第76章 对火星轨道变化问题的最后解释
第76章 对火星轨道变化问题的最后解释
作者君在作品相关中已经解释过这个问题，并在此列出相关参考文献中的一篇开源论文。
以下是文章内容：
long-term integrations and stability of plary orbits in our solar system
abstract
we present the results of very long-term numerical integrations of plary orbital motions over 109 -yr time-spans including all nine pls. a quick inspection of our numerical data shows that the plary 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 oscillations using a low-pass filter shows us the potentially diffusive character of terrestrial plary motion， especially that of mercury. the behaviour of the eccentricity of mercury in our integrations is qualitatively similar to the results from jacques laskar's secular perturbation theory (e.g. emax 0.35 over ± 4 gyr). however， there are no apparent secular increases of eccentricity or inclination in any orbital elements of the pls， 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 pls over the duration of ± 5 × 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 solar 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 played 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 solar 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 relation to the problem of plary motion in the solar system. actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our solar system.
among many definitions of stability， here we adopt the hill definition (gladman 1993): actually this is not a definition of stability， but of instability. we define a system as bing unstable when a close encounter occurs 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 larger hill radius. otherwise the system is defined as being stable. henceforward we state that our plary system is dynamically stable if no close encounter happens during the age of our solar system， about ±5 gyr. incidentally， this definition may be replaced by one in which an occurrence of any orbital crossing between either of a pair of pls takes place. this is because we know from experience that an orbital crossing is very likely to lead to a close encounter in plary and protoplary 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 pls in our solar 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 overlapping (murray & holman 1999; lecar， franklin & holman 2001). however， it would require integrating over an ensemble of plary systems including all nine pls for a period covering several 10 gyr to thoroughly understand the long-term evolution of plary 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 pls (sussman & wisdom 1988; kinoshita & nakai 1996). this is because the orbital periods of the outer pls are so much longer than those of the inner four pls 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 solar mass loss on the stability of plary orbits， they performed many integrations covering up to 1011 yr of the orbital motions of the four jovian pls. the initial orbital elements and masses of pls are the same as those of our solar 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 solar mass loss in the paper. consequently， they found that the crossing time-scale of plary 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 pls remain stable over 1010 yr， or perhaps longer. duncan & lissauer also performed four similar experiments on the orbital motion of seven pls (venus to neptune)， which cover a span of 109 yr. their experiments on the seven pls are not yetprehensive， but it seems that the terrestrial pls also remain stable during the integration period， maintaining almost regular oscillations.
on the other hand， in his accurate semi-analytical secular perturbation theory (laskar 1988)， laskar finds that large and irregular variations can appear in the eccentricities and inclinations of the terrestrial pls， especially of mercury and mars on a time-scale of several 109 yr (laskar 1996). the results of laskar's secular 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 plary orbits， covering a span of several 109 yr， and of two other integrations covering a span of ± 5 × 1010 yr. the total elapsed 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 solar system plary 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 plary orbital elements have been confined in a narrow region both in time and frequency domain， though plary 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 solar system plary motion. for readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (access )， where we show raw orbital elements， their low-pass filtered results， variation of delaunay elements and angular momentum deficit， and results of our simple time–frequency analysis on all of our integrations.
in section 2 we briefly explain 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 solar system plary motion is apparent both in plary 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 plary orbits using a low-pass filter and includes a discussion of angular momentum deficit. in section 5， we present a set of numerical integrations for the outer five pls that spans ± 5 × 1010 yr. in section 6 we also discuss the long-term stability of the plary 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 pls (n±1，2，3)， which is about 1/11 of the orbital period of the innermost pl (mercury). as for the determination of stepsize， we partly follow the previous numerical integration of all nine pls 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 accumulation of round-off error in theputation processes. in relation to this， wisdom & holman (1991) performed numerical integrations of the outer five plary 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 accurate enough， which partly justifies our method of determining the stepsize. however， since the eccentricity of jupiter (0.05) is much smaller than that of mercury (0.2)， we need some care when wepare these integrations simply in terms of stepsizes.
in the integration of the outer five pls (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 calculations of all nine pls (n±1，2，3)， and about 8000 000 d (21 903 yr) for the integration of the outer five pls (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 hadpleted all the calculations. see section 4.1 for more detail.
