INVESTlGACIÓNREVISTA MEXICANA DE FíSICA 47 (6) 525-531

DICIEMBRE 2001

The Sun-Earth-Moon problem in l1ewcoordinates, bidimensional caseA. Escalona Bucndíal and E. Piña Garza:.!

Departamelllo de FÚ'ica, Uniwrsidad Autónoma AlctHJ(w!itana./z.tapa!apaApartado postal 55534, 09340 Mh-ico, [).¡.~,MOLleo

e-II/(J i1: 1aheh@xmwm.II(lJII,IIIX, "lpge@xlIIlI/JII./f(lIl1.mx

Recibido e16 de marzo de 200J: ilCeplado el 14 de ag.o",lodc 2001

\Ve pre:-.elll a liN approach 10 the study of the bidimensional ea:-.e for the Slln-E<H"lh.~loon problem using !lew eoordinatcs intnxluccd by

Pilla ano J;lIléncz-i.ara. By adding the hypothesis that the Eartb-Moon sy:-.tellldescribe", a circular orhit amulld the Sun, we can eliminare(1m: 01' lhe coordinalcs. rhus rhe system is redlleeo into a problem wilh IWOdegree!'>of freedom

Kl'Yll'Imis: Three boJy problem; mooll theory; celestial mechanics

Pn:SClll:llllíl:-'una primera aproximación al estuJio del caso bidimcnsional del pruhlema Sol-Tierra-Luna ulilizando las nuevas coordenadaspre:-.ellladas por Piña y Jiméncz Lara, Al agregar la hipótesis de que el sislema Ticrra-Luna ue:-.cihe una órhita circular alrededor del Sol,pOlklTloSeliminar una de las coordenadas, de esta mancra el sistema qucda reducido a UIl prohlema Je dos grados de libertad.

/)e,w.,-i(lI()f"('S: Problema de tres cuerpos; teoría lunar. lllcdnica celesle

!'ACS: 40.01l.-j: 40.10.-h: %20.-n

where

FI(jURE l. L' Ang.h:.

(8)

( 1 1)

(9)(10)

( 12)

( 13)

(14 )

S,I" = ('';l;f'''''2.,: • .'j:~J')'

S,11~ ('';111' .';"21/' '';:11')'

S MST','o ,l. = 1'1 = pHi,

In. Sr = 111• S IJ = O.

where III = (In l' lI/:.!, //1.:\). And Eqs. (3 )-(5) become./

S,.MS" = O.

S MST J ',/1'1 = [= I/,Ri,

In this way, Eq. (2) becomes

/f/'1/1'2"),:~

111 [ + 111,'2 + 'lnJ'

Therc is anothcr coordinatc. an angle a, related to {heconflguratioll of lile syslcm. Thcre is not a simple geometri-cal interprclalÍan for this coordinate; the definition is given inEqs. (1CJ) :lnd (20), and we \ViIIlIlake a redefinition i; Sec, 4.

Equations (2)-(5) can be sllllllllarized d~fining t\Vo vec-lors

(3)(4)

(o)

1. !ntroduction

I.el 11IJ' I/I,:.!,In:l he the mas ses of Sun, Earth and !\1001land JI. (1. and ,. be the distances Earth-Moon, SUll-Moon, andSlIn-Earth. respectivcly, The new set uf coardinates is delinedin ;¡ nOll-incrtial rcference syslem, parallcl to the directionsof Ihe principal ¡nenia axes of the triangle described by lhethree bodies. \Vith the origin at lhe barycenler [1,2). Thelransfonnalioll of this frame inlo :'111 inertial system is madeby" rolalion matrix. r¡ = GSi (i = 1,2,3), Becallse theproblcm is restricted lo the bidimensional case, we need jllsllllle ElIler angle so

(

:("'/) - sin';; 0)G = Si'¡;") CO;~1/1 ~ . (1)

This coordinate 4} \ViIIenable liS to sludy the motion ofIhe ¡hree parlicles as a whole body (sec Fig. 1).

