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作者君在作品相关中其实已经解释过这个问题。
不过仍然有人质疑——“你说得太含糊了”，“火星轨道的变化比你想象要大得多！”
那好吧，既然作者君的简单解释不够有力，那咱们就看看严肃的东西，反正这本书写到现在，嚷嚷着本书bug一大堆，用初高中物理在书中挑刺的人也不少。
以下是文章内容：
long-term integrations and stability of pary orbits in our sor system
abstract
we present the results of very long-term numerical integrations of pary orbital motions over 109 -yr time-spans including all nine ps. a quick inspection of our numerical data shows that the pary motion， at least in our simple dynamical model， seems to be quite stable even over this very long time-span. a closer look at the lowest-frequency osciltions using a low-pass filter shows us the potentially diffusive character of terrestrial pary motion， especially that of mercury. the behaviour of the eentricity of mercury in our integrations is qualitatively simir to the results from jacques laskar's secur perturbation theory (e.g. emax～ 0.35 over ～± 4 gyr). however， there are no apparent secur increases of eentricity or inclination in any orbital elements of the ps， which may be revealed by still longer-term numerical integrations. we have also performed a couple of trial integrations including motions of the outer five ps over the duration of ± 5 x 1010 yr. the result indicates that the three major resonances in the neptune–pluto system have been maintained over the 1011-yr time-span.
1 introduction
1.1definition of the problem
the question of the stability of our sor system has been debated over several hundred years， since the era of newton. the problem has attracted many famous mathematicians over the years and has pyed a central role in the development of non-linear dynamics and chaos theory. however， we do not yet have a definite answer to the question of whether our sor system is stable or not. this is partly a result of the fact that the definition of the term ‘stability’ is vague when it is used in retion to the problem of pary motion in the sor system. actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our sor system.
among many definitions of stability， here we adopt the hill definition (gdman 1993): actually this is not a definition of stability， but of instability. we define a system as becoming unstable when a close encounter ours somewhere in the system， starting from a certain initial configuration (chambers， wetherill & boss 1996; ito & tanikawa 1999). a system is defined as experiencing a close encounter when two bodies approach one another within an area of the rger hill radius. otherwise the system is defined as being stable. henceforward we state that our pary system is dynamically stable if no close encounter happens during the age of our sor system， about ±5 gyr. incidentally， this definition may be repced by one in which an ourrence of any orbital crossing between either of a pair of ps takes pce. this is because we know from experience that an orbital crossing is very likely to lead to a close encounter in pary and protopary systems (yoshinaga， kokubo & makino 1999). of course this statement cannot be simply applied to systems with stable orbital resonances such as the neptune–pluto system.
1.2previous studies and aims of this research
in addition to the vagueness of the concept of stability， the ps in our sor system show a character typical of dynamical chaos (sussman & wisdom 1988， 1992). the cause of this chaotic behaviour is now partly understood as being a result of resonance overpping (murray & holman 1999; lecar， franklin & holman 2001). however， it would require integrating over an ensemble of pary systems including all nine ps for a period covering several 10 gyr to thoroughly understand the long-term evolution of pary orbits， since chaotic dynamical systems are characterized by their strong dependence on initial conditions.
from that point of view， many of the previous long-term numerical integrations included only the outer five ps (sussman & wisdom 1988; kinoshita & nakai 1996). this is because the orbital periods of the outer ps are so much longer than those of the inner four ps that it is much easier to follow the system for a given integration period. at present， the longest numerical integrations published in journals are those of duncan & lissauer (1998). although their main target was the effect of post-main-sequence sor mass loss on the stability of pary orbits， they performed many integrations covering up to ～1011 yr of the orbital motions of the four jovian ps. the initial orbital elements and masses of ps are the same as those of our sor system in duncan & lissauer's paper， but they decrease the mass of the sun gradually in their numerical experiments. this is because they consider the effect of post-main-sequence sor mass loss in the paper. consequently， they found that the crossing time-scale of pary orbits， which can be a typical indicator of the instability time-scale， is quite sensitive to the rate of mass decrease of the sun. when the mass of the sun is close to its present value， the jovian ps remain stable over 1010 yr， or perhaps longer. duncan & lissauer also performed four simir experiments on the orbital motion of seven ps (venus to neptune)， which cover a span of ～109 yr. their experiments on the seven ps are not yet comprehensive， but it seems that the terrestrial ps also remain stable during the integration period， maintaining almost regur osciltions.
on the other hand， in his aurate semi-analytical secur perturbation theory (laskar 1988)， laskar finds that rge and irregur variations can appear in the eentricities and inclinations of the terrestrial ps， especially of mercury and mars on a time-scale of several 109 yr (laskar 1996). the results of laskar's secur perturbation theory should be confirmed and investigated by fully numerical integrations.
