Showing papers in "Astrophysics and Space Science in 2007"

Modeling the emission processes in blazars

Abstract: Blazars are the most violent steady/recurrent sources of high-energy gamma-ray emission in the known Universe. They are prominent emitters of electromagnetic radiation throughout the entire electromagnetic spectrum. The observable radiation most likely originates in a relativistic jet oriented at a small angle with respect to the line of sight. This review starts out with a general overview of the phenomenology of blazars, including results from a recent multiwavelength observing campaign on 3C279. Subsequently, issues of modeling broadband spectra will be discussed. Spectral information alone is not sufficient to distinguish between competing models and to constrain essential parameters, in particular related to the primary particle acceleration and radiation mechanisms in the jet. Short-term spectral variability information may help to break such model degeneracies, which will require snap-shot spectral information on intraday time scales, which may soon be achievable for many blazars even in the gamma-ray regime with the upcoming GLAST mission and current advances in Atmospheric Cherenkov Telescope technology. In addition to pure leptonic and hadronic models of gamma-ray emission from blazars, leptonic/hadronic hybrid models are reviewed, and the recently developed hadronic synchrotron mirror model for TeV γ-ray flares which are not accompanied by simultaneous X-ray flares (“orphan TeV flares”) is revisited.

The Wide Field Spectrograph (WiFeS)

Abstract: This paper describes the Wide Field Spectrograph (WiFeS) under construction at the Research School of Astronomy and Astrophysics (RSAA) of the Australian National University (ANU) for the ANU 2.3 m telescope at the Siding Spring Observatory. WiFeS is a powerful integral field, double-beam, concentric, image-slicing spectrograph designed to deliver excellent throughput, wavelength stability, spectrophotometric performance and superb image quality along with wide spectral coverage throughout the 320–950 nm wavelength region. It provides a 25×38 arcsec field with 0.5 arcsec sampling along each of twenty five 38×1 arcsec slitlets. The output format is optimized to match the 4096×4096 pixel CCD detectors in each of two cameras individually optimized for the blue and the red ends of the spectrum, respectively. A process of “interleaved nod-and-shuffle” will be applied to permit quantum noise-limited sky subtraction. Using VPH gratings, spectral resolutions of 3000 and 7000 are provided. The full spectral range is covered in a single exposure at R=3000, and in two exposures in the R=7000 mode. The use of transmissive coated optics, VPH gratings and optimized mirror coatings ensures a throughput (including telescope atmosphere and detector) >30% over a wide spectral range. The concentric image-slicer design ensures an excellent and uniform image quality across the full field. To maximize scientific return, the whole instrument is configured for remote observing, pipeline data reduction, and the accumulation of calibration image libraries.

Magnetic fields in protoplanetary disks

Abstract: Magnetic fields likely play a key role in the dynamics and evolution of protoplanetary disks. They have the potential to efficiently transport angular momentum by MHD turbulence or via the magnetocentrifugal acceleration of outflows from the disk surface. Magnetically-driven mixing has implications for disk chemistry and evolution of the grain population, and the effective viscous response of the disk determines whether planets migrate inwards or outwards. However, the weak ionisation of protoplanetary disks means that magnetic fields may not be able to effectively couple to the matter. I examine the magnetic diffusivity in a minimum solar nebula model and present calculations of the ionisation equilibrium and magnetic diffusivity as a function of height from the disk midplane at radii of 1 and 5 AU. Dust grains tend to suppress magnetic coupling by soaking up electrons and ions from the gas phase and reducing the conductivity of the gas by many orders of magnitude. However, once grains have grown to a few microns in size their effect starts to wane and magnetic fields can begin to couple to the gas even at the disk midplane. Because ions are generally decoupled from the magnetic field by neutral collisions while electrons are not, the Hall effect tends to dominate the diffusion of the magnetic field when it is able to partially couple to the gas, except at the disk surfaces where the low density of neutrals permits the ions to remain attached to the field lines. For a standard population of 0.1 μm grains the active surface layers have a combined column Σactive≈2 g cm−2 at 1 AU; by the time grains have aggregated to 3 μm, Σactive≈80 g cm−2. Ionisation in the active layers is dominated by stellar X-rays. In the absence of grains, X-rays maintain magnetic coupling to 10% of the disk material at 1 AU (i.e. Σactive≈150 g cm−2). At 5 AU the Σactive≈Σtotal once grains have aggregated to 1 μm in size.

