Dynamics Of Accretion Disks Research Articles

Transient Phenomena of Inviscid Accretion Gasradiation Slim Disc in A Gravitational Potential after Adiabatic Perturbations of the Velocities

The formalisms for the analysis of the dynamics of slim inviscid accretion discs in a (also, non-Newtonian) gravitational potential are here developped; the role of the horizontal pressure and that of the entropy gradients are to be characterised. The adiabatic perturbations of the velocity are studied: the paradigms of the Lagrangian perturbations of the velocity are implemented. The perturbations of the slim disc is studied in the Lagrangean formalism of adiabatic perturbations of the velocities. The new conditions on the Eulerian variation of the gravitational potential are outlined. The equations of the transient phenomena are written.

Optical aspects of Born-Infeld BTZ black holes in massive gravity

We explore the dynamics of thin accretion disks, the radius of black hole shadows, observed intensities, and the visual characteristics of Born-Infeld BTZ black holes in massive gravity. We find out the relations for angular velocity, specific energy, and angular momentum of particles around the black hole. We observe that intense Born-Infeld electromagnetic effects lead to a reduction in the rotational motion of particles within the accretion disk, and the massive gravity slows down the orbital motion of these particles. We reveal that the influence of massive gravity parameter correlates with a reduction in the black hole’s shadow size, which suggests that massive gravity effects intensify the gravitational fields, thereby reducing the angular diameter of the shadows. On the other hand, a higher Born-Infeld parameter enlarges the black hole’s shadow, which manifests a visual relationship between the black hole’s physical dimensions and its gravitational influence. Moreover, we also uncover the optical characteristics of Born-Infeld BTZ black holes, which show that the Born-Infeld parameter greatly influences the electromagnetic field around the black hole, which affects energy distribution in the space. Finally, we observe that massive gravity significantly influences the spacetime structure near black holes, which is crucial for grasping gravitational lensing and the dynamics of accretion disks under such extreme conditions.

Proposing a physical mechanism to explain various observed sources of QPOs by simulating the dynamics of accretion disks around the black holes

We propose a mechanism to explain the low-frequency QPOs observed in X-ray binary systems and AGNs. To achieve this, we perturbed stable accretion disks around Kerr and EGB black holes at different angular velocities, revealing characteristics of shock waves and oscillations on the disk. By applying this perturbation to scenarios with varying alpha values for EGB black holes and different spin parameters for Kerr black holes, we numerically observed changes in the disk dynamic structure and its oscillations. Through various numerical models, we found that the formation of one- and two-armed spiral shock waves on the disk serves as a mechanism for generating QPOs. We compared the QPOs obtained from numerical calculations with the low-frequency QPOs observed in X-ray binary systems and AGN sources, finding high consistency with observations. We observed that the shock mechanism, leading to QPOs, explains the X-ray binaries and AGNs studied in this article. Our numerical findings indicate that QPOs are more strongly dependent on the EGB constant than on the black hole spin parameter. However, we highlighted that the primary impact on oscillations and QPOs is driven by the perturbation angular velocity. The results from the models showed that the perturbation asymptotic speed at V∞=0.2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{\\infty }=0.2$$\\end{document} independently generates QPO frequencies, regardless of the black hole spin parameter and the EGB coupling constant. Therefore, for the moderate value of V∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{\\infty }$$\\end{document}, a two-armed spiral shock wave formed around the black hole is suggested as a decisive mechanism in explaining low-frequency QPOs.

Discovery of a new IW And-type dwarf nova with both tilted disc and tidal instability

ABSTRACT IW And-type dwarf novae are anomalous Z Cam stars featured with outbursts happening during standstill states, which are not expected in the standard disc instability model. The physical mechanisms for these variations remain unclear. In this study, we report the discovery of a new candidate IW And-type dwarf nova J0652+2436, identified with its frequent outbursts from the slowly rising standstill states. Luckily, the Transiting Exoplanet Survey Satellite observations during a long standstill state and the earlier K2 observations give a chance to find the orbital and negative superhump period in the light curve of J0652+2436, allowing the measurement of its mass ratio of 0.366. This mass ratio is marginally possible for the tidal instability to set in according to previous smoothed particle hydrodynamics simulations. Thus, we propose that the outbursts in J0652+2436 are likely to be caused by the growing accretion disc during standstills, in favour of the previous hypothesis of the mechanisms lying in all IW And stars. We conclude that J0652+2436 might be the first IW And star with both a precessing tilted disc and tidal instability, which will be an important laboratory for studying the accretion disc dynamics and help understand the IW And phenomenon.

