Polytropic Model Research Articles

Stably stratified stars containing magnetic fields whose toroidal components are much larger than the poloidal ones in general relativity: A perturbation analysis

We construct the stably stratified magnetized stars within the framework of general relativity. The effects of magnetic fields on the structure of the star and spacetime are treated as perturbations of non-magnetized stars. By assuming ideal magnetohydrodynamics and employing one-parameter equations of state, we derive basic equations for describing stationary and axisymmetric stably stratified stars containing magnetic fields whose toroidal components are much larger than the poloidal ones. A number of the polytropic models are numerically calculated to investigate basic properties of the effects of magnetic fields on the stellar structure. According to the stability result obtained by Braithwaite, which remains a matter of conjecture for general magnetized stars, certain of the magnetized stars constructed in this study are possibly stable.

Constant heat outflow rates for air/surge vessel modelling during pressure transients

Air/surge vessels protect water pipelines during pressure transients. A Rational Heat Transfer (RHT) model and Polytropic (PVn=C) model exists for predicting the thermodynamic behaviour of the enclosed air in the vessel during pressure transients. The RHT model is not used today for design of air vessels because of the difficulty involved in the substitution of the correct form of the heat outflow rate. This work investigated through modelling and simulation, the constant heat outflow rates suitable for predicting the thermodynamic behaviour of the entrapped air. Initial laboratory data and experimental results obtained from literature were used. The results show that when constant heat outflow rate values of 0.7124 W and 0.98 W are assumed, the thermodynamic behaviour of the enclosed air closely matches those obtained from the Polytropic model at index values of 1.0 and 1.4 respectively. These rates are independent of the vessel size, and initial pressure transient event type.

Models of charged compact objects with generalized polytropic equation of state

In this work, we have studied the combined effect of charge and anisotropy on gravitational interaction of compact sources by making use of generalized polytropic equation of state (GPEoS). We have utilized four different values of polytropic index to ascertain the solution of Einstein–Maxwell field equations and develop a new class of spherically symmetric charged polytropic models. Further, we regain the masses of realistic strange stars 4U 1820-30, PSR J1614-2230, PSR J1903+327, Vela 4U and Vela X-1 that shows viability of the present study. Stability of presented models is analyzed by determining speed of sound that indicates the viability of newly generated models.

Stability of generalized polytropic models

In this paper, we have investigated the stability of a spherically symmetric object with charged anisotropic matter by using the concept of cracking. The cracking is a very intuitive technique to check the stability which is based on the analysis of the radial forces that appear on the system due to perturbations taking it out of its equilibrium state. For this, we have applied and studied the effect of local density perturbations to the hydrostatic equilibrium equation and on all the physical parameters with generalized polytropic equation of state. It is found that some of the generalized polytropic models exhibit cracking.

Study of Cylindrical Polytropes with Cosmological Constant

In this paper, we study cylindrically symmetric perfect fluid distribution in the presence of cosmological constant satisfying two cases of polytropic equation of state. The corresponding structure equations are formulated and solved through numerical technique. The resulting polytropic models turn out to be physically viable as they satisfy all the energy conditions. Finally, we analyze the stability of polytropes by applying perturbations on matter variables via polytropic constant as well as polytropic index and construct the force distribution function. It is found that compact object is stable for feasible choice of perturbed polytropic constant in each case while the perturbation in polytropic index yields stable results only for the first case.

Validation of On-board Polytropic Index Prediction Model for Compression Stroke of CI Engines Using Measurements of In-cylinder Local Heat Flux

Validation of On-board Polytropic Index Prediction Model for Compression Stroke of CI Engines Using Measurements of In-cylinder Local Heat Flux

Performance Evaluation of Model-Based Controller with On-board Polytropic Index Prediction Model for Compression Stroke under Steady Driving Condition of CI Engine

Performance Evaluation of Model-Based Controller with On-board Polytropic Index Prediction Model for Compression Stroke under Steady Driving Condition of CI Engine

Effect of interaction of the various modes on the radial velocity curves of the polytropic models of rotationally and tidally distorted pulsating variable stars

In our previous work, we developed a model to study the effects of rotation and/or tidal distortions on anharmonic radial oscillations and hence on the radial velocity curves of the polytropic models of pulsating variable stars. We considered the first three modes (fundamental and the next two higher modes) for the polytropic models of index 1.5 and 3.0 in that work. In the present paper, we are further extending our previous work to study the effect of the interaction of various modes on anharmonic radial oscillations and hence on radial velocity curves of the rotationally and/or tidally distorted polytropic models of pulsating variable stars. For this purpose, we have considered the following cases: (i) fundamental mode (ii) fundamental and the first mode, (iii) fundamental and the next two modes and finally (iv) fundamental and the next three higher modes of pulsation in our study. The objective of this paper is also to investigate whether the interaction of various modes affects the results of our previous study or not. The results of this study show that the interaction of the fundamental mode with higher modes appreciably changes the shape of the radial velocity curve of rotationally distorted and rotationally and tidally distorted polytropic models of pulsating variable stars.

