by Joachim Schubart, Astron. Rechen-Institut,
University of Heidelberg, Heidelberg, Germany
During the last 40 years many authors have published important papers on the evolution of orbits of asteroids at the 2/1 Jovian resonance of mean motion, see e.g. the review by Moons (1997) and the textbook by Morbidelli (2002). The problem of explaining the statistical gap in the distribution of the values of mean motion at this resonance has attracted much attention. I have mainly explored the evolution of orbits of Hilda type at the 3/2 resonance during this period. In the more recent time I have started a numerical study of fictitious orbits at the 2/1 resonance. Some of the results appear in the following sections. A comparison with analogous results on the motion of real asteroids at the 3/2 resonance is interesting. Other authors like Michtchenko and Ferraz-Mello (1995/96), Nesvorny' and Ferraz-Mello (1997) have already published comparative studies of motion at these two resonances. More recently Czech astronomers have studied the real population of asteroids at the 2/1, 3/2, and 4/3 resonances, see the paper by Brož and Vokrouhlicky' (2008). Here I can only add some details to the many important results that are available in the literature.
Since I have passed the age of 80 years, I have to be especially careful to avoid any systematical errors. Therefore I have only used methods and computer programs that are well known to me since many years. The results given in Sections 2 - 5 depend on numerical integration by an N-body program. I have used the program by Schubart and Stumpff (1966) for this, and I have retained the mass values recommended in this paper, so that it was easy to repeat former computations for checkout. Precise solutions result for orbits that show a quasi-periodic evolution in the intervals of time that are covered by the integrations. The sun, Jupiter, or Jupiter and Saturn, and a massless asteroid represent a three- or four-body system that is studied in two or three dimensions. I have applied digital filtering to some approximately quasi-periodic solutions using the method and formulas of Schubart and Bien (1984). This method is applicable, if the studied variable is very close to a quasi-periodic function of time in the interval of time that is used by the method.
a = semi-major axis, i = inclination, ln = longitude of node
e = eccentricity, lp = longitude of perihelion, lm = mean longitude
aj, ej, lpj, and lmj are the respective quantities of Jupiter,
Dlp = lp - lpj . * is the multiplication sign, ** indicates exponentiation
In a case of resonance of the mean motions of asteroid and Jupiter the ratio of these values is close to (p+q)/p with integer values p and q. p=q=1 refers to the 2/1 case. The slowly varying critical argument of a resonance with q=1 is given by crarg = (p+1) * lmj - p * lm - lp . This argument can show libration about a mean value in typical cases of resonance. TL designates the long period of such a process of libration of crarg. In the 2/1 case crarg depends on -lm with a factor one. This points to the importance of this case of resonance. If the argument Dlp shows revolutions, TP designates the period of such a revolution. TP is a long period as well.
In Sections 2 and 3 I report on results about secondary resonances. These results depend on the elliptic restricted problem sun, Jupiter, asteroid. Secondary resonances can appear in a case of resonance of mean motion, if the ratio of two long periods like TP and TL is close to a rational number. At first Lemaitre and Henrard (1990) have studied the most important secondary resonances in case of a 2/1 resonance of the orbits of the planar elliptic restricted problem. Michtchenko and Ferraz-Mello (1995/96) have listed other types of such resonances that appear in the three-dimensional restricted problem. In Section 2 I report on secondary resonances that refer to the ratios 3/2, 5/3, and 7/4 of the mean of the period TP and of the mean of the period TL within the 2/1 resonance of mean motion. Dealing with the 2/1 secondary resonance of these two long periods, I have found orbits that show rational ratios of long periods within this secondary resonance. This indicates the existence of "tertiary resonances" in case of the 2/1 resonance of mean motion.