Future Prospects for Constraining Black Hole Spacetime: Horizon-scale Variability of Astrophysical Jets

The Event Horizon Telescope (EHT) Collaboration has recently published the first horizon-scale images of the supermassive black holes M87* and Sgr A* and provided some first information on the physical conditions in their vicinity. The comparison between the observations and the three-dimensional general relativistic magnetohydrodynamic (GRMHD) simulations has enabled the EHT to set initial constraints on the properties of these black hole spacetimes. However, accurately distinguishing the properties of the accretion flow from those of the spacetime, most notably, the black hole mass and spin, remains challenging because of the degeneracies the emitted radiation suffers when varying the properties of the plasma and those of the spacetime. The next-generation EHT (ngEHT) observations are expected to remove some of these degeneracies by exploring the complex interplay between the disk–jet dynamics, which represents one of the most promising tools for extracting information on the black hole spin. By using GRMHD simulations of magnetically arrested disks and general relativistic radiative transfer (GRRT) calculations of the emitted radiation, we have studied the properties of the jet and the accretion disk dynamics on spatial scales that are comparable with the horizon. In this way, we are able to highlight that the radial and azimuthal dynamics of the jet are well correlated with the black hole spin. Based on the resolution and image reconstruction capabilities of the ngEHT observations of M87*, we can assess the detectability and associated uncertainty of this correlation. Overall, our results serve to assess the prospects for constraining the black hole spin with future EHT observations.

The Bardeen–Petterson effect, disc breaking, and the spin orientations of supermassive black hole binaries

ABSTRACT Supermassive black hole binaries are driven to merger by dynamical friction, loss-cone scattering of individual stars, disc migration, and gravitational wave emission. Two main formation scenarios are expected. Binaries that form in gas-poor galactic environments do not experience disc migration and likely enter the gravitational wave-dominated phase with roughly isotropic spin orientations. Comparatively, binaries that evolve in gas-rich galactic environments might experience prominent phases of disc accretion, where the Bardeen–Petterson effect acts to align the spins of the black holes with the orbital angular momentum of the disc. However, if the accretion disc breaks, alignment is expected to be strongly suppressed – a phenomenon that was recently shown to occur in a large portion of the parameter space. In this paper, we develop a semi-analytical model of joint gas-driven migration and spin alignment of supermassive black hole binaries taking into account the impact of disc breaking for the first time. Our model predicts the occurrence of distinct subpopulations of binaries depending on the efficiency of spin alignment. This implies that future gravitational wave observations of merging black holes could potentially be used to (i) discriminate between gas-rich and gas-poor hosts and (ii) constrain the dynamics of warped accretion discs.

Fully Kinetic Shearing-box Simulations of Magnetorotational Turbulence in 2D and 3D. I. Pair Plasmas

The magnetorotational instability (MRI) is a fundamental mechanism determining the macroscopic dynamics of astrophysical accretion disks. In collisionless accretion flows around supermassive black holes, MRI-driven plasma turbulence cascading to microscopic (i.e., kinetic) scales can result in enhanced angular-momentum transport and redistribution, nonthermal particle acceleration, and a two-temperature state where electrons and ions are heated unequally. However, this microscopic physics cannot be captured with standard magnetohydrodynamic (MHD) approaches typically employed to study the MRI. In this work, we explore the nonlinear development of MRI turbulence in a pair plasma, employing fully kinetic particle-in-cell (PIC) simulations in two and three dimensions. First, we thoroughly study the axisymmetric MRI with 2D simulations, explaining how and why the 2D geometry produces results that differ substantially from 3D MHD expectations. We then perform the largest (to date) 3D simulations, for which we employ a novel shearing-box approach, demonstrating that 3D PIC models can reproduce the mesoscale (i.e., MHD) MRI dynamics in sufficiently large runs. With our fully kinetic simulations, we are able to describe the nonthermal particle acceleration and angular-momentum transport driven by the collisionless MRI. Since these microscopic processes ultimately lead to the emission of potentially measurable radiation in accreting plasmas, our work is of prime importance to understand current and future observations from first principles, beyond the limitations imposed by fluid (MHD) models. While in this first study we focus on pair plasmas for simplicity, our results represent an essential step toward designing more realistic electron–ion simulations, on which we will focus in future work.

