non-Keplerian Motion Research Articles

Hot spots around Sagittarius A*. Joint fits to astrometry and polarimetry

Observations of Sagittarius A* ( in the near-infrared (NIR) show irregular flaring activity. Flares coincide with the astrometric rotation of the brightness centroid and with looping patterns in fractional linear polarization. These signatures can be explained with a model of a bright hot spot, transiently orbiting the black hole. We extend the capabilities of the existing algorithms to perform parameter estimation and model comparison in the Bayesian framework using NIR observations from the GRAVITY instrument, and simultaneously fitting the astrometric and polarimetric data. Using the numerical radiative transfer code we defined several geometric models describing a hot spot orbiting Sgr A*, threaded with a magnetic field, and emitting synchrotron radiation. We then explored the posterior space of our models with a nested sampling code We used Bayesian evidence to make comparisons between the models. We have used 11 models to sharpen our understanding of the importance of various aspects of the orbital model, such as non-Keplerian motion, hot-spot size, and off-equatorial orbit. All considered models converge to realizations that fit the data well, but the equatorial super-Keplerian model is favoured by the currently available NIR dataset. We have inferred an inclination of $ deg, which corroborates previous estimates, a preferred period of $ minutes, and an orbital radius of $ gravitational radii with the orbital velocity of $ times the Keplerian value. A hot spot with a diameter smaller than 5 gravitational radii is favoured. Black hole spin is not constrained well.

The Trajectory Prediction of Spacecraft under the Influence of Gyroscopic Effect Generated during Non-Keplerian Motion

Due to perturbation forces and control forces, trajectories of spacecraft around the Earth are usually non-Keplerian orbits, which may result in a gyroscopic effect. To meet the complex demands of space operations in the future, the trajectory prediction of spacecraft under the influence of the gyroscopic effect generated during non-Keplerian motion needs to be studied in depth. The paper investigated the trajectory of spacecraft under the gyroscopic effect generated during non-Keplerian motion. Firstly, according to the similarity between the spacecraft precession motion and the gyroscopic precession, as well as the definition of the “gyroscopic effect” of high-speed rotating bodies, the “gyroscopic effect” generated during the non-Keplerian motion of spacecraft around the earth was defined. Then, taking a continuous radial thrust orbit as an example, the dynamics equations of spacecraft under the influence of gyroscopic effect were deduced. Through theoretical analysis and numerical simulation, the trajectory of spacecraft under the influence of the gyroscopic effect generated during non-Keplerian motion was investigated. Finally, the paper simulated the examples and tested the performance of the proposed method. Simulation results show that a large gyroscopic moment may be generated in some non-Keplerian motion of the spacecraft. The greater the rotational angular velocity of the orbital plane, the greater the gyroscopic moment. Due to the gyroscopic effect, there is a significant deviation in the orbit and the orbital elements compared to those without considering the gyroscopic effect, which indicates that the influence of the gyroscopic effect generated during non-Keplerian motion on the orbit of the spacecraft cannot be ignored. It can be seen from the simulation results that the gyroscopic effect has a significant influence on the trajectory of spacecraft. In some special cases, the gyroscopic effect can be utilized reasonably to save fuel and realize low-energy orbit maneuver control technology in actual space missions; but the control should be considered for the spacecraft to bring it back to the desired orbit in most cases. It is necessary to study the trajectory of spacecraft under the influence of gyroscopic effect. The method and conclusions proposed can provide a theoretical reference for spacecraft trajectory prediction and future large-scale fast orbital maneuvers to meet the needs of complex space operations.

