Disturbing Function Research Articles

A semi-analytical model for coorbital motion

ABSTRACT A globally valid analytically averaged Hamiltonian model for the coorbital motion is hard to construct because the analytical expansions of the disturbing function usually diverge in the quasi-satellite domain that is close to collision singularity. In this paper, an analytically averaged model for the coorbital motion is proposed in case of the circular restricted three-body problem, which can describe properly the transitions that occur at small eccentricities and inclinations, such as the transition between the horseshoe orbit and the quasi-satellite orbit. With the help of the numerical averaging method, numerical experiments are carried out to show the validity and accuracy of the analytically averaged model. The averaged model proposed here can be easily extended to more complicated cases such as the elliptic three-body problem or the planetary three-body problem.

Analytical Study of the Co-orbital Motion in the Circular Restricted Three-body Problem

In the restricted three-body problem (RTBP), if a small body and a planet stably orbit around a central star with almost exactly the same semimajor axis, and thus almost the same mean motion, this phenomenon is called the co-orbital motion, or equivalently, the 1:1 mean motion resonance. The classical expansion of the disturbing function is divergent when the semimajor axis ratio of the small body to the planet is close to unity. Thus, most of the previous studies on the co-orbital dynamics were carried out through numerical integrations or semi-analytical approaches. In this work, we construct an analytical averaged model for the co-orbital motion in the framework of the circular RTBP. This model is valid in the entire coorbital region except in the vicinity of the collision singularity. The results of the analytical averaged model are in good agreement with the numerical averaged model even for moderate eccentricities and inclinations. The analytical model can reproduce the tadpole, horseshoe and quasi-satellite orbits common in the planar problem. Furthermore, the asymmetry of 1:1 resonance and the compound orbits (Icarus 137:293–314) in the general spatial problem can also be obtained from the analytical model.

Celmech: A Python Package for Celestial Mechanics

We present celmech, an open-source Python package designed to facilitate a wide variety of celestial mechanics calculations. The package allows users to formulate and integrate equations of motion incorporating user-specified terms from the classical disturbing function expansion of the interaction potential between pairs of planets. The code can be applied, for example, to isolate the contribution of particular resonances to a system’s dynamical evolution and develop simple analytical models with the minimum number of terms required to capture a particular dynamical phenomenon. Equations and expressions can be easily manipulated by leveraging the extensive symbolic mathematics capabilities of the sympy Python package. The celmech package is designed to interface seamlessly with the popular N-body code REBOUND to facilitate comparisons between calculation results and direct N-body integrations. The code is extensively documented, and numerous example Jupyter notebooks illustrating its use are available online.

An Accurate 3D Analytic Model for Exoplanetary Photometry, Radial Velocity, and Astrometry

Abstract We developed and provide AnalyticLC, a novel analytic method and code implementation for dynamical modeling of planetary systems, including non-coplanar interactions, based on a disturbing function expansion to fourth order in eccentricities and inclinations. AnalyticLC calculates the system dynamics in 3D and the resulting model light-curve, radial-velocity, and astrometry signatures, enabling simultaneous fitting of these data. We show that for a near-resonant chain of three planets, where the two super-periods are close to each other, the TTVs of the pair-wise interactions cannot be directly summed to give the full system TTVs because the super-periods themselves resonate. We derive the simultaneous three planets correction and include it in AnalyticLC. We compare the model computed by AnalyticLC to synthetic data generated by an N-body integrator, and evaluate its accuracy. Depending on the maximal order of expansion terms kept, AnalyticLC computation time can be up to an order of magnitude faster than the state-of-the-art published N-body integrator TTVFast, with a smaller enhancement seen at higher order. The advantage increases for long-term observations as our approach’s computation time does not depend on the time span of the data. Depending on the system parameters, the photometric accuracy is typically a few ppm, significantly smaller than Kepler’s and other observatories’ typical data uncertainty. Our highly efficient and accurate implementation allows full inversion of a large number of observed systems for planetary physical and orbital parameters, presented in a companion paper.

