对火星轨道变化问题的最后解释

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The variation of etricities and orbital inations for the inner four ps in the initial and final part of the iion N+1 is shown in Fig. 4. As expected， the character of the variation of pary orbital elements does not differ signifitly between the initial and final part of eategration， at least for Venus， Earth and Mars. The elements of Mercury， especially its etricity， seem to ge to a signifit extent. This is partly because the orbital time-scale of the p is the shortest of all the ps， which leads to a more rapid orbital evolution than other ps; the innermost p may be o instability. This result appears to be in some agreement with Laskar's (1994， 1996) expectations that rge and irregur variations appear in the etricities and inations of Mercury on a time-scale of several 109 yr. However， the effect of the possible instability of the orbit of Mercury may not fatally affect the global stability of the whole pary system owing to the small mass of Mercury. We will mention briefly the long-term orbital evolution of Mercury ter iion 4 using low-pass filtered orbital elements.

The orbital motion of the outer five ps seems rigorously stable and quite regur over this time-span (see also Se 5).

3.2 Time–frequency maps

Although the pary motion exhibits very long-term stability defined as the ence of close enter events， the chaotiature of pary dynamics ge the osciltory period and amplitude of pary orbital motion gradually over such long time-spans. Even such slight fluctuations of orbital variation in the frequenain， particurly in the case of Earth， potentially have a signifit effe its surface climate system through sor insotion variation (cf. Berger 1988).

To give an overview of the long-term ge in periodicity iary orbital motion， we performed many fast Fourier transformations (FFTs) along the time axis， and superposed the resulting periodgrams to draw two-dimensional time–frequency maps. The specific approach to drawing these time–frequency maps in this paper is very simple – much simpler than the wavelet analysis or Laskar's (1990， 1993) frequenalysis.

Divide the low-pass filtered orbital data into many fragments of the same length. The length of each data segment should be a multiple of 2 in order to apply the FFT.

Each fragment of the data has a rge overpping part: for example， wheh data begins from t=ti and ends at t=ti+T， the data segment ranges from ti+δT≤ti+δT+T， where δT?T. We tihis division until we reach a certain number N by whi+T reaches the total iioh.

ly an FFT to each of the data fragments， and obtain n frequency diagrams.

In each frequency diagram obtained above， the strength of periodicity be repced by a grey-scale (or colour) chart.

We perform the rept， and ect all the grey-scale (or colour) charts into one graph for eategration. The horizontal axis of these nehs should be the time， i.e. the starting times of each fragment of data (ti， where i= 1，…， n). The vertical axis represents the period (or frequency) of the osciltion of orbital elements.

We have adopted an FFT because of its overwhelming speed， sihe amount of numerical data to be deposed into frequenpos is terribly huge (several tens of Gbytes).

A typical example of the time–frequency map created by the above procedures is shown in a grey-scale diagram as Fig. 5， which shows the variation of periodicity in the etricity and ination of Earth in N+2 iion. In Fig. 5， the dark area shows that at the time indicated by the value on the abscissa， the periodicity indicated by the ordinate is strohan in the lighter area around it. We reize from this map that the periodicity of the etricity and ination of Earth only ges slightly over the entire period covered by the N+2 iion. This nearly regur trend is qualitatively the same in other iions and for other ps， although typical frequencies differ p by p and element by element.

4.2 Long-term exge of orbital energy and angur momentum

We calcute very long-periodic variation and exge of pary orbital energy and angur momentum using filtered Deunay elements L， G， H. G and H are equivalent to the pary orbital angur momentum and its vertical po per unit mass. L is reted to the pary orbital energy E per unit mass as E=?μ2/2L2. If the system is pletely linear， the orbital energy and the angur momentum in each frequency bin must be stant. Non-liy in the pary system cause an exge of energy and angur momentum in the frequenain. The amplitude of the lowest-frequency osciltion should increase if the system is unstable and breaks down gradually. However， such a symptom of instability is not promi in our long-term iions.

