News on HILDA ASTEROIDS
by Joachim Schubart, Astron.Rechen-Institut, Heidelberg, Germany
Index
1. Introduction
2. Low-Eccentricity Hilda Asteroids
2a. Four Hildas at Secondary Resonances
3. The Growing Family of (1911) Schubart
4. Other Recent Results
5. References: The papers by Schubart are available in the Internet
6. Appendix: Numerical values of characteristic parameters of Hildas
**NEW:
There is a separate continuation
News on HILDA ASTEROIDS, Part 2.
1. Introduction
Typical Hilda asteroids are captured in the 3/2 resonance of mean motion with
respect to that of Jupiter.
In numerical studies on the long-period evolution of Hilda-type orbits Schubart
(1982, 1991) has introduced three parameters that appear to characterize an
orbit during very long intervals of time, in analogy to the proper elements
of orbits of non-resonant asteroids. The following sections present values of
such parameters for recently numbered or corrected Hilda orbits. Some of these
orbits correspond to cases of special interest. Here the abbreviations for the
3 characteristic parameters of Schubart(1982) are:
Epm, Ip, and Ampl.
Epm and Ip are special parameters for the Hilda-type resonance. Epm is related
to the variations of eccentricity, in analogy to a proper eccentricity of non-
resonant motion. Ip is called proper inclination. The third parameter, Ampl,
gives the mean amplitude of a special librating angular argument. This special
argument differs from the usual definition of the critical argument of the 3/2
resonance due to the application of a transformation (Schubart, 1991). This
transformation removes an important part of the effects by the eccentricities
of Jupiter and Saturn. Its application allows to count some low-eccentricity
objects like (1256) Normannia with the librating Hilda asteroids.
Here the formulas of the transformation appear with different symbols,
* is the multiplication sign:
e' cos(lp' - lpj) = e cos(lp - lpj) - k * ej
e' sin(lp' - lpj) = e sin(lp - lpj)
crarg' = crarg + lp - lp' = 3 * lmj -2 * lm - lp'
with:
e = eccentricity, lp = longitude of perihelion, lm = mean longitude
ej, lpj, and lmj are the respective quantities of Jupiter
crarg = critical argument of the 3/2 resonance
k is a suitably fitted numerical constant
e', lp', crarg' are the respective transformed quantities of the asteroid
argp' = lp' - lnd : transformed argument of perihelion, if
lnd = longitude of ascending node
The low-eccentricity Hildas (1256) Normannia and (4196) Shuya have appeared
as special cases that are separated from the Hildas with eccentric orbits by
a wide gap in the distribution of the values of Epm (Schubart, 1991). The next
section demonstrates that this gap is not empty. In particular, four of the
orbits found in this range correspond to cases of secondary resonance studied
by fictitious orbits in 1988/89 (Schubart, 1990). In the Hilda group there are
subgroups of orbits that nearly agree in the values of all the 3 characteristic
parameters mentioned above (Schubart, 1991), in analogy to the well-known
families of asteroids. As shown below, the subgroup headed by (1911) is rather
populous, so that one can call it a family within the Hilda group. Another
section lists updated values of the three parameters for some Hildas, since
Schubart's(1982) values depend on one-opposition orbits in these cases.
The appendix lists values of Epm and Ampl of 347 Hilda-type orbits with numbers
less than 85000.
(1256) Normannia is the typical representative for some Hilda objects listed
below. Normannia shows circulation of the usual critical argument of the 3/2
resonance. However, this argument depends on its longitude of perihelion that
is influenced by strong effects from the eccentricities of Jupiter and Saturn.
The transformation mentioned above almost removes these effects from this
argument. The resulting special argument, here called the transformed critical
argument, shows libration, which is typical for the critical arguments of the
more eccentric Hilda orbits.
Schubart(1991) has studied the orbits of Normannia and of the analogous object
(4196) Shuya and derived values of Epm near 0.025 and of M(a), the mean value
of the semi-major axis given in au, near 3.90. These two orbits have appeared
to him as a small group that is well separated by a statistical gap (Schubart,
1990) from the majority of Hilda orbits with Epm greater or equal to
0.11 and M(a) near 3.96 or 3.97. Schubart(1990) pointed out that in this
apparent gap Hilda-type motion with an approximate 2/1 or 3/1 ratio of two
of the basic long periods is possible, and he found fictitious orbits that
represent such cases, now called secondary resonances. For the theoretical
importance of secondary resonances see, e.g., the review "Chaos in the Solar
System" by Lecar,Franklin et al.: Annual Review Astron. Astrophys. 39 (2001).
