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：

longternbsp;integrations and stability of plary orbits in our solar systebr>

abstract

we present the results of very longternbsp;nurical integrations of plary orbital tions over 109 yr tispans including all nine pls. a quick inspection of our nurical data shows that the plary tion， at least in our sile dynacal del， see to be quite stable even over this very long tispan. a closer look at the lowestfrequency oscillations using a lowpass filter shows us the potentially diffusive character of terrestrial plary tion， especially that of mercury. the behaviour of the entricity of mercury in our integrations is qualitatively silar to the results fronbsp;jacques laskar&039;s secular perturbation theory e.g. ex 0.35 over4 gyr. however， there are no apparent secular increases of entricity or inclination in any orbital elents of the pls， which y be revealed by still longerternbsp;nurical integrations. we have also perford a couple of trial integrations including tions of the outer five pls over the duration of51010 yr. the result indicates that the three jor resonances in the neptune–pluto systenbsp;have been intained over the 1011yr tispan.

1 introduction

1.1definition of the problebr>

the question of the stability of our solar systenbsp;has been debated over several hundred years， since the era of newton. the problenbsp;has attracted ny faus theticians over the years and has played a central role in the developnt of nonlinear dynacs and chaos theory. however， we do not yet have a definite answer to the question of whether our solar systenbsp;is stable or not. this is partly a result of the fact that the definition of the ternbsp;stability is vague when it is used in relation to the problenbsp;of plary tion in the solar syste actually it is not easy to give a clear， rigorous and physically aningful definition of the stability of our solar syste

ang ny definitions of stability， here we adopt the hill definition gladn 1993: actually this is not a definition of stability， but of instability. we define a systenbsp;as bing unstable when a close encounter urs sowhere in the syste starting fronbsp;a certain initial configuration chaers， wetherill & boss 1996; ito & tanikawa 1999. a systenbsp;is defined as experiencing a close encounter when two bodies approach one another within an area of the larger hill radius. otherwise the systenbsp;is defined as being stable. henceforward we state that our plary systenbsp;is dynacally stable if no close encounter happens during the age of our solar syste about 5 gyr. incidentally， this definition y be replaced by one in which an urrence of any orbital crossing between either of a pair of pls takes place. this is because we know fronbsp;experience that an orbital crossing is very likely to lead to a close encounter in plary and prlary syste yoshinaga， kokubo & makino 1999. of course this statent cannot be sily applied to syste with stable orbital resonances such as the neptune–pluto syste

1.2previous studies and ai of this research

in addition to the vagueness of the concept of stability， the pls in our solar systenbsp;show a character typical of dynacal chaos sussn & wisdonbsp;1988， 1992. the cause of this chaotic behaviour is now partly understood as being a result of resonance overlapping murray & holn 1999; lecar， franklin & holn 2001. however， it would require integrating over an ensele of plary syste including all nine pls for a period covering several 10 gyr to thoroughly understand the longternbsp;evolution of plary orbits， since chaotic dynacal syste are characterized by their strong dependence on initial conditions.

fronbsp;that point of view， ny of the previous longternbsp;nurical integrations included only the outer five pls sussn & wisdonbsp;1988; kinoshita & nakai 1996. this is because the orbital periods of the outer pls are so ch longer than those of the inner four pls that it is ch easier to follow the systenbsp;for a given integration period. at present， the longest nurical integrations published in journals are those of duncan & lissauer 1998. although their in target was the effect of postinsequence solar ss loss on the stability of plary orbits， they perford ny integrations covering up to 1011 yr of the orbital tions of the four jovian pls. the initial orbital elents and sses of pls are the sa as those of our solar systenbsp;in duncan & lissauer&039;s paper， but they decrease the ss of the sun gradually in their nurical experints. this is because they consider the effect of postinsequence solar ss loss in the paper. consequently， they found that the crossing tiscale of plary orbits， which can be a typical indicator of the instability tiscale， is quite sensitive to the rate of ss decrease of the sun. when the ss of the sun is close to its present value， the jovian pls rein stable over 1010 yr， or perhaps longer. duncan & lissauer also perford four silar experints on the orbital tion of seven pls venus to neptune， which cover a span of 109 yr. their experints on the seven pls are not yet&nbspprehensive， but it see that the terrestrial pls also rein stable during the integration period， intaining alst regular oscillations.

on the other hand， in his urate seanalytical secular perturbation theory laskar 1988， laskar finds that large and irregular variations can appear in the entricities and inclinations of the terrestrial pls， especially of mercury and mars on a tiscale of several 109 yr laskar 1996. the results of laskar&039;s secular perturbation theory should be confird and investigated by fully nurical integrations.

in this paper we present prelinary results of six longternbsp;nurical integrations on all nine plary orbits， covering a span of several 109 yr， and of two other integrations covering a span of51010 yr. the total elapsed ti for all integrations is re than 5 yr， using several dedicated pcs and workstations. one of the fundantal conclusions of our longternbsp;integrations is that solar systenbsp;plary tion see to be stable in ter of the hill stability ntioned above， at least over a tispan of4 gyr. actually， in our nurical integrations the systenbsp;was far re 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 elents have been confined in a narrow region both in ti and frequency doin， though plary tions are stochastic. since the purpose of this paper is to exhibit and overview the results of our longternbsp;nurical integrations， we show typical exale figures as evidence of the very longternbsp;stability of solar systenbsp;plary tion. for readers who have re specific and deeper interests in our nurical results， we have prepared a webpage ess ， where we show raw orbital elents， their lowpass filtered results， variation of delaunay elents and angular ntunbsp;deficit， and results of our sile ti–frequency analysis on all of our integrations.

in section 2 we briefly explain our dynacal del， nurical thod and initial conditions used in our integrations. section 3 is devoted to a description of the quick results of the nurical integrations. very longternbsp;stability of solar systenbsp;plary tion is apparent both in plary positions and orbital elents. a rough estition of nurical errors is also given. section 4 goes on to a discussion of the longestternbsp;variation of plary orbits using a lowpass filter and includes a discussion of angular ntunbsp;deficit. in section 5， we present a set of nurical integrations for the outer five pls that spans51010 yr. in section 6 we also discuss the longternbsp;stability of the plary tion and its possible cause.