Long-term dynamical evolution of Pallene (Saturn XXXIII) and its diffuse, dusty ring

ABSTRACT The distinctive set of Saturnian small satellites, Aegaeon, Methone, Anthe, and Pallene, constitutes an excellent laboratory to understand the evolution of systems immersed in co-orbital dusty rings/arcs, subjected to perturbations from larger satellites and non-gravitational forces. In this work, we carried out a comprehensive numerical exploration of the long-term evolution of Pallene and its ring. Through frequency map analysis, we characterized the current dynamical state around Pallene. A simple tidal evolution model serves to set a time frame for the current orbital configuration of the system. With detailed short- and long-term N-body simulations we determine whether Pallene is currently in resonance with one or more of six of Saturn’s major moons. We analysed a myriad of resonant arguments extracted from the direct and indirect parts of the disturbing function, finding that Pallene is not in mean motion resonance from the present up to 5 Myr into the future; none the less, some resonant arguments exhibit intervals of libration and circulation at different time-scales and moon pairings. We studied the dynamical evolution of micrometric particles forming the ring, considering gravitational and non-gravitational forces. Non-gravitational forces are responsible for particles vertical excursions and outward migration. By estimating the satellite’s mass production rate, we find that Pallene could be responsible for keeping its ring in steady-state only if it is mainly composed of large micrometre-sized particles. If mainly composed of particles with a few micrometres for which Pallene is the only source, the ring will spread out, both radially and vertically, until it finally disappears.

A new expansion of planetary disturbing function and applications to interior, co-orbital and exterior resonances with planets

In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination, and the harmonic arguments are linear combinations of the mean longitude, longitude of pericenter and longitude of ascending node of each mass. The resulting new expansion is valid for arbitrary inclinations and semimajor axis ratios. In the case of mean motion resonant configuration, the disturbing function can be easily averaged to produce the analytical expansion of resonant disturbing function. Based on the analytical expansion, the Hamiltonian model of mean motion resonances is formulated, and the resulting analytical developments are applied to Jupiter’s inner and co-orbital resonances and Neptune’s exterior resonances. Analytical expansion is validated by comparing the analytical results with the associated numerical outcomes.

Orbital Design and Control for Jupiter-Observation Spacecraft

This paper investigates the evolution of orbits around Jupiter and designs a sun-synchronous repeating ground track orbit. In the dynamical models, the leading terms of the Jupiter’s oblateness are J2 and J4 terms. A reasonable range of ground track repetition parameter Q is given and the best observation orbit elements are selected. Meanwhile, the disturbing function acting on the navigation spacecraft is the atmospheric drag and the third body. The law of altitude decay of the spacecraft’s semimajor orbit axis caused by the atmospheric drag is studied, and the inclination perturbation caused by the sun’s gravity is analyzed. This paper designs a semimajor axis compensation strategy to maintain the orbit’s repeatability and proposes an initial inclination prebiased strategy to limit the local time at the descending node in a permitted range. In particular, these two methods are combined in the context of sun-synchronous repeating ground track orbit for better observation of the surface of Jupiter.

Modelling the Triaxial Compression Behavior of Loess Using the Disturbed State Concept

This paper investigates the triaxial compression behaviour of Q3 loess soil and the construction of a constitutive model accounting for the structural effect of loess on the basis of the disturbed state concept. By analyzing the triaxial compression testing results, we have established a new disturbing function with respect to the volumetric and shear moduli parameters. A research into the evolution laws of the disturbing function was also conducted, followed by the construction of a constitutive model for loess soil as well as the verification of the constitutive model with model test results. The results indicate that the double‐parameter disturbing function evolves in an exponential form, capturing well the effect of moisture content and confining pressure on the loess structural behavior. The parameters of the constructed constitutive model based on the disturbed state are easy to be obtained and have clarified physical meanings. Considering the effectiveness in capturing the structural behavior of the loess, the constructed constitutive model has a great potential to be applied in the engineering practice in the loess area. The constructed constitutive model based on the disturbed state concept provides new ideas for the study of the structural constitutive model of loess, which is theoretically significant.

