A Journey into the Application of the Poincaré Map in the Circular Restricted Three-Body Problem

Abbasali, Ehsan; Ebrahimian, Fatemeh; Kosari, Amirreza; Bakhtiari, Majid

doi:10.22034/jsst.2025.1537

A Journey into the Application of the Poincaré Map in the Circular Restricted Three-Body Problem

Document Type : Original Research Paper

Authors

Ehsan Abbasali ¹

Fatemeh Ebrahimian ¹

Amirreza Kosari ¹

Majid Bakhtiari ²

¹ School of Aerospace Engineering, College of Interdisciplinary Science and Technologies, University of Tehran, Tehran, Iran

² School of Advanced Technologies. Iran University of Science and Technologies, Tehran, Iran

10.22034/jsst.2025.1537

Abstract

This article investigates the identification of periodic orbits and spacecraft attitude behavior in the Circular Restricted Three-Body Problem (CRTBP) using the Poincaré map as an innovative analytical tool. The CRTBP describes the motion of a spacecraft under the gravitational influence of two primary celestial bodies and presents challenges due to the absence of closed-form solutions for both orbital and attitude dynamics. To address this, the study employs the Poincaré map as an efficient numerical method to detect suitable initial conditions for periodic motion, thereby reducing mathematical complexity and improving computational performance. A notable innovation in this work is the integration of spacecraft attitude dynamics into the Poincaré-based analysis. Although the coupling is oneway, meaning attitude states do not influence the periodicity of orbital motion, they do affect the structure of the resulting Poincaré maps. The shape and position of islands on these maps vary with changes in the spacecraft’s inertia ratio, which demonstrates the method’s sensitivity to attitude behavior and its relevance to real-world applications. The proposed method is benchmarked against two classical techniques: the third-order approximation and Floquet theory-based approaches. Comparative analysis shows that the Poincaré map offers a superior balance of simplicity, computational efficiency, and accuracy. It achieves reliable identification of periodic orbits without requiring high-order models or intricate corrections. Overall, the study provides a novel and practical contribution to space mission design, offering a robust and accessible framework for analyzing trajectory and attitude dynamics in complex gravitational environments such as libration point missions.

Keywords

Poincaré

map

CRTBP

Periodic orbits

Three-body problem

Subjects

Aerospace Engineering

Article Title Persian

سفری به کاربرد نگاشت پوانکاره در مساله سه جسم محدود شده

Authors Persian

احسان عباسی ¹

فاطمه ابراهیمی ¹

امیررضا کوثری ¹

مجید بختیاری ²

¹ دانشکده مهندسی هوافضا، دانشکدگان علوم و فناوری‌های میان‌رشته‌ای، دانشگاه تهران، تهران، ایران

² دانشکده فناوری‌های نوین، دانشگاه علم و فناوری ایران، تهران، ایران

Abstract Persian

این مقاله به بررسی شناسایی مدارهای تناوبی و رفتار دینامیک وضعی فضاپیما در مسأله سه‌جسمی محدود دایروی (CRTBP) می‌پردازد و از نگاشت پوانکاره به‌عنوان یک ابزار تحلیلی نوآورانه استفاده می‌کند. مسأله CRTBP حرکت فضاپیما را تحت تأثیر گرانش دو جرم اصلی توصیف می‌کند و به دلیل نبود راه‌حل تحلیلی بسته برای دینامیک‌های مداری و وضعی، چالش‌هایی را ایجاد می‌نماید. برای غلبه بر این مسئله، این مطالعه از نگاشت پوانکاره به‌عنوان یک روش عددی کارآمد بهره می‌گیرد تا شرایط اولیه مناسب برای حرکت تناوبی را شناسایی کرده، پیچیدگی ریاضی را کاهش داده و عملکرد محاسباتی را بهبود بخشد. یکی از نوآوری‌های قابل توجه در این تحقیق، ترکیب دینامیک وضعی فضاپیما با تحلیل مبتنی بر نگاشت پوانکاره است. اگرچه این کوپلینگ یک‌طرفه است— به این معنی که حالت‌های وضعی بر تناوب مدار مداری تأثیر نمی‌گذارند— اما ساختار نگاشت‌های پوانکاره را تحت تأثیر قرار می‌دهد. شکل و موقعیت جزایر موجود در این نگاشت‌ها با تغییر در نسبت لختی فضاپیما تغییر می‌کنند، که حساسیت روش به رفتار وضعی و کاربردپذیری آن در سناریوهای واقعی را نشان می‌دهد. روش پیشنهادی با دو تکنیک کلاسیک — تقریب مرتبه سوم و روش مبتنی بر نظریه فلوکه — مقایسه شده است. این تحلیل تطبیقی نشان می‌دهد که نگاشت پوانکاره توازن مطلوبی میان سادگی، کارایی محاسباتی و دقت فراهم می‌آورد و قادر است مدارهای تناوبی را بدون نیاز به مدل‌های مرتبه بالا یا تصحیحات پیچیده با اطمینان شناسایی کند. در مجموع، این مطالعه سهمی نوآورانه و کاربردی در طراحی مأموریت‌های فضایی ارائه می‌دهد و چارچوبی قدرتمند و در دسترس برای تحلیل دینامیک مدار-وضیت است .

Keywords Persian

پاسخ های دوره ای: مدار

وضعیت کوپل شده: سطح مقطع: حل عددی: دینامیک چند جسمی

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