Rotational distortion of stars of arbitrary structure to fourth-order sectorial harmonics

An explicit form of the radial part of the exterior gravitational potential of a planet or star, in hydrostatic equilibrium is expanded in terms of fourth-order sectorial harmonics. In developing the theory of figure one seeks to express the Clairaut equation for an equipotential spheroid (level surface) in an explicit form which is an integral form to quantities of fourth-order sectorial harmonics.     

Rotational distortion of stars having arbitrary structure described by fourth-order sectorial harmonics

In a series of papers, the equilibrium configurations of highly rotating fluid bodies have been derived. The deformation of these inhomogeneous self-gravitating fluid, of arbitrary internal structure are due to centrifugation potential. These level surfaces are expressed in terms of fourth-order sectorial harmonics.In this paper, the main equations of the problem — such as the surface of the distorted body, the gravitational potential at an arbitrary point and the disturbing potential — have been expanded to the fourth-order in terms of the even-order sectorial harmonics.This work will hereafter be referred to as Paper I.     

Rotational distortion of polytropes by analytic approximation

This paper modifies the first-order perturbation theory of Chandrasekhar, for rotational distortion of polytropes. Comparison with numerical integrations by other authors demonstrates that the present analytic theory is as accurate as other published first-order theories. The present theory is in a form permitting rapid calculation of boundary shapes as a function of the rotation parameter,v, and the polytrope index,n. Results are presented for a critically rotating polytrope, for the casen=3.     

Rotational distortion of polytropes by analytic approximation,III

Paper II of this series was less rigorous than desirable, particularly in the fitting of internal and external potentials. A method is developed in this paper properly to fit the external and internal potentials on the distorted configuration boundary. Displacement of the fiting radius from the radius of a nonrotating configuration in general changes the distorted boundary location by an insignificant amount.A comparison of this theory with the numerical integrations of James (1964) shows excellent agreement to at least an angular velocity equalling 70% of critical rotation. The comparison is for polytropic indices of 1.0, 1.5, 2.0 and 3.0.     

Rotational distortion of polytropes by analytic approximation,II

Chandrasekhar's (1933) paper on rotational distortion of polytropes contained a perturbation term in the potential which was linear inv, the rotation parameter. The same paper, and subsequent papers by various authors, developed an analytic expression for the boundary also linear inv. The latter expression is equivalent to a two term Taylor series about the unperturbed boundary, and is in error by 12% near critical rotation, for a polytropic index 3.0. The boundary can be located directly from the functions representing density, potential, and the potential gradient. The boundary error by this procedure is 0.2% near critical rotation.     

On secular stability of rapidly rotating stars of arbitrary structure

The aim of the present paper will be to utilize Poincaré's criteria to investigate secular stability of self-gravitating configurations, of arbitrary structure, in the state of rapid rotation. The potential energy, a knowledge of which is necessary for application of these criteria, will be determined by an extension of Clairaut's method; and its evaluation in terms of suitably chosen generalized coordinates carried out explicitly to quantities of fourth order in superficial oblateness, for configurations of arbitrary internal structure.The method employed can, moreover, clearly be extended to attain accuracy of any order — at the expense of mere manipulative work which lends itself to machine automation; and the angular velocity of axial rotation can be an arbitrary function of position as well as of the time. An application of our results to homogeneous configurations in rigig-body rotation will be undertaken to demonstrate that our method, when applied to a case for which a closed solution exists, leads to results which are consistent with it.     

On secular stability of tidally distorted stars of arbitrary structure

The aim of the present paper will be to develop a theory which should make it possible to investigate secular stability of close binary systems, consisting of tidally-distorted components of arbitrary internal structure, by a minimization of the potential energy of the system as a whole. In the second section which follows brief introductory remarks, appropriate expressions for the total potential energy of a close binary will be formulated. Section 3 will be concerned mainly with the nature of the tide-generating potential, and its effects on the shape of each star. In Section 4, the amplitudes of partial tides raised by this potential will be specified, for stars of arbitrary structure, correctly to terms of second order in superficial distortion; and in Section 5 we shall investigate the effects of interaction between rotation and tides to the same degree of approximation. The concluding Section 6 will then contain an explicit formulation of different constituents adding up to the total potential energy of the system, which can be used as a basis for its secular stability by the methods outlined already in our previous investigation (Kopal, 1973).     

Rotational evolution of solar-type stars

The aim of the present investigation has been to consider rotational evolution of solar-type stars simulated by a polytropic model that possesses differential rotation of Clement's type. A properly defined reduction factor moderates the effects of such a rotation. The present treatment is based upon the general Eulerian equation, governing nonuniform (i.e., nonrigid-body) rotation, which has been set up in a previous investigation. Nonconservative terms, arising when stellar wind torque is under consideration, are taken into account. Data available for the viscosity of the Sun are used to construct a plausible viscosity model. Certain assumptions are made that remove the mathematical difficulties and simplify the physical ground. The obtained results are compared to corresponding estimates of recent observations.     

