Introduction Kirkwood Gaps Phil Peterman Physics 527 Computational Physics Project 1 Resonance phenomena, or tendencies for systems to oscillate at particular frequencies, occur in many distinct branches of physics. The mechanics of celestial bodies show a type of resonance that leads to instabilities of particular orbital trajectories. When two objects, which are orbiting around a common attractor, have a period that is related by a small fraction (eg. 1/2, 1/3), an enhancment of their mutual gravitational interaction will occur. Sometimes this additional interaction will lead to stable orbits where a small perturbation is self corrected, such as the moons of Jupiter which share the 1:2:4 orbital period relation. Most resonances however are the unstable type where the added interaction will distort the orbit until the resonance is destroyed. Kirkwood gaps are regions of the main belt asteroid field in our solar system where Jupiter has swept away asteroids that correspond to small fractions of Jupiter s orbital period. According to Wolfram s ScienceWorld website [1], the most pronounced Kirkwood gaps are the 2:1, 5:2, and 3:1 where the first integer is the number of complete cycles of the asteroid and the second is the number of complete cycles of Jupiter. My computational project was built to observe these Kirkwood gaps in a simulated asteroid field using the known value of the mass of Jupiter relative to the Sun. Simulation This java simulation is a modification of the Planet application used in chapter 5 of the textbook. A Planet class is used to simulate the large gas 1
giant Jupiter orbiting around our sun. The second class named Asteroid is for generating the dense asteroid field initially within the orbit of Jupiter. The third class is the target class KirkwoodApp which builds a simulation frame and a histogram frame to track the evolution of the asteroid field. The class which simulates Jupiter builds a numerical ODE solver based on the RK4 method of solving second order ordinary differential equations. The Planet class has a massplanet field value of 0.0009546 solar masses which is the approximate mass of Jupiter. Since the planet s mass is much larger than the mass of the asteroid field, the only force acting on the planet is that from the central force of the sun. The class Circle that draws the planet is modified to give a large, red circle. Both Asteroid and Planet classes implement the Drawable and ODE interfaces. The other planet like class is the Asteroid class. A single asteroid will feel a force from the Sun at the center and from the orbiting Jupiter. When computing the acceleration of each asteroid, the Planet s coordinate state is referenced. Since two different classes are transferring information inside of the same package Project1, the information must be accessed in a static way. Since there is only one instance of Planet, the code Planet.state[0] and Planet.state[2] are the Cartesian coordinates of Jupiter. The asteroid graphic is colored black and reduced in size as seen in figure 1. The target class KirkwoodApp is a subclass of the AbstractSimulation class and builds two frames: a PlotFrame (figure 1) of the planet and asteroids, and a HistogramFrame (figure 2) of the radial distance of each asteroid from the sun. After the target class has instantiated a Planet, it declares an array of Asteroids of size numasteroids (4000) which is set internally in 2
Figure 1: Initial Configuration the field section of the application. In the constructor section, the target class populates the asteroid array with 4000 instances of Asteroid. As the simulation progresses, the histogram refreshes the current radial distribution of asteroids by using the cleardata() method before rebuilding the plot. A method of the KirkwoodApp class is labeled randomizer and is used to randomly place an asteroid in the circular field. A random x coordinate is selected between -4 and 4 AUs along with the a y coordinate. The sought after Kirkwood gaps occur between 2 and 4 AUs, so an annulus is made by removing asteroids less than 2 and greater than 4 in radial distance from the sun. For circular orbits, the velocity v must equal 2π r. Dividing up the Cartesian plane in the usual four quadrants, the correct values of the initial 3
Figure 2: Initial Radial Distribution velocities are made by determining the x and y components of the velocity by using the atan function to find the angle with respect to the sun. From the plot of Kirkwood gaps found on NASA s Jet Propulsion Laboratory webpage (figure 3) [2], the resonances vary in full width at half maximum from 0.05 to 0.2 AUs. To properly resolve these structures, the bin width of the histrogramframe is set to 0.025 to yield at least two bins at the minimum resonance width. The density of asteroids was chosen to almost completely cover the region of interest, so when an instability arises a noticeable white gap will appear. The pixel radius of the asteroids was also chosen to maxi- 4
mize this effect. Of course, more asteroids can be initially populated in the annulus to bolster this effect but at the cost of a slower simulation rate. A systematic deviation of this simulation compared to the actual Kirkwood gaps data is that the real data shows a generally flat distribution of asteroids off resonance as a functin of radius while this simulation displays a linearly increasing distribution. This limitation is due to the manner at which the asteroids are randomly seeded. Inside the annulus, the asteroids are uniformly distributed resulting in an initial distribution that depends on the radius. A more robust and accurate simulation would seed the asteroid field with equal numbers of asteroids for a given radius. This limitation however does not restrict the observation of distinct gaps in the field at long times. Figure 3: NASA JPL Kirkwood Gaps Radial Distribution Results The most prominent orbital resonances of Jupiter have been experimen- 5
tally observed at 3.27 AU (2:1), 2.50 AU (3:1), 2.82 (5:2). These are the gaps I would like to see form on the simulation. After approximately 260 years (22 orbital cycles) of evolution, all three of these gaps are present and can be seen most easily in the histogram plot (figure 4) as well as visible white bands in the asteroid field (figure 5). Figure 4: Resonances at 260 Years of Evolution The full width at half maximum of the resonances are approximately 0.075 AUs for the 2.5 AU (3:1) gap, 0.05 AUs for the 2.82 (5:2) gap, and 0.25 AUs for the 3.27 AU (2:1) gap which is in excellent agreement with the JPL data. Another limitation of this simulation is from purely circular 6
Figure 5: Resonances at 260 Years of Evolution orbits. Jupiter has a small eccentricity (0.048775) so it oscillates back and forth radially by about 0.5 AU. This minor aberration of a perfectly circular orbit doesn t seem to play much of a role in clearing out asteroids whose period is commensurate by a small fraction. Conclusion This simulation was successfully able to reproduce an empirically derived phenomenon of Kirkwood gaps in the main belt asteroid field around our sun. This corroborates the hypothesis that the source of these gaps is indeed from the simple mathematical relation of small commensurate orbital periods. Potential improvements include generating an asteroid field density independent of radius and correcting for the small eccentricity of Jupiter. 7
Bibliography 1. http://scienceworld.wolfram.com/astronomy/asteroidbelt.html 2. http://en.wikipedia.org/wiki/kirkwood gaps 8