Using a semi-analytical approach, domains of possible motion for Trojan asteroids were established. It is shown that stable librating motion is possible for both high inclination and high eccentricity. Frequency distributions were also produced for real Trojan asteroids, against differing libration amplitudes and libration periods

Williams, I. P.

Yoshikawa, M.

Zagretdinov, R. V.

English

NASA Technical Reports Server

Asteroids, Comets, Meteors 1991, pp. 671-674
Lunar and Planetary Institute, Hoqston, 1992
_'_JS"_ "_ _j_ 671
it/l  I_I__
ON THE DYNAMICAL STRUCTURE OF THE TROJAN GROUP OF ASTEROIDS
R.V.Zagretdinov*, I.P.Williams,
Queen Mary and Wes_eld College, London, U.K.
M.Yoshikawa,
National Astronomical Observatory, Tokyo, Japan
Abstract
Using a semi-analytical approach, domains of possible motion for
Trojan asteroids have been established. It is shown that stable librating
motion is possible for both high inclination and high eccentricity.
frequency distributions have also been produced for real Trojan
asteroids, against differing libration amplitudes and libration periods.
Using a semi-analytical model described by Yoshikawa (1990), we consider the
possible motion of massless Trojan asteroids, that is asteroids situated close to the 1:1
resonance region with Jupiter, which is assumed to be moving on its actual elliptic
orbit. The asteroids are allowed to move on non-circular orbits in three dimensions. For
convenience we define the critical argument, if*, to be the mean longitude of Jupiter
rrfinus that of asteroid, and denote the longitude of perihelion of an asteroid minus that
of Jupiter by (_-_j)* .The variation of a*, the semi-major axis of an asteroid orbit as a
function of o* are obtained for specified values of the other parameters of the test orbit,
namely the proper eccentricity, e*, the proper inclination, i*, and (l_-I_j)* .Figure 1
shows the resulting twenty five plots when i* is zero with five different values of (_-
_j)*, 00, 900, 1800, 2700 and 3600 and five different values of e*, 0.02, 0.05, 0.10,
0.15 and 0.20 .By inspecting these plots, it is evident that stable libration motion is
possible in all cases provided the libration angle (maximum - minimum value of o*), D,
is less than about 500 . The results do not appear to be particularly dependent on either
the value of e* or on (li_-_j)* throughout the ranges investigated. In Figure 2, the value
of the proper inclination, i*, was increased to about 200, with minor variations about
this value, the other parameters taking the same values as in Figure 1. As can be seen,
the domain for stable libration is now generally larger than in Figure 1, especially at
low values of the proper eccentricity and sometimes even horseshoe orbits are found (
for example at e* = 0.1 and (_-_j)* = 900). It is also interesting to note that the size of
the stable libration region is larger when e*----0.2 than when e*---0.02 and that some
displacement of the libration centre from the Lagrangian equilibrium points (0*=600
and 3000) has occurred. Space does not permit the inclusion of diagrams indicating
areas of possible librafion for all sets of parameters investigated, however, even when
the proper inclination, i* is increased to 500, large domains of stable librating motion
still remain, though in this case, zones are more sensitive to the value of (laM_j)*. The
* Permanentaddress:Kazan State University,Kazan, U.S.S.R.
https://ntrs.nasa.gov/search.jsp?R=19930010080 2020-03-17T08:17:38+00:00Z
672 Asteroids, Comets, Meteors 1991
_0 [80 3+o 20 l_e 340 5o 160 3.4o 20 150 3#.o 20 150 3#o _ e* i*
5,60 5,60 O.ZO 0.00iii ,+o++d 5,_ 5,20 Q.iO O+Oe5.80 5.00 0+I0 0.004,80 4.BO O._ 5+00.+o _its o.oo5.oo- _ " 5.4o o.15 O+OO
5,4o
-- 5._o 5.20 o.15 o+00
*_ 5,00 5.00 0+15 0.00
4.8010 4.80 0.15 O.O0
i i_ 5+60 0+10 0.00
5,40 O. lO 0.00
c= e,, E80 0. I0 0.00
m 5,00 0+10 0.00
%P 4.80 O. IO 0.00
5.80-_ _ ' 5._ 0,05 0+00
'=--" +-"_))+,_,.t.t,_tTl--l_JJ+,- +000" 000
II 5+00 0,O5 °.00
% 4._ 0.65 0.06
4+_
i t . i , i 5+60 0._ 0.80
=.o_.. ,.'+ _Ol__iD))_(CoC.oi...-tDj)_tCotol- 5,0o0,°00
, ___ _ _ _ +ooo.. 0:00
._ 5.00 _ +.. 0.°t _o.o+
,. +-T-+o+<-r-++
('_-_ r)*= 0 (_-_j)* = 90 ('_-'_.,1" - 180 (_-_'j)* = 270 (_-'_j)" = 3GO
i_- 01o m*= o.o e.,= O,04g
FIo.I. The l:l resonance. Model A for eI = 0.048 and i*=O °. On _c. fight
side, the values of e* and i* ate shown asa function of a*.
iO 180 340 20 l°O 340 20 180 34+0 20 180 340 20 150 3_0 8" e*
5.++o_,_-_ _ _ :_ _ 5.°0 0.;_o
.= 5.00
° .801pi _ -m,pm_ +--T-_++._,.,°.,
oWN"°°"
,00"-0 o ++o_ o o o + + + u ,.,""0.,+o.,+!
"- ;._ _ +-T-+ ,--p--+_'80 o..
".5_))2 [tttoi--l,oJ)J)<,tto_i5+5o,o
,g 5.00 5.00 O, lO
*ff 4.80 4.80 0.10
, , . I . , +_-_-- _ 5+60 O+O$ i,.°
++ ,,,--'- _.,t___)_ ((((._m, Ot__j))_<,CCO_l_-_jj)_kkkOL_l-,.. o.o ,0.o
.,p 5,00
o_--..°+,o5"°_ (((_l__[ojjj) <,ttOL_l--_jjj_ktt_5.80°.,+o.o