2.4 error estimation
2.4.1 relative errors in total energy and angular momentum
according to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angular momentum)， our long-term numerical integrations seem to have been performed with very small errors. the averaged relative errors of total energy (109) and of total angular momentum (1011) have remained nearly constant throughout the integration period (fig. 1). the special startup procedure， warm start， would have reduced the averaged relative error in total energy by about one order of magnitude or more.
relative numerical error of the total angular 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 angular 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 secular numerical error in the total angular momentum， which should be rigorously preserved up to machine-ε precision.
2.4.2 error in plary longitudes
since the symplectic maps preserve total energy and total angular momentum of n-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the accuracy of numerical integrations， especially as a measure of the positional error of pls， i.e. the error in plary longitudes. to estimate the numerical error in the plary longitudes， we performed the following procedures. wepared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher accuracy than the main integrations. for this purpose， we performed a much more accurate integration with a stepsize of 0.125 d (1/64 of the main integrations) spanning 3 × 105 yr， starting with the same initial conditions as in the n1 integration. we consider that this test integration provides us with a ‘pseudo-true’ solution of plary orbital evolution. next， wepare the test integration with the main integration， n1. for the period of 3 × 105 yr， we see a difference in mean anomalies of the earth between the two integrations of 0.52°(in the case of the n1 integration). this difference can be extrapolated 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. similarly， 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. glance at the raw data
in this section we briefly review the long-term stability of plary orbital motion through some snapshots of raw numerical data. the orbital motion of pls indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of pls took place.
3.1 general description of the stability of plary orbits
first， we briefly look at the general character of the long-term stability of plary orbits. our interest here focuses particularly on the inner four terrestrial pls for which the orbital time-scales are much shorter than those of the outer five pls. as we can see clearly from the planar orbital configurations shown in figs 2 and 3， orbital positions of the terrestrial pls 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 pls 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 regular variations of plary orbital motion remain nearly the same as they are at present.
vertical view of the four inner plary orbits (from the z -axis direction) at the initial and final parts of the integrationsn±1. the axes units are au. the xy -plane is set to the invariant plane of solar system total angular momentum.(a) the initial part ofn+1 ( t = 0 to 0.0547 × 10 9 yr).(b) the final part ofn+1 ( t = 4.9339 × 10 8 to 4.9886 × 10 9 yr).(c) the initial part of n1 (t= 0 to 0.0547 × 109 yr).(d) the final part ofn1 ( t =3.9180 × 10 9 to 3.9727 × 10 9 yr). in each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 × 107 yr . solid lines in each panel denote the present orbits of the four terrestrial pls (taken from de245).
the variation of eccentricities and orbital inclinations for the inner four pls in the initial and final part of the integration n+1 is shown in fig. 4. as expected， the character of the variation of plary 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 eccentricity， seem to change to a significant extent. this is partly because the orbital time-scale of the pl is the shortest of all the pls， which leads to a more rapid orbital evolution than other pls; the innermost pl may be nearest to instability. this result appears to be in some agreement with laskar's (1994， 1996) expectations that large and irregular variations appear in the eccentricities 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 plary system owing to the small mass of mercury. we will mention briefly the long-term orbital evolution of mercury later in section 4 using low-pass filtered orbital elements.
the orbital motion of the outer five pls seems rigorously stable and quite regular over this time-span (see also section 5).