I have used starting values of (3789) Zhongguo and of two asteroids on eccentric orbits in a study on typical cases in the range of natural orbits at the 2/1 resonance. Since this study depends on a sun, Jupiter, Saturn model of the forces, the long-period evolution of the basic real orbits will be different. My results appear in Section 4. I report on mean values and amplitudes of variation of orbital parameters. In particular the influence of the periods TL and TP and of the comparatively long period of variation of ej is demonstrated. The (3789)-type orbit deviates from a quasi-periodic evolution. Evidence for this results from integrations that cover more than 100 millennia. I have also studied the long-period evolution of this orbit using fictitious orbits of the major planets, see Section 5. I have varied the starting values of the eccentricities of Jupiter and Saturn, or only of Jupiter neglecting the mass of Saturn.
Giffen (1973), following a proposal by Schubart (1968), has used simplified differential equations in a pioneering study on resonance cases of the planar sun, Jupiter, asteroid problem. The simplification consists in an averaging process that removes the only fast variable from the equations. The computer program used by Giffen (1973) is still available here. I have applied it to extend some of Giffen's results on the 2/1 resonance, especially on the change from quasi-periodic to chaotic types of evolution, if the values of eccentricity decrease. In case of some eccentric orbits I have studied the isolated effects by the periods TP and 1/2 TP. I present my results in Section 6.
In my former studies on the 3/2 resonance case (Schubart, 1991), I was able to remove an important part of the effects by the eccentricities of Jupiter and Saturn from e, lp, and crarg by a transformation to new variables e', lp', and crarg'. Its application allows to add some low-eccentricity objects like (1256) Normannia to the group of Hilda asteroids that show libration of crarg' at the 3/2 resonance. The 2/1 case is more complicated, so that an analogous attempt to remove these effects turns out to be more incomplete. However, here this transformation is necessary for the study of special low-eccentricity orbits (see the Appendix for an example). Furthermore it is useful for the simplification of curves demonstrated in graphs or for the derivation of characteristic mean values. Therefore I repeat the formulas here:
e' cos(lp' - lpj) = e cos(lp - lpj) - k * ej
e' sin(lp' - lpj) = e sin(lp - lpj)
crarg' = crarg + lp - lp'
k is a numerical constant that is suitably chosen for each of the graphs
that refer to a specific orbit. Let Dlp' be equal to lp' - lpj . Then:
e' cos Dlp' = e cos Dlp - k * ej
e' sin Dlp' = e sin Dlp
In an earlier paper, Schubart (1993), I have studied five fictitious low-eccentricity orbits that are designated by capital letters from A to E. The increasing sequence of the mean values of a of these orbits approaches the sunward border of the range of semi-major axis that is ruled by strong effects of secondary resonances within the 2/1 resonance. The starting values of e of these orbits are fitted to obtain a small mean amplitude of libration of crarg. Therefore the relation between the mean values of a and e of these orbits is similar to that of the "pericentric branch" of the 2/1 resonance, a branch of orbits that do not show effects of libration in a simplified model of the forces.
If demonstrated in an e versus a diagram, this branch turns off from the center of the resonance towards the sunward side in the low-eccentricity range as shown by Fig. 1 of Nesvorny' and Ferraz-Mello (1997). By analogous diagrams Figs. 3 and 7 of that paper demonstrate the range of strong effects by chaotic diffusion. This range is described by the starting values of a and e of the affected orbits. These orbits start with i = 0, but Jupiter and Saturn move on their real orbits. The pericentric branch enters this range at a border where secondary resonances are expected to be active. Here I am interested in the type of these resonances.
In my earlier paper the sequence of orbits A - E shows an increase of the ratio TP/TL from 1.1 to 1.44. I have extended the sequence to larger starting values of a, but then the evolution of the orbits changes from a quasi-periodic type to a wild chaotic evolution. According to these earlier results the border between the two types of evolution approximately corresponds to TP/TL = 3/2. The former studies, Schubart (1993), depend on a simplified Jupiter-Saturn model of the forces and demonstrate effects by the secular variation of ej. Now I use the elliptic restricted problem sun, Jupiter, asteroid for a study on the activity of secondary resonances with ratios TP/TL equal to or greater than 1.50 . I retain the starting values aj = 5.203345 au and ej = 0.047711 that I have used for orbits A - E, see Schubart (1994). The study of a special case refers to the circular restricted problem. I fit the starting values of the asteroid to obtain interesting examples of planar or three-dimensional motion. In several cases my examples show an additional type of libration with respect to a secondary resonance. TP refers to revolutions of Dlp or Dlp', but the period of revolution of another angular argument can give rise to a secondary resonance as well.