MRI-driven dynamo at very high magnetic Prandtl numbers

ABSTRACT The dynamo driven by the magnetorotational instability (MRI) is believed to play an important role in the dynamics of accretion discs and may also explain the origin of the extreme magnetic fields present in magnetars. Its saturation level is an important open question known to be particularly sensitive to the diffusive processes through the magnetic Prandtl number Pm (the ratio of viscosity to resistivity). Despite its relevance to proto-neutron stars and neutron star merger remnants, the numerically challenging regime of high Pm is still largely unknown. Using zero-net flux shearing box simulations in the incompressible approximation, we studied MRI-driven dynamos at unprecedentedly high values of Pm reaching 256. The simulations show that the stress and turbulent energies are proportional to Pm up to moderately high values (Pm ∼ 50). At higher Pm, they transition to a new regime consistent with a plateau independent of Pm for $\rm Pm \gtrsim 100$. This trend is independent of the Reynolds number, which may suggest an asymptotic regime where the energy injection and dissipation are independent of the diffusive processes. Interestingly, large values of Pm not only lead to intense small-scale magnetic fields but also to a more efficient dynamo at the largest scales of the box.

Shocks in Radiatively Driven Time-dependent, Relativistic Jets around Black Holes

We study time-dependent relativistic jets under the influence of the radiation field of the accretion disk. The accretion disk consists of an inner compact corona and an outer sub-Keplerian disk. The thermodynamics of the fluid is governed by a relativistic equation of state (EOS) for multispecies fluid that enables us to study the effect of composition on jet dynamics. Jets originate from the vicinity of the central black hole, where the effect of gravity is significant and traverses large distances where only special relativistic treatment is sufficient. So we have modified the flat metric to include the effect of gravity. In this modified relativistic framework we have developed a new total variation diminishing routine along with a multispecies EOS for the purpose. We show that the acceleration of jets crucially depends on flow composition. All the results presented are transonic in nature; starting from very low injection velocities, the jets can achieve high Lorentz factors. For sub-Eddington luminosities, lepton-dominated jets can be accelerated to Lorentz factors >50. The change in radiation field due to variation in the accretion disk dynamics will be propagated to the jet in a finite amount of time. Hence, any change in radiation field due to a change in disk configuration will affect the lower part of the jet before it affects the outer part. This can drive shock transition in the jet flow. Depending on the disk oscillation frequency, amplitude, and jet parameters, these shocks can collide with each other and may trigger shock cascades.

Thermal conduction effects on the accretion–ejection mechanism. Outflow process investigation

ABSTRACT Astrophysical jets emanating from different systems are one of the most spectacular and enigmatic phenomena pervading the Universe. These jets are typically bipolar and span hundreds of thousands of light years, some even longer than the diameter of our Milky Way. The study of the disc–jet systems is motivated by the observed correlation between ejection and accretion signatures and is still under debate. It was shown in our previous work the crucial role of thermal conduction in the dynamics of a thin viscous resistive accretion disc orbiting a central object and was provided an unprecedented wealth of discussion that has advanced our understanding of the inflow process. In this work, we expand our exploration by addressing the most outstanding basic questions concerning the launching, acceleration, and collimation processes of the jet in presence of thermal conduction. We also tackle in depth-analysis the effects of this physical ingredient on the time evolution of temperature and on mass fluxes such as inflow and outflow rates. We performed a series of 2.5-dimensional non-relativistic time-dependent numerical calculations of a disc–jet system using the PLUTO code. Our results revealed compelling evidence that thermal conduction contributes to launching a faster and more collimated jet. The mass extracted from the disc via the outflow channel is also affected by the presence of thermal conduction in the sense that the ejection efficiency is significantly improved.