The Magnificent Seven: Magnetic fields and surface temperature distributions

Abstract: Presently seven nearby radio-quiet isolated neutron stars discovered in ROSAT data and characterized by thermal X-ray spectra are known. They exhibit very similar properties and despite intensive searches their number remained constant since 2001 which led to their name “The Magnificent Seven”. Five of the stars exhibit pulsations in their X-ray flux with periods in the range of 3.4 s to 11.4 s. XMM-Newton observations revealed broad absorption lines in the X-ray spectra which are interpreted as cyclotron resonance absorption lines by protons or heavy ions and/or atomic transitions shifted to X-ray energies by strong magnetic fields of the order of 1013 G. New XMM-Newton observations indicate more complex X-ray spectra with multiple absorption lines. Pulse-phase spectroscopy of the best studied pulsars RX J0720.4-3125 and RBS 1223 reveals variations in derived emission temperature and absorption line depth with pulse phase. Moreover, RX J0720.4-3125 shows long-term spectral changes which are interpreted as due to free precession of the neutron star. Modeling of the pulse profiles of RX J0720.4-3125 and RBS 1223 provides information about the surface temperature distribution of the neutron stars indicating hot polar caps which have different temperatures, different sizes and are probably not located in antipodal positions.

X-ray observations of ultraluminous X-ray sources

Abstract: Ultraluminous X-ray sources (ULXs) are amongst the most intriguing of X-ray source classes. Their extreme luminosities—greater than 1039 erg s−1 in the 0.3–10 keV band alone—suggest either the presence of black holes larger than those regularly encountered in our own Galaxy (the Galactic centre excepted), or sources apparently radiating well above the Eddington limit. We review the insights afforded us by studies of their X-ray emission, focussing on what this reveals about the underlying compact object. In particular, we discuss recent deep observations of ULXs by the XMM-Newton observatory, and how the unprecedented data quality provided by this mission is starting to discriminate between the different physical models for these extraordinary X-ray emitters.

Fermi acceleration in astrophysical jets

Abstract: We consider the acceleration of energetic particles by Fermi processes (i.e., diffusive shock acceleration, second order Fermi acceleration, and gradual shear acceleration) in relativistic astrophysical jets, with particular attention given to recent progress in the field of viscous shear acceleration. We analyze the associated acceleration timescales and the resulting particle distributions, and discuss the relevance of these processes for the acceleration of charged particles in the jets of AGN, GRBs and microquasars, showing that multi-component powerlaw-type particle distributions are likely to occur.

Gamma rays from molecular clouds

Abstract: It is believed that the observed diffuse gamma-ray emission from the galactic plane is the result of interactions between cosmic rays and the interstellar gas. Such emission can be amplified if cosmic rays penetrate into dense molecular clouds. The propagation of cosmic rays inside a molecular cloud has been studied assuming an arbitrary energy and space dependent diffusion coefficient. If the diffusion coefficient inside the cloud is significantly smaller compared to the average one derived for the galactic disk, the observed gamma-ray spectrum appears harder than the cosmic ray spectrum, mainly due to the slower penetration of the low energy particles towards the core of the cloud. This may produce a great variety of gamma-ray spectra.

Isolated neutron stars: magnetic fields, distances,and spectra

Abstract: We present timing measurements, astrometry, and high-resolution spectra of a number of nearby, thermally emitting, isolated neutron stars. We use these to infer magnetic field strengths and distances, but also encounter a number of puzzles. We discuss three specific ones in detail: (i) For RX J0720.4-3125 and RX J1308.6+2127, the characteristic ages are in excess of 1 Myr, while their temperatures and kinematic ages indicate that they are much younger; (ii) For RX J1856.5-3754, the brightness temperature for the optical emission is in excess of that measured at X-ray wavelengths for reasonable neutron-star radii; (iii) For RX J0720.4-3125, the spectrum changed from an initially featureless state to one with an absorption feature, yet there was only a relatively small change in Teff. Furthermore, we attempt to see whether the spectra of all seven sources, in six of which absorption features have now been found, can be understood in the context of strongly magnetised hydrogen atmospheres. We find that the energies of the absorption features can be reproduced, but that it remains puzzling that, for J0720.4-3125 specifically, the spectrum was featureless in one state, and that, generally, the spectra do not have high-energy tails that are harder than the Wien-like ones obseved.