3.2 time–frequency maps
although the plary motion exhibits very long-term stability defined as the non-existence of close encounter events， the chaotic nature of plary dynamics can change the oscillatory period and amplitude of plary orbital motion gradually over such long time-spans. even such slight fluctuations of orbital variation in the frequency domain， particularly in the case of earth， can potentially have a significant effect on its surface climate system through solar insolation variation (cf. berger 1988).
to give an overview of the long-term change in periodicity in plary 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 large overlapping 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 replaced by a grey-scale (or colour) chart.
we perform the replacement， 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 oscillation of orbital elements.
we have adopted an fft because of its overwhelming speed， since the amount of numerical data to be dposed into frequencyponents 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 eccentricity 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 eccentricity and inclination of earth only changes slightly over the entire period covered by the n+2 integration. this nearly regular trend is qualitatively the same in other integrations and for other pls， although typical frequencies differ pl by pl and element by element.
4.2 long-term exchange of orbital energy and angular momentum
we calculate very long-periodic variation and exchange of plary orbital energy and angular momentum using filtered delaunay elements l， g， h. g and h are equivalent to the plary orbital angular momentum and its verticalponent per unit mass. l is related to the plary orbital energy e per unit mass as e=μ2/2l2. if the system ispletely linear， the orbital energy and the angular momentum in each frequency bin must be constant. non-linearity in the plary system can cause an exchange of energy and angular momentum in the frequency domain. the amplitude of the lowest-frequency oscillation 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 angular momentum of the four inner pls and all nine pls are shown for integration n+2. the upper three panels show the long-periodic variation of total energy (denoted ase- e0)， total angular momentum ( g- g0)， and the verticalponent ( h- h0) of the inner four pls calculated from the low-pass filtered delaunay 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 pls. the fluctuation shown in the lower panels is virtually entirely a result of the massive jovian pls.
paring the variations of energy and angular momentum of the inner four pls and all nine pls， it is apparent that the amplitudes of those of the inner pls are much smaller than those of all nine pls: the amplitudes of the outer five pls are much larger than those of the inner pls. this does not mean that the inner terrestrial plary subsystem is more stable than the outer one: this is simply a result of the relative smallness of the masses of the four terrestrial plspared with those of the outer jovian pls. another thing we notice is that the inner plary subsystem may be 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 irregular oscillations are more apparent than in the panels denoted astotal 9. actually， the fluctuations in theinner 4 panels are to a large extent as a result of the orbital variation of the mercury. however， we cannot neglect the contribution from other terrestrial pls， as we will see in subsequent sections.
4.4 long-term coupling of several neighbouring pl pairs
let us see some individual variations of plary orbital energy and angular momentum expressed by the low-pass filtered delaunay elements. figs 10 and 11 show long-term evolution of the orbital energy of each pl and the angular momentum in n+1 and n2 integrations. we notice that some pls form apparent pairs in terms of orbital energy and angular momentum exchange. in particular， venus and earth make a typical pair. in the figures， they show negative correlations in exchange of energy and positive correlations in exchange of angular momentum. the negative correlation in exchange of orbital energy means that the two pls form a closed dynamical system in terms of the orbital energy. the positive correlation in exchange of angular momentum means that the two pls 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 correlation in the angular momentum variation to the venus–earth system. mercury exhibits certain negative correlations in the angular momentum versus the venus–earth system， which seems to be a reaction caused by the conservation of angular momentum in the terrestrial plary subsystem.
it is not clear at the moment why the venus–earth pair exhibits a negative correlation in energy exchange and a positive correlation in angular momentum exchange. we may possibly explain this through observing the general fact that there are no secular terms in plary semimajor axes up to second-order perturbation theories (cf. brouwer & clemence 1961; boccaletti & pucacco 1998). this means that the plary orbital energy (which is directly related to the semimajor axis a) might be much less affected by perturbing pls than is the angular momentum exchange (which relates to e). hence， the eccentricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive correlation in the angular momentum exchange. on the other hand， the semimajor axes of venus and earth are less likely to be disturbed by the jovian pls. thus the energy exchange may be limited only within the venus–earth pair， which results in a negative correlation in the exchange of orbital energy in the pair.
as for the outer jovian plary subsystem， jupiter–saturn and uranus–neptune seem to make dynamical pairs. however， the strength of their coupling is not as strongpared with that of the venus–earth pair.
