arXiv subjects

Bhanu Kumar

Publications and source records attributed to Bhanu Kumar.

10 recordsLinked to original sources

arXiv2026

The Astrodynamics Primer on Cislunar and Translunar Space

The Earth-Moon environment is multiscale and strongly nonuniform, yet it is often discussed as a single cislunar regime. In reality, circumterrestrial space is partitioned by changes in perturbation hierarchy, by gateway topology, and secular and resonant structures that differ qualitatively across the domain. This paper develops a unified spatiographic description of that structure, combining perturbative theory, semi-analytical resonance cartography, restricted multi-body dynamics, and direct numerical mapping. The terrestrial-cislunar transition is sharpened through the Laplace radius, beyond which lunisolar torques overtake the classical oblateness-dominated picture. Earthward of the Moon, cislunar space divides into a secularly dominated inner zone and an outer zone structured by interior lunar mean-motion resonances. Near the Moon, circumlunar space forms a distinct dynamical enclave organized by the EM gateway geometry and the lunar SOI. Beyond the Moon, translunar space forms an outer circumterrestrial province in which the Moon acts as an interior perturber, the Sun remains an exterior perturber, and the resulting dynamics acquire a mixed lunisolar secular and resonant architecture before weakening outward toward heliocentric behavior. These results are synthesized through MEGNO and fate-class cartographies across six numerical map domains. The maps reveal where quasiperiodic confinement survives, where resonance overlap and gateway transport produce sticky residence and organized escape, and where solar forcing becomes a qualitative architectural ingredient. Placed alongside a curated catalog of satellites and debris, the framework provides a dynamical geography of this environments that clarifies the transitions among cislunar, circumlunar, and translunar motion and offers a more precise basis for interpreting stability, transport, and long-term Earth-bound behavior.

astro-ph.EP↗

arXiv2026

A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems

Given a 4D symplectic map $F_0$ that has a normally hyperbolic invariant cylinder foliated by invariant tori, those with rational rotation numbers are themselves foliated by subharmonic periodic orbits (SPOs). If $F_0$ is part of a perturbative family $F_\varepsilon$, one is often interested in computing those SPOs which persist for $\varepsilon >0$. Assuming that a persisting SPO of $F_0$ has been identified, in this paper, we develop a quasi-Newton method which solves for the SPO simultaneously with its Floquet vectors and multipliers. This in turn enables continuation by the perturbation parameter $\varepsilon$. The resulting SPO and Floquet vectors are then used to compute Taylor parameterizations of the SPO's weak stable and unstable manifolds, if they exist. Our quasi-Newton method is based on an adaptation of the parameterization method for invariant tori, with this paper being the first-ever to apply such a framework to directly compute periodic orbit points themselves. The new algorithm improves on efficiency compared to prior multi-shooting methods for SPOs, and notably applies to the case of stroboscopic maps of 2.5 DOF Hamiltonian flows resulting from periodic perturbations of 2 DOF systems. The tools have been successfully used for studies of resonant orbits in perturbed real-life celestial systems, the results of which are summarized as a demonstration of the methods' utility.

math.DS↗

arXiv2025

Cislunar Resonant Transport and Heteroclinic Pathways: From 3:1 to 2:1 to L1

Understanding the dynamical structure of cislunar space beyond geosynchronous orbit is critical for both lunar exploration and for high-Earth-orbiting trajectories. In this study, we investigate the role of mean-motion resonances and their associated heteroclinic connections in enabling natural semi-major axis transport in the Earth-Moon system. Working within the planar circular restricted three-body problem, we compute and analyze families of periodic orbits associated with the interior 4:1, 3:1, and 2:1 lunar resonances. These families exhibit a rich bifurcation structure, including transitions between prograde and retrograde branches and connections through collision orbits. We construct stable and unstable manifolds of the unstable resonant orbits using a perigee-based Poincar\'e map, and identify heteroclinic connections - both between resonant orbits and with lunar $L_1$ libration-point orbits - across a range of Jacobi constant values. Using a new generalized distance metric to quantify the closeness between trajectories, we establish operational times-of-flight for such heteroclinic-type orbit-to-orbit transfers. These connections reveal ballistic, zero-$\Delta v$ pathways that achieve major orbit changes within reasonable times-of-flight, thus defining a network of accessible semi-major axes. Our results provide a new dynamical framework for long-term spacecraft evolution and cislunar mission design, particularly in regimes where lunar gravity strongly perturbs distant circumterrestrial orbits.

astro-ph.EP↗

arXiv2025

Multi-shooting parameterization methods for invariant manifolds and heteroclinics of 2 DOF Hamiltonian Poincar\'e maps, with applications to celestial resonant dynamics

Studying 2 degree-of-freedom (DOF) Hamiltonian dynamical systems often involves the computation of stable & unstable manifolds of periodic orbits, due to the homoclinic & heteroclinic connections they can generate. Such study is generally facilitated by the use of a Poincar\'e section, on which the manifolds form 1D curves. A common method of computing such manifolds in the literature involves linear approximations of the manifolds, while the author's past work has developed a nonlinear manifold computation method under the assumption that the periodic orbit intersects the chosen Poincar\'e section only once. However, linear manifold approximations may require large amounts of numerical integration for globalization, while the single-intersection assumption of the previous nonlinear method often does not hold. In this paper, a parameterization method is developed and implemented for computing such stable and unstable manifolds even in the case of multiple periodic orbit intersections with a chosen Poincar\'e section. The method developed avoids the need to compose polynomials with Poincar\'e maps - a requirement of some previous related algorithms - by using an intermediate step involving fixed-time maps. The step yields curves near the chosen Poincar\'e section lying on the flow's periodic orbit manifolds, which are used to parameterize and compute the Poincar\'e map manifold curves themselves. These last curves and parameterizations in turn enable highly-accurate computation of heteroclinics between periodic orbits. The method has already been used for various studies of resonant dynamics in the planar circular restricted 3-body problem, which are briefly summarized in this paper, demonstrating the algorithm's utility for real-world investigations.

