SClTlVTIFIC REPORT NR. IO

September 1966

NASA-Research Grant Nr ~~~-52-046-0oi I 4 Principal Investigator and Contractor: Prof.F.Cap

7..~~~s~itutt ~ I if Theoretical Physics, Computer Department

University of Innsbruck, Innsbrubk, Austria,

Numerical Computation of Satellite Orbits

Using Lie Series. Comparison with other P'iethods.

by H. Knapp

This report, covering older work, gives a survey on the application

and the advafitages of the Lie series method in celestial mechanics.

It was reprinted on the basis of a request of Dr. Yilson, Applied

Mathematics Branch, EASA.

rf GPO PRICE $ I CFSTI PRICE(S) $ I

oN (ACCESSION NUMBER) (THRU)

5I? 40 I > A/J - / Microfiche (MF) c - (PAGE=! ---d U L i ff 853 July 65 (CATEGORY) -1-

Numerical Computation of Satellite,Orbits Using the Lie Series Method.

Comparison Yith Other Methods, by H. Knapp, Electronic Computer Depart- ment, University of Innsbruck, Austria.

Int ro du c t ion

Using the Lie series theory the formal solution of the astronomical n-body problem in a region where no collisions take place, is easy.

It could be demonstrated by a special example (J. Kovalevsky chose this example to test the Lie series method for celestial mechanics) +\ that after the transformation given by Groebner ' the Lie-series converge so rapidly that the method in its present form can be suc- cessfully employed for calculating the orbits in celestial mechanics.

This method of solution is particularly flexible and very general, and good estimates can be given since the theoretical expansions and estimations can be directly applied to general multi-body problems.

Chapter I

Presentation of the problems 0 I Prepsrati-

1) Coordinate system: Our calculations are based on the following coordinate system: Let the center of mass of the three celcfitial bo- dies be the origin, Due to the vanishingly small mass of the 8th moon of Jupiter, it lies on the connection line Sun - Jupiter. Let the x-axis indicate the direction of the ascending node of Jupiter for the year 1950, let the y-axis be rotated in the direction of

Jupiter motion by 90' relative to the x-axis in the Jupiter rrbital plane, let the z-axis be directed such that we have an orthogonal right-handed system. This coordinate system is then assumed to be an' inar5:ial system since only in such a, system Newton's law of gravita- tion l-olds in the simple form. This may be regarded as fulfilled with-

'2 b'?e accuracy of calculation required here (up to and inclusive of

-- ?th significant figJre of each step).

2) Zesignations: For reasons of simplicity we use vectors, thus, e.g;,..

I is a position vector * i, = {u, v, n-i _I is a velocity vector 33 .'. 11 xu + yv + zw is the scalar product -. -1 2 2 2' 1x1 7 t.:r + y f z is the absolute amount

r-, - p,*-q = &JW-ZV,i zu-xw, xv-yu) is the vector product

is the gradient symbol

, n. " I,.,. I

' L!i-i,hcmore, we use the following designations :

Sun Jupit cr 8th moon

-w -* 3 X x3 2 X 1 -+ 4 3 U U 3 2 1

m m m 3 2 1 1 constant and m = fMi -.01~3. i '~.?.lquantities occurring in our calculations are assumed. to be

;ii'l'crcntisble.The three celestial boiiies, the Sun, Jupiter and its

:>i!:.hth satellite are assumed to be replaced by mass points which are

71b'jec.t to gravitation according to Bewton's law.

Tile poPitions and velocities -3- of the three celestial bodies are given for the initial nom,,O?t t = t , 0 The 18 components of the vectors ? and (i = 1, 2, 3) are to be i zi determined as functions of time such that the mass points move accor- ding to the laws of a three-body problem.

