Showing papers on "Hydrostatic equilibrium published in 2014"
TL;DR: In this paper, a large, mass-limited sample of massive galaxy clusters from a high-resolution hydrodynamical cosmological simulation is used to show that acceleration of gas introduces biases in the hydrostatic mass estimate of galaxy clusters.
Abstract: The use of galaxy clusters as cosmological probes hinges on our ability to measure their masses accurately and with high precision. Hydrostatic mass is one of the most common methods for estimating the masses of individual galaxy clusters, which suffer from biases due to departures from hydrostatic equilibrium. Using a large, mass-limited sample of massive galaxy clusters from a high-resolution hydrodynamical cosmological simulation, in this work we show that in addition to turbulent and bulk gas velocities, acceleration of gas introduces biases in the hydrostatic mass estimate of galaxy clusters. In unrelaxed clusters, the acceleration bias is comparable to the bias due to non-thermal pressure associated with merger-induced turbulent and bulk gas motions. In relaxed clusters, the mean mass bias due to acceleration is small ( 3%), but the scatter in the mass bias can be reduced by accounting for gas acceleration. Additionally, this acceleration bias is greater in the outskirts of higher redshift clusters where mergers are more frequent and clusters are accreting more rapidly. Since gas acceleration cannot be observed directly, it introduces an irreducible bias for hydrostatic mass estimates. This acceleration bias places limits on how well we can recover cluster masses from future X-ray and microwave observations. We discuss implications for cluster mass estimates based on X-ray, Sunyaev-Zel'dovich effect, and gravitational lensing observations and their impact on cluster cosmology.
140 citations
TL;DR: In this paper, a time domain model is applied to a three-dimensional point absorber wave energy converter, where the relative velocities between the body and the waves increase, and the non-linear hydrostatic restoring moment is calculated by a cubic polynomial function fit to laboratory test results.
115 citations
TL;DR: In this article, the authors proposed a method to test the gravitational interactions in the outskirts of galaxy clusters by combining the gas and lensing measurements of the cluster. But their method is limited to the Coma cluster.
Abstract: We propose a novel method to test the gravitational interactions in the outskirts of galaxy clusters. When gravity is modified, this is typically accompanied by the introduction of an additional scalar degree of freedom, which mediates an attractive fifth force. The presence of an extra gravitational coupling, however, is tightly constrained by local measurements. In chameleon modifications of gravity, local tests can be evaded by employing a screening mechanism that suppresses the fifth force in dense environments. While the chameleon field may be screened in the interior of the cluster, its outer region can still be affected by the extra force, introducing a deviation between the hydrostatic and lensing mass of the cluster. Thus, the chameleon modification can be tested by combining the gas and lensing measurements of the cluster. We demonstrate the operability of our method with the Coma cluster, for which both a lensing measurement and gas observations from the X-ray surface brightness, the X-ray temperature, and the Sunyaev-Zel'dovich effect are available. Using the joint observational data set, we perform a Markov chain Monte Carlo analysis of the parameter space describing the different profiles in both the Newtonian and chameleon scenarios. We report competitive constraints on the chameleon field amplitude and its coupling strength to matter. In the case of f(R) gravity, corresponding to a specific choice of the coupling, we find an upper bound on the background field amplitude of |fR0| < 6 × 10−5, which is currently the tightest constraint on cosmological scales.
105 citations
TL;DR: In this paper, the authors studied the stability of locally isotropic spherical self-gravitating systems in f(R,T) gravity, where R is the Ricci curvature invariant and T is the trace of stress energy tensor.
Abstract: We study the factors affecting the stability of a locally isotropic spherical self-gravitating systems in f(R,T) gravity, where R is the Ricci curvature invariant and T is the trace of stress-energy tensor. Specifically, the collapse equation is obtained from conservation laws with non-null expansion scalar at Newtonian and post-Newtonian approximations. Initially, we consider the hydrostatic phase of the system which upon radial perturbation provides linearized field equations. This approach gives rise to specific instability constraints to ensure the collapsing behavior of the spherical isotropic fluid distribution. Finally, we discuss the role played by matter variables in this perspective.
94 citations
TL;DR: In this paper, the authors show that the initial disk of the Moon formed from a partially vaporized disk generated by a collision between the proto-Earth and a Mars-sized impactor, and they show the extent to which the properties of the disk can be inferred from smoothed Particle Hydrodynamic (SPH) simulations.