in this paper we present preliminary results of six long-term numerical integrations on all nine pary orbits， covering a span of several 109 yr， and of two other integrations covering a span of ± 5 x 1010 yr. the total epsed time for all integrations is more than 5 yr， using several dedicated pcs and workstations. one of the fundamental conclusions of our long-term integrations is that sor system pary motion seems to be stable in terms of the hill stability mentioned above， at least over a time-span of ± 4 gyr. actually， in our numerical integrations the system was far more stable than what is defined by the hill stability criterion: not only did no close encounter happen during the integration period， but also all the pary orbital elements have been confined in a narrow region both in time and frequency domain， though pary motions are stochastic. since the purpose of this paper is to exhibit and overview the results of our long-term numerical integrations， we show typical example figures as evidence of the very long-term stability of sor system pary motion. for readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (aess )， where we show raw orbital elements， their low-pass filtered results， variation of deunay elements and angur momentum deficit， and results of our simple time–frequency analysis on all of our integrations.
in section 2 we briefly expin our dynamical model， numerical method and initial conditions used in our integrations. section 3 is devoted to a description of the quick results of the numerical integrations. very long-term stability of sor system pary motion is apparent both in pary positions and orbital elements. a rough estimation of numerical errors is also given. section 4 goes on to a discussion of the longest-term variation of pary orbits using a low-pass filter and includes a discussion of angur momentum deficit. in section 5， we present a set of numerical integrations for the outer five ps that spans ± 5 x 1010 yr. in section 6 we also discuss the long-term stability of the pary motion and its possible cause.
2 description of the numerical integrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3 numerical method
we utilize a second-order wisdom–holman symplectic map as our main integration method (wisdom & holman 1991; kinoshita， yoshida & nakai 1991) with a special start-up procedure to reduce the truncation error of angle variables，‘warm start’(saha & tremaine 1992， 1994).
the stepsize for the numerical integrations is 8 d throughout all integrations of the nine ps (n±1，2，3)， which is about 1/11 of the orbital period of the innermost p (mercury). as for the determination of stepsize， we partly follow the previous numerical integration of all nine ps in sussman & wisdom (1988， 7.2 d) and saha & tremaine (1994， 225/32 d). we rounded the decimal part of the their stepsizes to 8 to make the stepsize a multiple of 2 in order to reduce the aumution of round-off error in the computation processes. in retion to this， wisdom & holman (1991) performed numerical integrations of the outer five pary orbits using the symplectic map with a stepsize of 400 d， 1/10.83 of the orbital period of jupiter. their result seems to be aurate enough， which partly justifies our method of determining the stepsize. however， since the eentricity of jupiter (～0.05) is much smaller than that of mercury (～0.2)， we need some care when we compare these integrations simply in terms of stepsizes.
in the integration of the outer five ps (f±)， we fixed the stepsize at 400 d.
we adopt gauss' f and g functions in the symplectic map together with the third-order halley method (danby 1992) as a solver for kepler equations. the number of maximum iterations we set in halley's method is 15， but they never reached the maximum in any of our integrations.
the interval of the data output is 200 000 d (～547 yr) for the calcutions of all nine ps (n±1，2，3)， and about 8000 000 d (～21 903 yr) for the integration of the outer five ps (f±).
although no output filtering was done when the numerical integrations were in process， we applied a low-pass filter to the raw orbital data after we had completed all the calcutions. see section 4.1 for more detail.
2.4 error estimation
2.4.1 retive errors in total energy and angur momentum
aording to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angur momentum)， our long-term numerical integrations seem to have been performed with very small errors. the averaged retive errors of total energy (～10?9) and of total angur momentum (～10?11) have remained nearly constant throughout the integration period (fig. 1). the special startup procedure， warm start， would have reduced the averaged retive error in total energy by about one order of magnitude or more.
retive numerical error of the total angur momentum δa/a0 and the total energy δe/e0 in our numerical integrationsn± 1，2，3， where δe and δa are the absolute change of the total energy and total angur momentum， respectively， ande0anda0are their initial values. the horizontal unit is gyr.
note that different operating systems， different mathematical libraries， and different hardware architectures result in different numerical errors， through the variations in round-off error handling and numerical algorithms. in the upper panel of fig. 1， we can recognize this situation in the secur numerical error in the total angur momentum， which should be rigorously preserved up to machine-e precision.
2.4.2 error in pary longitudes
since the symplectic maps preserve total energy and total angur momentum of n-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the auracy of numerical integrations， especially as a measure of the positional error of ps， i.e. the error in pary longitudes. to estimate the numerical error in the pary longitudes， we performed the following procedures. we compared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher auracy than the main integrations. for this purpose， we performed a much more aurate integration with a stepsize of 0.125 d (1/64 of the main integrations) spanning 3 x 105 yr， starting with the same initial conditions as in the n?1 integration. we consider that this test integration provides us with a ‘pseudo-true’ solution of pary orbital evolution. next， we compare the test integration with the main integration， n?1. for the period of 3 x 105 yr， we see a difference in mean anomalies of the earth between the two integrations of ～0.52°(in the case of the n?1 integration). this difference can be extrapoted to the value ～8700°， about 25 rotations of earth after 5 gyr， since the error of longitudes increases linearly with time in the symplectic map. simirly， the longitude error of pluto can be estimated as ～12°. this value for pluto is much better than the result in kinoshita & nakai (1996) where the difference is estimated as ～60°.
3 numerical results – i. gnce at the raw data
in this section we briefly review the long-term stability of pary orbital motion through some snapshots of raw numerical data. the orbital motion of ps indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of ps took pce.