Now the large set of numbered asteroids allows a search among the fainter
objects for orbits with Epm less than 0.09 and libration of the transformed
critical argument. The following table shows a recent result from the orbits
up to about (90000). Orbits with larger values of Epm are listed in an
Appendix together with these values. Epm , M(a) , and Ip
are introduced above. Values of Ip of the AstDys file mentioned in the
Appendix appear in some lines of the table. Ip is given in degrees. A round
value of Ip indicates irregular changes of inclination.
H, the brightness parameter of the lists of orbital
elements, indicates that Normannia is the largest body in this list. In the
set of recent additions the first 3 lines show further cases of similarity
to Normannia, but 15 other orbits are well distributed over the interval
of the gap conjectured in 1990. Even now the number of orbits corresponding
to this interval is comparatively small. Some of these orbits correspond to
the 2/1 or 3/1 cases of secondary resonance mentioned before and agree in
type to fictitious orbits studied earlier, see the respective
subsection 2a. Some details
follow in notes to the table.
Low-eccentricity Hildas librating in the transformed critical argument
Asteroid Epm M(a) Ip H Name Remarks
Status of 1991: (1256) 0.024 3.90 5.2 9.7 Normannia
(4196) .025 3.90 1.6 10.7 Shuya
Additions in
2002 - 2004: (8721) 0.022 3.89 6.7 11.2 AMOS
(78867) .024 3.89 1.6 13.9
(29574) .025 3.90 4.2 12.5
(7394) .034 3.92 8.4 11.1 Xanthomalitia
(17397) .040 3.93 2.6 13.4
(51865) .040 3.93 2.3 13.2
(32542) .051 3.94 1.3 12.9
(41351) .059 3.94 5.0 12.9 cf subsection 2a
(15417) .069 3.95 2.2 11.7 Babylon
(10889) .074 3.95 3.9 11.5
(90704) .074 3.95 0.6 13.3
(83900) .076 3.95 3.1 14.1
(47941) .080 3.95 3.2 13.2 cf subsection 2a
(55347) .082 3.95 9 14.5 cf subsection 2a
(15626) .083 3.95 1.1 12.1 cf subsection 2a
(11750) .087 3.95 3.7 12.1
(89903) .088 3.95 4.8 14.5
(62959) .089 3.96 4.4 14.7
Notes to the table
The table does not show related objects with circulation of the transformed
critical argument, like (334) Chicago or (9121) Stefanovalentini, that show
small values of M(a) similar to (8721) AMOS . The recent values of Epm and
M(a) are estimates from graphs derived from integrations over at least 8700 yr.
The attraction of Jupiter and Saturn is considered, in analogy to the method
of Schubart(1991). The starting elements correspond to editions of the database
'mpcorb.dat' of dates up to 2004 Sept. 30, prepared by Minor Planet Center,
or to M.P.C.52960 of 2004 Oct. 28 in case of (90704).
2a. Four Hildas at Secondary Resonances
The following table briefly describes 4 Hilda-type orbits with dominant
or strong influence of a secondary resonance. The table lists values of the
characteristic parameters, the value of the numerical constant k used
in the transformation, and indicates by symbols the special ratio
of two long periods that are involved in the secondary resonance.
TL : Period of libration of the transformed critical argument, crarg'
TA : Period of circulation of the transformed argument of perihelion, argp'
TP : Period of circulation of (lp' - lpj)
TQ : Period of circulation of 1/3 (2 argp' + lp'-lpj) , to avoid jumps in the
sequence of values of this argument, use changes by +/-120 deg.
The abbreviations for the 3 characteristic parameters are, as above,
Epm, Ip, and Ampl , Ip and Ampl are given in degrees.
Four Hildas at Secondary Resonances studied in 2003/04
Asteroid Epm Ip Ampl k Type of secondary resonance
(41351) 0.059 5.0 58 1.23 TA / TL = 2/1
(15626) .083 1.1 70 1.25 TP / TL = 3/1
(47941) .08 3.2 68 1.25 TQ / TL = 3/1
(55347) .08 9 1.25 TQ / TL = 3/1
Description of the evolution of the four orbits
(41351) with an inclination varying about 5 deg. is captured in a special
type of secondary resonance that is characterized by a 2/1 ratio of the
period of circulation of argp' and of TL. Schubart(1990) discovered this
type by studying fictitious orbits with small values of Epm in 1988. His
orbit A is analogous in the type of motion to (41351), and the former
description applies to the present case as well:
In a polar plot, see Fig.2 of the former paper, e' and argp' are the
polar coordinates. argp' is decreasing, and e' shows about two maxima per
revolution, if effects of short period are neglected. The curve is approximately
an ellipse with a changing direction of the major axis. However,
this direction, or the direction from the origin to one of the two maxima
of e' shows a permanent libration about the direction corresponding to
argp' = 0. In the present case this libration continues without significant
changes for 90000 yr at least, according to a forward computation with
sun, Jupiter, and Saturn as attracting bodies.
