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.     

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.     

On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC) ⁿ Hermite scheme for planetary N -body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N -body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N -body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC) ⁿ Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)² Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N  = 100,  m  = 10²⁵ g planetesimal system with the rms eccentricity 〈 e ²〉^1/2 ≲0.03.     

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.     

Ringlet–satellite interactions

We study the interaction of a satellite and a nearby ringlet on eccentric and inclined orbits. Secular torques originate from mean motion resonances and the secular interaction potential which represents the m  = 1 global modes of the ring. The torques act on the relative eccentricity and inclination. The resonances damp the relative eccentricity. The inclination instability owing to the resonances is turned off by a finite differential eccentricity of the order of 0.27 for nearly coplanar systems. The secular potential torque damps the eccentricity and inclination and does not affect the relative semi-major axis; also, it suppresses the inclination instability that persists at small differential eccentricities. The damping of the relative eccentricity and inclination forces an initially circular and planar small mass ringlet to reach the eccentricity and inclination of the satellite. When the planet is oblate, the interaction of the satellite damps the proper precession of a small mass ringlet so that it precesses at the satellite's rate independently of their relative distance. The oblateness of the primary modifies the long-term eccentricity and inclination magnitudes and introduces a constant shift in the apsidal and nodal lines of the ringlet with respect to those of the satellite. These results are applied to Saturn's F-ring, which orbits between the moons Prometheus and Pandora.     

Dynamics of the boxy elliptical galaxy NGC 1600

We use three-integral models to infer the distribution function (DF) of the boxy E3–E4 galaxy NGC 1600 from surface brightness and line-profile data on the minor and major axes. We assume axisymmetry and that the mass-to-light ratio is constant in the central ∼1 R _e. Stars in the resulting gravitational potential move mainly on regular orbits. We use an approximate third integral K from perturbation theory and write the DF as a sum of basis functions in the three integrals E , L _z and K . We then fit the projected moments of these basis functions to the kinematic observables and deprojected density, using a non-parametric algorithm. The deduced dynamical structure is radially anisotropic, with σ _θ σ _r ≈ σ _φ σ _r ≈0.7 on the major axis. Both on the minor axis and near the centre the velocity distribution is more isotropic; thus the model is flattened by equatorial radial orbits. The kinematic data are fitted without the need for a central black hole; the central mass determined previously from ground-based data therefore overestimates the actual black-hole mass. The mass-to-light ratio of the stars is M L _V =6  h ₅₀. The anisotropy structure of NGC 1600 with a radially anisotropic main body and more nearly isotropic centre is similar to that found recently in NGC 1399, 2434, 3379 and 6703, suggesting that this pattern may be common amongst massive elliptical galaxies. We discuss a possible merger origin of NGC 1600 in the light of these results.     

An explanation for long flares from extragalactic globular cluster X-ray sources

We compare orbits in a thin axisymmetric disc potential in Modified Newtonian Dynamics (MOND) with those in a thin disc plus near-spherical dark matter halo predicted by a ΛCDM cosmology. Remarkably, the amount of orbital precession in MOND is nearly identical to that which occurs in a mildly oblate CDM Galactic halo (potential flattening   q = 0.9  ), consistent with recent constraints from the Sagittarius stream. Since very flattened mass distributions in MOND produce rounder potentials than in standard Newtonian mechanics, we show that it will be very difficult to use the tidal debris from streams to distinguish between a MOND galaxy and a standard CDM galaxy with a mildly oblate halo.
If a galaxy can be found with either a prolate halo or one that is more oblate than   q ∼ 0.9  this would rule out MOND as a viable theory. Improved data from the leading arm of the Sagittarius dwarf – which samples the Galactic potential at large radii – could rule out MOND if the orbital pole precession can be determined to an accuracy of the order of  ±1°  .     

Slow modes in stellar systems with nearly harmonic potentials: I. Spoke approximation,radial orbit instability

