死在火星上 对火星轨道变化问题的最后解释

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。

以下是文章禸 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 over 4 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 of 5 1010 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 of 5 1010 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 of 4 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 spans 5 1010 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 3 105 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 3 105 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 ofn1 t 0 to 0.0547 10 9 yr.b the final part ofn1 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 fronbsp;de245.

the variation of entricities and orbital inclinations for the inner four pls in the initial and final part of the integration n1 is shown in fig. 4. as expected， the character of the variation of plary orbital elents does not differ significantly between the initial and final part of each integration， at least for venus， earth and mars. the elents of mercury， especially its entricity， seenbsp;to change to a significant extent. this is partly because the orbital tiscale of the pl is the shortest of all the pls， which leads to a re rapid orbital evolution than other pls; the innerst pl y be nearest to instability. this result appears to be in so agreent with laskar&039;s 1994， 1996 expectations that large and irregular variations appear in the entricities and inclinations of mercury on a tiscale of several 109 yr. however， the effect of the possible instability of the orbit of mercury y not fatally affect the global stability of the whole plary systenbsp;owing to the sll ss of mercury. we will ntion briefly the longternbsp;orbital evolution of mercury later in section 4 using lowpass filtered orbital elents.

the orbital tion of the outer five pls see rigorously stable and quite regular over this tispan see also section 5.

3.2 ti–frequency ps

although the plary tion exhibits very longternbsp;stability defined as the nonexistence of close encounter events， the chaotic nature of plary dynacs can change the oscillatory period and alitude of plary orbital tion gradually over such long tispans. even such slight fluctuations of orbital variation in the frequency doin， particularly in the case of earth， can potentially have a significant effect on its surface clite systenbsp;through solar insolation variation cf. berger 1988.

to give an overview of the longternbsp;change in periodicity in plary orbital tion， we perford ny fast fourier transfortions ffts along the ti axis， and superposed the resulting periodgra to draw twodinsional ti–frequency ps. the specific approach to drawing these ti–frequency ps in this paper is very sile – ch siler than the wavelet analysis or laskar&039;s 1990， 1993 frequency analysis.

divide the lowpass filtered orbital data into ny fragnts of the sa length. the length of each data segnt should be a ltiple of 2 in order to apply the fft.

each fragnt of the data has a large overlapping part: for exale， when the ith data begins fronbsp;tti and ends at ttit， the next data segnt ranges fronbsp;tiδttiδtt， where δtt. we continue this division until we reach a certain nuer n by which tnt reaches the total integration length.

we apply an fft to each of the data fragnts， and obtain n frequency diagra.

in each frequency diagranbsp;obtained above， the strength of periodicity can be replaced by a greyscale or colour chart.

we perfornbsp;the replacent， and connect all the greyscale or colour charts into one graph for each integration. the horizontal axis of these new graphs should be the ti， i.e. the starting tis of each fragnt of data ti， where i 1，…， n. the vertical axis represents the period or frequency of the oscillation of orbital elents.

we have adopted an fft because of its overwhelng speed， since the aunt of nurical data to be dposed into frequency&nbspponents is terribly huge several tens of gbytes.

a typical exale of the ti–frequency p created by the above procedures is shown in a greyscale diagranbsp;as fig. 5， which shows the variation of periodicity in the entricity and inclination of earth in n2 integration. in fig. 5， the dark area shows that at the ti 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 fronbsp;this p that the periodicity of the entricity and inclination of earth only changes slightly over the entire period covered by the n2 integration. this nearly regular trend is qualitatively the sa in other integrations and for other pls， although typical frequencies differ pl by pl and elent by elent.