94 citations
TL;DR: In this paper, an analytical model for intracluster non-thermal pressure in the virial region of relaxed clusters was developed based on insights gained from observations, numerical simulations, and theory of turbulence.
Abstract: Non-thermal pressure in the intracluster gas has been found ubiquitously in numerical simulations, and observed indirectly. In this paper we develop an analytical model for intracluster non-thermal pressure in the virial region of relaxed clusters. We write down and solve a firstorder di erential equation describing the evolution of non-thermal velocity dispersion. This equation is based on insights gained from observations, numerical simulations, and theory of turbulence. The non-thermal energy is sourced, in a self-similar fashion, by the mass growth of clusters via mergers and accretion, and dissipates with a time-scale determined by the turnover time of the largest turbulence eddies. Our model predicts a radial profile of nonthermal pressure for relaxed clusters. The non-thermal fraction increases with radius, redshift, and cluster mass, in agreement with numerical simulations. The radial dependence is due to a rapid increase of the dissipation time-scale with radii, and the mass and redshift dependence comes from the mass growth history. Combing our model for the non-thermal fraction with the Komatsu-Seljak model for the total pressure, we obtain thermal pressure profiles, and compute the hydrostatic mass bias. We find typically 10% bias for the hydrostatic mass enclosed within r500.
88 citations
TL;DR: In this article, the second-degree gravity field coefficients, the shape, the tidal Love number, the electric field, and the orientation of its rotation axis are used together to constrain the density profile assuming a four-layer interior model.
79 citations
TL;DR: The precise difference between rotational and orbital periods suggests an unexpected interior for one of Saturn’s moons, and it is argued that the satellite has either a large nonhydrostatic interior, or a hydrostatic one with an internal ocean beneath a thick icy shell.
Abstract: Like our Moon, the majority of the solar system’s satellites are locked in a 1:1 spin-orbit resonance; on average, these satellites show the same face toward the planet at a constant rotation rate equal to the satellite’s orbital rate In addition to the uniform rotational motion, physical librations (oscillations about an equilibrium) also occur The librations may contain signatures of the satellite’s internal properties Using stereophotogrammetry on Cassini Image Science Subsystem (ISS) images, we measured longitudinal physical forced librations of Saturn’s moon Mimas Our measurements confirm all the libration amplitudes calculated from the orbital dynamics, with one exception This amplitude depends mainly on Mimas’ internal structure and has an observed value of twice the predicted one, assuming hydrostatic equilibrium After considering various possible interior models of Mimas, we argue that the satellite has either a large nonhydrostatic interior, or a hydrostatic one with an internal ocean beneath a thick icy shell
74 citations
TL;DR: In this paper, a class of relativistic solutions with anisotropic fluid for compact stars in hydrostatic equilibrium is presented, and the interior space-time geometry considered here for compact objects are described by parameters namely, λ, k, A, R and n.
Abstract: We present a class of new relativistic solutions with anisotropic fluid for compact stars in hydrostatic equilibrium. The interior space-time geometry considered here for compact objects are described by parameters namely, λ, k, A, R and n. The values of the geometrical parameters are determined here for obtaining a class of physically viable stellar models. The energy-density, radial pressure and tangential pressure are finite and positive inside the anisotropic stars. Considering some stars of known mass we present stellar models which describe compact astrophysical objects with nuclear density.
69 citations
TL;DR: In this paper, the authors used the shape and gravity field of Vesta determined from observations of the Dawn spacecraft to place constraints on the asteroid's interior structure by minimizing the power of the residual gravity anomaly.
66 citations
TL;DR: In this article, the impact of chameleon-type f(R) gravity models on the properties of galaxy clusters and groups was investigated, and it was shown that velocity dispersions and intracluster medium temperatures are both increased by up to 1/3 in f(r) gravity in low mass halos, while the difference disappears in massive objects.