5 ± 5 × 1010-yr integrations of outer plary orbits
since the jovian plary masses are much larger than the terrestrial plary masses， we treat the jovian plary system as an independent plary system in terms of the study of its dynamical stability. hence， we added a couple of trial integrations that span ± 5 × 1010 yr， including only the outer five pls (the four jovian pls plus pluto). the results exhibit the rigorous stability of the outer plary system over this long time-span. orbital configurations (fig. 12)， and variation of eccentricities and inclinations (fig. 13) show this very long-term stability of the outer five pls in both the time and the frequency domains. although we do not show maps here， the typical frequency of the orbital oscillation of pluto and the other outer pls 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 relative numerical error in the total energy was 106 and that of the total angular momentum was 1010.
5.1 resonances in the neptune–pluto system
kinoshita & nakai (1996) integrated the outer five plary orbits over ± 5.5 × 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 λnp librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.
the argument of perihelion of pluto wp=θ2=pΩp librates around 90° with a period of about 3.8 × 106 yr. the dominant periodic variations of the eccentricity and inclination of pluto are synchronized with the libration of its argument of perihelion. this is anticipated in the secular perturbation theory constructed by kozai (1962).
the longitude of the node of pluto referred to the longitude of the node of neptune，θ3=ΩpΩn， circulates and the period of this circulation is equal to the period of θ2 libration. when θ3 bes zero， i.e. the longitudes of ascending nodes of neptune and pluto overlap， the inclination of pluto bes maximum， the eccentricity bes minimum and the argument of perihelion bes 90°. when θ3 bes 180°， the inclination of pluto bes minimum， the eccentricity bes maximum and the argument of perihelion bes 90° again. williams & benson (1971) anticipated this type of resonance， later confirmed by milani， nobili & carpino (1989).
an argument θ4=pn+ 3 (ΩpΩn) librates around 180° with a long period， 5.7 × 108 yr.
in our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain similar during the whole integration period (figs 14–16 ). however， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circulation 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 plary 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 secular) between any pair among the nine pls. 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 plary dynamical motion， but they are not so strong as to destroy the stable plary motion within the lifetime of the real solar system. the second feature， which we think is more important for the long-term stability of our plary system， is the difference in dynamical distance between terrestrial and jovian plary subsystems (ito & tanikawa 1999， 2001). when we measure plary separations by the mutual hill radii (r_)， separations among terrestrial pls are greater than 26rh， whereas those among jovian pls are less than 14rh. this difference is directly related to the difference between dynamical features of terrestrial and jovian pls. terrestrial pls have smaller masses， shorter orbital periods and wider dynamical separation. they are strongly perturbed by jovian pls that have larger masses， longer orbital periods and narrower dynamical separation. jovian pls are not perturbed by any other massive bodies.
the present terrestrial plary system is still being disturbed by the massive jovian pls. however， the wide separation and mutual interaction among the terrestrial pls renders the disturbance ineffective; the degree of disturbance by jovian pls is o(ej)(order of magnitude of the eccentricity of jupiter)， since the disturbance caused by jovian pls is a forced oscillation having an amplitude of o(ej). heightening of eccentricity， for example o(ej)0.05， is far from sufficient to provoke instability in the terrestrial pls having such a wide separation as 26rh. thus we assume that the present wide dynamical separation among terrestrial pls (> 26rh) is probably one of the most significant conditions for maintaining the stability of the plary system over a 109-yr time-span. our detailed analysis of the relationship between dynamical distance between pls and the instability time-scale of solar system plary motion is now on-going.
although our numerical integrations span the lifetime of the solar 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 plary dynamics.
