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 stabilityplaary orbitsour solar system

abstract

we present the resultsvery long-term numerical integrationsplaary orbital motions over 109 -yr time-spans including all nin inspectionour numerical data shows that the plaary motion，leastour simple dynamical model， seemsbe quite stable even over this very lon lookthe lowest-frequency oscillations using a low-pass filter showsthe potentially diffusive characterterrestrial plaary motion， especially that o behaviourthe eccentricitymercuryour integrationsqualitatively similarthe results from jacques laskar''s secular perturbation theory (e.g. emax 0.35 over ± 4 gyr). however， there areapparent secular increaseseccentricityinclinationany orbital elementsthe plas， which mayrevealedstill longer-term numerical i have also performed a coupletrial integrations including motionsthe outer five plas over the duration± 5 x 1010 yr. the result indicates that the three major resonancesthe neptunepluto system have been maintained over the 1011-yr time-span.

1 introduction

1.1definitionthe problem

the questionthe stabilityour solar system has been debated over several hundred years， since the era o problem has attracted many famous mathematicians over the years and has played a central rolethe developmentnon-linear dynamics and chao，do not yet have a definite answerthe questionwhether our solar systemstable opartly a resultthe fact that the definitionthe term ‘stability’vague whenis usedrelationthe problemplaary motionthe solais not easygive a clear， rigorous and physically meaningful definitionthe stabilityour solar system.

among many definitionsstability， hereadopt the hill definition (gladman 1993): actually thisnot a definitionstability， but define a systembeing unstable when a close encounter occurs somewherethe system， starting from a certain initial configuration (chambers， wetherill & boss 1996; ito & tanikawa 1999). a systemdefinedexperiencing a close encounter when two bodies approach one another withinareathe larger hil the systemdefinedbeinstate that our plaary systemdynamically stableno close encounter happens during the ageour solar system， about ±， this definition mayreplacedonewhichoccurrenceany orbital crossing between eithera pairplas takebecauseknow from experience thatorbital crossingvery likelyleada close encounterplaary and protoplaary systems (yoshinaga， kokubo & makino 1999).course this statement cannotsimply appliedsystems with stable orbital resonances suchthe neptunepluto system.

1.2previous studies and aimsthis research

in additionthe vaguenessthe conceptstability， the plasour solar system show a character typicaldynamical chaos (sussman & wisdom 1988， 1992). the causethis chaotic behaviournow partly understoodbeing a resultresonance overlapping (murray & holman 1999; lecar， franklin & holman 2001). however，would require integrating overensembleplaary systems including all nine plas for a period covering severalgyrthoroughly understand the long-term evolutionplaary orbits， since chaotic dynamical systems are characterizedtheir strong dependenceinitial conditions.

from that pointview， manythe previous long-term numerical integrations included only the outer five plas (sussman & wisdom 1988; kinoshita & nakai 1996). thisbecause the orbital periodsthe outer plas aremuch longer than thosethe inner four plas thatis much easierfollow the system for a given integratio present， the longest numerical integrations publishedjournals are thoseduncan & lissauer (1998). although their main target was the effectpost-main-sequence solar mass lossthe stabilityplaary orbits， they performed many integrations coveringto 1011of the orbital motionsthe four jovia initial orbital elements and massesplas are the samethoseour solar systemduncan & lissauer''s paper， but they decrease the massthe sun graduallytheir numerical because they consider the effectpost-main-sequence solar mass lossth， they found that the crossing time-scaleplaary orbits， which cana typical indicatorthe instability time-scale，quite sensitivethe ratemass decreaseth the massthe suncloseits present value， the jovian plas remain stable over 1010 yr，perhap & lissauer also performed four similar experimentsthe orbital motionseven plas (venusneptune)， which cover a span109 yr. their experimentsthe seven plas are not yet prehensive， butseems that the terrestrial plas also remain stable during the integration period， maintaining almost regular oscillations.

on the other hand，his accurate semi-analytical secular perturbation theory (laskar 1988)， laskar finds that large and irregular variations can appearthe eccentricities and inclinationsthe terrestrial plas， especiallymercury and marsa time-scaleseveral 109(laskar 1996). the resultslaskar''s secular perturbation theory shouldconfirmed and investigatedfully numerical integrations.

in this paperpresent preliminary resultssix long-term numerical integrationsall nine plaary orbits， covering a spanseveral 109 yr， andtwo other integrations covering a span± 5 x 1010 yr. the total elapsed time for all integrationsmore than 5 yr， using several dedicated pcs and wthe fundamental conclusionsour long-term integrationsthat solar system plaary motion seemsbe stabletermsthe hill stability mentioned above，least over a time-span± ，our numerical integrations the system was far more stable than whatdefinedthe hill stability criterion: not only didclose encounter happen during the integration period， but also all the plaary orbital elements have been confineda narrow region bothtime and frequency domain， though plaary motionsthe purposethis paperto exhibit and overview the resultsour long-term numerical integrations，show typical example figuresevidencethe very long-term stabilitysolar system plaar readers who have more specific and deeper interestsour numerical results，have prepared a webpage (access )， whereshow raw orbital elements， their low-pass filtered results， variationdelaunay elements and angular momentum deficit， and resultsour simple timefrequency analysisallour integrations.

in section 2briefly explain our dynamical model， numerical method and initial conditions usedour i 3devoteda descriptionthe quick resultsthe numerical i long-term stabilitysolar system plaary motionapparent bothplaary positions and orbita estimationnumerical errorsals 4 goesto a discussionthe longest-term variationplaary orbits using a low-pass filter and includes a discussionangular mometu section 5，present a setnumerical integrations for the outer five plas that spans ± 5 x 1010 yr.section 6also discuss the long-term stabilitythe plaary motion and its possible cause.

