三体轨道数值模拟及理想天体重力收敛范围研究
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摘要
本文讨论了基于理想模型三体轨道运动规律问题,并给出了一个完整的三体问题的数值解决方案;另外还研究了与引力有关的基于刚体模型的天体表面引力收敛问题。首先总结了经典的二体轨道运动方程和经典积分及其结论,在此基础之上利用二体质心系为参考系和新的参数给出了二体的运动方程和解析解。研究了轨道运动方程的数值计算方法——龙格-库塔算法,通过二体微分运动方程的数值解和解析解结果进行对比来检验数值方法的可靠性。深入研究了基于质点系理想动力学理论的三体问题的力学特点及三体运动方程的常用表达方式,分析了几种三体运动方程的表示方法和特点。选择了以绝对坐标为参考系的三体运动方程作为数值计算依据,重点给出了三体系统完整的数值解决方法,利用解一阶微分方程组的数值方法来解三体轨道微分运动方程,并分析了数值方法的结果及其运用方法。分析了数值计算过程中必须注意的若干问题。
另外本文基于理想、天体规模的刚体数学模型,利用刚体转动力学理论和引潮力力学理论,对天体表面的重力收敛问题进行了研究。即研究天体表面物质所受到的重力指向天体中心点的集合所形成的形状。深入讨论了几种不同性质的力学方式如天体自转、引潮力、进动及章动等,所对应的重力收敛特性、。给出了由各种性质的力共同作用形成的一般天体的重力收敛规律,及标准重力收敛。给出一些典型天体的重心离散度数值,并指出重力收敛的研究方向和存在的问题。
A issues about the orbital modeling of three bodies which are based on ideal rigid body. And therefore give rise to a integrate numerical method of three body problem. The gravity convergence of rigid body is studied based on rigid body model too. At first, the approach with classical equation of two-body orbital motion and integral approach as well as its conclusions are summed up, on this basis, an absolute coordinate and the new parameters in which centroid of two body are taken as reference system are used to give the two-body motion equations and analytical solution. After pre-processing method with a two-body motion equation is studied, the numerical solution of differential motion equations for the two-body and results with the classical analytical solution are compared and analyzed. Mass point dynamics is the basic principium of of all the orbit equations. Runge-kutta numerical is used here. The mechanical characteristics for the issues of three-body, the commonly used expression for the equations of motion, analyze the characteristics of several representations are fully studied. In order to facilitate numerical calculation for the three-body motion equation, the absolute coordinate system is selected in the three-body motion equation as the basis for numerical calculation, a complete numerical method is given with focusing on a three-body system, and a numerical method for solving a first-order differential equations which is known as Runge - Kutta method to solve this celestial orbit differential equations of motion, in which some examples are given ,and the results of numerical methods and its application methods are analyzed,as well as it is emphasized in the numerical results to discover and analyze the initial value in numerical calculation and given initial value and the impact of the trend of the calculated results. In the numerical calculation of the analysis process a number of issues are given for which it should pay attention in processing.
Another issue in the paper is the study of gravity convergence. Gravity convergence is A set made up of the points where the gravity directions of substances on the surface of the ideal celestial body point to. Several different kinds of mechanical methods such as celestial rotation, tidal force, precession and nutation, etc as well as its corresponding gravity convergence shape and movement characteristics by gravity are discussed. The gravity convergence shape of common ideal celestial body and standard gravity convergence shape which are formed by the nature of the forces from a variety of general objects are given. Gravity dispersion is proposed as well as a limit of amount for the size of the gravity convergence. Some dispersion values of gravity direction for the typical ideal celestial body is given. Some defects and directions which need to be studied go a step further is proposed and discussed in the paper.