3.1 general description of the stability of pary orbits
first， we briefly look at the general character of the long-term stability of pary orbits. our interest here focuses particurly on the inner four terrestrial ps for which the orbital time-scales are much shorter than those of the outer five ps. as we can see clearly from the pnar orbital configurations shown in figs 2 and 3， orbital positions of the terrestrial ps differ little between the initial and final part of each numerical integration， which spans several gyr. the solid lines denoting the present orbits of the ps lie almost within the swarm of dots even in the final part of integrations (b) and (d). this indicates that throughout the entire integration period the almost regur variations of pary orbital motion remain nearly the same as they are at present.
vertical view of the four inner pary orbits (from the z -axis direction) at the initial and final parts of the integrationsn±1. the axes units are au. the xy -pne is set to the invariant pne of sor system total angur momentum.(a) the initial part ofn+1 ( t = 0 to 0.0547 x 10 9 yr).(b) the final part ofn+1 ( t = 4.9339 x 10 8 to 4.9886 x 10 9 yr).(c) the initial part of n?1 (t= 0 to ?0.0547 x 109 yr).(d) the final part ofn?1 ( t =?3.9180 x 10 9 to ?3.9727 x 10 9 yr). in each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 x 107 yr . solid lines in each panel denote the present orbits of the four terrestrial ps (taken from de245).
the variation of eentricities and orbital inclinations for the inner four ps in the initial and final part of the integration n+1 is shown in fig. 4. as expected， the character of the variation of pary orbital elements does not differ significantly between the initial and final part of each integration， at least for venus， earth and mars. the elements of mercury， especially its eentricity， seem to change to a significant 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 nearest to instability. this result appears to be in some agreement with laskar's (1994， 1996) expectations that rge and irregur variations appear in the eentricities and inclinations 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 in section 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 section 5).
3.2 time–frequency maps
although the pary motion exhibits very long-term stability defined as the non-existence of close encounter events， the chaotic nature of pary dynamics can change the osciltory period and amplitude of pary orbital motion gradually over such long time-spans. even such slight fluctuations of orbital variation in the frequency domain， particurly in the case of earth， can potentially have a significant effect on its surface climate system through sor insotion variation (cf. berger 1988).
to give an overview of the long-term change in periodicity in pary 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) frequency analysis.
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， when the ith data begins from t=ti and ends at t=ti+t， the next data segment ranges from ti+δt≤ti+δt+t， where δt?t. we continue this division until we reach a certain number n by which tn+t reaches the total integration length.
we apply an fft to each of the data fragments， and obtain n frequency diagrams.
in each frequency diagram obtained above， the strength of periodicity can be repced by a grey-scale (or colour) chart.
we perform the repcement， and connect all the grey-scale (or colour) charts into one graph for each integration. the horizontal axis of these new graphs 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， since the amount of numerical data to be decomposed into frequency components 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 eentricity and inclination of earth in n+2 integration. 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 stronger than in the lighter area around it. we can recognize from this map that the periodicity of the eentricity and inclination of earth only changes slightly over the entire period covered by the n+2 integration. this nearly regur trend is qualitatively the same in other integrations and for other ps， although typical frequencies differ p by p and element by element.
4.2 long-term exchange of orbital energy and angur momentum
we calcute very long-periodic variation and exchange 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 component per unit mass. l is reted to the pary orbital energy e per unit mass as e=?μ2/2l2. if the system is completely linear， the orbital energy and the angur momentum in each frequency bin must be constant. non-linearity in the pary system can cause an exchange of energy and angur momentum in the frequency domain. 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 prominent in our long-term integrations.
in fig. 7， the total orbital energy and angur momentum of the four inner ps and all nine ps are shown for integration 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 component ( h- h0) of the inner four ps calcuted from the low-pass filtered deunay elements.e0， g0， h0 denote the 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 jovian ps.
comparing 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 inner terrestrial pary subsystem is more stable than the outer one: this is simply a result of the retive smallness of the masses of the four terrestrial ps compared with those of the outer jovian ps. another thing we notice is that the inner pary subsystem may become unstable more rapidly than the outer one because of its shorter orbital time-scales. this can be seen in the panels denoted asinner 4 in fig. 7 where the longer-periodic and irregur osciltions are more apparent than in the panels denoted astotal 9. actually， the fluctuations in theinner 4 panels are to a rge extent as a result of the orbital variation of the mercury. however， we cannot neglect the contribution from other terrestrial ps， as we will see in subsequent sections.
4.4 long-term coupling of several neighbouring 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 each p and the angur momentum in n+1 and n?2 integrations. we notice that some ps form apparent pairs in terms of orbital energy and angur momentum exchange. in particur， venus and earth make a typical pair. in the figures， they show negative corretions in exchange of energy and positive corretions in exchange of angur momentum. the negative corretion in exchange of orbital energy means that the two ps form a closed dynamical system in terms of the orbital energy. the positive corretion in exchange of angur momentum means that the two ps are simultaneously under certain long-term perturbations. candidates for perturbers are jupiter and saturn. also in fig. 11， we can see that mars shows a positive corretion in the angur momentum variation to the venus–earth system. mercury exhibits certain negative corretions in the angur momentum versus the venus–earth system， which seems to be a reaction caused by the conservation of angur momentum in the terrestrial pary subsystem.
