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the tegration of the outer five plas (f&psn;)， we fixed the stepsize at 400 d

we adopt gas f and g functions the syplectic ap tother with the third-order halley thod (danby 1992) as a lver for kepler eations the nuber of axiu iterations we set halleys thod is 15， but they never reached the axiu any of our tegrations

the terval of the data output is 200 000 d (&si;547 yr) for the calculations of all ne plas (n&psn;1，2，3)， and about 8000 000 d (&si;21 903 yr) for the tegration of the outer five plas (f&psn;)

although no output filterg was done when the nurical tegrations were process， we applied a low-pass filter to the raw orbital data after we had pleted all the calculations see section 41 for ore detail

24 error estiation

241 relative errors total energy and angular ontu

aordg to one of the basic properties of syplectic tegrators， which nserve the physically nservative antities well (total orbital energy and angular ontu)， our long-ter nurical tegrations see to have been perford with very sall errors the averad relative errors of total energy (&si;10&;9) and of total angular ontu (&si;10&;11) have reaed nearly nstant throughout the tegration period (fig 1) the special startup procedure， war start， would have reduced the averad relative error total energy by about one order of agnitude or ore

relative nurical error of the total angular ontu δa/a0 and the total energy δe/e0 our nurical tegrationsn&psn; 1，2