另外本文基于理想、天体规模的刚体数学模型,利用刚体转动力学理论和引潮力力学理论,对天体表面的重力收敛问题进行了研究。即研究天体表面物质所受到的重力指向天体中心点的集合所形成的形状。深入讨论了几种不同性质的力学方式如天体自转、引潮力、进动及章动等,所对应的重力收敛特性、。给出了由各种性质的力共同作用形成的一般天体的重力收敛规律,及标准重力收敛。给出一些典型天体的重心离散度数值,并指出重力收敛的研究方向和存在的问题。
A issues about the orbital modeling of three bodies which are based on ideal rigid body. And therefore give rise to a integrate numerical method of three body problem. The gravity convergence of rigid body is studied based on rigid body model too. At first, the approach with classical equation of two-body orbital motion and integral approach as well as its conclusions are summed up, on this basis, an absolute coordinate and the new parameters in which centroid of two body are taken as reference system are used to give the two-body motion equations and analytical solution. After pre-processing method with a two-body motion equation is studied, the numerical solution of differential motion equations for the two-body and results with the classical analytical solution are compared and analyzed. Mass point dynamics is the basic principium of of all the orbit equations. Runge-kutta numerical is used here. The mechanical characteristics for the issues of three-body, the commonly used expression for the equations of motion, analyze the characteristics of several representations are fully studied. In order to facilitate numerical calculation for the three-body motion equation, the absolute coordinate system is selected in the three-body motion equation as the basis for numerical calculation, a complete numerical method is given with focusing on a three-body system, and a numerical method for solving a first-order differential equations which is known as Runge - Kutta method to solve this celestial orbit differential equations of motion, in which some examples are given ,and the results of numerical methods and its application methods are analyzed,as well as it is emphasized in the numerical results to discover and analyze the initial value in numerical calculation and given initial value and the impact of the trend of the calculated results. In the numerical calculation of the analysis process a number of issues are given for which it should pay attention in processing.
Another issue in the paper is the study of gravity convergence. Gravity convergence is A set made up of the points where the gravity directions of substances on the surface of the ideal celestial body point to. Several different kinds of mechanical methods such as celestial rotation, tidal force, precession and nutation, etc as well as its corresponding gravity convergence shape and movement characteristics by gravity are discussed. The gravity convergence shape of common ideal celestial body and standard gravity convergence shape which are formed by the nature of the forces from a variety of general objects are given. Gravity dispersion is proposed as well as a limit of amount for the size of the gravity convergence. Some dispersion values of gravity direction for the typical ideal celestial body is given. Some defects and directions which need to be studied go a step further is proposed and discussed in the paper.
引文
Aarseth S J. Direct methods for N-body simulations in Multiple Time Scales ed. J U Brackbill & B I Cohen. New York: Academic Press,1985.377
Adam M. Dziewonski and Don L. Anderson, Preliminary reference Earth model, Physics of the Earth and Planetary Interiors, 25 297-356, 1981
Bursa, M., Variations of the earth gravity field due to the free nutation, Stadia Geophys. Et Geod., No 2, 1972.
Barnes, D. F., Gravity changes during the Alaska Earthquake, J. G. R., Vol. 71, No 2, 1966.
Cartwrite, D. E., Tayler, R. J., New computations of the tidegeneration potential, G. J. R. A. S. , Vol. 23, 1971.
陈刚,易维勇,孙付平.地球质心的运动的频谱分析方法.测绘学院学报. 2002, 19 F.普雷斯等,地球,北京,科学出版社,1986。
陈益惠、杜品仁等,北京、四川、西藏地区重力固体潮研究,中国科学,24卷,2期,1981年
陈运泰, 1975年海城地震和1976年唐山地震前后的重力变化,地震学报, 2卷, 1期,1980年。
Chambers J E. A hybrid symplectic integrator that permits close encounters between massive bodies. Mon Not Roy Astron Soc, 304:793-799,1999
Drewes, H., Investigation on vertical crustal movements in Venezuelan Andeao by gravimetric methods, Applications of Geodesy to Geodynamics, OSU., 1978.
