Resonances of low orders in the planetary system of HD37124

The full set of published radial velocity data (52 measurements from Keck + 58 ones from ELODIE + 17 ones from CORALIE) for the star HD37124 is analysed. Two families of dynamically stable high-eccentricity orbital solutions for the planetary system are found. In the first one, the outer planets c and d are trapped in the 2/1 mean-motion resonance. The second family of solutions corresponds to the 5/2 mean-motion resonance between these planets. In both families, the planets are locked in (or close to) an apsidal corotation resonance. In the case of the 2/1 MMR, it is an asymmetric apsidal corotation (with the difference between the longitudes of periastra Δω ~ 60°), whereas in the case of the 5/2 MMR it is a symmetric antialigned one (Δω = 180°). It remains also possible that the two outer planets are not trapped in an orbital resonance. Then their orbital eccentricities should be relatively small (less than, say, 0.15) and the ratio of their orbital periods is unlikely to exceed 2.3 − 2.5.     

First ground‐based photometry and light‐curve analysis of the eccentric eclipsing binary V744 Cas

The first ground‐based BVR photometric observations of the recently discovered eclipsing binary V744 Cas are presented. From these measurements, timings for two primary and one secondary minima have been calculated. The light curves of the system were analyzed by using the Wilson‐Devinney program. The analysis shows that the system is detached with two similar components of spectral type A2V, and the orbit is eccentric (e = 0.0662 ± 0.0005). The longitude of the periastron (ω) was found significantly different for two different light curves (ours and that of Hipparcos), which is strongly suggestive of an apsidal motion with a period of about 425 ± 68 yr. This makes V744 Cas an important candidate for studies of apsidal motions. The first estimate of the absolute dimensions place the system close to the terminal age of the main sequence (TAMS) in the HR diagram. The distance from the spectroscopic parallax (d = 740 ± 10 pc) was found to be slightly larger than the Hipparcos distance of d = 610 ± 400 pc. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of the darkforce on the extra-anomalous apsidal precession of solar planets

We demonstrate that, while the proposed Gravitational Dark-force Theory (of Nyambuya (New Astron. 67:1, 2019b) here-in Paper II) predicts an extra-anomalous apsidal precession for Solar planets due to the gravitational dark-force on the orbits of these planets, the predicted extra-anomalous apsidal precession is so small—so much that—it can not account for the observed extra-anomalous apsidal precession of Solar planets. This null result is important in that it informs us that whatever may be the cause of the extra-anomalous apsidal precession, it is not the proposed gravitational dark-force.     

Optimization analysis of the light curves of eccentric eclipsing binary systems

The iterative optimization procedures described in a previous paper by the author have been extended to cover unrectified light curves of eccentric eclipsing binary systems. Two stars have been studied in detail: namely, Coronae Borealis and V477 Cygni. In both cases improved element sets have been obtained. Moreover, in the case of V477 Cygni the obtained values of the longitude of periastron for three different light curves are found to provide a useful basis for discussion of the apsidal motion of this system.     

A Three Body Solution for the DI Her System

The DI Herculis system has been extensively studied over the past few decades because its observed rate of apsidal advance is less than a quarter of that which is expected from its physical and orbital properties. Work by Khaliullin et al. (1991) proposed that this slow rate of apsidal advance is a result of the presence of a third (stellar mass) body orbiting the system, however, observations by Guinan et al. (1994) severely restrict the orbital properties of such a solution. We show that a planetary mass object in a highly inclined orbit relative to the binary is capable of producing the observed apsidal motion, while remaining within the bounds of the most recent set of observations. A wide range of stable solutions are possible.     

Theory of satellite orbit-orbit resonance

On the basis of the strong mathematical and physical parallels between orbit-orbit and spin-orbit resonances, the dynamics of mutual orbit perturbations between two satellites about a massive planet are examined, exploiting an approach previously adopted in the study of spin-orbit coupling. The satellites are assumed to have arbitrary mass ratio and to move in non-intersecting orbits of arbitrary size and eccentricity. Resonances are found to exist when the mean orbital periods are commensurable with respect to some rotating axis, which condition also involves the apsidal and nodal motions of both satellites. In any resonant state the satellites are effectively trapped in separate potential wells, and a single variable is found to describe the simultaneous librations of both satellites. The librations in longitude are 180° out-of-phase, with fixed amplitude ratio that depends only on their relative masses and semimajor axes. At the same time the stroboscopic longitude of conjunction also librates about the commensurate axis with the same period. The theory is applicable to Saturn's resonant pairs Titan-Hyperion and Mimas-Tethys, and in these cases our calculated libration periods are in reasonably good agreement with the observed periods.This research supported under a grant from the California Institute of Technology President's Fund and NASA Contract NAS 7-100.     

Satellite resonance with longitude-dependent gravity—II Effects involving the eccentricity

For a satellite in a nominally circular orbit at arbitrary inclination whose mean motion is commensurable with the Earth's rotation, the dependence of gravity on longitude leads to a resonant variation in eccentricity as well as the long-period oscillation in longitude. Provided forces capable of processing perigee are present, it is shown that the change in eccentricity for a satellite captured in librational resonance is not secular but periodic.

