Oblateness Coefficient Research Articles

Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System

Physical properties of the Sun (orientation of rotation axis, oblateness coefficient J2⊙, and change rate of the gravitational parameter μ˙⊙) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and Trans-Neptunian objects (TNOs). Among the many kinds of observations used to determine the orbits and physical properties of the bodies, the most important for our study are precise interplanetary ranging data: Earth–Mercury ranges from MESSENGER spacecraft and Earth–Mars ranges from Odyssey and MRO. The findings allow us to improve the model of the Sun in modern planetary ephemerides. First, the dynamically determined direction of the Sun’s pole is ≈2° off the visible axis of rotation of the Sun’s surface, which is corroborated by present knowledge of the Sun’s interior. Second, the change rate of the Sun’s gravitational parameter is found to be smaller (in absolute value) than the nominal value derived from the estimate of mass loss through radiation and solar wind. Possible interpretations are discussed.

Orbital dynamics in the Hill problem with oblateness

In this paper, we investigate the impact of the oblateness of the secondary body on the motion of a test particle in the near vicinity of the secondary body, specifically in the context of the extended version of the planar Hill problem. To achieve this objective, we conduct a comprehensive survey by systematically classifying the initial conditions of trajectories and scanning the phase space using two-dimensional (2D) maps on various planes. We then numerically integrate the starting conditions on these maps and classify the final states of the test particles as either bounded or unbounded. Bounded orbits are further subclassified as collision orbits and regular (or chaotic), while unbounded orbits are sub-classified according to the sector of the (x,y) plane through which the particle escapes the potential well. Our results reveal that increasing values of the oblateness coefficient reduce the number of escaping orbits and limit the extension of islands of regular motion (in the Liouville–Arnold sense). Regular bounded motion occurs only at larger distances from the secondary body.

The Neptunian gravity estimated from the motion of Triton based on astrometric observations

Context. Accurate gravity and ephemerides estimations for Neptune and its satellites are necessary for the forthcoming deep space exploration missions targeting its system. In addition, these estimations are also meaningful for the modeling of Neptune's interior and for solar system dynamics studies. The body of astrometric data concerning Triton has been accumulating for about two hundred years, but more accurate updates to the ephemerides of this moon and revisions to the relevant gravity parameters would be possible with more precise astrometric data. The new Gaia catalog of star positions plus observations from Voyager 2 and the Hubble Space Telescope provide such a basis for high-precision astrometry and to complement and extend the existing body of data. Aims. We aim to report integrated orbital fits for Triton based on all the available astrometric data from 1847 to 2020, including observations from Earth-based telescopes, Voyager 2, and the Hubble Space Telescope. We also estimate the Neptunian gravity using the motion of Triton. Methods. Triton’s orbital solution was determined by a weighted least-squares method to fit the model to the most complete astrometric data set to date. The DOP853 algorithm was adopted in the numerical integration calculations. For the dynamical model parameters, our orbital model for Triton is similar to the NEP081 but with an update. The perturbations from the inner satellites (Naiad, Thalassa, Despina, Galatea, Larissa, Proteus, and Hippocamp) were considered by adding corrections to J2 and J4 for Neptune. As the gravitational oblateness coefficient of Neptune is correlated with its orientation, the pole parameters were thus kept fixed in the integration when estimating Neptunian gravity. A Monte Carlo analysis was performed, however, to obtain reliable accuracy estimations and to assess the uncertainty of pole parameters on the results’ formal error. Results. We provide a new orbit and dynamical model values for Triton. The estimated accuracy of the model we built and updated fit all the astrometric data. The RMS of the residuals was 0.074 arcsec in the right ascension and 0.071 arcsec in declination. The RMS was 0.102 arcsec for X and 0.139 arcsec for Y in differential coordinates. The RMS for the position angle was 0.834 degrees, and the angular separation distance was 0.257 arcsec for the data collected before 1960. The orbit of Triton was well determined with the orbit differences from NEP081 and NEP097 (so far the latest Triton ephemerides from Jet Propulsion Laboratory) as being less than 300 km (about 15 mas) during the observation coverage period of this study. The large body of astrometric data for Triton over a time interval from 1847 to 2020 was used to constrain its position at the initial epoch, allowing us to reduce formal uncertainty to about 3 km. Based on the most complete weighting astrometric observations of Triton, the estimated mass of the Neptune system is GMs = 6 836 525.210 ±19.526 km3 s−2. Our revised gravity model yields J2 = 3401.655 ±1.850 × 10−6 and J4 = −33.294 ±10.000 × 10−6. The astrometric observations showed little sensitivity to GMs and J4, but acted well on J2. A Monte Carlo method was used to analyze the error caused by a variation in the pole parameters and showed that J2 = 3401.655 ± 3.994 × 10−6 was a more realistic error.

