RADIALLY ADAPTIVE EVALUATION OF THE SPHERICAL HARMONIC GRAVITY SERIES FOR NUMERICAL ORBITAL PROPAGATION

Transcription

1 AAS RADIALLY ADAPTIVE EVALUATION OF THE SPHERICAL HARMONIC GRAVITY SERIES FOR NUMERICAL ORBITAL PROPAGATION Austin B. Probe, * Brent Macomber,* Julie I. Read,* Robyn M. Woollands,* and John L. Junkins Evaluation of the Spherical Harmonic Series for gravity is one of the most computationally intensive requirements for high-accuracy orbital propagation. The Earth s gravity is non-uniform and these perturbative effects must be incorporated into orbital propagation models to ensure an accurate numerical approximation is computed. However, considering the radial nonlinearity of the force-field allows for the judicious selection of the harmonic series degree and order as a method of reducing computational cost, without sacrificing accuracy. This paper details a method of radial adaptation for the spherical harmonic series, supporting analysis demonstrating its accuracy, and some characteristic results. INTRODUCTION The spherical harmonic series, which theoretically represents gravity as a solution of Laplace s equation, is routinely used as method for high-fidelity gravity calculation in precision astrodynamics. Due to the computationally intensive nature of this series, when carried to a high degree and order expansion, it is frequently one of the most burdensome elements of high-accuracy orbital propagation. Despite this high cost, the non-uniform elements of the Earth s gravity field must be considered to achieve accurate numerical representations of orbital trajectories. The standard spherical harmonic calculation requires that the user select the fixed degree and order to be used a-priori. This is generally a heuristic choice based on the user s desired level of accuracy and their available computational resources. The selected degree and order are then used to compute values for the normalized Associated Legendre Polynomials for the geocentric latitude ϕ (Eqn. 1-5). P 0,0 = 1 P 1,0 = 3 cos(φ) P 1,1 = 3 sin(φ) (1) (2) (3) * Graduate Researcher, Aerospace Engineering, Texas A&M University, 702 H.R. Bright Building M.S Distinguished Professor, department, Aerospace Engineering, Texas A&M University, 702 H.R. Bright Building M.S

2 P n,n = 2n+1 2n sin(φ) P n 1,n 1 n > 1 P n,m = (2n+1)(2n 1) (n+m)(n m) cos(φ) P n 1,m (2n+1)(m2 (n 1) 2 ) (2n 3)(m n)(n+m) P n 2,m m < n (4) (5) Once obtained, the normalized Associated Legendre Polynomials can be evaluated with the radius (r), longitude (λ), and the constant spherical harmonic coefficients (C n,m and S n,m ) for the selected gravity model to produce the gravitational potential (Eqn. 6). 1 U(r, φ, λ) = μ n r (R e r ) n=2 n P n,m ( C n,m cos(λ) + S n,m sin (λ)) m=0 (6) Selection of the fixed degree and order in this fashion can be problematic. It is difficult to make a selection appropriate for the application without a clear knowledge of the relationship between the degree and order and the resulting accuracy for the series. Ideally, the order would be selected to so that the accuracy of the gravitational approximation calculated would match the desired tolerance for propagation and the physical acceleration accuracy of the other models being used. To avoid using an inaccurate gravity approximation conservative degree and order specifications are frequently chosen. This can result in significant wasted computation time, because as shown in Fig. 1 the computational cost of the spherical harmonic model increases quadratically with the degree. 2 Fig. 1: Relative Cost of Gravity Series Computation Additionally, the Earth s gravity is non-uniform and the standard method of degree and order selection does not account for these changes. A spacecraft in an eccentric orbit with a perigee radius of 1.2 Earth radii experiences far greater perturbative gravitational effects compared with when it is at apogee, say 7 Earth radii. Close to the Earth where the gravitational acceleration changes more abruptly it is essential to consider the high order, computationally expensive, gravity model. As the strength of the gravitational field decreases radially, the changes are much more subtle and we can anticipate that a lower fidelity gravity model is all that is required. The traditional method of selecting the degree for the spherical harmonic gravity, must account for the worst-case scenario at