In Ihe rotaling frame, the conditioll that lhe orie:in is atlile baryccnter can be \\.Titten ~

L:.mj~i= O. (2)

Tlle panicular selection of the reference systeIll makcs Iheinerlia Illtltrix lO be diagonal, this can be wrilten in the fol~lowin,g equilliolls:

Ikc:lu ....e tllree particles conform ti plane, just lwo 01'thelIlolllcnts of inenia are independent: [:~ = l[ + ["l' Pii1a:lIld Jiménel-Lara define t\VOcoordinates with dimensions ofk'ngth Jll, n:.! relatcd ta these lllalllents of inertia [2]

J, = ¡d¡;, (6)

THE SUN-EARHI.MOON PROBLEM IN NEW COORDINATES, BIDI¡\lENSIONAL CASE

Lel a ami h bC'two vectors. in a spac~ of mi.lSScs. orthog-ollal to JII ami orthogonal to each oth~r (Ihe last condilion isarhitrary hltt simplitics the calculalions).

a . 111 = h . JlI = él • h = (J,

wherc matrix II depends on lhe masses of the paJ1icles 011-Iy. ¡,c.

-filIal !JI)-fII:!fl.,l!J.2 . (22)

-11I:lrt.I!J:l., .,

I/I.-:'(l~., ,1

., .,nl"'¡(/j

2. EIllIali()lIs ()f JII()li()lI(15 )

(J)(J .

1I1;l

1)

f/I'2

(J('/1,

M= O11

\\'here

(17) where w is thc angular velm:ity. I is the malrix of inertia, and

(23)

(25)

(26)

(27)(28)

/''2 = I,R'2'

¡'ff = I,(f?"f t trj);,- - '21/[(1nl~';I.

Thc corrcsponding canonicall1lol1lents are

The lagrangian of lhe systcllI is given by [3]

RestriclL'd lO Ihe bidilllL'J1sional case wby E'Is. (12)-( 14). (19) ""d (20) we gel

L = ~[ili + iI; + (Jli + 11;)(,,' + J,') - 'iR,n,,,J,j- F. (24)

( 16)

( 18)

(21 )

( 19)

(20)

a~Iar = 1',

hMhT = Il.

T = (J.al\Ih

(/") ( ni si"',, + I!j cos'" )/I.!. = 13 Ri ('os:! rT + nj sill:! rT ,

rol 2(Rj-Ui)sillrTCOSa

Sinl"L ooth Sr and S!I are orthogonal to lll, lhey musl belinear cOlllhinalions of a and b, Assuming

Piil<l and Jilll¿ncz-Lara define a sllch that [2]

S, = af(:!cosa + hU:!silla,

S'I = -aRt silla + hUI cosa,

Inlhis way, Eqs. (11 )-( 14) are satislicd.Si tlct'

1" = Is, - s:l'. ,i' = Is" - s,I'. r' = Is, - s,I'.lile rebtion tor lile new coordinates and the distances IJ, 11.;lIld 1" L';1llht: \vritlen as

So \\-'c gCl thL' hal1liltonian

1 [ "H = - Pi + 1', +2/, .(29)

/', = (l,(37)

(36)

(35)

¡JI'iJR'2

:lRj + F!f

(Ui - nj)'_->!!.J..¡) J'- ",

/'

¡JI'Da '

/1, ( ") :W, + lIi- /)- + p~ .._--" "(1" n').ll' II - ll~

a, :,>Ri + R~+')--=.fJ 1) -- /' " "(Ui - un.l

1',

(.'3)

(.11 )

(.12)

(.10)

/'

( , JI')l' "1' n nRI + ~ v + - l¡ll',:!Fj]

1'(1Ii - un',.

¡):!N., -=

AmI Ihe t'quatiolls of l1lolioll

1',nI -=t'

iJl'0.\

ByEq.(21)

(ui sill'" + 1,', <'OS'" )n ., .. , ~.,u-.-',u:; :-illl- (1 + Ni l'OS- (f ,

').\ " ,.'2 (/{0. - H¡) SlIl (J ("os a

(.18)

Rn'. "'-In. 1-'1.\. -l7 (6) (200l) 525-5~1

A. ESCALONA lJUENOiA ANO E. P¡:':A (I"I{Zt\527

wllL'n: X can be any ol' (r, ni' or n:!. p~.'., is a constant ol'11111tion. in ract it is the anglll:ir lllOlllcntulll of the syslem.

3. SUII-Earth.MoolI case

\Ve lIeed an explicit form 1'01'a and h in order to calculateth\? lllalrix B. Since f\la is onhogonal lo h it must he a lineall"\lI11hinatloll of a and 111

Ma = ;¡:<l+ Jllll.FIGUJ{E") Sllll-Earth.fvlo011 syslern.

