Retrograde precession of stellar orbits and gravitational loss-cone instability in galactic centers

We consider disk and spherical subsystems of stars with nearly radial orbits under conditions when the well-known radial orbit instability is not possible. This requires that the precession of stellar orbits be retrograde, i.e., in the direction opposite to the orbital rotation of stars. We show that an instability that is an analogue of the loss-cone instability known in plasma physics can then develop in the presence of a “loss cone” in the angular momentum distribution of stars, which ensures a deficit or even absence of stars with low angular momenta. Examples of systems with a loss cone are the centers of galaxies or star clusters with massive black holes. The instability can produce a flux of stars onto the galactic center, i.e., it can serve as a mechanism of fueling the nuclear activity of galaxies. Mathematically, the problem is reduced to analyzing simple characteristic equations that describe small perturbations in a disk and a sphere of radially highly elongated stellar orbits. In turn, these characteristics equations are derived through a number of successive simplifications of the general linearized Vlasov equations (i.e., the system that includes the collisionless Boltzmann kinetic equation and the Poisson equation) in action—angle variables.     

Unstable non-radial modes in radial pulsators: theory and an example

We study the possibility of the excitation of non-radial oscillations in classical pulsating stars. The stability of an RR Lyrae model is examined through non-adiabatic non-radial calculations. We also explore stability in the presence of non-linear coupling between radial and non-radial modes of nearly identical frequency.   In our model, a large number of unstable low-degree (ℓ = 1,2) modes have frequencies in the vicinity of unstable radial mode frequencies. The growth rates of such modes, however, are considerably smaller than those of the radial modes. We also recover an earlier result that at higher degrees (ℓ = 6–12) there are modes trapped in the envelope with growth rates similar to those of radial modes.   Subsequently, monomode radial pulsation of this model is considered. The destabilizing effect of the 1:1 resonance between the radial mode and nearby non-radial modes of low degrees is studied, with the assumption that the excited radial mode saturates the linear instability of all other modes. The instability depends on the radial mode amplitude, the frequency difference, the damping rate of the non-radial mode, and the strength of the non-linear coupling between the modes considered. At the pulsation amplitudes typical for RR Lyrae stars, the instability of the monomode radial pulsation and the concomitant resonant excitation of some non-radial oscillation modes is found to be very likely.     

The plasma radiation of flare kernels

We survey the mathematics of non-linear Hamiltonian oscillations with emphasis being laid on the more recently discovered Kolmogorov instability. In the context of radial adiabatic oscillations of stars this formalism predicts a Kolmogorov instability even at low oscillation energies, provided that sufficiently high linear asymptotic modes have been excited. Numerical analysis confirms the occurrence of this instability. It is found to show up already among the lowest order modes, although high surface amplitudes are then required (¦δr¦/R ～ 0.5 for an unstable fundamental mode - first harmonic coupling). On the basis of numerical evidence we conjecture that in the Kolmogorov unstable regime the enhanced coupling due to internal resonance effects leads to an equipartition of energy over all interacting degrees of freedom. We also indicate that the power spectrum of such oscillations is expected to display two components: A very broad band of overlapping pseudo-linear frequency peaks spread out over the asymptotic range, and a strictly non-linear l/f-noise type component close to the frequency origin. It is finally argued that the Kolmogorov instability is likely to occur among non-linearly coupled non-radial stellar modes at a surface amplitude much lower than in the radial case. This lends support to the view that this instability might be operative among the solar oscillations.     

Seismic signatures of strange stars with crust

We study acoustic oscillations (eigenfrequencies, velocity distributions, damping times) of normal crusts of strange stars. These oscillations are very specific because of huge density jump at the interface between the normal crust and the strange matter core. The oscillation problem is shown to be self-similar. For a low (but non-zero) multipolarity l , the fundamental mode (without radial nodes) has a frequency of ∼300 Hz and mostly horizontal oscillation velocity; other pressure modes have frequencies ≳20 kHz and almost radial oscillation velocities. The latter modes are similar to radial oscillations (having approximately the same frequencies and radial velocity profiles). The oscillation spectrum of strange stars with crust differs from the spectrum of neutron stars. If detected, acoustic oscillations would allow one to discriminate between strange stars with crust and neutron stars and constrain the mass and radius of the star.     