——以上文段引自 ito， t.& tanikawa， k. long-term integrations and stability of plary orbits in our solar 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已经解释过这个问题，并在此列出相关参考文献中的一篇开源论文。
以下是文章内容：
long-term integrations and stability of plary orbits in our solar system
abstract
we present the results of very long-term numerical integrations of plary orbital motions over 109 -yr time-spans including all nine pls. a quick inspection of our numerical data shows that the plary 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 oscillations using a low-pass filter shows us the potentially diffusive character of terrestrial plary motion， especially that of mercury. the behaviour of the eccentricity of mercury in our integrations is qualitatively similar to the results from jacques laskar's secular perturbation theory (e.g. emax 0.35 over ± 4 gyr). however， there are no apparent secular increases of eccentricity or inclination in any orbital elements of the pls， 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 pls over the duration of ± 5 × 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 solar 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 played 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 solar 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 relation to the problem of plary motion in the solar system. actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our solar system.
among many definitions of stability， here we adopt the hill definition (gladman 1993): actually this is not a definition of stability， but of instability. we define a system as bing unstable when a close encounter occurs 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 larger hill radius. otherwise the system is defined as being stable. henceforward we state that our plary system is dynamically stable if no close encounter happens during the age of our solar system， about ±5 gyr. incidentally， this definition may be replaced by one in which an occurrence of any orbital crossing between either of a pair of pls takes place. this is because we know from experience that an orbital crossing is very likely to lead to a close encounter in plary and protoplary 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 pls in our solar 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 overlapping (murray & holman 1999; lecar， franklin & holman 2001). however， it would require integrating over an ensemble of plary systems including all nine pls for a period covering several 10 gyr to thoroughly understand the long-term evolution of plary 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 pls (sussman & wisdom 1988; kinoshita & nakai 1996). this is because the orbital periods of the outer pls are so much longer than those of the inner four pls 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 solar mass loss on the stability of plary orbits， they performed many integrations covering up to 1011 yr of the orbital motions of the four jovian pls. the initial orbital elements and masses of pls are the same as those of our solar 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 solar mass loss in the paper. consequently， they found that the crossing time-scale of plary 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 pls remain stable over 1010 yr， or perhaps longer. duncan & lissauer also performed four similar experiments on the orbital motion of seven pls (venus to neptune)， which cover a span of 109 yr. their experiments on the seven pls are not yetprehensive， but it seems that the terrestrial pls also remain stable during the integration period， maintaining almost regular oscillations.
on the other hand， in his accurate semi-analytical secular perturbation theory (laskar 1988)， laskar finds that large and irregular variations can appear in the eccentricities and inclinations of the terrestrial pls， especially of mercury and mars on a time-scale of several 109 yr (laskar 1996). the results of laskar's secular 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 plary orbits， covering a span of several 109 yr， and of two other integrations covering a span of ± 5 × 1010 yr. the total elapsed 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 solar system plary 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 plary orbital elements have been confined in a narrow region both in time and frequency domain， though plary 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 solar system plary motion. for readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (access )， where we show raw orbital elements， their low-pass filtered results， variation of delaunay elements and angular momentum deficit， and results of our simple time–frequency analysis on all of our integrations.
in section 2 we briefly explain 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 solar system plary motion is apparent both in plary 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 plary orbits using a low-pass filter and includes a discussion of angular momentum deficit. in section 5， we present a set of numerical integrations for the outer five pls that spans ± 5 × 1010 yr. in section 6 we also discuss the long-term stability of the plary 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 pls (n±1，2，3)， which is about 1/11 of the orbital period of the innermost pl (mercury). as for the determination of stepsize， we partly follow the previous numerical integration of all nine pls 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 accumulation of round-off error in theputation processes. in relation to this， wisdom & holman (1991) performed numerical integrations of the outer five plary 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 accurate enough， which partly justifies our method of determining the stepsize. however， since the eccentricity of jupiter (0.05) is much smaller than that of mercury (0.2)， we need some care when wepare these integrations simply in terms of stepsizes.
in the integration of the outer five pls (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 calculations of all nine pls (n±1，2，3)， and about 8000 000 d (21 903 yr) for the integration of the outer five pls (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 hadpleted all the calculations. see section 4.1 for more detail.