Let xi be equal to Dlp' or to another argument. Fig. 1 of Schubart (2009) refers to a Hilda-type orbit, but demonstrates a useful way to show the influence of a secondary resonance: e' sin xi is plotted versus e' cos xi , so that a polar graph for e' and xi appears. One of the periods of interest is given by the revolution of xi, the other one, TL, causes the main maxima and minima of the polar radius, e', unless the amplitude of libration of crarg is very small. If the ratio of the two periods is a rational number, the maxima of e' reappear at the same directions xi after one, or after several revolutions of xi. If the ratio is only close to such a number, each of these directions can vary and possibly librate about a mean direction. Such a libration is an additional process that appears together with the libration of crarg. Fig. 1 of Schubart (2009) contains a demonstration of such a process: An approximately circular curve shows outward extensions due to the variation of the polar radius. In that case the extensions reappear at nearly the same directions after only one revolution along the curve. In this way the figure of this curve rotates forward and backward, but does not leave the vicinity of a mean orientation.
I have mentioned the extension of orbits A - E to orbits with larger
starting values of a. These orbits lead to the range of interest, but now
I consider only orbits with vanishing or small inclination, i, with respect
to Jupiter's orbit. At first I put i = 0 and report on orbits with TP/TL
close to a rational number. Here are my results:
There are orbits that show additional libration according to the secondary
resonances given by TP/TL equal to 3/2, 5/3, and 7/4 . The last ratio shows
that 7 TL is nearly equal to 4 TP. Then 4 revolutions along the curve in a
polar diagram are necessary to demonstrate the approximate reappearance of
the directions that correspond to extremes of the polar radius. This will
cause difficulties for orbits with a small amplitude of libration of crarg,
but this amplitude equals about 30 deg for my examples. To center the curves
with respect to the origin, I use the small value k = 0.2 in the transformation
to e' and Dlp'. In case of all the three rational ratios several cycles of
an additional libration with small amplitude are covered by the integration.
If all the points are plotted, a broadened curve represents the mean
orientation. For TP/TL = 3/2, 5/3, 7/4 the observed periods of the additional
libration equal 35, 30, and 20 millennia and approximate mean values of e'
are 0.062, 0.068, 0.072, respectively.
Now I put i = 5 deg and vary the other starting values of the orbit that corresponds to TP/TL = 3/2. I study the variations of curves in a polar graph of e' and xi = 1/3 (Dlp' + 2 lp' - 2 ln) or of e' and 3 xi . ln designates the longitude of the ascending node. Small changes of the starting values of a and e of the case TP/TL = 3/2 are sufficient to show that additional libration can occur about a secondary resonance given by a 3/2 ratio of the period of revolution of xi and of TL. Since this case appears in the vicinity of the other one, and since ej varies to larger values in nature, I do not expect real orbits with small values of e and i and with TP/TL at 1.50 . I have searched for real orbits in a suitable range of TP/TL, but the values do not exceed 1.47 . The Appendix lists results about seven orbits found in this search. In case of Hilda-type orbits arguments xi depending on lp' - ln lead to dominant effects by secondary resonances as mentioned by Schubart (2009), but the preceding results point to the importance of rational ratios of TP/TL in the range of interest.