2 descriptionthe 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。）

2.3 numerical method

we utilize a second-order wisdomholman symplectic mapour main integration method (wisdom & holman 1991; kinoshita， yoshida & nakai 1991) with a special start-up procedurereduce the truncation errorangle variables，‘warm start’(saha & tremaine 1992， 1994).

the stepsize for the numerical integrations8 d throughout all integrationsthe nine plas (n±1，2，3)， whichabout 1/11the orbital periodthe innermost pla (mercury).for the determinationstepsize，partly follow the previous numerical integrationall nine plassussman & wisdom (1988， 7.2 d) and saha & tremaine (1994， 225/32 d).rounded the decimal partthe their stepsizes8make the stepsize a multiple2orderreduce the accumulationround-off errorthe putatio relationthis， wisdom & holman (1991) performed numerical integrationsthe outer five plaary orbits using the symplectic map with a stepsize400 d， 1/10.83the orbital period o result seemsbe accurate enough， which partly justifies our methoddetermining th， since the eccentricityjupiter (0.05)much smaller than thatmercury (0.2)，need some care whenpare these integrations simplytermsstepsizes.

in the integrationthe outer five plas (f±)，fixed the stepsize400 d.

we adopt gauss'' f and g functionsthe symplectic map together with the third-order halley method (danby 1992)a solver for keple numbermaximum iterationssethalley''s method15， but they never reached the maximumanyour integrations.

the intervalthe data output200 000 d (547 yr) for the calculationsall nine plas (n±1，2，3)， and about 8000 000 d (21 903 yr) for the integrationthe outer five plas (f±).

althoughoutput filtering was done when the numerical integrations wereprocess，applied a low-pass filterthe raw orbital data afterhad pleted all the c section 4.1 for more detail.

2.4 error estimation

2.4.1 relative errorstotal energy and angular momentum

accordingohe basic propertiessymplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angular momentum)， our long-term numerical integrations seemhave been performed with very smal averaged relative errorstotal energy (109) andtotal 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 errortotal energyabout one ordermagnitudemore.

relative numerical errorthe total angular momentum δa/a0 and the total energy δe/e0our numerical integrationsn± 1，2，3， whereandare the absolute changethe total energy and total angular momentum， respectively， ande0anda0are their initia horizontal unitgyr.

note that different operating systems， different mathematical libraries， and different hardware architectures resultdifferent numerical errors， through the variationsround-off error handling and numerica the upper panel o，can recognize this situationthe secular numerical errorthe total angular momentum， which shouldrigorously preservedto machine-e precision.

2.4.2 errorplaary longitudes

since the symplectic maps preserve total energy and total angular momentumn-body dynamical systems inherently well， the degreetheir preservation may nota good measurethe accuracynumerical integrations， especiallya measurethe positional errorplas， i.e. the errorplaar estimate the numerical errorthe plaary longitudes，performed the followin pared the resultour main long-term integrations with some test integrations， which span much shorter periods but with much higher accuracy than the main i this purpose，performed a much more accurate integration with a stepsize0.125 d (1/64the main integrations) spanning 3 x 105 yr， starting with the same initial conditionsin the consider that this test integration provideswith a ‘pseudo-true’ solutionplaary orbita，pare the test integration with the main integration， n1. for the period3 x 105 yr，see a differencemean anomaliesthe earth between the two integrations0.52°(in the casetheintegration). this difference canextrapolatedthe value 8700°， aboutrotationsearth after 5 gyr， since the errorlongitudes increases linearly with timethe symplecti， the longitude errorpluto canestimated12°. this value for plutomuch better than the resultkinoshita & nakai (1996) where the differenceestimated60°.

3 numerical results i. glancethe raw data

in this sectionbriefly review the long-term stabilityplaary orbital motion through some snapshotsraw numerica orbital motionplas indicates long-term stabilityallour numerical integrations:orbital crossings nor close encounters between any pairplas took place.

3.1 general descriptionthe stabilityplaary orbits

first，briefly lookthe general characterthe long-term stabilityplaar interest here focuses particularlythe inner four terrestrial plas for which the orbital time-scales are much shorter than thosethe outer fivcan see clearly from the planar orbital configurations shownfigs 2 and 3， orbital positionsthe terrestrial plas differ little between the initial and final parteach numerical integration， which spans severa solid lines denoting the present orbitsthe plas lie almost within the swarmdots eventhe final partintegrations (b) and (d). this indicates that throughout the entire integration period the almost regular variationsplaary orbital motion remain nearly the samethey arepresent.

vertical viewthe four inner plaary orbits (from the z -axis direction)the initial and final partsthe integrationsn±1. the axes units are au. the-planesetthe invariant planesolar system total angular momentum.(a) the initial part ofn+1 ( t = 00.0547 x9 yr).(b) the final part ofn+1 ( t = 4.9339 x84.9886 x9 yr).(c) the initial partn1 (t= 00.0547 x 109 yr).(d) the final part ofn1 ( t =3.9180 x93.9727 x9 yr).each panel， a total23 684 points are plotted withintervalabout 2190over 5.47 x 107. solid lineseach panel denote the present orbitsthe four terrestrial plas (taken from de245).