Orbital Debris Mathematical Modeling

Back in 1992, I left my job at the NASA Johnson Space Center, working for the orbital debris program, for a position in academia. Fortunately, NASA has continued to employ me as a consultant, and in that venue I have been able to work on many interesting theoretical problems involving debris in earth orbit. I plan to include more descriptions and graphics on these topics as time permits. Most recently, I have been developing software to simulate the light curves that result from telescopic observations of sunlight reflecting off of orbiting debris pieces, while they rotate. This is a fascinating problem! Below, on the left, is an animation of the specular and diffuse light reflected from a rotating debris fragment. The spherical plot in the lower right panel of the figure represents the specular reflection intensities for all possible orientations of the object, for a given solar phase angle. All possible orientations have been discretized using equal-area partitions of the unit quaternion (the four-dimensional unit sphere concept invented by Sir William Rowan Hamilton in the mid 1800's), where a dot, color-coded by reflected intensity, has been placed at the center of each equal-area partition. Only three dimensions of the quaternion are shown in the projection. The yellow trajectory represents the trajectory through orientation-space taken by the rotating object. The image on the right is a frame from a simulated video of the rotation of a different fragment. Mathematical models of these objects are created either synthetically, or from the output of a HandyScan laser scanner at the Johnson Space Center in Houston. The simulations of light curves are then compared with curves measured in the Optical Measurements Center at JSC by the skillful work of the University of Houston doctoral candidate Heather Cowardin. The laser scans are generated by Nicole Hill, also at JSC. The objects are rotated under artificial sunlight by a robotic arm, and observed by a digital camera.

I

Depicting the debris collision probabilities in the three-dimensional space above the earth in a visually helpful manner is a challenge. I call the graphic below a "windmill plot". It depicts the collision probabilities between two orbiting and precessing satellites above the earth (the colorful ball in the center) by color-coding of the cells on the blade-like panels. The red points represent the locations of simulated collisions generated probabilistically.