4.2 longternbsp;exchange of orbital energy and angular ntubr>

we calculate very longperiodic variation and exchange of plary orbital energy and angular ntunbsp;using filtered delaunay elents l， g， h. g and h are equivalent to the plary orbital angular ntunbsp;and its vertical&nbspponent per unit ss. l is related to the plary orbital energy e per unit ss as e22l2. if the systenbsp;is&nbsppletely linear， the orbital energy and the angular ntunbsp;in each frequency bin st be constant. nonlinearity in the plary systenbsp;can cause an exchange of energy and angular ntunbsp;in the frequency doin. the alitude of the lowestfrequency oscillation should increase if the systenbsp;is unstable and breaks down gradually. however， such a sytonbsp;of instability is not pronent in our longternbsp;integrations.

in fig. 7， the total orbital energy and angular ntunbsp;of the four inner pls and all nine pls are shown for integration n2. the upper three panels show the longperiodic variation of total energy denoted ase e0， total angular ntunbsp; g g0， and the vertical&nbspponent h h0 of the inner four pls calculated fronbsp;the lowpass filtered delaunay elents.e0， g0， h0 denote the initial values of each quantity. the absolute difference fronbsp;the initial values is plotted in the panels. the lower three panels in each figure showee0，gg0 andhh0 of the total of nine pls. the fluctuation shown in the lower panels is virtually entirely a result of the ssive jovian pls.

coaring the variations of energy and angular ntunbsp;of the inner four pls and all nine pls， it is apparent that the alitudes of those of the inner pls are ch sller than those of all nine pls: the alitudes of the outer five pls are ch larger than those of the inner pls. this does not an that the inner terrestrial plary subsystenbsp;is re stable than the outer one: this is sily a result of the relative sllness of the sses of the four terrestrial pls&nbsppared with those of the outer jovian pls. another thing we notice is that the inner plary subsystenbsp;y be unstable re rapidly than the outer one because of its shorter orbital tiscales. this can be seen in the panels denoted asinner 4 in fig. 7 where the longerperiodic and irregular oscillations are re 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 fronbsp;other terrestrial pls， as we will see in subsequent sections.

4.4 longternbsp;coupling of several neighbouring pl pairs

let us see so individual variations of plary orbital energy and angular ntunbsp;expressed by the lowpass filtered delaunay elents. figs 10 and 11 show longternbsp;evolution of the orbital energy of each pl and the angular ntunbsp;in n1 and n2 integrations. we notice that so pls fornbsp;apparent pairs in ter of orbital energy and angular ntunbsp;exchange. in particular， venus and earth ke a typical pair. in the figures， they show negative correlations in exchange of energy and positive correlations in exchange of angular ntu the negative correlation in exchange of orbital energy ans that the two pls fornbsp;a closed dynacal systenbsp;in ter of the orbital energy. the positive correlation in exchange of angular ntunbsp;ans that the two pls are siltaneously under certain longternbsp;perturbations. candidates for perturbers are jupiter and saturn. also in fig. 11， we can see that mars shows a positive correlation in the angular ntunbsp;variation to the venus–earth syste mercury exhibits certain negative correlations in the angular ntunbsp;versus the venus–earth syste which see to be a reaction caused by the conservation of angular ntunbsp;in the terrestrial plary subsyste

it is not clear at the nt why the venus–earth pair exhibits a negative correlation in energy exchange and a positive correlation in angular ntunbsp;exchange. we y possibly explain this through observing the general fact that there are no secular ter in plary sejor axes up to secondorder perturbation theories cf. brouwer & clence 1961; baletti & puco 1998. this ans that the plary orbital energy which is directly related to the sejor axis a ght be ch less affected by perturbing pls than is the angular ntunbsp;exchange which relates to e. hence， the entricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive correlation in the angular ntunbsp;exchange. on the other hand， the sejor axes of venus and earth are less likely to be disturbed by the jovian pls. thus the energy exchange y be lited 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 subsyste jupiter–saturn and uranus–neptune seenbsp;to ke dynacal pairs. however， the strength of their coupling is not as strong&nbsppared with that of the venus–earth pair.

5 5 1010yr integrations of outer plary orbits

since the jovian plary sses are ch larger than the terrestrial plary sses， we treat the jovian plary systenbsp;as an independent plary systenbsp;in ter of the study of its dynacal 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 systenbsp;over this long tispan. orbital configurations fig. 12， and variation of entricities and inclinations fig. 13 show this very longternbsp;stability of the outer five pls in both the ti and the frequency doins. although we do not show ps here， the typical frequency of the orbital oscillation of pluto and the other outer pls is alst constant during these very longternbsp;integration periods， which is denstrated in the ti–frequency ps on our webpage.

in these two integrations， the relative nurical error in the total energy was 106 and that of the total angular ntunbsp;was 1010.