AN AYURVEDIC OVERVIEW OF RAKTAPRADAR- A CONCEPTUAL APPROACH

Reproductive capability in a young woman begins at the point of menarche, which is the beginning of cyclic uterine bleeding in the anatomically and physiologically normal female. Menarche marks the beginning of an important stage in a young woman’s physical reproductive maturation and development. Even before the onset of this entirely natural but potentially disturbing function, a young woman’s early psychological reactions to menstruation, and probably also her lifelong view, can be influenced by the accuracy of her information and the degree of empathy with which this knowledge has been conveyed to her. Many women, perhaps appropriately, conclude that any departure from their personal menstrual experience is abnormal, and they will seek treatment for these departures. Conversely, some women accept or perhaps ignore even significant variations in their menstrual function, sometimes to the extent that serious health impairment occurs (e.g., severe iron-deficiency anemia). Therefore in this study an effort has been put forth to make a conceptual study covering almost all the aspect of Raktapradar as per Ayurvedic and modern.

Dynamics of retrograde $1/n$ mean motion resonances: the $1/{-2}$, $1/{-3}$ cases

In this paper, we investigate the dynamics of the exterior retrograde $1/n$ resonances within the framework of the planar circular restricted three-body problem (PCRTBP), taking the $1/{-2}$ and $1/{-3}$ resonances as examples. We prove that there is no asymmetric libration in retrograde $1/n$ resonances using analytical, numerical integrations, and semi-analytical methods, by mutual authenticating. For retrograde $1/n$ resonances, we calculate the magnitudes of the first- and second-harmonics of the expansion. The analytical results showed that the second-order harmonics in the expansion of the disturbing function can be neglected, which are the leading cause of the existence of the asymmetric libration in prograde $1/n$ resonances. Our results also conform to the latest qualitative criterion for the appearance of asymmetric libations proposed by Namouni and Morais (2016). And our analytical results are verified well by numerical integrations. By semi-analytical theory, we generate a series of phase-space portraits in the $e$ – $\phi $ polar plane for the full range of eccentricity to confirm the absence of the asymmetric librations in retrograde $1/n$ resonances. It is the first time the absence of the asymmetric librations in retrograde $1/n$ resonances is studied. We also analyze the dynamics of the pericentric and apocentric librations of the retrograde $1/n$ resonances. The stable resonant libration zones of the $1/{-2}$ , $1/{-3}$ , $1/{-4}$ and $1/{-5}$ resonances are illustrated in the $a$ – $e$ plane. Our research here reveals the differences between the dynamics of retrograde and prograde $1/n$ resonances.

Huaier Restrains Cholangiocarcinoma Progression in vitro and in vivo Through Modulating lncRNA TP73-AS1 and Inducing Oxidative Stress.

PurposeHuaier, the fruiting body of Trametes robiniophila Murr, is a kind of traditional Chinese medicine. Recently, many studies have confirmed that Huaier has antitumor effects on various malignancies. Moreover, studies have demonstrated that long noncoding RNAs play an important regulatory role in the occurrence and progression of malignancies. Our present study was to explore whether Huaier has a potential antitumor effect in cholangiocarcinoma and reveal the relationship between lncRNAs and Huaier-induced tumor inhibition.MethodsMicroarray assay was performed to identify the candidate lncRNAs regulated by Huaier. Quantitative real-time PCR was applied to assess the effect of Huaier on TP73-AS1 expression. The effect of Huaier on the cell viability, proliferation, migration and invasion was evaluated by CCK-8, colony formation, wound healing and Transwell assays, respectively. The ratio of cell apoptosis was determined using AO/EB, Hoechst 33342 and flow cytometry. The effect of Huaier on oxidative stress was revealed using DCFH-DA, mito-SOX, JC-1 probes and Western blotting. In addition, the effect of Huaier on tumor growth and metastasis was explored using subcutaneous tumor model and lung metastatic tumor model in nude mice.ResultsIn vitro, Huaier inhibited the proliferation, migration and invasion of cholangiocarcinoma cells by down-regulating TP73-AS1 and induced apoptosis through mitochondrial apoptotic pathway. In vivo, Huaier suppressed the growth and metastasis of cholangiocarcinoma by modulating the expression of proliferation and EMT-associated proteins.ConclusionHuaier could inhibit cell proliferation, invasion and metastasis by modulating the expression of TP73-AS1, meanwhile promote apoptosis of CCA cells through disturbing mitochondrial function, inducing oxidative stress and activating caspases in vitro. In addition, Huaier could suppress tumor growth and metastasis by regulating the expression of proliferation and EMT-related proteins. In the meantime, Huaier prolonged the survival of nude mice in lung metastatic model with acceptable drug safety.

