拉格朗日/高斯无奇点卫星运动方程推导与分析
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摘要
针对卫星轨道理论中的奇点问题,对拉格朗日/高斯无奇点卫星运动方程做了深入的分析和探索,从原始拉格朗日和高斯运动方程及其物理意义出发,考虑圆轨道、圆赤道轨道和赤道轨道3种奇点情况,推导了一种新的拉格朗日/高斯无奇点卫星运动方程,并探讨了方程的连续性。该方程消除了零因子,解决了卫星运动方程的奇点问题。
Aiming at the singularity problem in satellite orbit theory,the singularity-free Lagrangian/Gaussian equations of motion is analyzed.Considering the original and physical meaning of the Lagrangian and Gaussian equations of motion,a new Lagrangian/Gaussian singularity-free disturbed equations of motion is proposed and then discussed in three cases:the circular orbit,equatorial orbit,circular and equatorial orbit.Besides,the continuity of these equations is explored.The proposed equations eliminate the zero factor and in this way the singularity problem in the orbital mechanics is solved.
Aiming at the singularity problem in satellite orbit theory,the singularity-free Lagrangian/Gaussian equations of motion is analyzed.Considering the original and physical meaning of the Lagrangian and Gaussian equations of motion,a new Lagrangian/Gaussian singularity-free disturbed equations of motion is proposed and then discussed in three cases:the circular orbit,equatorial orbit,circular and equatorial orbit.Besides,the continuity of these equations is explored.The proposed equations eliminate the zero factor and in this way the singularity problem in the orbital mechanics is solved.
引文
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[17]韩星远,向开恒,王海红.第一类无奇点变量的广播星历参数拟合算法[J].航天器工程,2011,20(4):54-59.HAN Xingyuan,XIANG Kaiheng,WANG Haihong.Research on Broadcast Ephemeris Parameters Fitting Algorithm Based on the First Class of No Singularity Variables[J].Spacecraft Engineering,2011,20(4):54-59.
[18]张中凯,杜兰,旦增曲英,等.基于第二类无奇点根数的改进根数[J].测绘科学技术学报,2012,29(4):257-261.ZHANG Zhongkai,DU Lan,DAN Zengquying,et al.Improved Elements Based on Second Class of No-singularity Variables[J].Journal of Geomatics Science and Technology,2012,29(4):257-261.
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[20]BATTIN R H.An Introduction to the Mathematics and Methods of Astrodynamics[M].New York:American Institute of Aeronautics and Astronautics.1987.
[21]XU Guochang,XU Jia.On the Singularity Problem in Orbital Mechanics[J].Monthly Notices of the Royal Astronomical Society,2013,429(2):1139-1148.
[22]XU Guochang,XU Jia.Orbits:2nd Order Singularity-free Solutions[M].Berlin:Springer,2013.
[23]XU G,LV Z P,SHEN Y Z,et al.A Mathematical Derivation of Singularity-free Lagrange Equations of Planetary Motion,Special Issue for Celebration 80th Birthday of Academician Houze Xu[J].Journal of Surveying and Mapping,2014.
[24]许国昌,陈武,沈云中,等.高斯无奇点卫星运动方程的数学推导——谨以本文恭贺师兄欧吉坤教授七十寿辰[J].导航定位学报,2015,3(3):5-12.XU Guochang,CHEN Wu,SHEN Yunzhong,et al.A Mathematical Derivation of Singularity-free Gaussian Equations of Planetary Motion[J].Journal of Navigation and Positioning,2015,3(3):5-12.
[25]杜玉军.卫星轨道的三维可视化程序设计[D].武汉:武汉大学,2008.DU Yujun.3D Visualization Programming of Satellite Orbiting[D].Wuhan:Wuhan University,2008.
[2]DIACU F.The Solution of The N-body Problem[J].The Mathematical Intelligencer,1996,18(3):66-70.
[3]DIACU F,HOLMES P.Celestial Encounters:the Origins of Chaos and Stability[M].Princeton,NJ:Princeton University Press,1996.
[4]WANG Qiudong.The Global Solution of the N-body Problem[J].Celestial Mechanics and Dynamical Astronomy,1990,50(1):73-88.
[5]XU Guochang.Sciences of Geodesy-I:Advances and Future Directions[M].Berlin:Springer,2010:105-154.
[6]BROUWER D,CLEMENCE G M.Methods of Celestial Mechanics[M].Burlington,MA:Elsevier,2013.
[7]CHOBOTOV V A.Orbital Mechanics[M].3rd ed.Washington DC:AIAA,2002.
