News on HILDA ASTEROIDS

      by  Joachim Schubart, Astron.Rechen-Institut, Heidelberg, Germany
                                  s24@ix.urz.uni-heidelberg.de

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.

2. Low-Eccentricity Hilda Asteroids

(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.

3. The Growing Family of (1911) Schubart

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.

4. Other Recent Results:

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.


5. References

Scanned copies of the following papers are available by the Web pages of Astronomisches Rechen-Institut Heidelberg: www.ari.uni-heidelberg.de/publikationen/ 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. My e-mail address: s24@ix.urz.uni-heidelberg.de


6. Appendix: Numerical values

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