Line Of Apsides Research Articles

Rotation of the apsidal line in the eclipsing system V1059 Cep

Three light curves of the V1059 Cep eclipsing binary obtained in 2012, 2013 and 2021 were analyzed. We found the rapid apsidal motion in this system at the ratedw/dt= 7.2°per year. Assuming the equality of observed and theoretical apsidal motion rates we estimated physical characteristics of the components based on the data on their temperatures from the literature. According to our calculations the components are two very similar stars of the B7 V–B7.5 V spectral type with masses≈(3.4 ± 0.3)M⊙and age 180 ± 30 million years.

Measuring a Gravitomagnetic Effect with the Triple Pulsar PSR J0337+1715

Measuring a Gravitomagnetic Effect with the Triple Pulsar PSR J0337+1715

Primordial Orbital Alignment of Sednoids

We examined the past history of the three most detached trans-Neptunian objects (TNOs)—Sedna, 2012 VP113, and Leleakuhonua (2015 TG387)—the three clearest members of the dynamical class known as sednoids, with high perihelia distances q. By integrating backward their nominal (and a set of cloned) orbits for the solar system’s age, we surprisingly find that the only time all their apsidal lines tightly cluster was 4.5 Gyr ago, at perihelion longitude ϖ of 200°. This “primordial alignment” is independent of the observational biases that contribute to the current on-sky clustering in the large-semimajor-axis Kuiper Belt. If future sednoid discoveries confirm these findings, this strongly argues for an initial event during the planet formation epoch, which imprinted this particular apsidal orientation on the early detached TNO population. Their apsidal orientations were then subsequently modified only by the simple precession from the four giant planets (and weakly by the galactic tide). If other sednoids also cluster around the same primordial value, various models suggesting a still present planet in the outer solar system would be incompatible with this alignment. We inspected two scenarios that could potentially explain the primordial alignment. First, a rogue planet model (where another massive planet raises perihelia near its own longitude until ejection) naturally produces this signature. Alternatively, a close stellar passage early in solar system history raises perihelia, but it is poor at creating strong apsidal clustering. We show that all other known 35 < q < 55 au TNOs are either too perturbed or orbits are still too uncertain to provide evidence for or against this paradigm.

Rotation of the Apsidal Line in the Eclipsing System V1059 Cep

Rotation of the Apsidal Line in the Eclipsing System V1059 Cep

The Influence of Cold Jupiters in the Formation of Close-in Planets. I. Planetesimal Transport

The formation of a cold Jupiter (CJ) is expected to quench the influx of pebbles and the migration of cores interior to its orbit, thus limiting the efficiency of rocky planet formation either by pebble accretion and/or orbital migration. Observations, however, show that the presence of outer CJs (>1 au and ≳0.3M Jup) correlates with the presence of inner super-Earths (at <1 au). This observation may simply be a result of an enhanced initial reservoir of solids in the nebula required to form a CJ or a yet-to-be-determined mechanism assisted by the presence of the CJ. In this work, we focus on the latter alternative and study the orbital transport of planetesimals interior to a slightly eccentric (∼0.05) CJ subject to the gravity and drag from a viscously evolving gaseous disk. We find that a secular resonance sweeping inward through the disk gradually transports rings of planetesimals when their drag-assisted orbital decay is faster than the speed of the resonance scanning. This snowplow-like process leads to large concentration (boosted by a factor of ∼10–100) of size-segregated planetesimal rings with aligned apsidal lines, making their expected collisions less destructive, due to their reduced velocity dispersion. This process is efficient for a wide range of α-disk models (and thus disk lifetimes) and Jovian masses, peaking for values ∼1–5M Jup, which are typical of observed CJs in radial velocity surveys. Overall, our work highlights the major role that the disk's gravity may have on the orbital redistribution of planetesimals, depicting a novel avenue by which CJs may enhance the formation of inner planetary systems, including super-Earths and perhaps even warm and hot Jupiters.