The blazar sequence: validity and predictions

Abstract: The “blazar sequence” posits that the most powerful BL Lacertae objects and flat-spectrum radio quasars should have relatively small synchrotron peak frequencies, ν peak, and that the least powerful such objects should have the highest ν peak values. This would have strong implications for our understanding of jet formation and physics and the possible detection of powerful, moderately high-redshift TeV blazars. I review the validity of the blazar sequence by using the results of very recent surveys and compare its detailed predictions against observational data. I find that the blazar sequence in its simplest form is ruled out. However, powerful flat-spectrum radio quasars appear not to reach the ν peak typical of BL Lacs. This could indeed be related to some sort of sequence, although it cannot be excluded that it is instead due to a selection effect.

Resonant Compton upscattering in anomalous X-ray pulsars

Abstract: A significant new development in the study of Anomalous X-ray Pulsars (AXPs) has been the recent discovery by INTEGRAL and RXTE of flat, hard X-ray components in three AXPs. These non-thermal spectral components differ dramatically from the steeper quasi-power-law tails seen in the classic X-ray band in these sources. A prime candidate mechanism for generating this new component is resonant, magnetic Compton upscattering. This process is very efficient in the strong magnetic fields present in AXPs. Here an introductory exploration of an inner magnetospheric model for upscattering of surface thermal X-rays in AXPs is offered, preparing the way for an investigation of whether such resonant upscattering can explain the 20–150 keV spectra seen by INTEGRAL. Characteristically flat emission spectra produced by non-thermal electrons injected in the emission region are computed using collision integrals. A relativistic QED scattering cross section is employed so that Klein–Nishina reductions are influential in determining the photon spectra and fluxes. Spectral results depend strongly on the magnetospheric locale of the scattering and the observer’s orientation, which couple directly to the angular distributions of photons sampled.

The trigonometric parallax of the neutron star Geminga

Abstract: We obtained a series of four observations of the isolated neutron star Geminga over an 18 month period using the Advanced Camera for Surveys (ACS) Wide Field Camera (WFC) on the Hubble Space Telescope in order to determine its trigonometric parallax. We find the parallax π=4.0±1.3 mas, corresponding to a distance to Geminga of 250 −62 +120 pc, a result 60% larger than the previously published value. The proper motion is 178.2±1.8 mas/year. In this paper, we describe the analysis techniques in detail since the amplitude of the parallactic shift is smaller than the camera’s pixel size. We fit each star in the images with an appropriate effective PSF and applied a distortion correction to generate stellar positions accurate to 0.01 pixels (∼0.5 mas). The 134 stars common to all images serve to establish a reference frame for alignment of the image series. Our observations were made around the times of maximum parallactic shift. We discuss the implications of this new distance measurement for the inferred radius of Geminga, and the neutron star equation of state.

Recent progress on anomalous X-ray pulsars

Abstract: I review recent observational progress on Anomalous X-ray Pulsars, with an emphasis on timing, variability, and spectra. Highlighted results include the recent timing and flux stabilization of the notoriously unstable AXP 1E 1048.1–5937, the remarkable glitches seen in two AXPs, and the newly recognized variety of AXP variability types, including outbursts, bursts, flares, and pulse profile changes. I also discuss recent discoveries regarding AXP spectra, including their surprising hard X-ray and far-infrared emission, as well as the pulsed radio emission seen in one source. Much has been learned about these enigmatic objects over the past few years, with the pace of discoveries remaining steady. However additional work on both observational and theoretical fronts is needed before we have a comprehensive understanding of AXPs and their place in the zoo of manifestations of young neutron stars.

Nonextensivity and the power-law distributions for the systems with self-gravitating long-range interactions

Abstract: By a natural nonextensive generalization of the conservation of energy in the q-kinetic theory, we study the nonextensivity and the power-law distributions for the many-body systems with the self-gravitating long-range interactions. It is shown that the power-law distributions describe the long-range nature of the interactions and the non-local correlations within the self-gravitating system with the inhomogeneous velocity dispersion. A relation is established between the nonextensive parameter q≠1 and the measurable quantities of the self-gravitating system: the velocity dispersion and the mass density. Correspondingly, the nonextensive parameter q can be uniquely determined from the microscopic dynamical equation and thus the physical interpretation of q different from unity can be clearly presented. We derive a nonlinear differential equation for the radial density dependence of the self-gravitating system with the inhomogeneous velocity dispersion, which can correctly describe the density distribution for the dark matter in the above physical situation. We also apply this q-kinetic approach to analyze the nonextensivity of self-gravitating collisionless systems and self-gravitating gaseous dynamical systems, giving the power-law distributions the clear physical meaning.