There are secondary resonances arising from a ratio 4/3 as well, but the
related processes of additional libration are very slow and indicate a weak
influence of these resonances. An orbit with TP/TL at 4/3 needs about 150
millenia to complete a cycle of such a libration. It is possible to study
a related case with xi = lp - ln and with a starting value i = 5 deg on the
basis of the circular restricted problem, but in that case about 400 millennia
are necessary to complete a cycle of additional libration. Lemaitre and
Henrard (1990) have mentioned the secondary resonances with TP/TL close to
3/2 or 4/3, and that these are comparatively weak in comparison with those at
2/1, 3/1, and so on. However I think that the 3/2 case is important for real
orbits in its vicinity.
My results about secondary resonances of type 2/1 and 3/1 appear in the next
section.
In Section 2 I have described processes of additional libration that can appear together with a secondary resonance given by a rational ratio of the periods TP and TL. I have mentioned a way to demonstrate such a process by a graph. Here I call the period corresponding to one cycle of this type of additional libration TC. Large values of TC have resulted for the examples of the preceding section, but comparatively small values appear in studies of orbits with TP = 2 TL. Now I use again the elliptic restricted problem and the starting values of Jupiter's orbit given in the preceding section. I vary the starting values of a from 3.244 to 3.266 au and those of e from 0.07 to 0.14 to explore the case TP = 2 TL. The small body moves in the orbital plane of Jupiter. Some examples show the type of additional libration mentioned above. In one case TC equals 3 TP = 6 TL. Therefore the succeeding cycles of additional libration differ only by effects of short period from the initial cycles. According to a proposal by Lemaitre and Henrard (1990) I use the designation tertiary resonance for cases within a secondary resonance that show a rational ratio of TC and TP.
The orbit with TC/TP = 3/1 starts with a = 3.244 au, e = 0.136 , crarg = 0.012 deg, lmj = 248.271 deg, lp = lpj = 0 . The corresponding curve in a polar plot for e and lp closes after three revolutions of lp, or after one cycle of TC, if effects of short period are neglected. The following cycles do not deviate from an evolution along a mean curve that corresponds to the beginning. The amplitude of the libration of crarg varies between 20 and 45 deg during a cycle of TC.
Another orbit shows an additional libration with respect to a secondary resonance of type TP/TL = 3/1 . This orbit starts with a = 3.25 au, e = 0.1404, Dlp = 165 deg, crarg = 0.012 deg, and one cycle of TC corresponds to about 3.5 revolutions of lp. This example indicates the possible existence of a tertiary resonance with TC/TP = 7/2 within a 3/1 secondary resonance.
I have selected starting values of the real asteroids (3789) Zhongguo, (28459), and (78801) for a study of three orbits that approximate the motion of the real objects in the near future. A file of orbital elements prepared by the Minor Planet Center in 2004 is the source. The starting epoch is the standard date 2004 July 14.0 . Jupiter and Saturn are the attracting planets, their orbital elements are available from an almanac. The elements are transformed to the initial orbital plane of Jupiter as the plane of reference. The step length of the numerical integrations equals 4.44 days. Forward and backward integrations use a sufficiently large number of decimal digits and can proceed for more than 10**7 steps. Computations that cover 10 or 20 millennia are useful for a qualitative study of effects by the periods TL and TP. However the difference of the two main frequencies of the secular variations of ej and lpj corresponds to a period of about 54 millennia, here called Tsc. This period is present in the basic sun-Jupiter-Saturn problem and gives rise to strong variations of the eccentricities of the major planets. ej varies between 0.03 and 0.06. Effects by the period TP depend on the amount of ej, since Dlp describes a revolution with respect to lpj and the effects by lpj vanish together with ej. Therefore it is interesting to study the influence of Tsc on the evolution of the three orbits now called 3789, 28459, 78801.
The results of the numerical integrations are stored at equally spaced intervals. In Table 1 I collect some characteristic mean values that refer to the initial parts of the integrations. The values of i given there are equal to the synthetic proper inclinations derived by Kneževic' and Milani (2000) for the respective real asteroids.