Non-adiabatic linear pulsation models for low-mass helium stars

Non-adiabatic linear pulsation models have been calculated for low-mass stars with effective temperatures between 16 000 and 35 000 K, and with surface gravities in the range 3⊙, X =0.00, Z =0.02. It is shown that the Z -bump instability persists to low masses ( M ∼0.4 M_⊙) but is suppressed either by a reduction in metallicity Z or by a selective enhancement of the carbon abundance. An unexpected result is the discovery that Z -bump instability persists at hydrogen abundances X >0.3, although the position of the red edge is sensitive to X . We have found that non-radial pulsations are also excited in the same instability region as radial pulsations.
The implications of these results for individual low-mass helium stars are discussed. It is concluded that Z -bump driven pulsations (radial and/or non-radial) may be excited in some helium-rich subdwarf B stars, representing a possible major extension to the class of variable stars represented by the prototype V652 Her.     

The β Cephei instability strip

We carried out a series of linear stability analyses of the radial and low-degree non-radial p modes for stellar models with initial masses of     . The stellar models were computed by using convective overshoot distance     , 0.25 and 0.40  H _P. Our numerical results show that the β Cephei instability strip forms a horn-shaped region pointing upwards near the main sequence on the Hertzsprung–Russell diagram (HRD). The lower part of the instability strip for the radial modes join the zero-age main-sequence (ZAMS) at     , while the top of the instability strip extends up to     . The instability strip for the non-radial modes is even wider. The overall instability strip is dominated by the radial and non-radial fundamental modes. The first overtone (the radial-order index     is also pulsationally unstable. We have shown that the β Cephei stability is almost independent of the overshoot parameter d _over used for the stellar models, while it depends critically on the metal abundance. With decreasing metal abundance, the instability region shrinks and eventually disappears for     .     

Gravitational loss-cone instability in stellar systems with retrograde orbit precession

We study spherical and disc clusters in a near-Keplerian potential of galactic centres or massive black holes. In such a potential orbit precession is commonly retrograde, that is, the direction of the orbit precession is opposite to the orbital motion. It is assumed that stellar systems consist of nearly-radial orbits. We show that if there is a loss-cone at low angular momentum (e.g. due to consumption of stars by a black hole), an instability similar to loss-cone instability in plasma may occur. The gravitational loss-cone instability is expected to enhance black hole feeding rates. For spherical systems, the instability is possible for the number of spherical harmonics   l ≥ 3  . If there is some amount of counter-rotating stars in flattened systems, they generally exhibit the instability independent of azimuthal number m . The results are compared with those obtained recently by Tremaine for distribution functions monotonically increasing with angular momentum.
The analysis is based on simple characteristic equations describing small perturbations in a disc or a sphere of stellar orbits highly elongated in radius. These characteristic equations are derived from the linearized Vlasov equations (combining the collisionless Boltzmann kinetic equation and the Poisson equation), using the action-angle variables. We use two techniques for analysing the characteristic equations: the first one is based on preliminary finding of neutral modes, and the second one employs a counterpart of the plasma Penrose–Nyquist criterion for disc and spherical gravitational systems.     

Nonlinearity of Nonradial Modes in Evolved Stars

We show that in evolved stars, even at relatively low surfaceamplitudes, nonradial modes become strongly nonlinear in thehydrogen shell source, where the Brunt-Väisälä frequency hasits absolute maximum. The measure of nonlinearity is the productof horizontal displacement times the radial wavenumber, . It becomes large already in evolved -Scuti stars.This nonlinearity presents a major problem for interpretations ofamplitude modulation in RR Lyrae stars in terms of nonradial modeexcitation.     

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.     