New phase-coherent measurements of pulsar braking indices

Abstract: Pulsar braking indices offer insight into the physics that underlies pulsar spin-down. Only five braking indices have been measured via phase-coherent timing; all measured values are less than 3, the value expected from magnetic dipole radiation. Here we present new measurements for three of the five pulsar braking indices, obtained with phase-coherent timing for PSRs J1846-0258 (n=2.65±0.01), B1509-58 (n=2.839±0.001) and B0540-69 (n=2.140±0.009). We discuss the implications of these results and possible physical explanations for them.

Anisotropic Bianchi type-I models with constant deceleration parameter in general relativity

Abstract: A special law of variation for Hubble’s parameter is presented in a spatially homogeneous and anisotropic Bianchi type-I space-time that yields a constant value of deceleration parameter. Using the law of variation for Hubble’s parameter, exact solutions of Einstein’s field equations are obtained for Bianchi-I space-time filled with perfect fluid in two different cases where the universe exhibits power-law and exponential expansion. It is found that the solutions are consistent with the recent observations of type Ia supernovae. A detailed study of physical and kinematical properties of the models is carried out.

Two decades of pulsar timing of Vela

Abstract: Pulsar timing at the Mt Pleasant observatory has focused on Vela, which can be tracked for 18 hours of the day. These nearly continuous timing records extend over 24 years allowing a greater insight into details of timing noise, micro glitches and other more exotic effects. In particular we report the glitch parameters of the 2004 event, along with the reconfirmation that the spin up for the Vela pulsar occurs instantaneously to the accuracy of the data. This places a lower limit of about 30 seconds for the acceleration of the pulsar to the new rotational frequency. We also confirm of the low braking index for Vela, and the continued fall in the DM for this pulsar.

Bridging the gap between stellar-mass black holes and ultraluminous X-ray sources

Abstract: The X-ray spectral and timing properties of ultraluminous X-ray sources (ULXs) have many similarities with the very high state of stellar-mass black holes (power-law dominated, at accretion rates greater than the Eddington rate). On the other hand, their cool disk components, large characteristic inner-disk radii and low characteristic timescales have been interpreted as evidence of black hole masses ∼1000 M ⊙ (intermediate-mass black holes). Here we re-examine the physical interpretation of the cool disk model, in the context of accretion states of stellar-mass black holes. In particular, XTE J1550–564 can be considered the missing link between ULXs and stellar-mass black holes, because it exhibits a high-accretion-rate, low-disk-temperature state (ultraluminous branch). On the ultraluminous branch, the accretion rate is positively correlated with the disk truncation radius and the bolometric disk luminosity, while it is anti-correlated with the peak temperature and the frequency of quasi-periodic-oscillations. Two prototypical ULXs (NGC 1313 X-1 and X-2) also seem to move along that branch. We use a phenomenological model to show how the different range of spectral and timing parameters found in the two classes of accreting black holes depends on both their masses and accretion rates. We suggest that ULXs are consistent with black hole masses ∼50–100 M ⊙, moderately inefficiently accreting at ≈20 times Eddington.

Suprathermal particle distributions in space physics: Kappa distributions and entropy

Abstract: The energization of a charged test-particle of mass m in contact with a large ensemble of charged particles of mass M at equilibrium is studied with the Fokker-Planck equation for Coulomb collisions and a quasi-linear diffusion operator for wave-particle interactions. The features of the nonequilibrium steady state velocity distribution of the test-particle system is studied as a function of the mass ratio m/M, and the relative strengths of the wave-particle interactions and Coulomb collisions. It is shown that the steady distribution function is not necessarily a Kappa distribution. The temperature of heavy minor ions given by the model is shown to vary linearly with the mass ratio as observed in the solar wind. The time evolution of the distribution function with and without the energization by wave-particle interactions is calculated and it is demonstrated that the Kullback relative entropy rather than the Tsallis nonextensive entropy rationalizes the results obtained.