Table 1. Characteristic values
Orbit 3789 28459 78801
a (AU) 3.272 3.276 3.275
e' 0.189 0.301 0.350
i (deg) 1.8 1.5 7.3
Ampl (deg) 63 95 30
Alp (deg) 6 4 24
TL (yr) 425 373 423
TP (yr) 3100 7500 9100
k 0.10 0.51 -0.95
Notes to Table 1. Ampl is a mean value of the amplitude of libration of
crarg. Alp equals a mean value of the amplitude of a variation that is
superimposed on the linear regression of Dlp. This variation follows the
period TP. TL causes a comparatively small variation in Dlp. Mean values
of a, e', TL, and TP are listed as well. See the text for i and k.
In Table 1
k is the constant used in the transformation from e, Dlp to e', Dlp'. If
e sin Dlp is plotted
versus e cos Dlp, a polar diagram of e and Dlp results.
Then a suitable choice of k leads to a centered polar curve of e' and Dlp'.
However, if the resulting polar curve of e' and Dlp' of orbit 78801 is studied
in rotating polar coordinates that counterbalance the mean motion of Dlp',
the new polar angle shows a large amplitude of a variation caused by TP.
This example clearly demonstrates that the transformation to e', lp' does
not remove all of the important effects by the eccentricities of the major
planets. In Section 6 I show that the higher harmonics of the frequency
of TP are important for studies of curves in case of the 2/1 resonance.
The period TL causes the libration of crarg and variations of e that are shifted in phase, but also oscillations of the values of the semi-major axis. All the three orbits show remarkable changes of these oscillations and of the length of TL during a cycle of the much longer period TP, if a is plotted versus time. The amplitude of the oscillations of a is largest and TL is short near Dlp = 0. The opposite extremes occur at Dlp = 180 deg. The changes of amplitude of the analogous variations of crarg and e during a cycle of TP are less simple. At the initial part of the integration on 3789 the extreme values of TL are 370 and 500 yr. Strong changes of this kind are typical for the 2/1 resonance.
Let an angular argument revolve according to the period TL. Then it will
show an important oscillation of its angular velocity during a cycle of TP.
Here I use a numerical way to introduce such an argument called beta.
I plot (crarg - corr) versus csc * (a - am) to generate a curve. By corr
I remove an important part of the variations of crarg by TP. am is the mean
value of a. If the scale factor csc is suitably chosen, the two arguments
of the plot oscillate about zero with nearly the same amplitude according to
TL, but the oscillations differ in phase by about 90 deg. Therefore the
sequence of the plotted points revolves around the origin according to TL
and remains in the vicinity of a mean circle. The polar angle beta that gives
the direction from the origin to the points of the generated curve is the
subject of study. beta shows large positive and negative deviations from a
linear progression with time during a cycle of TP. At the beginning of the
computations the deviations rise to about 80, 130, and 280 deg in case of
3789, 28459, and 78801, respectively. I note that at the 3/2 resonance the
analogous deviations do not exceed 10 deg in case of (153) Hilda.
I have introduced Tsc, the period of the secular variations of ej. At present Tsc gives rise to a secular increase of ej. Let T be the time counted in millennia from the starting epoch. Tsc equals 54 in this scale. ej will reach a maximum value of 0.06 at T = 13. The next minimum of 0.03 follows at T = 40. Together with these changes Tsc causes a periodic variation with an amplitude of about 20 deg in lpj. This variation is superimposed on the slow linear progression of lpj and is expected in Dlp according to definition. Indeed Dlp of 78801 shows a corresponding variation with nearly the same amplitude. Analogous variations due to Tsc affect the mean longitudes, lm, with smaller amplitudes. Furthermore the amplitude of a variation of Dlp due to TP described in Table 1 turns out to vary during a cycle of Tsc in a more extended integration. These and analogous effects by Tsc appear, if variations of short period are removed by digital filtering. The maximum values of the deviations of beta from a linear progression mentioned before are approximately proportional to ej and are especially large at T = 13.