Abstract: We investigate the impact of chameleon-type f(R) gravity models on the properties of galaxy clusters and groups. Our f(R) simulations follow for the first time also the hydrodynamics of the intracluster and intragroup medium. This allows us to assess how f(R) gravity alters the X-ray scaling relations of clusters and how hydrostatic and dynamical mass estimates are biased when modifications of gravity are ignored in their determination. We find that velocity dispersions and intracluster medium temperatures are both increased by up to 1/3 in f(R) gravity in low-mass halos, while the difference disappears in massive objects. The mass scale of the transition depends on the background value f_R0 of the scalar degree of freedom. These changes in temperature and velocity dispersion alter the mass-temperature and X-ray luminosity-temperature scaling relations and bias dynamical and hydrostatic mass estimates that do not explicitly account for modified gravity towards higher values. Recently, a relative enhancement of X-ray compared to weak lensing masses was found by the Planck Collaboration (2013). We demonstrate that an explanation for this offset may be provided by modified gravity and the associated bias effects, which interestingly are of the required size. Finally, we find that the abundance of subhalos at fixed cluster mass is only weakly affected by f(R) gravity.
TL;DR: In this article, the gravity wave drag and momentum fluxes produced by gravity waves generated in flow over orography are reviewed, focusing on adiabatic conditions without phase transitions or radiation effects, and steady mean incoming flow.
Abstract: The drag and momentum fluxes produced by gravity waves generated in flow over orography are reviewed, focusing on adiabatic conditions without phase transitions or radiation effects, and steady mean incoming flow. The orographic gravity wave drag is first introduced in its simplest possible form, for inviscid, linearized, non-rotating flow with the Boussinesq and hydrostatic approximations, and constant wind and static stability. Subsequently, the contributions made by previous authors (primarily using theory and numerical simulations) to elucidate how the drag is affected by additional physical processes are surveyed. These include the effect of orography anisotropy, vertical wind shear, total and partial critical levels, vertical wave reflection and resonance, non-hydrostatic effects and trapped lee waves, rotation and nonlinearity. Frictional and boundary layer effects are also briefly mentioned. A better understanding of all of these aspects is important for guiding the improvement of drag parametrization schemes.
TL;DR: In this article, a unified, idealised baroclinic instability test case is proposed for both deep and shallow-atmosphere models, which is suitable for models employing a pressure- or height-based vertical coordinate.
Abstract: Idealised studies of key dynamical features of the atmosphere provide insight into the behaviour of atmospheric models. A very important, well understood, aspect of midlatitude dynamics is baroclinic instability. This can be idealised by perturbing a vertically sheared basic state in geostrophic and hydrostatic balance. An unstable wave mode then results with exponential growth (due to linear dynamics) in time until, eventually, nonlinear effects dominate and the wave breaks.
A new, unified, idealised baroclinic instability test case is proposed. This improves on previous ones in three ways. First, it is suitable for both deep- and shallow-atmosphere models. Second, the constant surface pressure and zero surface geopotential of the basic state makes it particularly well-suited for models employing a pressure- or height-based vertical coordinate. Third, the wave triggering mechanism selectively perturbs the rotational component of the flow; this, together with a vertical tapering, significantly improves dynamic balance.
TL;DR: In this paper, the column physics of a mesoscale-resolution global atmospheric model that simulates realistic hurricane frequency statistics and then coupling it to rotating hydrostatic dynamics in doubly periodic domains are studied.
Abstract: Rotating radiative–convective equilibrium is studied by extracting the column physics of a mesoscale-resolution global atmospheric model that simulates realistic hurricane frequency statistics and then coupling it to rotating hydrostatic dynamics in doubly periodic domains. The parameter study helps in understanding the tropical cyclones simulated in the global model and also provides a reference point for analogous studies with cloud-resolving models.The authors first examine the sensitivity of the equilibrium achieved in a large square domain (2 × 104 km on a side) to sea surface temperature, ambient rotation rate, and surface drag coefficient. In such a large domain, multiple tropical cyclones exist simultaneously. The size and intensity of these tropical cyclones are investigated.The variation of rotating radiative–convective equilibrium with domain size is also studied. As domain size increases, the equilibrium evolves through four regimes: a single tropical depression, an intermittent tropic...
TL;DR: In this paper, the authors calculate the radial profile of compactness and curvature (square root of the full contraction of the Weyl tensor) within a neutron star and determine the domain not probed by the Solar System tests of GR.
