Power-law Disk Research Articles

Evidence of hard power-law spectral cutoff and disc reflection features from the X-ray transient XTE J1739–285

ABSTRACT We report on the nearly simultaneous NICER and NuSTAR observations of the known X-ray transient XTE J1739–285. These observations provide the first sensitive hard X-ray spectrum of this neutron star X-ray transient. The source was observed on 2020 February 19 in the hard spectral state with a luminosity of 0.007 of the Eddington limit. The broadband 1−70 keV NICER and NuSTAR observation clearly detects a cutoff of the hard spectral component around 34−40 keV when the continuum is fitted by a soft thermal component and a hard power-law component. This feature has been detected for the first time in this source. Moreover, the spectrum shows evidence for disc reflection – a relativistically broadened Fe K α line around 5−8 keV and a Compton hump in the 10−20 keV energy band. The accretion disc reflection features have not been identified before from this source. Through accretion disc reflection modelling, we constrain the radius of the inner disc to be $R_{\rm in}=3.1_{-0.5}^{+1.8}R_{\rm ISCO}$ for the first time. In addition, we find a low inclination, i ∼ 33°. Assuming the magnetosphere is responsible for such truncation of the inner accretion disc above the stellar surface, we establish an upper limit of 6.2 × 108 G on the magnetic field at the poles.

Forming pressure traps at the snow line to isolate isotopic reservoirs in the absence of a planet

Context.Pressure maxima are regions in protoplanetary disks in which pebbles can be trapped because the regions have no local pressure gradient. These regions could be ideal places in which planetesimals might be formed or to isotopic reservoirs might be isolated. Observations of protoplanetary disks show that dusty ring structures are common, and pressure maxima are sometimes invoked as a possible explanation. In our Solar System, pressure bumps have been suggested as a possible mechanism for separating reservoirs with different nucleosynthetic compositions that are identified among chondrites and iron meteorites. In this paper, we detail a mechanism by which pressure maxima form just inward of the snow line in stratified disks (with a dead zone and an active layer). This mechanism does not require the presence of a planet.Aims.We investigate the conditions for the formation of pressure maxima using a vertically averagedαviscosity model and release of water vapor at the snow line.Methods.We considered a 1Dαdisk model. Using a combination of analytical and numerical investigations, we explored the range of conditions for a pressure maximum to form inside the dead zone and just inward of the snow line.Results.When the vertically averagedαis a decreasing function of the surface density, then the release of water vapor at the snow line decreases the sound velocity, and a pressure bump appears in turn. This requires a constant inflow of icy pebbles with a ratio of the pebble influx to gas influx >0.6 for a power-law disk with a 1% ice-to-gas ratio, and >1.8 for a disk with an ice-to-gas ratio ~0.3%. If these conditions are met, then a pressure maximum appears just inward of the snow line due to a process that couples the dead and active layers at the evaporation front. The pressure bump survives as long as the icy pebble flux is high enough. The formation of the pressure bump is triggered by the decrease in sound velocity inward of the snow line through the release of water vapor.Conclusions.This mechanism is promising for isolating early reservoirs carrying different isotopic signatures in the Solar System and for promoting dry planetesimal formation inward of the snow line, provided the vertically averaged description of a dead zone is valid.

Global Spiral Density Wave Modes in Protoplanetary Disks: Morphology of Spiral Arms

Abstract We analyze two-armed global spiral density wave modes generated by gravitational instability in razor-thin, nonviscous, self-gravitating protoplanetary disks to understand the dependence of spiral arm morphology (pitch angle α and amplitude) on various disk conditions. The morphologies of the resulting spiral density wave modes closely resemble observations. Their pitch angles and pattern speeds are insensitive to the boundary conditions adopted. Gaussian disks exhibit more tightly wound spirals (smaller pitch angle) than power-law disks under the same conditions. We find that at a fixed disk-to-star mass ratio (M d /M *), the pitch angle increases with average Toomre’s stability parameter ( ) or average disk aspect ratio ( ). For a given , density wave modes with higher M d /M * have larger pitch angles, while the behavior reverses for a given . The interdependence between pitch angle and disk properties can be roughly approximated by , where c s is the sound speed. Our gravitational instability-excited spiral density waves can be distinguished from planet-launched spirals: (1) massive cool disks have spiral pitch angle falling with radius, while low-mass hot disks have spiral pitch angle rising with radius; (2) the profile of spiral amplitude presents several dips and bumps. We propose that gravitational instability-excited density waves can serve as an alternative scenario to explain the observed spiral arms in self-gravitating protoplanetary disks.