The anatomy of a magnetar: XMM monitoring of the transient anomalous X-ray pulsar XTE J1810-197

Abstract: We present the latest results from a multi-epoch timing and spectral study of the Transient Anomalous X-ray Pulsar XTE J1810–197. We have acquired seven observations of this pulsar with the Newton X-ray Multi-mirror Mission (XMM-Newton) over the course of two and a half years, to follow the spectral evolution as the source fades from outburst. The spectrum is arguably best characterized by a two-temperature blackbody whose luminosities are decreasing exponentially with τ1=870 d and τ2=280 d, respectively. The temperatures of these components are currently cooling at a rate of 22% per year from a nearly constant value recorded at earlier epochs of kT1=0.25 keV and kT2=0.67 keV, respectively. The new data show that the temperature T1 and luminosity of that component have nearly returned to their historic quiescent levels and that its pulsed fraction, which has steadily decreased with time, is now consistent with the previous lack of detected pulsations in quiescence. We also summarize the detections of radio emission from XTE J1810–197, the first confirmed for any AXP. We consider possible models for the emission geometry and mechanisms of XTE J1810–197.

Studying millisecond pulsars in X-rays

Abstract: Millisecond pulsars represent an evolutionarily distinct group among rotation-powered pulsars. Outside the radio band, the soft X-ray range (∼0.1–10 keV) is most suitable for studying radiative mechanisms operating in these fascinating objects. X-ray observations revealed diverse properties of emission from millisecond pulsars. For the most of them, the bulk of radiation is of a thermal origin, emitted from small spots (polar caps) on the neutron star surface heated by relativistic particles produced in pulsar acceleration zones. On the other hand, a few other very fast rotating pulsars exhibit almost pure nonthermal emission generated, most probably, in pulsar magnetospheres. There are also examples of nonthermal emission detected from X-ray nebulae powered by millisecond pulsars, as well as from pulsar winds shocked in binary systems with millisecond pulsars as companions. These and other most important results obtained from X-ray observations of millisecond pulsars are reviewed in this paper, as well as results from the search for millisecond pulsations in X-ray flux of the radio-quite neutron star RX J1856.5-3754.

Variable modified Chaplygin gas and accelerating universe

Abstract: In this letter, I have proposed a model of variable modified Chaplygin gas and shown its role in accelerating phase of the universe. I have shown that the equation of state of this model is valid from the radiation era to quiessence model. The graphical representations of statefinder parameters characterize different phase of evolution of the universe. All results presented in the letter concerns the case k=0.

Long gamma-ray burst progenitors: boundary conditions and binary models

Abstract: The observed association of Long Gamma-Ray Bursts (LGRBs) with peculiar Type Ic supernovae gives support to Woosley‘s collapsar/hypernova model, in which the GRB is produced by the collapse of the rapidly rotating core of a massive star to a black hole. The association of LGRBs with small star-forming galaxies suggests low-metallicity to be a condition for a massive star to evolve to the collapsar stage. Both completely-mixed single star models and binary star models are possible. In binary models the progenitor of the GRB is a massive helium star with a close companion. We find that tidal synchronization during core-helium burning is reached on a short timescale (less than a few millennia). However, the strong core-envelope coupling in the subsequent evolutionary stages is likely to rule out helium stars with main-sequence companions as progenitors of hypernovae/GRBs. On the other hand, helium stars in close binaries with a neutron-star or black-hole companion can, despite the strong core-envelope coupling in the post-helium burning phase, retain sufficient core angular momentum to produce a hypernova/GRB.

Gamma rays from halos around stars and the Sun

Abstract: Inverse Compton (IC) scattering by relativistic electrons produces a major component of the diffuse emission from the Galaxy. The photon fields involved are the cosmic microwave background and the interstellar radiation field (ISRF) from stars and dust. Calculations of the inverse Compton distribution have usually assumed a smooth ISRF, but in fact a large part of the Galactic luminosity comes from the most luminous stars, which are rare. Therefore we expect the ISRF, and hence the inverse Compton emission, to be clumpy at some level, which could be detectable by instruments such as GLAST. Even individual nearby luminous stars could be detectable assuming just the normal cosmic-ray electron spectrum. We present the basic formalism required and give possible candidate stars to be detected and make predictions for GLAST. Then we apply the formalism to the OB associations and the Sun, showing that the IC emission produced is not negligible compared to the sensitivity of current or coming detectors. We estimate that the gamma-ray flux from the halo around the Sun contributes to the diffuse background emission at the few percent level.