4.80 i 34020 I ii_-_540 _9-_ifilmll 0_1_-_0 340 _ _0l'_'_mm lm_mm34_O 4"_ 0"05 20"4
f f I" f
(_,___),. 0 ('_'-_'l)*- 90 (;r-_'l)"m IB0 (I-'_._) t- l?O (I-IZ)'- 3613
i*
19.6
1i711
i0,0
80.1
80.4_
10+6
ll+i
50.0
_0.2
20.4
19.0
19.0
lO, O
EO, l
_0.4
+* lO.O _ = 0+0 e+- 0.04819 m
FIO.2.ModelA for to*=0° and/*=20 °.
Asteroids, Comets, Meteors 1991 673
0.26
0.20
0.1_
t_
0.05
0.00
0 •
n
x I989 6Q
30
i x • v "_, "h. "_,,x
x 4" ,P x" % _e_,
-- x ,;'_v v"% x ,_. "e_...,
vv N •
• v x vv w v x
It I I I I , vx_ i , I
_0e 20" _0m 40* fl0 e 60o ?0°
LIBRA'FIO_T.,U,_UTUDE
FIo.3. DLs_buL[onof properececnu-ici_(e*) vsh'brat;one.mpU_,']¢(D) for
73 numbcreA (•) and 88 uanum_w,d (_) Trojans.
80"
n o ' i , n • t . _
2O
1o
o
o"
3O
20
10
146'
UIII_TIOI_ARPUTUD8
FIO.4,Frcquctwy distribution of libration amplitude (D) of Trojans.
g t
166 v 18fi_"
I_R_TIOH P_RIOD
FIO.5.Frequency distribution ofh'braflon period of Trojans.
U_3
80"
175 _"
674 Asteroids, Comets, Meteors 1991
results do not appear to be sensitive to the value of the argument of perihelion. The
dimensions of the domain for stable Iibrating motion thus appears to be large,
encompassing a wide range of values for eccentricity and inclination. It is clear that the
real Trojan problem can't be meaningfully investigated in the context of the restricted
circular planar _ body problem, _ _= :T .... .... _ _ _ _
Using an analytical theory (see Garfinkel, 1977, Zagretdinov, 1986) the amplitudes
and periods of libration for 73 numbered and 88 unnumbered Trojan asteroids have
been calculated. For this calculation, the proper inclination i* was=est_maied by
referring the osculating asteroid orbit to the Jovian orbital plane. The proper
eccentricity, e*, was obtained using the procedure described in Shoemaker et al (1989).
Figure 3 displays some of these results, being a plot of proper eccentricity against
libration amplitude.The curve defines the boundary of the stability region obtained by
Shoemaker et al (1989). Also shown is an earlier limit to stability obtained by Rabe
(1967) It is seen than while many asteroids lie in the unstable region as defined by
Rabe, almost all of the 161 Trojan asteroids investigated lie in a region where
Shoemaker et al (1989) had indicated that such motion was stable. Two lie just outside
this region, (1989 UC5 and 1988 RNB) while 1989BQ (now numbered 4835) lies a
very considerable distance into the forbidden zone.
Figure 4 shows the frequency with which Trojan asteroids with various libration
amplitudes actually occur, the bin size being 40. The distrbution for all Trojan asteroids
shows a strong maximum at a libration amplitude of 300-400, but the corresponding
peak does not appear to be present when only the numbered asteroids are
considered.There is also evidence for a drop in the distribution when D exceeds 500.
Eleven asteroids have a libration amplitude greater than 500, while three have an
amplitude of over 600_ _ : _ ....
Finally, Figure 5 shows the frequency of occurrence of the Trojan asteroids with
various libration periods, the bin size being one year.There is a clear peak in the
distribution at a period of just under 150 years, when all multiple opposition Trojans
are included, but the distribution is much flatter when only the numbered Trojans are
included. Both figures 4 and 5 indicate the importance of obtaining a complete a sample
of Trojans as soon as possible, there are gaps in the distribution corresponding to a 1:1
resonance with the orbital period of Neptune and also the 14:1 resonance with the
orbital period of Jupiter, but the distribution in this region is so sparsely populated that
it is not possible to say whether these gaps have any significance.There is also a
suggestion of a gap coincident with the 13:1 orbital resonance with Jupiter, while a
strong gap exists at 150 years which does not correspond to any obvious
resonance.There are insufficient asteroids present in the sample for us to be able to
comment positively on the hypothesis by Garfinkel (1977) that "internal" resonance
may play a role in determining the distribution of Trojan asteroids
_i
J
References
Garfinkel, B., 1977, Theory of the Trojan asteroids. Part I, Astron. J., 82, 368-379.
Rabe, E., 1967, Third order stability of the long-period Trojan librations., Astron. L,
72, 10-17.
Shoemaker, E.M., Shoemaker, C.S. and Wolfe, R.F., 1989, Trojan asteroids:
Populations, Dynamical structure and Origin of the IA and 1_,5swarms, In
Asteroids II, eds. R.P. Binzel,T. Gehrels and M.S. Matthews, Univ. of
Arizona Press, pp 487-523.
Yoshikawa, M., 1990, Motions of Asteroids at the Kirkwood Gaps. I. On the 3:1
resonance with Jftpiter., Icarus, 87, 78-102.
Zagretdinov, R.V., 1986, On the theory of Trojan asteroids motion., Kinem.
Phys. Neb. Tel. 2, 3, 68-74 (in Russian).


On the dynamical structure of the Trojan group of asteroids

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