3) Units:

Unit length 1 L = 1 estronomical unit = ’1495,04200 . lo 10 cm unit time 1 d = 1 mean solar day unit velocity 1 Ld’l -._.. uu;t EZZZ 1 u = mass of thc Sun In these units the gravitational constant f assumes the numerical v~~lue: “1 f = 0,29591220828559 . lom3 p-’ L3 do2 mass numbers : = 0,295912208 . 10 -3 L3 d-2 m3 m2 = 0,282532864 . lo-6 L3 d-2 = m3 : 1047,555 m = o (vanishingly small as compared to m2 and mz) 1 J

4) Equations of motion of the mechanical system: According to the general theorems of mechanics we obtain the following system of diffe- rential equations for the three-l,?y prolslem: ** 1

i

I I

= f ‘iMk 1,xith u -L--r rik = /Zi- -+Ixkt i

(the dot denotes differentiation “ith respect to the time t)

Let the operator belonging to the diffcrential equations (1.1) be designated by D; ------~------~------~-~----- *) This and all othcr numericzl values are tclken from R pnper by J. Kovalevsky. Since ne zre concerned with the txplanetion of thc method rather than wltn The vzl~e~-th~rsclvec th~. Frablcm 3f their accuracy is of minor importoccc. **) See Y, Groebner, Die Lie liethen und ihre Anwendungen, p.71 ff, Since m = 0 it has the following form: 1

5) Known integrals of the system:

Law of conservetion of energy

Law of conservation of angular momentum :

Conservation of center of grnvity :

4 1 * + m 2 ) with m = m,

xs = ;;; b2X2 33 2 + m3

is the position of the center of mass of the three bodies. 2, Since D xs = 0 and owing to the spccicl selection of the coordinate

system zs = 0 is vnlid for ?,11 times*): the center of gravity rests in the origin of thc coordinate system. Hence we have:

+ = m25?2 m 332 o (1.5) 2s = 0 md 3 - 0 or : s- * -4 = m 22u + m3u3 0.

4-+- The nine components of the vectors x s9 Us, P and the constant energy

(1.3) are the 10 algebraic integrals of the problem. Uith these 10 4 -., relations between the 18 unknown components of the vectors xi and Ui

(i = 1, 2, 3) the number of unkno-tm functions could bt reduced to eight.

In our example the conservntion l~,mfor energy and angular momentum

*) This choice does not restrict generality. See VI. Groebner, Die Lie Freihen und ihre Snvendungen p. 75 can be easily eliminated and the motion can then be described by only two position- and tr;o velocity vectors: 4x and -9*x u and 3 S m' s m'

6) Transformation of variables :

Due to (I .5) this transformation is al-mys reversible :

The converted operator (1.2) has the following form:

3 3 a

x -x X

S m - +m 12 - 213 m S m S i J L

f 2 Formulation of the problem

',?e now hcve to integrate the system of differential equctions

4 4 x =u S 9

which belongs to the operator (1.8) under the initial conditions -u-

3 which are to be calculated from the initial conditions x.(t ) arid 10 4 u.(t ) for i = 1, 2, 3 according to the formulas (1.6). 10 The solution can be eesily obtcined by Lie series:

If f(t) is an arbitrary function holomorphic in the neighborhooq

44 of t = t of the twelve sought components of the vectors 7 , xm9 Us) 0 S and 2 then the Lie series m'

holds.

The superscript zero denotes thnt after application of the operc-tor

-9-4 4 D instead of the variable components of 2s' xm' u s' and um the cam- *(') +(') .;*,('I,and arc ponents of the constant initial values x s 'xm ' s m to substituted. The trajectories are obtnined by writing dowr, this a forrnulc for the vectors x (t) 2nd ? (t) and by analytically continu- S m in8 the series. In this form, the solution:: can, however, not bc used for numerical purposcs since the series converge too weakly. (This hns been distinctly shown by J. Kovnlcvsky in a comparison with the Col-icll method). Hence n transformation is necessary: First, we determine cbn approximzte orbit which is then corrected by a perturbntion cnlculn- tion.

