Optimized Spacecraft Hovering Position for Painting an Asteroid

1. Optimized Spacecraft Hovering Position for Painting an Asteroid S. Ge, N. SatakGraduate Students, Texas A&M University D.C. Hyland, Professor of Aerospace Engineering, College of Engineering, Professor of Physics, College of Science, Texas A&M University ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ International Symposium on Near Earth Asteroids April 4-6, 2011

2. Outline Introduction Algorithm Results Future Work

3. Intro: Yarkovsky Effect D. Vokrouhlicky, A. Milani, and S. R. Chesley. “Yarkovsky Effect on Small Near-Earth Asteroids: Mathematical Formulation and Examples”, Icarus 148, 118-138 (2000). 11

4. Intro: YORP Effect Rubincam, David. “Radiative Spin-up and Spin-down of Small Asteroids”, Icarus 148, 2-11 (2000).

5. Motivation Assuming a model for an asteroid, develop a method to find the most effective areas to change the asteroid’s albedo.

6. Required Surface Area Coverage J. Giorgini, et al. “Predicting the Earth Encounters of (99942) Apophis”, Icarus 193: 1-19 (2008).

7. Assumptions Constant temperature Lambertian surface Constant solar direction

8. Algorithm: Part 1 N r c = speed of light Fs = solar intensity σ = Stefan-Boltzmann constant T = temperature (assume constant 400 K) 1. Generate a polyhedron model of asteroid of radius 135 m, nominal radius of Apophis. 2. Acquire the three vectors that describes the x-y-z positions of the ith triangular surface and find its centroid. 3. Find the normal vector N and the normalized force vector f of the ith triangular surface.

9. Algorithm: Part 2 4. Find the area vector dA of the ith triangular surface. Considering that the area to be covered is a constant circle while the triangular surfaces are of varying sizes, a method was used to find the intersection area of the circle and the triangle. 5. Knowing the normal vector, normalized force vector, and the area vector, find the forces, torques and normalized torques of the ith triangular surface. 6. Repeat steps 2-4 for all surfaces to account for all surfaces. For each surface, there is a different radius vector, normal vector N, and area vector dA.

10. Algorithm: Part 3 Asteroid is tilted at an angle to z-axis z y x Asteroid rotates around x-axis 7. Repeat steps 2-5 for all time steps in one orbit. Sum up the forces, torques, and normalized torques.

11. Last Step 8. Find the combinations of surfaces that give the area required to be covered and produce the maximum torques and normalized torques in x, y, and z direction.

12. Maximum Force Changes

13. Maximum Torque Changes

14. Significance of YORP Effect Torque effect is negligible compared to force. Deimos-like Ida-like

15. Distribution of Effective Areas … 18.5 m VS For this model, a collection of smaller surfaces with same area as one large surface were found to be less effective. .5% of surface area ~ 1070 m2

16. Body-fixed Hovering Thrust required to maintain spacecraft position relative to the body. Minimize the thrust by choosing the optimal height for the spacecraft to hover. Equations of motion for system in uniformly rotating, small-body Cartesian coordinate frame For this model, body-fixed hovering is more advantageous since the distribution of effective areas is concentrated.

17. Future Work • Optimize the altitude for the thrust • Use Dr. Hyland’s rubble pile model for the asteroid geometry • Include orbital parameter changes • Radial vector Rs from sun to asteroid • Distance from sun to asteroid will affect temperature of surface • Orientation of orbital plane will change spin axis relative to sun • More realistic model • Non-symmetrical center of mass for asteroid • Non-uniform thermal distribution of asteroid surface