A torsion-based solution to the hyperbolic regime of the J_2-problem

A popular intermediary in the theory of artificial satellites is obtained after the elimination of parallactic terms from the J_2-problem Hamiltonian. The resulting quasi-Keplerian system is in turn converted into the Kepler problem by a torsion. When this reduction process is applied to unbounded orbits, the solution is made of Keplerian hyperbolae. For this last case, we show that the torsion-based solution provides an effective alternative to the Keplerian approximation customarily used in flyby computations. Also, we check that the extension of the torsion-based solution to higher orders of the oblateness coefficient yields the expected convergence of asymptotic solutions to the true orbit.

Regularization of the Hill four-body problem with oblate bodies

We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the 4th, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the smallest massive body passes through the zero value.

Stability analysis of first order resonant periodic orbit

In this work, the perturbed restricted three–body problem is investigated numerically. The problem is applied to three real systems: Saturn–Hyperion, Saturn–Titan, and Earth–Moon, for analyzing the stability of first order resonant periodic orbits. In particular, the nature of periodic orbits is studied for all three systems, where their masses ratios represent small, moderate and large values. Using different types of numerical techniques, we have identified how the parameter of mass ratio, the Jacobi constant, and the oblateness coefficient affect the geometrical properties, and the periodic solutions of system.

Laplace-like resonances with tidal effects

The first three Galilean satellites of Jupiter, Io, Europa, and Ganymede, move in a dynamical configuration known as the Laplace resonance, which is characterized by a 2:1 ratio of the rates of variation in the mean longitudes of Io-Europa and a 2:1 ratio of Europa-Ganymede. We refer to this configuration as a 2:1&2:1 resonance. We generalize the Laplace resonance among three satellites, S1, S2, and S3, by considering different ratios of the mean-longitude variations. These resonances, which we call Laplace-like, are classified as first order in the cases of the 2:1&2:1, 3:2&3:2, and 2:1&3:2 resonances, second order in the case of the 3:1&3:1 resonance, and mixed order in the case of the 2:1&3:1 resonance. We consider a model that includes the gravitational interaction with the central body together with the effect due to its oblateness, the mutual gravitational influence of the satellites S1, S2, and S3 and the secular gravitational effect of a fourth satellite S4, which plays the role of Callisto in the Galilean system. In addition, we consider the dissipative effect due to the tidal torque between the inner satellite and the central body. We investigate these Laplace-like resonances by studying different aspects: (i) we study the survival of the resonances when the dissipation is included, taking two different expressions for the dissipative effect in the case of a fast- or a slowly rotating central body, (ii) we investigate the behavior of the Laplace-like resonances when some parameters are varied, specifically, the oblateness coefficient, the semimajor axes, and the eccentricities of the satellites, (iii) we analyze the linear stability of first-order resonances for different values of the parameters, and (iv) we also include the full gravitational interaction with S4 to analyze its possible capture into resonance. The results show a marked difference between first-, second-, and mixed-order resonances, which might find applications when the evolutionary history of the satellites in the Solar System are studied, and also in possible actual configurations of extrasolar planetary systems.