In Fig. 7， the total orbital energy and angur momentum of the four inner ps and all nine ps are shown for iion N+2. The upper three panels show the long-periodic variation of total energy (denoted asE- E0)， total angur momentum ( G- G0)， and the vertical po ( H- H0) of the inner four ps calcuted from the low-pass filtered Deunay elements.E0， G0， H0 dehe initial values of each quantity. The absolute difference from the initial values is plotted in the panels. The lower three panels in each figure showE-E0，G-G0 andH-H0 of the total of nine ps. The fluctuation shown in the lower panels is virtually entirely a result of the massive jovias.

paring the variations of energy and angur momentum of the inner four ps and all nine ps， it is apparent that the amplitudes of those of the inner ps are much smaller than those of all nine ps: the amplitudes of the outer five ps are much rger than those of the inner ps. This does not mean that the ierrestrial pary subsystem is more stable thaer ohis is simply a result of the retive smallness of the masses of the four terrestrial ps pared with those of the outer jovias. Ahiice is that the inner pary subsystem may bee unstable more rapidly thaer one because of its shorter orbital time-scales. This be seen in the panels denoted asinner 4 in Fig. 7 where the longer-periodid irregur osciltions are more apparent than in the panels denoted astotal 9. Actually， the fluctuations in theinner 4 panels are te extent as a result of the orbital variation of the Mercury. However， we ot he tribution from other terrestrial ps， as we will see in subsequeions.

4.4 Long-term coupling of several neighb p pairs

Let us see some individual variations of pary orbital energy and angur momentum expressed by the low-pass filtered Deunay elements. Figs 10 and 11 show long-term evolution of the orbital energy of eaet and the angur momentum in N+1 and N?2 iions. We notice that some ps form apparent pairs in terms of orbital energy and angur momentum exge. In particur， Venus ah make a typical pair. In the figures， they show ive corretions in exge of energy and positive corretions in exge of angur momentum. The ive corretion in exge of orbital energy means that the two ps form a closed dynamical system in terms of the orbital energy. The positive corretion in exge of angur momentum means that the two ps are simultaneously under certain long-term perturbations. didates for perturbers are Jupiter and Saturn. Also in Fig. 11， we see that Mars shoositive corretion in the angur momentum variation to the Veh system. Mercury exhibits certaiive corretions in the angur momentum versus the Veh system， which seems to be a rea caused by the servation of angur momentum ierrestrial pary subsystem.

It is not clear at the moment why the Veh pair exhibits a ive corretion in energy exge and a positive corretion in angur momentum exge. ossibly expin this through the general fact that there are no secur terms iary semimajor axes up to sed-order perturbation theories (cf. Brouwer & Clemence 1961; Boccaletti & Pucacco 1998). This means that the pary orbital energy (which is directly reted to the semimajor axis a) might be much less affected by perturbing phan is the angur momentum exge (which retes to e). Hehe etricities of Venus ah be disturbed easily by Jupiter and Saturn， which results in a positive corretion in the angur momentum exge. Oher hand， the semimajor axes of Venus ah are less likely to be disturbed by the jovias. Thus the energy exge may be limited only within the Veh pair， which results in a ive corretion in the exge of orbital energy in the pair.

As for the outer joviaary subsystem， Jupiter–Saturn and Uranus–uo make dynamical pairs. However， the strength of their coupling is not as strong pared with that of the Veh pair.

5 ± 5 × 1010-yr iions of outer pary orbits

Sihe joviaary masses are much rger thaerrestrial pary masses， we treat the joviaary system as an indepe pary system in terms of the study of its dynamical stability. Hence， we added a couple of trial iions that span ± 5 × 1010 yr， including only the outer five ps (the four jovias plus Pluto). The results exhibit the rigorous stability of the outer pary system over this long time-span. Orbital figurations (Fig. 12)， and variation of etricities and inations (Fig. 13) show this very long-term stability of the outer five ps in both the time and the frequenains. Although we do not shos here， the typical frequency of the orbital osciltion of Pluto and the other outer ps is almost stant during these very long-term iion periods， which is demonstrated iime–frequency maps on our webpage.

Iwo iions， the retive numerical error ial energy was ～10?6 and that of the total angur momentum was ～10?10.