Chapter I1 Solution of the problem: . 3 Sun - Jupiter as an unperturbed two-body problem

1) Splitting. of the operator: "/e shall now split D into c om- ponents :

= (3.1) D D S + 5 where 429 m -+a (3.2) I) = u e- 3 xs zr- s als I'jiRI 3 - while the remnining terms of the opc-rztor (1.8) are denote D.

~~ ~~ 4 2) Calculation of 2 (t): The partial operntor D out of the total S S operztor 3 will solsly mt, if in the plzce of functions depending

only on 4x and 4u but not depending on 4x and are substituted S S' m zm, into the final formula (2.2). Thus, ve have, for instance,

and the problem visu;?lized by the partial operator Ds can bc solved

separately. 'Je nay say: The varic.bles -ax and 4u are separated from S S -' xm and 2m since they do not depend on these. - Ds is, however, the operator of the unperturbed two-body problem Sun - Jupiter. We shall

give the solution together imth the respective numerical data in

Chapter 111.

$ 4 Construction of the approximative orbit

of the eigh-k sctellite of Jupiter

1) -Further splitting of the operator: It xould be most natursl to

split up 5 in such 2. way thc2t its essential part again is the opera-

tor of a two-body problem in this case of the fictive two-body prob-

kn Jupfter - satellite. Rather voluminous intermediate calculations,

:-:hich may be a large source of occuculnting rounding errors, are re-

-uired for the deternination of the Kepier ellipse as an approximative

orbit (particularly in the reversal of Kepler's cquntion!). In order

to zvoid these TX hcve decided on cq,lculating with a simpler, although

less eccuratc approximntive orbit.

',?e shall split the operctor where

4 The perturbation function h has the form m

i 11 with

-z 6= m I (4.4') 4 d m I1

4 2) Rough estimntion of th't ordcr of rn_2gnitude of 6 : ---- m

--z --i (a) if x md -x respcztively, are substituted in the plrxe of S m' 4 4 ~1 and b in thc form212

3 me obtnin for 6 an expansion into q" scrics by means of which thc order of mcgnitndc cc2n bc estimatcd more eFsily th,m by means of the 4 expression (4.4') for 6 which contn-ins diffcrences of approximtcly equal orders :

If we consider the first two terns of the series jointly and observe that -, d

(gA>4.95 L 2nd 0.05 lfo.25 L, vi11 have in the most unfnvo- Lf!; rn we rnblc cme

4 * (b) bn is less fzvornble to hmdle. If we trmsform 6 in such i-T LI a msy that the Kepler ellipse rcl0,l;ions cnter the formulc zs an zpproximntivr orbit rre find thzt

n4.05'70 -6 Ldm3. )Atl

vhere t = t - tq is the lcngth of the concerned step of calculntion.

Ho:-rever, we shzll not go into these dctzils.

3) F!els,tive orbit- of the sF"tel1itc ~ithrespect to Jupiter

''re shzll first ricglect h in comyzrison to D since then also the m m' variables 2 and < =re sep;.r-,teu i,om 4x_ 2nd -4u . In this way, the a n 7 S problem reprceente?L 5y tht opr;i'r?tor D %:I.;: he soived sep--

owing to (4.7) x~d(4.8). lie should note, however, that extrdmcly unf2-vorable condltlons hr?vc been c?zsumed in thest3 estimztions; the

figures in (4.7) and (4.8) will be sm-ller in generol!

The solution of the systems of diffcrcnticl equztions

(4.9)

* '8' 2-4 a with the opcrator Dmc - uno -a? cxma -aii and with the initial ma ma values for the moment t me obtnined in the form of the rather clmplc 0 approximative orbit (ellipse)

(The additionn.1 subscript nu is to indicnte that these approxinativC2

-3 4 functions, in difference from the sought exact solutions x 2nd u of m m the original three-body problem.)

The connection with tise t is Evident; the reversal of n Kepler equntion is superfluous.

5 Solution of the thrcr:-body problem by means of thc

given npproximntivc orbit; perturbation cr,lculus.