Abstract: Neutron stars are thought to be excellent laboratories for determining the equation of state (EoS) of cold dense matter. Their strong gravity suggests that they can also be used to constrain gravity models. The two observables of neutron stars---mass and radius (M-R)---both depend on the choice of EoS and relativistic gravity, meaning that neutron stars cannot be simultaneously good laboratories for both of these questions. A measurement of mass and/or radius would constrain the less well known physics input. The most common assumption---namely, that M-R measurements can be used to constrain the EoS---presumes that general relativity (GR) is the ultimate model of gravity in the classical regime. We calculate the radial profile of compactness and curvature (square root of the full contraction of the Weyl tensor) within a neutron star and determine the domain not probed by the Solar System tests of GR. We find that, except for a tiny sphere of radius less than a millimeter at the center, the curvature is several orders of magnitude above the values present in Solar System tests. The compactness is beyond the solar surface value for $rg10\text{ }\text{ }\mathrm{m}$, and increases by 5 orders of magnitude towards the surface. With the density being only an order of magnitude higher than that probed by nuclear scattering experiments, our results suggest that the employment of GR as the theory of gravity describing the hydrostatic equilibrium of the neutron stars is a rather remarkable extrapolation from the regime of tested validity, as opposed to that of EoS models. Our ignorance of gravity within neutron stars suggests that a measurement of mass and/or radius constrains gravity rather than the EoS, and given that the EoS has yet to be determined by nucleon scattering experiments, M-R measurements cannot tightly constrain the gravity models either. Near the surface the curvature and compactness attain their largest values, while the EoS in this region is fairly well known. This renders the crust as the best site to look for deviations from GR.
TL;DR: In this article, an interior structure model satisfying simultaneously the surface gravity and long-wavelength topography was developed to investigate the compensation mechanisms, and the long-term mechanical stability of the internal mass anomalies by computing the relaxation rate of each internal interface in response to interface mass load.
TL;DR: In this article, the authors used thermal radiometry and visible photometry to constrain the size, shape, and albedo of the large Kuiper belt object Haumea.
Abstract: We use thermal radiometry and visible photometry to constrain the size, shape, and albedo of the large Kuiper belt object Haumea. The correlation between the visible and thermal photometry demonstrates that Haumea’s high amplitude and quickly varying optical light curve is indeed due to Haumea’s extreme shape, rather than large scale albedo variations. However, the well-sampled high precision visible data we present does require longitudinal surface heterogeneity to account for the shape of lightcurve. The thermal emission from Haumea is consistent with the expected Jacobi ellipsoid shape of a rapidly rotating body in hydrostatic equilibrium. The best Jacobi ellipsoid fit to the visible photometry implies a triaxial ellipsoid with axes of length 1,920 × 1,540 × 990 km and density 2.6 g cm ^(−3) , as found by Lellouch et al. (A&A, 518:L147, 2010. doi:10.1051/0004-6361/201014648). While the thermal and visible data cannot uniquely constrain the full non-spherical shape of Haumea, the match between the predicted and measured thermal flux for a dense Jacobi ellipsoid suggests that Haumea is indeed one of the densest objects in the Kuiper belt.
TL;DR: The multilayer model for hydrostatic pressure is approximate by using a polynomial viscosity matrix finite volume scheme and it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles.
Abstract: In this work we present a multilayer approach to the solution of non-stationary 3D Navier–Stokes equations. We use piecewise smooth weak solutions. We approximate the velocity by a piecewise constant (in z) horizontal velocity and a linear (in z) vertical velocity in each layer, possibly discontinuous across layer interfaces. The multilayer approach is deduced by using the variational formulation and by considering a reduced family of test functions. The procedure naturally provides the mass and momentum interfaces conditions. The mass and momentum conservation across interfaces is formulated via normal flux jump conditions. The jump conditions associated to momentum conservation are formulated by means of an approximation of the vertical derivative of the velocity that appears in the stress tensor. We approximate the multilayer model for hydrostatic pressure, by using a polynomial viscosity matrix finite volume scheme and we present some numerical tests that show the main advantages of the model: it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles.
TL;DR: In this paper, Chandra X-ray observations of the core of the galaxy cluster PKS 0745-191 were used to obtain thermodynamical cluster profiles, under the assumptions of hydrostatic equilibrium and spherical symmetry.