Modeling Apsidal Motion in Eclipsing Binaries Using ELC

Apsidal motion is the precession of the line of apsides in the orbit of a binary star due to perturbations from General Relativity (GR), tides, or third-body interactions. The rate of precession due to tidal effects depends on the interior structures of the stars, and as a result, binaries in which this precession occurs are of great interest. Apsidal motion is observed through the analysis of eclipse times, which reveal small changes in the average interval between successive primary and secondary eclipses, taking all available observed times of eclipse and yielding an estimate of the apsidal rate. Given that this is a single observed quantity, various degeneracies are unavoidably present. Ideally, one would have a model that predicts eclipse times given the orbital and stellar parameters. These parameters for a given binary could then be computed using least squares, provided a suitably large number of eclipse times. Here we use the eclipsing light curve (ELC) program as such a model. The Newtonian equations of motion with additional force terms accounting for GR contributions and tidal distortions are integrated, yielding precise sky positions as a function of time. Times of mid-eclipse and instantaneous orbital elements are computed as a function of time. In this paper, we outline the method and compare numerically computed apsidal rates with standard formulae using a set of 15 binaries based on real systems. For our simulated systems, the derived apsidal rates agree with the standard formula.

Orbital precession of stars in the Galactic Centre

ABSTRACT The region around the centre of our Galaxy is very dense of stars. The kinematics of inner moving stars in the Galaxy (the so-called S-stars) has been deeply studied by different research groups leading to the conclusion of the existence of a very compact object (Sgr A*, likely a supermassive black hole) responsible for their high speed. Here, we start from the observational evidence of orbital apsidal line precession for the S2 (also called S0-2) star to investigate on a theoretical side what level of quality in such regime of relatively strong gravitational field is reached in the orbit angular precession determination when using a direct orbital integration of the star motion subjected to an acceleration computed in the post-Newtonian (PN) scheme up to different orders. This approach, although approximated and limited to particle speed not exceeding ∼ 0.3c, allows the inclusion of various effects, like that of a possible spin of the central massive object. Our results show that the inclusion of PN terms above the standard 1PN term (the one corresponding to the classic Einstein–Schwarzschild estimate of pericenter advance) is compulsory to determine angular precession at sufficient level of accuracy for those penetrating stars that would allow to pick contemporary the value of the mass and of the spin of a rotating (Kerr-like) supermassive black hole (SMBH). We discuss how future observational data, together with a proper modelization, could allow the determination of both mass and spin of the SMBH of our Galaxy.

Tidally Tilted Pulsations in HD 265435, a Subdwarf B Star with a Close White Dwarf Companion

Abstract Tidally tilted pulsators (TTPs) are an intriguing new class of oscillating stars in binary systems; in such stars, the pulsation axis coincides with the line of apsides, or tidal axis, of the binary. All three TTPs discovered so far have been δ Scuti stars. In this Letter, we report the first conclusive discovery of tidally tilted pulsations in a subdwarf B (sdB) star. HD 265435 is an sdB–white dwarf binary with a 1.65 hr period that has been identified and characterized as the nearest potential Type Ia supernova progenitor. Using TESS 20 s cadence data from Sectors 44 and 45, we show that the pulsation axis of the sdB star has been tidally tilted into the orbital plane and aligned with the tidal axis of the binary. We identify 31 independent pulsation frequencies, 27 of which have between 1 and 7 sidebands separated by the orbital frequency (ν orb) or multiples thereof. Using the observed amplitude and phase variability due to tidal tilting, we assign ℓ and m values to most of the observed oscillation modes and use these mode identifications to generate preliminary asteroseismic constraints. Our work significantly expands our understanding of TTPs, as we now know that (i) they can be found in stars other than δ Scuti pulsators, especially highly evolved stars that have lost their H-rich envelopes, and (ii) tidally tilted pulsations can be used to probe the interiors of stars in very tight binaries.