修正的粘滞律及有磁吸积盘的稳定性研究

本文采用修正的粘滞定律及磁流体力学研究了薄吸积盘内区及外区的稳定性问题。运用微扰方法导出了色散方程，分析了四种情况下吸积盘的不稳定性，结果表明：在同时考虑磁场和修正的粘滞律时，吸积盘中存在着三种振荡模式，其中粘滞模式总是稳定的，磁声速模式（包括向里、向外传播两种模式）通常是不稳定的。这些结果为解释ＢＬ Ｌａｃ天体、Ｓｅｙｆｅｒｔ星系、类星体等活动星系核的光变现象提供了理论依据。     

Local stellar stability. II

The existence proof of continuous spectra of eigenvaluess developed in the framework of the function space ofq-regularizations (Perdang, 1976) is extended in this paper by relaxing the severe restrictions previously imposed o the mathematical structure of the stellar stability equations. It is stressed that these local modes depend on the variable system in terms of which the linearized stellar structure equations are set up. We therefore search for a systematic procedure to select the most satisfactory system to analyze Local Stability. Our procedure is illustrated in great detail in the case of nonradial adiabatic stability. Moreover when applied to nonadiabatic perturbations it reveals the existence of two new types of local instability which seem to prevail in the majority of stars in a thermonuclear burning phase: (a) a nonrdial local secular instability; (b) a radial local nuclear instability. Numerical test calculations exhibit that the latter helps us to understand certain evolutionary features of stars, in particular it provides an interpretation of Hayashiet al.'s (1962) rule.     

Secular instability of g-modes in rotating neutron stars

Gravitational radiation tends to drive gravity modes in rotating neutron stars to become unstable. For an inviscid star, the instability sets in when the rotation frequency is about 0.7 times the corresponding mode frequency of the non-rotating star. Neutron stars with spin frequencies ≳100 Hz are susceptible to this instability, with a growth time of the order of years. However, it is likely that viscous dissipation suppresses the instability except for a narrow range of temperatures around 10⁹ K. We also show that the viscosity-driven instability of g-modes is absent.     

自引力磁均分和磁守恒吸积盘模型的稳定性分析

对自引力磁均分和磁守恒两种模型的径向振荡稳定性的研究表明：在同时考虑自引力和磁场作用的情况下,吸积盘在磁均分和磁守恒模型中均存在着三种振荡模式,其中粘滞模式总是稳定的,磁声模式是不稳定的,中性模式在图中较长波段范围趋于稳定,在较短波段范围是不稳定的．这些结果有利于解释活动天体的Mark421和Pks2155—304的长周期光变现象．同时阐明了自引力在两种模型中对三种模式的影响有相同的趋势,而对磁场则分别起着相反的作用．这个结论表明磁均分吸积盘模型在解释光变现象时更为有利．     

Slow modes in stellar systems with nearly harmonic potentials: II. Gravitational loss-cone instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). We consider star clusters with monoenergetic distribution functions that monotonically increase with angular momentum in the entire range of angular momenta (from purely radial orbits to circular ones) or have a growing region only at low angular momenta. In these cases, there are orbits with a retrograde precession, i.e., in a direction opposite to the orbital rotation of the star. The presence of a gravitational loss-cone instability, which is also observed in systems of 1: 1 orbits in near-Keplerian potentials, is associated with such orbits. In contrast to 1: 1 systems, the loss-cone instability takes place even for distribution functions monotonically increasing with angular momentum, including those for systems with circular orbits. The regions of phase space with retrograde orbits do not disappear when the distribution function is smeared in energy. We investigate the influence of a weak inhomogeneity of a heavy halo with a density that decreases with distance from the center.     

V473 Lyrae: a Blazhko-effect Cepheid?

The amplitude spectra of the light curves of RR Lyrae Blazhko stars can often be typified by symmetrical triplets. It is shown that the same applies to V473 Lyr. Furthermore, the star is probably a radial pulsator – as is thought to be the case for the RR Lyrae stars – thus ruling out rotational splitting of a purely non-radial mode.     

