Triangular Points Research Articles

Two-maneuver transfers from the collinear L2 point to the triangular L4 point in the planar Earth-Moon system

China’s Chang’E-4 (CE-4) probe has started its exploration on the far side of the Moon thanks to the relay satellite---Queqiao---which is around the Earth-Moon collinear libration point L2 and provides continuous communication links between the lander and ground stations. The triangular libration points of the Earth-Moon system, L4 and L5, have long been considered as potential locations for future space applications. As a possible extension of Queqiao or other future missions around L2, the work presented here contributes to constructing transfers from the collinear libration point (L2) to the triangular libration points. Taking the L4 point as an example, two types of two-maneuver transfers from a planar Lyapunov orbit around L2 to the periodic orbit around L4 are investigated in the planar Earth-Moon system and compared with each other, including the type-I transfer utilizing invariant manifolds along with powered lunar gravity assist and the type-II transfer using the Jacobi integral. Our studies indicate that type-I transfer saves more Δv cost than type-II transfer. For the cases studied by us, the minimum Δv cost is about 0.1636 km/s for type-I transfer, with a corresponding transfer time of 169.84 days. For type-II transfer and the cases studied by us, the corresponding values are 0.4745 km/s and 137.19 days. At the end of the paper, the type-I transfer is generalized (with some modifications) to the ephemeris model and some example transfer trajectories are given. Although only the transfer from L2 to L4 is studied in this paper, the same orbit design strategy can be generalized to the transfer to the triangular libration point L5, and the transfers from the collinear libration point L1 to triangular libration points.

Capturing an Asteroid via Triangular Libration Points

This paper develops a new asteroid capture mechanism via triangular libration points based on their stable structure and chaotic motion nearby. Different from collinear libration points, the triangular libration points are linearly stable surrounded by long- and short-period orbit families in the planar case. First, their dynamical structure is explored by the Lie series method and an analytical approximation of long-period orbits is proposed. Then, massive escaping trajectories from the parking orbits are generated by a chaotic-assisted method on the finite time Lyapunov exponent map. The trajectories from the asteroid’s natural orbit to the Earth–moon system are constructed by modified Lambert transfers. Next, the fuel consumption for all potential near-Earth asteroids is evaluated ergodically in both perihelion and aphelion cases based on their initial condition, which guides the selection of the target. Finally, it is applied to 20 asteroids, and the minimum fuel consumption () appears capturing asteroid 2015DU. Furthermore, a postcapture maneuver strategy based on a Poincaré section is proposed using long-period orbits to adjust the parking orbits and to satisfy the mission requirements.

On the secondary’s rotation in a synchronous binary asteroid

ABSTRACT This article studies the secondary’s rotation in a synchronous binary asteroid system in which the secondary enters the 1:1 spin-orbit resonance. The model used is the planar full two-body problem, composed of a spherical primary plus a triaxial ellipsoid secondary. Compared with classical spin-orbit work, there are two differences: (1) influence of the secondary’s rotation on the mutual orbit is considered and (2) instead of the Hamiltonian approach, the approach of periodic orbits is adopted. Our studies find the following. (1) The genealogy of the two families of periodic orbits is the same as that of the families around triangular libration points in the restricted three-body problem. That is, the long-period family terminates on to a short-period orbit travelling N times. (2) In the limiting case where the secondary’s mass is negligible, our results can be reduced to classical spin-orbit theory, by equating the long-period orbit with free libration and the short-period orbit with the forced libration caused by orbit eccentricity. However, the two models show obvious differences when the secondary’s mass is non-negligible. (3) By studying the stability of periodic orbits for a specific binary asteroid system, we are able to obtain the maximum libration amplitude of the secondary (which is usually less than 90°) and the maximum mutual orbit eccentricity that does not break the secondary’s synchronous state. We also find an anti-correlation between the secondary’s libration amplitude and the orbit eccentricity. The (65803) Didymos system is taken as an example to show the results.

A new species of Limnophyes Eaton (Diptera: Chironomidae) from Xizang, China.

A new species of the genus Limnophyes Eaton, 1875 from Xizang, China is described and figured based on adult males. Limnophyes nudus sp. nov. belongs to the minimus group and can be easily separated from other species by antenna with 13 flagellomeres, higher AR of 0.85-1.00, bluntly triangular anal point, phallapodeme with ring-like appendix, inferior volsella bare and gonostylus with bubble-like inner margin.