3 perigee, meaning that significant wasted computation is introduced for the rest of the orbit, especially near apogee. Various efforts have been made to develop new methods of accurately approximating gravitational potential that are more efficient in terms of the memory and computation required. However, some of these methods still require the selection of various heuristic elements and do not address the potential wasted calculation resulting in changes from the force field. While these methods have some advantages, they lack the heuristic and operational heritage of the spherical harmonic model. This means that their implementation would require significant changes to existing methods and substantial testing and verification. 3,4,5 RADIALLY ADAPTIVE SPHERICAL HARMONIC EVALUATION The ideal implementation of a spherical harmonic gravitational approximation would allow the specification of the required accuracy by the user and provide an approximate solution that meets that requirement. This would reduce the effort needed from the user to select a degree and order while also avoiding inaccurate approximations and wasted computation. Careful consideration of the character of the gravitational force-field and analysis of the spherical harmonic gravity approximation allows for the judicious selection of the series degree and order as a method of reducing computational cost, without sacrificing accuracy. To achieve this adaptation of degree and order for the spherical harmonic series based on radius and required accuracy can be applied to ensure the appropriate level of system dynamics is captured with the minimum computational cost. Adaptation Reference Study To determine the appropriate way to adapt the function order based on the orbital radius and required accuracy a study was performed examining the size of each of the component terms of the spherical harmonic series as a function of the series degree and order. An examination of the marginal contributions for the various series components for a single evaluation of the EGM2008 series is shown in Fig This demonstrates that while the marginal contribution generally decreases with series degree and order it does not do so monotonically. Fig. 2: Marginal contributions of gravity terms as a function of order and magnitude

4 To determine the maximum possible contribution from each degree we identify the maximum value of the Associated Legendre Function for each degree and order. This occurs for the P n,0 value when the geocentric latitude is 0, as can be seen from Eqn Fig. 3 shows the variation maximum Associated Legendre Function value as a function of degree and latitude. Taking this as the maximum contribution from the Associated Legendre Function at each degree and order, the maximum marginal component of gravitational potential can be obtained by multiplying it by the greater of C n,m or S n,m within the summation in Eqn. 6. Fig. 3: Maximum Associated Legendre Function component as a function of order and geocentric latitude This maximum perturbation value from within the summation for each degree and order is recorded. This process is then repeated at a series of radii chosen using densified cosine sampling depicted in Fig. 4. The transformation between the densified cosine sampling and ordinary cosine sampling of the radius is known, and it permits the orthogonality conditions of Chebyshev polynomial approximation to be satisfied with physically much denser radial sampling nearest the surface of the Earth. This device allows the required order as a function of a given acceleration accuracy and radius (see Fig. 5 & 6) to be approximated as an orthogonal Chebyshev polynomial using gravity evaluations on a significantly reduced number of radial evaluation points. 5

5 Fig. 4: Comparison of Uniform and Cosine-Like Sampling for Series Evaluation Comparing the maximum perturbation value to the required level of accuracy allows you to select the appropriate order for each evaluation. Fig. 5 shows the resulting relationship between the acceptable error tolerance, the orbital radius, and the required spherical harmonic gravity order (with a limit of 50) and Fig. 6 shows the acceptable error tolerance, the orbital radius, and the required spherical harmonic gravity order on a logarithmic scale for the entire series. This surface can be easily inverted to obtain the required gravity expansion order (N O) as a function of two variables (required accuracy of acceleration, and radial distance from the center of the Earth). Thus, for specified acceleration accuracy, each call of the gravity routine is inherently adaptive based on the radial distance, and this scheme is implementable to achieve increased efficiency with any existing algorithm to compute a spherical harmonic gravity field. Fig. 5: Required spherical harmonic gravity degree as a function of accuracy of the gravitational acceleration vector

