Spacecraft landing on small bodies pass through regions where conventional gravitation formulations using exterior spherical harmonics are inaccurate. An investigation shows that a formulation using interior solid spherical harmonics might be satisfactory. Interior spherical harmonic expansions are usable inside an imaginary, empty sphere. For this application, such a sphere could be positioned in empty space above the intended landing site and rotating with the body. When the spacecraft is inside this sphere, the interior harmonic expansion would be used instead of the conventional, exterior harmonic expansion.
Coefficients can be determined by a least-squares fit to gravitation measurements synthesized from conventional formulations. Due to their unfamiliarity, recurrences for interior, as well as exterior, expansions are derived. Hotine’s technique for partial derivatives of exterior spherical harmonics is extended to interior harmonics.
This work was done by Robert A. Werner of Caltech for NASA’s Jet Propulsion Laboratory. For more information, download the Technical Support Package (free white paper) at www.techbriefs.com/tsp under the Information Sciences category. NPO-46697
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Evaluating Descent and Ascent Trajectories Near Non-Spherical Bodies
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Overview
The document titled "Evaluating Descent and Ascent Trajectories Near Non-Spherical Bodies" is a technical memorandum from the Jet Propulsion Laboratory (JPL), authored by R. A. Werner in May 2010. It addresses the challenges faced by spacecraft when landing on small celestial bodies, which often have irregular shapes and gravitational fields that deviate from conventional models.
The abstract highlights that traditional gravitational formulations can be inadequate in these scenarios. To overcome this limitation, the document proposes a novel approach using interior solid spherical harmonics. This method aims to provide a more accurate representation of the gravitational field by fitting coefficients derived from gravitation measurements. The use of least-squares fitting is emphasized as a technique to synthesize these measurements, allowing for a more precise modeling of the gravitational environment around non-spherical bodies.
The document also discusses the mathematical foundations of gravitational field modeling, referencing earlier works, such as those by Kaula in 1959, which established the use of spherical harmonics for representing the Earth's gravitational field. This foundational knowledge is crucial for extending the application of spherical harmonics to other celestial bodies, particularly those that are not spherical in shape.
Additionally, the memorandum outlines the derivation of recurrences for both interior and exterior expansions of spherical harmonics, which are essential for calculating gravitational effects in various regions around these bodies. The document extends Hotine’s technique for partial derivatives of exterior spherical harmonics to include interior harmonics, thereby enhancing the analytical tools available for researchers and engineers working in this field.
The research is sponsored by the Interplanetary Network Directorate Technology Program Office at JPL, under NASA's auspices, indicating its significance in the broader context of space exploration and technology development. The document serves as a resource for engineers and scientists involved in mission planning and trajectory analysis, providing insights into the complexities of navigating and operating spacecraft in the vicinity of non-spherical celestial bodies.
In summary, this technical memorandum presents a comprehensive approach to improving gravitational modeling for spacecraft operations on small bodies, emphasizing the need for innovative methodologies to ensure successful landings and operations in challenging environments.