[8]XU Guochang,XU Tianhe,YEH T K,et al.Analytical Solution of a Satellite Orbit Disturbed by Lunar and Solar Gravitation[J].Monthly Notices of the Royal Astronomical Society,2011,410(1):645-653.
[9]XU Yan,YANG Yuanxi,ZHANG Qin,et al.Solar Oblateness and Mercury’s Perihelion Precession[J].Monthly Notices of the Royal Astronomical Society,2011,415(4):3335-3343.
[10]XU Guochang,XU Jia.On Orbital Disturbing Effects of the Solar Radiation[J].Monthly Notices of the Royal Astronomical Society,2013,432(1):584-588.
[11]张兵兵,聂琳娟,吴汤婷,等.SWARM卫星简化动力学厘米级精密定轨[J].测绘学报,2016,45(11):1278-1284.DOI:10.11947/j.AGCS.2016.20160284.ZHANG Bingbing,NIE Linjuan,WU Tangting,et al.Centimeter Precise Orbit Determination for SWARM Satellite via Reduced-dynamic Method[J].Acta Geodaetica et Cartographica Sinica,2016,45(11):1278-1284.DOI:10.11947/j.AGCS.2016.20160284.
[12]邹贤才,李建成,姜卫平,等.卫星重力资料分析的同解法研究及其仿真[J].测绘学报,2010,39(4):344-348.ZOU Xiancai,LI Jiancheng,JIANG Weiping,et al.Research on the Simultaneous Solution Method for Satellite Gravity Data Analysis and Its Simulation[J].Acta Geodaetica et Cartographica Sinica,2010,39(4):344-348.
[13]HAVEL K.N-body Gravitational Problem:Unrestricted Solution[M].Brampton,ON,Canada:Grevyt Press,2008.
[14]BATTIN R H.An Introduction to the Mathematics and Methods of Astrodynamics[M].Reston,VA:AIAA,1999.
[15]KAULA W M.Theory of Satellite Geodesy:Applications of Satellites to Geodesy[M].Mineola,NY:Dover Publications Inc,2000.
[16]XU Guochang.Orbits[M].Berlin:Springer,2008.
[17]韩星远,向开恒,王海红.第一类无奇点变量的广播星历参数拟合算法[J].航天器工程,2011,20(4):54-59.HAN Xingyuan,XIANG Kaiheng,WANG Haihong.Research on Broadcast Ephemeris Parameters Fitting Algorithm Based on the First Class of No Singularity Variables[J].Spacecraft Engineering,2011,20(4):54-59.
[18]张中凯,杜兰,旦增曲英,等.基于第二类无奇点根数的改进根数[J].测绘科学技术学报,2012,29(4):257-261.ZHANG Zhongkai,DU Lan,DAN Zengquying,et al.Improved Elements Based on Second Class of No-singularity Variables[J].Journal of Geomatics Science and Technology,2012,29(4):257-261.
[19]BROUCKE R A,CEFOLA P J.On the Equinoctial Orbit Elements[J].Celestial Mechanics,1972,5(3):303-310.
[20]BATTIN R H.An Introduction to the Mathematics and Methods of Astrodynamics[M].New York:American Institute of Aeronautics and Astronautics.1987.
[21]XU Guochang,XU Jia.On the Singularity Problem in Orbital Mechanics[J].Monthly Notices of the Royal Astronomical Society,2013,429(2):1139-1148.
[22]XU Guochang,XU Jia.Orbits:2nd Order Singularity-free Solutions[M].Berlin:Springer,2013.
[23]XU G,LV Z P,SHEN Y Z,et al.A Mathematical Derivation of Singularity-free Lagrange Equations of Planetary Motion,Special Issue for Celebration 80th Birthday of Academician Houze Xu[J].Journal of Surveying and Mapping,2014.
[24]许国昌,陈武,沈云中,等.高斯无奇点卫星运动方程的数学推导——谨以本文恭贺师兄欧吉坤教授七十寿辰[J].导航定位学报,2015,3(3):5-12.XU Guochang,CHEN Wu,SHEN Yunzhong,et al.A Mathematical Derivation of Singularity-free Gaussian Equations of Planetary Motion[J].Journal of Navigation and Positioning,2015,3(3):5-12.
[25]杜玉军.卫星轨道的三维可视化程序设计[D].武汉:武汉大学,2008.DU Yujun.3D Visualization Programming of Satellite Orbiting[D].Wuhan:Wuhan University,2008.
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