On the average, there are two main maxima of e' per revolution of argp'.
Since these two maxima are caused by two cycles of TL, the above ratio results.
In 1988 and the following years the type of orbit A of Schubart(1990)
seemed to be "missing in nature" (Schubart, 1994). However, the evolution
of the orbit of (41351) is similar to that of orbit A, although the
amplitude of the libration of the direction from the origin to one of
the two maxima of e' is larger in the present case.
(15626) moves in an orbit of small inclination that develops under a
strong influence of the resonance given by a 3/1 ratio of the periods TP
and TL. If e' and (lp'-lpj) are the polar coordinates in a polar plot,
e' shows about three main maxima caused by TL per one retrograde
revolution of (lp'-lpj). In a forward computation with sun, Jupiter,
and Saturn, the directions from the origin to the three main maxima
of e' appear to be almost fixed during an interval of 25000 yr, one
of these directions corresponding to (lp'-lpj) = -15 deg.
However, in a continuation of the computation these directions show
a slow direct rotation by about 180 deg. during the following 65000 yr.
A backward computation covering 23000 yr indicates a similar rotation.
Therefore this orbit is analogous in type to Schubart's(1990) orbit C,
but in case of the present orbit there is no visible tendency of a
long-period increase or decrease in the values of e' and inclination.
Note: The eccentricity of Jupiter reaches its secular maximum value
in the first interval of 25000 yr.
The last two orbits are analogous in type to Schubart's(1990) orbit B.
The preceding description applies to these cases in the main points, if
TQ and the corresponding angular argument are inserted. However, there are
specific differences. In case of (47941) the curve with the three main maxima of
e' librates about a mean orientation with a large amplitude, perhaps a temporary
process. In an extended forward integration of the orbit of (55347) a continuing
decrease of the mean inclination appears, and the effects caused by the period
TL in crarg' and e' develop to very small amplitudes, so that the three
characteristic main maxima of e' in the curve become less important with
increasing time and finally disappear between other small effects.
In the Hilda group Schubart(1991) has found subgroups of orbits that nearly
agree in the values of all three characteristic parameters introduced above.
The asteroid (1911) Schubart heads the most convincing subgroup that
now turns out to be populous and represents a family within the Hilda group,
in analogy to the well-known families of non-resonant asteroids. This is
demonstrated by the following table.
Again the abbreviations for the 3 characteristic parameters of Schubart(1982)
are: Epm, Ip, and Ampl , Ip and Ampl are given in degrees.
The subgroup headed by (1911) Schubart.
Asteroid Epm Ip Ampl Note
Status of 1982: (1911) 0.190 2.9 27
P-L 2554 .193 2.8 26 1 , now: (19034)
Additions 1991: (3923) 0.195 2.9 23
(4230) .196 2.9 24
(4255) .196 2.8 26
Additions 2000: (8550) 0.188 2.8 31
(8913) .197 2.9 26
(9829) .192 2.8 24
(12307) .191 2.9 28
(13035) .195 2.9 23
(15545) .195 2.9 26
(15615) .197 3.0 23
1981 EK47 .196 2.9 28 now: (30764)
1999 ND26 .197 2.8 26 now: (18036)
More distant
neighbors : (8376) 0.189 2.8 38
1988 SL2 .194 2.8 35 now: (85162)
Note:
1. More reliable orbital elements have resulted from the identification
P-L 2554 = 1999 RP10, but the corresponding changes in the characteristic
parameters are zero or negligible in this context.
Remarks to 'Additions 2000':
A copy of the Asteroid Orbital Elements Database 'astorb.dat' by Dr.Edward
Bowell, Lowell Observatory, of 2000 Jan.29 allowed the selection of suitable
Hilda asteroids. Preliminary values of the characteristic parameters have
resulted from a procedure that resembles the one of Schubart(1982).
Three additions correspond to 'mpcorb.dat' of 2000 June 19, prepared by
Minor Planet Center.
Five Numbered Hildas with P-L Numbers
Reliable osculating elements of the recently numbered orbits are available
for the five Hildas with P-L numbers studied by Schubart(1982). The new
elements allow a correction of Schubart's values of the three characteristic
parameters determined from one-opposition orbits in 1982.