1 ) Trnnsformntion of thc solution (2 .2) : 'Jith the ne:v symbol

(5.1) D 1 = D_0 + D m

3 Expmdi ng (D, +A,) , ordering ?-ccording to the positions of ,!.l:,47and applying the exchange theorem to the Lie scrics, one obtains the for- mu12 (siehc :-!.Groebner; Dii Lie-lieihcln und ihre Anwendungen p. 92,

Forme1 (12.3e))

which is very importr-nt for thc subsequcnt cslculntions. This formula - 11 - expresses how the approximative solution f (t) has to be modified in a order to yield a solution of the original problem. The expression

means that A Daf has to be calculr?ted first, and thnt then the com- m -a ponents of x and; hove to be substituted by the components of the m rn approximative solution -3x (2) and 4u (q). ma ma

2) Expansion of the essenti3,l terms in the series (5.3): We shall now substitute the requircd speci2l functions -fx,(t) and -+u,(t) in the

Y place of the general functions f(t) in formula (5.3). - In the sub- sequent numerical computrtion ve shcll have to break the corresponding series 2nd to confine ourselves to the essentinl terms. Of course, the accurzcy of the result mny be increased to any degree if morc terms are taken into cccount. In the present instance, the following appro- ximations may be sufficient :

-3 t x (t) = x (t) +I (t -T) 2 (T)d? + ma t ma 0

with

(5.5)

Naturally, the formulas (5.4) are of use only 2,s long as the time space It - t } is chosen so sncll thnt the further terms of the 0 series may be neglected according to the required sccurzcy. (It is obvious that t may never be outside the region of convergence of the series.) 6 Estimation ol thc: zrro,r due to breaking off thc sEriLs

-4 1) Region of validity of the fornulas (5.4): We know from formula *

(5.4) that it is the solution O? -the problen (2.1) nithin a cert,ofr region Of the t-plene. Within this region, the solution functions con- structed by means of formula (5.4) have to satisfy the differentin1

d 4 equations (2,l). If xm(t) nnd um(t) are cclculated from (5.4), one ob t aim

where a-? --+ R(t) =& a=o =j t 0

Compnrison of (6.1 ) with (2.1 ) yields

We shall make use of this in order to determine the order of magnitude:

4 of the expression R( t ). VIith the abbreviation

where

and with the aid of formula (4.5)*) me obtain

\2Zma2 + -2 dl *) The series converges for --- < 1, which is certninly ful- 1; 1; 12 ma filled in a region where formula (5.3) represents the solutions, when \t - to/ = \At\ is chosen sufficiently small Substitution of (6.4) in (4.6) yields

so that

r 4 m (6.8) \R(t)imax

.-.

K(t) varies between

By virtue of

and with (6.8) me obtain in the most unfavorable case the following estimate for the order of magnitude of Ii *,R(t)l :

This estimate is critical for 1 - (t - t ) 2K(t) = 0, vhich means near 0 the perijove for It - to1 %’21 d

-+ *) The terms linezr in !E! ,?-resufficient in estimating the order of magnihde. I near the apo jove for It - to! 220 d I so that, ns it WRS to be expected, the mcgnitude of the region of convergence of formula (5.3) depends strongly on the distance betvecn the two celesticl bodies, Formula (5.3) is vnlid in any cc.se for a time space of at least 20 days.

In numerically evnlunting the formula it will be desirnblc to chose the interval rnther long. Onc h?,s to bc c'?reful, however, not to come close to the edge of thc region of convergence since then the rP"pid convergonce of thi: series, which is desired in practice, will no lon- ger be given.