There are corresponding resonance effects for a satellite in a nominally equatorial but eccentric orbit. Here the commensurability condition is that the longitudes of the apses shall be nearly repetitive relative to the rotating Earth. There will be a long-period oscillation in longitude which can take the form of either a libration (trapped) or a circulation (free), and there will also be an oscillation of the orbital plane having the same period as the precession of perigee relative to inertial space.     

Triple systems showing apsidal motion

Binary systems showing both apsidal motion and light travel time (LTT) effects which cause orbital period changes in close binaries were studied. 15 triple systems showing apsidal motion were found by searching the literature, and a table including the important parameters of these systems was constructed. Six of the systems given in this table were selected and observed photometrically. Existence of both apsidal motion and LTT effects in all selected systems was investigated by means of the analysis of their eclipse times. The mean observed internal structure constants,  log = k _2,obs  , and contributions to the apsidal motion from the theory of General Relativity and the third/fourth bodies were calculated. The masses of the third/fourth bodies and some characteristics of their orbits were also calculated.     

The Apsidal Antialignment of the HD 82943 System

We perform numerical simulations to explore the dynamical evolution of the HD 82943 planetary system. By simulating diverse planetary configurations, we find two mechanisms of stabilizing the system: the 2:1 mean motion resonance (MMR) between the two planets can act as the first mechanism for all stable orbits. The second mechanism is a dynamical antialignment of the apsidal lines of the orbiting planets, which implies that the difference of the periastron longitudes ₃ librates about 180° in the simulations. We also use a semi-analytical model to explain the numerical results for the system under study.     

Passing through resonance: The excitation and dissipation of the lunar free libration in longitude

The acceleration of the mean lunar longitude has a small effect on the periods of most terms in a Fourier expansion of the longitude. There are several planetary perturbation terms that have small amplitudes, but whose periods are close to the resonant period of the lunar libration in longitude. Some of these terms are moving toward resonance, some are moving away from resonance, and the periods of those terms that do not include the Delaunay variables in their arguments are not moving. Because of its acceleration of longitude, the Moon is receding from the Earth, so the magnitude of the restoring torque that the Earth exerts on the rotating Moon is gradually attenuating; thus resonance itself is moving, but at a much slower rate than the periods of the accelerating planetary perturbations. There are five planetary perturbation terms from the ELP-2000 Ephemeris (with amplitudes of 0.00001 or greater) that have passed through resonance in the past two million years. One of them is of special interest because it appears to be the excitation source of a supposed free libration in longitude that has been detected by the lunar laser ranging experiment. The amplitude of the term is only 0.00021 but it could be the source of the 1 amplitude free libration term if the viscoelastic properties of the Moon are similar to those of the Earth.     

Stationary Orbits in Resonant Extrasolar Planetary Systems

We present a catalog of stable and unstable apsidal corotation resonance (ACR) for the resonant planar planetary three-body problem, including both symmetric and asymmetric solutions. Calculations are performed with a new approach based on a numerical determination of the averaged Hamiltonian function. It has the advantage of being very simple to use and, with the exception of the immediate vicinity of the collision curve, yields precise results for any values of the eccentricities and semimajor axes. The present catalog includes results for the 3/2, 3/1, and 4/1 mean-motion resonances. The 5/1 and 5/2 commensurabilities are also discussed briefly. These results complement our previous results for the 2/1 (Beaugé et al. 2006, MNRAS, 365, 1160–1170), and give a broad picture of the structure of many important planetary resonances.     

An o-c diagram analysis of PV cassiopeiae

Basing on recent and old photoelectric times of minimum, a new slightly longer period of the apsidal motion for an eclipsing binary PV Cas has been obtained. A difference between observed and theoretical periods of the apsidal motion estimated on the base of Claret and Gimenez models growed up to 30 years is found.     

Apsidal Motion in Binaries: Rotation of the Components

A sample of 51 separated binary systems with measured apsidal periods and rotational velocities of the components is examined. The ranges of the angles of inclination of the equatorial planes of the components to the orbital plane are estimated for these systems. The observed apsidal velocities can be explained by assuming that the axes of rotation of the stars are nonorthogonal to the orbital plane in roughly 47% of the systems (24 of the 51) and the rotation of the components is not synchronized with the orbital motion in roughly 59% of the systems (30 of 51). Nonorthogonality and nonsynchrony are defined as deviations from 90° and a synchronized angular velocity, respectively, at levels of 1 or more.     

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto–Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain on the semimajor axis, eccentricity and inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericentre or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericentre or longitude of node.     