5.1 Resonances in the une–Pluto system

Kinoshita & Nakai (1996) ied the outer five pary orbits over ± 5.5 × 109 yr . They found that four major resonances betweeune and Pluto are maintained during the whole iion period， and that the resonances may be the main causes of the stability of the orbit of Pluto. The major four resonances found in previous research are as follows. In the following description，λ dehe mean longitude，Ω is the longitude of the asding node and ? is the longitude of perihelion. Subscripts P and e Pluto aune.

Mean motion resoweeune and Pluto (3:2). The critical argument θ1= 3 λP? 2 λN??P librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.

The argument of perihelion of Pluto ωP=θ2=?P?ΩP librates around 90° with a period of about 3.8 × 106 yr. The dominant periodic variations of the etricity and ination of Pluto are synized with the libration of its argument of perihelion. This is anticipated in the secur perturbation theory structed by Kozai (1962).

The longitude of the node of Pluto referred to the longitude of the node of une，θ3=ΩP?ΩN， circutes and the period of this circution is equal to the period of θ2 libration. When θ3 bees zero， i.e. the longitudes of asding nodes of une and Pluto overp， the ination of Pluto bees maximum， the etricity bees minimum and the argument of perihelion bees 90°. When θ3 bees 180°， the ination of Pluto bees minimum， the etricity bees maximum and the argument of perihelion bees 90° again. Williams & Benson (1971) anticipated this type of resoer firmed by Mini， Nobili & Carpino (1989).

An argument θ4=?P??N+ 3 (ΩP?ΩN) librates around 180° with a long period，～ 5.7 × 108 yr.

In our numerical iions， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain simir during the whole iion period (Figs 14–16 ). However， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circution over a 1010-yr time-scale (Fig. 17). This is an iing fact that Kinoshita & Nakai's (1995， 1996) shorter iions were not able to disclose.

6 Discussion

What kind of dynamical meism maintains this long-term stability of the pary system? We immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no signifit lower-order resonances (mean motion and secur) between any pair among the nine ps. Jupiter and Saturn are close to a 5:2 mean motion resohe famous ‘great inequality’)， but not just in the resonance zone. Higher-order resonances may cause the chaotiature of the pary dynamical motion， but they are not s as to destroy the stable pary motion within the lifetime of the real Sor system. The sed feature， which we think is more important for the long-term stability of our pary system， is the differen dynamical distaween terrestrial and joviaary subsystems (Ito & Tanikawa 1999， 2001). When we measure pary separations by the mutual Hill radii (R_)， separations among terrestrial ps are greater than 26RH， whereas those among jovias are less than 14RH. This difference is directly reted to the differeween dynamical features of terrestrial and jovias. Terrestrial ps have smaller masses， shorter orbital periods and wider dynamical separation. They are strongly perturbed by joviahat have rger masses， longer orbital periods and narrower dynamical separation. Jovias are not perturbed by any other massive bodies.

The present terrestrial pary system is still being disturbed by the massive jovias. However， the wide separation and mutual iion among the terrestrial ps rehe disturbaneffective; the degree of disturbance by jovias is O(eJ)(order of magnitude of the etricity of Jupiter)， sihe disturbance caused by jovias is a forced osciltion having an amplitude of O(eJ). Heightening of etricity， for example O(eJ)～0.05， is far from suffit to provoke instability ierrestrial ps having such a wide separation as 26RH. Thus we assume that the present wide dynamical separation among terrestrial ps (> 26RH) is probably one of the most signifit ditions for maintaining the stability of the pary system over a 109-yr time-span. Our detailed analysis of the retionship between dynamical distawees and the instability time-scale of Sor system pary motion is now on-going.

Although our numerical iions span the lifetime of the Sor system， the number of iions is far from suffit to fill the initial phase space. It is necessary to perform more and more numerical iions to firm and examine iail the long-term stability of our pary dynamics.

——以上文段引自 Ito， T.& Tanikawa， K. Long-term iions and stability of pary orbits in our Sor System. Mon. Not. R. Astron. Soc. 336， 483–500 (2002)

这只是作者君参考的一篇文章，关于太阳系的稳定性。

还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《Nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。