The corrected parameters, again designated by
Epm, Ip, and Ampl, have resulted from a procedure corresponding to Schubart
(1991). In four of the five cases the corrections are significant. According
to these results, a reliable determination of Epm and Ampl cannot result from
a one-opposition orbit of a Hilda asteroid.
The following values of the three characteristic parameters depend on
osculating elements taken from the edition of the database 'mpcorb.dat'
of 2002 April 13, prepared by Minor Planet Center. Ip and Ampl
are given in degrees:
Asteroid Epm Ip Ampl Note
(26761) Stromboli = 2033 P-L .179 4.4 56 1
(19034) Santorini = 2554 P-L .192 2.8 26 2
(39382) Opportunity = 2696 P-L .166 1.9 60 1
(37452) Spirit = 4282 P-L .160 8.5 39 1
(17305) Caniff = 4652 P-L .171 7.3 64 1
Notes:
1. The corrections of the former values of Epm and Ampl are significant.
2. 2554 P-L: The former determination is confirmed in this case.
Scanned copies of the following papers are available by the Web pages
of Astronomisches Rechen-Institut Heidelberg:
https://wwwadd.zah.uni-heidelberg.de/publikationen/authors/index.htm
Schubart J., 1982, Three characteristic parameters of orbits of Hilda-type
asteroids. Astron. Astrophys. 114, 200-204
Schubart J., 1990, The low-eccentricity gap at the Hilda group of asteroids.
Asteroids, Comets, Meteors III, eds. C.-I. Lagerkvist, H. Rickman,
B.A. Lindblad, M. Lindgren, Uppsala University, pp. 171-174
Schubart J., 1991, Additional results on orbits of Hilda-type asteroids.
Astron. Astrophys. 241, 297-302
A special paper on details lists further references:
Schubart J., 1994, Orbits of real and fictitious asteroids studied by numerical
integration. Astron. Astrophys. Suppl. Ser. 104, 391-399
Author's Request
I retired in 1993. Possibly some recent work on the above topics did not come
to my attention. I shall be grateful for any comments that refer to them.
The following lists show the values of the characteristic parameters Epm
and Ampl of 347 Hilda asteroids. k, the numerical constant introduced in
section 1, appears between the two parameters.
Reliable values of proper inclination of most Hildas are available in the
list of synthetic proper elements computed by Z. Knezevic and A.Milani,
and presented in the
files of AstDys.
All the values given here refer to Hilda-type orbits with numbers less than
85000 and with values of Epm greater than 0.090 . Cf section 2 for orbits
with smaller values of Epm. The values of the list have resulted from a
computer program that simulates Schubart's former procedures of derivation.
The program reads files of orbital elements of an extended interval and
is able to derive all the values of Epm and k, and most values of Ampl.
Some of the values of k were adjusted to come close to an empirical relation
between Epm and k. Ampl results from a generalization of Schubart's (1982)
procedure, but a special treatment by digital filtering is necessary to
separate the effects of the period of libration from other effects of similar
amplitude, if Ampl is small. (39266) with Ampl = 1 deg is an example.
Some of the values of Ampl are integer numbers. This indicates the presence
of small, slowly acting effects noticed by the program during the process of
derivation of Ampl. See the remarks at the end of the appendix for details
about this.
The values of the following two lists depend on elements taken from editions
of the database 'mpcorb.dat' of April 2002 and May 2004, prepared by Minor
Planet Center.