I have plotted crarg of 3789 versus time to study the variation of the amplitude of the libration during a cycle of Tsc. The period TP causes changes of amplitude, but the amount of these changes varies according to Tsc. It is large together with ej at T = 13 and comparatively small at T = 40. The changes of the length of TL induced by TP are clearly visible in the plot. They appear to be smaller at T = 40 as well. It is interesting to isolate the influence of the single frequencies that contribute to the libration of crarg. Let nu1 be the main frequency corresponding to TL and nu2 correspond to TP. Then the expected frequencies nu are given by nu = nu1 + nu2 * j , with positive or negative integers j. I isolate by the method of filtering the variations that arise from some of these frequencies. Then equidistant oscillations with a slowly changing amplitude appear in a plot against time in each case. The total changes during a cycle of Tsc are significant. While the amplitude corresponding to nu1 varies with a well defined minimum at T = 13, the studied nearby frequencies show a maximum of amplitude at this time.
I have studied orbit 3789 by more extended integrations, since the large
value of the Lyapunov Characteristic Exponent derived by Kneževic' and
Milani (2000) for the corresponding real asteroid indicates a visible
non-quasi-periodic evolution of this orbit in the comparatively near future.
Indeed, in the interval from T = -200 to 200 there is evidence for such
an evolution. To demonstrate this, I have plotted a simplified function
of lm only at moments of time with values of crarg close to zero. I have
restricted the deviation from zero to 1.5 deg. In this way most of the
oscillation of lm due to TL does not appear.
I have subtracted from lm
a linear function of time to compensate the progression on the average,
and I have added a term proportional to ej sin Dlp to reduce an effect by TP.
If the result is plotted versus time in the above interval, the curve shows
only small effects of scattering. The result varies in an interval of about
300 deg with strong but slow changes of slope which I cannot explain by
effects due to the period Tsc. These changes indicate deviations from a
quasi-periodic type of evolution for the studied orbit, since otherwise
they do not appear. However in such cases the result of a long-period
integration corresponds to the starting values only in a first part. I have
shown this by an example in a former paper, Schubart (1993).
If the variations of lp are studied in the same way, opposite
changes of slope appear, since lp + lm = 2 lmj - crarg and crarg
is close to zero for the plotted points. lmj does not introduce
changes of slope of the observed kind.
I have applied the procedure described at the end of the preceding section to examples of orbits of the attracting planets that do not occur in our present solar system, but I did not change the mass values. The evolution of orbit 3789 is the subject of the studies that refer to planar n-body problems. The starting values of Section 4 are used with the following changes: The inclinations change to zero. The eccentricities of the massive bodies change by the multiplication with a common factor c. Now let both Jupiter and Saturn be attracting. Then the long period Tsc that rules the variation of ej depends on c. The length of Tsc changes from 54 to 66 millennia, if c goes down from 1 to 0.5 . I expect changes by the variation of c in other periods of the major planets as well, but I have not studied these changes. The amount of the variation of ej changes downward together with c.
Now I report on the long-period effects in the progression of lm of 3789 that result in the varied sun - Jupiter - Saturn problems. Again I consider intervals of time (in millennia) from -200 to 200 and plots versus time of the simplified function of lm that I have introduced in Section 4. At first I put c = 1, so that the change with respect to Section 4 is restricted to the starting inclinations. Then strong changes of slope of the curve in the plot again appear, but in a completely different way, if compared with the analogous result of Section 4. However, only quasi-periodic oscillations of small amplitude appear in a plot that refers to c = 0.5 . Since the mean progression of lm is compensated in the plotted function, the values remain in an interval of 10 deg.
