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A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth. [ABSTRACT FROM AUTHOR]

The work presents a study of the 1:1 resonance case in a hierarchical quadruple stellar system of the 2+2 type. The resonance appears if orbital periods of both binaries are approximately equal. It is assumed that both periods are significantly shorter than the period of principal orbit of one binary with respect to the other. In these circumstances, the problem can be treated as three independent Kepler problems perturbed by mutual gravitational interactions. By means of canonical perturbation methods, the planar problem is reduced to a secular system with 1 degree of freedom involving a resonance angle (the difference of mean longitudes of the binaries) and its conjugate momentum (involving the ratio of orbital period in one binary to the period of principal orbit). The resonant model is supplemented with short periodic perturbations expressions, and verified by the comparison with numerical integration of the original equations of motion. Estimates of the binaries periods variations indicate that the effect is rather weak, but possibly detectible if it occurs in a moderately compact system. However, the analysis of resonance capture scenarios implies that the 1:1 resonance should be exceptional amongst the 2+2 quadruples. [ABSTRACT FROM AUTHOR]

The invariance of the Lagrangian under time translations and rotations in Kepler's problem yields the conservation laws related to the energy and angular momentum. Noether's theorem reveals that these same symmetries furnish generalized forms of the first integrals in a special non-conservative case, which approximates various physical models. The system is perturbed by a biparametric acceleration with components along the tangential and normal directions. A similarity transformation reduces the biparametric disturbance to a simpler uniparametric forcing along the velocity vector. The solvability conditions of this new problem are discussed, and closed-form solutions for the integrable cases are provided. Thanks to the conservation of a generalized energy, the orbits are classified as elliptic, parabolic, and hyperbolic. Keplerian orbits appear naturally as particular solutions to the problem. After characterizing the orbits independently, a unified form of the solution is built based on theWeierstrass elliptic functions. The new trajectories involve fundamental curves such as cardioids and logarithmic, sinusoidal, and Cotes' spirals. These orbits can represent the motion of particles perturbed by solar radiation pressure, of spacecraft with continuous-thrust propulsion, and some instances of Schwarzschild geodesics. Finally, the problem is connected with other known integrable systems in celestial mechanics. [ABSTRACT FROM AUTHOR]

We present results of our spectroscopic observations of nine detached eclipsing binaries (DEBs), selected from the Kepler Eclipsing Binary Catalog, that only show one set of spectral lines. Radial velocities (RVs) were calculated from the high-resolution spectra obtained with the HIgh-Dispersion Echelle Spectrograph (HIDES) instrument, attached to the 1.88-m telescope at the Okayama Astrophysical Observatory, and from the public Apache Point Observatory Galactic Evolution Experiment archive. In our sample, we found five single-lined binaries, with one component dominating the spectrum. The orbital and light-curve solutions were found for four of them, and compared with isochrones, in order to estimate absolute physical parameters and evolutionary status of the components. For the fifth case, we only update the orbital parameters, and estimate the properties of the unseen star. Two other systems show orbital motion with a period known from the eclipse timing variations (ETVs). For these we obtained parameters of outer orbits, by translating the ETVs to RVs of the centre of mass of the eclipsing binary, and combining with the RVs of the outer star. Of the two remaining ones, one is most likely a blend of a faint background DEB with a bright foreground star, which lines we see in the spectra, and the last case is possibly a quadruple bearing a sub-stellar mass object. Where possible, we compare our results with literature, especially with results from asteroseismology. We also report possible detections of solar-like oscillations in our RVs. [ABSTRACT FROM AUTHOR]

We apply one of the exactly symplectic integrators, which we call HB15, of Hernandez & Bertschinger, along with theKepler problem solver ofWisdom&Hernandez, to solve planetary system N-body problems. We compare the method to Wisdom-Holman (WH) methods in the MERCURY software package, the MERCURY switching integrator, and others and find HB15 to be the most efficient method or tied for the most efficient method in many cases. Unlike WH, HB15 solved N-body problems exhibiting close encounters with small, acceptable error, although frequent encounters slowed the code. Switching maps like MERCURY change between two methods and are not exactly symplectic. We carry out careful tests on their properties and suggest that they must be used with caution. We then use different integrators to solve a three-body problem consisting of a binary planet orbiting a star. For all tested tolerances and time steps, MERCURY unbinds the binary after 0 to 25 years. However, in the solutions of HB15, a time-symmetric HERMITE code, and a symplectic Yoshida method, the binary remains bound for >1000 years. The methods' solutions are qualitatively different, despite small errors in the first integrals in most cases. Several checks suggest that the qualitative binary behaviour of HB15's solution is correct. The Bulirsch-Stoer and Radau methods in the MERCURY package also unbind the binary before a time of 50 years, suggesting that this dynamical error is due to a MERCURY bug. [ABSTRACT FROM AUTHOR]

We investigate different amplitude scaling relations adopted for the asteroseismology of stars that show solar-like oscillations. Amplitudes are among the most challenging asteroseismic quantities to handle because of the large uncertainties that arise in measuring the background level in the star's power spectrum. We present results computed by means of a Bayesian inference on a sample of 1640 stars observed with Kepler, spanning from main sequence to red giant stars, for 12 models used for amplitude predictions and exploiting recently well-calibrated effective temperatures from Sloan Digital Sky Survey photometry. We test the candidate amplitude scaling relations by means of a Bayesian model comparison. We find the model having a separate dependence upon the mass of the stars to be largely the most favoured one. The differences among models and the differences seen in their free parameters from early to late phases of stellar evolution are also highlighted. [ABSTRACT FROM AUTHOR]