Solvjn~ rOl"a, orlhogonality wilh 111leads lO

which is ,1 qlladratic equalion 1'01':r. Vector b obeys the sameI:qllation. so, taking the higgcr solutioll, :1:(1' for a ami the

",111;i1In, .1'", for h . .tI" and .111, can be calculated using Eqs. (16)dllll (17l.1I:spcctively.

¡:nr Ihe SlIn.Earth-,\loon case If/I » 1H:.!, the matrix Ucan h\.' appra:\imated by

h) (J' lakes this valllc at tlrst ami last quarter wherc,(/:::/' » JI, thus

This allows to Illtlkc an estimation: J(:! > :J550R1.lile tri;lll~1e conformed by ti1\.' thrcc bodies is very nar-

1'0\••..'. and ]¡¡ys 011the.r axis in the rotating frame, with the Sunar Ihe.1: > ti .•.•ide and the Earth-l\loon couple at the oppositcside (see Hg. 2).

.,...., Jrn211l:J :!= ----R.).In'l + 11t3 -

"2 ...., III:.! + 1fI:~ :.!/' = ----R,.J'I/.:.!III,:~

= 11.I/I~.: 1It~)+--'-

1IJ:.! - .1' f/l.:f - ~r

In Ihis fonn, Piña ami Jiméllcz-Lara gd the foJlowingrcslllts 11. :2}:

{¿"11(1" = ;11'('10111 -' = G.:.12SGo.

ItI:.!

,,) (1 I¡¡kes its lllaXillllllll allt!minilllum vallles at full t\"IOOIl,lIld ne\\' l\1001l respeclively, oscillating aroul1d ;:¡cons-tanl valllc

(.12)(J::: (1" + 171,

This 171 \';lriahle describes slllal] oscillations arallnd zera,r!lcn \Ve \.'an rewrite El]. (21) as

4. Redefillill~ a

Considering tllat the vallle 01' (1 has small oscillationsaroul1d !TI¡ \Ve redefine

n' = Be.

so \ve gel Ihe disl;lnces in terms 01' 17l' Detlning

+0(_1 ).TIII

11

11

-1f2 f2Il/.:.! 11':1

B = 11 f2111 :1f2 11

III:.!

lhe transfol'lllalioll llltltrix depends 011(J'q only:

( ,(( IS- (1"

e ::: S1Il2 al)

2 SIn a n ( US (1"

. ,",111- (J'1l

.,("os- (1"

-.~"'iIl 17" ('C)San )

SII¡ 17" C'( IS (J'", .. )

("os. (1" - SII¡- (1••

(45)

\Hiting it in terJ1lS oI" the llIasses

This causes a' . h' i- (J. our Ihis condition is not relevanl aslong as Eqs. (16)-( IH) rcmain satislicd.

(

111, 111." - J1II.,III,,)e -= ---- 1/1:1 111"2 Jm:!1II::. (46)

/11.21-//1:: :!~ -'2JII/:!TlI:1 1/1."1-/11:1

111mder lO recove,- Eqs. ( 1<))aJHi (20) in tcrms 01' (] l' \'cc-fors a alld h are modifted too

a' = a ("lIS (1" + h sin (J'", (47)

Hel'.Me.\. Fú. -l7 «(l) (2001) 525-531

5.28 TItE SlJN.EARTlI-MOON PROBLEM IN NEW COOR[>INATES. BIDI.\1ENSIO,"JAL CASE

Thl"n:fore. for Ihe Sun.Eanll-/'vloon case \Ve llave

I;wm l!lis POilll '''''e call this malrix. B and we call r7 lo rr l'

lhe only dlange in lile hamiltonian ami the equalions 01' 1110-¡ioll is lhe L'.\plicil fOl"ln of lhe Illiltrix n.

j\'lultiplying Eq. (21) al lhe right side by the inverse ma-tri.\

11(1H'2+'111,:¡)'2

IIJITI'2'I1:l

13'= ..//1/'21T1:¡mi +0(_1) (49)

lll'.! + 'I':l JrII'2111:~

fl/.11111

.,JIII'.!'Il.;j

III~

J'''"}.III':f-111':1

(54)

(55)

(56)

q, we

111., + 111. '" ( 11',')- , ., 1"- = n-, 1 + -.,

~ - Rj

)/11"2"':1" .,--~-V = /li.1111 + Ift:\

tall '2a =

1"1 < 1I.(III1I:l.

