A new class of relativistic stellar models

Einstein field equations for a static and spherically symmetric perfect fluid are considered. A formulation given by Patiño and Rago is used to obtain a class of nine solutions, two of them are Tolman solutions I, IV and the remaining seven are new. The solutions are the correct ones corresponding to expressions derived by Patiño and Rago which have been shown by Knutsen to be incorrect. Similar to Tolman solution IV each of the new solutions satisfies energy conditions inside a sphere in some range of two independent parameters. Besides, each solution could be matched to the exterior Schwarzschild solution at a boundary where the pressure vanishes and thus the solutions constitute a class of new physically reasonable stellar models.     

General vacuum-universe solutions in Brans-Dicke cosmology

By a rescalation of the scalar field ? of the Jordan-Brans and Dicke cosmology, the general solutions of the Friedmannian ‘vacuum’ Universe are obtained. Only the flat space solution was previously known. Each solution is caracterized by the sign of the second time derivative of the rescaled field ψ≡?R ³ (R being the scale factor of the Robertson-Walker line-element): \(\ddot \psi\) = 0 (flat space), \(\ddot \psi\) < 0 (closed space), and \(\ddot \psi\) > 0 (open space), so that the solutions are mutually exclusive. Of these, the open space one is damped-oscillatory andR attains its absolute minimum, equal to zero, in only one of the two ‘extreme’ cycles. Otherwise,R min remains positive. If the ?-field is dominant near the singularity, these solutions may have physical significance. Also obtained, by the method mentioned above, is the general flat space solution for a ‘dust’ Universe and from it a closed space ‘dust’ solution. Both were found before by different authors, each one using a different method and, therefore, seemed up to now unrelated.     

Comparison of the classical and the global solutions of the Ideal Resonance Problem

The Ideal Resonance Problem is defined by the Hamiltonian $$F = B(y) + 2\varepsilon A(y) \sin ^2 x,\varepsilon \ll 1.$$ The classical solution of the Problem, expanded in powers of ε, carries the derivativeB′ as a divisor and is, therefore, singular at the zero ofB′, associated with resonance. With α denoting theresonance parameter, defined by $$\alpha \equiv - B'/|4AB''|^{1/2} \mu ,\mu = \varepsilon ^{1/2} ,$$ it is shown here that the classical solution is valid only for $$\alpha ^2 \geqslant 0(1/\mu ).$$ In contrast, the global solution (Garfinkelet al., 1971), expanded in powers ofμ=ε^1/2, removes the classical singularity atB′=0, and is valid for all α. It is also shown here that the classical solution is an asymptotic approximation, for largeα ², of the global solution expanded in powers ofα ^?2. This result leads to simplified expressions for resonancewidth and resonantamplification. The two solutions are compared with regard to their general behavior and their accuracy. It is noted that the global solution represents a perturbed simple pendulum, while the classical solution is the limiting case of a pendulum in a state offast circulation.     

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

In this paper, we consider the elliptic collinear solutions of the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler–Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler–Moulton collinear solution of n-bodies splits into \((n-1)\) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2-body problem at Kepler elliptic orbit, and each of the other \((n-2)\) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martínez et al. on 3-body Euler solutions in 2004–2006. As an example, we carry out the detailed derivation of the linear stability for an elliptic Euler–Moulton solution of the 4-body problem with two small masses in the middle.     

An extended Ideal Resonance Problem

An Extended Resonance Problem is defined by the Hamiltonian, $$F = B(y) + 2\mu ^2 A(y)[\sin x + \lambda (y)]^2 \mu<< 1,\lambda = O(\mu ).$$ It is noted here that the phase-plane trajectories exhibit adouble libration, enclosing two centers, for the initial conditions of motion satisfying the inequality $$1 - |\lambda |< |\alpha |< 1 + |\lambda |,$$ where α is the usualresonance parameter. A first order solution for the case of double libration is constructed here by a generalization of the procedure previously used in solving the Ideal Resonance Problem with λ=0. The solution furnishes a reference orbit for a Perturbed Ideal Problem if a double libration occurs as a result of perturbations.     

Solving Kepler’s equation via Smale’s \alpha -theory

We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler’s equation $E-e\sin (E)=M$ for any $e\in [0,1)$ and $M\in [0,\pi ]$ . Our solution is guaranteed, via Smale’s $\alpha $ -theory, to converge to the actual solution $E$ through Newton’s method at quadratic speed, i.e. the $n$ -th iteration produces a value $E_n$ such that $|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|$ . The formula provided for $\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$ , where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $[0,1)\times [0,\pi ]$ if only rational functions are allowed in each branch.     