Multiple Rings of Millimeter Dust Emission in the HD 15115 Debris Disk

Abstract We present observations of the HD 15115 debris disk from the Atacama Large Millimeter/submillimeter Array (ALMA) at 1.3 mm that capture this intriguing system with the highest resolution (0.″6 or 29 au) at millimeter wavelengths to date. This new ALMA image shows evidence for two rings in the disk separated by a cleared gap. By fitting models directly to the observed visibilities within a Markov Chain Monte Carlo framework, we are able to characterize the millimeter continuum emission and place robust constraints on the disk structure and geometry. In the best-fit model of a power-law disk with a Gaussian gap, the disk inner and outer edges are at 43.9 ± 5.8 au (0.″89 ± 0.″12) and 92.2 ± 2.4 au (1.″88 ± 0.″49), respectively, with a gap located at 58.9 ± 4.5 au (1.″2 ± 0.″10) with a fractional depth of 0.88 ± 0.10 and a width of 13.8 ± 5.6 au (0.″28 ± 0.″11). Because we do not see any evidence at millimeter wavelengths for the dramatic east–west asymmetry seen in scattered light, we conclude that this feature most likely results from a mechanism that only affects small grains. Using dynamical modeling and our constraints on the gap properties, we are able to estimate a mass for the possible planet sculpting the gap to be 0.16 ± 0.06 M Jup.

Revisiting the case of R Monocerotis: Is CO removed at R < 20 au?

To our knowledge, R Mon is the only B0 star in which a gaseous Keplerian disk has been detected. However, there is some controversy about the spectral type of R Mon. Some authors propose that it could be a later B8e star, where disks are more common. Our goal is to re-evaluate the R Mon spectral type and characterize its protoplanetary disk. The spectral type of R Mon has been re-evaluated using the available continuum data and UVES emission lines. We used a power-law disk model to fit previous 12CO 1→0 and 2→1 interferometric observations and the PACS CO data to investigate the disk structure. Interferometric detections of 13CO J=1→0, HCO+ 1→0, and CN 1→0 lines using the IRAM Plateau de Bure Interferometer (PdBI) are presented. The HCN 1→0 line was not detected. Our analysis confirms that R Mon is a B0 star. The disk model compatible with the 12CO 1→0 and 2→1 interferometric observations falls short of predicting the observed fluxes of the 14<J u <31 PACS lines; this is consistent with the scenario in which some contribution to these lines is coming from a warm envelope and/or UV-illuminated outflow walls. More interestingly, the upper limits to the fluxes of the J u >31 CO lines suggest the existence of a region empty of CO at R≲20 au in the proto-planetary disk. The intense emission of the HCO+ and CN lines shows the strong influence of UV photons on gas chemistry. The observations gathered in this paper are consistent with the presence of a transition disk with a cavity of R in ≳20 au around R Mon. This size is similar to the photoevaporation radius that supports the interpretation that UV photoevaporation is main disk dispersal mechanism in massive stars.

Numerical computation of gravitational field of infinitely-thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33

AbstractWe developed a numerical method[-70pt] to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’ algorithm. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density.

Saving Super-Earths: Interplay between Pebble Accretion and Type I Migration

Abstract Overcoming type I migration and preventing low-mass planets from spiralling into the central star is a long-studied topic. It is well known that outward migration is possible in viscously heated disks relatively close to the central star because the entropy gradient can be sufficiently steep for the positive corotation torque to overcome the negative Lindblad torque. Yet efficiently trapping planets in this region remains elusive. Here we study disk conditions that yield outward migration for low-mass planets under specific planet migration prescriptions. In a steady-state disk model with a constant α-viscosity, outward migration is only possible when the negative temperature gradient exceeds ∼0.87. We derive an implicit relation for the highest mass at which outward migration is possible as a function of viscosity and disk scale height. We apply these criteria, using a simple power-law disk model, to planets that have reached their pebble isolation mass after an episode of rapid accretion. It is possible to trap planets with the pebble isolation mass farther than the inner edge of the disk provided that α crit ≳ 0.004 for disks older than 1 Myr. In very young disks, the high temperature causes the planets to grow to masses exceeding the maximum for outward migration. As the disk evolves, these more massive planets often reach the central star, generally only toward the end of the disk lifetime. Saving super-Earths is therefore a delicate interplay between disk viscosity, the opacity profile, and the temperature gradient in the viscously heated inner disk.

