1. U N C L A S S I F I E D Operated by the Los Alamos National Security, LLC for the DOE/NNSA IMPACT Project Drag coefficients of Low Earth Orbit satellites computed with the Direct Simulation Monte Carlo method Andrew Walker, ISR-1 LA-UR 12-24986

2. U N C L A S S I F I E D Outline Motivation Direct Simulation Monte Carlo (DSMC) method Closed-form solutions for drag coefficients Gas-surface interaction models – Maxwell’s model – Diffuse reflection with incomplete accommodation – Cercignani-Lampis-Lord (CLL) model Fitting DSMC simulations with closed-form solutions

3. U N C L A S S I F I E D Motivation Many empirical atmospheric models infer the atmospheric density from satellite drag – Some models assume a constant value of 2.2 for all satellites – The drag coefficient can vary a great deal from the assumed value of 2.2 depending on the satellite geometry, atmospheric and surface temperatures, speed of the satellite, surface composition, and gas- surface interaction Without physically realistic drag coefficients, the forward propagation of LEO satellites is inaccurate – Inaccurate tracking of LEO satellites can lead to large uncertainties in the probability of collisions between satellites

4. U N C L A S S I F I E D Direct Simulation Monte Carlo (DSMC) DSMC is a stochastic particle method that can solve gas dynamics from continuum to free molecular conditions – DSMC is especially useful for solving rarefied gas dynamic problems where the Navier-Stokes equations break down and solving the Boltzmann equation can be expensive – DSMC is valid throughout the continuum regime but becomes prohibitively expensive compared to the Navier-Stokes equations Knudsen Number, Kn = λ/L 0 0.010.11 10100∞ Euler Eqns. Navier-Stokes Eqns. Boltzmann Equation / Direct Simulation Monte Carlo Inviscid Limit Free Molecular Limit

5. U N C L A S S I F I E D Direct Simulation Monte Carlo (DSMC) Particle movement and collisions are decoupled based on the dilute gas approximation – Movement is performed by applying F=ma – Collisions are allowed to occur between molecules in the same cell Collisions Movement Possible Collision Partners

6. U N C L A S S I F I E D Direct Simulation Monte Carlo (DSMC) These drag coefficient calculations utilize NASA’s DSMC Analysis Code (DAC) – Parallel – 3-dimensional – Adaptive timestep and spatial grid Freestream Boundary DAC Flowfield

7. U N C L A S S I F I E D Closed-form Solutions

8. U N C L A S S I F I E D Closed-form Solutions

9. U N C L A S S I F I E D Gas-surface interaction models Specular Reflection Incident Velocity, V i Reflected Velocity, V r Diffuse Reflection

10. U N C L A S S I F I E D Gas-surface interaction models

11. U N C L A S S I F I E D Gas-surface interaction models Figure from Cercignani and Lampis (1971)

12. U N C L A S S I F I E D Local Sensitivity Analysis

13. U N C L A S S I F I E D Geometries Investigated Four geometries have been investigated thus far: Flat Plate Cube Cuboid Sphere

14. U N C L A S S I F I E D Sensitivity Analysis – Satellite Velocity

15. U N C L A S S I F I E D Sensitivity Analysis – Surface Temperature

16. U N C L A S S I F I E D Sensitivity Analysis – Atm. Temperature

17. U N C L A S S I F I E D Sensitivity Analysis – Number Density The closed-form solutions assume free molecular flow DAC CLL simulations show this assumption breaks down across all geometries for number densities above ~10 16 m -3 (with a 1 m satellite length scale) This corresponds to an altitude of ~200 km or above

18. U N C L A S S I F I E D Sensitivity Analysis – Tang. Acc. Coefficient

19. U N C L A S S I F I E D Sensitivity Analysis – Norm. Acc. Coefficient

20. U N C L A S S I F I E D Sensitivity Analysis – Norm. Acc. Coefficient

21. U N C L A S S I F I E D Conclusions

22. U N C L A S S I F I E D Future Work Thus far, only simple geometries where the closed-form solution is known have been investigated – Allows for verification of the DAC CLL model vs. closed-form solution Use DAC CLL model to find empirical closed-form fits to realistic and complicated satellite geometries (e.g. CHAMP) Recreate Langmuir isotherm fit for normal energy accommodation coefficient (Pilinski et al. 2010) with the GITM physics-based atmospheric model Perform global sensitivity analysis with Latin Hypercube sampling