In f¡rst ami lasl (juaner. where a ~ ti and l' ~

rccovt'r

Dividing Eq. (53) by lht' ditTerellce 01' Eqs. (511 and (52)' ••..e gel

\Ve can Illa~e the estim<llion: In full 01" new Moon.JI:! = q'.! + ,.".2- 2(jl". f ¡Ul '1.a == :t.tJ,OOO,j.jGG. therefore wehave

+0 (_1_), (50)lit 1

JIII"].'II:1

~r¿

1//"].+11I:1 1//"2 111 :l

13-'= )/1/"2111.:11 11

III'.! + 111:1

JlI'.! -11£3-1

I/I'.!+I/I:~

., . 1 .,'JHís1I1 a+R1.cos-a, (51)

5. Rcco\'cry 01' a two hody problcl11

(57)1 [ 11' (n')]-1 1+'2--i+() --+ ./1, 11, [1,

I

(/1; - 11j)'

The appro.\imated equations of motion for n1 and ti) are

1 [" 1'; +. 1';, ,p"p" _/1,] \"" «8)11 = - 1',- +P;+ --,--+ ", +._,'2,1 - /(} R"}. 1?1

lile lerll\ R1/Uj is ()(Itl-lo). so Ihe hamillOnian can beap¡R()XimaICd by

The differencc in three ordel"s of magnitudc between R I<lne! H:! allows 10 Illake :-'Olllt:'approximations:

(53)

(52)

(' .')'2 U} - Rj siu a {'osa.

, , l/l.) -/1/..\ .,rr - 1"- - - . p- =11I'"2 + 1//:\

1, -:! -

f',; + P,~,l/ni

,P(1Pt" nI , [/11'1 + 1J/.:~ sill:! a ({2;111:: 1 {2;H"!. 1) ("os'.! a+h--.----IL(Jl/llll/..)l/I ..~ -------., + --+--,-

l/ni H"!. - _. )'/II"}.ll1-:l 11' l¡r' 11/.".2(ti 11/:\ rl 11I"].+ IIt:\

({2;".,1 2'" 1) ,ir,'" ( 1 1 ) 2 sil!" nlS."]+ _- _ + _. _ _ + - - - ----- . (59)In:~ {j"J \ l/I'.! ,..1 III'.! + 1/1:: (r( ,.:~ /1/1 + "':1

. 1'. ( P 11 )¡" = ~ 1 + '2--!!...-' .

¡tIl" P", 11-,(60)

in this ,,,:ay. when H'.! » NI lhl" equations for R'1 and l/J canbe I"educcd 10 Ihose rOl" ¡¡ lw() hody problclll. Therefore, wecan expecl that lhe behavior dcscribed for Ul alllll/J consisl'iin slllall oscillations •.lrolllHi a kcplerian orbie

hom Ihe available astronomical dala H,5] \I,.'e hnd that(l1T/ Pl, < O,lIt) L thus ,\'e can ~ISSllIllC thal

; '" I 'l"I.¡J = --:;:;-. (61)

I,[lj

\Vilh Ihe S;UIlt: appro.\illlatioll in Eq. (59), whcn (1 =: ()¡¡lid ') ~ (j ,ve ohlain

., :'/>:, 11I1 ~.I.,I/I.'I ("1.) + /11,) ,t (' V'''"}..''':I _ .

IIHJ - r ni )1111'113(62)

,\ ESCALONA BUENOíA ANO 1:'.. PIÑA GARZA529

Ld H be lhe Jacobi vector, \',:hich points from the barycenternI"lhe Eanh-t\loon systclll lo lhe Sun {ti] (sec f-ig. 3). Fromlhe fcolllctry of the system, \"C can writc a relation of t!lis\ t:.'('lorand vcclors p. q. and r; in lhe inertial refercnce syslcm

6. Eliminating ~

P = J':'1 - l':!,

q = 1", - r~= R - np,

m,

Ih})

(64)

.