No. Asteroid Epm k Ampl
1 ( 153) 0.1719 1.36 18.7
2 ( 190) 0.1693 1.38 40.4
3 ( 361) 0.2044 1.48 46.1
4 ( 499) 0.2002 1.44 61
5 ( 748) 0.1682 1.36 43.9
6 ( 958) 0.1711 1.38 47.5
7 ( 1038) 0.1618 1.42 56.8
8 ( 1162) 0.1401 1.34 50.7
9 ( 1180) 0.1665 1.35 39.3
10 ( 1202) 0.1243 1.31 65.0
11 ( 1212) 0.2300 1.47 25.3
12 ( 1268) 0.1334 1.32 48.5
13 ( 1269) 0.1245 1.32 88.8
14 ( 1345) 0.2022 1.44 28.9
15 ( 1439) 0.1749 1.39 48.4
16 ( 1512) 0.1933 1.41 47.1
17 ( 1529) 0.1518 1.37 67.1
18 ( 1578) 0.2015 1.44 60
19 ( 1746) 0.1388 1.32 23.5
20 ( 1748) 0.1767 1.39 64.3
21 ( 1754) 0.1912 1.44 48.8
22 ( 1877) 0.2039 1.44 36.6
23 ( 1902) 0.1882 1.40 11.8
24 ( 1911) 0.1894 1.39 26.9
25 ( 1941) 0.2162 1.45 49.6
26 ( 2067) 0.1750 1.38 33.6
27 ( 2246) 0.1506 1.34 37.2
28 ( 2312) 0.1099 1.30 36.6
29 ( 2483) 0.2463 1.55 23.0
30 ( 2624) 0.1098 1.28 31.3
31 ( 2760) 0.1771 1.37 51.5
32 ( 2959) 0.2121 1.47 41.1
33 ( 3134) 0.1821 1.42 33.4
34 ( 3202) 0.1263 1.35 61.4
35 ( 3254) 0.1103 1.30 43.0
36 ( 3290) 0.1956 1.40 31.6
37 ( 3415) 0.1856 1.37 5.0
38 ( 3514) 0.1255 1.34 52.0
39 ( 3557) 0.1724 1.37 61.0
40 ( 3561) 0.1302 1.32 18.4
41 ( 3571) 0.1249 1.31 44.4
42 ( 3577) 0.1929 1.43 48.9
43 ( 3655) 0.1553 1.37 75.8
44 ( 3694) 0.1317 1.32 58.9
45 ( 3843) 0.1159 1.30 72.1
46 ( 3923) 0.1949 1.40 22.8
47 ( 3990) 0.1667 1.37 36.1
48 ( 4230) 0.1956 1.39 23.7
49 ( 4255) 0.1961 1.40 25.7
50 ( 4317) 0.2128 1.46 38.3
51 ( 4446) 0.2703 1.60 29.3
52 ( 4495) 0.1228 1.35 89.8
53 ( 4757) 0.1363 1.30 27.2
54 ( 5368) 0.1310 1.31 28.8
55 ( 5439) 0.1234 1.33 74.6
56 ( 5603) 0.1292 1.31 58.7
57 ( 5661) 0.1960 1.44 39.2
58 ( 5711) 0.1392 1.33 32.9
59 ( 5928) 0.1586 1.36 19.0
60 ( 6124) 0.1904 1.43 35.0
61 ( 6237) 0.1108 1.29 58.6
62 ( 6984) 0.1992 1.44 22.7
63 ( 7027) 0.1873 1.41 34.0
64 ( 7174) 0.2047 1.43 50.1
65 ( 7284) 0.1657 1.36 55.3
66 ( 8086) 0.1797 1.38 37.8
67 ( 8130) 0.2142 1.45 50.3
68 ( 8376) 0.1884 1.40 38.7
69 ( 8550) 0.1887 1.40 30.9
70 ( 8551) 0.1484 1.36 42.5
71 ( 8743) 0.1702 1.38 48.9
72 ( 8913) 0.1962 1.39 26.3
73 ( 8915) 0.0969 1.28 38.1
74 ( 9661) 0.1743 1.40 39.8
75 ( 9829) 0.1906 1.39 23.9
76 (10063) 0.1119 1.29 45.7
77 (10296) 0.2199 1.46 47.5
78 (10331) 0.1837 1.39 40.5
79 (10608) 0.2167 1.44 46.5
80 (10610) 0.1524 1.33 20.9
81 (10632) 0.1364 1.36 60.3
82 (11175) 0.1372 1.33 9.4
83 (11249) 0.1907 1.43 34.8
84 (11274) 0.1749 1.38 73.4
85 (11388) 0.1550 1.36 55.6
86 (11410) 0.1455 1.35 36.9
87 (11542) 0.2003 1.43 44.4
88 (11739) 0.1758 1.48 57.1
89 (11951) 0.1575 1.35 41.9
90 (12006) 0.1400 1.37 60.9
91 (12307) 0.1919 1.40 29.4
92 (12896) 0.2163 1.47 41.7
93 (12920) 0.1942 1.43 49.5
94 (13035) 0.1942 1.39 23.5
95 (13317) 0.1514 1.33 28.0
96 (13381) 0.1953 1.40 32.6
97 (13504) 0.2165 1.50 40.7
98 (13897) 0.1355 1.36 65.2
99 (14195) 0.1866 1.40 29.0
100 (14569) 0.2188 1.48 39.9
101 (14669) 0.1339 1.32 43.1