方俊,固体潮,科学出版社, 1984。
方俊,重力测量与地球形状学(下册),科学出版社, 1975。
Farrell, W. E., Deformation of the earth by surface loads, Rev. Geophys. And Space Physics, Vol 10, 1972
傅承义,陈运泰,祁贵仲.地球物理学基础,科学出版社, 1985
盖保民,地球演化第三卷,中国科学技术出版社,1996
黄润乾.恒星物理,科学出版社, 1998
蒋玉立,位理论基础中国地质大学(北京)1980
金旭,固体地球物理学基础吉林大学出版社,2003
李林森,韩晓明.地心引力常数变化原因的分析和讨论.宁夏大学学报(自然科学版), 2005, 26 (2): 135-138
李瑞浩,重力学引论地震出版社1988
李晓文,李维亮,周秀骥,中国近30年太阳辐射状况研究,应用气象学报,9(1):24-31,1998
李宗伟、肖兴华.天体物理学,高等教育出版社,北京,2000
刘迎春、刘林,月球卫星的精密定轨,空间科学学报,22(3):249-255,2002
刘学富.基础天文学,高等教育出版社2004
廖新浩、刘林,辛算法在限制性三体问题研究中的应用,计算物理,12(1):102-108,1995
马宗晋等,大陆地震构造的基本特征,大陆地震活动和地震预报国际学术论文集,北京,地震出版社,1984
Montag H. Geocenter variations derived by different satellite methods. IERS Technical Note 25. Observatoire de Paris, 1999, 75-81.
Pavls E C. Fortnightly Resolution geocenter series: A combined Analysis of Lageos 1 and 2 SLR data(1993~96). IERS Technical note 25. Observatoire de Paris, 1999, 75-81.
彭芳麟等,理论力学计算机模拟清华大学出版社2002
P.Melchior. The Tides of the planet Earth, Pergamon press, 1978
强元棨.经典力学(上、下册),科学出版社,2003
秦显平,杨元喜.用SLR数据导出的地心运动结果.测绘学报. 2003,
R King et al. Serveying with GPS. 1985 DUMLER. BONN
Reid, R. F., The elastic-rebound theory of earthquakes, Univ. Calif. Pub. Bull. Dept. Geol. Sci., 6, 413, 1910
Rapp, R. H., The Earth gravity field, Geophys. Surveys, No 2, 1981.
苏云荪.理论力学.高等教育出版社, 1990
Sweatman W. Journ Comp Phys,1994,111:110
藤吉文等,岩石圈物理学科学出版社,2004
吴晓梅,Kuiper带天体轨道稳定性的数值模拟,西南大学学报(自然科学版),29(7):32-37,2007
王庆宾,周媛,魏忠邦.地球质心的运动对卫星轨道的影响.测绘学院学报.
许厚泽、陈振邦、杨怀冰,海洋潮汐对重力潮汐观测的影响,地球物理学报, 25卷,2期,1982年。
叶叔华等,天文地球动力学山东科技出版社,2000
郁丽忠,郑学塘,李林森.太阳质量损失对行星轨道的影响.空间科学学报. 1994
易照华.天体力学基础,南京大学出版社,1993
杨远玲、聂清香、吴晓梅等,N体问题的几种数值算法比较,计算物理, 23(5): 599-603,2006
占腊生,何娟美,叶艺林等,太阳活动周期的小波分析,天文学报,47(2):166-174,2006曾融
生,固体地球物理学导论,北京,科学出版社,1984
赵凯华罗蔚茵,热学高等教育出版社1998
赵凯华、罗蔚茵,新概念物理教程力学(第二版),高等教育出版社,2004
周旭华,高布锡.地心的变化及原因.地球物理学报. 2000, 43(2): 160-165,2004
周淑贞等,气象学与气候学第三版高等教育出版社1997
张志刚等, Matlab与数学实验中国铁道出版社2004
朱慈墭.天文学教程(上、下册),高等教育出版社,2003
Adam M. Dziewonski and Don L. Anderson, Preliminary reference Earth model, Physics of the Earth and Planetary Interiors, 25 297-356, 1981
Bursa, M., Variations of the earth gravity field due to the free nutation, Stadia Geophys. Et Geod., No 2, 1972.
Barnes, D. F., Gravity changes during the Alaska Earthquake, J. G. R., Vol. 71, No 2, 1966.
Cartwrite, D. E., Tayler, R. J., New computations of the tidegeneration potential, G. J. R. A. S. , Vol. 23, 1971.
陈刚,易维勇,孙付平.地球质心的运动的频谱分析方法.测绘学院学报. 2002, 19 F.普雷斯等,地球,北京,科学出版社,1986。
陈益惠、杜品仁等,北京、四川、西藏地区重力固体潮研究,中国科学,24卷,2期,1981年
陈运泰, 1975年海城地震和1976年唐山地震前后的重力变化,地震学报, 2卷, 1期,1980年。
Chambers J E. A hybrid symplectic integrator that permits close encounters between massive bodies. Mon Not Roy Astron Soc, 304:793-799,1999
Drewes, H., Investigation on vertical crustal movements in Venezuelan Andeao by gravimetric methods, Applications of Geodesy to Geodynamics, OSU., 1978.
