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。

那么作者君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 ary orbits in our sr system

abstract

we present the results of very long-term numerical integrations of ary orbital motions over 109 -yr time-spans including all nine s. a quick inspection of our numerical data shows that the ary 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 oscitions using a low-pass filter shows us the potentially diffusive character of terrestrial ary motion， especially that of mercury. the behaviour of the entricity of mercury in our integrations is qualitatively simr to the results from jacquesskar's secr perturbation theory (e.g. emax～ 0.35 over ～± 4 gyr). however， there are no apparent secr increases of entricity or inclination in any orbital elements of the s， 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 s 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 sr 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 yed 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 sr 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 rtion to the problem of ary motion in the sr system. actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our sr system.

among many definitions of stability， here we adopt the hill definition (dman 1993): actually this is not a definition of stability， but of instability. we define a system as being unstable when a close encounter urs 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 therger hill radius. otherwise the system is defined as being stable. henceforward we state that our ary system is dynamically stable if no close encounter happens during the age of our sr system， about ±5 gyr. incidentally， this definition may be reced by one in which an urrence of any orbital crossing between either of a pair of s takes ce. this is because we know from experience that an orbital crossing is very likely to lead to a close encounter in ary and protoary 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 s in our sr 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 ovepping (murray & holman 1999; lecar， franklin & holman 2001). however， it would require integrating over an ensemble of ary systems including all nine s for a period covering several 10 gyr to thoroughly understand the long-term evolution of ary 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 s (sussman & wisdom 1988; kinoshita & nakai 1996). this is because the orbital periods of the outer s are so much longer than those of the inner four s 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 sr mass loss on the stability of ary orbits， they performed many integrations covering up to ～1011 yr of the orbital motions of the four jovian s. the initial orbital elements and masses of s are the same as those of our sr 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 sr mass loss in the paper. consequently， they found that the crossing time-scale of ary 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 s remain stable over 1010 yr， or perhaps longer. duncan & lissauer also performed four simr experiments on the orbital motion of seven s (venus to neptune)， which cover a span of ～109 yr. their experiments on the seven s are not yet prehensive， but it seems that the terrestrial s also remain stable during the integration period， maintaining almost regr oscitions.

on the other hand， in his urate semi-analytical secr perturbation theory skar 1988)，skar finds thatrge and irregr variations can appear in the entricities and inclinations of the terrestrial s， especially of mercury and mars on a time-scale of several 109 yr skar 1996). the results ofskar's secr 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 ary orbits， covering a span of several 109 yr， and of two other integrations covering a span of ± 5 x 1010 yr. the total psed 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 sr system ary 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 ary orbital elements have been confined in a narrow region both in time and frequency domain， though ary 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 sr system ary motion. for readers who have more specific and deeper interests in our numerical results， we have prepared a webpage (ess )， where we show raw orbital elements， their low-pass filtered results， variation of dunay elements and angr momentum deficit， and results of our simple time–frequency analysis on all of our integrations.

in section 2 we briefly exin 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 sr system ary motion is apparent both in ary 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 ary orbits using a low-pass filter and includes a discussion of angr momentum deficit. in section 5， we present a set of numerical integrations for the outer five s that spans ± 5 x 1010 yr. in section 6 we also discuss the long-term stability of the ary 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 s (n±1，2，3)， which is about 1/11 of the orbital period of the innermost (mercury). as for the determination of stepsize， we partly follow the previous numerical integration of all nine s 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 umtion of round-off error in the putation processes. in rtion to this， wisdom & holman (1991) performed numerical integrations of the outer five ary 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 urate enough， which partly justifies our method of determining the stepsize. however， since the entricity of jupiter (～0.05) is much smaller than that of mercury (～0.2)， we need some care when we pare these integrations simply in terms of stepsizes.

in the integration of the outer five s (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 calctions of all nine s (n±1，2，3)， and about 8000 000 d (～21 903 yr) for the integration of the outer five s (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 pleted all the calctions. see section 4.1 for more detail.

2.4 error estimation

2.4.1 rtive errors in total energy and angr momentum

ording to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angr momentum)， our long-term numerical integrations seem to have been performed with very small errors. the averaged rtive errors of total energy (～10?9) and of total angr momentum (～10?11) have remained nearly constant throughout the integration period (fig. 1). the special startup procedure， warm start， would have reduced the averaged rtive error in total energy by about one order of magnitude or more.

rtive numerical error of the total angr 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 angr 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 secr numerical error in the total angr momentum， which should be rigorously preserved up to machine-e precision.

2.4.2 error in ary longitudes

since the symplectic maps preserve total energy and total angr momentum of n-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the uracy of numerical integrations， especially as a measure of the positional error of s， i.e. the error in ary longitudes. to estimate the numerical error in the ary longitudes， we performed the following procedures. we pared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher uracy than the main integrations. for this purpose， we performed a much more urate 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 ary orbital evolution. next， we pare 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 extrapted 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. simrly， 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. nce at the raw data

in this section we briefly review the long-term stability of ary orbital motion through some snapshots of raw numerical data. the orbital motion of s indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of s took ce.

3.1 general description of the stability of ary orbits

first， we briefly look at the general character of the long-term stability of ary orbits. our interest here focuses particrly on the inner four terrestrial s for which the orbital time-scales are much shorter than those of the outer five s. as we can see clearly from the nar orbital configurations shown in figs 2 and 3， orbital positions of the terrestrial s 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 s 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 regr variations of ary orbital motion remain nearly the same as they are at present.

vertical view of the four inner ary orbits (from the z -axis direction) at the initial and final parts of the integrationsn±1. the axes units are au. the xy -ne is set to the invariant ne of sr system total angr 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 s (taken from de245).