it is not clear at the moment why the venus–earth pair exhibits a negative corretion in energy exchange and a positive corretion in angur momentum exchange. we may possibly expin this through observing the general fact that there are no secur terms in pary semimajor axes up to second-order perturbation theories (cf. brouwer & clemence 1961; boaletti & pucao 1998). this means that the pary orbital energy (which is directly reted to the semimajor axis a) might be much less affected by perturbing ps than is the angur momentum exchange (which retes to e). hence， the eentricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive corretion in the angur momentum exchange. on the other hand， the semimajor axes of venus and earth are less likely to be disturbed by the jovian ps. thus the energy exchange may be limited only within the venus–earth pair， which results in a negative corretion in the exchange of orbital energy in the pair.
as for the outer jovian pary subsystem， jupiter–saturn and uranus–neptune seem to make dynamical pairs. however， the strength of their coupling is not as strong compared with that of the venus–earth pair.
5 ± 5 x 1010-yr integrations of outer pary orbits
since the jovian pary masses are much rger than the terrestrial pary masses， we treat the jovian pary system as an independent pary system in terms of the study of its dynamical stability. hence， we added a couple of trial integrations that span ± 5 x 1010 yr， including only the outer five ps (the four jovian ps plus pluto). the results exhibit the rigorous stability of the outer pary system over this long time-span. orbital configurations (fig. 12)， and variation of eentricities and inclinations (fig. 13) show this very long-term stability of the outer five ps in both the time and the frequency domains. although we do not show maps here， the typical frequency of the orbital osciltion of pluto and the other outer ps is almost constant during these very long-term integration periods， which is demonstrated in the time–frequency maps on our webpage.
in these two integrations， the retive numerical error in the total energy was ～10?6 and that of the total angur momentum was ～10?10.
5.1 resonances in the neptune–pluto system
kinoshita & nakai (1996) integrated the outer five pary orbits over ± 5.5 x 109 yr . they found that four major resonances between neptune and pluto are maintained during the whole integration 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 denotes the mean longitude，Ω is the longitude of the ascending node and ? is the longitude of perihelion. subscripts p and n denote pluto and neptune.
mean motion resonance between neptune 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 x 104 yr.
the argument of perihelion of pluto wp=θ2=?p?Ωp librates around 90° with a period of about 3.8 x 106 yr. the dominant periodic variations of the eentricity and inclination of pluto are synchronized with the libration of its argument of perihelion. this is anticipated in the secur perturbation theory constructed by kozai (1962).
the longitude of the node of pluto referred to the longitude of the node of neptune，θ3=Ωp?Ωn， circutes and the period of this circution is equal to the period of θ2 libration. when θ3 becomes zero， i.e. the longitudes of ascending nodes of neptune and pluto overp， the inclination of pluto becomes maximum， the eentricity becomes minimum and the argument of perihelion becomes 90°. when θ3 becomes 180°， the inclination of pluto becomes minimum， the eentricity becomes maximum and the argument of perihelion becomes 90° again. williams & benson (1971) anticipated this type of resonance， ter confirmed by mini， nobili & carpino (1989).
an argument θ4=?p??n+ 3 (Ωp?Ωn) librates around 180° with a long period，～ 5.7 x 108 yr.
in our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain simir during the whole integration 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 interesting fact that kinoshita & nakai's (1995， 1996) shorter integrations were not able to disclose.
6 discussion
what kind of dynamical mechanism maintains this long-term stability of the pary system? we can immediately think of two major features that may be responsible for the long-term stability. first， there seem to be no significant 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 resonance (the famous ‘great inequality’)， but not just in the resonance zone. higher-order resonances may cause the chaotic nature of the pary dynamical motion， but they are not so strong as to destroy the stable pary motion within the lifetime of the real sor system. the second feature， which we think is more important for the long-term stability of our pary system， is the difference in dynamical distance between terrestrial and jovian pary 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 jovian ps are less than 14rh. this difference is directly reted to the difference between dynamical features of terrestrial and jovian ps. terrestrial ps have smaller masses， shorter orbital periods and wider dynamical separation. they are strongly perturbed by jovian ps that have rger masses， longer orbital periods and narrower dynamical separation. jovian ps are not perturbed by any other massive bodies.
the present terrestrial pary system is still being disturbed by the massive jovian ps. however， the wide separation and mutual interaction among the terrestrial ps renders the disturbance ineffective; the degree of disturbance by jovian ps is o(ej)(order of magnitude of the eentricity of jupiter)， since the disturbance caused by jovian ps is a forced osciltion having an amplitude of o(ej). heightening of eentricity， for example o(ej)～0.05， is far from sufficient to provoke instability in the terrestrial 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 significant conditions for maintaining the stability of the pary system over a 109-yr time-span. our detailed analysis of the retionship between dynamical distance between ps and the instability time-scale of sor system pary motion is now on-going.
although our numerical integrations span the lifetime of the sor system， the number of integrations is far from sufficient to fill the initial phase space. it is necessary to perform more and more numerical integrations to confirm and examine in detail the long-term stability of our pary dynamics.