An optimization approach to search for quasi-critical inclinations for direct and retrograde orbits

When only the secular terms due to the central body oblateness are considered in the gravitational potential, the critical inclination classical values are about 63.43° and 116.56° for the direct and retrograde orbits, respectively. If sectoral terms are included into disturbing function, the equations of motion become coupled, then searching for critical inclinations is not a trivial problem. Thus, for the purpose of overcoming this difficulty and to shed light on the effect of inserting the C22 sectoral hamornic in the Hamiltonian function, the quasi-critical inclinations concept is revised and its solution is proposed as an optimization problem. In this sense, the present article makes the following contributions to the quasi-critical inclination problem: (i) applying the nonlinear optimization tools to solve this problem; (ii) through this technique, obtaining quasi-critical inclinations for the retrograde case for the first time; (iii) using this technique to find direct and retrograde orbits around Io, a very important body to be studied in the future.

Effects of higher-order multipoles of the lunar disturbing potential on elongated orbits in cislunar space

For an Earth satellite in cislunar space, the effects of the third or higher-order harmonics in the solar disturbing function are negligible. For lunar perturbations, however, these terms become increasingly important as the semimajor axis increases. We investigate the effects of these higher-order multipole moments on circular, moderate, and highly elliptical orbits, where the semimajor axis is a relatively large fraction (~20%) of the Moon’s one. We specifically characterize the regions of cislunar space where the octupole-order approximation, often used in celestial and astrophysical dynamics for studying the stability and fates of hierarchical planetary systems, is actually a valid truncation of the gravitational interactions.

An analytically treatable model of long-term dynamics in a mean motion resonance with coexisting resonant modes

We examine a 2-DOF Hamiltonian system modeling some previously unexplored dynamical effects in first-order mean motion resonances in the spatial circular restricted three-body problem “star-planet-asteroid.” In distinction from the well-known integrable model of Sessin and Ferraz-Mello (Celest Mech Dyn Astron 32:307–332, 1984) and Wisdom (Celest Mech Dyn Astron 38:175–180, 1986), in our analysis, we kept more terms in the expansion for the disturbing function. This allowed us to study the phenomena caused by coexisting resonant modes, that were lost in the integrable model. In addition, we identified the areas of dynamical chaos in the phase space of the considered system. Despite the non-integrability of our model, we were able to compute analytically many quantities characterizing its dynamical behavior. Finally, we illustrate some of our results through the examples of the known resonant Kuiper belt objects.

Nodal Precession in Closely Spaced Planet Pairs

Abstract Planet–planet perturbations can cause planets’ orbital elements to change on secular timescales. Previous work has evaluated the nodal precession rate for planets in the limit of low α (semimajor axis ratio; 0 < α ≤ 1). Our simulations show that systems at high α (or low period ratio), similar to multiplanet systems found in the Kepler survey, have a nodal precession rate that is more strongly dependent on eccentricity and inclination. We present a complete expansion of the nodal precession rate to fourth order in the disturbing function and show that this analytical solution much better describes the simulated N-body behavior of high-α planet pairs; at α ≈ 0.5, the higher-order solution, on average, reduces the median analytical error by a factor of 7.5 from linear theory and 6.2 from a similar expansion assuming low α. We set limits on eccentricity and inclination where the theory is precisely validated by N-body integrations, which can be useful in future secular treatments of planetary systems.