Neutral-charged-particle Collisions as the Mechanism for Accretion Disk Angular Momentum Transport

The matter in an accretion disk must lose angular momentum when moving radially inwards but how this works has long been a mystery. By calculating the trajectories of individual colliding neutrals, ions, and electrons in a weakly ionized 2D plasma containing gravitational and magnetic fields, we numerically simulate accretion disk dynamics at the particle level. As predicted by Lagrangian mechanics, the fundamental conserved global quantity is the total canonical angular momentum, not the ordinary angular momentum. When the Kepler angular velocity and the magnetic field have opposite polarity, collisions between neutrals and charged particles cause: (i) ions to move radially inwards, (ii) electrons to move radially outwards, (iii) neutrals to lose ordinary angular momentum, and (iv) charged particles to gain canonical angular momentum. Neutrals thus spiral inward due to their decrease of ordinary angular momentum while the accumulation of ions at small radius and accumulation of electrons at large radius produces a radially outward electric field. In 3D, this radial electric field would drive an out-of-plane poloidal current that produces the magnetic forces that drive bidirectional astrophysical jets. Because this neutral angular momentum loss depends only on neutrals colliding with charged particles, it should be ubiquitous. Quantitative scaling of the model using plausible disk density, temperature, and magnetic field strength gives an accretion rate of 3 × 10−8 solar mass per year, which is in good agreement with observed accretion rates.

Multi-mission view of extragalactic black hole X-ray binaries LMC X-1 and LMC X-3: Evolution of broadband spectral features

Extragalactic black hole X-ray binaries LMC X-1 and LMC X-3 are the persistent sources, usually found in soft spectral state. We present the results from spectral and timing study of these sources using MAXI, NICER, NuSTAR and AstroSat observations carried out during 2014-2020. Study of long-term MAXI lightcurve shows that the fractional variability of flux in 2-10keV is moderate ($\sim$20%) in LMCX-1 and high ($\sim$50%) in LMCX-3. The energy spectra of both sources are characterized by a disc blackbody and a Comptonization component with LMCX-1 having an additional Fe-line emission feature. NICER (0.3-10keV), NuSTAR (3.0-40keV) and AstroSat (0.5-20keV) spectral analysis collectively show that LMC X-1 remained in the soft state (disc flux contribution $f_{disc}>80$%, photon index$\Gamma\sim2.06-4.08$) throughout 2014-2020. Mass accretion rate, $\dot{M}$ of LMC X-1 calculated from bolometric luminosity (0.1-50keV) is found to be within $0.07-0.24\dot{M}_{Edd}$ (Eddington mass accretion rate). Although LMC X-3 remained in the soft state ($f_{disc}>95\%,\Gamma\sim2.3$) during most of the time, it exhibits transition into intermediate ($f_{disc}=47-73\%,\Gamma\sim2.02-2.36$) and hard state ($f_{disc}\sim26\%,\Gamma\sim1.6$). $\dot{M}$ of LMC X-3 through different spectral states varies within $0.01-0.42\dot{M}_{Edd}$. Temporal study show that the Power Density Spectra (PDS) in 0.3-10keV follow a red-noise with rms of 2% for LMC X-1 and in case of LMC X-3, it is 0.08-2.35% during the soft state, but relatively high in the intermediate(3.05-4.91%) and hard state($\sim$17.06%). From continuum-fitting method we constrain spin of LMC X-1 and LMC X-3 to be within 0.85-0.94 and 0.16-0.33 respectively and from Fe-line fitting method, spin of LMC X-1 is found to be 0.93-0.94. Finally, we discuss the implication of our findings in the context of accretion disc dynamics around the vicinity of the BHs.

Unravelling the foretime of GRS 1915+105 using AstroSat observations: Wide-band spectral and temporal characteristics