——以上文段引自 ito， t.& tanikawa， k. long-term integrations 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。
作者君在作品相关中其实已经解释过这个问题。
不过仍然有人质疑——“你说得太含糊了”，“火星轨道的变化比你想象要大得多！”
那好吧，既然作者君的简单解释不够有力，那咱们就看看严肃的东西，反正这本书写到现在，嚷嚷着本书bug一大堆，用初高中物理在书中挑刺的人也不少。
以下是文章内容：
long-term integrations and stability of pary orbits in our sor system
abstract
we present the results of very long-term numerical integrations of pary orbital motions over 109 -yr time-spans including all nine ps. a quick inspection of our numerical data shows that the pary motion， at least in our simple dynamical model， seems to be quite stable even over this very long time-span. a closer look at the lowest-frequency osciltions using a low-pass filter shows us the potentially diffusive character of terrestrial pary motion， especially that of mercury. the behaviour of the eentricity of mercury in our integrations is qualitatively simir to the results from jacques laskar's secur perturbation theory (e.g. emax～ 0.35 over ～± 4 gyr). however， there are no apparent secur increases of eentricity or inclination in any orbital elements of the ps， which may be revealed by still longer-term numerical integrations. we have also performed a couple of trial integrations including motions of the outer five ps over the duration of ± 5 x 1010 yr. the result indicates that the three major resonances in the neptune–pluto system have been maintained over the 1011-yr time-span.
1 introduction
1.1definition of the problem
the question of the stability of our sor system has been debated over several hundred years， since the era of newton. the problem has attracted many famous mathematicians over the years and has pyed a central role in the development of non-linear dynamics and chaos theory. however， we do not yet have a definite answer to the question of whether our sor system is stable or not. this is partly a result of the fact that the definition of the term ‘stability’ is vague when it is used in retion to the problem of pary motion in the sor system. actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our sor system.
among many definitions of stability， here we adopt the hill definition (gdman 1993): actually this is not a definition of stability， but of instability. we define a system as becoming unstable when a close encounter ours somewhere in the system， starting from a certain initial configuration (chambers， wetherill & boss 1996; ito & tanikawa 1999). a system is defined as experiencing a close encounter when two bodies approach one another within an area of the rger hill radius. otherwise the system is defined as being stable. henceforward we state that our pary system is dynamically stable if no close encounter happens during the age of our sor system， about ±5 gyr. incidentally， this definition may be repced by one in which an ourrence of any orbital crossing between either of a pair of ps takes pce. this is because we know from experience that an orbital crossing is very likely to lead to a close encounter in pary and protopary systems (yoshinaga， kokubo & makino 1999). of course this statement cannot be simply applied to systems with stable orbital resonances such as the neptune–pluto system.
1.2previous studies and aims of this research
in addition to the vagueness of the concept of stability， the ps in our sor system show a character typical of dynamical chaos (sussman & wisdom 1988， 1992). the cause of this chaotic behaviour is now partly understood as being a result of resonance overpping (murray & holman 1999; lecar， franklin & holman 2001). however， it would require integrating over an ensemble of pary systems including all nine ps for a period covering several 10 gyr to thoroughly understand the long-term evolution of pary orbits， since chaotic dynamical systems are characterized by their strong dependence on initial conditions.
from that point of view， many of the previous long-term numerical integrations included only the outer five ps (sussman & wisdom 1988; kinoshita & nakai 1996). this is because the orbital periods of the outer ps are so much longer than those of the inner four ps that it is much easier to follow the system for a given integration period. at present， the longest numerical integrations published in journals are those of duncan & lissauer (1998). although their main target was the effect of post-main-sequence sor mass loss on the stability of pary orbits， they performed many integrations covering up to ～1011 yr of the orbital motions of the four jovian ps. the initial orbital elements and masses of ps are the same as those of our sor system in duncan & lissauer's paper， but they decrease the mass of the sun gradually in their numerical experiments. this is because they consider the effect of post-main-sequence sor mass loss in the paper. consequently， they found that the crossing time-scale of pary orbits， which can be a typical indicator of the instability time-scale， is quite sensitive to the rate of mass decrease of the sun. when the mass of the sun is close to its present value， the jovian ps remain stable over 1010 yr， or perhaps longer. duncan & lissauer also performed four simir experiments on the orbital motion of seven ps (venus to neptune)， which cover a span of ～109 yr. their experiments on the seven ps are not yet comprehensive， but it seems that the terrestrial ps also remain stable during the integration period， maintaining almost regur osciltions.
on the other hand， in his aurate semi-analytical secur perturbation theory (laskar 1988)， laskar finds that rge and irregur variations can appear in the eentricities and inclinations of the terrestrial ps， especially of mercury and mars on a time-scale of several 109 yr (laskar 1996). the results of laskar's secur perturbation theory should be confirmed and investigated by fully numerical integrations.