Fitting the light curves of Sagittarius A* with a hot-spot model

Context. Sagittarius A* (Sgr A*) exhibits frequent flaring activity across the electromagnetic spectrum. Signatures of an orbiting hot spot have been identified in the polarized millimeter wavelength light curves observed with ALMA in 2017 immediately after an X-ray flare. The nature of these hot spots remains uncertain. Aims. We expanded existing theoretical hot-spot models created to describe the Sgr A* polarized emission at millimeter wavelengths. We sampled the posterior space, identifying best-fitting parameters and characterizing uncertainties. Methods. Using the numerical radiative transfer code ipole, we defined a semi-analytical model describing a ball of plasma orbiting Sgr A*, threaded with a magnetic field and emitting synchrotron radiation. We then explored the posterior space in the Bayesian framework of dynesty. We fit the static background emission separately, using a radiatively inefficient accretion flow model. Results. We considered eight models with a varying level of complexity, distinguished by choices regarding dynamically important cooling, non-Keplerian motion, and magnetic field polarity. All models converge to realizations that fit the data, but one model without cooling, non-Keplerian motion, and magnetic field pointing toward us improved the fit significantly and also matched the observed circular polarization. Conclusions. Our models represent observational data well and allow testing various effects in a systematic manner. From our analysis, we have inferred an inclination of ∼155 − 160 deg, which corroborates previous estimates, a preferred period of ∼90 min, and an orbital radius of 9 − 12.0 gravitational radii. Our non-Keplerian models indicate a preference for an orbital velocity of 0.6–0.9 times the Keplerian value. Last, all our models agree on a high dimensionless spin value (a* > 0.8), but the impact of spin on the corresponding light curves is subdominant with respect to other parameters.

Beyond Point Masses. II. Non-Keplerian Shape Effects Are Detectable in Several TNO Binaries

About 40 trans-Neptunian binaries (TNBs) have fully determined orbits with about 10 others being solved except for breaking the mirror ambiguity. Despite decades of study, almost all TNBs have only ever been analyzed with a model that assumes perfect Keplerian motion (e.g., two point masses). In reality, all TNB systems are non-Keplerian due to nonspherical shapes, possible presence of undetected system components, and/or solar perturbations. In this work, we focus on identifying candidates for detectable non-Keplerian motion based on sample of 45 well-characterized binaries. We use MultiMoon, a non-Keplerian Bayesian inference tool, to analyze published relative astrometry allowing for nonspherical shapes of each TNB system’s primary. We first reproduce the results of previous Keplerian fitting efforts with MultiMoon, which serves as a comparison for the non-Keplerian fits and confirms that these fits are not biased by the assumption of a Keplerian orbit. We unambiguously detect non-Keplerian motion in eight TNB systems across a range of primary radii, mutual orbit separations, and system masses. As a proof of concept for non-Keplerian fitting, we perform detailed fits for (66652) Borasisi-Pabu, possibly revealing a J 2 ≈ 0.44, implying Borasisi (and/or Pabu) may be a contact binary or an unresolved compact binary. However, full confirmation of this result will require new observations. This work begins the next generation of TNB analyses that go beyond the point mass assumption to provide unique and valuable information on the physical properties of TNBs with implications for their formation and evolution.

The period bouncer system SDSS J105754.25+275947.5: First radial velocity study

Abstract We report the first radial velocity spectroscopic study of the eclipsing period bouncer SDSS J105754.25+275947.5. Together with eclipse light curve modeling, we redetermined the system parameters and studied the accretion disk structure. We confirm that the system contains a white dwarf with MWD = 0.83(3) M⊙ and an effective temperature of 11 500(400) K. The mass of the secondary is M2 = 0.056 M⊙ with an effective temperature of T2 = 2100 K or below. The system inclination is i = 84$fdg$3(6). The data is in good agreement with our determination of K1 = 33(4) km s−1. We estimate the mass transfer rate as $\dot{M}=1.9(2)\times 10^{-11}$M⊙yr −1. Based on an analysis of the SDSS and OSIRIS spectra, we conclude that the optical continuum is formed predominantly by the radiation from the white dwarf. The contribution of the accretion disk is low and originates from the outer part of the disk. The Balmer emission lines are formed in a plasma with log N0 = 12.7 [cm−1] and a kinetic temperature of T∼ 10,000 K. The size of the disk, where the emission lines are formed, expands up to Rd, out = 0.29 R⊙. The inner part of the emission line forming region goes down to Rd, in ≈ 2RWD . The Doppler tomography and trailed spectra shows the presence of a hot spot and a clumpy structure in the disk, with variable intensity along the disk position angle. There is an extended region at the side opposite the hot spot with two bright clumps caused more probably by non-Keplerian motion there.

Qualitative study of ballistic capture at Mars via Lagrangian descriptors

Lagrangian descriptors reveal the dynamical skeleton governing transport mechanisms of a generic flow. In doing so, they unveil geometrical structures in the phase space that separate regions with different qualitative behavior. This work investigates to what extent Lagrangian descriptors provide information about non-Keplerian motion in Mars proximity, which is modeled under the planar elliptic restricted three-body problem. We propose a novel technique to reveal ballistic capture orbits extracting separatrices of the phase space highlighted by Lagrangian descriptor scalar fields. The Roberts’ operator to approximate the gradient is used to detect the edges in the fields. Results demonstrate the chaos indicator ability to distinguish sets of initial conditions exhibiting different dynamics, including ballistic capture ones. Separatrices are validated against reference weak stability boundary derived on similar integration intervals. Compared to other techniques, Lagrangian descriptors provide dynamics insight bypassing the propagation of the variational equations.