Global searches of frozen orbits around an oblate Earth-like planet

A frozen orbit is beneficial for observation owing to its stationary apsidal line. The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J2 and limited high-order terms, which cannot meet the stringent demands of all missions. In this study, the gravitational field is expanded to J15 terms and the Hamiltonian canonical form described by the Delaunay variables is used. The zonal harmonic coefficients of the Earth are chosen as the sample. Short-periodic terms are eliminated based on the Hori–Lie transformation. An algorithm is developed to solve all equilibrium points of the Hamiltonian function. A stable frozen orbit with an argument of perigee that equals neither 90◦ nor 270◦ is first reported in this paper. The local stability and topology of the equilibrium points are obtained from their eigenvalues. The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods. The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case. The analytical results can be applied to other Earth-like planets and asteroids.

Fast and accurate estimation of fuel-optimal trajectories to Near-Earth Asteroids

This paper proposes an improved method for the preliminary evaluation of minimum-propellant trajectories to Near-Earth Asteroids (NEAs). The method applies to missions from Earth to asteroids with small eccentricity and inclination. A planar and a plane-change problem can be distinguished. In the planar problem, the solution assumes that multiple burn arcs are performed in correspondence of the apsides of the target asteroid in order to change the initial spacecraft orbit (i.e., Earth’s orbit) into the target one. The number of arcs is established once the time of flight is given (1 burn at each apsis per revolution, 1 revolution per year can be assumed). The length and propellant consumption of each arc to attain the required changes of semi-major axis and eccentricity are computed by a procedure based on Edelbaum’s approximation, which is well-suited to the problem at hand, as eccentricity changes are expected to be small for feasible missions. No numerical integration is required, but only the numerical solution of a three-unknown algebraic system is needed, making the procedure extremely fast. Plane change is taken into account assuming a constant out-of-plane thrust angle during each burn. A previous simple formulation used an averaged thrust effect over one revolution and neglected the fact that plane changes are more effective at the nodes. Several improvements are here introduced, which greatly increase the method accuracy. The influences of the eccentricity change, the angle between the asteroid line of nodes and line of apsides, and the expected length of the arc are considered: In fact, when the eccentricity is small, the thrust arc can be performed at the nodes where the inclination is efficiently changed, with little penalty in the planar maneuver. An efficient plane change is also performed when the angle between the asteroid line of nodes and line of apsides is small and/or the length of the arc is large, because, in this case, the node is comprised in the apsidal burn. A simple corrective formula accounts for this effect. The new method shows remarkable accuracy. The results comparison with solutions obtained with an indirect optimization method for a set of more than 60 NEAs shows a 0.95 correlation coefficient in the propellant masses. The estimation error is below 10% for 75% of the targets, below 15% for 95% of the targets, and always below 20%.

Long-term orbit stability of the Apollo 11 “Eagle” Lunar Module Ascent Stage

The Apollo 11 “Eagle” Lunar Module ascent stage was abandoned in lunar orbit after the historic landing in 1969. Its fate is unknown. Numerical analysis described here provides evidence that this object might have remained in lunar orbit to the present day. The simulations show a periodic variation in eccentricity of the orbit, correlated to the selenographic longitude of the apsidal line. The rate of apsidal precession is correlated to eccentricity. These two factors appear to interact to stabilize the orbit over the long term.

Detecting general relativistic orbital precession in transiting hot Jupiters

ABSTRACT Both classical and relativistic weak-field and slow-motion perturbations to planetary orbits can be treated as perturbative corrections to the Keplerian model. In particular, tidal forces and General Relativity (GR) induce small precession rates of the apsidal line. Accurate measurements of these effects in transiting exoplanets could be used to test GR and to gain information about the planetary interiors. Unfortunately, models for transiting planets have a high degree of degeneracy in the orbital parameters that, combined to the uncertainties of photometric transit observations, results in large errors on the determinations of the argument of periastron and precludes a direct evaluation of the apsidal line precession. Moreover, tidal and GR precession time-scales are many order of magnitudes larger than orbital periods, so that on the observational time-spans required to cumulate a precession signal enough strong to be detected, even small systematic errors in transit ephemerides add up to cancel out the tiny variations due to precession. Here we present a more feasible solution to detect tidal and GR precession rates through the observation of variations of the time interval (Δτ) between primary and secondary transits of hot Jupiters and propose the most promising target for such detection, WASP-14 b. For this planet we expect a cumulated $\Delta \tau \, \approx$ −250 s, due to tidal and relativistic precession, since its first photometric observations.