Radial Surface Density Profiles of Gas and Dust in the Debris Disk around 49 Ceti

Abstract We present ∼0.″4 resolution images of CO(3–2) and associated continuum emission from the gas-bearing debris disk around the nearby A star 49 Ceti, observed with the Atacama Large Millimeter/Submillimeter Array (ALMA). We analyze the ALMA visibilities in tandem with the broadband spectral energy distribution to measure the radial surface density profiles of dust and gas emission from the system. The dust surface density decreases with radius between ∼100 and 310 au, with a marginally significant enhancement of surface density at a radius of ∼110 au. The SED requires an inner disk of small grains in addition to the outer disk of larger grains resolved by ALMA. The gas disk exhibits a surface density profile that increases with radius, contrary to most previous spatially resolved observations of circumstellar gas disks. While ∼80% of the CO flux is well described by an axisymmetric power-law disk in Keplerian rotation about the central star, residuals at ∼20% of the peak flux exhibit a departure from axisymmetry suggestive of spiral arms or a warp in the gas disk. The radial extent of the gas disk (∼220 au) is smaller than that of the dust disk (∼300 au), consistent with recent observations of other gas-bearing debris disks. While there are so far only three broad debris disks with well characterized radial dust profiles at millimeter wavelengths, 49 Ceti’s disk shows a markedly different structure from two radially resolved gas-poor debris disks, implying that the physical processes generating and sculpting the gas and dust are fundamentally different.

Chemistry in disks

(abridged) We used the IRAM 30-m to perform a sensitive wideband survey of 30 protoplanetary disks in the Taurus Auriga region. We simultaneously observed HCO$^+$(3-2), HCN(3-2), C$_2$H(3-2), CS(5-4), and two transitions of SO. We combine the results with a previous survey which observed $^{13}$CO (2-1), CN(2-1), two o-H$_2$CO lines and one of SO. We use available interferometric data to derive excitation temperatures of CN and C$_2$H in several sources. We determine characteristic sizes of the gas disks and column densities of all molecules using a parametric power-law disk model. Our study is mostly sensitive to molecules at 200-400 au from the stars. We compare the derived column densities to the predictions of an extensive gas-grain chemical disk model, under conditions representative of T Tauri disks. This survey provides 20 new detections of HCO$^+$ in disks, 18 in HCN, 11 in C$_2$H, 8 in CS and 4 in SO. HCO$^+$ is detected in almost all sources, and its J=3-2 line is essentially optically thick, providing good estimates of the disk radii. The other transitions are (at least partially) optically thin. Variations of the column density ratios do not correlate with any specific property of the star or disk. Disks around Herbig Ae stars appear less rich in molecules than those around T Tauri stars, although the sample remains small. SO is only found in the (presumably younger) embedded objects, perhaps reflecting an evolution of the S chemistry due to increasing depletion with time. Overall, the molecular column densities, and in particular the CN/HCN and CN/C$_2$H ratios, are well reproduced by gas-grain chemistry in cold disks. This study provides a census of simple molecules in disks of radii $> 200-300$ au. Extending that to smaller disks, or searching for less abundant or more complex molecules requires a much more sensitive facility, i.e. NOEMA and ALMA.

A self-similar solution for thermal disc winds

We derive a self-similar description for the 2D streamline topology and flow structure of an axi-symmetric, thermally driven wind originating from a disc in which the density is a power law function of radius. Our scale-free solution is strictly only valid in the absence of gravity or centrifugal support; comparison with 2D hydrodynamic simulations of winds from Keplerian discs however demonstrates that the scale-free solution is a good approximation also in the outer regions of such discs, and can provide a reasonable description even for launch radii well within the gravitational radius of the flow. Although other authors have considered the flow properties along streamlines whose geometry has been specified in advance, this is the first isothermal calculation in which the flow geometry and variation of flow variables along streamlines is determined self-consistently. It is found that the flow trajectory is very sensitive to the power-law index of radial density variation in the disc: the steeper the density gradient, the stronger is the curvature of streamlines close to the flow base that is required in order to maintain momentum balance perpendicular to the flow. Steeper disc density profiles are also associated with more rapid acceleration, and a faster fall-off of density, with height above the disc plane. The derivation of a set of simple governing equations for the flow structure of thermal winds from the outer regions of power law discs offers the possibility of deriving flow observables without having to resort to hydrodynamical simulation.