Kinematic Evidence of an Embedded Protoplanet in HD 142666 Identified by Machine Learning

Observations of protoplanetary disks have shown that forming exoplanets leave characteristic imprints on the gas and dust of the disk. In the gas, these forming exoplanets cause deviations from Keplerian motion, which can be detected through molecular line observations. Our previous work has shown that machine learning can correctly determine if a planet is present in these disks. Using our machine-learning models, we identify strong, localized non-Keplerian motion within the disk HD 142666. Subsequent hydrodynamics simulations of a system with a 5 M J planet at 75 au recreate the kinematic structure. By currently established standards in the field, we conclude that HD 142666 hosts a planet. This work represents a first step toward using machine learning to identify previously overlooked non-Keplerian features in protoplanetary disks.

Kinematical Constraint on Eccentricity in the Protoplanetary Disk MWC 758 with ALMA

We analyzed the archival data of the 13CO and C18O J = 3 − 2 emission lines in the protoplanetary disk around MWC 758 obtained with the Atacama Large Millimeter/submillimeter Array to discuss possible planet–disk interaction and non-Keplerian motion in the disk. We performed fitting of a Keplerian disk model to the observational data and measured the velocity deviations from the Keplerian rotation. We found significant velocity deviations around the inner cavity in the MWC 758 disk. We examined several possibilities that may cause the velocity deviations, such as pressure gradient, height of the emitting layer, infall motion, inner warp, and eccentricity in the disk. We found that the combination of an eccentric orbital motion with eccentricity of 0.1 ± 0.04 at the radius of the inner cavity and an infalling flow best explains the observed velocity deviations. Our kinematically constrained eccentricity of the gas orbital motion close to the inner cavity is consistent with the eccentricity of the dust ring around the inner cavity measured in the submillimeter continuum emission. Our results hint at strong dust–gas coupling around the inner cavity and presence of a gas giant planet inside the inner cavity in the MWC 758 disk.

Observational Signature of Tightly Wound Spirals Driven by Buoyancy Resonances in Protoplanetary Disks

Abstract Besides the spirals induced by the Lindblad resonances, planets can generate a family of tightly wound spirals through buoyancy resonances. The excitation of buoyancy resonances depends on the thermal relaxation timescale of the gas. By computing timescales of various processes associated with thermal relaxation, namely, radiation, diffusion, and gas–dust collision, we show that the thermal relaxation in protoplanetary disks’ surface layers (Z/R ≳ 0.1) and outer disks (R ≳ 100 au) is limited by infrequent gas–dust collisions. The use of the isothermal equation of state or rapid cooling, common in protoplanetary disk simulations, is therefore not justified. Using three-dimensional hydrodynamic simulations, we show that the collision-limited slow thermal relaxation provides favorable conditions for buoyancy resonances to develop. Buoyancy resonances produce predominantly vertical motions, whose magnitude at the 12CO emission surface is of the order of 100 m s−1 for Jovian-mass planets, sufficiently large to detect using molecular line observations with ALMA. We generate synthetic observations and describe characteristic features of buoyancy resonances in Keplerian-subtracted moment maps and velocity channel maps. Based on the morphology and magnitude of the perturbation, we propose that the tightly wound spirals observed in TW Hya could be driven by a (sub-)Jovian-mass planet at 90 au. We discuss how non-Keplerian motions driven by buoyancy resonances can be distinguished from those driven by other origins. We argue that observations of multiple lines tracing different heights, with sufficiently high spatial/spectral resolution and sensitivity to separate the emission arising from the near and far sides of the disk, will help constrain the origin of non-Keplerian motions.

