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 an addition to the author's studies about circumbinary planets of binaries that revolve on a circular orbit. Some examples of the sun-Jupiter-asteroid problem follow.
My earlier studies of resonant motion of small circumbinary planets have indicated an approximately quasi-periodic evolution of many orbits. Here I resume the study of examples that are based on the planar circular restricted three-body problem. I demonstrate by extended numerical integrations and by use of graphs that the evolution of the respective orbits appears to be ruled by two constant basic periods. For this I store coordinates of the small body at intervals that equal the length of the shorter period. Then a plot in a suitable system of rotating coordinates only shows effects by the other period. I apply the same method to some resonant orbits of the sun-Jupiter-asteroid problem.
At first I study circumbinary orbits that refer to a 4/1 or 5/1 ratio of the orbital periods to that of the binary system. The former results are available in the internet: P-type planets 1and P-type planets 2, respectively. The 5/1 results have appeared in print (Schubart, 2017). I refer to the components of the binary by their masses, m1 and m2, and to the planet by m3 = 0. In case of mr = m2/m1 I consider the three values mr = 1.0, 0.5, and 0.2. Please remind that mr is equal to the ratio of the masses in this text. The simultaneous numerical integration proceeds in barycentric rectangular coordinates. Then m2 moves opposite to m1 on a circular orbit. All the bodies revolve in the same direct way. The distance from m1 to m2 equals 1 au.
For the numerical integration I use the N-body program by Schubart and Stumpff (1966) that works with a constant step length. I have retained the value of the solar mass plus mass sum of the four inner planets recommended in this paper, and I put the sum m1 + m2 equal to two times this value. Then in case of mr = 1 the masses of the two components are very close to solar mass. 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 is the respective longitude of m2. After that I transform the rectangular coordinates to a rotating system so that m1 and m2 remain at the starting positions. I call the resulting coordinates x and y. The results are stored at equally spaced intervals.
Let the resonance ratio be p/1 with p = 4 or 5. Then the librating angular element of a circumbinary orbit equals s1 = lm2 - lp - p*M + c. c is a constant, * is the multiplication sign. I call the mean periods of revolution of M and lm2-lp T1 and T2, respectively. TL is the long period that rules the process of libration. Since s1 librates, T1 = p*T2. T1 and TL are the basic periods to be studied. At first I need an accurate value of T1. Values of the elements refer to very long intervals, since I use up to 3 million steps of integration. I use plots of arguments versus time. In this way I get an estimate of the mean period of revolution of lp. The one of lm2 is known. Then I correct the resulting value of T2 and get a fairly accurate value of T1.
I want to show that stored values of x and y result from phases of another period, if I fix the phase of T1 to a selected value. For this I put the step of integration equal to T1/600, but I only store every sixhundredth result of a step, beginning with the starting values of x and y or with the values of step 121. In this way I get up to 5000 pairs of the coordinates that refer to the same phase of T1 and I expect each pair to refer to a special phase of a second period. Unless the ratio of the two periods equals the ratio of two small integers these phases will densely cover the range of the second period. Then a plot of y versus x will show a closed curve covered by points. I have used this method with the starting values of 14 of the former examples of librating orbits, omitting the ones with a vanishing or very small amplitude due to TL in s1. In all but one of these cases the plots show a single closed curve by densely positioned points. A small scattering of the points about the curve is easily removed by a final adjustment of T1. Extended and narrow curves appear and can resemble a crescent or banana in shape. The edges of some curves almost coincide or intersect each other in the central part.
I have combined four plots in a separately stored figure:
Fg1. I use different sizes in the x and y scales to get a suitable
presentation of the curves. The two graphs on the left side refer to orbit No.21
of
P-type planets 2. It is an example of the 5/1 resonance. In case of the
upper graph the phase of T1 corresponds to the starting values that lead to a
symmetry in the presented curve. The two graphs show small parts of the large
domain that is densely filled if all the results of the integration are plotted.
The graph on the upper right refers to orbit No.24 of the same set. This is a
special case since TL/T1 equals 19/1. In this graph the points cluster about
more than 19 positions. A more complicated situation is indicated.
I have applied digital filtering using the method and formulas of Schubart and
Bien (1984) to remove the effects of TL and of the short periods from s1. It
turns out that the period 4*TL and its fractions give rise to small variations
in s1. Apparently 4*TL is the second basic period in this case. This allows
many clusters of points. If the graph is divided into small areas about
selected clusters the points appear to be grouped along small curves.
An example appears in the lower right of
Fg1. The area is an enlargement of a cluster that appears on the x axis in
the upper right of the figure.
According to the graphs two basic periods rule the evolution of the orbits and
T1 does not change during very long periods. Earlier studies have shown that TL
remains constant.
In 1963 I have studied the sun-Jupiter-asteroid problem on the basis of the circular restricted problem (Schubart, 1964) but by use of simplified equations of motion. I have obtained results about the processes of libration in cases of resonant motion. Now the fast computers allow extended integrations of the unchanged differential equations. I have applied the procedure described above 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 longitude of m2 = Jupiter that revolves slower than lm, the mean longitude of m3 = asteroid; m1 = sun. The eccentricity of m2 equals zero. lp is the longitude of perihelion of m3. Let T3 be the mean period of revolution of lm - lm2. Then the period of libration, TL, and T3 are the expected basic periods. The way of deriving the value of T3 and of fixing the phase of T3 in the set of stored coordinates is analogous to the one described above. I have integrated two examples of each case of resonance. Single closed curves appear in the plots of the stored coordinates of the four examples.
Finally I have studied two orbits of the Trojan-type resonance. Then 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 36 to 107
deg. The mean period of revolution of the mean anomaly of the asteroid is the
second basic period that replaces T3. Another application of the above procedure
leads to plots of the coordinates that show single closed curves.
Again the graphs indicate that the evolution of the considered orbits is ruled by
two basic periods, at least during a long period of time.
Schubart J., 1964, Long-period effects in nearly commensurable cases of the
restricted three-body problem. Smithsonian Astrophysical Observatory,
Research in space science, Special report No. 149
Schubart J., 2017, Libration of arguments of circumbinary-planet orbits at
resonance. Celest Mech Dyn Astr 128, 295-301
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 July 24