Constraining the physical structure of the inner few 100 AU scales of deeply embedded low-mass protostars

(Abridged) The physical structure of deeply-embedded low-mass protostars (Class 0) on scales of less than 300 AU is still poorly constrained. Determining this is crucial for understanding the physical and chemical evolution from cores to disks. In this study two models of the emission, a Gaussian disk intensity distribution and a parametrized power-law disk model, are fitted to sub-arcsecond resolution interferometric continuum observations of five Class 0 sources, including one source with a confirmed Keplerian disk. For reference, a spherically symmetric single power-law envelope is fitted to the larger scale ($\sim$1000 AU) emission and investigated further for one of the sources on smaller scales. A thin disk model can approximate the emission and physical structure in the inner few 100 AU scales of the studied deeply-embedded low-mass protostars and paves the way for analysis of a larger sample with ALMA. While the disk radii agree with previous estimates the masses are different for some of the sources studied. Assuming a typical temperature distribution, the fractional amount of mass in the disk above 100 K varies in between 7% to 30%. Kinematic data are needed to determine the presence of any Keplerian disk. Using previous observations of p-H$_2^{18}$O, we estimate the relative gas phase water abundances roughly an order of magnitude higher than previously inferred when both warm and cold H$_2$ was used as reference. A spherically symmetric single power-law envelope model fails to simultaneously reproduce both the small and large scale emission.

Numerical computation of gravitational field of infinitely thin axisymmetric disc with arbitrary surface mass density profile and its application to preliminary study of rotation curve of M33

We developed a numerical method[-70pt] to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’ algorithm. By using the new method, we show the rotation curves of some non-trivial discs: (i) truncated power-law discs, (ii) discs with a non-negligible center hole, (iii) truncated Mestel discs with edge-softening, (iv) double power-law discs, (v) exponentially-damped power-law discs, and (vi) an exponential disc with a sinusoidal modulation of the density profile. Also, we present a couple of model fittings to the observed rotation curve of M33: (i) the standard deconvolution by assuming a spherical distributin of the dark matter and (ii) a direct fit of infinitely-thin disc mass with a double power-law distribution of the surface mass density.

RESOLVING PROTOPLANETARY DISKS AT MILLIMETER WAVELENGTHS WITH CARMA

We present continuum observations at 1.3 and 2.7 mm using the Combined Array for Research in Millimeter-wave Astronomy (CARMA) toward six protoplanetary disks in the Taurus molecular cloud: CI Tau, DL Tau, DO Tau, FT Tau, Haro 6-13, and HL Tau. We constrain physical properties of the disks with Bayesian inference using two disk models; flared power-law disk model and flared accretion disk model. Comparing the physical properties, we find that the more extended disks are less flared and that the dust opacity spectral index (beta) is smaller in the less massive disks. In addition, disks with a steeper mid-plane density gradient have a smaller beta, which suggests that grains grow and radially move. Furthermore, we compare the two disk models quantitatively and find that the accretion disk model provides a better fit overall. We also discuss the possibilities of substructures on three extended protoplanetary disks.

DISCOVERY OF RESOLVED DEBRIS DISK AROUND HD 131835

We report the discovery of the resolved disk around HD 131835 and present the analysis and modeling of its thermal emission. HD 131835 is a ~15 Myr A2 star in the Scorpius-Centaurus OB association at a distance of 122.7 +16.2 -12.8 parsec. The extended disk has been detected to ~1.5" (200 AU) at 11.7 {\mu}m and 18.3 {\mu}m with T-ReCS on Gemini South. The disk is inclined at an angle of ~75{\deg} with the position angle of ~61{\deg}. The flux of HD 131835 system is 49.3+-7.6 mJy and 84+-45 mJy at 11.7 {\mu}m and 18.3 {\mu}m respectively. A model with three grain populations gives a satisfactory fit to both the spectral energy distribution and the images simultaneously. This best-fit model is composed of a hot continuous power-law disk and two rings. We characterized the grain temperature profile and found that the grains in all three populations are emitting at temperatures higher than blackbodies. In particular, the grains in the continuous disk are unusually warm; even when considering small graphite particles as the composition.

