相对论长期岁差

唐凯

(中国科学院上海天文台上海200030)

相对论长期岁差

唐凯†

(中国科学院上海天文台上海200030)

长期岁差描述在长时间范围内黄道和赤道的长周期运动.根据Vondr´ak等人(2011)的工作,我们将周期大于1万年的进动归为岁差,其他短周期序列则在章动模型中考虑.本文主要工作是在相对论框架下计算长期岁差.相对于P03岁差只适用于历元J2000.0附近几百年的范围,这种新理论能更好地体现出岁差长周期的本质.之前的所有相关工作与广义相对论理论并不完全相符.他们仅仅考虑主要的相对论效应:太阳1阶后牛顿改正和测地岁差.且处理测地岁差的方式也只是在牛顿解的基础上加相对论改正.

最近Klioner、Gerlach和So ff el(2010)建立了地球的相对论自转理论.在Klioner等人(2003)给出的后牛顿自转方程基础上,他们还考虑如何计算后牛顿惯性矩,如何处理多个相对论参考系及不同时间系统和相对论尺度化问题.此外测地岁差/章动也以更自然的方式得到处理.这个地球自转理论符合广义相对论原理,也使得我们能在更严格的相对论框架下计算地球长期岁差.

我们的目标是计算J2000.0前后100万年的地球岁差.其中黄道岁差的计算与前人工作一样,通过数值的方法得到.赤道岁差部分则基于前面提到的相对论地球自转理论,通过数值的方法对Klioner等人(2003)给出的后牛顿刚体自转方程积分,然后再根据潮汐耗散作用对结果进行修正.于是可以得到岁差表达式,其中包含一个线性项和20到30个周期项.在J2000.0前后2 000 yr的范围内,与P03相比误差仅几个角秒,同时也与其他长期岁差理论相符.最后我们计算分析了岁差中的相对论影响.

本文将对这种相对论长期岁差模型做具体介绍.第1章简要回顾相关历史背景,并阐述本工作的目的.第2到5章则给出岁差的具体计算过程.第2章主要描述我们完成的准辛积分器结构.根据太阳系动力学模型,使用SABA4型辛积分器,并对积分器做一定调整以处理潮汐耗散及舍入误差等问题.太阳1阶后牛顿效应也考虑其中.第3章首先给出我们所用的频谱分析方法,然后对地月质心轨道做数值处理得到黄道岁差参数表达式.第4章主要介绍Klioner等人(2010)提出的后牛顿地球自转理论.第5章在相对论框架下积分自转方程,得到赤道岁差表达式,并分析讨论了相对论效应对岁差的影响.最后1章是工作总结与展望.

A long-term precession represents a secular motion of the ecliptic and the equator in a long time interval.With Vondr´ak et al.(2011),we assume that precession covers all periods longer than 100 centuries,while the shorter ones are included in the nutation.Thisthesis deals with the long-term precession in a relativistic framework.Compared with the P03 precession theory which is only valid for several centuries around the epoch J2000.0, the new theory better re fl ects the realistic long-term behavior of precession.All previous works are not fully consistent with General Relativity.They only consider the dominant relativistic corrections:the first-order post-Newtonian corrections due to the Sun and the geodetic precession.Their standard way to account for the geodetic precession is to solve the purely Newtonian equations of rotational motion and add the geodetic precession as a correction to the solution.

Recently,Klioner,Gerlach,and So ff el(2010)have constructed a relativistic theory of Earth’s rotation.According to the post-Newtonian equations of rotational motion given by Klioner et al.(2003),they explain how to calculate the relativistic inertial torque,and discuss how to deal with di ff erent relativistic reference systems as well as various time scales and relativistic scalings.The geodetic precession and nutation are also taken into account in a natural way.This theory of Earth’s rotation is consistent with General Relativity. This approach allows us to obtain the long-term precession of the Earth in a more rigorous relativistic framework.

Our goal is to obtain the relativistic Earth’s precession from−1Myr to 1Myr around J2000.0.The precession of the ecliptic is obtained by numerical integration as in most previous works.The precession of the equator,which is calculated with the relativistic theory of Earth’s rotation as mentioned above,is also derived numerically.This part of work starts with a post-Newtonian rigid-multipole formalism that has been published by Klioner et al.(2003).Then the equations are integrated numerically,and the results are modi fied due to the effect of tidal dissipation.Approximations for the precession are derived and expressed in form of a linear term plus 20–30 periodic terms.Compared with P03,the di ff erence is only several arcseconds in an interval of 2000 years around J2000.0.The results are consistent with other long-term precession theories.Finally,the relativistic effects of precession are analyzed.

In this thesis,the models for the relativistic long-term precession of the Earth are given. Chapter 1 brie fl y introduces some historical background and the aim of our work.Chapters 2 to 5 give the way to calculate the precession in detail.Chapter 2 is about the structure of a quasi symplectic integrator which was developed by ourselves.According to our dynamical model of the solar system,the numerical integrator is based on the symplectic SABA4 scheme,and some tricks are used to treat the problems of tidal dissipation,close encounters, and round-o fferrors.The first-order post-Newtonian effects related with the Sun are also included.In Chapter 3,we describe the frequency analysis method.Some algorithms are applied to the orbits of Earth-Moon barycenter to get approximations for the precession of the ecliptic parameters.Chapter 4 mainly introduces the post-Newtonian theory of Earth’s rotation by Klioner et al.(2010).Based on Chapter 4,the equations of rotational motion with rigorous treatment of relativistic effects are integrated,and expressions for the precession of the equator are obtained in Chapter 5.The in fl uences of these relativistic effects on the precession are obtained and discussed in the end.The last Chapter contains conclusions and prospects for the future work.

A Relativistic Long-term Precession of the Earth

TANG Kai
(Shanghai Astronomical Observatory,Chinese Academy of Sciences,Shanghai 200030)

10.15940/j.cnki.0001-5245.2016.03.013 博士学位论文摘要选登

†2014-07-06获得博士学位,导师:上海天文台唐正宏研究员,陶金河研究员,德雷斯顿工业大学Michael So ff el教授;tangkai@shao.ac.cn