Pointing at fossils: pXRF identification of a fossilized shell projectile point

A shell artifact recovered during wet-screening of cultural sediments from CA-SDI-39 (San Diego, CA) had several unusual properties. Fashioned into a triangular projectile point using lithic flaking techniques, the artifact was unlike any other shell tools in the associated assemblage or reported elsewhere in California. Unlike other shell tools in the assemblage, the artifact was not flakey, and appeared to have been formed from a very large shell. This combination of characteristics suggested possible origins from fossilized shells in Pliocene-Pleistocene aged marine terraces. Destructive sampling was not possible, so pXRF was used to investigate chemical evidence for fossilization, or other possible treatments. Results suggest that this artifact was formed from shell that formed under substantially different water conditions than present, evidencing recrystallization consistent with fossil or subfossil shell, but an incomplete mineral conversion.

Accuracy of registrations between cone-beam computed tomography and conventional computed tomography images and dose mapping methods in RaySearch software for the bladder during brachytherapy of cervical cancer patients.

PurposeThe aim of the study was to assess selected methods of image registration available in the RaySearch software and their impact on the accuracy of mapping of doses deposited in the bladder during brachytherapy (BRT) of cervical cancer in images used during external beam radiotherapy (EBRT).Material and methodsThe study was based on data from ten patients. Cone-beam computed tomography (CBCT) images (BRT) were aligned with CT images (EBRT) using four registration methods: Reg_1 (rigid), Reg_2a, Reg_2b (hybrid), and Reg_3 (biomechanical). Image mapping accuracy was evaluated based on bladder’s anatomy. Sørensen-Dice coefficient (DSC) values were analyzed for all the registrations. Discrepancies between triangular mesh points set on the basis of bladder contours were analyzed. Dose distributions from BRT were transformed according to registration results and mapped on CT images. Original BRT doses deposited in 2 cm3 volume of the bladder were compared to those transformed and associated with bladder’s volume determined on CT images.ResultsMean DSC values amounted to 0.36 (Reg_1), 0.87 and 0.88 (Reg_2a and Reg_2b), and 0.97 (Reg_3). Significant differences were found between DSC for the following comparisons: Reg_3/Reg_1 (p = 0.001), Reg_2a/Reg_1 (p = 0.011), and Reg_2b/Reg_1 (p = 0.014). The lowest discrepancies between triangular mesh points were for Reg_3 (p < 0.001, Reg_3 vs. Reg_1, and p = 0.039, Reg_3 vs. Reg_2b). Finally, the lowest discrepancies between the original and transformed doses were found for Reg_3. Nevertheless, only 5 out of 10 observations for Reg_3 yielded error of less than 5%.ConclusionsBiomechanical registration (Reg_3) enabled the most accurate alignment between CBCT and CT images. Satisfactory registration results of anatomical structures do not guarantee a correct mapping of primary BRT doses on the bladder delineated on CT images during EBRT. The results of dose transformation based on biomechanical registration had an error of less than 5% for only 50% of the observations.

Systematic survey of the dynamics of Uranus Trojans

Context. The discovered Uranus Trojan (UT) 2011 QF99 and several candidate UTs have been reported to be in unstable orbits. This implies that the stability region around the triangular Lagrange points L4 and L5 of Uranus should be very limited. Aims. In this paper, we aim to locate the stability region for UTs and find out the dynamical mechanisms responsible for the structures in the phase space. The null detection of primordial UTs also needs to be explained. Methods. Using the spectral number as the stability indicator, we constructed the dynamical maps on the (a0, i0) plane. The proper frequencies of UTs were determined precisely with a frequency analysis method that allows us to depict the resonance web via a semi-analytical method. We simulated radial migration by introducing an artificial force acting on planets to mimic the capture of UTs. Results. We find two main stability regions: a low-inclination (0° −14°) and a high-inclination regime (32° −59°). There is also an instability strip in each of these regions at 9° and 51°, respectively. These strips are supposed to be related with g − 2g5 + g7 = 0 and ν8 secular resonances. All stability regions are in the tadpole regime and no stable horseshoe orbits exist for UTs. The lack of moderate-inclined UTs is caused by the ν5 and ν7 secular resonances, which could excite the eccentricity of orbits. The fine structures in the dynamical maps are shaped by high-degree secular resonances and secondary resonances. Surprisingly, the libration centre of UTs changes with the initial inclination, and we prove it is related to the quasi 1:2 mean motion resonance (MMR) between Uranus and Neptune. However, this quasi-resonance has an ignorable influence on the long-term stability of UTs in the current planetary configuration. About 36.3% and 0.4% of the pre-formed orbits survive fast and slow migrations with migrating timescales of 1 and 10 Myr, respectively, most of which are in high inclination. Since low-inclined UTs are more likely to survive the age of the solar system, they make up 77% of all such long-life orbits by the end of the migration, making a total fraction up to 4.06 × 10−3 and 9.07 × 10−5 of the original population for fast and slow migrations, respectively. The chaotic capture, just like depletion, results from secondary resonances when Uranus and Neptune cross their mutual MMRs. However, the captured orbits are too hot to survive until today. Conclusions. About 3.81% UTs are able to survive the age of the solar system, among which 95.5% are on low-inclined orbits with i0 &lt; 7.5°. However, the depletion of planetary migration seems to prevent a large fraction of such orbits, especially for the slow migration model. Based on the widely adopted migration models, a swarm of UTs at the beginning of the smooth outward migration is expected and a fast migration is favoured if any primordial UTs are detected.