Using a consistent perturbation theory for collisionless disk-like and spherical star clusters, we construct a theory of slow modes for systems having an extended central region with a nearly harmonic potential due to the presence of a fairly homogeneous (on the scales of the stellar system) heavy, dynamically passive halo. In such systems, the stellar orbits are slowly precessing, centrally symmetric ellipses (2: 1 orbits). Depending on the density distribution in the system and the degree of halo inhomogeneity, the orbit precession can be both prograde and retrograde, in contrast to systems with 1: 1 elliptical orbits where the precession is unequivocally retrograde. In the first paper, we show that in the case where at least some of the orbits have a prograde precession and the stellar distribution function is a decreasing function of angular momentum, an instability that turns into the well-known radial orbit instability in the limit of low angular momenta can develop in the system. We also explore the question of whether the so-called spoke approximation, a simplified version of the slow mode approximation, is applicable for investigating the instability of stellar systems with highly elongated orbits. Highly elongated orbits in clusters with nonsingular gravitational potentials are known to be also slowly precessing 2: 1 ellipses. This explains the attempts to use the spoke approximation in finding the spectrum of slow modes with frequencies of the order of the orbit precession rate. We show that, in contrast to the previously accepted view, the dependence of the precession rate on angular momentum can differ significantly from a linear one even in a narrow range of variation of the distribution function in angular momentum. Nevertheless, using a proper precession curve in the spoke approximation allows us to partially “rehabilitate” the spoke approach, i.e., to correctly determine the instability growth rate, at least in the principal (O(α _T^−1/2) order of the perturbation theory in dimensionless small parameter α _T, which characterizes the width of the distribution function in angular momentum near radial orbits.     

Dynamic portrait of the planetary 2/1 mean-motion resonance – I. Systems with a more massive outer planet

We analyse the global structure of the phase space of the planar planetary 2/1 mean-motion resonance in cases where the outer planet is more massive than its inner companion. Inside the resonant domain, we show the existence of two families of periodic orbits, one associated to the librational motion of resonant angle (σ-family) and the other related to the circulatory motion of the difference in longitudes of pericentre (  Δϖ  -family). The well-known apsidal corotation resonances (ACR) appear as intersections between both families. A complex web of secondary resonances is also detected for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system.
The construction of dynamical maps for various values of the total angular momentum shows the evolution of the families of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems. For low–moderate eccentricities, several different stable modes exist outside the ACR. For larger eccentricities, however, all stable solutions are associated to oscillations around the stationary solutions.
Finally, we present a possible link between these stable families and the process of resonance capture, identifying the most probable routes from the secular region to the resonant domain, and discussing how the final resonant configuration may be affected by the extension of the chaotic layer around the resonance region.     

Optical spectroscopy of X-Mega targets – II. The massive double-lined O-type binary HD 93205

A new high-quality set of orbital parameters for the O-type spectroscopic binary HD 93205 has been obtained combining échelle and coudé CCD observations. The radial velocity orbits derived from the He  ii λ 4686 Å (primary component) and He  i λ 4471 Å (secondary component) absorption lines yield semi-amplitudes of 133±2 and 314±2 km s⁻¹ for each binary component, resulting in minimum masses of 31 and 13 M_⊙ ( q =0.42) . We also confirm for the binary components the spectral classification of O3 V+ O8 V previously assigned. Assuming for the O8 V component a 'normal' mass of 22–25 M_⊙ we would derive for the primary O3 V a mass of 'only' 52–60 M_⊙ and an inclination of about 55° for the orbital plane. We have also determined for the first time a period of apsidal motion for this system, namely 185±16 yr using all available radial velocity data sets of HD 93205 (from 1975 to 1999). Phase-locked variations of the X-ray emission of HD 93205 consisting of a rise of the observed X-ray flux near periastron passage are also discussed.     

Phase-space mixing and the merging of cusps

Collisionless stellar systems are driven towards equilibrium by mixing of phase-space elements. I show that the excess-mass function     [where     is the coarse-grained distribution function] always decreases on mixing . D ( f ) gives the excess mass from values of     . This novel form of the mixing theorem extends the maximum phase-space density argument to all values of f . The excess-mass function can be computed from N -body simulations and is additive: the excess mass of a combination of non-overlapping systems is the sum of their individual D ( f ). I propose a novel interpretation for the coarse-grained distribution function, which avoids conceptual problems with the mixing theorem.
As an example application, I show that for self-gravitating cusps (  ρ∝ r ^−γ  as   r → 0  ) the excess mass   D ∝ f ^{−2(3−γ)/(6−γ)}  as   f →∞  , i.e. steeper cusps are less mixed than shallower ones, independent of the shape of surfaces of constant density or details of the distribution function (e.g. anisotropy). This property, together with the additivity of D ( f ) and the mixing theorem, implies that a merger remnant cannot have a cusp steeper than the steepest of its progenitors. Furthermore, I argue that the cusp of the remnant should not be shallower either, implying that the steepest cusp always survives.     