Injection of Inner Oort Cloud Objects into the Distant Kuiper Belt by Planet Nine

Abstract The outer solar system exhibits an anomalous pattern of orbital clustering, characterized by an approximate alignment of the apsidal lines and angular momentum vectors of distant, long-term stable Kuiper Belt objects. One explanation for this dynamical confinement is the existence of a yet-undetected planetary-mass object, “Planet Nine (P9).” Previous work has shown that trans-Neptunian objects, that originate within the scattered disk population of the Kuiper Belt, can be corralled into orbital alignment by Planet Nine’s gravity over ∼Gyr timescales, and characteristic P9 parameters have been derived by matching the properties of a synthetic Kuiper Belt generated within numerical simulations to the available observational data. In this work, we show that an additional dynamical process is in play within the framework of the Planet Nine hypothesis, and demonstrate that P9-induced dynamical evolution facilitates orbital variations within the otherwise dynamically frozen inner Oort cloud. As a result of this evolution, inner Oort cloud bodies can acquire orbits characteristic of the distant scattered disk, implying that if Planet Nine exists, the observed census of long-period trans-Neptunian objects is comprised of a mixture of Oort cloud and Kuiper Belt objects. Our simulations further show that although inward-injected inner Oort cloud objects exhibit P9-driven orbital confinement, the degree of clustering is weaker than that of objects originating within the Kuiper Belt. Cumulatively, our results suggest that a more eccentric Planet Nine is likely necessary to explain the data than previously thought.

Linear and non-linear tidal oscillations and mode identification in the eccentric binary system KIC 3858884

ABSTRACT The phoebe code was used to analyse the Kepler light-curve and to estimate the physical and geometrical parameters of a rare pulsating binary system, KIC 3858884. The analysis indicated that the system is composed of two detached and very similar main-sequence A-type stars, in a highly eccentric orbit with e = 0.47. After disentangling the binarity effect, the residual data were subjected to Fourier frequency decomposition using period04 software. The resulting frequency spectrum consists of two moderately high-amplitude nearby frequencies, F1 = 7.232199 d−1 and $F2=7.472889\, \mathrm{d}^{-1}$, which were attributed to δ Scuti-type pulsations. In addition, 18 frequencies were identified that were exact harmonics of the orbital frequency $f_{\rm orb}= 0.038533\, \mathrm{d}^{-1}$, and also 53 anharmonics. However, it was found that many of these anharmonic frequencies coupled together non-linearly to give harmonic modes of pulsation. Furthermore, some existing theoretical models of the tidal oscillations were numerically verified in general binary systems through estimations of various modal characteristics, for example mode quantum numbers ${n, l, m,}$ energies Ei, threshold energies Ei,th, damping rates γi, growth rates Γi and stability criteria, etc. The evolution of the stars in the binary system was compared with some similar single pulsating stars on the Hertzsprung–Russell diagram and it was concluded that the evolution of a single star is more rapid. Finally, the observed rate of apsidal line displacement was estimated through eclipse timing variation analysis as Uobs = 74745.2 ± 2566 yr. This was compared with the theoretically calculated rate of the line of apsides motion, UTheo = 73588 ± 2298 yr, and found to be in good agreement within errors, hence verifying general relativity theory once again.