in this paper we present preliminary results of six long-term numerical integrations on all nine pary orbits， covering a span of several 109 yr， and of two other integrations covering a span of ± 5 x 1010 yr. the total epsed time for all integrations is more than 5 yr， using several dedicated pcs and workstations. one of the fundamental conclusions of our long-term integrations is that sor system pary motion seems to be stable in terms of the hill stability mentioned above， at least over a time-span of ± 4 gyr. actually， in our numerical integrations the system was far more stable than what is defined by the hill stability criterion: not only did no close encounter happen during the integration period， but also all the pary orbital elements have been confined in a narrow region both in time and frequency domain， though pary motions are stochastic. since the purpose of this paper is to exhibit and overview the results of our long-term numerical integrations， we show typical example figures as evidence of the very long-term stability of sor system pary motion. for readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (aess )， where we show raw orbital elements， their low-pass filtered results， variation of deunay elements and angur momentum deficit， and results of our simple time–frequency analysis on all of our integrations.
in section 2 we briefly expin our dynamical model， numerical method and initial conditions used in our integrations. section 3 is devoted to a description of the quick results of the numerical integrations. very long-term stability of sor system pary motion is apparent both in pary positions and orbital elements. a rough estimation of numerical errors is also given. section 4 goes on to a discussion of the longest-term variation of pary orbits using a low-pass filter and includes a discussion of angur momentum deficit. in section 5， we present a set of numerical integrations for the outer five ps that spans ± 5 x 1010 yr. in section 6 we also discuss the long-term stability of the pary motion and its possible cause.
2 description of the numerical integrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3 numerical method
we utilize a second-order wisdom–holman symplectic map as our main integration method (wisdom & holman 1991; kinoshita， yoshida & nakai 1991) with a special start-up procedure to reduce the truncation error of angle variables，‘warm start’(saha & tremaine 1992， 1994).
the stepsize for the numerical integrations is 8 d throughout all integrations of the nine ps (n±1，2，3)， which is about 1/11 of the orbital period of the innermost p (mercury). as for the determination of stepsize， we partly follow the previous numerical integration of all nine ps in sussman & wisdom (1988， 7.2 d) and saha & tremaine (1994， 225/32 d). we rounded the decimal part of the their stepsizes to 8 to make the stepsize a multiple of 2 in order to reduce the aumution of round-off error in the computation processes. in retion to this， wisdom & holman (1991) performed numerical integrations of the outer five pary orbits using the symplectic map with a stepsize of 400 d， 1/10.83 of the orbital period of jupiter. their result seems to be aurate enough， which partly justifies our method of determining the stepsize. however， since the eentricity of jupiter (～0.05) is much smaller than that of mercury (～0.2)， we need some care when we compare these integrations simply in terms of stepsizes.
in the integration of the outer five ps (f±)， we fixed the stepsize at 400 d.
we adopt gauss' f and g functions in the symplectic map together with the third-order halley method (danby 1992) as a solver for kepler equations. the number of maximum iterations we set in halley's method is 15， but they never reached the maximum in any of our integrations.
the interval of the data output is 200 000 d (～547 yr) for the calcutions of all nine ps (n±1，2，3)， and about 8000 000 d (～21 903 yr) for the integration of the outer five ps (f±).
although no output filtering was done when the numerical integrations were in process， we applied a low-pass filter to the raw orbital data after we had completed all the calcutions. see section 4.1 for more detail.
2.4 error estimation
2.4.1 retive errors in total energy and angur momentum
aording to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angur momentum)， our long-term numerical integrations seem to have been performed with very small errors. the averaged retive errors of total energy (～10?9) and of total angur momentum (～10?11) have remained nearly constant throughout the integration period (fig. 1). the special startup procedure， warm start， would have reduced the averaged retive error in total energy by about one order of magnitude or more.
retive numerical error of the total angur momentum δa/a0 and the total energy δe/e0 in our numerical integrationsn± 1，2，3， where δe and δa are the absolute change of the total energy and total angur momentum， respectively， ande0anda0are their initial values. the horizontal unit is gyr.
note that different operating systems， different mathematical libraries， and different hardware architectures result in different numerical errors， through the variations in round-off error handling and numerical algorithms. in the upper panel of fig. 1， we can recognize this situation in the secur numerical error in the total angur momentum， which should be rigorously preserved up to machine-e precision.
2.4.2 error in pary longitudes
since the symplectic maps preserve total energy and total angur momentum of n-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the auracy of numerical integrations， especially as a measure of the positional error of ps， i.e. the error in pary longitudes. to estimate the numerical error in the pary longitudes， we performed the following procedures. we compared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher auracy than the main integrations. for this purpose， we performed a much more aurate integration with a stepsize of 0.125 d (1/64 of the main integrations) spanning 3 x 105 yr， starting with the same initial conditions as in the n?1 integration. we consider that this test integration provides us with a ‘pseudo-true’ solution of pary orbital evolution. next， we compare the test integration with the main integration， n?1. for the period of 3 x 105 yr， we see a difference in mean anomalies of the earth between the two integrations of ～0.52°(in the case of the n?1 integration). this difference can be extrapoted to the value ～8700°， about 25 rotations of earth after 5 gyr， since the error of longitudes increases linearly with time in the symplectic map. simirly， the longitude error of pluto can be estimated as ～12°. this value for pluto is much better than the result in kinoshita & nakai (1996) where the difference is estimated as ～60°.
3 numerical results – i. gnce at the raw data
in this section we briefly review the long-term stability of pary orbital motion through some snapshots of raw numerical data. the orbital motion of ps indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of ps took pce.