AHerschelPACS survey of the dust and gas in Upper Scorpius disks

We present results of far-infrared photometric observations with Herschel PACS of a sample of Upper Scorpius stars, with a detection rate of previously known disk-bearing K and M stars at 70, 100, and 160 micron of 71%, 56%, and 50%, respectively. We fit power-law disk models to the spectral energy distributions of K & M stars with infrared excesses, and have found that while many disks extend in to the sublimation radius, the dust has settled to lower scale heights than in disks of the less evolved Taurus-Auriga population, and have much reduced dust masses. We also conducted Herschel PACS observations for far-infrared line emission and JCMT observations for millimeter CO lines. Among B and A stars, 0 of 5 debris disk hosts exhibit gas line emission, and among K and M stars, only 2 of 14 dusty disk hosts are detected. The OI 63 micron and CII 157 micron lines are detected toward [PZ99] J160421.7-213028 and [PBB2002] J161420.3-190648, which were found in millimeter photometry to host two of the most massive dust disks remaining in the region. Comparison of the OI line emission and 63 micron continuum to that of Taurus sources suggests the emission in the former source is dominated by the disk, while in the other there is a significant contribution from a jet. The low dust masses found by disk modeling and low number of gas line detections suggest that few stars in Upper Scorpius retain sufficient quantities of material for giant planet formation. By the age of Upper Scorpius, giant planet formation is essentially complete.

PROBING THE RADIAL TEMPERATURE STRUCTURE OF PROTOPLANETARY DISKS WITH HERSCHEL /HIFI

Herschel/HIFI spectroscopic observations of CO J=10-9, CO J=16-15 and [CII] towards HD 100546 are presented. The objective is to resolve the velocity profile of the lines to address the emitting region of the transitions and directly probe the distribution of warm gas in the disk. The spectra reveal double-peaked CO line profiles centered on the systemic velocity, consistent with a disk origin. The J=16-15 line profile is broader than that of the J=10-9 line, which in turn is broader than those of lower J transitions (6-5, 3-2, observed with APEX), thus showing a clear temperature gradient of the gas with radius. A power-law flat disk model is used to fit the CO line profiles and the CO rotational ladder simultaneously, yielding a temperature of T_0=1100 \pm 350 K (at r_0 = 13 AU) and an index of q=0.85 \pm 0.1 for the temperature radial gradient. This indicates that the gas has a steeper radial temperature gradient than the dust (mean q_{dust} ~ 0.5), providing further proof of the thermal decoupling of gas and dust at the disk heights where the CO lines form. The [CII] line profile shows a strong single-peaked profile red-shifted by 0.5 km s-1 compared to the systemic velocity. We conclude that the bulk of the [CII] emission has a non-disk origin (e.g., remnant envelope or diffuse cloud).

Linear and non-linear evolution of the vertical shear instability in accretion discs

We analyse the stability and non-linear dynamics of power-law accretion disc models. These have mid-plane densities that follow radial power laws and have either temperature or entropy distributions that are strict power-law functions of cylindrical radius, R. We employ two different hydrodynamic codes to perform high-resolution 2D axisymmetric and 3D simulations that examine the long-term evolution of the disc models as a function of the power-law indices of the temperature or entropy, the disc scaleheight, the thermal relaxation time of the fluid and the disc viscosity. We present an accompanying stability analysis of the problem, based on asymptotic methods, that we use to guide our interpretation of the simulation results. We find that axisymmetric disc models whose temperature or entropy profiles cause the equilibrium angular velocity to vary with height are unstable to the growth of perturbations whose most obvious character is modes with horizontal and vertical wavenumbers that satisfy |kR/kZ| ≫ 1. Instability occurs only when the thermodynamic response of the fluid is isothermal, or the thermal evolution time is comparable to or shorter than the local dynamical time-scale. These discs appear to exhibit the Goldreich–Schubert–Fricke or ‘vertical shear’ linear instability. Closer inspection of the simulation results uncovers the growth of two distinct modes. The first are characterized by very short radial wavelength perturbations that grow rapidly at high latitudes in the disc, and descend down towards the mid-plane on longer time-scales. We refer to these as ‘finger modes’ because they display kR/kZ ≫ 1. The second appear at slightly later times in the main body of the disc, including near the mid-plane. These ‘body modes’ have somewhat longer radial wavelengths. Early on they manifest themselves as fundamental breathing modes, but quickly become corrugation modes as symmetry about the mid-plane is broken. The corrugation modes are a prominent feature of the non-linear saturated state, leading to strong vertical oscillation of the disc mid-plane. In a viscous disc with aspect ratio H/r = 0.05, instability is found to operate when the viscosity parameter α < 4 × 10−4. In three dimensions the instability generates a quasi-turbulent flow, and the associated Reynolds stress produces a fluctuating effective viscosity coefficient whose mean value reaches α ∼ 10−3 by the end of the simulation. The evolution and saturation of the vertical shear instability in astrophysical disc models which include realistic treatments of the thermal physics has yet to be examined. Should it occur on either global or local scales, however, our results suggest that it will have significant consequences for their internal dynamics, transport properties and observational appearance.