Towards mode selection in δ Scuti stars: regularities in observed and theoretical frequency spectra

Only a fraction of the theoretically predicted non-radial pulsation modes have so far been observed in δ Scuti stars. Nevertheless, the large number of frequencies detected in recent photometric studies of selected δ Scuti stars allow us to look for regularities in the frequency spacing of modes. Mode identifications are used to interpret these results.
Statistical analyses of several δ Scuti stars (FG Vir, 44 Tau, BL Cam and others) show that the photometrically observed frequencies are not distributed at random, but that the excited non-radial modes cluster around the frequencies of the radial modes over many radial orders.
The observed regularities can be partly explained by modes trapped in the stellar envelope. This mode selection mechanism was proposed by Dziembowski & Królikowska and shown to be efficient for  ℓ= 1  modes. New pulsation model calculations confirm the observed regularities.
We present the s – f diagram, which compares the average separation of the radial frequencies ( s ) with the frequency of the lowest frequency unstable radial mode ( f ). This provides an estimate for the  log  g   value of the observed star, if we assume that the centres of the observed frequency clusters correspond to the radial mode frequencies. This assumption is confirmed by examples of well-studied δ Scuti variables in which radial modes were definitely identified.     

Discovery of A New Class of Pulsating Stars: Gravity-mode Pulsators among Subdwarf B Stars

During the course of an ongoing CCD monitoring program to investigate low-level light variations in subdwarf B (sdB) stars, weserendipitously discovered a new class of multimode pulsators withperiods of the order of an hour. These periods are a factor of tenlonger than those of previously known multimode sdB pulsators (EC14026 stars), implying the new pulsations are due to gravity modes rather than pressure modes. The iron opacity instability that drives the short period EC 14026 stars is effective in hot sdB's. Thelong period pulsators are found only among cooler sdB stars, wherethey are surprisingly common. The mechanism responsible for excitingthe deeper g-modes in cool sdB's is currently unknown, but thetemperature and gravity range in which these stars occur must be animportant clue. We present the first observational results for thisnew class of pulsating sdB stars, and discuss some possible implications.     

Relativistic and Newtonian diskoseismology

The normal mode oscillations of thin accretion disks around black holes and other compact objects are analyzed and contrasted with those in stars. For black holes, the most robust modes are gravitationally trapped near the radius at which the radial epicyclic frequency is maximum. Their eigenfrequencies depend mainly on the mass and angular momentum of the black hole. The fundamental g-mode has recently been seen in numerical simulations of black hole accretion disks. For stars such as white dwarfs, the modes are trapped near the inner boundary (magnetospheric or stellar) of the accretion disk. Their eigenfrequencies are approximately multiples of the (Keplerian) angular velocity of the inner edge of the disk. The relevance of these modes to the high frequency quasi-periodic oscillations observed in the power spectra of accreting binaries will be discussed. In contrast to most stellar oscillations, most of these modes are unstable in the presence of viscosity (if the turbulent viscosity induced by the magnetorotational instability acts hydrodynamically).     

Dynamo-generated magnetic fields at the surface of a massive star

Spruit has shown that an astrophysical dynamo can operate in the non-convective material of a differentially rotating star as a result of a particular instability in the magnetic field (the Tayler instability). By assuming that the dynamo operates in a state of marginal instability, Spruit has obtained formulae which predict the equilibrium strengths of azimuthal and radial field components in terms of local physical quantities. Here, we apply Spruit's formulae to our previously published models of rotating massive stars in order to estimate Tayler dynamo field strengths. There are no free parameters in Spruit's formulae. In our models of 10- and  50-M_⊙  stars on the zero-age main sequence, we find internal azimuthal fields of up to 1 MG, and internal radial components of a few kG. Evolved models contain weaker fields. In order to obtain estimates of the field strength at the stellar surface, we examine the conditions under which the Tayler dynamo fields are subject to magnetic buoyancy. We find that conditions for Tayler instability overlap with those for buoyancy at intermediate to high magnetic latitudes. This suggests that fields emerge at the surface of a massive star between magnetic latitudes of about 45° and the poles. We attempt to estimate the strength of the field which emerges at the surface of a massive star. Although these estimates are very rough, we find that the surface field strengths overlap with values which have been reported recently for line-of-sight fields in several O and B stars.