Long-term evolution and lifetime analysis of geostationary transfer orbits with solar radiation pressure

The dynamical evolution of geostationary transfer orbits (GTOs) is a fundamental issue for the space debris mitigation in this orbit region. In this paper, the long-term dynamical evolution and orbital lifetime of low-inclination GTOs are investigated based on the Semi-analytic Tool for End of Life Analysis software (STELA) with the solar radiation pressure (SRP) and Earth's shadow taken into account. We carry out numerical investigations of long-term orbital evolution of objects with area-to-mass ratio (A/m) between 0.02 m2/kg and 1 m2/kg in the GTO region. Their dynamical orbital lifetime is recorded and presented in the Ω–ω (right ascension of ascending node–argument of perigee) plane to investigate its dependence on the initial conditions and physical parameters, particularly the initial azimuth of the Sun with respect to the apsidal line and the value of A/m. Based on these results, we further analyze the effect of SRP on the long-term dynamical evolution, as well as the solar apsidal resonances. The results are helpful to enrich the knowledge about long-term evolution and orbital lifetimes of GTO objects with relatively high A/m as well as to promote uncatalogued objects observation and identification.

Mutual Gravitational Energy of Gaussian Rings and the Problem of Perturbations in Celestial Mechanics

A new approach to the study of long-period and secular perturbations in celestial mechanics is developed. In contrast to the traditional use of the apparatus of the perturbing Lagrange function, we rely on the mutual potential energy of elliptic Gaussian rings. This approach is important due to the fact that instead of averaging the expression for the perturbing Lagrange function obtained in a very complicated way, it is methodologically simpler to immediately calculate the mutual energy of Gaussian rings. In this paper, we consider the problem for two Gaussian rings with one common focus, with small eccentricities, a small angle of mutual inclination, and an arbitrary angle between the lines of apsides. An expression for the mutual energy of such a system of rings is obtained in the form of a series up to the terms of the 4th order of smallness inclusively. This expression is used to derive and solve a system of differential equations describing the evolution of rings in an ecliptic reference frame. The method is used for a detailed study of the two-planetary Sun–Jupiter–Saturn problem. The results complement and refine the results of other authors. The new expression of the perturbing function can be applied not only to the planetary problem, in which all inclinations should be small, but also to the problem with nonplanetary rings found around small celestial bodies.

The single-sided pulsator CO Camelopardalis

ABSTRACT CO Cam (TIC 160268882) is the second ‘single-sided pulsator’ to be discovered. These are stars where one hemisphere pulsates with a significantly higher amplitude than the other side of the star. CO Cam is a binary star comprised of an Am δ Sct primary star with Teff = 7070 ± 150 K, and a spectroscopically undetected G main-sequence secondary star. The dominant pulsating side of the primary star is centred on the L1 point. We have modelled the spectral energy distribution combined with radial velocities, and independently the TESS light curve combined with radial velocities. Both of these give excellent agreement and robust system parameters for both stars. The δ Sct star is an oblique pulsator with at least four low radial overtone (probably) f modes with the pulsation axis coinciding with the tidal axis of the star, the line of apsides. Preliminary theoretical modelling indicates that the modes must produce much larger flux perturbations near the L1 point, although this is difficult to understand because the pulsating star does not come near to filling its Roche lobe. More detailed models of distorted pulsating stars should be developed. These newly discovered single-sided pulsators offer new opportunities for astrophysical inference from stars that are oblique pulsators in close binary stars.

Order–disorder phase transition in black hole star clusters – III. A mono-energetic cluster

ABSTRACT Supermassive black holes at the centres of galaxies are often surrounded by dense star clusters. For a wide range of cluster properties and orbital radii the resonant relaxation times in these clusters are much shorter than the Hubble time. Since resonant relaxation conserves semimajor axes, these clusters should be in the maximum-entropy state consistent with the given semimajor axis distribution. We determine these maximum-entropy equilibria in a simplified model in which all of the stars have the same semimajor axes. We find that the cluster exhibits a phase transition from a disordered, spherical, high-temperature equilibrium to an ordered low-temperature equilibrium in which the stellar orbits have a preferred orientation or line of apsides. Here ‘temperature’ is a measure of the non-Keplerian or self-gravitational energy of the cluster; in the spherical state, temperature is a function of the rms eccentricity of the stars. We explore a simple two-parameter model of black hole star clusters – the two parameters are semimajor axis and black hole mass – and find that clusters are susceptible to the lopsided phase transition over a range of ∼102 in semimajor axis, mostly for black hole masses $\lesssim 10^{7.5}\, \mathrm{M}_{\odot }$.