2.4 error estimation
2.4.1 relative errors in total energy and angular momentum
according to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angular momentum)， our long-term numerical integrations seem to have been performed with very small errors. the averaged relative errors of total energy (109) and of total angular momentum (1011) have remained nearly constant throughout the integration period (fig. 1). the special startup procedure， warm start， would have reduced the averaged relative error in total energy by about one order of magnitude or more.
relative numerical error of the total angular 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 angular 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 secular numerical error in the total angular momentum， which should be rigorously preserved up to machine-ε precision.
2.4.2 error in plary longitudes
since the symplectic maps preserve total energy and total angular momentum of n-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the accuracy of numerical integrations， especially as a measure of the positional error of pls， i.e. the error in plary longitudes. to estimate the numerical error in the plary longitudes， we performed the following procedures. wepared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher accuracy than the main integrations. for this purpose， we performed a much more accurate integration with a stepsize of 0.125 d (1/64 of the main integrations) spanning 3 × 105 yr， starting with the same initial conditions as in the n1 integration. we consider that this test integration provides us with a ‘pseudo-true’ solution of plary orbital evolution. next， wepare the test integration with the main integration， n1. for the period of 3 × 105 yr， we see a difference in mean anomalies of the earth between the two integrations of 0.52°(in the case of the n1 integration). this difference can be extrapolated 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. similarly， 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. glance at the raw data
in this section we briefly review the long-term stability of plary orbital motion through some snapshots of raw numerical data. the orbital motion of pls indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of pls took place.
3.1 general description of the stability of plary orbits
first， we briefly look at the general character of the long-term stability of plary orbits. our interest here focuses particularly on the inner four terrestrial pls for which the orbital time-scales are much shorter than those of the outer five pls. as we can see clearly from the planar orbital configurations shown in figs 2 and 3， orbital positions of the terrestrial pls 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 pls 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 regular variations of plary orbital motion remain nearly the same as they are at present.
vertical view of the four inner plary orbits (from the z -axis direction) at the initial and final parts of the integrationsn±1. the axes units are au. the xy -plane is set to the invariant plane of solar system total angular momentum.(a) the initial part ofn+1 ( t = 0 to 0.0547 × 10 9 yr).(b) the final part ofn+1 ( t = 4.9339 × 10 8 to 4.9886 × 10 9 yr).(c) the initial part of n1 (t= 0 to 0.0547 × 109 yr).(d) the final part ofn1 ( t =3.9180 × 10 9 to 3.9727 × 10 9 yr). in each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 × 107 yr . solid lines in each panel denote the present orbits of the four terrestrial pls (taken from de245).
the variation of eccentricities and orbital inclinations for the inner four pls in the initial and final part of the integration n+1 is shown in fig. 4. as expected， the character of the variation of plary 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 eccentricity， seem to change to a significant extent. this is partly because the orbital time-scale of the pl is the shortest of all the pls， which leads to a more rapid orbital evolution than other pls; the innermost pl may be nearest to instability. this result appears to be in some agreement with laskar's (1994， 1996) expectations that large and irregular variations appear in the eccentricities 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 plary system owing to the small mass of mercury. we will mention briefly the long-term orbital evolution of mercury later in section 4 using low-pass filtered orbital elements.
the orbital motion of the outer five pls seems rigorously stable and quite regular over this time-span (see also section 5).