2) Residue of the series nftcr the second perturbpetion integral; choice of proper step length L. t: The comprehensive deliberations which hp-vc been made to estimate tnc CxprLssion

have shown thnt the stcp length neL& ncvGr be shorter thcn 0.3 d if the error due to the brc?king-off of the series in postulated in one -1 1 step of ccwlculntionto cmount to not more than 5.10 L in the ccsc

\- of 2' and to not more than 5.10 -I3 Ld-' in the c~scof u , m 1 I m 1 Moreover, one may conclude thot the breaking-off error after the second pcrturbction integr,n,l in first approximation amounts to

t3 t+ (6.12) j R (I)&:- - F. (t) CP b 4P 0

4 in the case of and to um' ~ - 15 -

-3 iii the case of x Therefors, thcse qumtities may be calculated at I2 Y x, the end of each step"'. After this one may determine the step length p?rmissiblc at the prescribed cccurrtcy,

In pr-ct?cc one will alw?.yti stny somcnhnt below the accurccy limit, but nill cnlc;zl?-t,t sever21 stcns of equal length. Only when cpprocching

ihis limit one will reduce the step lensth R little (or increase it if

L'LAC zbsolutc cmomts oi the cxprisslorLs (6.12) 2nd (6.13) h,o.ve dropped bclov sone certain value). If this is sensibly done by the conputcr

--IL,L 7n-nLlCi" -I"uhth;nr* "LilLlLb tc dc but tc 24;uct thL lCZ?r;.+il ef thi, first stfyl- 0 ---

n3~ioa~ly,this is of p~z';lcul~rsignific~nce for czlculation of rocket

tra,Ectoricz (when the. r approxim~,tecourse is knc-irn, 2nd when esti- i-n+io-ls nczording to thc ~.'~cvc'ptttcrn c-.n be nede only for short scc-

tio;ic of tile trnjecto:-y).

3) Proyg?

L

-> 1::~. 1::~. formed, cnd if thcse qumtitics are mJed cs corrections to x 3 m nnS L. rcspcctivcly, one will obtnin iTiprovcd solutions. A checking m' c?-lculntion, also to ten d;,;its, hrs shJ;:,l th,ot after 30 steps the rcsulb for 2 is exectly chc snmt ?-s tht obtained when two pertur- n 3 Sc,tion in-Legrpelsvier' t?-ken into zcco-urt. The result for u m differed hut i:i~Lgn

(brecking-off after the second pcrturbntion integral) will be tcrmei! - in-; f 1. J. For the error qucntities

we obtain the rccurrcuce formulns

2, in which 5 dcnotzs thc! nmount of t1:e error in x rt the n-th step, n - m due to brep,king-off thi series, the ,o.aount of the brcp,king-off 'n -3 error in the series for urn rafter thc n-th step.

L (i.e. the maximum of this expression in thc! time intLrvnl of thc. n-til step of cnlculpution).

The solution of the recurrence formulr-s niny bc written str,?ight- forwcrd, if n good part of the pnth is computed with thi: snmc step - length if the breaking-off p in the formulns (&tl, errors i and < i (6.15) are replaced by their maximum vnlues and and if p is 5 s, n replaced by the rnaximun P. Thus, -. 17 -

where

a 1 I--- = (I + P) (1 -q7 C1Q.ttr) (e

' p = tLpP - y (l+P)i/_\tU1 1-, 1 (6.18) k= 22 P2 - (l+F) 7c q = [CP - 5 (1-8-7) ?(ir'\,ct(ii< 1. {. .. 1

pi and yi arc the constwts of thc general solution of thc recurrence formulas which make th; ?dapt,o.t_iOR to tht initial conditions Fossiblc.

With p* being the error of t?ic initi,-.l d,-.,ti. of our cdculztion in .-$ -- ix,l and q* the error of tiic initill C;,ntl, in \u,l we hr?ve the rela- tions

6) 7 ------_--Cnlculr,tion of the pcrtGrbntion integrals

It would be an i?r~fullot of nork to cvniuate generally the integrals - 10 -

occurring in (5.3). ie rzthcr go nnotner wVyrhich yields the intc- * grals in question with sufficient accurccy. \-'e lnbcl the vrc.?llkpa'krn

€unctions

for the 4 equidistmt instcnts of time*>

(7.3) to, t +h, to+2h, to+3h where

At t - to h = - (7.4) -3 -3'

2nd with the aid of the differentinting scheme of the table

2 3 7 g,( T-1 4R,(yi) a g,(?;.) 9 gJr>

. we repl-xe the function g by the Newton interpolation polynomial. cc (7)

