Concluding Remarks

The Battin, Gooding, and Sun algorithms are able to converge for nearly 100% of all orbit combinations. Both the Gooding and Sun algorithms have approximately the same perfor-mance, while the Battin algorithm is consistently slower. The universal variable method is conceptually the easiest method to understand and is the most commonly discussed method in the literature, but it lacks the robustness of the other three methods. If the efficiency of the algorithm is the most important factor, both Gooding’s and Sun’s methods, as implemented here, are likely the best to use.

The performance for each Lambert algorithm, when run on the GPU, has been compared.

Each Lambert solution algorithm is able to compute tens of millions of solutions per second, with the Sun’ method having the best performance at nearly 32 million solutions per second.

This represents an increase in performance of two orders of magnitude when utilizing GPUs over standard CPU algorithms. While even the grid search CPU run times are not high when compared with common high performance computing algorithms, mission optimization algo-rithms often require run times ranging from hours to multiple days. By offloading computations for solutions to Lambert’s problem to GPU(s) and sufficiently parallelizing optimization algo-rithms, performance increases of up to two orders of magnitude can be realized when compared with standard CPU algorithms. This will allow mission designers to quickly compute complex trajectories when evaluating potential mission architectures.

CHAPTER 3. ROBOTIC AND HUMAN

EXPLORATION/DEFLECTION MISSION DESIGN FOR ASTEROID 99942 APOPHIS

In this chapter both robotic and human return missions to the asteroid Apophis are designed and analyzed. These mission are used to validate the solutions to Lambert’s algorithm from the previous chapter, as well as other astrodynamics algorithms such as: solutions to Kepler’s problems, date conversions, coordinate conversions, etc. The missions analysis and design performed in this chapter is done through and exhaustive grid search. The algorithms developed for this search are able to determine optimal launch dates and launch windows for simple rendezvous, direct intercept, and Earth return asteroid missions. These exhaustive search algorithms can then be used to evaluate the global convergence of trajectories found with the hybrid GNLP algorithm. This helps determine which options and algorithms should be used to determine optimal trajectories with the hybrid GNLP algorithm.

3.1 Introduction

Asteroids and comets have collided with the Earth in the past and will do so again in the future. Throughout Earth’s history these collisions have had a significant role in shaping Earth’s biological and geological histories. One major example of this is the extinction of the dinosaurs, which is widely believed to have been caused by the collision of an asteroid or comet.

In recent years, near-Earth objects (NEOs) have also collided with the Earth, the most notable example in recent history is an impact in Siberia, known as the Tunguska event. This impact is estimated to have released an explosive energy of approximately 3 − 5 megatons. While, the

impact occurred in a sparsely populated area, such an impact in a highly populated area would be extremely devastating.

Of all the NEO’s found to date, the asteroid 99942 Apophis is considered one of the most potentially hazardous NEOs and has received much attention from the planetary defense com-munity. However, an impact from Apophis does appear unlikely, with an estimated impact probability of approximately four-in-a-million in 2036. On April 13, 2029, Apophis will pass by the Earth, inside the limits of geostationary orbit. If Apophis passes through a relatively small, approximately 600-meter keyhole, impact will occur on April 13, 2036. A fictional scenario in which Apophis passes through a keyhole in 2029 and impacts with the Earth in 2036 is studied in this chapter. The purpose of this chapter is to perform the mission analysis and design for robotic and human exploration mission to Apophis, using the software algorithms developed for the Asteroid Deflection Research Center (ADRC). Possible launch windows, trajectories, and accompanying ∆V’s for both robotic rendezvous and human piloted return missions prior to the April 13, 2029 Earth-Apophis close encounter will be analyzed. In addition, mission analysis and design will be performed for robotic and human piloted missions for the fictional scenario in which Apophis passes through a keyhole on April 13, 2029, resulting in an impact on April 13, 2036. The orbital current estimated orbital elements of Apophis, the fictional orbital elements, and the estimated physical characteristics can be found in Tables 3.1(a), 3.1(b), and 3.1(c), respectively [26]. For the fictional Apophis mission, launch windows will be determined throughout the 7-year period (keyhole passage through impact), which allow sufficient time for a fictional high-energy nuclear deflection mission.

A preliminary Interplanetary Ballistic Missile (IPBM) architecture, designed by the ADRC, will be used as the reference robotic space system throughout this chapter. The IPBM archi-tecture is similar in design to the ADRC’s hypervelocity asteroid intercept vehicle (HAIV).

The most capable IPBM architecture, uses the Delta-IV Heavy launch vehicle, and is capable of a total ∆V of 4 km/s, carrying a 1500-kg nuclear payload. In addition, the reference depar-ture orbit for the robotic mission analysis, used when determining the Earth-depardepar-ture ∆V, is assumed to be a geostationary transfer orbit [27]. Using this baseline IPBM architecture

and ∆V capabilities, launch windows for both the pre-2029 and post-2029 missions have been determined in [28].