SURVIVABILITY ANALYSIS IN THE SHADOW OF
APOLLO: PART I – CISLUNAR DEBRIS PROPAGATION
AND SPACECRAFT VULNERABILITY
By Mr. Nathan Boone and Maj. Robert Bettinger
Image Courtesy of Pixabay
Although the reality of routine travel and sustained operations and habitation in space likely remains several decades away, the building blocks of space exploration and space system development necessary to realize this reality are currently being laid by peer, near-peer, and emerging space-faring nations. Traditionally, global space operations were notionally limited to near-Earth space, with mission altitudes extending to geosynchronous or even highly elliptical orbits. During the 2010s, however, space operations began moving beyond this paradigm to encompass cislunar space with reinvigorated U.S. initiatives to return to the Moon via the Artemis program, planned commercial space projects, and an accelerated international push toward further lunar exploration.
Moreover, recent international activity includes plans to develop a joint Chinese-Russian base at the lunar south pole in the 2036–2045 time frame , China’s Chang’e-5 lunar sample-return mission in 2020 , Israel’s attempted lunar surface mission in 2019 , and China’s Chang’e-4 far-side lunar mission in 2018. Of particular note, the Chang’e-4 mission was accompanied by the Queqiao communications relay satellite orbiting the L2 Earth-Moon Lagrange point . Most likely then, international missions in cislunar space will increase throughout the 2020s, with a likewise increase in the number of spacecraft operating in this region as scientific exploration expands, space system technology evolves, and the lunar economy emerges and develops.
Most people are familiar with terms such as “low Earth orbit” and “geostationary orbit,” but what exactly is “cislunar”? Measured in hundreds of thousands of kilometers, the term represents the toroidal volume of space extending from approximately twice the geosynchronous altitude to orbits that extend beyond the average Earth-Moon distance. This altitude includes the Earth-Moon Lagrange points, a series of stable and unstable locations generated within a multibody gravitational field.
Unlike near-Earth space, the cislunar domain features unique challenges to space operations related to the spatial magnitude and weather considerations of the environment. Furthermore, as cislunar space becomes increasingly competitive, congested, and contested in the coming decades, the pursuance of sustained cislunar space operations will necessitate a renewed examination of spacecraft survivability and the capability of spacecraft to avoid and/or withstand the natural and manmade environmental risks posed by this “new,” ultimate high ground.
This article is the first in a two-part series provided to help readers better understand spacecraft survivability in cislunar space. Presented here is emerging research into spacecraft mission risk with respect to space debris propagation and kinetic impact. The second part, which will be published in a future issue of Aircraft Survivability, will discuss emerging research into the solar-induced environmental phenomena of spacecraft charging.
NEW OPPORTUNITIES AND NEW RISKS
The U.S. Air Force (USAF) has increasingly expressed interest in the strategic opportunities presented by space between the Earth and the Moon. This space, known as cislunar space, offers to serve as a new “high ground” for space operations, allowing a positional and logistic advantage over other space-based assets. That said, with new opportunities also come new risks. As more spacecraft operate in cislunar space, a catastrophic spacecraft mishap in this region becomes more likely, with each mishap generating a significant number of debris fragments.
Many studies have investigated the threat of debris events in lower Earth orbits. These concerns have grown with recent anti-satellite (ASAT) tests performed by China in 2007  and by India in 2019 , which generated debris clouds that continue to threaten other spacecraft. Several satellites have also suffered failures that led to breakup events, including the National Oceanic and Atmospheric Administration (NOAA) 16 weather satellite in 2015, the U.S. Air Force’s Defense Meteorological Satellite Program (DMSP) satellite in 2016, and the Japanese Hitomi spacecraft in2016 . The 2009 collision of Cosmos 2251 and Iridium 33 also created two large fragment clouds . These fragmentation events could lead to a cascading series of collisions, eventually leaving certain orbits so full of debris that they are unusable, a phenomenon known as “Kessler Syndrome” .
While the threat of debris in orbits near Earth has been heavily studied, few prior studies have investigated the effects of a debris event in cislunar space. A catastrophic spacecraft mishap in cislunar space could pose risks to other operational spacecraft by creating a cloud of debris particles that intersect much lower orbits. Debris particles produced by a catastrophic spacecraft mishap could also begin to “pool” in certain stable orbits in cislunar space, potentially creating significant debris hazards there. Furthermore, the lack of atmospheric drag in cislunar space could lead to debris that remains in space indefinitely, potentially creating a permanent threat to cislunar spacecraft.
In support of the study of potentially damaging cislunar debris effects, there are four main aspects of debris simulations: the trajectory generation model, the reference trajectories for the notional spacecraft, the parameters of the catastrophic spacecraft mishap, and the spacecraft survivability models. The models used for each component of the simulation are discussed in turn in the following paragraphs.