*) This is nrbitrr>,ry! The functions could ?,s well be lnbcled more finely (in thc case of lm-gc step lengths this might bc necessciryb nnturnlly, the integrnl formula (7.6) would then hnve to be changed). But since the step length hzs to bc chosen short anyhow in order to keep the brenking-3ff errors, lov, and since it is evident that few but finely graded steps involve just 2s much work as more steps with 0- coarser grcding, thcrc is no rcrtson to lnbcl the functions more finely since the errors due to the chosen interpolation do not reach the amount of the breaking-off errors. This can be demonstrntcd thL most rapidly by c?vlculnting forth -.nd back vrith differcnt step lengths The difference 0t)g (t ) we defined as ao

We hnve then

When calculcting back,bt (md ni~o;I> has to b? taken negative. The difference 4'ga(to) nre cnlcyl: tsltl <-mr,l thcir definition (7.5) nlso

in this cese.

1) Initial instant: Tiriing b:gins fi-gm Oct. 29, 1958 - the Ju1i.m

dey 2429200.5 - and continuc s il; d:::;s,

2) Relative motion of tha su1; nzid Jupiter: ta-buletion of 3x (t): - S for the instants

the corresponding vnlues of E v ?"'?re to be detormined by inversion of the Kepler equction

(8.2) E, - E sin E, -3 y,t,, + 1.1. - 20 -

Numerical values :

= 0.0484011060 (eccentricity)

1~.= 0.001450215293 d-' (mean notion) (F.3) 1 M = 50645944315 (mean anomaly) [to= 0 (cc..lendar day)

The solution of (8.2) with respect to EV is most ensily achieved by iteration of Nentonls approximr.tc formulpv for solving equctions :

where E *) is a value which ?vpproximntely satisfies Cq. (8.2), nnd \;I EvII is an improvcd approximntc vnlue. Formula (8.4) has to b; itcrc- ted until E,, = Evv satisfies Eq. (8.2) with a given accurccy.

Then, x (t,) can be cr?lculnted from the resulting values of E : 9 u 0.015676901-4.1866~6655 sin -0.323895551 cos Eu. EV (8.5) xs(tv)= i-0.251333487-0.323515939 sin Ev+5.192722630 cos E" L to i 5) Initial data for t'-ie orbit of the moon: Computation is to be carried out with the mass numbcrs of page 3

m2 = 0.2825328640'10-~L~d-* (84 = 0.2~5~122080~10'3L3d'2 L3 and with the values for the relF-tivc position and the rclntive v(.tlO- city of the moon, corresponding to the instant to:

*) The value of E c0rrcspondin.g to the prcccding instrat t is v -1 U-I best taken as the initial value 0: E (starting from E = 5.615994607) VI 01 4) Approximate orbit for Jupiter's moon: We first calculate

m2 . = ImImj*

Then, the position of the moon on its approximate orbit nt the in-

stants (8.1) is found from the forauls:

The velocity of the moon on its ?wpproxim?wteorbit must bc- knovn on1.r

for the end point t = t +At of the intervd: 3 0-

e c (8.10) -5.u (t,) = -x(') c sin c(t3-to)! + ?(') cos lc(t3-to)7 1. ma m - m

4,

5) Computntion of the pcrturb .L'7n integrds: Now, thc functions 6 j Lh m 2 cnd f ? mn(t) must be tFdbulp,ted for the instants (8.1) from thi. formulns*> .

c - 22 -

With the aid of the differences betaeen these tables, obtained from

(7.5), we are able to calculate the perturbation integrals:

(8.12) 0

6) Formulas of solution: Thc perturbation integr3,ls (8.12) are used

to correct the npproximi?-te sol.utlons (8.9) and (8.10):

t +at 10 -z =X )dT mfi (T

t

--3 =U ma

Now, we replzce t ky t +(At) in a11 the formul

(8.7). Again, At can be newly chosen.