Gravitational radiation from approaching neutron stars and the rotational explosion mechanism for collapsing supernovae

We consider the evolution of a neutron star binary system under the effect of two factors: gravitational radiation and mass transfer between the components. Gravitational radiation is specified under the justified assumption of a circular orbit and point masses and in the approximation of a weak gravitational field at nonrelativistic velocities of the binary components. During the first evolutionary phase determined only by gravitational radiation, the neutron stars approach each other according to a simple analytical solution. The second evolutionary phase begins at the time of Roche-lobe filling by the low-mass component, when the second factor, mass transfer as a result of mass loss by the latter, also begins to affect the evolution. Under the simplest assumptions of conservative mass transfer and exact equality between the Roche-lobe radius and the radius of the low-mass neutron star, it is still possible to extend the analytical solution of the problem of evolution to its second phase. We present this complete solution at both phases and, in particular, give theoretical light curves of gravitational radiation that depend only on two dimensionless parameters (m _t and δ ₀). Based on the solution found, we analyze the theoretical gravitational signals from SN 1987A; this analysis includes the hypothesis about the rotational explosion mechanism for collapsing supernovae.     

Cosmological electrovac model in Bertotti-Robinson space-time

Cosmological electrovac field equations are studied in Bertotti-Robinson-type space-time, and a class of cosmological solutions is obtained. The nature of the electromagnetic fields and singularities of the solution is studied. A technique is established to generate these solutions from a known vacuum solution with a non-zero cosmological constant.     

Spherical magnetic vortex in a uniform gravity field: A new exact solution and its applications for modeling solar flares and coronal spiders

The well-known Chandrasekhar-Prendergastmagnetostatic solution for a sphericalmagnetic vortex with axial symmetry squeezed externally by a potential magnetic field is of considerable interest both for describing the magnetic field of a star as a whole and for modeling solar active phenomena (flares, coronal spiders, etc.). This solution is generalized to the case of a uniform gravity field. In contrast to the Chandrasekhar-Prendergast model, the dependence of the plasma density in the spherical vortex on magnetic flux appears in the new solution. This expands considerably the class of magnetoplasma equilibria being analyzed and allows new scenarios for energy release in solar flares and coronal spiders to be proposed.     

Orbites periodiques de satellites

In a previous paper (Stellmacher, 1981, hereafter mentioned as Paper I), we have given an algorithm for the construction of periodic orbits in a rotating frame, for satellites around an oblate planet. In the present paper, we apply this theory to the Mimas-Tethys case; we obtain the following results:

1. Without resonance, it is possible to find a rotating system in which the solution is a periodic one. The angular velocity of this rotating frame is calculated as function of the masses of the two satellites.
2. Including the resonant terms and assuming an exact commensurability of the implied frequencies, we demonstrate that the condition for periodic solutions in the rotating system as defined in (a) is: the initial position of the satellites at conjunction lies on an axis defined by (Ω₁+Ω₂)/2 or (Ω₁+Ω₂)/2 + π/2;Ω₁ and Ω₂ are the longitudes of the ascending nodes of the satellite's orbits. The solution still is a periodic one, thus all the conjunction occur in either axis.
3. In the Mimas Tethys case there is only approximately commensurability between these frequencies. The two satellites are considered as oscillators whose amplitudes and phases are functions of time. The equation of the libration can be established; we find the usual form, but for each satellite the generating solution is a periodic solution (as defined in Paper I), but not a Keplerian one. It follows a determination of the masses which slightly differs from that given by Kozai (1957), when the same values of the observed quantities are used for calculations.
4. The equation of the libration is: $$\ddot z + n_1^2 h^2 \sin z + n_1 q\dot z\sin z = 0$$

     

Free time minimizers for the three-body problem

Free time minimizers of the action (called “semi-static” solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120–131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton–Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton’s three-body problem which is asymptotic to Lagrange’s parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange’s solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209–227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler’s central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.     

Bianchi type-II Bulk viscous string cosmological model in self-creation theory of gravitation

A spatially homogeneous and anisotropic LRS Bianchi type-II space-time is considered in the frame work of second self-creation theory of gravitation proposed by Barber (Gen. Relativ. Gravit. 14:117, 1982) in the presence of bulk viscous fluid containing one dimensional cosmic strings. A determinate solution of the field equations is presented using special variation for Hubble’s parameter given by Berman (Nuovo Cimento B 74:182, 1983) and some physically plausible conditions. The solution represents a bulk viscous string model in the second self-creation cosmology. We have also discussed some physical and kinematical properties of the model.     

A second-order solution of the Ideal Resonance Problem by Lie series

A second-order libration solution of theIdeal Resonance Problem is construeted using a Lie-series perturbation technique. The Ideal Resonance Problem is characterized by the equations $$\begin{gathered} - F = B(x) + 2\mu ^2 A(x)sin^2 y, \hfill \\ \dot x = - Fy,\dot y = Fx, \hfill \\ \end{gathered} $$ together with the property thatB _x vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initial-value problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished.     