Development of On-board Polytropic Index Prediction Model for Injection Timing Optimization of Diesel Engines

Diesel engines are required to reduce exhaust emissions during real-world operations. In this regard, a new control concept called model-based control has been explored. Unlike the conventional method of relying on steady-state measurements, model-based control allows cycle-by-cycle optimization of control inputs based on physical principles. Existing models for combustion control have been using empirical equations to predict polytropic index for the compression stroke for estimation of in-cylinder pressure and temperature at fuel injection. Therefore, in this study, a polytropic index prediction model was developed in MATLAB to maintain the engine performance under transient conditions and to reduce the required number of experiments. The model includes a heat loss model and a gas flow model to consider the effect of wall heat transfer and gas flows inside the cylinder. The computational load of the model was reduced through discretization of a single engine cycle into several calculation points. The model was validated against numerical simulation results under steady conditions first, and then applied to transient conditions for more realistic operational conditions. The model estimated the polytropic index with average errors under steady and transient conditions with 0.22% and 0.37%, respectively. Finally, the calculation time of the model was evaluated to be 50.6 μs. It was concluded the model can be implemented on a model-based controller in the future.

The antesonic condition for the explosion of core-collapse supernovae – I. Spherically symmetric polytropic models: stability and wind emergence

Shock revival in core-collapse supernovae (CCSNe) may be due to the neutrino mechanism. While it is known that in a neutrino-powered CCSN, explosion begins when the neutrino luminosity of the proto-neutron star exceeds a critical value, the physics of this condition in time-dependent, multidimensional simulations are not fully understood. \citet{Pejcha2012} found that an `antesonic condition' exists for time-steady spherically symmetric models, potentially giving a physical explanation for the critical curve observed in simulations. In this paper, we extend that analysis to time-dependent, spherically symmetric polytropic models. We verify the critical antesonic condition in our simulations, showing that models exceeding it drive transonic winds whereas models below it exhibit steady accretion. In addition, we find that (1) high spatial resolution is needed for accurate determination of the antesonic ratio and shock radius at the critical curve, and that low resolution simulations systematically underpredict these quantities, making explosion more difficult at lower resolution; (2) there is an important physical connection between the critical mass accretion rate at explosion and the mass loss rate of the post-explosion wind: the two are directly proportional at criticality, implying that, at criticality, the wind kinetic power is tied directly to the accretion power; (3) the value of the post-shock adiabatic index $\Gamma$ has a large effect on the length and time scales of the post-bounce evolution of the explosion larger values of $\Gamma$ result in a longer transition from the accretion to wind phases.

Cracking in anisotropic polytropic models

This paper is devoted to examine the cracking of spherically symmetric anisotropic fluid configuration for polytropic equation of state. For this purpose, we formulate the corresponding field equations as well as generalized Tolman–Oppenheimer–Volkoff equation. We introduce density perturbations in matter variables and then construct the force distribution function. In order to examine the occurrence of cracking/overturning, we consider two models corresponding to two values of the polytropic index. It is found that the first model exhibits overturning for the considered values of polytropic constant while the second model neither exhibits cracking nor overturning for larger values of polytropic constant.

An adaptive ALE scheme for non-ideal compressible fluid dynamics over dynamic unstructured meshes

This paper investigates the application of mesh adaptation techniques in the Non-Ideal Compressible Fluid Dynamic (NICFD) regime, a region near the vapor-liquid saturation curve where the flow behavior significantly departs from the ideal gas model, as indicated by a value of the fundamental derivative of gasdynamics less than one. A recent interpolation-free finite-volume adaptive scheme is exploited to modify the grid connectivity in a conservative way, and the governing equations for compressible inviscid flows are solved within the Arbitrary Lagrangian Eulerian framework by including special fictitious fluxes representing volume modifications due to mesh adaptation.The absence of interpolation of the solution to the new grid prevents spurious oscillations that may make the solution of the flow field in the NICFD regime more difficult and less robust.Non-ideal gas effects are taken into account by adopting the polytropic Peng-Robinson thermodynamic model. The numerical results focus on the problem of a piston moving in a tube filled with siloxane $\mathrm{MD_4M}$, a simple configuration which can be the core of experimental research activities aiming at investigating the thermodynamic behavior of NICFD flows. Several numerical tests involving different piston movements and initial states in 2D and 3D assess the capability of the proposed adaption technique to correctly capture compression and expansion waves, as well as the generation and propagation of shock waves, in the NICFD and in the non-classical regime.