方俊,固体潮,科学出版社, 1984。
方俊,重力测量与地球形状学(下册),科学出版社, 1975。
Farrell, W. E., Deformation of the earth by surface loads, Rev. Geophys. And Space Physics, Vol 10, 1972
傅承义,陈运泰,祁贵仲.地球物理学基础,科学出版社, 1985
盖保民,地球演化第三卷,中国科学技术出版社,1996
黄润乾.恒星物理,科学出版社, 1998
蒋玉立,位理论基础中国地质大学(北京)1980
金旭,固体地球物理学基础吉林大学出版社,2003
李林森,韩晓明.地心引力常数变化原因的分析和讨论.宁夏大学学报(自然科学版), 2005, 26 (2): 135-138
李瑞浩,重力学引论地震出版社1988
李晓文,李维亮,周秀骥,中国近30年太阳辐射状况研究,应用气象学报,9(1):24-31,1998
李宗伟、肖兴华.天体物理学,高等教育出版社,北京,2000
刘迎春、刘林,月球卫星的精密定轨,空间科学学报,22(3):249-255,2002
刘学富.基础天文学,高等教育出版社2004
廖新浩、刘林,辛算法在限制性三体问题研究中的应用,计算物理,12(1):102-108,1995
马宗晋等,大陆地震构造的基本特征,大陆地震活动和地震预报国际学术论文集,北京,地震出版社,1984
Montag H. Geocenter variations derived by different satellite methods. IERS Technical Note 25. Observatoire de Paris, 1999, 75-81.
Pavls E C. Fortnightly Resolution geocenter series: A combined Analysis of Lageos 1 and 2 SLR data(1993~96). IERS Technical note 25. Observatoire de Paris, 1999, 75-81.
彭芳麟等,理论力学计算机模拟清华大学出版社2002
P.Melchior. The Tides of the planet Earth, Pergamon press, 1978
强元棨.经典力学(上、下册),科学出版社,2003
秦显平,杨元喜.用SLR数据导出的地心运动结果.测绘学报. 2003,
R King et al. Serveying with GPS. 1985 DUMLER. BONN
Reid, R. F., The elastic-rebound theory of earthquakes, Univ. Calif. Pub. Bull. Dept. Geol. Sci., 6, 413, 1910
Rapp, R. H., The Earth gravity field, Geophys. Surveys, No 2, 1981.
苏云荪.理论力学.高等教育出版社, 1990
Sweatman W. Journ Comp Phys,1994,111:110
藤吉文等,岩石圈物理学科学出版社,2004
吴晓梅,Kuiper带天体轨道稳定性的数值模拟,西南大学学报(自然科学版),29(7):32-37,2007
王庆宾,周媛,魏忠邦.地球质心的运动对卫星轨道的影响.测绘学院学报.
许厚泽、陈振邦、杨怀冰,海洋潮汐对重力潮汐观测的影响,地球物理学报, 25卷,2期,1982年。
叶叔华等,天文地球动力学山东科技出版社,2000
郁丽忠,郑学塘,李林森.太阳质量损失对行星轨道的影响.空间科学学报. 1994
易照华.天体力学基础,南京大学出版社,1993
杨远玲、聂清香、吴晓梅等,N体问题的几种数值算法比较,计算物理, 23(5): 599-603,2006
占腊生,何娟美,叶艺林等,太阳活动周期的小波分析,天文学报,47(2):166-174,2006曾融
生,固体地球物理学导论,北京,科学出版社,1984
赵凯华罗蔚茵,热学高等教育出版社1998
赵凯华、罗蔚茵,新概念物理教程力学(第二版),高等教育出版社,2004
周旭华,高布锡.地心的变化及原因.地球物理学报. 2000, 43(2): 160-165,2004
周淑贞等,气象学与气候学第三版高等教育出版社1997
张志刚等, Matlab与数学实验中国铁道出版社2004
朱慈墭.天文学教程(上、下册),高等教育出版社,2003
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