Gravitational energy, momentum and angular momentum of Schwarzschild Anti-de Sitter space-times in the teleparallel equivalent of general relativity

We give a class of spherically symmetric-Anti de Sitter (Ads), exact solution in the teleparallel equivalent of general relativity (TEGR). The solution depends on an arbitrary function F(R)\mathcal{F}(R) and reproduce the metric of Schwarzschild Ads space-time. In the context of the Hamiltonian formulation of the TEGR we compute the gravitational energy of this class. The calculation is carried out by means of an expression for the energy of the gravitational field that naturally arises from the integral form of the constraint equations of the formalism. We show that the form of the energy depends on the arbitrary function. We make a constrain on this arbitrary function to give the correct form of energy.     

Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach

This paper presents a new theory of the dynamical tides of celestial bodies. It is founded on a Newtonian creep instead of the classical delaying approach of the standard viscoelastic theories and the results of the theory derive mainly from the solution of a non-homogeneous ordinary differential equation. Lags appear in the solution but as quantities determined from the solution of the equation and are not arbitrary external quantities plugged in an elastic model. The resulting lags of the tide components are increasing functions of their frequencies (as in Darwin’s theory), but not small quantities. The amplitudes of the tide components depend on the viscosity of the body and on their frequencies; they are not constants. The resulting stationary rotations (often called pseudo-synchronous) have an excess velocity roughly proportional to $6ne^2/(\chi ^2+\chi ^{-2})$ ( $\chi $ is the mean-motion in units of one critical frequency—the relaxation factor—inversely proportional to the viscosity) instead of the exact $6ne^2$ of standard theories. The dissipation in the pseudo-synchronous solution is inversely proportional to $(\chi +\chi ^{-1})$ ; thus, in the inviscid limit, it is roughly proportional to the frequency (as in standard theories), but that behavior is inverted when the viscosity is high and the tide frequency larger than the critical frequency. For free rotating bodies, the dissipation is given by the same law, but now $\chi $ is the frequency of the semi-diurnal tide in units of the critical frequency. This approach fails, however, to reproduce the actual tidal lags on Earth. In this case, to reconcile theory and observations, we need to assume the existence of an elastic tide superposed to the creeping tide. The theory is applied to several Solar System and extrasolar bodies and currently available data are used to estimate the relaxation factor $\gamma $ (i.e. the critical frequency) of these bodies.     

Action-minimizing solutions of the one-dimensional <Emphasis Type="Italic">N</Emphasis>-body problem

We supplement the following result of C. Marchal on the Newtonian N-body problem: A path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution when the dimension d of the physical space \({\mathbb {R}}^d\) satisfies \(d\ge 2\). The focus of this paper is on the fixed-ends problem for the one-dimensional Newtonian N-body problem. We prove that a path minimizing the action functional in the set of paths joining two given configurations and having all the time the same order is always a true (collision-free) solution. Considering the one-dimensional N-body problem with equal masses, we prove that (i) collision instants are isolated for a path minimizing the action functional between two given configurations, (ii) if the particles at two endpoints have the same order, then the path minimizing the action functional is always a true (collision-free) solution and (iii) when the particles at two endpoints have different order, although there must be collisions for any path, we can prove that there are at most \(N! - 1\) collisions for any action-minimizing path.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Eddington's method

An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz., $$B_v \left( T \right) = b_0 + b_1 e^{ - \beta \tau } $$ is obtained by Eddington's method. is obtained by Eddington's method.     

Einstein–Maxwell Field Equation in Isotropic Coordinates: An Application to Modeling Superdense Star

We present a charged analogue of Pant et al. (2010, Astrophys. Space Sci., 330, 353) solution of the general relativistic field equations in isotropic coordinates by using simple form of electric intensity E that involve charge parameter K. Our solution is well behaved in all respects for all values of X lying in the range 0 <X≤ 0.11, K lying in the range 4 <K≤ 6.2 and Schwarzschild compactness parameter u lying in the range 0 <u≤ 0.247. Since our solution is well behaved for wide ranges of the parameters, we can model many different types of ultra-cold compact stars like quark stars and neutron stars. We have shown that corresponding to X = 0.077 and K = 6.13 for which u = 0.2051 and by assuming surface density ρ _b =4.6888×10¹⁴ g cm ^?3 the mass and radius are found to be 1.509M _⊙, 10.906 km respectively which match with the observed values of mass 1.51M _⊙ and radius 10.90 km of the quark star XTE J1739-217. The well behaved class of relativistic stellar models obtained in this work might have astrophysical significance in the study of more realistic internal structures of compact stars.