Dynamical models for secular evolution of navigation satellites

In this work, two dynamical models are formulated to describe the secular dynamics of navigation satellites moving in the medium Earth orbit (MEO) and geosynchronous orbit (GSO) regions. In the dynamical models, the leading terms of the Earth’s oblateness and the luni-solar gravitational perturbations are considered. For convenience, the orbits of the Sun and the Moon are described in the geocentric ecliptic reference frame, where the regression of nodal line and precession of apsidal line of the lunar orbit can be approximated as linear functions of time. The disturbing function acting on navigation satellites is analytically averaged over the mean motions of both the satellite and the third body (the Sun or the Moon). Explicit expressions of the averaged disturbing function are provided in the geocentric ecliptic and equatorial reference frames, corresponding to averaged model 1 and averaged model 2, respectively. It is found that there are seven resonant arguments in averaged model 1, while there are thirty-two resonant arguments in averaged model 2. The associated resonance curves corresponding to these resonant arguments in each model form the dynamical backbone in the phase space, organizing secular behavior of navigation satellites. At last, the averaged models are numerically compared to the associated non-averaged model, and simulation results indicate that (a) the averaged models formulated in the geocentric ecliptic and equatorial reference frames are identical, and (b) both of these two averaged models are applicable in predicting secular behavior of navigation satellites.

The Orbital Elements of Venus in Medieval Islamic Astronomy: Interaction Between Traditions and the Accuracy of Observations

The orbital elements of each planet are the eccentricity and the direction of the apsidal line of its orbit defined by the ecliptic longitude of either of its apses, i.e., the two points on its orbit where the planet is either furthest from or closest to the Earth, which are called the planet’s apogee and perigee. In the geocentric view of the solar system, the eccentricity of Venus is a bit less than half of the solar one, and its apogee is located behind that of the Sun. Ptolemy correctly found that the apogee of Venus is behind that of the Sun, but determined the eccentricity of Venus to be exactly half the solar one. In the Indian Midnight System of Āryabhaṭa (b. ad 476), the eccentricity of Venus is assumed to be half the solar one, and also the longitudes of their apogees are assumed to be the same. This hypothesis became prevalent in early medieval Middle Eastern astronomy (ad 800–1000), where its adoption resulted in large errors of more than 10° in the values for the longitude of the apogee of Venus adopted by Yaḥyā b. Abī Manṣūr (d. ad 830), al-Battānī (d. ad 929), and Ibn Yūnus (d. ad 1007). In Western Islamic astronomy, it was used in combination with Ibn al-Zarqālluh’s (d. ad 1100) solar model with variable eccentricity, which only by coincidence resulted in accurate values for the eccentricity of Venus. In late Islamic Middle Eastern astronomy (from ad 1000 onwards), Āryabhaṭa’s hypothesis gradually lost its dominance. Ibn al-A‘lam (d. ad 985) seems to have been the first Islamic astronomer who rejected it. Late Eastern Islamic astronomers from the middle of the thirteenth century onwards arrived at the correct understanding that the eccentricity of Venus should be somewhat less than half of the solar one. Its most accurate medieval value was measured in the Samarqand observatory in the fifteenth century. Also, the values for the longitude of the apogee of Venus show a significant improvement in late Middle Eastern Islamic works, reaching an accuracy better than a degree in Khāzinī’s Mu‘tabar zīj, Ibn al-Fahhād’s ‘Alā’ī zīj, the Īlkhānī zīj, and Ulugh Beg’s Sulṭānī zīj.