Circular Restricted Three-Body Problem (CR3BP)
The CR3BP may be used to model the motion of a spacecraft of negligible mass moving in the Earth-Moon gravitational system, assuming that the Earth and Moon move in coplanar circular orbits about their barycenter with a constant angular velocity. The equations of motion are given in the coordinates of the Earth-Moon barycentric rotating reference frame shown in Figure 1.
The nondimensionalized, second-order equations of motion in the rotating reference frame of the CR3BP are expressed as the following:
and the value of the dimensionless parameter μ is 0.01215. The CR3BP employs canonical units rather than standard dimensional units due to the scale of astronomical motion in the Earth-Moon system. As a result, the Distance Unit (DU) is defined as the distance between the Earth and the Moon, or 384,400 km. Likewise, the Time Unit (TU) is defined as the time for the system to complete one 2π orbit, or 375,190 s . Equilibrium points in the CR3BP are computed by first setting the accelerations and velocities in equations 1–3 equal to zero. Solving for the positions based on this zero-condition gives the five Lagrange points shown in Figure 2 and the accompanying two-dimensional coordinates for these points in Table 1.
Bi-Circular Restricted Four-Body Problem (BCR4BP)
Although the CR3BP provides significant insight into the dynamics of the Earth-Moon system, a model incorporating the gravitational influence of the Sun is preferable for long-term trajectory modeling. The BCR4BP is an extension of the CR3BP, which includes solar gravity and assumes that the Earth and Moon move in circular orbits about their barycenter. The barycenter itself revolves in a circular orbit about the Sun-Earth-Moon barycenter, with all orbits assumed to be coplanar in geometry. A diagram of the BCR4BP is shown in Figure 3.
The second-order nondimensional equations of motion of the BCR4BP in the Earth-Moon rotating frame are as follows:
The angular velocity of the system about the Sun-Earth-Moon barycenter, ω1, is given by the following expression:
With respect to the Sun-Earth-Moon barycenter, the coordinates of the Sun are computed by
with the distance from the Sun to the particle given by
The distances to the Earth and Moon are given in equations 4 and 5, respectively.
Catastrophic Mishap Model
The NOAA 16 satellite battery explosion was used as the model for the spacecraft catastrophic mishaps . The two parameters of interest in the simulation are the mass distribution of particles released in the explosion and the change in velocity (ΔV) given to each particle by the explosion. The particle masses generated in the explosion were modelled by fitting a lognormal probability distribution to the observed mass distribution in the NOAA 16 explosion, then selecting random numbers from this distribution to assign each particle mass. The NOAA 16 explosion mass histogram, with the lognormal probability distribution overlaid, is shown in Figure 4. In the simulation, particle masses were selected randomly from the lognormal distribution until the combined mass of the particles matched the mass of the original NOAA 16 satellite of 1,457 kg. Therefore, the total number of particles simulated is random and changes with each simulation.
Particle ΔV was determined from the particle mass using the average kinetic energy given to particles by the NOAA 16 explosion. All particles are assumed to be given the same kinetic energy by the explosion, and the explosion is assumed to be omnidirectional. These assumptions produce simulated explosion velocity vectors such as those shown in Figure 5.
Spacecraft Survivability Model
A method for determining the survivability of a spacecraft in a cloud of particles is the Poisson approach shown by Ball , wherein the number of hits to the spacecraft is a random variable, with an expected number of hits E. If the cloud of debris fragments is treated as a spray of M penetrators within a volume VS, then the penetrator spray density ρ is
where VS is the volume of a spherical “danger zone” surrounding the spacecraft, within which the particle density is calculated. For the present research, this danger zone has a radius of 10,000 km. Once ρ has been calculated through debris propagation modeling, the expected number of hits on a spacecraft’s “hazard zone,” considered to have a volume of VHZ, is given by
As in the spacecraft fragmentation study by Bettinger and Hess , VHZ defines a sphere around the spacecraft such that any particle that enters this volume is considered to have hit the spacecraft. The hazard zone is like an error ellipsoid around the spacecraft and avoids numerical precision issues related to determining the exact locations of the spacecraft and debris particles at cislunar scales. For the present research, the hazard zone has a radius of 500 m.
With a probability of kill given a hit of PK|H, or the probability that a particle will destroy the spacecraft given that it strikes it, the instantaneous probability of system hazard is given by
This is the result of the spacecraft survivability study. For each simulation, the value of PHZ is tracked over time. The total probability of hazard during a time interval t0 to tf is the area under the PHZ curve with respect to time:
Probability of Kill With a Hit
As shown in equation 14, the probability of hazard PHZ depends on the chosen probability of kill with a hit PK|H. For this research, two separate models are applied to calculate the PK|H. First, a simplified model that includes only particle mass as a factor in determining PK|H is presented. Second, a more robust model that includes both particle mass and velocity is developed using ballistic limit equations for spacecraft shields. Both models are used to calculate survivability in this research, and the PHZ results for each model are compared.