Five dimensional string cosmological models in a scalar-tensor theory of gravitation

Abstract: A five dimensional FRW cosmological space-time is considered in the scalar-tensor theory of gravitation proposed by Saez and Ballester (Phys. Lett. A 113:467, 2003) in the presence of a perfect fluid source. Cosmological models corresponding to stiff fluid, disordered radiation, dust and false vacuum are obtained. Some physical and kinematical properties of each of the models are also studied.

The effect of magnetic fields on the formation of circumstellar discs around young stars

Abstract: We present first results of our simulations of magnetic fields in the formation of single and binary stars using a recently developed method for incorporating Magnetohydrodynamics (MHD) into the Smoothed Particle Hydrodynamics (SPH) method. An overview of the method is presented before discussing the effect of magnetic fields on the formation of circumstellar discs around young stars. We find that the presence of magnetic fields during the disc formation process can lead to significantly smaller and less massive discs which are much less prone to gravitational instability. Similarly in the case of binary star formation we find that magnetic fields, overall, suppress fragmentation. However these effects are found to be largely driven by magnetic pressure. The relative importance of magnetic tension is dependent on the orientation of the field with respect to the rotation axis, but can, with the right orientation, lead to a dilution of the magnetic pressure-driven suppression of fragmentation.

Bianchi type-II space-times with constant deceleration parameter in self creation cosmology

Abstract: The properties of locally rotationally symmetric Bianchi type-II perfect fluid space-times are analyzed in Barber’s second self-creation theory by using a special law of variation for Hubble’s parameter that yields a constant value of deceleration parameter. By assuming the equation of state p=γ ρ, many new solutions are obtained for different era—Zel’dovich, radiation, vacuum and vacuum energy dominated. The solutions with power-law and exponential expansion are discussed. A detailed study of geometrical and physical parameters is carried out. The nature of singularity is also clarified in each case.

The complex X-ray spectrum of the isolated neutron star RBS1223

Abstract: We present a first analysis of a deep X-ray spectrum of the isolated neutron star RBS1223 obtained with XMM-Newton. Spectral data from four new monitoring observations in 2005/2006 were combined with archival observations obtained in 2003 and 2004 to form a spin-phase averaged spectrum containing 290 000 EPIC-pn photons. This spectrum shows higher complexity than its predecessors, and can be parameterised with two Gaussian absorption lines superimposed on a blackbody. The line centers, E2≃2E1, could be regarded as supporting the cyclotron interpretation of the absorption features in a field B∼4×1013 G. The flux ratio of those lines does not support this interpretation. Hence, either feature might be of truly atomic origin.

Synchrotron emission in the fast cooling regime: which spectra can be explained?

Abstract: We consider the synchrotron emission from relativistic shocks assuming that the radiating electrons cool rapidly (either through synchrotron or any other radiation mechanism). It is shown that the theory of synchrotron emission in the fast cooling regime can account for a wide range of spectral shapes. In particular, the magnetic field, which decays behind the shock front, brings enough flexibility to the theory to explain the majority of gamma-ray burst spectra even in the parameter-free fast cooling regime. Also, we discuss whether location of the peak in observed spectral energy distributions of gamma-ray bursts and active galactic nuclei can be made consistent with predictions of diffusive shock acceleration theory, and find that the answer is negative. This result is a strong indication that a particle injection mechanism, other than the standard shock acceleration, works in relativistic shocks.

Nonlinear stability in the generalised photogravitational restricted three body problem with Poynting-Robertson drag

Abstract: The nonlinear stability of triangular equilibrium points has been discussed in the generalised photogravitational restricted three body problem with Poynting-Robertson drag. The problem is generalised in the sense that smaller primary is supposed to be an oblate spheroid. The bigger primary is considered as radiating. We have performed first and second order normalization of the Hamiltonian of the problem. We have applied KAM theorem to examine the condition of non-linear stability. We have found three critical mass ratios. Finally we conclude that triangular points are stable in the nonlinear sense except three critical mass ratios at which KAM theorem fails.