Keplerian motion on a carousel: a research teaching project

Italian ministerial programs have introduced the teaching of modern physics in the last year of high school (students of 18 years). The textbooks also introduce, obviously in an elementary way, some current problems as, for example, the mysterious galactic rotation curves. It is explained that the cause of their non-Keplerian motion is due to the presence of a non-luminous form of matter and, for this reason, called Dark Matter. To stimulate the curiosity of students who have an aptitude for research, it might be useful to observe that “dark particles” are not provided by the standard model of particle physics and most cosmologists believe in their existence only because they very well explain a series of experimental observations. For this reason, in our opinion, it is interesting to highlight that, until a dark particle is found, it is not wrong to look for an alternative explanation of the experimental data. For example, to simulate the strange behavior of galaxies rotation using elementary and well-known concepts, we have built with students, during a series of lectures in the classroom, a Galilean composition between a Keplerian motion and a very slow circular motion.

A Modified Relativistic Model of Gravitational Propagation

A novel model of gravity is proposed and developed by modifying general relativity through propagating the gravitational field in an entirely analogous way to that of electromagnetic fields. It is therefore not a purely geometric model of gravitation, but is self-consistent, having clear causality and has the benefit of being inherently compatible with unified field theories. This model reproduces the observed almost constant rotational velocities of many galaxies as well as other large scale non-Keplerian motion. This is achieved without assuming the existence of dark matter and is made possible by modelling a rapidly rotating central star which with the inclusion of a velocity induced Doppler shift (of gravity) generates a highly anisotropic and intense, sheet like gravitational field. At extremely high gravitational fields this model remains real and finite i.e. does not generate a black hole, instead it asymptotically approaches a field limit below which light may escape. This is due to the inclusion of self-interaction of gravity in vacuum leading to a non-li nearity in the propagation of gravitational energy i.e. the effects of a gravitational field upon itself. This model is implemented computationally using an iterative finite element model. On the scale of our solar system these corrections are small and are shown not to be in obvious disagreement with high precision solar system tests.

IFU spectroscopy of 10 early-type galactic nuclei – III. Properties of the circumnuclear gas emission

Many Early-type galaxies (ETG) have ionized gas emission in their centres that extends to scales of ~ 1kpc. The majority of such objects are classified as LINERs, but the nature of their ionizing source is still not clear. The kinematics associated with these gaseous structures usually shows deviations from a pure rotational motion due to non-gravitational effects or to non-axisymmetric potentials. This is the third of a series of papers that describes a sample of 10 nearby and massive ETG observed with the Gemini Multi-Object Spectrograph in Integral Field mode installed on the Gemini-South telescope. In paper II, we performed spectral synthesis to subtract the stellar components from the data cubes of the sample galaxies in order to study their nuclear spectra. Here, we analyse the circumnuclear gas emission (scales of ~ 100 pc) of the sample galaxies. Circumnuclear gas emission was detected in seven galaxies, all of them classified as LINERs. Pure gaseous discs are found in three galaxies. In two objects, gaseous discs are probably present, but their kinematics are affected by non-Keplerian motions. In IC 5181, we detected a spiral structure of gas that may be caused either by a non-axisymmetric potential or by an outflow together with a gaseous disc. In NGC 3136, an ionization bicone is present in addition to five compact structures with LINER-like emission. In galaxies with a gaseous disc, we found that ionizing photons emitted by an AGN are not enough to explain the observed Ha flux along this structure. On the other hand, the Ha flux distribution and equivalent width along the direction perpendicular the gaseous disc suggest the presence of low-velocity ionized gas emission which seem to be related to the nuclear activity. We propose a scenario for LINER-like circumnuclear regions where a low-velocity ionization cone is formed by a collimating agent aligned with the gaseous disc.

Numerical orbit integration based on Lie series with use of parallel computing techniques

This article outlines necessary steps to perform numerical orbit integrations based on a Lie series approach. Its implementation requires an efficient evaluation of resulting series coefficients. As an example we treat the classical main problem in satellite orbit calculation (J2 only) and the case of a 4×4-gravity field. All calculations were performed in very high precision with up to 100 significant digits. In comparison to independent third party computations this approach led to superior results referring to the verifiable constancy of various integrals of motion. To achieve a performance similar to classical numerical integrations in terms of acceptable computing time, at least for non-Keplerian motion problems, we exploited parallel computing capabilities. For our examples, run times were improved by several orders of magnitude, depending on the actual chosen precision level (up to a factor of 50,000 in case of double precision). Here we present the mathematical framework of the proposed orbital integration scheme as well as the work flow for its application in a multi-core, parallel computing environment.