ABSTRACT We present a comprehensive study of GRS 1915+105 in wide-energy band (0.5–60 keV) using AstroSat – SXT and LAXPC observations during the period of 2016–2019. The MAXI X-ray light curve of the source shows rise and decay profiles similar to canonical outbursting black holes. However, the source does not follow the exemplary ‘q’-diagram in the hardness–intensity diagram. Model independent analysis of light curves suggest that GRS 1915+105 displays various types of variability classes (δ, χ, ρ, κ, ω, and γ). We also report possible transitions from one class to another (χ → ρ, ρ → κ via an ‘unknown’ class and ω → γ → ω + γ) within a few hours duration. Detailed ‘spectrotemporal’ analysis indicates a gradual increase in the photon index (Γ) from 1.83 to 3.8, disc temperature (kTin) from 1.33 to 2.67 keV, and quasi-periodic oscillation frequency (ν) from 4 to 5.64 Hz during the rise, while the parameters decrease to Γ ∼1.18, kTin ∼1.18 keV, and ν ∼1.38 Hz, respectively, in the decline phase. The source shows maximum bolometric luminosity (Lbol) during the peak at ∼36 per cent of Eddington luminosity (LEdd), and a minimum of ∼2.4 per cent of LEdd during the decay phase. Further evolution of the source towards an obscured low-luminosity (Lbol of ∼1 per cent LEdd) phase, with a decrease in the intrinsic bolometric luminosity of the source due to obscuration, has also been indicated from our analysis. The implication of our results are discussed in the context of accretion disc dynamics around the black hole.

On the role of $$\varLambda $$ on accretion disks

The dynamics of accretion disk is considered taking into account the Lense-Thirring precession in the presence of cosmological constant $\Lambda$ of Schwarzschild-de Sitter. The nodal and apsidal frequencies are obtained, and the role of $\Lambda$ is revealed in their properties, including the consequences for the Bardeen-Petterson effect.

Water megamaser emission in hard X-ray selected AGN

Context. Water megamaser emission at 22 GHz has proven to be a powerful tool for astrophysical studies of active galactic nuclei (AGN) because it allows an accurate determination of the mass of the central black hole and of the accretion disc geometry and dynamics. However, after searches among thousands of galaxies, only about 200 of them have shown such spectroscopic features, most of them of uncertain classification. In addition, the physical and geometrical conditions under which a maser activates are still unknown. Aims. We characterize the occurrence of water maser emission in an unbiased sample of AGN by investigating the relation with the X-ray properties and the possible favourable geometry that is required to detect water maser. Methods. We searched for 22 GHz maser emission in a hard X-ray selected sample of AGN, taken from the INTEGRAL/IBIS survey above 20 keV. Only half of the 380 sources in the sample have water maser data. We also considered a volume-limited sub-sample of 87 sources, for which we obtained new observations with the Green Bank and Effelsberg telescopes (for 35 sources). We detected one new maser and increased its radio coverage to 75%. Results. The detection rate of water maser emission in the total sample is 15 ± 3%. This fraction increases to 19 ± 5% for the complete sub-sample, especially when we consider type 2 (22 ± 5% and 31 ± 10% for the total and complete samples, respectively) and Compton-thick AGN (56 ± 18% and 50 ± 35% for the total and complete samples, respectively). No correlation is found between water maser and X-ray luminosity. We note that all types of masers (disc and jet) are associated with hard X-ray selected AGN. Conclusions. These results demonstrate that the hard X–ray selection may significantly enhance the maser detection efficiency over comparably large optical or infrared surveys. A possible decline in detection fraction with increasing luminosity might suggest that an extremely luminous nuclear environment does not favour maser emission. The large fraction of CT AGN with water maser emission could be explained in terms of geometrical effects. The maser medium would then be the very edge-on portion of the obscuring medium.

Changing look active galactic nuclei in the MaNGA survey

ABSTRACT Changing look active galactic nuclei (CLAGNs) are rare cases of AGNs, where the continuum flux increases/decreases and the broad emission lines appear/disappear within short time-scales. These extreme changes challenge our understanding of accretion disc dynamics. We present a sample of four new CLAGNs at 0.026 < z < 0.107, which were found by cross-matching the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey with AGNs from the SDSS spectroscopic data base. Our results show that the selection criteria of $\gt \mathopen |0.5\mathclose |$ mag change in SDSS-g band is ineffective at lower redshifts. This is, in part, due to the fact that the g band is probing a different part of the AGNs spectral energy distribution at these redshifts. The bigger issue is that the galaxy continuum dominates the spectrum and thus overwhelms any variation of the AGNs continuum that might contribute to photometric variations. This suggests the need to use different methods for future low-redshift CLAGNs searches.

