Welcome to the Zoo Rings Website, the home for orbits of a small point-mass about a large and massive ring of mass M, and radius R.  The page is maintained by Dr. Joseph O. West, department of Physics, Indiana State University.

The system of interest shows a very large variety of closed orbits, even when the small mass is restricted to move in the x-z plane.  As discussed in a recent paper (R Wesely Tobin and J West "Closed orbits about a massive thin ring," Eur. J. Phys. 27 (2006) 215-223), an initial investigation of the system resulted in only a start at an enumeration of the possible closed orbit forms, families, and family groups.  As such, we issue an invitation to high school and undergraduate research groups to continue the search for additional orbital forms.   To aid in such a quest, the executable program used for the calculations can be downloaded here.  The program is straightforward to use, requiring that the user choose the initial conditions of the point mass including initial position (x, z) and initial velocity (vx, vy), and then choosing an upper limit on the time the simulation should run.  A description of the program and the instructions can also be downloaded here.  The output of the program is in a file with comma delimited data that can be opened with a spreadsheet program, such as Excel, so that one can then view the trajectory of the small mass.


Instructions for and description of the program in .PDF format.

Executable Program


If a group finds a orbit, or family of orbits, they can email the initial conditions to Dr. West, and if the results are confirmed, and they are new, the the new orbit will be posted here.  Those finding the new orbit may name it, although Dr. West reserves the right to change the name, if he deems it necessary.

Figures from the paper {This will become a link to jump to that section of the page}

Two-D figures not included in the paper {This will become a link to jump to that section of the page}

Three-D views of some of the orbits {This will become a link to jump to that section of the page}

New Orbits From High School and Undergraduate Research Groups {This will become a link to jump to that section of the page}


Figures from the Paper.

The orbits are shown with Name: initial conditions, where the initial condition are of the form [x(0), z(0), vx(0), vz(0)].  To view a larger image of the graphs, simply click on the thumbnails.

Fig. 1.  The geometry of the ring and point-mass.


Fig. 2a. Figure 8: [0, 0, 0.422, 0.422.]                           Fig. 2b. Omega [0, 0.5, 0.34, 0 ]


Fig. 2c. Pole [0.8, 0, 0, 0.42 ]                                                                 Fig. 2d. Hybrid [0, 0.6, 0.185, 0]

 Fig. 3. Period vs. Radius for Pole Orbits


Fig. 4. The characterization plot, this shows the initial conditions of the form [x = 0, z, vx, vz = 0] for which stable orbits have been found.  Family groups lie along the lines.  It is helpful in organizing the data, and in recognizing the difference between families that appear to be similar in shape, but actually belong to different families.  This is the case for the Omega1 and Omega2 families.


Fig. 5a. Tulip [0, 1.95, 0.0836, 0]                                                                 Fig. 5b. Ram [0, 0.35, 0.159, 0]


Fig. 5c. Pretzel [0, 0.50, 0.985, 0]                                                                Fig. 5d. Six-Leaf [0, 2.0, 0.4595, 0]


Fig. 5e. One-Wing Butterfly [0, 0, 0.2076, 1.0]                                               Fig. 5f. Five-Wing Butterfly [0, 2.0, 0.2, 0]

Fig. 5g. Fish [1.2, 0, 0, 0.4]

Fig. 5h.  An example THREE-D orbit.  Using a modified version of the program listed above.  No attempt has been made to find a closed orbit,

just one that is stable.  The initial conditions are: xo = 2.0, yo = 0, zo = 0, vxo =0, vyo = 0.5, vzo = 0.5


Two-D figures by West and Tobin not included in the paper


Boomerang  [0.6, 1.146, 0, 0]                                                    Cowboy Hat  [0, 0.575, 0.4031, 0] **Not really stable.

Crossed Omega [1.3, 0, 0, 0.3036]


Falling Leaf Right [0.6, 1.2, 0, 0] **Not a closed orbit.              Falling Leaf Left  [0.6, 1.0, 0, 0] **Not a closed orbit.


Eleven Lobe Crown  [0.6, 1.4, 0, 0]                 Fifteen Lobe Crown  [0.6, 1.398, 0, 0]                          Fancy Crown  [0.6, 1.38, 0, 0]


Three-D Views of a few of the Two-D orbits found by West and Tobin


One-Wing Butterfly                Double Omega Mobius                Hybrid                                        Omega 1                                Pretzel


Ram                                        Six-Leaf                                        Spirograph                                Tulip (x < R)



New Orbits From High School and Undergraduate Research Groups

J. West (faculty, department of physics, Indiana State University) and R. Tobin (junior, at the time, Indiana State University) Fall 2003.

Crossed Omega [  ]

This is actually an orbit from the paper, but it is here to show the format that will be used for new submissions.