7) Precautionary measures tnken to avoid unnecessary rounding errors :

Since the SIE 2002 computer of the TH Aachcn, :rith which our numerici-1

comnutations ncre mndc, usurtlly c?.lcul-tc-s with no more than 10 digits,

some precnutionnry mensurcs h-?d to be tp.kc;n to eliminate rounding

errors : ------______------*) The contribution of this integral mnnifcsts itself only with greet steps, but the situcztinn is c~,boutthe spmc ns in the foregoing footnott. - 23 -

a) Prior to c~rcomputations we reduced the qurtntity ?!I (nnd E ) by 0 a f:lctor of 2n in order to nr-intcin thL nc-mlj- !;I< I for sone nun-

dred dF.ys. Thus, the l~khdigit cannot be lost during the invcrsion of

the ICepler equntion;

b) Instend of t +3h we c?lt:rnys cnlcul-ted t +At since h is equal to 0 0 only within roundir,g errors so thrt a ncticerblc Gnmr nipht -?DZ-? bt3 in the time counting;

c> "'hen calculating solutions from (8.13), we first deternined ths

suzi of the perturbation integr::ls and then added the approximte solu-

tion. In this rray, the rounding error of the additions enters the re-

sult only once.

.;. : y Results i

Trizl computations mnde s3 fm and zxpericnce gathered from thx.

The following trial calculations 7tcre nnee :

a) The first informative cornnutations aith different stcps (on.- step

forncrd and one step bnckr-rard) hzve shown th;rt forpiulc (7.6) is sdffi-

ciently a.ccurste 2nd thct the step constitent vith thc considerations

in 46, 2) is ap7roxinntEly Id.

b) loo steps were cdcul,-.ted frovrnrd md b::ckr-mrd vrithbt = Id. *)

This mas the most import.-nt put of our cclculntions since thcy could

be compared with other results.

J. Kovclevsky pointed out that his -12-digit coFyutntiona, carried out

by Cornell's method with an IBM 650 conputer, tcok 10 sec. for each

operation and that the devi3tions- in the coordinates nnd velocites,

obt9,ined when cclculnting rith (At) = 5d loo dq-s formrd and back- -1 0 -10 w,-,rd (i.e. in 40 operations) mere less than 50.10 L find 10o.10 L

respectively (unit not given).

*) The relevnnt section of the t,-..ble nrzy be scen from the enclosed table of dzta - 24 -

-Vre obtzined the folloning results by this method: -lo-digit computation with an SIE 2002 conputer took 2 sec. for each operp-tion (the printing of four lines of dnt? r.fter c-mh operation, which was neccssary for informp,tive purposes but could be omitted later, took 1.6 sec.). When cF-lculnting nith the step (&t)=ld loo dcys forward and backward (i.e., in 200 operations), the deviations in the coordinctes and velocities were less than 15.1 o-'~--L and 1.2'10-~~L d", respec- tively, On the bnsis of this result End with tht' aid of the (still very rough) estimate it could be shown in 4 6, 3) that the errors in the clnalytic continuation nt At=ld accounted for no more thcn 50 $ of the values indicnted, whereas the rencining deviations were due to the rounding errors. The styme conput?,tion crith dt=2d yielded deviations in the coordin?.tes and velocities of less th;.,n 26.10 -1 0 L nnd 4.1o-I 'Ld", respectively. 'Phe rem-ining test time was used for informative compu- tations with greater steps (3d, 5d, lod). Here, the break-off errors ti were Elready nocznble.V As u result of these computations, rve came to -? .c the conclusion that thc expressions R (t) and R(t) might be used for l a correction (cf.j6).

c) Integrp-tion wrls performed from At=ld (then o.8d, o.6d, o.4d) beyond the nc:ilrest dist,zr.cc bctmcen Jupiter nnd the moon, and thL time . left ims used for bnckwnrd calculp-tion. The values obtained agein agreed very well. In order to save timG, o~lytwc lines of values were prin- ted.

d) The modification mentioncd in thc. footnote p. 15 wns cablP,tc?d.