Contribution to the search for binaries among Am stars – VII. Orbital elements of seven new spectroscopic binaries, implications on tidal effects

We present the results of a radial-velocity study of seven Am stars (HD 3970, 35035, 93946, 151746, 153286, 204751 and 224002) observed at the Observatoire de Haute-Provence (OHP) and the Cambridge Observatories with CORAVEL instruments. We find that these systems are single-lined spectroscopic binaries whose orbital elements are determined for the first time. Among this sample, HD 35035 and 153286 have long periods, with   P = 2.8  and 9.5 yr, respectively, which is rather unusual for Am stars. Four systems have orbits with large eccentricities (with   e ≥ 0.4  ). Physical parameters are inferred from this study for the primaries of those systems.
We then investigate the influence of tidal interaction, which has already led to the synchronism of the primaries and/or to the circularization of the orbits of some systems belonging to this sample. We extend this study to the list of 33 objects studied in this series of papers and derive values of the critical fractional radii   r = R / a   for circularization and synchronization of Am-type binaries. We find that the stars with   r ≳ 0.15  are orbiting on circular orbits and that synchronism is likely for all components with   r ≳ 0.20  .     

Some properties of a two-body system under the influence of the Galactic tidal field

In this paper the effect of the Galactic tidal field on a Sun–comet pair will be considered when the comet is situated in the Oort cloud and planetary perturbations can be neglected. First, two averaged models were created, one of which can be solved analytically in terms of Jacobi elliptic functions. In the latter system, switching between libration and circulation of the argument of perihelion is prohibited. The non-averaged equations of motion are integrated numerically in order to determine the regions of the ( e ,  i ) phase space in which chaotic orbits can be found, and an effort is made to explain why the chaotic orbits manifest in these regions only. It is evident that for moderate values of semimajor axis, a ∼50 000 au , chaotic orbits are found for ( e <0.15 , 40°≤ i ≤140°) as determined by integrating the evolution of the comet over a period of 10⁴ orbits. These regions of chaos increase in size with increasing semimajor axis. The typical e-folding times for these orbits range from around 600 Myr to 1 Gyr and thus are of little practical interest, as the time-scales for chaos arising from passing stars are much shorter. As a result of Galactic rotation, the chaotic regions in ( e ,  i ) phase space are not symmetric for prograde and retrograde orbits.     

Precession dynamics in spin-orbit coupling: A unified theory

Taking into account the entire spectrum of potential harmonics, equations are developed for the secular motions of the node and pericenter referred to the invariant plane in a two-body problem when one is a sphere and the other a spinning asymmetrical rigid body (e. g., Sun-planet system, or point satellite-planet system). Oblateness precession of satellite orbits and equinoctial precession of planets are shown to be merely opposite extreme cases of the single phenomenon of precession in spin-orbit coupling, in which the determining parameter is the ratio of orbital to spin angular momentum (h₀/h_s). The critical inclination for apsidal motion also depends on this ratio, varying from 63.43° in the one extreme when (h₀/h_s)1, to 90° in the opposite extreme when (h₀/h_s)1. Application is made to the Earth-Sun-Moon system.     

LO Pegasi: an investigation of multiband optical polarization

We present BVR polarimetric study of the cool active star LO Pegasi (LO Peg) for the first time. LO Peg was found to be highly polarized among the cool active stars. Our observations yield average values of polarization in LO Peg:   P_B = 0.387 ± 0.004 per cent, θ_B= 88°± 1°; P_V = 0.351 ± 0.004 per cent, θ_V= 91°± 1°  and   P_R = 0.335 ± 0.003 per cent, θ_R= 91°± 1°  . Both the degree of polarization and the position angle are found to be variable. The semi-amplitude of the polarization variability in B, V and R bands is found to be  0.18 ± 0.02, 0.13 ± 0.01  and  0.10 ± 0.02  per cent, respectively. We suggest that the levels of polarization observed in LO Peg could be the result of scattering of an anisotropic stellar radiation field by an optically thin circumstellar envelope or scattering of the stellar radiation by prominence-like structures.     

New light on the search for low-metallicity galaxies – I. The N2 calibrator

We present a simple metallicity estimator based on the logarithmic [N  ii ]   λ 6584/H α   ratio, hereafter N2, which we envisage will become very useful for ranking galaxies in a metallicity sequence from redshift survey-quality data even for moderately low spectral resolution.
We have calibrated the N2 estimator using a compilation of H  ii galaxies having accurate oxygen abundances, plus photoionization models covering a wide range of abundances. The comparison of models and observations indicates that both primary and secondary nitrogen are important for the relevant range of metallicities.
The N2 estimator follows a linear relation with log(O/H) that holds for the whole abundance range covered by the sample, from approximately  1/50th  to twice the Solar value  [7.2<12+log(O/H)<9.1]  . We suggest that the ([S  ii ]   λλ 6717,6731/H α )  ratio (hereafter S2) can also be used as a rough metallicity indicator. Because of its large scatter the S2 estimator will be useful only in systems with very low metallicity, where [N  ii ] λ 6584 is not detected or in low-resolution spectra where [N  ii ] λ 6584 is blended with H α .     