2 description of the nurical 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 nurical thod

we utilize a secondorder wisdoholn sylectic p as our in integration thod wisdonbsp;& holn 1991; kinoshita， yoshida & nakai 1991 with a special startup procedure to reduce the truncation error of angle variables，warnbsp;startsaha & treine 1992， 1994.

the stepsize for the nurical integrations is 8 d throughout all integrations of the nine pls n1，2，3， which is about 111 of the orbital period of the innerst pl mercury. as for the deternation of stepsize， we partly follow the previous nurical integration of all nine pls in sussn & wisdonbsp;1988， 7.2 d and saha & treine 1994， 22532 d. we rounded the decil part of the their stepsizes to 8 to ke the stepsize a ltiple of 2 in order to reduce the ulation of roundoff error in the&nbspputation processes. in relation to this， wisdonbsp;& holn 1991 perford nurical integrations of the outer five plary orbits using the sylectic p with a stepsize of 400 d， 110.83 of the orbital period of jupiter. their result see to be urate enough， which partly justifies our thod of deterning the stepsize. however， since the entricity of jupiter 0.05 is ch sller than that of mercury 0.2， we need so care when we&nbsppare these integrations sily in ter of stepsizes.

in the integration of the outer five pls f， we fixed the stepsize at 400 d.

we adopt gauss&039; f and g functions in the sylectic p together with the thirdorder halley thod danby 1992 as a solver for kepler equations. the nuer of xinbsp;iterations we set in halley&039;s thod is 15， but they never reached the xinbsp;in any of our integrations.

the interval of the data output is 200 000 d 547 yr for the calculations of all nine pls n1，2，3， 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 nurical integrations were in process， we applied a lowpass filter to the raw orbital data after we had&nbsppleted all the calculations. see section 4.1 for re detail.

2.4 error estition

2.4.1 relative errors in total energy and angular ntubr>

ording to one of the basic properties of sylectic integrators， which conserve the physically conservative quantities well total orbital energy and angular ntu our longternbsp;nurical integrations seenbsp;to have been perford with very sll errors. the averaged relative errors of total energy 109 and of total angular ntunbsp;1011 have reined nearly constant throughout the integration period fig. 1. the special startup procedure， warnbsp;start， would have reduced the averaged relative error in total energy by about one order of gnitude or re.

relative nurical error of the total angular ntunbsp;δaa0 and the total energy δee0 in our nurical integrationsn 1，2，3， where δe and δa are the absolute change of the total energy and total angular ntu respectively， ande0anda0are their initial values. the horizontal unit is gyr.

note that different operating syste， different thetical libraries， and different hardware architectures result in different nurical errors， through the variations in roundoff error handling and nurical algorith. in the upper panel of fig. 1， we can recognize this situation in the secular nurical error in the total angular ntu which should be rigorously preserved up to chinee precision.

2.4.2 error in plary longitudes

since the sylectic ps preserve total energy and total angular ntunbsp;of nbody dynacal syste inherently well， the degree of their preservation y not be a good asure of the uracy of nurical integrations， especially as a asure of the positional error of pls， i.e. the error in plary longitudes. to estite the nurical error in the plary longitudes， we perford the following procedures. we&nbsppared the result of our in longternbsp;integrations with so test integrations， which span ch shorter periods but with ch higher uracy than the in integrations. for this purpose， we perford a ch re urate integration with a stepsize of 0.125 d 164 of the in integrations spanning 3105 yr， starting with the sa initial conditions as in the n1 integration. we consider that this test integration provides us with a pseudotrue solution of plary orbital evolution. next， we&nbsppare the test integration with the in integration， n1. for the period of 3105 yr， we see a difference in an anolies of the earth between the two integrations of 0.52in 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 ti in the sylectic p. silarly， the longitude error of pluto can be estited as 12. this value for pluto is ch better than the result in kinoshita & nakai 1996 where the difference is estited as 60.

3 nurical results – i. glance at the raw data

in this section we briefly review the longternbsp;stability of plary orbital tion through so snapshots of raw nurical data. the orbital tion of pls indicates longternbsp;stability in all of our nurical 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 longternbsp;stability of plary orbits. our interest here focuses particularly on the inner four terrestrial pls for which the orbital tiscales are ch shorter than those of the outer five pls. as we can see clearly fronbsp;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 nurical integration， which spans several gyr. the solid lines denoting the present orbits of the pls lie alst within the swarnbsp;of dots even in the final part of integrations b and d. this indicates that throughout the entire integration period the alst regular variations of plary orbital tion rein nearly the sa as they are at present.

vertical view of the four inner plary orbits fronbsp;the z axis direction at the initial and final parts of the integrationsn1. the axes units are au. the xy plane is set to the invariant plane of solar systenbsp;total angular ntua the initial part ofn1t0 to 0.054710 9 yr.b the final part ofn1t4.933910 8 to 4.988610 9 yr.c the initial part of n1 t 0 to 0.0547109 yr.d the final part ofn1t 3.918010 9 to 3.972710 9 yr. in each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47107 yr . solid lines in each panel denote the present orbits of the four terrestrial pls taken fronbsp;de245.