3.2 time–frequency maps
although the plary motion exhibits very long-term stability defined as the non-existence of close encounter events， the chaotic nature of plary dynamics can change the oscillatory period and amplitude of plary orbital motion gradually over such long time-spans. even such slight fluctuations of orbital variation in the frequency domain， particularly in the case of earth， can potentially have a significant effect on its surface climate system through solar insolation variation (cf. berger 1988).
to give an overview of the long-term change in periodicity in plary 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 large overlapping 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 replaced by a grey-scale (or colour) chart.
we perform the replacement， 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 oscillation of orbital elements.
we have adopted an fft because of its overwhelming speed， since the amount of numerical data to be dposed into frequencyponents 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 eccentricity 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 eccentricity and inclination of earth only changes slightly over the entire period covered by the n+2 integration. this nearly regular trend is qualitatively the same in other integrations and for other pls， although typical frequencies differ pl by pl and element by element.
4.2 long-term exchange of orbital energy and angular momentum
we calculate very long-periodic variation and exchange of plary orbital energy and angular momentum using filtered delaunay elements l， g， h. g and h are equivalent to the plary orbital angular momentum and its verticalponent per unit mass. l is related to the plary orbital energy e per unit mass as e=μ2/2l2. if the system ispletely linear， the orbital energy and the angular momentum in each frequency bin must be constant. non-linearity in the plary system can cause an exchange of energy and angular momentum in the frequency domain. the amplitude of the lowest-frequency oscillation 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 angular momentum of the four inner pls and all nine pls are shown for integration n+2. the upper three panels show the long-periodic variation of total energy (denoted ase- e0)， total angular momentum ( g- g0)， and the verticalponent ( h- h0) of the inner four pls calculated from the low-pass filtered delaunay 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 pls. the fluctuation shown in the lower panels is virtually entirely a result of the massive jovian pls.
paring the variations of energy and angular momentum of the inner four pls and all nine pls， it is apparent that the amplitudes of those of the inner pls are much smaller than those of all nine pls: the amplitudes of the outer five pls are much larger than those of the inner pls. this does not mean that the inner terrestrial plary subsystem is more stable than the outer one: this is simply a result of the relative smallness of the masses of the four terrestrial plspared with those of the outer jovian pls. another thing we notice is that the inner plary subsystem may be 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 irregular oscillations are more apparent than in the panels denoted astotal 9. actually， the fluctuations in theinner 4 panels are to a large extent as a result of the orbital variation of the mercury. however， we cannot neglect the contribution from other terrestrial pls， as we will see in subsequent sections.
4.4 long-term coupling of several neighbouring pl pairs
let us see some individual variations of plary orbital energy and angular momentum expressed by the low-pass filtered delaunay elements. figs 10 and 11 show long-term evolution of the orbital energy of each pl and the angular momentum in n+1 and n2 integrations. we notice that some pls form apparent pairs in terms of orbital energy and angular momentum exchange. in particular， venus and earth make a typical pair. in the figures， they show negative correlations in exchange of energy and positive correlations in exchange of angular momentum. the negative correlation in exchange of orbital energy means that the two pls form a closed dynamical system in terms of the orbital energy. the positive correlation in exchange of angular momentum means that the two pls 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 correlation in the angular momentum variation to the venus–earth system. mercury exhibits certain negative correlations in the angular momentum versus the venus–earth system， which seems to be a reaction caused by the conservation of angular momentum in the terrestrial plary subsystem.
it is not clear at the moment why the venus–earth pair exhibits a negative correlation in energy exchange and a positive correlation in angular momentum exchange. we may possibly explain this through observing the general fact that there are no secular terms in plary semimajor axes up to second-order perturbation theories (cf. brouwer & clemence 1961; boccaletti & pucacco 1998). this means that the plary orbital energy (which is directly related to the semimajor axis a) might be much less affected by perturbing pls than is the angular momentum exchange (which relates to e). hence， the eccentricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive correlation in the angular momentum exchange. on the other hand， the semimajor axes of venus and earth are less likely to be disturbed by the jovian pls. thus the energy exchange may be limited only within the venus–earth pair， which results in a negative correlation in the exchange of orbital energy in the pair.
as for the outer jovian plary subsystem， jupiter–saturn and uranus–neptune seem to make dynamical pairs. however， the strength of their coupling is not as strongpared with that of the venus–earth pair.
