Polytropic Stellar Models Research Articles

Influence of a mass transfer stability criterion on double white dwarf populations

Context. Mass transfer stability is a key issue in studies of binary evolution. Critical mass ratios for dynamically stable mass transfer have been analyzed on the basis of an adiabatic mass loss model, finding that the donor stars on the giant branches tend to be more stable than that based on the composite polytropic stellar model. Double white dwarfs (DWDs) are of great importance in many fields and their properties would be significantly affected under the new mass transfer stability criterion. Aims. We seek to investigate the influence of mass transfer stability on the formation and properties of DWD populations and discuss the implications in supernova Type Ia (SN Ia) and gravitational wave (GW) sources. Methods. We performed a series of binary population synthesis, adopting the critical mass ratios from the adiabatic mass loss model (i.e., Ge’s model) and that of the composite polytropic model, respectively. In each simulation, 5 × 106 binaries were included and evolved from zero-age main sequence to the end of their evolution and the DWDs were gradually obtained. Results. For Ge’s model, most of the DWDs are produced from the stable non-conservative Roche lobe (RL) overflow, along with a common-envelope (CE) ejection channel (RL+CE channel), regardless of the CE ejection efficiency, αCE. Conversely, the results of the polytropic model strongly depend on the adopted value of αCE. We find DWDs produced from the RL+CE channel have comparable WD masses and the mass ratio distribution peaks at around 1. Based on the magnitude-limited sample of DWDs, the space densities for the detectable DWDs and those with extremely low-mass WD (ELM WD) companions in Ge’s model is: 1347 kpc−3 and 473 kpc−3, respectively, which is close to what has been shown in observations. On the other hand, the polytropic model overpredicts space density of DWDs by a factor of about 2−3. We also find that the results of DWD merger rate distribution per Galaxy in Ge’s model reproduce the observations better than that of the polytropic model, and the merger rate of DWDs with ELM WD companions in the Galaxy is about 1.8 × 10−3 yr−1 in Ge’s model. This result is comparable to the observation estimation of 2 × 10−3 yr−1. The findings from Ge’s model predict a Galactic SN Ia rate of ∼6 × 10−3 yr−1 from DWDs, supporting observations of (5.4 ± 1.2)×10−3 yr−1. For the fiducial model of αCE = 1, the number of detectable GW sources in the polytropic model is larger than that in Ge’s model by about 35%. Conclusions. We confirm that mass transfer stability plays an important role in the formation and properties of DWD populations as well as in the progenitors of SNe Ia and detectable GW sources. The results of Ge’s model support the observational DWD merger rate distribution per Galaxy and the space density of DWDs in the Galaxy.

Viscous and centrifugal instabilities of massive stars

ABSTRACT Massive stars exhibit a variety of instabilities, many of which are poorly understood. We explore instabilities induced by centrifugal forces and angular momentum transport in massive rotating stars. First, we derive and numerically solve linearized oscillation equations for adiabatic radial modes in polytropic stellar models. In the presence of differential rotation, we show that centrifugal and Coriolis forces combined with viscous angular momentum transport can excite stellar pulsation modes, under both low- or high-viscosity conditions. In the low-viscosity limit, which is common in real stars, we demonstrate how to compute mode growth/damping rates via a work integral. Finally, we build realistic rotating $30\, \mathrm{M}_\odot$ star models and show that overstable (growing) radial modes are predicted to exist for most of the star’s life, in the absence of non-adiabatic effects. Peak growth rates are predicted to occur while the star is crossing the Hertzsprung–Russell gap, though non-adiabatic damping may dominate over viscous driving, depending on the effective viscosity produced by convective and/or magnetic torques. Viscous instability could be a new mechanism to drive massive star pulsations and is possibly related to instabilities of luminous blue variable stars.

Oscillations of differentially rotating and tidally distorted gaseous spheres : Effect of non-uniform density

The original Strum-Liouville type of equation is tuned suitably to study the pulsational properties of gaseous spheres of non-uniform densities from centre to surface, which are non-rotating, have solid body rotation or are differentially rotating as well as those which are varying by tidal effects. The eign-frequencies of radial oscillations have been computed numerically for certain polytropic stellar models of non-uniform densities. In order to analyse the effect of non-radial modes of oscillations, the linearised system of differential equations that govern non-radial modes of oscillations, have also been developed and solved for certain polytropic models. It has been observed that with the exception of purely radial oscillations, which are unstable only when γ<4/3 (γ is the ratio of molar specific heats), there also exist non-radial modes of oscillations. It is necessary to consider radial as well as non-radial nodes of oscillations in order to ascertain the stabilities of the existence of any assumed model.