5.1 resonances in the neptune–pluto systebr>

kinoshita & nakai 1996 integrated the outer five plary orbits over 5.5 109 yr . they found that four jor resonances between neptune and pluto are intained during the whole integration period， and that the resonances y be the in causes of the stability of the orbit of pluto. the jor four resonances found in previous research are as follows. in the following description，λ denotes the an longitude， is the longitude of the ascending node and is the longitude of perihelion. subscripts p and n denote pluto and neptune.

mean tion resonance between neptune and pluto 3:2. the critical argunt θ1 3 λp 2 λnp librates around 180 with an alitude of about 80 and a libration period of about 2 104 yr.

the argunt of perihelion of pluto pθ2pp librates around 90 with a period of about 3.8 106 yr. the donant periodic variations of the entricity and inclination of pluto are synchronized with the libration of its argunt 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，θ3pn， 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 xi the entricity bes ninbsp;and the argunt of perihelion bes 90. when θ3 bes 180， the inclination of pluto bes ni the entricity bes xinbsp;and the argunt of perihelion bes 90 again. willia & benson 1971 anticipated this type of resonance， later confird by milani， nobili & carpino 1989.

an argunt θ4pn 3 pn librates around 180 with a long period， 5.7 108 yr.

in our nurical integrations， the resonances i–iii are well intained， and variation of the critical argunts θ1，θ2，θ3 rein silar during the whole integration period figs 14–16 . however， the fourth resonance iv appears to be different: the critical argunt θ4 alternates libration and circulation over a 1010yr tiscale fig. 17. this is an interesting fact that kinoshita & nakai&039;s 1995， 1996 shorter integrations were not able to disclose.

6 discussion

what kind of dynacal chanisnbsp;intains this longternbsp;stability of the plary systenbsp;we can iediately think of two jor features that y be responsible for the longternbsp;stability. first， there seenbsp;to be no significant lowerorder resonances an tion and secular between any pair ang the nine pls. jupiter and saturn are close to a 5:2 an tion resonance the faus great inequality， but not just in the resonance zone. higherorder resonances y cause the chaotic nature of the plary dynacal tion， but they are not so strong as to destroy the stable plary tion within the lifeti of the real solar syste the second feature， which we think is re iortant for the longternbsp;stability of our plary syste is the difference in dynacal distance between terrestrial and jovian plary subsyste ito & tanikawa 1999， 2001. when we asure plary separations by the tual hill radii r， separations ang terrestrial pls are greater than 26rh， whereas those ang jovian pls are less than 14rh. this difference is directly related to the difference between dynacal features of terrestrial and jovian pls. terrestrial pls have sller sses， shorter orbital periods and wider dynacal separation. they are strongly perturbed by jovian pls that have larger sses， longer orbital periods and narrower dynacal separation. jovian pls are not perturbed by any other ssive bodies.

the present terrestrial plary systenbsp;is still being disturbed by the ssive jovian pls. however， the wide separation and tual interaction ang the terrestrial pls renders the disturbance ineffective; the degree of disturbance by jovian pls is oejorder of gnitude of the entricity of jupiter， since the disturbance caused by jovian pls is a forced oscillation having an alitude of oej. heightening of entricity， for exale oej0.05， is far fronbsp;sufficient to provoke instability in the terrestrial pls having such a wide separation as 26rh. thus we assu that the present wide dynacal separation ang terrestrial pls > 26rh is probably one of the st significant conditions for intaining the stability of the plary systenbsp;over a 109yr tispan. our detailed analysis of the relationship between dynacal distance between pls and the instability tiscale of solar systenbsp;plary tion is now ongoing.

although our nurical integrations span the lifeti of the solar syste the nuer of integrations is far fronbsp;sufficient to fill the initial phase space. it is necessary to perfornbsp;re and re nurical integrations to confirnbsp;and exane in detail the longternbsp;stability of our plary dynacs.

以上文段引自 ito， t.& tanikawa， k. longternbsp;integrations and stability of plary orbits in our solar syste mon. not. r. astron. soc. 336， 483–500 2002

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