At the same time, the printing commnnds rcrc distributed more conve- niently in order to stcp thc co,?.putcr for n shorter time. CRlculntion and printing took ?,bout 2 SCC. for one operation so that the printing process was hardly interrupted. 2) Influcnce of errors: Thc rcsults can ba falsified in four v?

a) by calculating with an insufficient number of protective places.

Rounding errors mny CRUS^ serious errors unless they ?re sm-?,ller thnn

the break-off errors from the vcry outset;

b) by using too grcat steps. If s dGfinitc number cf terns is us~d.,

the required rapid convergence of series ccn be achieved only if thc

steF at is reduced; c) by successively performing mmy, sufficiently accurate oper-;,ticl-,r (if at is definitely cl~osen, the excessivrly strong propagation of th:

break-off error can bc- elininatcd only by alloping for further terms

of (5.3). This means, however, that the break-off error is reduoed

simultaneously, Beduction of the step alor,e is not very advantageous

since the required number of operations increases simultaneously, cf.

(6.16) ff.);

d) by inexact tabulation of the functions appearing in the pertur-

bation integrals, which can be avoided either by a more exact tabula-

tion or by reducing the step.

The rounding errors show a random character, r:.hereas the other three

error sources reside in the method, however, they can all be controlled:

in b) by observing the increase of (6.11) and by redilcing the step

in time:

in c) with the aid of the estinate (6.16) which can be improved

since we have always taken the maxima of the absolute values of the

quantities involved,

in d) by calculating forward and backward (random sampling) and,

if necessary, by reducing the step.

"Then choosing the step At , it is necessary that conflicting require-

ments be compensated: - -~_~- - - - 26 - ..

Results of given accuracy are to be obtained with thegreatest possible

step and the least possible number of operations. The modification mentioned in $6 is very helpful in this respect, since it makes it

6 possible to allow for the essential part of the rests of series with- out determining the required perturbation integrals. Finally, it should be stressed that we have dealt only with a special example and that our method can also be used for the numerical solution of general many- body problems. The elaboration of our method is still under way, and we , hope that we shall soon be able ttc achieve even better results.

Notes on the table of data

Since the data were originally printed only for the purpose of obtai- ning information on the efficiency of our method, we expressed the numbers in the way they were stored in the computer. The comma was omitted. The last two figures of each numer are the so-called charac-

teristics of the values representee? as floating-point numbers (charac-

teristic = exponent + 50; the point of the computer is put behind the

sign). The decimal numer fo.7, for example, corresponds to the floa-

ting-point number i-700 ooo ooo 050. Another disadvantage of the tables

is that the printed numerical vzlues are not clearly arranged. After

each operation the values were printed in the following four-line .

arrangement (dimensions arc given in brackets): 8

time t r-l:-dj, step At [d;, components of X(t) rLT, KL] -4 I"(t)\ componclnts of Tm(t) fL d-' 2

The numbers in the third line give information only on the order of + magnitude of the expression R,(t) (~ecalculated only with ten digits

and several digits verc lost in the course of celcul:j.tion, especiczlly - 27 - .e *' during the determinetion of the difference between two approximately

h equal numbers from formula (6.11)): The first two figures and the

characteristic are valid at most, while the other digits are insigmi- b f icant .

Table of data

!\;e do not le?roduce the full table vihich covers 24 pages. Anyone who

is intereatedto have a copy should write to the author.

A short summary reads

time [dj stepld'- 3! L 2

0.00000 + 0l.00000 - 165921387450 + 201577536050

1.00000 + 01 .ooooo - 18571 1957150 + 201288442250 . . . . 9 .

?Q. 00000 + 01,00000 - 129514535750 + 158151 320350

loo. 00000 + 01 .ooooo - 128523006850 + 1~~550010150

99.00000 - 07 ,00000 - 129514535750 + 1581 51 320350 D .

0.00000 - 03.00000 - 185921386050 + 201577534650 .

A