102 (14845) 0.1918 1.40 20.3
103 (15068) 0.2068 1.44 50.0
104 (15231) 0.2136 1.45 38.9
105 (15278) 0.1746 1.41 56.6
106 (15373) 0.1196 1.30 8.3
(15376) see below
107 (15426) 0.1057 1.31 54.6
108 (15505) 0.1774 1.40 62.2
109 (15540) 0.1875 1.40 34.9
110 (15545) 0.1950 1.39 27.0
111 (15615) 0.1971 1.39 23.3
112 (15638) 0.1434 1.34 48.3
113 (15671) 0.1683 1.40 65.9
114 (15783) 0.2014 1.42 48.9
115 (16232) 0.1626 1.37 35.0
116 (16843) 0.1794 1.37 31.6
117 (16915) 0.1925 1.41 45.9
118 (16927) 0.1448 1.35 49.9
119 (16970) 0.1380 1.34 28.7
120 (17212) 0.2043 1.43 23.5
121 (17305) 0.1710 1.41 64.0
122 (17428) 0.1657 1.37 44.9
123 (17867) 0.1357 1.31 29.6
124 (18036) 0.1961 1.40 25.7
125 (19034) 0.1922 1.39 25.5
(19752) see below
126 (20038) 0.1993 1.46 31.8
127 (20628) 0.1928 1.45 59.7
128 (20630) 0.2302 1.49 45.4
129 (20640) 0.1667 1.36 11.4
130 (21047) 0.1358 1.31 39.1
131 (21128) 0.1696 1.39 58.0
132 (21804) 0.1928 1.42 60
133 (21930) 0.1823 1.37 28.5
134 (22058) 0.1611 1.41 70.2
135 (22070) 0.2168 1.48 16.0
136 (22647) 0.1806 1.38 30.8
137 (22699) 0.1795 1.37 27.5
138 (23174) 0.1833 1.42 55.6
139 (23186) 0.2110 1.47 52.8
(23301) see below
140 (23405) 0.1200 1.34 62
141 (24701) 0.2028 1.47 47.5
142 (25800) 0.1993 1.38 35.9
143 (25869) 0.2089 1.49 26.0
144 (26761) 0.1787 1.36 55.9
145 (26929) 0.2205 1.44 46.7
146 (27561) 0.1452 1.35 35.0
147 (28918) 0.1784 1.38 6.1
148 (29053) 0.0944 1.28 48.9
149 (29433) 0.1033 1.29 35.5
150 (29591) 0.1225 1.29 26.6
151 (29944) 0.1884 1.41 23.7
152 (29973) 0.1808 1.37 27.9
153 (30435) 0.2259 1.46 48.5
154 (30764) 0.1953 1.39 29.5
155 (31020) 0.1943 1.40 38.3
156 (31097) 0.1660 1.41 40.9
157 (31284) 0.1862 1.40 15.8
158 (31338) 0.1289 1.32 33.2
159 (31817) 0.1613 1.36 36.3
160 (32395) 0.2073 1.43 38.7
161 (32455) 0.2007 1.41 18.2
(32460) see below
162 (32724) 0.1626 1.38 33.3
163 (33753) 0.1643 1.33 37.5
164 (34919) 0.1855 1.45 29.5
165 (35016) 0.1695 1.36 21.2
166 (35630) 0.1940 1.41 19.8
167 (36182) 0.1463 1.34 53.0
168 (36274) 0.1755 1.41 45.6
169 (36941) 0.0975 1.27 37.0
170 (37155) 0.1189 1.33 40.9
171 (37452) 0.1597 1.35 38.9
172 (37578) 0.1974 1.42 63.4
173 (37590) 0.1933 1.40 37.3
174 (38046) 0.1807 1.37 29.0
175 (38292) 0.2074 1.44 56.9
176 (38470) 0.1456 1.36 42.5
177 (38553) 0.1737 1.40 57.5
178 (38579) 0.1403 1.34 47.4
179 (38613) 0.1360 1.35 12
180 (38684) 0.1877 1.39 23.8
181 (38701) 0.1909 1.39 26.4
182 (38709) 0.2491 1.54 29.9
183 (38830) 0.1747 1.40 47.0
184 (39266) 0.2475 1.52 1.2
185 (39282) 0.2014 1.41 14.3
186 (39294) 0.1229 1.32 62.8
187 (39301) 0.1757 1.39 52.5
188 (39382) 0.1667 1.34 61
189 (39405) 0.1976 1.39 26.1
190 (39415) 0.2058 1.45 56.3
191 (39427) 0.1749 1.37 34.7
Continuation of June 2004
No. Asteroid Epm k Ampl
1 (40227) 0.2015 1.43 26.7
2 (40238) 0.1818 1.41 75.3
3 (40246) 0.2046 1.42 40.8
4 (41278) 0.1941 1.41 24.9