Location and stability of the triangular Lagrange points in photo-gravitational elliptic restricted three body problem with the more massive primary as an oblate spheroid

The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit  for stability remains the same.  Â

Stability of triangular equilibrium points of oblate infinitesimal influenced by oblate and radiating primaries in Elliptic Restricted Three Body Problem based on Floquet’s theory

This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of μ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal.

Single impulsive transfer to the Earth–Moon triangular point L4 in a bicircular model

Abstract This paper investigates a single impulsive strategy of transferring a spacecraft from the Earth to the Earth–Moon triangular point L4 in the Sun–Earth–Moon bicircular model (BCM). The motion of the spacecraft at L4 is influenced by the Sun's perturbation and different Sun's initial phase angles can cause different motion. The motion that can easily escape the vicinity of L4 in a backward propagation is utilized to design the single impulsive transfer to L4 and then the window of easily approach (WOEA) to L4 is defined by analyzing the change in the Jacobi energy of the motion. Accordingly, the perigee of the motion can be obtained by propagating initial state of the spacecraft at L4 backwards. The transfer whose perigee height is approximated to that of the parking low Earth orbit (LEO) can be utilized as an initial guess, and therefore the single impulsive transfer can be calculated by correcting the Sun's initial phase angle and the flight time. Meanwhile, by comparing the changes in the Jacobi energy and the energy of the spacecraft, it can be found that the lunar flyby enables the single impulsive transfer to be accomplished. Since only one maneuver is required during an entire transfer, this transfer strategy may be more suitable for the small spacecraft that has a limited propulsion capability.

Road map to L4/L5 with a solar sail

This paper explores the capability of solar sails to transfer a probe from the displaced Sun-Earth L1 and L2 libration points to the region of practical stability (RPS) around the triangular equilibrium points L4 and L5. If the sailcraft arrives inside the RPS with zero synodical velocity, it will remain there with minor station keeping requirements. Moreover, the location of the RPS is ideal for space weather missions as the Sun can be observed from a different angle compared to spacecraft orbiting the L1 point. The unstable manifolds of the displaced L1 and L2 points come close to these regions providing opportunities for natural transfer trajectories. By varying the solar sail orientation along the manifold, the dynamics can be altered and simple transfer trajectories that reach the RPS with few sail maneuvers are enabled. However, these trajectories are not optimal from a transfer time perspective, but can serve as suitable initial guesses for a direct optimization method to find minimum-time transfers between the displaced L1 and L2 points and the RPS at L4 and L5.

Fragments of the Moon Formation: Geophysical Consequences of the Giant Impact

The most likely scenario for the Moon formation is given by calculations of the Giant Impact (GI) of the ProtoEarth with a protoplanet with a mass close to Mars. During the GI, gases and silicate fragments with a total mass of about 55 to 70% of the mass of the Moon go to infinity, but infinity for the runaway particles is infinity in the terrestrial reference frame. In the Solar System, these fragments go into finite orbits with periods both less and more than a year. The most important feature of their orbits is that they all pass through the region of the Earth’s orbit where the GI occurred. A concentrated gas–dust flow was formed there with a fading intensity; it existed for about a million years. Numerically solving the three-body problem yields numerous fragment trajectories. The probabilities of fragments colliding with the Earth and the Moon are estimated as a function of time after the GI. The scenario of fragments falling into the L4 and L5 triangular Lagrange points and the trajectories of their evolution is considered. The most important geophysical consequence of these collisions was the formation of the terrestrial atmosphere and ocean from the concentrated gas–dust flow.

Effects of Triaxiality and Radiation Pressure on Existence of Resonance Stability of Triangular Equilibrium Points in Elliptical Restricted Three-Body Problem

This paper deals with the effects of radiation and triaxiality on linear stability of the triangular librations points for small value of eccentricity ‘\(e\)’ in presence of resonance. The study is carried out for various values of radiation pressure and Triaxiality parameter of both Primaries. It is found that the parametric resonance is only possible at the resonance frequency \(\omega_2 =1/2\) in elliptical case.