Frozen Orbits Construction for a Lunar Solar Sail

Frozen orbit is an attractive option for orbital design owing to its characteristics (its argument of pericenter and eccentricity are kept constant on an average). Solar sails are attractive solutions for massive and expensive missions. However, the solar radiation pressure effect represents an additional force on the solar sail that may greatly affect its orbital behavior in the long run. Thus, this force must be included as a perturbation force in the dynamical model for more accuracy. This study shows the calculations of initial conditions for a lunar solar sail frozen orbit. The disturbing function of the problem was developed to include the lunar gravitational field that is characterized by uneven mass distribution, third body perturbation, and the effect of solar radiation. An averaging technique was used to reduce the dynamical problem to a long period system. Lagrange planetary equations were utilized to formulate the rate of change of the argument of pericenter and eccentricity. Using the reduced system, frozen orbits for the Moon sail orbiter were constructed. The resulting frozen orbits are shown by two 3Dsurface (semimajor, eccentricity, inclination) figures. To simplify the analysis, we showed inclination–eccentricity contours for different values of semi-major axis, argument of pericenter, and values of sail lightness number.

Resonance libration and width at arbitrary inclination

ABSTRACTWe apply the analytical disturbing function for arbitrary inclination derived in our previous work to characterize resonant width and libration of mean motion resonances at arbitrary inclination obtained from direct numerical simulations of the three-body problem. We examine the 2:1 and 3:1 inner Jupiter and 1:2 and 1:3 outer Neptune resonances and their possible asymmetric librations using a new analytical pendulum model of resonance that includes the simultaneous libration of multiple arguments and their second harmonics. The numerically derived resonance separatrices are obtained using the mean exponential growth factor of nearby orbits (megno chaos indicator). We find that the analytical and numerical estimates are in agreement and that resonance width is determined by the first few fundamental resonance modes that librate simultaneously on the resonant time-scale. Our results demonstrate that the new pendulum model may be used to ascertain resonance width analytically, and more generally, that the disturbing function for arbitrary inclination is a powerful analytical tool that describes resonance dynamics of low as well as high inclination asteroids in the Solar system.

Nonsingular recursion formulas for third-body perturbations in mean vectorial elements

The description of the long-term dynamics of highly elliptic orbits under third-body perturbations may require an expansion of the disturbing function in series of the semi-major axes ratio up to higher orders. To avoid dealing with long series in trigonometric functions, we refer the motion to the apsidal frame and efficiently remove the short-period effects of this expansion in vectorial form up to an arbitrary order. We then provide the variation equations of the two fundamental vectors of the Keplerian motion by analogous vectorial recurrences, which are free from singularities and take a compact form useful for the numerical propagation of the flow in mean elements.

Computing Perturbations in the Two-Planetary Three-Body Problem with Masses Varying Non-isotropically at Different Rates

The classical problem of three bodies of variable masses is considered in the case when two of the bodies are protoplanets and all the masses vary non-isotropically at different rates. The problem is analyzed in the framework of the planetary perturbation theory in terms of the osculating elements of aperiodic motion on quasi-conic sections. An algorithm for symbolic computation of the disturbing function and its expansion into power series in terms of the eccentricities and inclinations is discussed in detail. Differential equations describing the long-term evolution of the orbital parameters are derived in the form of Lagrange’s planetary equations. All the relevant calculations are done with the computer algebra system Wolfram Mathematica.

An Integrable Model for the Dynamics of Planetary Mean-motion Resonances

Abstract I consider the dynamics of mean-motion resonances (MMRs) between pairs of coplanar planets and derive a new integrable Hamiltonian model for the planets’ resonant motion. The new model generalizes integrable Hamiltonians previously derived for first-order resonances to the case of higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such truncated expansion is necessary to derive an integrable model. This enables the new model to accurately capture the dynamics of both first- and higher-order resonances for eccentricities up to orbit crossing. I demonstrate that predictions of the new integrable model agree well with numerical integrations of resonant planet pairs. Finally, I explore the secular evolution of resonant planets’ eccentricities. I show that the secular dynamics are governed by conservation of an angular-momentum-deficit-like quantity. I also demonstrate that secular frequencies depend on planets’ resonant libration amplitude and this generally gives rise to a secular resonance inside the MMR at large libration amplitudes. The integrable model derived in this work can serve as a framework for analyzing the dynamics of planetary MMRs in a wide variety of contexts.