5 (41283) 0.1552 1.36 36.1
6 (41365) 0.1621 1.39 46.0
7 (41419) 0.1025 1.28 42.3
8 (41488) 0.1523 1.34 33.6
9 (42167) 0.1989 1.46 63
10 (42190) 0.1456 1.35 27.5
11 (42237) 0.1245 1.35 72.6
12 (43818) 0.2060 1.45 65.0
13 (43940) 0.1603 1.34 27.2
14 (44549) 0.1234 1.30 46.1
(45739) see below
15 (45850) 0.1166 1.35 50.3
16 (45862) 0.1111 1.29 39.5
17 (46302) 0.2179 1.46 42.5
18 (46629) 0.1816 1.36 31.5
19 (47907) 0.2243 1.45 14.7
20 (48342) 0.1173 1.31 75.3
21 (48529) 0.1971 1.44 42.5
22 (48881) 0.1202 1.34 68.4
23 (51178) 0.1708 1.36 16.3
24 (51284) 0.2255 1.46 49.9
25 (51298) 0.1623 1.35 42.2
26 (51349) 0.2078 1.45 38.7
27 (51838) 0.2290 1.46 42.2
28 (51874) 0.2191 1.49 38.9
29 (51885) 0.1930 1.42 51.9
30 (51888) 0.1843 1.36 28.3
31 (51930) 0.1755 1.38 20.0
32 (51983) 0.1164 1.31 3.7
33 (52016) 0.2084 1.42 40.3
34 (52068) 0.2066 1.45 31.3
35 (52079) 0.1883 1.43 61
36 (52702) 0.1886 1.41 48.2
37 (54514) 0.2243 1.55 58.6
38 (54599) 0.0991 1.27 43.2
39 (54628) 0.2028 1.43 54.6
40 (54630) 0.1489 1.34 26.3
41 (54631) 0.1184 1.33 62.4
42 (54644) 0.1251 1.33 46.7
43 (54657) 0.1535 1.33 25.0
44 (55196) 0.1895 1.42 60.8
45 (55439) 0.1781 1.37 26.9
46 (55498) 0.2122 1.42 26.5
47 (55505) 0.1919 1.38 33.0
48 (56982) 0.1889 1.42 34.2
49 (56985) 0.1990 1.39 28.2
50 (56996) 0.1764 1.37 38.9
51 (57027) 0.1121 1.29 19.9
52 (57759) 0.2003 1.39 26.5
53 (58095) 0.1415 1.36 68.8
54 (58188) 0.0950 1.28 41.4
55 (58279) 0.1773 1.36 44.4
56 (58353) 0.1445 1.35 39.8
57 (59050) 0.1877 1.39 33.1
58 (59079) 0.1572 1.36 48.0
59 (59112) 0.1474 1.34 46.9
60 (60232) 0.1519 1.35 36.6
61 (60318) 0.1523 1.36 56.5
62 (60381) 0.1365 1.34 34.1
63 (60398) 0.2059 1.43 50.0
64 (61042) 0.2062 1.44 30.9
65 (62145) 0.1475 1.35 19.8
66 (62241) 0.1871 1.41 33.7
67 (62244) 0.1922 1.42 59.8
68 (62408) 0.1980 1.41 25.9
69 (62489) 0.1990 1.39 24.3
70 (62820) 0.1620 1.36 45.8
71 (63184) 0.1160 1.31 60.4
72 (63249) 0.1349 1.39 69.4
73 (63293) 0.1702 1.34 42.0
74 (63488) 0.1072 1.29 25.1
75 (63491) 0.1964 1.39 37.5
76 (64390) 0.2394 1.51 46.9
77 (64739) 0.2079 1.46 59.3
78 (64823) 0.2153 1.45 47.8
79 (65236) 0.1707 1.39 49.1
80 (65244) 0.1992 1.41 54.1
81 (65374) 0.2071 1.41 13.7
82 (65389) 0.1934 1.38 25.1
83 (65821) 0.1860 1.42 64
84 (65859) 0.2133 1.43 11.0
85 (65989) 0.2267 1.49 34.6
86 (66187) 0.1980 1.45 59.7
87 (66227) 0.2159 1.43 18.6
88 (67203) 0.1833 1.37 63.3
89 (67246) 0.1626 1.43 53.3
90 (67340) 0.1621 1.36 31.5
91 (67368) 0.2083 1.42 13.8
92 (68247) 0.1928 1.40 25.0
93 (68287) 0.2256 1.50 34.0
94 (68374) 0.1383 1.35 53.3
95 (68402) 0.1271 1.32 45.6
96 (68933) 0.1626 1.34 70.6
97 (69302) 0.1938 1.40 32.6
98 (69417) 0.1555 1.36 65.8
99 (69566) 0.1989 1.41 24.5
100 (73418) 0.2215 1.54 38.9
101 (73436) 0.1332 1.33 45.8
102 (73455) 0.1692 1.39 53.3
103 (73457) 0.1970 1.44 44.1
104 (73458) 0.1297 1.32 33.0
105 (73475) 0.1783 1.36 49.6