Partial stellar explosions – ejected mass and minimal energy

ABSTRACT Many massive stars appear to undergo enhanced mass-loss during late stages of their evolution. In some cases, the ejected mass likely originates from non-terminal explosive outbursts, rather than continuous winds. Here we study the dependence of the ejecta mass, mej, on the energy budget E of an explosion deep within the star, using both analytical arguments and numerical hydrodynamics simulations. Focusing on polytropic stellar models, we find that for explosion energies smaller than the stellar binding energy, the ejected mass scales as $m_{\rm ej} \propto E^{\varepsilon _{\rm m}}$, where εm = 2.4–3.0 depending on the polytropic index. The loss of energy due to shock breakout emission near the stellar edge leads to the existence of a minimal mass-shedding explosion energy, corresponding to a minimal ejecta mass. For a wide range of progenitors, from Wolf–Rayet stars to red supergiants (RSGs), we find a similar limiting energy of $E_{\rm min} \approx 10^{46}\!-\!10^{47} \rm \, erg$, almost independent of the stellar radius. The corresponding minimal ejecta mass varies considerably across different progenitors, ranging from ${\sim } 10^{-8} \, \rm M_\odot$ in compact stars, up to ${\sim } 10^{-2} \, \rm M_\odot$ in RSGs. We discuss implications of our results for pre-supernova outbursts driven by wave heating, and complications caused by the non-constant opacity and adiabatic index of realistic stars.

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.

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.

Current and future white dwarf mass-radius constraints on varying fundamental couplings and unification scenarios

We discuss the feasibility of using astrophysical observations of white dwarfs as probes of fundamental physics. We quantify the effects of varying fundamental couplings on the white dwarf mass-radius relation in a broad class of unification scenarios, both for the simple case of a polytropic stellar structure model and for more general models. Independent measurements of the mass and radius, together with direct spectroscopic measurements of the fine-structure constant in white dwarf atmospheres lead to constraints on combinations of the two phenomenological parameters describing the underlying unification scenario (one of which is related to the strong sector of the theory while the other is related to the electroweak sector). While currently available measurements do not yet provide stringent constraints, we show that forthcoming improvements, expected for example from the Gaia satellite, can break parameter degeneracies and lead to constraints that ideally complement those obtained from local laboratory tests using atomic clocks.

Asymptotic theory of gravity modes in rotating stars

Context. The seismology of early-type stars is limited by our incomplete understanding of gravito-inertial modes. Aims. We develop a short-wavelength asymptotic analysis for gravito-inertial modes in rotating stars. Methods. The Wentzel-Kramers-Brillouin approximation was applied to the equations governing adiabatic small perturbations about a model of a uniformly rotating barotropic star. Results. A general eikonal equation, including the effect of the centrifugal deformation, is derived. The dynamics of axisymmetric gravito-inertial rays is solved numerically for polytropic stellar models of increasing rotation and analysed by describing the structure of the phase space. Three different types of phase-space structures are distinguished. The first type results from the continuous evolution of structures of the non-rotating integrable phase space. It is predominant in the low-frequency part of the phase space. The second type of structures are island chains associated with stable periodic rays. The third type of structures are large chaotic regions that can be related to the envelope minimum of the Brunt-V\"ais\"al\"a frequency. Conclusions. Gravito-inertial modes are expected to follow this classification, in which the frequency spectrum is a superposition of sub-spectra associated with these different types of phase-space structures. The detailed confrontation between the predictions of this ray-based asymptotic theory and numerically computed modes will be presented in a companion paper.

Stellar Structure and Evolution With Varying Fundamental Couplings

We discuss the feasibility of using solar-type main-sequence stars as probes of fundamental physics and unification. We use a simple polytropic stellar structure model and study its sensitivity to variations of the gravitational, strong and electroweak coupling constants in the context of unification scenarios. We quantify the sensitivity of the Sun's interior temperature to these variations, finding |Δα/α| < 1.3×10-4 for a ‘canonical’ choice of unification scenario, and discuss prospects for future improvements. Further details can be found in [1].

