相對論長期歲差

唐凱

(中國科學(xué)院上海天文臺上海200030)

相對論長期歲差

唐凱?

(中國科學(xué)院上海天文臺上海200030)

長期歲差描述在長時(shí)間范圍內(nèi)黃道和赤道的長周期運(yùn)動.根據(jù)Vondr′ak等人(2011)的工作,我們將周期大于1萬年的進(jìn)動歸為歲差,其他短周期序列則在章動模型中考慮.本文主要工作是在相對論框架下計(jì)算長期歲差.相對于P03歲差只適用于歷元J2000.0附近幾百年的范圍,這種新理論能更好地體現(xiàn)出歲差長周期的本質(zhì).之前的所有相關(guān)工作與廣義相對論理論并不完全相符.他們僅僅考慮主要的相對論效應(yīng):太陽1階后牛頓改正和測地歲差.且處理測地歲差的方式也只是在牛頓解的基礎(chǔ)上加相對論改正.

最近Klioner、Gerlach和So ff el(2010)建立了地球的相對論自轉(zhuǎn)理論.在Klioner等人(2003)給出的后牛頓自轉(zhuǎn)方程基礎(chǔ)上,他們還考慮如何計(jì)算后牛頓慣性矩,如何處理多個(gè)相對論參考系及不同時(shí)間系統(tǒng)和相對論尺度化問題.此外測地歲差/章動也以更自然的方式得到處理.這個(gè)地球自轉(zhuǎn)理論符合廣義相對論原理,也使得我們能在更嚴(yán)格的相對論框架下計(jì)算地球長期歲差.

我們的目標(biāo)是計(jì)算J2000.0前后100萬年的地球歲差.其中黃道歲差的計(jì)算與前人工作一樣,通過數(shù)值的方法得到.赤道歲差部分則基于前面提到的相對論地球自轉(zhuǎn)理論,通過數(shù)值的方法對Klioner等人(2003)給出的后牛頓剛體自轉(zhuǎn)方程積分,然后再根據(jù)潮汐耗散作用對結(jié)果進(jìn)行修正.于是可以得到歲差表達(dá)式,其中包含一個(gè)線性項(xiàng)和20到30個(gè)周期項(xiàng).在J2000.0前后2 000 yr的范圍內(nèi),與P03相比誤差僅幾個(gè)角秒,同時(shí)也與其他長期歲差理論相符.最后我們計(jì)算分析了歲差中的相對論影響.

本文將對這種相對論長期歲差模型做具體介紹.第1章簡要回顧相關(guān)歷史背景,并闡述本工作的目的.第2到5章則給出歲差的具體計(jì)算過程.第2章主要描述我們完成的準(zhǔn)辛積分器結(jié)構(gòu).根據(jù)太陽系動力學(xué)模型,使用SABA4型辛積分器,并對積分器做一定調(diào)整以處理潮汐耗散及舍入誤差等問題.太陽1階后牛頓效應(yīng)也考慮其中.第3章首先給出我們所用的頻譜分析方法,然后對地月質(zhì)心軌道做數(shù)值處理得到黃道歲差參數(shù)表達(dá)式.第4章主要介紹Klioner等人(2010)提出的后牛頓地球自轉(zhuǎn)理論.第5章在相對論框架下積分自轉(zhuǎn)方程,得到赤道歲差表達(dá)式,并分析討論了相對論效應(yīng)對歲差的影響.最后1章是工作總結(jié)與展望.

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 博士學(xué)位論文摘要選登

?2014-07-06獲得博士學(xué)位,導(dǎo)師:上海天文臺唐正宏研究員,陶金河研究員,德雷斯頓工業(yè)大學(xué)Michael So ff el教授;tangkai@shao.ac.cn