Shock location and CME 3D reconstruction of a solar type II radio burst with LOFAR

Context. Type II radio bursts are evidence of shocks in the solar atmosphere and inner heliosphere that emit radio waves ranging from sub-meter to kilometer lengths. These shocks may be associated with coronal mass ejections (CMEs) and reach speeds higher than the local magnetosonic speed. Radio imaging of decameter wavelengths (20–90 MHz) is now possible with the Low Frequency Array (LOFAR), opening a new radio window in which to study coronal shocks that leave the inner solar corona and enter the interplanetary medium and to understand their association with CMEs. Aims. To this end, we study a coronal shock associated with a CME and type II radio burst to determine the locations at which the radio emission is generated, and we investigate the origin of the band-splitting phenomenon. Methods. Thetype II shock source-positions and spectra were obtained using 91 simultaneous tied-array beams of LOFAR, and the CME was observed by the Large Angle and Spectrometric Coronagraph (LASCO) on board the Solar and Heliospheric Observatory (SOHO) and by the COR2A coronagraph of the SECCHI instruments on board the Solar Terrestrial Relation Observatory(STEREO). The 3D structure was inferred using triangulation of the coronographic observations. Coronal magnetic fields were obtained from a 3D magnetohydrodynamics (MHD) polytropic model using the photospheric fields measured by the Heliospheric Imager (HMI) on board the Solar Dynamic Observatory (SDO) as lower boundary. Results. The type II radio source of the coronal shock observed between 50 and 70 MHz was found to be located at the expanding flank of the CME, where the shock geometry is quasi-perpendicular with θBn ~ 70°. The type II radio burst showed first and second harmonic emission; the second harmonic source was cospatial with the first harmonic source to within the observational uncertainty. This suggests that radio wave propagation does not alter the apparent location of the harmonic source. The sources of the two split bands were also found to be cospatial within the observational uncertainty, in agreement with the interpretation that split bands are simultaneous radio emission from upstream and downstream of the shock front. The fast magnetosonic Mach number derived from this interpretation was found to lie in the range 1.3–1.5. The fast magnetosonic Mach numbers derived from modelling the CME and the coronal magnetic field around the type II source were found to lie in the range 1.4–1.6.

Asymptotic theory of gravity modes in rotating stars

Context. Differential rotation has a strong influence on stellar internal dynamics and evolution, notably by triggering hydrodynamical instabilities, by interacting with the magnetic field, and more generally by inducing transport of angular momentum and chemical elements. Moreover, it modifies the way waves propagate in stellar interiors and thus the frequency spectrum of these waves, the regions they probe, and the transport they generate. Aims. We investigate the impact of a general differential rotation (both in radius and latitude) on the propagation of axisymmetric gravito-inertial waves. Methods. We use a small-wavelength approximation to obtain a local dispersion relation for these waves. We then describe the propagation of waves thanks to a ray model that follows a Hamiltonian formalism. Finally, we numerically probe the properties of these gravito-inertial rays for different regimes of radial and latitudinal differential rotation. Results. We derive a local dispersion relation that includes the effect of a general differential rotation. Subsequently, considering a polytropic stellar model, we observe that differential rotation allows for a large variety of resonant cavities that can be probed by gravito-inertial waves. We identify that for some regimes of frequency and differential rotation, the properties of gravito-inertial rays are similar to those found in the uniformly rotating case. Furthermore, we also find new regimes specific to differential rotation, where the dynamics of rays is chaotic. Conclusions. As a consequence, we expect modes to follow the same trend. Some parts of oscillation spectra corresponding to regimes similar to those of the uniformly rotating case would exhibit regular patterns, while parts corresponding to the new regimes would be mostly constituted of chaotic modes with a spectrum rather characterised by a generic statistical distribution.

Effects of rotation and tidal distortions on the shapes of radial velocity curves of polytropic models of pulsating variable stars

Anharmonic oscillations of rotating stars have been studied by various authors in literature to explain the observed features of certain variable stars. However, there is no study available in literature that has discussed the combined effect of rotation and tidal distortions on the anharmonic oscillations of stars. In this paper, we have created a model to determine the effect of rotation and tidal distortions on the anharmonic radial oscillations associated with various polytropic models of pulsating variable stars. For this study we have used the theory of Rosseland to obtain the anharmonic pulsation equation for rotationally and tidally distorted polytropicmodels of pulsating variable stars. The main objective of this study is to investigate the effect of rotation and tidal distortions on the shapes of the radial velocity curves for rotationally and tidally distorted polytropic models of pulsating variable stars. The results of the present study show that the rotational effects cause more deviations in the shapes of radial velocity curves of pulsating variable stars as compared to tidal effects.

