by Joachim Schubart,
retired staff member of Astron. Rechen-Institut,
ZAH, University of Heidelberg, Heidelberg, Germany
schubart@ari.uni-heidelberg.de
The present text is a second addition to the author's theoretical studies
about orbits of circumbinary planets. Examples of the sun-Jupiter-asteroid
problem follow.
Several preceding articles that refer to these subjects
are
available by an index
Here I continue my preceding report on studies about the long-period evolution of resonant orbits of small circumbinary planets, now called Part 1. Part 1 that refers to the planar circular restricted problem is available in the internet: Evolution of orbits. In this continuation I try to demonstrate the way of action of basic periods on special orbits of the planar elliptic restricted 3-body problem. Now an additional period is active: TP, the mean period of revolution of the longitude of pericenter of the small body with respect to that of the second massive body. Therefore the method used in Part 1 is only applicable with modifications. It is necessary to remove all the short-period effects from the results of numerical integration in general. This is not necessary if the amplitude of the effects due to the period of libration of the critical argument, TL, is very small and negligible. Again I consider two basic mean periods. I store orbital elements of the small body at intervals that equal a value of the length of the shorter period. If that length turns out to be constant I want to find out whether the stored values depend on the longer period in a periodic way. Only a part of the studied circumbinary orbits shows this type of evolution. Then I apply the same method to some resonant orbits of the sun-Jupiter-asteroid problem. These orbits resemble natural examples and all of them show the wanted type of evolution.
I resume my earlier studies of circumbinary orbits that refer to a 5/1 ratio of the orbital periods to that of the binary system. The former results are available in the internet: P-type planets 2. Here I restrict myself to the examples with an eccentricity of the orbit of the massive bodies that equals 0.1 since the ratio TP/TL of the periods TP and TL introduced above is comparatively large in such cases. The method of deriving orbital elements equals the one used in Part 1. I refer to the components of the binary by their masses, m1 and m2, and to the planet by m3 = 0. The studies refer to three values of the ratio m2/m1. For the numerical integration I use the N-body program by Schubart and Stumpff (1966) that works with a constant step length. The result consists in barycentric rectangular coordinates and velocities of m3. These are transformed to osculating barycentric orbital elements by the assumption that the combined mass m1 + m2 is concentrated at the barycenter. I shall use the following elements of m3: the mean longitude, lm, the longitude of pericenter, lp, and the mean anomaly, M = lm - lp. lm2 and lp2 are the respective longitudes of m2.
The method of Part 1 is not applicable if three basic periods are active. In that case I remove the short-period effects by digital filtering using the method and formulas of Schubart and Bien (1984). I assume that these effects are given by a sum of periodic terms and that the respective frequencies are equal to linear combinations of the frequencies corresponding to the mean period of revolution of M, TM, and of TL and TP. Special combinations can correspond to long periods that pass the filter but will cause only small effects. This assumption allows the application of the method and a suitable filter truly removes the short-period effects from the numerical eccentricity, e, and from the librating critical argument, s = lm2 - 5*lm + 3*lp + lp2 .
Now TL is the shorter period and TP the long one. Only a constant value of the shorter period leads to a successful application of the method of Part 1. Therefore I study the effects in s due to TL. I start with the length of a preliminary value of TL and store the values of s at equidistant intervals that equal this length. I use a special filter that eliminates both short-period and long-period effects from the sequence of these values so that only effects by TL are expected. If the resulting values are plotted versus time a part of the curve of variation of s appears. I vary the value of TL so that smaller and smaller parts of the curve appear. Finally the plotted points either vary about a constant level or remain constant on such a level. The latter case means that the shorter period is constant. Now I apply this method to examples of my former study P-type planets 2. I retain the numbers of the orbits. Orbits 1, 6, and 11 show large amplitudes of libration of s and are close to the limit of permanent libration. In these cases the plotted points deviate from a constant level and indicate oscillations of very long period. Apparently there are additional frequencies close to the main frequency of TL that pass the filter and cause small effects. Variability of TL is indicated. The respective orbits evolve in a complicated way.
Orbits 7 and 12 show a smaller amplitude of libration. In addition I have studied an orbit similar to orbit 2 but with a slightly larger amplitude of libration. TL appears to be constant in these three cases. Now it is meaningful to find out whether the filtered values of s and e are periodic functions of TP if the phase of TL is fixed. I store these values at intervals that equal TL so that the pairs of s and e correspond to a selected phase of TL but to different phases of TP that fill the range of TP in general. Then a plot of e versus s shows points that are distributed along a closed curve. In analogy to Part 1 this indicates that the stored values are periodic functions of TP if the phase of TL is fixed. I have used two choices of the phase of TL for each of the three orbits and found closed curves.
Orbits 3, 8, and 13 are listed with a zero amplitude of libration due to TL. By a small change of a starting value an additional decrease of this amplitude has lead to a very small value so that effects by TL are negligible. Due to this only the two basic periods TM and TP are active. TM, the shorter period, appears to be constant and the method of Part 1 is applicable without use of a filtering process. Again I have fixed the phase of the shorter period and plotted e versus s. The points appear to be situated on closed curves in the three cases. According to the results presented in this section I assume that orbits of the studied problem with a sufficiently small amplitude of the effects due to TL are ruled by the basic periods TM, TL, and TP in general. Orbits close to a secondary resonance will show a different evolution.
In analogy to the respective section of Part 1 I have studied the sun-Jupiter-asteroid problem but now on the basis of the elliptic restricted problem. I have applied the procedure described in the preceding section to examples of the inner Jovian 2/1 and 3/2 cases of resonance. The librating argument equals s2 = lm2 - lp - p*(lm - lm2) with p = 1 or 2, respectively. lm2 is the mean longitude of m2 = Jupiter that revolves slower than lm, the mean longitude of m3 = asteroid; m1 = sun. The eccentricity of m2 equals 0.048 . lp is the longitude of perihelion of m3. After removal of the short-period effects TL shows a constant mean value and TP is the long period. Filtered values of s2 and e that correspond to a selected phase of TL are stored. Then a plot of e versus s2 demonstrates that the points are distributed along a closed curve. I have studied two examples of each case of resonance and I have used starting values that approximate the conditions of real asteroids. Again I assume that the evolution of the examples is ruled by three basic periods.
Finally I have studied two orbits of the Trojan-type resonance. Now s3=lm-lm2 is the argument that librates according to the period TL. The arguments librate about 60 deg. The larger range covered by an argument extends from 42 to 86 deg. Again the short-period terms are removed. TL and TP represent the short and the long period in an application of the above procedure. The corresponding values of e and s3 lead to points in a plot and the points are distributed along a closed curve.
Schubart J. and Bien R., 1984, An Application of Labrouste's method to
quasi-periodic asteroidal motion. Celest Mech 34, 443-452
Schubart J. and Stumpff P., 1966, On an N-body program of high accuracy, etc.
Veroeffentl. Astron. Rechen-Inst. Heidelberg 18, 1-31
Last modified: 2020 Dec. 30