3.1 general description of the stability of pary orbits
first， we briefly look at the general character of the long-term stability of pary orbits. our interest here focuses particurly on the inner four terrestrial ps for which the orbital time-scales are much shorter than those of the outer five ps. as we can see clearly from the pnar orbital configurations shown in figs 2 and 3， orbital positions of the terrestrial ps differ little between the initial and final part of each numerical integration， which spans several gyr. the solid lines denoting the present orbits of the ps lie almost within the swarm of dots even in the final part of integrations (b) and (d). this indicates that throughout the entire integration period the almost regur variations of pary orbital motion remain nearly the same as they are at present.
vertical view of the four inner pary orbits (from the z -axis direction) at the initial and final parts of the integrationsn±1. the axes units are au. the xy -pne is set to the invariant pne of sor system total angur momentum.(a) the initial part ofn+1 ( t = 0 to 0.0547 x 10 9 yr).(b) the final part ofn+1 ( t = 4.9339 x 10 8 to 4.9886 x 10 9 yr).(c) the initial part of n?1 (t= 0 to ?0.0547 x 109 yr).(d) the final part ofn?1 ( t =?3.9180 x 10 9 to ?3.9727 x 10 9 yr). in each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 x 107 yr . solid lines in each panel denote the present orbits of the four terrestrial ps (taken from de245).
the variation of eentricities and orbital inclinations for the inner four ps in the initial and final part of the integration n+1 is shown in fig. 4. as expected， the character of the variation of pary orbital elements does not differ significantly between the initial and final part of each integration， at least for venus， earth and mars. the elements of mercury， especially its eentricity， seem to change to a significant 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 nearest to instability. this result appears to be in some agreement with laskar's (1994， 1996) expectations that rge and irregur variations appear in the eentricities and inclinations 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 in section 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 section 5).
3.2 time–frequency maps
although the pary motion exhibits very long-term stability defined as the non-existence of close encounter events， the chaotic nature of pary dynamics can change the osciltory period and amplitude of pary orbital motion gradually over such long time-spans. even such slight fluctuations of orbital variation in the frequency domain， particurly in the case of earth， can potentially have a significant effect on its surface climate system through sor insotion variation (cf. berger 1988).
to give an overview of the long-term change in periodicity in pary 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) frequency analysis.
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， when the ith data begins from t=ti and ends at t=ti+t， the next data segment ranges from ti+δt≤ti+δt+t， where δt?t. we continue this division until we reach a certain number n by which tn+t reaches the total integration length.
we apply an fft to each of the data fragments， and obtain n frequency diagrams.
in each frequency diagram obtained above， the strength of periodicity can be repced by a grey-scale (or colour) chart.
we perform the repcement， and connect all the grey-scale (or colour) charts into one graph for each integration. the horizontal axis of these new graphs 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， since the amount of numerical data to be decomposed into frequency components 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 eentricity and inclination of earth in n+2 integration. 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 stronger than in the lighter area around it. we can recognize from this map that the periodicity of the eentricity and inclination of earth only changes slightly over the entire period covered by the n+2 integration. this nearly regur trend is qualitatively the same in other integrations and for other ps， although typical frequencies differ p by p and element by element.
4.2 long-term exchange of orbital energy and angur momentum
we calcute very long-periodic variation and exchange 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 component per unit mass. l is reted to the pary orbital energy e per unit mass as e=?μ2/2l2. if the system is completely linear， the orbital energy and the angur momentum in each frequency bin must be constant. non-linearity in the pary system can cause an exchange of energy and angur momentum in the frequency domain. 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 prominent in our long-term integrations.
in fig. 7， the total orbital energy and angur momentum of the four inner ps and all nine ps are shown for integration 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 component ( h- h0) of the inner four ps calcuted from the low-pass filtered deunay elements.e0， g0， h0 denote the 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 jovian ps.
comparing 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 inner terrestrial pary subsystem is more stable than the outer one: this is simply a result of the retive smallness of the masses of the four terrestrial ps compared with those of the outer jovian ps. another thing we notice is that the inner pary subsystem may become unstable more rapidly than the outer one because of its shorter orbital time-scales. this can be seen in the panels denoted asinner 4 in fig. 7 where the longer-periodic and irregur osciltions are more apparent than in the panels denoted astotal 9. actually， the fluctuations in theinner 4 panels are to a rge extent as a result of the orbital variation of the mercury. however， we cannot neglect the contribution from other terrestrial ps， as we will see in subsequent sections.
4.4 long-term coupling of several neighbouring 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 each p and the angur momentum in n+1 and n?2 integrations. we notice that some ps form apparent pairs in terms of orbital energy and angur momentum exchange. in particur， venus and earth make a typical pair. in the figures， they show negative corretions in exchange of energy and positive corretions in exchange of angur momentum. the negative corretion in exchange of orbital energy means that the two ps form a closed dynamical system in terms of the orbital energy. the positive corretion in exchange of angur momentum means that the two ps are simultaneously under certain long-term perturbations. candidates for perturbers are jupiter and saturn. also in fig. 11， we can see that mars shows a positive corretion in the angur momentum variation to the venus–earth system. mercury exhibits certain negative corretions in the angur momentum versus the venus–earth system， which seems to be a reaction caused by the conservation of angur momentum in the terrestrial pary subsystem.