Rotational parameters of strange stars in comparison with neutron stars

I study stellar structures, i.e. the mass, the radius, the moment of inertia and the oblateness parameter at different spin frequencies for strange stars and neutron stars in a comparative manner. I also calculate the values of the radii of the marginally stable orbits and Keplerian orbital frequencies. By equating kHz QPO frequencies to Keplerian orbital frequencies, I find corresponding orbital radii. Knowledge about these parameters might be useful in further modeling of the observed features from LMXBs with advanced and improved future techniques for observations and data analysis.     

Light variations of spotted stars: Second approximation

In the framework of the direct photometric problem we obtained an analytical representation of the light variations of spotted stars adopting quadratic limb-darkening law. The btained results can be a basis for the solution of the inverse photometric problem for spotted late FK Com-type giants.     

Rotational distortion on the surfaces of Jupiter and Saturn

Having formulated the Clairaut second-order differential equations up to the fourth order in superficial distortion due to Hensen's coefficients in the previous article (El-Sharawyet al., 1989 III, hereafter denotes by SM3), we are now in a position to solve them. In this paper we shall discuss the methods of solving the Clairaut theory, to give an explicit form about the distortion of the surfaces of Jupiter and Saturn, numerically up to the fourth-order.     

Axisymmetric modes of rotating relativistic stars in the Cowling approximation

Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core collapse, crust- and core-quakes or binary mergers, and could become detectable in either gravitational waves or high-energy radiation. Here, we present a comprehensive study of all low-order axisymmetric modes of uniformly and rapidly rotating relativistic stars. Initial stationary configurations are appropriately perturbed and are numerically evolved using an axisymmetric, non-linear relativistic hydrodynamics code, assuming time-independence of the gravitational field (Cowling approximation). The simulations are performed using a high-resolution shock-capturing finite-difference scheme accurate enough to maintain the initial rotation law for a large number of rotational periods, even for stars at the mass-shedding limit. Through Fourier transforms of the time evolution of selected fluid variables, we compute the frequencies of quasi-radial and non-radial modes with spherical harmonic indices l =0 , 1, 2 and 3, for a sequence of rotating stars from the non-rotating limit to the mass-shedding limit. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50 per cent of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear. In addition to the above modes, several axisymmetric inertial modes are also excited in our numerical evolutions.     

Rotational evolution of protoneutron stars with hyperons: spin up or not?

We study the evolution of a rigidly rotating protoneutron star (PNS) with hyperons and nucleons or solely nucleons in its core due to the escape of trapped neutrinos. As the neutrinos escape, the core nucleonic neutron star (NS) expands and the stellar rotation slows. After the neutrinos escape, the range of the spin periods is narrower than the initial one, but the distribution is still nearly uniform. A PNS with hyperons, at the late stage of its evolution, keeps shrinking and spinning up until all the trapped neutrinos escape. Consequently, the distribution of the stellar initial spin periods is skewed towards shorter periods. If the hyperonic star is metastable, its rotational frequency accelerates distinguishedly before it collapses to a black hole.     

On the internal structure of stars

A new model of the internal structure of certain types of celestial bodies is proposed. It is based on the concept that some neutron stars might have been formed earlier than all other type of stars, at an early stage of expansion of the universe, directly from continuous cosmic matter. Under such conditions, a neutron star after forming becomes an efficient center for the accretion of cosmic plasma. The plasma streams falling onto the neutron star carry magnetic fields with them that are created in the process (by thermoelectric currents and the dynamo process) and pack the fields tightly around the star. After a certain time, an extended and strongly magnetized plasma layer is formed around the neutron star. As a result, a stellar configuration is formed with an outer layer, mass, radius, and luminosity similar to those of an ordinary star. In the magnetized part of such a configuration, the gravitational attraction of the masses is compensated for by a magnetic pressure gradient, while the plasma is confifned by the magnetic field itself. Numerical estimates corroborate the possibility that magnetized stars exist. The radii and masses of the magnetized spheres of such stars are considerably less than the radii and masses of the corresponding configurations, so in observations they should not differ from ordinary stars: the outer layers (intermediate layer, photosphere, and chromosphere) of the magnetized configuration are the same as for an ordinary star. Structural differences may appear in the inner regions, however, involving magnetic activity and neutrino luminosity, for example.     

On the r-mode spectrum of relativistic stars in the low-frequency approximation

The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time-independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for     , it is nevertheless also possible to find discrete mode solutions (the r modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time-dependent equations. For stellar models admitting a physical r-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of l it seems that in certain cases there are no mode solutions at all.     

Presupernova structure of massive stars

Issues concerning the structure and evolution of core collapse progenitor stars, and stellar evolution in general, are discussed with an emphasis on interior evolution. We discuss some recent results that address quantifying the uncertainties inherent in modern stellar evolution calculations, and we describe a research effort aimed at investigating the transport and mixing processes associated with stellar turbulence, which is arguably the greatest source of uncertainty in supernova progenitor structure, besides mass loss, at the time of core collapse. We highlight the important role played by precision observations of stellar parameters in constraining theoretical models, as well as the physical insight that can be garnered from three-dimensional hydrodynamic simulation.     

Light variations of spotted stars in case of arbitrary limb-darkening law

In the framework of the direct photometric problem we obtained analytic expressions describing the light variations of spotted stars in case of arbitrary limb-darkening law. These could be used as a basis for solution of the inverse photometric problem for arbitrary types of rotating stars characterized by surface spots.