In case of c = 0.75 oscillations of small amplitude are again visible, but a small change of the mean slope results from a comparison of the forward and backward integrations: the slope turns upward. If I go forward by changing c from 0.8 to 0.85 and 0.9 , I observe analogous changes of the mean slope. These changes are not much larger, but the oscillations that are superimposed on the mean slope are comparatively large in case of c = 0.9 . Possibly some effect of resonance leads to visible effects. Finally plots corresponding to c = 0.93 and 0.95 show strong changes of the slope in analogy to the case c = 1 .
At the 3/2 resonance most of the Hilda asteroids do not show such strong changes. I have applied the above procedure to three-dimensional starting values of (153) Hilda, Jupiter, and Saturn with values of c as large as 1.25 and 1.40, but that causes only minor changes of slope in the plots of the simplified function of lm.
Returning to orbit 3789, in a special study I have neglected the attraction of Saturn. This study refers to the planar elliptic restricted three-body problem. Now the starting values change by a variation of ej to 0.06 and 0.075. The second value corresponds to a fictitious orbit of Jupiter, since ej does not exceed 0.06 during a cycle of Tsc. However, in both cases plots of the function of lm do not show the strong changes of slope that have appeared in the more general case with c = 1. Apparently effects in the evolution of the orbits of Jupiter and Saturn that depend on c give rise to these strong changes in the general case.
The following results depend on simplified differential equations of the planar elliptic restricted problem sun, Jupiter, asteroid. Differential equations for four variables that are functions of a, e, lp, and crarg are simplified by an averaging process that removes a fast variable from the equations. Schubart (1968) has published the basic formulas. The numerical integration of the simplified equations can proceed with a large step length and the result does not depend on very large frequencies. Giffen (1973) has used this procedure. Here I extend some of his results that refer to the 2/1 resonance. Then I apply the procedure to starting values of special orbits. aj, ej, and lpj = 0 are constant, Dlp = lp.
At first I use the mass value of Jupiter and the values of aj and ej proposed by Giffen (1973). Ten fictitious orbits start with crarg = lp = 0 . Table 2 shows the respective starting values of a and e. Orbit No.1 is equal to Giffen's typical example of a non-quasi-periodic evolution of an orbit, see Fig.21 of his paper. Orbits 2 to 10 show a general decrease of the starting value of e. According to Giffen's results one can expect a nearly quasi-periodic evolution for orbits 2 and 3 at least.
Table 2. Starting values of a and e
No. 1 2 3 4 5 6 7 8 9 10
a 3.254 3.26 3.2 3.26 3.26 3.26 3.26 3.26 3.255 3.255
e 0.14 0.3 0.3 0.275 0.25 0.225 0.2 0.18 0.18 0.16
Let x and y be given by x = e cos lp and y = e sin lp . The result of the integration of the simplified equations leads to a stored sequence of values of e and lp, or of x and y. Giffen has plotted y versus x and drawn the closed curves that result from his study on periodic orbits. Elliptic curves that are not centered but symmetric with respect to the x axis result for his examples of large eccentricity. My study refers to non-periodic orbits, but to plots of the same kind that are polar diagrams of e and lp. By an application of digital filtering to the stored values of x or y of orbits 2 - 5 it turns out that the frequency corresponding to TP is well separated from all the larger frequencies. If the influence of these frequencies is removed from x and y by filtering and the result is plotted as above, a precise ellipse appears that is symmetric with respect to the x axis, but the center does not coincide with the origin. A point on the curve rotates in the retrograde direction according to period TP. A centered ellipse appears, if a band of frequencies around the value corresponding to 1/2 TP is isolated in x and y. If the point on the non-centered ellipse passes the x axis with a maximum value of x, the faster point on the respective centered ellipse does so as well.
I list some results on orbit 2: TL varies about 426 yr, TP equals 8210 yr,
26 deg is a mean value of the amplitude of libration of crarg. The amplitudes
of the isolated effects by TP equal 0.330 in x and 0.314 in y. The center of
the corresponding curve in a polar diagram is displaced by 0.088 from the
origin towards the negative x axis. The effects by 1/2 TP lead to a curve
with a mean radius 0.04. The transformation from e, lp to e', lp' can remove
a displacement, but not these effects and analogous effects by 1/3 TP.
