surface density distribution favor in situ star formation in gaseous accretion disks as the origin of the young stars.

Characterizing the gravitational instability in cooling accretion discs

In this paper we perform a systematic analysis of the structure induced by the onset of gravitational instabilities in cooling gaseous accretion discs. It is well known that for low enough cooling rates the disc reaches a quasi-steady configuration, with the instability saturating at a finite amplitude such that the disc is kept close to marginal stability. We analyse the dependence of the saturation amplitude on the imposed cooling rate, and we find that it scales with the inverse square root of the cooling parameter β = t cool /t dyn . This indicates that the heating rate induced by the instability is proportional to the energy density of the density waves excited by the disc self-gravity. In particular, we find that at saturation the energy dissipated per dynamical time by weak shocks due to the gravitational perturbations is of the order of 20 per cent of the wave energy. We further perform a Fourier analysis of the disc structure, and subsequently determine the dominant radial and azimuthal wavenumbers of the density waves. While the number of spiral arms (corresponding to the azimuthal wavenumber) is fairly constant with radius, we find that the disc displays a range of radial wavenumbers whose mean increases with increasing radius. The dominant modes closely match the locally most unstable wavelength as predicted by linear perturbation analysis. As a consequence, we numerically demonstrate that the density waves excited in relatively low-mass discs M disc /M * ∼ 0.1 are always close to corotation, deviating from it by approximately 10 per cent. This result can be understood in terms of the constancy of the Doppler-shifted phase Mach number of the flow - the pattern speed self-adjusts so that the flow into spiral arms is always sonic. This has profound effects on the degree to which the extraction of energy and angular momentum from the mean flow through density waves can be modelled as a viscous process. Our results thus provide (i) a detailed description of how the self-regulation mechanism is established for low cooling rates, (ii) a clarification of the conditions required for describing the transport induced by self-gravity through an effective viscosity, (iii) an estimate of the maximum amplitude of the density perturbation before fragmentation takes place and finally (iv) a simple recipe to estimate the density perturbation in different thermal regimes.