A measurement of the faint source correlation function in the GOODS and UDF surveys

We present a stable procedure for defining and measuring the two point angular autocorrelation function,   w (θ) =[θ/θ₀( V )]^−Γ  , of faint  (25 < V < 29)  , barely resolved and unresolved sources in the Hubble Space Telescope Great Observatories Origins Deep Survey and Ultra Deep Field data sets. We construct catalogues that include close pairs and faint detections. We show, for the first time, that, on subarcsec scales, the correlation function exceeds unity. This correlation function is well fit by a power law with index  Γ≈ 2.5  and a  θ₀= 10^{−0.1( V −25.8)} arcsec  . This is very different from the values of  Γ≈ 0.7  and  θ₀( r ) = 10^{−0.4( r −21.5)} arcsec  associated with the gravitational clustering of brighter galaxies. This observed clustering probably reflects the presence of giant star-forming regions within galactic-scale potential wells. Its measurement enables a new approach to measuring the redshift distribution of the faintest sources in the sky.     

Kinematics of SDSS subdwarfs: structure and substructure of the Milky Way halo

We construct a new sample of ∼1700 solar neighbourhood halo subdwarfs from the Sloan Digital Sky Survey (SDSS), selected using a reduced proper-motion diagram. Radial velocities come from the SDSS spectra and proper motions from the light-motion curve catalogue of Bramich et al. Using a photometric parallax relation to estimate distances gives us the full phase-space coordinates. Typical velocity errors are in the range  30–50 km s⁻¹  . This halo sample is one of the largest constructed to date and the disc contamination is at a level of ≲1 per cent. This enables us to calculate the halo velocity dispersion to excellent accuracy. We find that the velocity dispersion tensor is aligned in spherical polar coordinates and that  (σ _r , σ_φ, σ_θ) = (143 ± 2, 82 ± 2, 77 ± 2) km s⁻¹  . The stellar halo exhibits no net rotation, although the distribution of   v _φ  shows tentative evidence for asymmetry. The kinematics are consistent with a mildly flattened stellar density falling with distance like   r ^−3.75  .
Using the full phase-space coordinates, we look for signs of kinematic substructure in the stellar halo. We find evidence for four discrete overdensities localized in angular momentum and suggest that they may be possible accretion remnants. The most prominent is the solar neighbourhood stream previously identified by Helmi et al., but the remaining three are new. One of these overdensities is potentially associated with a group of four globular clusters (NGC 5466, NGC 6934, M2 and M13) and raises the possibility that these could have been accreted as part of a much larger progenitor.     

Lensing, reddening and extinction effects of Mg ii absorbers from z= 0.4 to 2

Using a sample of almost 7000 strong Mg  ii absorbers with   W ₀ > 1 Å  and  0.4 < z < 2.2  detected in the SDSS DR4 data set, we investigate the gravitational lensing and dust extinction effects they induce on background quasars. After carefully quantifying several selection biases, we isolate the reddening effects as a function of redshift and absorber rest equivalent width, W ₀. We find the amount of dust to increase with cosmic time as  τ( z ) ∝ (1 + z )^−1.1±0.4  , following the evolution of cosmic star density or integrated star formation rate. We measure the reddening effects over a factor of 30 in E ( B − V ) and we find that  τ∝ ( W ₀)^1.9±0.1  , providing us with an important scaling for theoretical modelling of metal absorbers. We also measure the dust-to-metal ratio and find it similar to that of the Milky Way. In contrast to previous studies, we do not detect any gravitational magnification by Mg  ii systems. We measure the upper limit  μ < 1.10  and discuss the origin of the discrepancy. Finally, we estimate the fraction of absorbers missed due to extinction effects and show that it rises from 1 to 50 per cent in the range  1 < W ₀ < 6 Å  . We parametrize this effect and provide a correction for recovering the intrinsic  ∂ N /∂ W ₀  distribution.     

Cosmological perturbation theory and the spherical collapse model — II. Non-Gaussian initial conditions

In Paper I of this series, we introduced the spherical collapse (SC) approximation in Lagrangian space as a way of estimating the cumulants ξ _J of density fluctuations in cosmological perturbation theory (PT). Within this approximation, the dynamics is decoupled from the statistics of the initial conditions, so we are able to present here the cumulants for generic non-Gaussian initial conditions, which can be estimated to arbitrary order including the smoothing effects. The SC model turns out to recover the exact leading-order non-linear contributions up to terms involving non-local integrals of the J -point functions. We argue that for the hierarchical ratios S _J , these non-local terms are subdominant and tend to compensate each other. The resulting predictions show a non-trivial time evolution that can be used to discriminate between models of structure formation. We compare these analytic results with non-Gaussian N -body simulations, which turn out to be in very good agreement up to scales where σ ≲ 1.