A modified “Type I migration” model for propeller moons in Saturn's rings

We propose a mechanism for the observed non-keplerian motion (Tiscareno et al., 2010a) of “propeller” moons embedded in Saturn's rings. Our mechanism, in which radial variations in surface density – external to, and unaffected by, the embedded moon – result in an equilibrium semimajor axis for the moon due to “Type I” angular momentum exchange (Crida et al., 2010), provides a good fit to the observations. Future observations should distinguish between our model and others recently proposed.

CARE AND FEEDING OF FROGS

"Propellers" are features in Saturn's A ring associated with moonlets that open partial gaps. They exhibit non-Keplerian motion (Tiscareno 2010); the longitude residuals of the best-observed propeller, "Bl\'eriot," appear consistent with a sinusoid of period ~4 years. Pan and Chiang (2010) proposed that propeller moonlets librate in "frog resonances" with co-orbiting ring material. By analogy with the restricted three-body problem, they treated the co-orbital material as stationary in the rotating frame and neglected non-co-orbital material. Here we use simple numerical experiments to extend the frog model, including feedback due to the gap's motion, and drag associated with the Lindblad disk torques that cause Type I migration. Because the moonlet creates the gap, we expect the gap centroid to track the moonlet, but only after a time delay t_diff, the time for a ring particle to travel from conjunction with the moonlet to the end of the gap. We find that frog librations can persist only if t_diff exceeds the frog libration period P_lib, and if damping from Lindblad torques balances driving from co-orbital torques. If t_diff << P_lib, then the libration amplitude damps to zero. In the case of Bl\'eriot, the frog resonance model can reproduce the observed libration period P_lib ~ 4 yr. However, our simple feedback prescription suggests that Bl\'eriot's t_diff ~ 0.01P_lib, which is inconsistent with the observed libration amplitude of 260 km. We urge more accurate treatments of feedback to test the assumptions of our toy models.

THE STRUCTURE AND DYNAMICS OF MOLECULAR GAS IN PLANET-FORMING ZONES: A CRIRES SPECTRO-ASTROMETRIC SURVEY

We present a spectro-astrometric survey of molecular gas in the inner regions of 16 protoplanetary disks using CRIRES, the high resolution infrared imaging spectrometer on the Very Large Telescope. Spectro-astrometry with CRIRES measures the spatial extent of line emission to sub-milliarcsecond precision, or <0.2 AU at the distance of the observed targets. The sample consists of gas-rich disks surrounding stars with spectral types ranging from K to A. The properties of the spectro-astrometric signals divide the sources into two distinct phenomenological classes: one that shows clear Keplerian astrometric spectra, and one in which the astrometric signatures are dominated by gas with strong non-Keplerian (radial) motions. Similarly to the near-infrared continuum emission, as determined by interferometry, we find that the size of the CO line emitting region in the Keplerian sources obeys a size-luminosity relation as $R_CO L_*^0.5. The non-Keplerian spectro-astrometric signatures are likely indicative of the presence of wide-angle disk winds. The central feature of the winds is a strong sub-Keplerian velocity field due to conservation of angular momentum as the wind pressure drives the gas outwards. We construct a parametrized 2-dimensional disk+wind model that reproduces the observed characteristics the observed CO spectra and astrometry. The modeled winds indicate mass-loss rates of >10^-10 to 10^-8 Msol/yr. We suggest a unifying model in which all disks have slow molecular winds, but where the magnitude of the mass-loss rate determines the degree to which the mid-infrared molecular lines are dominated by the wind relative to the Keplerian disk surface.

THE PROPELLER AND THE FROG

"Propellers" in planetary rings are disturbances in ring material excited by moonlets that open only partial gaps. We describe a new type of co-orbital resonance that can explain the observed non-Keplerian motions of propellers. The resonance is between the moonlet underlying the propeller, and co-orbiting ring particles downstream of the moonlet where the gap closes. The moonlet librates within the gap about an equilibrium point established by co-orbiting material and stabilized by the Coriolis force. In the limit of small libration amplitude, the libration period scales linearly with the gap azimuthal width and inversely as the square root of the co-orbital mass. The new resonance recalls but is distinct from conventional horseshoe and tadpole orbits; we call it the "frog" resonance, after the relevant term in equine hoof anatomy. For a ring surface density and gap geometry appropriate for the propeller Bl\'eriot in Saturn's A ring, our theory predicts a libration period of ~4 years, similar to the ~3.7 year period over which Bl\'eriot's orbital longitude is observed to vary. These librations should be subtracted from the longitude data before any inferences about moonlet migration are made.