Locations of Lagrangian points and periodic orbits around triangular points in the photo gravitational elliptic restricted three-body problem with oblateness

Locations of the Lagrangian points are computed and periodic orbits are studied around the triangular points in the photogravitational elliptic restricted three-body problem (ER3BP) by considering the more massive primary as the source of radiation and smaller primary as an oblate spheroid. A new mean motion taken from Sharma et al. [13] is used to study the effect of radiation pressure and oblateness of the primaries. The critical mass parameter  that bifurcates periodic orbits from non-periodic orbits tends to reduce with radiation pressure and oblateness. The transition curves defining stable region of orbits are drawn for different values of radiation pressure and oblateness using the analytical method of Bennet [14]. Tadpole orbits with long- and short- periodic oscillations are obtained for Sun-Jupiter and Sun-Saturn systems. Â

A Modified Targeting Strategy for Station-Keeping of Libration Point Orbits in the Real Earth-Moon System

In this paper, a modified targeting strategy is developed for missions on libration point orbits (LPOs) in the real Earth-Moon system. In order to simulate a station-keeping procedure in a dynamic model as realistic as possible, LPOs generated in the circular restricted three-body problem (CRTBP) are discretized and reconverged in a geocentric inertial system for later simulations. After that, based on the dynamic property of the state transition matrix, a modified strategy of selecting target points for station-keeping is presented to reduce maneuver costs. By considering both the solar gravity and radiation pressure in a nominal LPO design, station-keeping simulations about fuel consumption for real LPOs around both collinear and triangular libration points are performed in a high-fidelity ephemeris model. Results show the effectivity of the modified strategy with total maneuver costs reduced by greater than 10% for maintaining triangular LPOs.

Forced motions around triangular libration points by solar radiation pressure in a binary asteroid system

By using two tri-axial ellipsoids to approximate the two asteroids, forced orbits around triangular libration points of the binary asteroid system (BAS) induced by solar radiation pressure are studied. The work is firstly carried out in the doubly synchronous binary asteroid system (DSBAS). The results show that the amplitude of the forced periodic orbit can be large, even for small to moderate surface area-to-mass ratios of the spacecraft. The position, amplitude, and stability of these forced periodic orbits are influenced by the asteroids' non-spherical terms. Also, the stability of them may be different, depending on the Sun's motion direction w.r.t. to the BAS's orbit motion direction. This study is then generalized to the asynchronous and synchronous BAS (ABAS and SBAS, respectively). The forced orbits in the complete system are quasi-periodic orbits around the forced periodic orbit of the averaged system.

Effect on L&lt;sub&gt;4,5&lt;/sub&gt; in the ER3BP when Both Primaries are Radiating with Oblateness up to Zonal Harmonic J&lt;sub&gt;4&lt;/sub&gt;

This study examines the triangular points in the elliptic restricted three-body problem when both primaries are sources of radiation as well as oblate spheroids with oblateness up to zonal harmonic J4. The positions of triangular points and their critical mass ratio are seen to be affected by the eccentricity, semi major axis, radiation and oblateness of both primaries up to zonal harmonic J4. We highlight the effects of the said parameters on the locations of the triangular points of 61 CYGNI and STRUVE 2398. The triangular points of these systems are found to be unstable.

Upper triangular operator matrices and limit points of the essential spectrum

In this paper, we investigate the limit points set of essential spectrum of upper triangular operator matrices We prove that accσe(MC) ∪ W = accσe(A) ∪ accσe(B) where W is the union of certain holes in accσe(MC), which happen to be subsets of accσe(B) ∩ accσe(A). Also, several sufficient conditions for accσe(MC) = accσe(A) ∪ accσe(B) holds are given.

Effect on L&lt;sub&gt;4,5&lt;/sub&gt; in the ER3BP when Both Primaries are Radiating with Oblateness up to Zonal Harmonic J&lt;sub&gt;4&lt;/sub&gt;

This study examines the triangular points in the elliptic restricted three-body problem when both primaries are sources of radiation as well as oblate spheroids with oblateness up to zonal harmonic J4. The positions of triangular points and their critical mass ratio are seen to be affected by the eccentricity, semi major axis, radiation and oblateness of both primaries up to zonal harmonic J4. We highlight the effects of the said parameters on the locations of the triangular points of 61 CYGNI and STRUVE 2398. The triangular points of these systems are found to be unstable.

Basic Bifurcation Scenarios in Neighborhoods of Boundaries of Stability Regions of Libration Points in the Three-Body Problem

In this paper, we construct stability regions (in the linear approximation) of triangular libration points for the planar, restricted, elliptic three-body problem and examine bifurcations that occur when parameters of the system pass through the boundaries of these regions. A new scheme for the construction of stability regions is presented, which leads to approximation formulas describing these boundaries. We prove that on one part of the boundary, the main scenario of bifurcation is the appearance of nonstationary 4π-periodic solutions that are close to a triangular libration point, whereas on the other part, the main scenario is the appearance of quasiperiodic solutions.