6 RESULTS Fig. 6: Required spherical harmonic gravity degree as a function of accuracy of the gravitational acceleration vector Once the required order surface is obtained, it is then possible to fit a function to this surface and use it to guide the selection of the order that should be used for each spherical harmonic gravity evaluation during orbital propagation. Fig. 7 shows the resulting required gravity degree around a highly eccentric orbit that is functionally equivalent to evaluating a (40 40) gravity field everywhere. At the beginning and end of the orbit when the spacecraft is near perigee the full (40 40) model is required, but as the craft moves further away from the earth the adaptation allow for a lower degree and order to be used.

7 Fig. 7: Variation in the required gravity degree for a high eccentricity orbit (limit 40). This radially adaptive gravity results in a lower computational cost for a given orbit as shown by the total number of equivalent gravity evaluations, or the comparison of the cost of each evaluation used with respect to the cost of a selected maximum allowed order. The equivalent N gravity cost is defined as follows: E = 1 ( ) 2 + Const., where N is the order selected, N max is 2 N max the maximum allowed order, and the constant represents the function overhead. The computational savings of this method are a function of the orbit selected and the maximum allowed order of the spherical harmonic series selected for the propagation. Highly eccentric orbits allow for the adaptation to use a lower gravity order for more evaluations, thus reducing the cost. Selecting a high maximum allowed order for use means that the drop in degree with adaptation can be greater. To illustrate the advantages of the radial adaptation RK45 was used to propagate a series of orbits with meter level precision (required accuracy ~10-8 ) with perigee altitudes at 1.05 Earth radii and varying eccentricities using both a fixed degree and order of (100x100) and the radially adaptive method. Fig. 8 shows the relative computation time required and how it relates to eccentricity. Fig. 9 shows that the relative Hamiltonian conservation for the 0.8 eccentricity orbit for the radially adaptive model and the fixed degree and order (100x100) are essentially equivalent, illustrating that the accuracy for the two is comparable.

8 Fig. 8: Relative evaluation time as a function of eccentricity Fig. 9: Hamiltonian conservation comparison for radially adaptive propagation vs. fixed degree and order of 100 CONCLUSION This method for radial adaptation of the spherical harmonic function allows for automatic selection of the required degree and order based on user specified accuracy, reducing the probability of wasted computation or inaccurate solutions. This method can greatly accelerate orbital propagation that utilizes the spherical harmonic series, especially for highly eccentric orbits. Additionally, radial adaptation can be integrated into current spherical harmonic code with minimal modifications.

9 REFERENCES 1 Gottlieb, R. G., Fast Gravity, Gravity Partials, Normalized Gravity, Gravity Gradient Torque and Magnetic Field: Derivation, Code and Data, Tech. Rep. NASA Contractor Report , NASA Lyndon B. Johnson Space Center, Houston, TX, February Beylkin, G. and Cramer, R., Toward Multiresolution Estimation and Efficient Representation of Gravitational Fields, Celestial Mechanics and Dynamical Astronomy, Vol. 84, pp , Arora, N., Russell, R.P., Efficient Interpolation of the GRACE GGM03C Gravity Model, Paper AAS , Kauai, HA, Feb Jones, Brandon A., Efficient Models for the Evaluation and Estimation of the Gravity Field, Ph.D. dissertation, University of Colorado at Boulder, CO, Bani Younes, A., Orthogonal Polynomial Approximation in Higher Dimensions: Applications in Astrodynamics., Ph.D. dissertation, Texas A&M Univ, College Station, TX, Pavlis, N. K., Holmes, S. A., Kenyon, S. C., and Factor, J. K., An Earth Gravitational Model to Degree 2160: EGM2008, in Proceedings of the European Geosciences Union General Assembly, Vienna, Austria, April