Possible Molecular Spiral Arms in the Protoplanetary Disk of AB Aurigae

The circumstellar dust disk of the Herbig Ae star AB Aur has been found to exhibit complex spiral-like structures in the near-IR image obtained with the Subaru Telescope. We present maps of the disk in both 12CO (3-2) and dust continuum at 345 GHz with the Submillimeter Array at an angular resolution of 10 × 07 (144 × 100 AU). The continuum emission traces a dust disk with a central depression and a maximum overall dimension of 450 AU (FWHM). This dust disk exhibits several distinct peaks that appear to coincide with bright features in the near-IR image, in particular the brightest inner spiral arm. The CO emission traces a rotating gas disk of size 530 × 330 AU with a deprojected maximum velocity of 2.8 km s-1 at 450 AU. In contrast with the dust disk, the gas disk exhibits an intensity peak at the stellar position. Furthermore, the CO emission in several velocity channels traces the innermost spiral arm seen in the near-IR. We compare the observed spatial-kinematic structure of the CO emission to a simple model of a disk in Keplerian rotation and find that only the emission tracing the main spiral arm clearly lies outside the confines of our model. This emission has a net outward radial motion compared with the radial velocity predicted by the model at the location of the main spiral arms. The disk of AB Aur is therefore quite different from the Keplerian disks seen around many Herbig Ae stars. The spiral-like structures of the disk with non-Keplerian motions we revealed in 12CO (3-2), together with the central depression of the dust disk, could be explained to be driven by the possible existence of a giant planet forming in the disk.

Sub-arcsec imaging of the AB Aur molecular disk and envelope at millimeter wavelengths: a non Keplerian disk

We present sub-arcsecond images of AB Auriga obtained with the IRAM Plateau de Bure interferometer in the isotopologues of CO, and in continuum at 3 and 1.3 mm. These observations allow us to trace the structure of the circumstellar material of AB Aur in regions where optical and IR imaging is impossible because of the emission from the star. These images reveal that the environment of AB Aur is widely different from the proto-planetary disks that surround T Tauri stars like DM Tau and LkCa15 or HAeBe stars like MWC 480 in several aspects. Instead of being centrally peaked, the continuum emission is dominated by a bright, asymmetric (spiral-like) feature at about 140 AU from the central star. Little emission is associated with the star itself. The molecular emission shows that AB Aur is surrounded by a very extended flattened structure (“disk”), which is rotating around the star. Bright molecular emission is also found towards the continuum “spiral”. The large-scale molecular structure suggests the AB Aur disk is inclined between 23 and 43 degrees, but the strong asymmetry of the continuum and molecular emission prevents an accurate determination of the inclination of the inner parts. Analysis of the emission in terms of a Keplerian disk provides a reasonable fit to the data, but fails to give a consistent picture because the inclinations determined from , , and do not agree. The mass predicted for the central star in such Keplerian models is in the range 0.9 to 1.2 , much smaller than the expected 2.2 from the spectral type of AB Aur. Better and more consistent fits to the , data are obtained by relaxing the Keplerian hypothesis. We find significant non-Keplerian motion, with a best fit exponent for the rotation velocity law of 0.41 ± 0.01, but no evidence for radial motion. The disk has an inner hole about 70 AU in radius. The disk is warm and shows no evidence of depletion of CO. The dust properties suggest that the dust is less evolved than in typical T Tauri disks. Both the spiral-like feature and the departure from purely Keplerian motion indicates the AB Aur disk is not in quasi-equilibrium. Disk self-gravity is insufficient to create the perturbation. This behavior may be related either to an early phase of star formation in which the Keplerian regime is not yet fully established and/or to a disturbance of yet unknown origin. An alternate, but unproven, possibility is that of a low mass companion located about 40 AU from AB Aur.

Origin of non-Keplerian motions of masers in NGC 1068

We demonstrate that the “sub-Keplerian” rotation curve of maser spots in NGC 1068 can be explained by the gravitational attraction of the disc orbiting the central black hole. Possible parameters matching observations are: black hole mass , disc outer edge ≳1.3 pc, aspect ratio , surface density , and disc mass . The physical conditions required for the excitation of masers are fulfilled, and the outer disc would stand in a gravitationally marginally stable state.