it is not clear at the moment why the venus–earth pair exhibits a negative corretion in energy exchange and a positive corretion in angur momentum exchange. we may possibly expin this through observing the general fact that there are no secur terms in pary semimajor axes up to second-order perturbation theories (cf. brouwer & clemence 1961; boaletti & pucao 1998). this means that the pary orbital energy (which is directly reted to the semimajor axis a) might be much less affected by perturbing ps than is the angur momentum exchange (which retes to e). hence， the eentricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive corretion in the angur momentum exchange. on the other hand， the semimajor axes of venus and earth are less likely to be disturbed by the jovian ps. thus the energy exchange may be limited only within the venus–earth pair， which results in a negative corretion in the exchange of orbital energy in the pair.
as for the outer jovian pary subsystem， jupiter–saturn and uranus–neptune seem to make dynamical pairs. however， the strength of their coupling is not as strong compared with that of the venus–earth pair.
5 ± 5 x 1010-yr integrations of outer pary orbits
since the jovian pary masses are much rger than the terrestrial pary masses， we treat the jovian pary system as an independent pary system in terms of the study of its dynamical stability. hence， we added a couple of trial integrations that span ± 5 x 1010 yr， including only the outer five ps (the four jovian ps plus pluto). the results exhibit the rigorous stability of the outer pary system over this long time-span. orbital configurations (fig. 12)， and variation of eentricities and inclinations (fig. 13) show this very long-term stability of the outer five ps in both the time and the frequency domains. although we do not show maps here， the typical frequency of the orbital osciltion of pluto and the other outer ps is almost constant during these very long-term integration periods， which is demonstrated in the time–frequency maps on our webpage.
in these two integrations， the retive numerical error in the total energy was ～10?6 and that of the total angur momentum was ～10?10.
5.1 resonances in the neptune–pluto system
kinoshita & nakai (1996) integrated the outer five pary orbits over ± 5.5 x 109 yr . they found that four major resonances between neptune and pluto are maintained during the whole integration 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 denotes the mean longitude，Ω is the longitude of the ascending node and ? is the longitude of perihelion. subscripts p and n denote pluto and neptune.
mean motion resonance between neptune 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 x 104 yr.
the argument of perihelion of pluto wp=θ2=?p?Ωp librates around 90° with a period of about 3.8 x 106 yr. the dominant periodic variations of the eentricity and inclination of pluto are synchronized with the libration of its argument of perihelion. this is anticipated in the secur perturbation theory constructed by kozai (1962).
the longitude of the node of pluto referred to the longitude of the node of neptune，θ3=Ωp?Ωn， circutes and the period of this circution is equal to the period of θ2 libration. when θ3 becomes zero， i.e. the longitudes of ascending nodes of neptune and pluto overp， the inclination of pluto becomes maximum， the eentricity becomes minimum and the argument of perihelion becomes 90°. when θ3 becomes 180°， the inclination of pluto becomes minimum， the eentricity becomes maximum and the argument of perihelion becomes 90° again. williams & benson (1971) anticipated this type of resonance， ter confirmed by mini， nobili & carpino (1989).
an argument θ4=?p??n+ 3 (Ωp?Ωn) librates around 180° with a long period，～ 5.7 x 108 yr.
in our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain simir during the whole integration 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 interesting fact that kinoshita & nakai's (1995， 1996) shorter integrations were not able to disclose.
6 discussion
what kind of dynamical mechanism maintains this long-term stability of the pary system? we can immediately think of two major features that may be responsible for the long-term stability. first， there seem to be no significant 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 resonance (the famous ‘great inequality’)， but not just in the resonance zone. higher-order resonances may cause the chaotic nature of the pary dynamical motion， but they are not so strong as to destroy the stable pary motion within the lifetime of the real sor system. the second feature， which we think is more important for the long-term stability of our pary system， is the difference in dynamical distance between terrestrial and jovian pary 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 jovian ps are less than 14rh. this difference is directly reted to the difference between dynamical features of terrestrial and jovian ps. terrestrial ps have smaller masses， shorter orbital periods and wider dynamical separation. they are strongly perturbed by jovian ps that have rger masses， longer orbital periods and narrower dynamical separation. jovian ps are not perturbed by any other massive bodies.
the present terrestrial pary system is still being disturbed by the massive jovian ps. however， the wide separation and mutual interaction among the terrestrial ps renders the disturbance ineffective; the degree of disturbance by jovian ps is o(ej)(order of magnitude of the eentricity of jupiter)， since the disturbance caused by jovian ps is a forced osciltion having an amplitude of o(ej). heightening of eentricity， for example o(ej)～0.05， is far from sufficient to provoke instability in the terrestrial 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 significant conditions for maintaining the stability of the pary system over a 109-yr time-span. our detailed analysis of the retionship between dynamical distance between ps and the instability time-scale of sor system pary motion is now on-going.
although our numerical integrations span the lifetime of the sor system， the number of integrations is far from sufficient to fill the initial phase space. it is necessary to perform more and more numerical integrations to confirm and examine in detail the long-term stability of our pary dynamics.
——以上文段引自 ito， t.& tanikawa， k. long-term integrations 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。