math.DS↗

arXiv2025

Cislunar Mean-Motion Resonances: Definitions, Widths, and Comparisons with Resonant Satellites

Lunar mean-motion resonances (MMRs) significantly shape cislunar dynamics beyond GEO, forming stable-unstable orbit pairs with corresponding intermingled chaotic and regular regions. The resonance zone is rigorously defined using the separatrix of unstable resonant periodic orbits surrounding stable quasi-periodic regions. Our study leverages the planar, circular, restricted three-body problem (PCR3BP) to estimate the (stable) resonance widths and (unstable) chaotic resonance zones of influence of the 2:1 and 3:1 MMRs across various Jacobi constants, employing a Poincar\'e map at perigee and presenting findings in easily interpretable geocentric orbital elements. An analysis of the semi-major axis versus eccentricity plane reveals broader regions of resonance influence than those predicted by semi-analytical models based on the perturbed Kepler problem. A comparison with high-fidelity 3-dimensional ephemeris propagation of several spacecraft - TESS, IBEX, and Spektr-R - in these regions is made, which shows good agreement with the simplified CR3BP model.

astro-ph.EP↗

arXiv2023

4th Body-Induced Secondary Resonance Overlapping Inside Unstable Resonant Orbit Families: a Jupiter-Ganymede 4:3 + Europa Case Study

The overlapping of mean-motion resonances is useful for low or zero-propellant space mission design, but while most related prior work uses a planar CRTBP model, tours of multi-moon systems require using resonances affected by two moons. In this case study, we investigate Jupiter-Ganymede unstable 4:3 resonant orbits in a concentric circular restricted 4-body Jupiter-Europa-Ganymede model. We show that despite their high order, secondary resonances between the 4:3 orbits and Europa have a large effect, including 11/34, 12/37, 23/71, and 25/77. Computing newly generated objects inside the secondary resonances definitively confirms their overlap, which causes a complete structural change of the higher-energy unstable 4:3 orbits whose manifolds are most useful for low-TOF orbit transfers. We believe this phenomenon is general, with major implications for resonant orbit use in tour design.

astro-ph.EP↗

arXiv2021

Rapid GPU-Assisted Search and Parameterization-Based Refinement and Continuation of Connections between Tori in Periodically Perturbed Planar Circular Restricted 3-Body Problems

When the planar circular restricted 3-body problem (PCRTBP) is periodically perturbed, as occurs in many useful astrodynamics models, most unstable periodic orbits persist as whiskered tori. Intersections between stable and unstable manifolds of such tori provide natural heteroclinic pathways enabling spacecraft to greatly modify their orbits without using propellant. However, the 2D Poincar\'e sections used in PCRTBP studies no longer work to find these intersections. Thus, in this study, we develop new fast methods to search for and compute such heteroclinics. First, the dynamics are used to restrict the intersection search to only certain manifold subsets, greatly reducing the required computational effort. Next, we present a massively parallel procedure for carrying out this search by representing the manifolds as discrete meshes and adapting methods from computer graphics collision detection algorithms. Implementing the method in Julia and OpenCL, we obtain a 5-7x speedup by leveraging GPUs versus CPU-only execution. Finally, we show how to use manifold parameterizations to refine the approximate intersections found in the mesh search to very high accuracy, as well as to numerically continue the connections through families of tori; the families' Whitney differentiability enables interpolation of needed parameterizations. The ability to very rapidly find a heteroclinic intersection between tori of fixed frequencies thus allows the systematic exploration of intersections for tori of nearby frequencies as well, yielding a variety of potential zero-fuel spacecraft trajectories. We demonstrate the tools on the Jupiter-Europa planar elliptic RTBP.

math.DS↗

arXiv2021

Computation and Analysis of Jupiter-Europa and Jupiter-Ganymede Resonant Orbits in the Planar Concentric Circular Restricted 4-Body Problem

Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In this study, we compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-body problem (CCR4BP). Motivated by the 2:1 Laplace resonance between Europa and Ganymede's orbits, we then attempt the continuation of a Jupiter-Europa 3:4 resonant orbit from the CCR4BP into the Jupiter-Ganymede PCRTBP. We strongly believe that the resulting dynamical object is a KAM torus lying near but not on the 3:2 Jupiter-Ganymede resonance.

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arXiv2021

High-Order Resonant Orbit Manifold Expansions For Mission Design In the Planar Circular Restricted 3-Body Problem

In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections. In this paper, we study hyperbolic resonant periodic orbits in the planar circular restricted 3-body problem, and transfer trajectories between them, by: 1) determining where to search for resonant periodic orbits; 2) developing and implementing a parameterization method for accurate computation of their invariant manifolds as Taylor series; and 3) developing a procedure to compute intersections of the computed stable and unstable manifolds. We develop and implement algorithms that accomplish these three goals, and demonstrate their application to the problem of transferring between resonances in the Jupiter-Europa system.

math.DS↗

arXiv2021

Rapid and Accurate Methods for Computing Whiskered Tori and their Manifolds in Periodically Perturbed Planar Circular Restricted 3-Body Problems

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology improves on efficiency and accuracy compared to prior studies, and applies to a variety of periodic perturbations. We demonstrate the tools near resonances in the planar elliptic RTBP.

math.DS↗