Abstract: We present Chandra X-ray Observatory observations of the core of the galaxy cluster PKS 0745-191. Its centre shows X-ray cavities caused by AGN feedback and cold fronts with an associated spiral structure. The cavity energetics imply they are powerful enough to compensate for cooling. Despite the evidence for AGN feedback, the Chandra and XMM-RGS X-ray spectra are consistent with a few hundred solar masses per year cooling out of the X-ray phase, sufficient to power the emission line nebula. The coolest X-ray emitting gas and brightest nebula emission is offset by around 5 kpc from the radio and X-ray nucleus. Although the cluster has a regular appearance, its core shows density, temperature and pressure deviations over the inner 100 kpc, likely associated with the cold fronts. After correcting for ellipticity and projection effects, we estimate density fluctuations of ~4 per cent, while temperature, pressure and entropy have variations of 10-12 per cent. We describe a new code, MBPROJ, able to accurately obtain thermodynamical cluster profiles, under the assumptions of hydrostatic equilibrium and spherical symmetry. The forward-fitting code compares model to observed profiles using Markov Chain Monte Carlo and is applicable to surveys, operating on 1000 or fewer counts. In PKS0745 a very low gravitational acceleration is preferred within 40 kpc radius from the core, indicating a lack of hydrostatic equilibrium, deviations from spherical symmetry or non-thermal sources of pressure.
TL;DR: In this article, the influence of the tilt and recess shape on the static and dynamic performance characteristics of the hydrostatic thrust pad bearing system having Rabinowitsch fluid model lubricant was numerically analyzed.
TL;DR: In this paper, an integrated heat-fluid-solid coupling model of the hydrostatic spindle system is built to simulate the heat generation process and the fluid-structure conjugate heat transfer.
Abstract: The temperature rise of an ultra-precision machine tool has a great impact on machining accuracy. Meanwhile, the hydrostatic spindle system is the main internal heat source of the machine tool, which consists of a hydrostatic spindle and a direct current motor. Therefore, it is very significant to study the thermal behaviors of the hydrostatic spindle system. In this paper, an integrated heat-fluid–solid coupling model of the hydrostatic spindle system is built to simulate the heat generation process and the fluid–structure conjugate heat transfer. Then a finite volume element method (FVEM) is proposed by combining the advantages of the finite volume method (FVM) and the finite element method (FEM) with consideration of the interaction of the temperature field, thermal deformation, and eccentricity. Based on the proposed model and method, the thermal characteristics of the hydrostatic spindle system are studied by the two-way heat-fluid–solid coupling analysis. The temperature variations obtained by the simulation agree well with the experimental results, which validate the proposed model and method.
TL;DR: In this article, the exact analytical potential of homogeneous ellipsoids was used to obtain recursive analytical solutions and an exact numerical method for the hydrostatic equilibrium shape problem of multi-layer planets and synchronous moons.
Abstract: The hydrostatic equilibrium of multi-layer bodies lacks a satisfactory theoretical treatment despite its wide range of applicability. Here we show that by using the exact analytical potential of homogeneous ellipsoids we can obtain recursive analytical solutions and an exact numerical method for the hydrostatic equilibrium shape problem of multi-layer planets and synchronous moons. The recursive solutions rely on the series expansion of the potential in terms of the polar and equatorial shape eccentricities, while the numerical method uses the exact potential expression. These solutions can be used to infer the interior structure of planets and synchronous moons from their observed shape, rotation, and gravity. When applied to the dwarf planet Ceres, we show that it is most likely a differentiated body with an icy crust of equatorial thickness 30-90 km and a rocky core of density 2.4-3.1 g cm–3. For synchronous moons, we show that the J 2/C 22 10/3 and the (b – c)/(a – c) 1/4 ratios have significant corrections of order Ω2/(πGρ), with important implications for how their gravitational coefficients are determined from fly-by radio science data and for how we assess their hydrostatic equilibrium state.
TL;DR: In this paper, the authors derived the modified forms of Tolman-Oppenheimer-Volkoff (TOV) equations for a generic function of $f(G)$ gravity.
Abstract: Based on a stringy inspired Gauss-Bonnet (GB) modification of classical gravity, we constructed a model for neutron stars. We derived the modified forms of Tolman-Oppenheimer-Volkoff (TOV) equations for a generic function of $f(G)$ gravity. The hydrostatic equations remained unchanged but the dynamical equations for metric functions are modified due to the effects of GB term.