106 (73654) 0.2182 1.47 49.0
107 (73769) 0.1999 1.40 27.4
108 (73886) 0.1858 1.41 23.5
109 (74051) 0.1521 1.34 67.9
110 (74054) 0.2161 1.46 53.1
111 (76750) 0.2079 1.41 54.5
112 (76805) 0.1347 1.31 7.5
113 (76810) 0.1289 1.32 18.3
114 (76811) 0.1407 1.34 8.8
115 (76822) 0.2011 1.39 32.6
116 (76831) 0.2389 1.53 46.8
117 (77734) 0.1251 1.31 27.2
118 (77820) 0.1899 1.40 23.2
119 (77884) 0.1527 1.35 29.2
120 (77892) 0.2194 1.47 38.1
121 (77893) 0.1295 1.31 24.8
122 (77895) 0.1954 1.41 65.3
123 (77903) 0.2183 1.46 46.4
124 (77905) 0.1721 1.38 36.2
125 (77910) 0.1980 1.41 46.9
126 (78133) 0.2262 1.48 35.8
127 (78159) 0.1105 1.28 37.8
128 (78470) 0.1998 1.45 40.0
129 (78477) 0.1109 1.33 46.2
(78809) see below
130 (78815) 0.1244 1.33 42.9
131 (79096) 0.1936 1.40 26.8
132 (79097) 0.1888 1.43 66.4
133 (79190) 0.1484 1.33 4.6
134 (79439) 0.2111 1.49 65.2
135 (79515) 0.1764 1.37 9.2
136 (79724) 0.2003 1.41 33.9
137 (82011) 0.1890 1.39 29.3
138 (82023) 0.1029 1.28 37.8
139 (82041) 0.2155 1.46 16.7
140 (82043) 0.1325 1.34 61.9
141 (82044) 0.1900 1.44 54.9
142 (83722) 0.2012 1.45 25.5
143 (83801) 0.2130 1.49 55.9
144 (83804) 0.1914 1.40 32.3
145 (83867) 0.2004 1.45 13.3
146 (83877) 0.2271 1.48 37.9
147 (83903) 0.1476 1.30 17.6
148 (83916) 0.1935 1.39 32.2
149 (84011) 0.2061 1.47 69.8
150 (84103) 0.1145 1.26 54.7
Objects with a large inclination or osculating eccentricity
151 (15376) 0.2340 1.52 40.7
152 (19752) 0.1781 1.41 53.1
153 (23301) 0.1896 1.43 51.0
154 (32460) 0.2599 1.57 28.1
155 (45739) 0.2469 1.54 42.0
156 (78809) 0.2604 1.57 21.5
Notes to the six last objects:
The present osculating eccentricities of four of these objects are greater
than or close to 0.30, but there are other objects like (4446) with
similar or greater values of Epm.
In case of (19752) and (23301) the values of the inclination with respect
to Jupiter's orbit are greater than 20 deg, which is unusual for a Hilda
asteroid.
Further notes and remarks to the appendix:
All the values of Ampl listed above are less than or equal to 90 deg.
These values depend on an interval of about 23000 yr that is covered by
the basic integration, but 1/3 or 1/2 of this interval is sufficient to
approximate Ampl in almost all the cases. The cases of exception show
deviations of 2 or 3 deg in the values derived from such a smaller interval.
In such a case the appendix shows a rounded integer value of Ampl.
(499) Venusia and (1578) Kirkwood are the first representatives among these
cases. According to tests with the two objects, these small deviations appear
to result from slowly acting effects that show up in extended simultaneous
integrations of the four-body problem Sun, Jupiter, Saturn, asteroid. For a study about this
go to the separate continuation
News on HILDA ASTEROIDS, Part 2.
Last modified: 2004 Dec. 23, Links inserted: June 2005