5 ± 5 × 1010-yr integrations of outer plary orbits
since the jovian plary masses are much larger than the terrestrial plary masses， we treat the jovian plary system as an independent plary system in terms of the study of its dynamical stability. hence， we added a couple of trial integrations that span ± 5 × 1010 yr， including only the outer five pls (the four jovian pls plus pluto). the results exhibit the rigorous stability of the outer plary system over this long time-span. orbital configurations (fig. 12)， and variation of eccentricities and inclinations (fig. 13) show this very long-term stability of the outer five pls in both the time and the frequency domains. although we do not show maps here， the typical frequency of the orbital oscillation of pluto and the other outer pls 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 relative numerical error in the total energy was 106 and that of the total angular momentum was 1010.
5.1 resonances in the neptune–pluto system
kinoshita & nakai (1996) integrated the outer five plary orbits over ± 5.5 × 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 λnp librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.
the argument of perihelion of pluto wp=θ2=pΩp librates around 90° with a period of about 3.8 × 106 yr. the dominant periodic variations of the eccentricity and inclination of pluto are synchronized with the libration of its argument of perihelion. this is anticipated in the secular perturbation theory constructed by kozai (1962).
the longitude of the node of pluto referred to the longitude of the node of neptune，θ3=ΩpΩn， circulates and the period of this circulation is equal to the period of θ2 libration. when θ3 bes zero， i.e. the longitudes of ascending nodes of neptune and pluto overlap， the inclination of pluto bes maximum， the eccentricity bes minimum and the argument of perihelion bes 90°. when θ3 bes 180°， the inclination of pluto bes minimum， the eccentricity bes maximum and the argument of perihelion bes 90° again. williams & benson (1971) anticipated this type of resonance， later confirmed by milani， nobili & carpino (1989).
an argument θ4=pn+ 3 (ΩpΩn) librates around 180° with a long period， 5.7 × 108 yr.
in our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain similar during the whole integration period (figs 14–16 ). however， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circulation 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 plary 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 secular) between any pair among the nine pls. 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 plary dynamical motion， but they are not so strong as to destroy the stable plary motion within the lifetime of the real solar system. the second feature， which we think is more important for the long-term stability of our plary system， is the difference in dynamical distance between terrestrial and jovian plary subsystems (ito & tanikawa 1999， 2001). when we measure plary separations by the mutual hill radii (r_)， separations among terrestrial pls are greater than 26rh， whereas those among jovian pls are less than 14rh. this difference is directly related to the difference between dynamical features of terrestrial and jovian pls. terrestrial pls have smaller masses， shorter orbital periods and wider dynamical separation. they are strongly perturbed by jovian pls that have larger masses， longer orbital periods and narrower dynamical separation. jovian pls are not perturbed by any other massive bodies.
the present terrestrial plary system is still being disturbed by the massive jovian pls. however， the wide separation and mutual interaction among the terrestrial pls renders the disturbance ineffective; the degree of disturbance by jovian pls is o(ej)(order of magnitude of the eccentricity of jupiter)， since the disturbance caused by jovian pls is a forced oscillation having an amplitude of o(ej). heightening of eccentricity， for example o(ej)0.05， is far from sufficient to provoke instability in the terrestrial pls having such a wide separation as 26rh. thus we assume that the present wide dynamical separation among terrestrial pls (> 26rh) is probably one of the most significant conditions for maintaining the stability of the plary system over a 109-yr time-span. our detailed analysis of the relationship between dynamical distance between pls and the instability time-scale of solar system plary motion is now on-going.
although our numerical integrations span the lifetime of the solar 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 plary dynamics.
——以上文段引自 ito， t.& tanikawa， k. long-term integrations and stability of plary orbits in our solar 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。