TL;DR: In this article, the authors presented a new characterization technique based on buoyancy loss measurement under hydrostatic pressure. But this technique is not suitable for testing synthetic foams under real conditions of use, as there is no standard procedure for testing material under pure hydrostatic compression.
TL;DR: In this paper, a new technique for modeling flow over orography is introduced that guarantees curl-free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity.
Abstract: Steep orography can cause noisy solutions and instability in models of the atmosphere. A new technique for modeling flow over orography is introduced that guarantees curl-free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity. This mimetic property leads to better hydrostatic balance and better energy conservation on test cases using terrain-following grids. Curl-free gradients are achieved by using the covariant components of velocity over orography rather than the usual horizontal and vertical components.In addition, gravity and acoustic waves are treated implicitly without the need for mean and perturbation variables or a hydrostatic reference profile. This enables a straightforward description of the implicit treatment of gravity waves.Results are presented of a resting atmosphere over orography and the curl-free pressure gradient formulation is advantageous. Results of gravity waves over orography are insensitive to the placement of t...
TL;DR: In this paper, a pseudo-incompressible model for moist atmospheric flows is presented, which is derived from a fully compressible system by assuming that the pressure perturbations are small.
TL;DR: In this article, the authors discuss dynamical instability of non-adiabatic anisotropic collapse of spherical self-gravitating systems through collapse equation in Palatini f(R) gravity.
Abstract: We discuss dynamical instability of non-adiabatic anisotropic collapse of spherical self-gravitating systems through collapse equation in Palatini f(R) gravity. We take R+ϵR
n
model and assume hydrostatic equilibrium of celestial object at large past time such that T(−∞)=0. Considering perturbation from hydrostatic phase, and linearizing the dynamical as well as field equations, we evaluate instability constraints at Newtonian as well as post-Newtonian approximations. We conclude that pressure and heat flow assist collapsing phenomenon while Palatini f(R) dark source terms affect dynamics of self-gravitating object due to its non-attractive nature through stiffness parameter.
TL;DR: In this article, a new set of equilibrium solutions for rotating and pressure truncated filaments were obtained, and these new equilibrium solutions were found to present both radial and projected column density shallower than their Ostriker-like counterparts.
Abstract: The physicalproperties ofthe so-calledOstrikerisothermal,non-rotatingfilament havebeen classically used as benchmark to interpret the stability of the filaments observedin nearby clouds. However, such static picture seems to contrast with the more dynam-ical state observed in different filaments. In order to explore the physical conditionsof filaments under realistic conditions, in this work we theoretically investigate howthe equilibrium structure of a filament changes in a rotating configuration. To do so,we solve the hydrostatic equilibrium equation assuming both uniform and differentialrotations independently. We obtain a new set of equilibrium solutions for rotating andpressure truncated filaments. These new equilibrium solutions are found to presentboth radial and projected column density profiles shallower than their Ostriker-likecounterparts. Moreover, and for rotational periods similar to those found in the obser-vations, the centrifugal forces present in these filaments are also able to sustain largeamounts of mass (larger than the mass attained by the Ostriker filament) withoutbeing necessary unstable. Our results indicate that further analysis on the physicalstate of star-forming filaments should take into account rotational effects as stabilizingagents against gravityKeywords: stars: formation – ISM: clouds – ISM: kinematics and dynamics – ISM:structure
TL;DR: In this paper, the stability regions of a non-static restricted class of axially symmetric spacetime with anisotropic matter distribution were analyzed and dynamical as well as collapse equations were explored at Newtonian and post-Newtonian regimes.
Abstract: In this paper, we analyze stability regions of a non-static restricted class of axially symmetric spacetime with anisotropic matter distribution. We consider f(R)=R+ϵR2 model and assume hydrostatic equilibrium of the axial self-gravitating system at large past time. Considering perturbation from hydrostatic phase, we develop dynamical as well as collapse equations and explore dynamical instabilities at Newtonian and post-Newtonian regimes. It is concluded with the help of stiffness parameter, Γ1, that radial profile of physical parameters like pressure anisotropy, energy density and higher curvature terms of the f(R) model affect the instability ranges.