A study on the positions and velocity sensitivities in the restricted three-body problem with radiating and oblate primaries

This paper investigates triangular points' positions and velocity sensitivities in the circular restricted three-body problem framework for Luyten 726–8 and Achird systems. We observe that the possible boundary regions for both systems depend on the value of the Jacobian constant C and the oblateness coefficients (A1&A2). As the oblateness coefficients increase with other fixed parameters, the Jacobian constant value increases, making it possible to connect the primary regions. The Poincare's Surface of Section of the two binary systems has been utilized to display their sensitivity to change positions and velocities. With certain initial conditions, the systems possess regular orbits, however, the orbits become irregular when these initial conditions are changed. An increase in oblateness coefficients results in the singularities and stiffness of the non-linear system. Considering the range of stability and instability given as 0<μ<μc for stable triangular points and μc≤μ≤12 for unstable triangular points, we have established that the triangular points are unstable for the binary systems: Luyten 726–8 and Achird. In our analysis, for each set of values, there exists at least one complex root with a positive real part. Hence in the sense of Lyapunov, the triangular points are unstable for the systems as mentioned earlier.

A note on commensurability induced by oblateness at L3

t is known that there exists a near one-to-one commensurability ratio between the planar angular frequencies (s1,2,3) and the corresponding angular frequency (sz) in the z-direction at the three collinear points (L1,2,3) in the three-dimensional restricted three-body problem. It is significant for small and practically important values of the mass parameter (μ). In this note we have used a new mean motion expression which includes the secular effects of oblateness on argument of perigee, right ascension of ascending node and mean anomaly[1], which was used in the studies[2,3,4,5]. We establish that there is ono-to-one commensurability at the external points L3 (to the right of the more massive primary), when the more massive primary is considered as an oblate spheroid with its equatorial plane coincident with the plane of motion. This study will be useful in generating the halo orbits at L3. For Saturn-Titan system, the values of the μ and oblateness coefficient (A1) are 0.000236695 and 0.000039653936, respectively [1].It is interesting to note that the value of A1 obtained for one-to-one commensurability is 0.0000344978 for this system, which is very close to the actual value. Thus, halo orbits of small size can be generated in Saturn-Titan system at L3.

Stability of generalized elliptic restricted four body problem with radiation and oblateness effects

This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter.

Rotation and figure evolution in the creep tide theory: a new approach and application to Mercury

This paper deals with the rotation and figure evolution of a planet near the 3/2 spin–orbit resonance and the exploration of a new formulation of the creep tide theory (Folonier et al. in Celest Mech Dyn Astron 130:78, 2018). This new formulation is composed by a system of differential equations for the figure and the rotation of the body simultaneously (which is the same system of equations used in Folonier et al. 2018), different from the original one (Ferraz-Mello in Celest Mech Dyn Astron 116:109–140, 2013; Celest Mech Dyn Astron 122:359–389, 2015a. arXiv: 1505.05384) in which rotation and figure were considered separately. The time evolution of the figure of the body is studied for both the 3/2 and 2/1 spin–orbit resonances. Moreover, we provide a method to determine the relaxation factor $$\gamma $$ of non-rigid homogeneous bodies whose endpoint of rotational evolution from tidal interactions is the 3/2 spin–orbit resonance, provided that (i) an initially faster rotation is assumed and (ii) no permanent components of the flattenings of the body existed at the time of the capture in the 3/2 spin–orbit resonance. The method is applied to Mercury, since it is currently trapped in a 3/2 spin–orbit resonance with its orbital motion and we obtain $$4.8 \times 10^{-8}$$ s$$^{-1} \le \gamma \le 4.8 \times 10^{-9}$$ $$\mathrm{s}^{-1}$$. The equatorial prolateness and polar oblateness coefficients obtained for Mercury’s figure with such range of values of $$\gamma $$ are the same as the ones given by the Darwin–Kaula model (Matsuyama and Nimmo in J Geophys Res 114, E01010, 2009). However, comparing the values of the flattenings obtained for such range of $$\gamma $$ with those obtained from MESSENGER’s measurements (Perry et al. in Geophys. Res. Lett. 42, 6951–6958, 2015), we see that the current values for Mercury’s equatorial prolateness and polar oblateness are 2–3 orders of magnitude larger than the values given by the tidal theories.