Regular oscillation sub-spectrum of rapidly rotating stars

We present an asymptotic theory that describes regular frequency spacings of pressure modes in rapidly rotating stars. We use an asymptotic method based on an approximate solution of the pressure wave equation constructed from a stable periodic solution of the ray limit. The approximate solution has a Gaussian envelope around the stable ray, and its quantization yields the frequency spectrum. We construct semi-analytical formulas for regular frequency spacings and mode spatial distributions of a subclass of pressure modes in rapidly rotating stars. The results of these formulas are in good agreement with numerical data for oscillations in polytropic stellar models. The regular frequency spacings depend explicitly on internal properties of the star, and their computation for different rotation rates gives new insights on the evolution of mode frequencies with rotation.

Stellar test of the physics of unification

We discuss the feasibility of using solar-type main-sequence stars as probes of fundamental physics and unification. We use a simple polytropic stellar structure model and study its sensitivity to variations of the gravitational, strong and electroweak coupling constants in the context of unification scenarios. We quantify the sensitivity of the Sun's interior temperature to these variations, finding $|\Delta\alpha/\alpha|\lesssim1.3\times10^{-4}$ for a 'canonical' choice of unification scenario, and discuss prospects for future improvements.

F-Mode Instability in Relativistic Neutron Stars

We present the first calculation of the basic properties of the f-mode instability in rapidly rotating relativistic neutron stars, adopting the Cowling approximation. By accounting for dissipation in neutron star matter, i.e., shear or bulk viscosity and superfluid mutual friction, we calculate the associated instability window. For our specific stellar model, a relativistic polytrope, we obtain a minimum gravitational growth time scale (for the dominant ℓ=m=4 mode) of the order of 10(3)-10(4) s near the Kepler frequency Ω(K) while the instability is active above ∼0.92 Ω(K) and for temperatures ∼(10(9)-2×10(10)) K, characteristic of newborn neutron stars.

Collapse of differentially rotating neutron stars and cosmic censorship

We present new results on the dynamics and gravitational-wave emission from the collapse of differentially rotating neutron stars. We have considered a number of polytropic stellar models having different values of the dimensionless angular momentum $J/{M}^{2}$, where $J$ and $M$ are the asymptotic angular momentum and mass of the star, respectively. For neutron stars with $J/{M}^{2}&lt;1$, i.e. ``sub-Kerr'' models, we were able to find models that are dynamically unstable and that collapse promptly to a rotating black hole. Both the dynamics of the collapse and the consequent emission of gravitational waves resemble those seen for uniformly rotating stars, although with an overall decrease in the efficiency of gravitational-wave emission. For stellar models with $J/{M}^{2}&gt;1$, i.e. ``supra-Kerr'' models, on the other hand, we were not able to find models that are dynamically unstable and all of the computed supra-Kerr models were found to be far from the stability threshold. For these models a gravitational collapse is possible only after a very severe and artificial reduction of the pressure, which then leads to a torus developing nonaxisymmetric instabilities and eventually contracting to a stable axisymmetric stellar configuration. While this does not exclude the possibility that a naked singularity can be produced by the collapse of a differentially rotating star, it also suggests that cosmic censorship is not violated and that generic conditions for a supra-Kerr progenitor do not lead to a naked singularity.

Reissner–Nordström and Charged Gas Spheres

This paper studies polytropic gas spheres with some innovations. The main idea, already advanced in the context of neutral, homogeneous, polytropic stellar models, is to base the theory firmly on a variational principle. Another essential novelty is that the mass distribution extends to infinity, the boundary between bulk and atmosphere being defined by an abrupt change in the polytropic index, triggered by the density. The logical next step is to include the effect of radiation, which is a very significant complication since a full treatment would have to include an account of ionization, thus fields representing electrons, ions, photons, gravitons and neutral atoms as well. In way of preparation, we consider models that are charged but homogeneous, involving only gravity, electromagnetism and a single scalar field that represents both the mass and the electric charge; in short, a non-neutral plasma. While this work only represents a stage in the development of a theory of stars, without direct application to physical systems, it does shed some light on the meaning of the Reissner–Nordström solution of the modified Einstein–Maxwell equations. But the main point of the paper is a suggestion about the proper place of the photon gas in a theory of stellar structure and other plasmas, with an application to a simple system; it is proposed to treat the photon gas as part of the dynamics.

Moments of inertia of relativistic magnetized stars

We consider principal moments of inertia of axisymmetric, magnetically deformed stars in the context of general relativity. The general expression for the moment of inertia with respect to the symmetric axis is obtained. The numerical estimates are derived for several polytropic stellar models. We find that the values of the principal moments of inertia are modified by a factor of 2 at most from Newtonian estimates.

