Introduction
The behavior of planets revolving around a star is very interesting. Especially the number of planets are more than two and their orbits are very close. In case of two planets system, it seems likely that their behaviors are rather stable comparing to those of three planets system when the eccentricity of the planets are tiny little. However, if radius of both planets are enough big one planet can be swung away from the system. In case of three planets system, planets exhibit dramatical changes in both radius and eccentricity. This is because the three planets case is a typical example of Three Body Non-Linear System. Their semimajor axis and eccentricity exhibit CHAOTICAL change and how the planets behave is very sensitive to the initial condition.
Comparing 2-Dimensional and 3-Dimensional simulations, it seems that a planet in 2-Dimensional case can be swung away more easily than that of 3-Dimansional case. This is because, in 2-Dimensional case, the planets are restricted in the same plane and they can have more chance to strongly intercat each other than 3-Dimensional case.
Simulation Program
The program is written in C Language, and its core part, which calculates Gravitatinal Interactions among Planets, is very simple; it is only about 50 lines. Simulation is performed in 3 dimensional space, if you don't intentionally signify 2-D option. A phase angle, which determines the direction of orbits in X-Y plane, is automatically provided at random. And a inclination, which decides how much angle the orbit is rotated toward Z direction, is automatically provided at random following Gausian density function. For the inclination, you can specify SIGMA value as the option.
There are two kind of programs. One, named Giant, performs just numerical calculation and outputs numerical data. While the program is doing a job, you have to just wait for the end of the job. All parameters and options are to be signified as the command line parameter. Another, named TkGiant, performs simulation with GUI and you can have a look how planets revolve around a star as well as numerical data. If you are interested about only taking numerical data, using Giant is preferable because TkGiant comsumes CPU time for performing some GUI.
Both package are archived by TAR and compressed by GZIP. Packages contain C source file and executable command for Windows. In case of TkGiant package, it contains also Tcl/Tk script file which manages user interface for performing simulations and document file, TkGiant.txt, which is written about extension of Tcl/Tk commands and variables. On both UNIX and Windows systems, you can use completely same C source code and Tcl/Tk script and there is NO switches for compiling source codes.
| Giant: | Numerical simulation version | [TAR&GZIP], | [Directory] | |
| TkGiant: | GUI simulation version | [TAR&GZIP], | [Directory] | |
| Screen Shots => |
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Results of Simulations
Here I show several examples how the planets' orbital parameters change. In case of 2 DIMENSIONAL simulation, it's likely that both semimajor axis and eccentricity changes rapidly and a planet is rather easily swung away out of the system after the eccentricities become larger. However, in case of 3 DIMENSIONAL simulation, orbital parameters change more softly than those of 2 Dimensional case and it takes long time for a planet to be swung away out of the system because the chance of strong interaction among planets decrese due to the inclination of orbits.
In case of three planets system, system exhibits more CHAOTICAL behavior than two planets system. And you can have interesting Poincare Sections in case of 3 DIMENSIONAL simulation.
Note, data, which are shown below, for two planets case are taken setting Spatial Resolution = 0.01 [AU] and data for three planets case are taken with Spatial Resolution = 0.001 [AU].
| Semimajor Axis | Eccentricity | |||
|---|---|---|---|---|
| 2-D Simulation | Two Planets Cases | a1=5.0, e1=0.0, p1=49.8, a2=9.0, e2=0.0, p2=55.6 |  |  |
| a1=5.0, e1=0.0, p1=-89.9, a2=7.0, e2=0.0, p2=18.1 |  |  | ||
| Three Planets Cases | a1=5.0, e1=0.0, p1=-30.6, a2=7.0, e2=0.0, p2=-44.2, a3=9.0, e3=0.0, p3=-58.6 |   |   | |
| 3-D Simulation | Two Planets Cases | a1=5.0, e1=0.0, p1=-88.8, i1=-1.3, a2=9.0, e2=0.0, p2=14.1, i2=26.2 |  |  |
| a1=5.0, e1=0.0, p1=-16.3, i1=21.7, a2=7.0, e2=0.0, p2=18.1, i2=-15.4 |  |  | ||
| a1=5.0, e1=0.0, p1=-46.2, i1=2.4, a2=7.0, e2=0.093, p2=18.1, i2=-4.1 |  |  | ||
| a1=7.0, e1=0.0, p1=-26.4, i1=-3.2, a2=9.0, e2=0.094, p2=-81.9, i2=-3.1 |  |  | ||
| Three Planets Cases | a1=5.0, e1=0.0, p1=75.5, i1=-3.0, a2=7.0, e2=0.0, p2=-16.4, i2=-1.0, a3=9.0, e3=0.0, p3=54.2, i3=3.6 |   |  | |
| a1=5.0, e1=0.0, p1=28.7, i1=1.2, a2=7.0, e2=0.0, p2=-12.5, i2=3.4, a3=9.0, e3=0.0, p3=-44.6, i3=-1.3 |  |  | ||
| Radius - Z Axis at apsidal point | X - Z Axis at Y=-0 | Y - Z Axis at X=-0 | ||
|---|---|---|---|---|
| Two Planets Cases | a1=5.0, e1=0.0, p1=-88.8, i1=-1.3, a2=9.0, e2=0.0, p2=14.1, i2=26.2 |  |  |  |
| a1=5.0, e1=0.0, p1=-16.3, i1=21.7, a2=7.0, e2=0.0, p2=18.1, i2=-15.4 |   |   |   | |
| a1=5.0, e1=0.0, p1=-46.2, i1=2.4, a2=7.0, e2=0.093, p2=18.1, i2=-4.1 |   |   |   | |
| a1=7.0, e1=0.0, p1=-26.4, i1=-3.2, a2=9.0, e2=0.094, p2=-81.9, i2=-3.1 |   |   |   | |
| Three Planets Cases | a1=5.0, e1=0.0, p1=75.5, i1=-3.0, a2=7.0, e2=0.0, p2=-16.4, i2=-1.0, a3=9.0, e3=0.0, p3=54.2, i3=3.6 |    |    |    |
| a1=5.0, e1=0.0, p1=28.7, i1=1.2, a2=7.0, e2=0.0, p2=-12.5, i2=3.4, a3=9.0, e3=0.0, p3=-44.6, i3=-1.3 |   |   |   | |