Orbits and resonances of the regular moons of Neptune

We report integrated orbital fits for the inner regular moons of Neptune based on the most complete astrometric data set to date, with observations from Earth-based telescopes, Voyager 2, and the Hubble Space Telescope covering 1981–2016. We summarize the results in terms of state vectors, mean orbital elements, and orbital uncertainties. The estimated masses of the two innermost moons, Naiad and Thalassa, are GMNaiad = 0.0080 ± 0.0043 km3 s−2 and GMThalassa = 0.0236 ± 0.0064 km3 s−2, corresponding to densities of 0.80 ± 0.48 g cm−3 and 1.23 ± 0.43 g cm−3, respectively. Our analysis shows that Naiad and Thalassa are locked in an unusual type of orbital resonance. The resonant argument 73λ̇Thalassa−69λ̇Naiad−4Ω̇Naiad≈0 librates around 180° with an average amplitude of ~66° and a period of ~1.9 years for the nominal set of masses. This is the first fourth-order resonance discovered between the moons of the outer planets. More high precision astrometry is needed to better constrain the masses of Naiad and Thalassa, and consequently, the amplitude and the period of libration. We also report on a 13:11 near-resonance of Hippocamp and Proteus, which may lead to a mass estimate of Proteus provided that there are future observations of Hippocamp. Our fit yielded a value for Neptune's oblateness coefficient of J2 = 3409.1 ± 2.9 × 10−6.

Orbit classification in the Copenhagen problem with oblate primaries

AbstractThis article aims to numerically investigate the orbital dynamics of the planar circular‐restricted three‐body problem, where the two primary bodies have equal masses and an oblate spheroid shape. By numerically integrating several large sets of initial conditions of both retrograde and prograde orbits, we classify them into three main categories: (a) bounded (regular or chaotic), (b) escaping, and (c) collision orbits. We reveal the motion of the test particle by presenting color‐coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy (or equivalently of the Jacobi constant) as well as of the oblateness coefficient. Our numerical results are compared with related previous ones, corresponding to the classical version of the planar‐restricted three‐body problem. The presented outcomes have practical physical applications as they can be used for modeling the motion of a massless test particle in the gravitational field of a binary system of two oblate exoplanets.

Analytic orbit theory with any arbitrary spherical harmonic as the dominant perturbation

In the gravitational potential of Earth, the oblateness term is the dominant perturbation, with its coefficient at least three orders of magnitude greater than that of any other zonal or tesseral spherical harmonic. Therefore, analytic orbit theories (or satellite theories) are developed using the Keplerian Hamiltonian as the unperturbed solution, oblateness term as the first-order and the remaining spherical harmonics as the second-order perturbations. These orbit theories are generally constructed by applying multiple near-identity canonical transformations to the perturbed Hamiltonian in conjunction with averaging to obtain a secular Hamiltonian, from which the short-period and long-period terms are removed. If the oblateness term is the only first-order perturbation, then the long-period terms appear only in the second- or higher-order terms of the single-averaged Hamiltonian, from which the short-period terms are removed. These second-order long-period terms are separated from the Hamiltonian by the first-order generating function using a second canonical transformation. This results in a secular Hamiltonian dependent only on the momenta. However, in the case of other gravitational bodies with more deformed shapes compared to Earth such as moons and asteroids, the oblateness coefficient may have the same order of magnitude as some of the higher spherical harmonic coefficients. If these higher harmonics are treated as the first-order perturbation along with the oblateness term, then the long-period terms appear in the first-order single-averaged Hamiltonian. These first-order long-period terms cannot be separated using the generating function in the conventional way. This problem occurs because the zeroth-order Hamiltonian, i.e., the Keplerian part, is degenerate in the angular momentum. In this paper, a new approach to the long-period transformation is proposed to resolve this issue and obtain a fully analytic orbit theory when for the perturbing gravitational body, any arbitrary zonal or tesseral harmonic is the dominant perturbation. The proposed theory is closed form in the eccentricity as well. It is applied to predict the motion of artificial satellites for the two test cases: a lunar orbiter and a satellite of 433 Eros asteroid. The prediction accuracy is validated against the numerical propagation using a force model with $$6\times 6$$ gravity field.

Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem

This paper studies numerically the existence of collinear and non-collinear equilibrium points and their linear stability in the frame work of photogravitational circular restricted four-body problem with Stokes drag acting as a dissipative force and considering the first primary as a radiating body and the second primary as an oblate spheroid. The mass of the fourth body is assumed to be infinitesimal and does not affect the motion of the three primaries which are always at the vertices of an equilateral triangle (Lagrangian configuration). It is found that at constant dissipative force, and a simultaneous increase in both radiation pressure and oblateness coefficients, the curves that define the path of the motion of the infinitesimal body are found to shrink due to shifts in the positions of equilibrium points. All the collinear and non-collinear equilibrium points are found to be linearly unstable under the combined effect of radiation pressure, oblateness and Stokes drag. The energy integral is seen to be time dependent due to the presence of the drag force. More so the dynamic property of the system is investigated with the help of Lyapunov characteristic exponents (LCEs). It is found that the system is chaotic as the trajectories locally diverge from each other and the equilibrium points are chaotic attractors. We justified the relevance of the model in astronomy by applying it to a stellar system (Gliese 667C) and another found that both the existence and stability of the equilibrium points of any restricted few body system greatly depend of the value of the mass parameter.

On the classification of orbits in the three-dimensional Copenhagen problem with oblate primaries

The character of motion for the three-dimensional circular restricted three-body problem with oblate primaries is investigated. The orbits of the test particle are classified into four types: non-escaping regular orbits around the primaries, trapped chaotic (or sticky) orbits, escaping orbits that pass over the Lagrange saddle points L2 and L3, and orbits that lead the test particle to collide with one of the primary bodies. We numerically explore the motion of the test particle by presenting color-coded diagrams, where the initial conditions are mapped to the orbit type and studied as a function of the total orbital energy, the initial value of the z-coordinate and the oblateness coefficient. The fraction of the collision orbits, measured on the color-coded diagrams, show an algebraic dependence on the oblateness coefficient, which can be derived by simple arguments.

Halo orbit of regularized circular restricted three-body problem with radiation pressure and oblateness

In this paper, computation of the halo orbit for the KS-regularized photogravitational circular restricted three-body problem is carried out. This work extends the idea of Srivastava et al. (Astrophys. Space Sci. 362: 49, 2017) which only concentrated on the (i) regularization of the 3D-governing equations of motion, and (ii) validation of the modeling for small out-of-plane amplitude ( $$A_z =110000$$ km) assuming the third-order analytical approximation as an initial guess with and without differential correction. This motivated us to compute the halo orbits for the large out-of-plane amplitudes and to study their stability analysis for the regularized motion. The stability indices are described as a function of out-of-plane amplitude, mass reduction factor and oblateness coefficient. Three different Sun–planet systems: the Sun–Earth, Sun–Mars and the Sun–Jupiter are chosen in this study. Stable halo orbits do not exist around the $$L_{1}$$ point, however, around the $$L_{2}$$ point stable halo orbits are found for the considered systems.

Comparing the basins of attraction for several methods in the circular Sitnikov problem with spheroid primaries

The circular Sitnikov problem, where the two primary bodies are prolate or oblate spheroids, is numerically investigated. In particular, the basins of convergence on the complex plane are revealed by using a large collection of numerical methods of several order. We consider four cases, regarding the value of the oblateness coefficient which determines the nature of the roots (attractors) of the system. For all cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distribution of the iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical computations indicate that most of the iterative schemes provide relatively similar convergence structures on the complex plane. However, there are some numerical methods for which the corresponding basins of attraction are extremely complicated with highly fractal basin boundaries. Moreover, it is proved that the efficiency strongly varies between the numerical methods.

On the Newton–Raphson basins of convergence of the out-of-plane equilibrium points in the Copenhagen problem with oblate primaries

The Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton–Raphson basins of convergence, related to the out-of-plane equilibrium points. The evolution of the position of the libration points is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two-dimensional planes, are revealed by using the multivariate Newton–Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry of the basins of convergence. The convergence regions are also related with the required number of iterations and also with the corresponding probability distributions. The degree of the fractality is also determined by calculating the fractal dimension and the basin entropy of the convergence planes.

Investigating the Basins of Convergence in the Circular Sitnikov Three-Body Problem with Non-spherical Primaries

In this work we numerically explore the Newton-Raphson basins of convergence, related to the equilibrium points, in the Sitnikov three-body problem with non-spherical primaries. The evolution of the position of the roots is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two dimensional planes, are revealed by using the classical Newton-Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry as well as the overall properties of the convergence regions. The basins of convergence are also related with the required number of iterations and also with the corresponding probability distributions.