FH;UKE 4. Surface 01'seclion 1:.

lllotion are invariant under the transformation

(70)

Thcl"e are t\\'o inval'iant planes lInder this (ransformation:(R¡.f'¡./'" = O) and (U, = O.a.f'"). Weusethelatteras•.1 surface of scction, callcd ~~ (sce Fig. 4). On this surfacethe hamiltonian is reduced to

1"= 1", -I"¡ = -H.-lip. ((1:;)

\\'hcre (l = 1I1,:!./(In:! + m:~), (j = 1fI:J(1fI.2 + 1It::¡). As ,1

consequcncc

- f'} + 2¡L(H - F).,.' = R' + 13'1>" + 21Jp . n.

(h6)

(67)

I ( ,11=;- I'¡-+2't

fmm which, \Ve can \\TiIC

p:! + p~n <-

ni

¡>j -+- f'; + I'J,) .[{j + \ , (71 )

(72)

Suhstitution in El). (5)) leads ro

uf sin:! a + Ri eos"!. (J = 1/1"2 + '11/:1 n:!..JlI/11/l:1

(hS)

By Ihis procedure, giving Il and P1¡.' and uSlllgEq. (69), 1'1 is determined for cach point on ¿ except fora signo \Ve define a Poincaré Illapping

(73)

As lhe cccclllricity of Earth's orbit around Ihe Sun is....mal!. abaut (J.Oli. in arder 10 make a tirst approximation lO

rhe prohh:m. we add the hypothesis

fl = ('(llIstilll!..

Thc 1Il0tiOll 01"the Earth-Moon systcm around the Sun is rc-duced 10 a llniforIn circular motion, meanwhile the conngura-lion oftlle lriangle confonned by the three bodies is describedby U [ ¡(nd (J. \Vith this idea, the smne used by Hill [6.7], lhe~y~telll is reduced inlO a problem wilh Iwo degrees of free-domo

Thl'refore we can wrile R1 in lerl11s 01' U. /l¡ aud a:

" (III,+II¡JR' JI"') 1d., = -~--~ - I SlIl a --,-o- JII1!,'H3 ("os a

(69)

as the integralion of the equatiolls af mol ion starting from ini-tial condition~ on ~. \\.'ith PJ > O, unlil the solutions intersectIhe sallle surface, with PI > O.

Transformation lo is an il1volulioll. that is 16 = l. Thusfhc rc\"crsibilify propcrty can be \vrinen [8J as

(74)

Dcflning file involulions: 1) = 1\1/ lo we get a transfornlJ~tiolls group 011¿

M}-' = 1I (75)1 "

lj+~ = M}I,., (76)

1j _~_= I}M'. (77)

A sYllllllclry tille is the set 01' points on the surface ofscclion invarianl lIndel':ln involulion

7. S)'nllnctr~' Hnes r} = {S E LV: = liS}. (78)'I'hi."i is a reversible system, so wc can apply the technique ofth!..' SYIlllllctl'Y lines. The hamiltonian and the equalions 01"

The propcrties 01' these sYllllllctry lines are given by Ihetransfortl1<.llions group. Firsl of all, Ihey are related with cach

R{'\,. AleJ:. Fú. 47 «1) (2001) 525-5~ 1

530 THE SUN-EARTH-MOON PROOLEM IN NEw COORDINATES, OIDIMENSIONAL CASE

• •

R¡ (AU)000'

FI(jllRE 5. Pc:riodil.: orhib.

FH¡UR.I: 7. V"IUI.?<'llf a.

• r, üOOJ)

• í0000,

r. \"." \

• ~\

• \' "1 • " ••~~, ID.y)

.O(Ol~

-"IXJJJ\~

•

r.

•

f;. O,uUo

'.

olher by

(79)

1,[', = ['2)_'" (80)

rhen. \\'L' l';lIl gl't ¡he whole family 01'symmetry ¡ines. :-.tarting:

trolll ro dlHl I'J'T\\'o reslllts lllakc tbis furmalisllI ver)' llseful:

(/) :\ poillt in Ihe intcrscctioll oJ 1WO s)'lll1J1ctry lillesS El', n 1', is a periodic poinl

s=;\¡lr'ls (XII

;l1ld ih pt'riod is Dile of lhe divisors of l.i - 1.:1.

h) An l'\'l'1l pL'riod orbit has 1\\'0 poinls 011r" or {\Vnpoints()1l \'1. ¡\nd ;In odd periad orbit has une point 011 )'11 andolle poill! 011 l' 1 (sec f"ig. 5).