The Bardeen–Petterson effect in accreting supermassive black hole binaries: a systematic approach

ABSTRACT Disc-driven migration is a key evolutionary stage of supermassive black hole binaries hosted in gas-rich galaxies. Besides promoting the inspiral, viscous interactions tend to align the spins of the black holes with the orbital angular momentum of the disc. We present a critical and systematic investigation of this problem, also known as the Bardeen–Petterson effect. We design a new iterative scheme to solve the non-linear dynamics of warped accretion discs under the influence of both relativistic frame dragging and binary companion. We characterize the impact of the disc ‘critical obliquity’, which marks regions of the parameter space where stationary solutions do not exist. We find that black hole spins reach either complete alignment or a critical configuration. Reaching the critical obliquity might imply that the disc breaks as observed in hydrodynamical simulations. Our findings are important to predict the spin configurations with which supermassive black hole binaries enter their gravitational-wave driven regime and become detectable by LISA.

Thermal Conduction Effects on the Accretion–Ejection Mechanism. I. Accretion Flow Investigation

Abstract Understanding the accretion of matter around compact objects such as young stars or black holes remains one of the most interesting problems in astrophysics. Recent observations detected the accretion of matter of several young stars from their circumstellar disk and proposed the existence of the X-ray emission at very high temperature. The hot gases surrounding these systems are likely to proceed under the collisionless regime, which indicates that the thermal conduction has a relevant contribution in the transport of energy along the accretion disks. Notwithstanding its importance, this physical ingredient is largely neglected in previous time-dependent studies of accretion–ejection structure. More research is still necessary before obtaining a complete picture of such a phenomenon. In this series, we have included the three nonideal effects together, viscosity, resistivity, and thermal conduction, to explore the physical mechanism of the disk–jet system. We carry out extensive time-dependent simulations of the accretion disk and jet using the PLUTO code. The mean field approach is used in this work by taking into account the evolution of the disk and jet simultaneously. We show in this first paper how the thermal conduction plays a crucial role in the accretion disk dynamics by presenting a detailed discussion of its effects on the evolution of basic facets of the inflow properties with a special emphasis on the equatorial plane and disk surface. Our model also shows strong indications of the ejection efficiency improvement in the presence of thermal conduction.

Influence of Ohmic and ambipolar heating on the thermal structure of accretion discs

We investigate dynamics of accretion discs of young stars with fossil large-scale magnetic field. Our magneto-gas-dynamic (MHD) model of the accretion discs includes equations of Shakura and Sunyaev, induction equation, equations of thermal and collisional ionization. Induction equation takes into account Ohmic and magnetic ambipolar diffusion, magnetic buoyancy. We also consider the influence of Ohmic and ambipolar heating on thermal structure of the accretion discs. We analyse the influence of considered dissipative MHD effects on the temperature of the accretion discs around classical T Tauri star. The simulations show that Ohmic and ambipolar heating operate near the borders of the region with low ionization fraction (`dead' zone). Temperature grows by $\sim 1000$ K near the inner boundary of the `dead' zone, $r\approx (0.5-1)$ au, and by $\sim 100$ K near its outer boundary, $r\approx (30-50)$ au.

Effective sound speed in relativistic accretion discs around Schwarzschild black holes

For low angular momentum axially symmetric accretion flow maintained in hydrostatic equilibrium along the vertical direction, the value of the Mach number at the critical points deviates from unity, resulting in the non-isomorphism of the critical and the sonic points. This introduces several undesirable complexities while analytically dealing with the stationary integral accretion solutions and the corresponding phase portraits. We propose that the introduction of an effective dynamical sound speed may resolve the issue in an elegant way. We linear perturb the full spacetime-dependent general relativistic Euler and the continuity equations governing the structure and the dynamics of accretion disc in vertical equilibrium around Schwarzschild black holes and identify the sonic metric embedded within the stationary background flow. Such metric describes the propagation of the linear acoustic perturbation inside the accretion flow. We construct the wave equation corresponding to that acoustic perturbation and find the speed of propagation of such perturbation. We finally show that the ordinary thermodynamic sound speed should be substituted by the speed of propagation of the linear acoustic wave which has been obtained through the dynamical perturbation. Such substitution will make the value of Mach number at the critical point to be equal to unity. Use of the aforementioned effective sound speed will lead to a modified stationary disc structure where the critical and the sonic points will be identical.