In case of orbit 3 the amplitude of libration is large and can reach 141 deg,
the periods and the amplitudes in x and y are smaller, and the displacement
is directed towards the positive x axis. Like orbit 2, orbits 4 - 10 show
comparatively small mean amplitudes of libration.
To make the action by TL visible in the polar diagram, I isolate the effects in x and y of a sufficiently large band of frequencies around the one of the mean value of TL. If the result is plotted as before, the point generates a curve that consists of narrow extensions that reach out in different directions from the vicinity of the origin. The extensions are due to the variation of e caused by TL, the polar angle rotates according to period TP and causes the change of the directions. If after many cycles of TP this curve nearly closes and then nearly repeats the initial part of the plot, and if the isolated effects by TP lead to the plot of a precise ellipse as described above for orbits 2 - 5, I assume that the respective orbit evolves in a nearly quasi-periodic way in the considered interval. Due to the application of digital filtering this interval is much smaller than the interval of 60 millennia covered by most of the basic integrations.
As expected from Giffen's (1973) results, orbit 1 evolves in a different way. In this case an attempt to show the isolated effects by TP in a polar plot leads to a sequence of points that fill a wide elliptic ring. However there is evidence for a nearly quasi-periodic evolution of orbits 2 - 5. In case of orbits 6 and 7 small deviations from a precise ellipse appear in a plot derived from the isolated effects by TP. This turns out for orbit 8 as well, but orbit 8 is apparently captured in a secondary resonance: TP and the mean of TL are close to a 6/1 ratio and the figure resulting from the isolated effects by TL librates with its directions about a mean orientation. Since the sequence of starting values of e has reached the range of secondary resonances, it is no surprise that orbits 9 and 10 clearly deviate from a quasi-periodic evolution.
Now I apply the same procedure to starting values of some other orbits, using suitable values of the mass of Jupiter. Two of the three orbits of Section 4 are similar in type to orbits 2 and 3 and show analogous results here. However in case of orbit 3789 the frequency of 1/2 TP is surrounded by other frequencies and an attempt to demonstrate its effects in an x - y diagram does not lead to a precise ellipse. At the 3/2 resonance, (153) Hilda has an orbit with similar mean eccentricity, but precise ellipses appear in such diagrams, if the methods described in this section are applied to Schubart's (1968) starting values of Hilda in a study of the effects by TP and 1/2 TP. These ellipses appear to be circles. The radius of the circle that results for 1/2 TP equals only about 4 percent of the radius found for TP.
Brož M. and Vokrouhlicky' D., 2008, Asteroid families in the first-order
resonances with Jupiter. Mon. Not. R. Astron. Soc. 390, 715-732
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Author's Request
I retired from office in 1993. Possibly some recent work on the above topics
did not come to my attention. I shall be grateful for any news about such work.
Table 3. Seven real orbits
Orbit 128136 159289 167683 172058 223012 243192 258833
a (AU) 3.253 3.253 3.255 3.252 3.255 3.253 3.254
e' 0.054 0.054 0.060 0.054 0.058 0.056 0.058
A' 0.038 0.048 0.008 0.031 0.007 0.040 0.030
i (deg) 4.4 2.0 1.5 1.1 4.1 0.6 5.5
Ampl (deg) 62 79 11 48 10 60 45
TL (yr) 405 403 423 403 419 414 418
TP (yr) 552 555 620 543 605 587 584
TP/TL 1.36 1.38 1.47 1.35 1.44 1.42 1.40
Notes to Table 3. The values of a and i are equal to
the
synthetic proper elements derived by Novakovic, Kneževic
and Milani and presented in the Internet.
Mean values appear in the other lines. A' describes the amplitude of the
variations of e' around the given mean value. Ampl: amplitude of libration
of crarg', TL: period of this libration, TP: period of the retrograde
revolution of Dlp'.