Accretion flow diagnostics with X-ray spectral timing: the hard state of SWIFT J1753.5-0127

Recent XMM–Newton studies of X-ray variability in the hard states of black hole X-ray binaries (BHXRBs) indicate that the variability is generated in the ‘standard’ optically thick accretion disc that is responsible for the multi-colour blackbody emission. The variability originates in the disc as mass-accretion fluctuations and propagates through the disc to ‘light up’ inner disc regions, eventually modulating the power-law emission that is produced relatively centrally. Both the covariance spectra and time-lags that cover the soft bands strongly support this scenario. Here, we present a comparative spectral-timing study of XMM–Newton data from the BHXRB SWIFT J1753.5−0127 in a bright 2009 hard state with that from the significantly fainter 2006 hard state to show for the first time the change in disc spectral-timing properties associated with a global increase in both the accretion rate and the relative contribution of the disc emission to the bolometric luminosity. We show that, although there is strong evidence for intrinsic disc variability in the more luminous hard state, the disc variability amplitude is suppressed relative to that of the power-law emission, which contrasts with the behaviour at lower luminosities where the disc variability is slightly enhanced when compared with the power-law variations. Furthermore, in the higher luminosity data the disc variability below 0.6 keV becomes incoherent with the power-law and higher energy disc emission at frequencies below 0.5 Hz, in contrast with the coherent variations seen in the 2006 data. We explain these differences and the associated complex lags in the 2009 data in terms of the fluctuating disc model, where the increase in accretion rate seen in 2009 leads to more pronounced and extended disc emission. If the variable signals are generated at small radii in the disc, the variability of disc emission can be naturally suppressed by the fraction of unmodulated disc emission arising from larger radii. Furthermore, the drop in coherence can be produced by disc accretion fluctuations arising at larger radii which are viscously damped and hence unable to propagate to the inner, power-law emitting region.

The Newtonian potential of thin disks

The one-dimensional, ordinary differential equation (ODE) by Hur\'e & Hersant (2007) that satisfies the midplane gravitational potential of truncated, flat power-law disks is extended to the whole physical space. It is shown that thickness effects (i.e. non-flatness) can be easily accounted for by implementing an appropriate "softening length" $\lambda$. The solution of this "softened ODE" has the following properties: i) it is regular at the edges (finite radial accelerations), ii) it possesses the correct long-range properties, iii) it matches the Newtonian potential of a geometrically thin disk very well, and iv) it tends continuously to the flat disk solution in the limit $\lambda \rightarrow 0$. As illustrated by many examples, the ODE, subject to exact Dirichlet conditions, can be solved numerically with efficiency for any given colatitude at second-order from center to infinity using radial mapping. This approach is therefore particularly well-suited to generating grids of gravitational forces in order to study particles moving under the field of a gravitating disk as found in various contexts (active nuclei, stellar systems, young stellar objects). Extension to non-power-law surface density profiles is straightforward through superposition. Grids can be produced upon request.

AGN disks and black holes on the weighting scales

We exploit our formula for the gravitational potential of finite size, power-law disks to derive a general expression linking the mass of the black hole in active galactic nuclei (AGN), the mass of the surrounding disk, its surface density profile (through the power index s), and the differential rotation law. We find that the global rotation curve v(R) of the disk in centrifugal balance does not obey a power law of the cylindrical radius R (except in the confusing case s = -2 that mimics a Keplerian motion), and discuss the local velocity index. This formula can help to understand how, from position-velocity diagrams, mass is shared between the disk and the black hole. To this purpose, we have checked the idea by generating a sample of synthetic data with different levels of Gaussian noise, added in radius. It turns out that, when observations are spread over a large radial domain and exhibit low dispersion (standard deviation less than 10% typically), the disk properties (mass and s-parameter) and black hole mass can be deduced from a non linear fit of kinematic data plotted on a (R, Rv 2)-diagram. For a deviation higher than 10%, masses are estimated fairly well from a linear regression (corresponding to the zeroth-order treatment of the formula), but the power index s is no longer accessible. We have applied the model to 7 AGN disks whose rotation has already been probed through water maser emission. For NGC3393 and UGC3789, the masses seem well constrained through the linear approach. For IC1481, the power-law exponent s can even be deduced. Because the model is scale-free, it applies to any kind of star/disk system. Extension to disks around young stars showing deviation from Keplerian motion is thus straightforward.