1. Figure 2: Definition of the potential
Comparison of gravity fields derived using ground radar and spacecraft observations for close proximity operations around asteroid Itokawa
Shankar N. Ramaseri Chandra* and R. Fevig* *Department of Space Studies, University of North Dakota, Grand Forks, ND 58202
ABSTRACT
The study of minor planets and asteroids will help us understand their composition for in-situ resource extraction and also to reveal the secrets of solar system formation. Due to the presence of undetermined mass distributions, asteroid bodies exhibit complex gravitational fields. Hence, an efficient modeling of their gravity field is important for close proximity spacecraft studies. The Hayabusa spacecraft which visited the asteroid Itokawa modeled its gravity field during close proximity. Radar observations of this asteroid also suggest a shape and gravity field model. In this work, we study the differences in the above gravity models by modeling close proximity spacecraft scenarios in STK. The dynamics of the spacecraft help us to determine which of these gravity models is suitable for close orbit studies for Itokawa.
The orbits in yellow are simulated with the radar-derived gravity field (RG) and the blue ones with the satellite derived gravity field (SG).The satellites with 0.75 km semi-major axis have stable retrograde obits while compared to prograde orbits (ref. Fig. 4). For the satellites with a 0.5km initial semi-major axis the prograde satellite orbits using both the RG and SG models are unstable, while the retrograde satellite orbit with RG is partially stable compared to SG (ref. Fig 5). The retrograde satellite with SG is ejected out of the system and the satellite with RG crashed into the asteroid.
INTRODUCTION
Near Earth Objects (NEOs) have now been targets of space exploration for two decades. The high interest in studying NEOs is due to the fact that they can provide in-situ resources for deep space travel, to better understand how to mitigate the threats from potential collisions with Earth and most importantly, to study the history and evolution of the solar system.
Although, ground based radar observations help to study these objects, close proximity operations help to properly analyze the interiors, mine, relocate or mitigate the threat by a NEO. There have been many spacecraft missions to asteroids like 4 Vesta, 1 Ceres and Itokawa. Operation of these spacecraft around these asteroids were challenging because of their irregular shapes and tenuous, irregular gravity fields. Hence, the initial characterization of the gravity field is important, and the failure to perform the mapping with the desired precision may result in mission failure.
Figure 3: Shape models of Itokawa derived by ground radar (left) and spacecraft observations (right)
Fig 8 & 9: Variations in ‘a’ and ‘e’ for 0.5km initial semi-major axis
Figure 5: Left: Unstable prograde orbits with 0.5km semi-major axis around Itokawa. Right: Partially stable orbits with 0.5km semi-major axis around Itokawa.
The values of ‘a’ and ‘e’ change very drastically in couple of hours of orbit epoch. In contrast, the retrograde satellites are partially stable for a day and show the same trend as prograde satellites. It is observed that the retrograde satellite with RG orbited the asteroid for longer duration while the satellite with SG showed chaotic trend with increase in ‘a’ and ‘e’. Similar trends occur for orbits with 0.75km semi-major axes.
In this study, Itokawa was chosen because of its relatively small size compared to other asteroids visited by spacecraft, and because of the complex maneuvers that were executed in the close-proximity environment.
The radar observations of Itokawa displayed a near tri-axial ellipsoid shape with semi axes x x km in x, y and z directions respectively [1].
A radar gravity model (RG) using the spherical harmonic approach was determined by Scheeres et al., 2004 assuming a constant density of 2.5 gm/cm3 [2]. Hayabusa spacecraft observations indicated Itokawa is a rubble-pile with a density of 1.98 gm/cm3 and the semi-axes of x x km. The shape obtained by satellite observations showed Itokawa as potato shaped with two lobes attached to one another [3]. The spherical harmonic spacecraft gravity field (SG) was derived using the shape model by Scheeres et al., 2006 [3]. In this project, we studied the orbits (change in eccentricity (e) and semi-major axis (a)) of satellites around Itokawa using both the above mentioned gravity fields.
CONCLUSIONS
RESULTS
Retrograde orbits are more stable and suitable for close proximity operations than prograde orbits around Itokawa (as shown by Scheeres, 2006)
Satellites under the influence of radar derived gravity field have shown stable orbits around Itokawa.
Satellites under the influence of spacecraft (Hayabusa) derived gravity field have experienced chaotic behavior.
Hence, radar derived gravity field is not sufficient to perform near surface orbit operations due to less precision of shape and gravity.
The satellite derived gravity field shown to be more precise than the radar derived gravity model due to higher precision in shape model and gravity field.
In future, we plan to model the gravity field using Ellipsoidal Harmonic Model (EHM), Multi Resolution Quadrature Spheres (MRQS), etc., with more point densities and also taking into account the density variations in the asteroid.
For the above orbit simulations, the variations of semi-major axis (a) and eccentricity (e) on x-axis are graphed over time on y-axis using report and graph manager tool.
NEO ENVIRONMENT
Figure 1: Landing sequence during the first touch down attempt by the Hayabusa spacecraft on asteroid Itokawa.
The comparison of gravity fields is done with scenarios involving close proximity orbit operations around Itokawa using AGI’s System Tool Kit (STK) program. The steps involved in creating and analyzing the spacecraft trajectories include:
1. Shape model of Itokawa: The shape of Itokawa was modeled as a tri-axial ellipsoid using the component browser in Systems Tool Kit (STK).
2. Satellites: Two sets of satellites having prograde and retrograde orbits respectively with each set containing 3 satellites with semi-major axes 1.0km, 0.75km, 0.5km were simulated for two weeks around Itokawa.
3. Gravity Fields: The gravity models derived by Scheeres in [1] and [2] are incorporated into satellite models to simulate close proximity operations.
The most primitive and common model of gravity estimation is the spherical harmonic gravity field model often called the “exterior gravity model”. It represents the gravity field as a linear combination of an infinite number of orthogonal functions on the sphere.
𝑈 𝑠𝑝(𝑒) 𝑟,𝜆,𝛷 = µ 𝑅 𝑙=0 ∞ ( 𝑅 𝑟 ) 𝑙 { 𝑚=0 𝑙 [ 𝐶 𝑙𝑚 cos 𝑚𝜆 + 𝑆 𝑙𝑚 sin 𝑚𝜆 ] 𝑃 𝑙𝑚 ( sin 𝛽)}
REFERENCES
[1] Ostro, S.J. et al. (2004) Meteoritics & Planet. Sci., 39(3), [2] Scheeres, D.J et al., (2004) AIAA [3] Scheeres, D.J et al., (2006) AIAA
Where
𝒓,𝝀,𝜱 are the radius of a point outside the Earth (satellite), latitude and the longitude respectively,
R is the radius of Earth,
µ is the gravitational parameter,
𝐶 𝑙𝑚 and 𝑆 𝑙𝑚 are the normalized harmonic coefficients of the geopotential of degree l and order m and,
𝑃 𝑙𝑚 are the normalized Legendre polynomials and associated functions.
ACKNOWLEDGEMENTS
Fig 6 & 7: Variations in ‘a’ and ‘e’ for 0.75km initial semi-major axis
Figure 4: Left: Partially stable prograde orbits with 0.75km semi-major axis around Itokawa. Right: Stable orbits with 0.75km semi-major axis around Itokawa.
We would like to thank and acknowledge the following for their help in making this research possible:
Department of Space Studies
NSF grant through ND EPSCoR
Figure 2: Definition of the potential