Gravitational Radiation Limit on the Spin of Young Neutron Stars

A newly discovered instability in rotating neutron stars, driven by gravitational radiation reaction acting on the stars' $r$-modes, is shown here to set an upper limit on the spin rate of young neutron stars. We calculate the timescales for growth of linear perturbations due to gravitational radiation reaction, and for dissipation by shear and bulk viscosity, working to second order in a slow-rotation expansion within a Newtonian polytropic stellar model. The results are very temperature-sensitive: in hot neutron stars ($T>10^9$ K), the lowest-order $r$-modes are unstable, while in colder stars they are damped by viscosity. These calculations have a number of interesting astrophysical implications. First, the $r$-mode instability will spin down a newly born neutron star to a period close to the initial period inferred for the Crab pulsar, probably between 10 and 20 ms. Second, as an initially rapidly rotating star star spins down, an energy equivalent to roughly 1% of a solar mass is radiated as gravitational waves, which makes the process an interesting source for detectable gravitational waves. Third, the $r$-mode instability rules out the scenario whereby millisecond pulsars are formed by accretion-induced collapse of a white dwarf: the new star would be hot enough to spin down to much slower rates.

A new numerical approach to the oscillation modes of relativistic stars

The oscillation modes of a simple polytropic stellar model are studied. Using a new numerical approach (based on integration for complex coordinates) to the problem for the stellar exterior we have computed the eigenfrequencies of the highly damped w-modes. The results obtained agree well with recent ones of Leins, Nollert and Soffel (1993) Specifically, we are able to explain why several modes in this regime of the complex frequency plane could not be identified within the WKB approach of Kokkotas and Schutz (1992). Furthermore, we have established that the ``kink'' that was a prominent feature of the spectra of Kokkotas and Schutz, but did not appear in the results of Leins {\em et al.}, was a numerical artefact. Using our new numerical code we are also able to compute, for the first time, several of the slowly damped (p) modes for the considered stellar models. For very compact stars we find, somewhat surprisingly, that the damping of these modes does not decrease monotonically as one proceeds to higher oscillation frequencies. The existence of low-order modes that damp away much faster than anticipated may have implications for questions regarding stellar stability and the lifetime of gravitational-wave sources. The present results illustrate the accuracy and reliability of the complex-coordinate method and indicate that the method could prove to be of great use also in problems involving rotating stars. There is no apparent reason why the complex-coordinate approach should not extend to rotating stars, whereas it is accepted that all previous methods will fail to do so.

Normal modes of relativistic stars in slow rotation limit

view Abstract Citations (45) References (26) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Normal Modes of Relativistic Stars in Slow Rotation Limit Kojima, Yasufumi Abstract Rotational shifts of normal frequencies in polytropic stellar models are calculated within the framework of general relativity. The stellar rotation is assumed to be slow and the first-order rotational effects are included to the eigenfrequencies of the nonrotating stars. Unlike in Newtonian theory, the normal frequencies of nonradial oscillations of slowly rotating configurations are complex numbers. The real and imaginary parts correspond to the oscillatory and damping rates of the mode. The frequency and decay rate of the corotating f-mode increase with the increase of the stellar angular velocity. On the other hand, those of the counterrotating mode decrease. The counterrotating mode eventually becomes unstable beyond a critical angular velocity. The critical angular velocity cannot be determined exactly but can be estimated by extrapolating the behavior of the normal mode of a slowly rotating star. The critical value is smaller for larger azimuthal wavenumber and harder equation of state. The relativistic effects also tend to diminish the critical value. Publication: The Astrophysical Journal Pub Date: September 1993 DOI: 10.1086/173073 Bibcode: 1993ApJ...414..247K Keywords: Neutron Stars; Relativistic Theory; Stellar Models; Stellar Oscillations; Stellar Rotation; Angular Velocity; Boundary Conditions; Variational Principles; Astrophysics; RELATIVITY; STARS: NEUTRON; STARS: OSCILLATIONS; STARS: ROTATION full text sources ADS |

A note on large amplitude non-linear radial oscillation in polytropic stellar models

The periodic solutions of inharmonic oscillators describing the systematics of large amplitude single mode radial pulsator have been obtained using modified Lindstedt-Poincare method for different values of the amplitude,a.