This redut't's lhe scarching rOl" periadic orbirs into scarch-ing ror intl'rsl..'ctiolls 01' lhe symlllclry lines with ro and r 1"

In Ibis I.'ase, in ~. lhe horizontal axis (Frr = O) is in-\arian! 1111lkr111\: Iransformation 11). so this is Ihe s)'lllIlIell"Y

I¡ne 1'0'

FI(¡UR.I; X. SYlllllldry linc:-..

p. u .••••.•uo

.-

X. Thl' ¡Jcriodic orhits

From tl1e da1<l of october of 1999 [.IJ. whcll Ihe mean Earth-Sun di:-.l;llltT "'<1:-' U.D970 ::i: O.()()'2(i AU, ami Ihc hypolhe-"i" n == 1 AU. \vc calculate succcssive vallles fol' UI ami (r.

In Fi~",. () ;lnd 7 v.c can sec Ihe extreme values of (1 and Ihc zc-ro •..' ..•nI' Ji 1 ;11 full ~ 100n anl! ne\V r..IOOIl, Ihe 9th and '2.~,lhd;¡y:-.rc"pcclivl..'ly. The phasc locking. betwcell R1 and (1 is evidenl11l Eq.I~7l. 1111..' conditions R1 ::::. (l, 1'., ;:::O imply ir ::::.(J. solhe /L'rol':-' n( nI match ",¡lh Ihc t:xtrcmal vallle:-. 01"(1 in ;lll)perio(\ic orhil OH 1'11'

I;rolll aSlr<Hl\lmical dala 1,1,;)] we make an estimalioll orIhe I..'nc r¡;y ami an,8ular 1I10mcntum orthe syslcm thal \VC need

F\(¡tTR.E 9. POllearé lll:lppill,8.

fOI"lhe Poincaré mapping 1'v1. Figure 8 shows lhe sYlllmetl'Ylines l' 1 10 r l' it is ca ..•)' lo :-.ec lhal alt of lhclll inlerSCCI ro inIwn POilllS, indicaling lhe presence o •.t\\..o period.! points.

hgurc l) •.•hows a PoilKaré mapping conllrming lhe exis-SUlllilL"IlCL" 01" 1\\'0 st;¡hk per;odic nrbits. The pcriod of thellrhit inlllL'¡T > (l ..•iliL' i" ".li',If;.-, days. very clo...e ~othe side-real Illol1ll1. :li'.:~I:ll(¡(¡ day ..•.ThL" pcriod uf lhe moit in lhe(1 < \l side is 17A 78 days. I~lis would be lhe OI"bil of a 1\10011

w;th n:trogradc 1Il0vt'1Ilcnl.

1</'1'.¡\In. F/\ . .t7 (6) COOI) 525-531

A. ESCALONA BUENDIA A!'\D E. PISA (lARZA 531

1. E. Piii,l. Ce/e,\"{. i\kch. 7'" t 1991.J) 1()J.

2. E. Pil-laamI L. Jiméncz Lara. C('it'st. M/'ch. ((o he puhlishcd).

:~. E. Pilla, /Jil/(ÍlIliclI lit> Ro/acirmn (UniVl:fSidad Autónoma¡\klrnpolit,lllil-I ztapalapa. Mé:":.ico. 1(96).

1. ¡\I/I/I/rio l(jI.)(j tlellnstitl/lO tll' ;\'\'tI"{Jl/0I11¡(1 dt' !tI UNA"" (lJni-wrsldad N,u:íollal Autónoma de i\1é:":.Íl'o, ,1\1éXI(O, 11)1)<)).

S. ¡\mll/rio 2()()() dd Inslit/110 dl' A\/rorlOlfll'a dl' la UNAA1 (Uni-versidad Nacional Autónoma de :\1éxico_ Méx.ico, 20(0).

(j. D. Brouwcr ,1111.1 (J. Hori, 1'lIy.ü('.\' 1l1/l1,h/lDl101IlY oJ lhe Moo/l

cditL'd hy ZdL'IlCCKopill. (I\CiKlclllic Prcss. Ncw York, 19(2),

7. M. C1utz\\'ilkr. Rn Atod. I'''n, 70 (199R) SS9.

s E. Pub anJ L. JillléllCl Lara. I'hniw /J 2() (ll)~O) 3(1),

R('\'. I\kr. ¡:t~."'7 (6) (2()OI) )25-:')31