The zonal distribution of hydrogen in the Jovian atmosphere

The Voyager ultraviolet spectrometer disclosed strong longitude variation in the midlatitude Lyman alpha brightness of Jupiter. Minimum brightness of 16 and 14.4 kR were observed from Voyagers 1 and 2, respectively, with the intensity rising to peaks of 21 and 19.6 kR at a longitude near 110°. Observations of Jovian Lyman alpha, made with the International Ultraviolet Explorer (IUE) beginning in December 1978, and continuing through January 1982, also show a region of persistently enhanced but variable flux near a longitude, λ, of 100°; however, IUE measured brightnesses are consistently lower than those of Voyager. Although the Lyman alpha flux from the “normal” region of the plant between λ 200 and 300° remained nearly constant during the period of the IUE observations, that from the “perturbed” region centered on λ 110° varied by ±25% from the mean. The sources of Lyman alpha flux include resonance scattering of solar and interplanetary Lyman alpha, and excitation by charged particle precipitation. That portion of the dayside flux due to charged particle excitation has been variously estimated at between 2.3 and 7 kR. About 1 kR of the dayside flux is due to resonance scattering of the sky background. It is assumed that H and an absorber (CH₄) are distributed above the homopause according to the local height distribution of temperature. The daytime equation of radiative transfer is solved to determine the longitudinal distribution of freely scattering atomic hydrogen that would account for the observed flux. This daytime solution shows that if the hydrogen bulge is the result of localized heating and a consequent increase in scale height, the temperature in the perturbed region must be about 100°K warmer than that in the normal region. The nightside Lyman alpha brightness exhibits a longitude variation very similar to that on the dayside. The H distribution derived from the dayside solution is used with the nightside flux to estimate the longitude variation of particle precipitation on the nightside.     

A New Apsidal Motion Study of DI Her

B and V photometry of the eccentric eclipsing binary DI Her is reported, with the aim of contributing to a more accurate determination of the timings required for the ongoing apsidal motion studies. The resulting apsidal motion rate is close to the one reported earlier by Khodykin and Volkov (1989), Guinan, Marshal and Maloney (IBVS 4101), and Yildiz et al. (2000). This revised version was published online in July 2006 with corrections to the Cover Date.     

Latitudinal librations of Mercury with a fluid core

In the framework of the space missions to Mercury, an accurate model of rotation is needed. Librations around the 3:2 spin-orbit resonance as well as latitudinal librations have to be predicted with the best possible accuracy. In this paper, we use a Hamiltonian analysis and numerical integrations to study the librations of Mercury, both in longitude and latitude. Due to the proximity of the period of the free libration in longitude to the orbital period of Jupiter, the 88-day and 11.86-year contributions dominate Mercury’s libration in longitude (with the Hermean parameters chosen). The amplitude of the libration in latitude is much smaller (under 1 arcsec) and should not be detected by the space missions. Nevertheless, we point out that this amplitude could be much larger (up to several tens of arcsec) if the free period related to the libration in latitude approaches the period of the Jupiter-Saturn Great Inequality (883 years). Given the large uncertainties on the planetary parameters, this new resonant forcing on Mercury’s libration in latitude should be borne in mind.     

An analytical expression for apsidal superhump precession and comparisons with numerical simulations and dwarf nova observations

We compare analytical expressions of precession rates from apsidal (positive) superhumps in close binary systems with numerical disc simulation results and observed values. In the analytical expressions, we include both the dynamical effects on the precession of the disc and effects caused by pressure forces that have been theorized to provide a retrograde effect (i.e. slowing) on the prograde disc precession. We establish new limits on density wave pitch angle to a normalized disc sound speed 60≥Ω_orb  d  tan  i / c >2.214 . Using average values for the density wave pitch angle i and speed of sound c , we find good correlation between numerical simulations and the analytical expression for the apsidal superhump period excess, which includes both the prograde and retrograde effects, for mass ratios of 0.025≤ q ≤0.33 . We also show good correlations with the four known eclipsing systems, OY Car, Z Cha, HT Cas, and WZ Sge. Our analytical expression for apsidal superhump period excess as a function of orbital period is consistent with the trend found in observed systems.     

An extension of Newton's apsidal precession theorem

Newton's apsidal precession theorem in Proposition 45 of Book I of the 'Principia' has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ( N , e ), where N is the index for the centripetal force law, for varying eccentricity e . The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of   N = 2  which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives  ∂θ/∂ N   and  ∂θ/∂ h   [where h ( N , e ) is the angular momentum] are analytically expressed in the neighbourhood of   N = 2  (case of the inverse square law). The value of  ∂θ/∂ N   is computed numerically as well for  1 ≤ N < 3  . The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for   N = 2  , can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The   N = 1  case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high-eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.     

On the apsidal motion of BP Vulpeculae

BP Vulpeculae is a bright eclipsing binary system showing apsidal motion. It was found in an earlier study that it shows retrograde apsidal motion which contradicts theory. In this paper we present the first BV light curve of the system and its light curve solution as well as seven new times of the minima from the years 1959–1963. This way we could expanded the baseline of the investigation to five decades. Based on this longer baseline we concluded that the apsidal motion is prograde agreeing with the theoretical expectations and its period is about 365 years and the determined internal structure constant is close to the theoretically expected one.