Anisotropic charged generalized polytropic models

In this paper, we found some new anisotropic charged models admitting generalized polytropic equation of state with spherically symmetry. An analytic solution of the Einstein–Maxwell field equations is obtained through the transformation introduced by Durgapal and Banerji (Phys. Rev. D 27:328, 1983). The physical viability of solutions corresponding to polytropic index $\eta =1/2$ , $2/3$ , 1, 2 is analyzed graphically. For this, we plot physical quantities such as radial and tangential pressure, anisotropy, speed of sound which demonstrated that these models achieve all the considerable physical conditions required for a relativistic star. Further, it is mentioned here that previous results for anisotropic charged matter with linear, quadratic and polytropic equation of state can be retrieved.

A novel second-order thermal model of Stirling engines with consideration of losses due to the speed of the crack system

Very accurate second-order thermal models have been developed for the thermal simulation of Stirling engines in recent years. One of the last ones is the comprehensive polytropic model of Stirling engine called the CPMS model. The accuracy of the CPMS model was found to be sufficient for the nominal operation of a prototype Stirling engine known as the GPU-3 engine. Nevertheless, the accuracy of the CPMS model was drastically reduced at high rotational speeds of the engine. In this paper, power loss and pressure change due to the inertial force of the crank system were integrated into the CPMS thermal model in order to compensate inaccuracy of the CPMS model at high rotational speeds. Moreover, the effect of rotational speed on the gas temperature in heater and cooler was also incorporated. A precise model for evaluating the mechanical friction loss was also employed and compared with the simple frictional model of the simple frictional model used in the CPMS. The model was examined on the GPU-3 engine, and it was found that it has superior accuracy compared to the previous thermal model over the entire working regime of the GPU-3 engine.

Quantitative analysis of a non-adiabatic process undergone by a gas in a piston-cylinder assembly

The influence of the finite values of piston speed as well as the thermal conductivity and capacity of the cylinder and piston walls on the connection between the thermodynamic parameters is investigated on a quantitative level. There are two possible qualitatively different types of evolution of thermodynamic processes for both the compression and expansion of gas due to the presence of heat transfer through the confining shell. Among them there may be cases where gas compressing (expanding) in the cylinder will cool (heat) during a certain interval of its volumes. Moreover, due to the presence of the dissipative process of heat transfer through the cylinder walls, the compression and expansion of gas should pass along the different T–V curves. The analytical expressions for such processes in one important limiting case are also derived. We show that the polytropic model is actually only able to take into account the process of absorption of heat by the walls and not the dissipative heat transfer through the walls. In the presence of heat transfer an exactly polytropic process is possible only under very specific conditions.

A physical model of mass ejection in failed supernovae

During the core collapse of massive stars, the formation of the protoneutron star is accompanied by the emission of a significant amount of mass-energy ($\sim 0.3 \, M_{\odot}$) in the form of neutrinos. This mass-energy loss generates an outward-propagating pressure wave that steepens into a shock near the stellar surface, potentially powering a weak transient associated with an otherwise-failed supernova. We analytically investigate this mass-loss-induced wave generation and propagation. Heuristic arguments provide an accurate estimate of the amount of energy contained in the outgoing sound pulse. We then develop a general formalism for analyzing the response of the star to centrally concentrated mass loss in linear perturbation theory. To build intuition, we apply this formalism to polytropic stellar models, finding qualitative and quantitative agreement with simulations and heuristic arguments. We also apply our results to realistic pre-collapse massive star progenitors (both giants and compact stars). Our analytic results for the sound pulse energy, excitation radius, and steepening in the stellar envelope are in good agreement with full time-dependent hydrodynamic simulations. We show that {prior} to the sound pulses arrival at the stellar photosphere, the photosphere has already reached velocities $\sim 20-100 \%$ of the local sound speed, thus likely modestly decreasing the stellar effective temperature prior to the star disappearing. Our results provide important constraints on the physical properties and observational appearance of failed supernovae.

Effects of mass variation on structures of differentially rotating polytropic stars

A method is proposed for determining equilibrium structures and various physical parameters of differentially rotating polytropic models of stars, taking into account the effect of mass variation inside the star and on its equipotential surfaces. The law of differential rotation has been assumed to be the form of ω2(s)=b1+b2s2+b3s4. The proposed method utilizes the averaging approach of Kippenhahn and Thomas and concepts of Roche-equipotential to incorporate the effects of differential rotation on the equilibrium structures of polytropic stellar models. Mathematical expressions of determining the equipotential surfaces, volume, surface area and other physical parameters are also obtained under the effects of mass variation inside the stars. Some significant conclusions are also drawn.