Once the particle masses have been determined through the spacecraft catastrophic mishap model, PK|H can be determined based on the severity of damage impacts from particles of those masses are likely to cause to a spacecraft. The first PK|H model uses a variable PK|H that depends on particle mass. A logistic curve was used to model the dependence of the PK|H on the mass of particles. Logistic curves are often used to measure vulnerabilities in other applications, including the likelihood of bridge collapse in earthquakes of varying magnitudes [13, 14]. Studies of the damage to spacecraft caused by particle impacts of various sizes, such as that by Elvidge , can aid in determining the parameters of the logistic curve for the present research. The assumed model for the logistic curve for PK|H is shown in Figure 6. The equation for this logistic curve is
where m is particle mass and values chosen for the parameters A, B, C, K, Q, and v are given in Table 2.
RESULTS: CISLUNAR DEBRIS CASE STUDIES
The following sections provide highlights of the results of the cislunar debris case studies simulated through this research effort. Each case study examines a catastrophic spacecraft mishap in a particular pre-explosion trajectory. The effects of the mishap are analyzed in terms of the trajectories of the resulting debris particles and the threats to one or more notional spacecraft operating elsewhere in cislunar space, with risks quantified using the Poisson survivability model. The four case studies presented include simulation of catastrophic mishaps at the collinear. Earth-Moon Lagrange points L1 and L2, during an Apollo-like transfer, at the stable Earth-Moon Lagrange points L4 and L5, and in lunar orbit.
Explosions at the Collinear Lagrange Points L1 and L2
Several recent missions in cislunar space have sought to use the collinear Earth-Moon Lagrange points to complete their missions. For example, China’s recent Chang’e-4 far-side lunar lander was accompanied by the Queqiao relay satellite orbiting the Earth-Moon L2 Lagrange point . The L2 point is an appealing location for communications spacecraft because it enables constant communication with both the far side of the Moon and the Earth. The manned Lunar Orbital Platform-Gateway (LOP-G, or Gateway) station will also use the dynamics of the Earth-Moon Lagrange points when it becomes operational. The Gateway is a crucial part of NASA’s Artemis program to return humans to the Moon and will use a Near Rectilinear Halo Orbit (NRHO) to enable exploration of the Moon. NRHOs are families of halo orbits about the L1 and L2 Lagrange points that are nearly stable, enable constant communication with the Earth, and pass close to the Moon .
In this case study, explosions were simulated at the L1 and L2 Lagrange points; and the resulting risks to the Lunar Gateway, spacecraft at L1 or L2, and a spacecraft conducting a transfer to the Moon were analyzed. The results of this case study are summarized in Tables 3 and 4. The transfer simulation runs were conducted for the duration of the transfer from the Earth to the Moon, while the other simulations lasted 50 days.
This case study showed risks to other regions of cislunar space due to debris that fills the L1 manifold and moves toward the Earth or Moon. The L1 manifold acts as a sort of “highway” that transports objects to and from the L1 point and lunar region; and in this case study, debris circulated through the L1 manifold throughout the simulations. Compared to a catastrophic spacecraft mishap at L2, a mishap at L1 created more risk to lunar spacecraft, such as the NASA’s planned Lunar Gateway space station, due to the significant number of particles that move toward the lunar region following the mishap. Although the risks to other notional cislunar spacecraft are low in each case, the debris following a mishap at either L1 or L2 would circulate cislunar space indefinitely due to the lack of atmospheric drag, thus potentially creating long-term debris hazards.
Mishap During an Apollo-Like Transfer
One spacecraft mishap has already occurred in cislunar space, namely the explosion on the Apollo 13 spacecraft. The next case study analyzed in this research sought to determine if a significant debris-generating event during an Earth-Moon transfer would pose any risk to spacecraft near Earth if it occurred today. The results of this case study in terms of the number of debris particles that end the 50-day simulation with a perigee within geostationary (GEO) altitude, or 35,786 km, are summarized in Table 5. Note that the survivability model was not applied for this case study to instead focus on risks to the crowded environments near Earth.
Of the case studies examined in this research, a catastrophic spacecraft mishap during an Apollo-like transfer results in the greatest risk to spacecraft near Earth, with hundreds of debris particles coming within geosynchronous altitude at high relative velocities during each perigee passage. The particles would likely remain in space for a long time (much longer than the 50-day simulation) due to their high apogees that would decay slowly due to atmospheric drag. Therefore, this type of debris event may be the most concerning of the case studies due to the potential for long-term risks to crowded orbits near Earth.
Mishap at the Stable Earth-Moon Lagrange Points
The Earth-Moon Lagrange points L4 and L5 are stable in the restricted three-body problem, and studies have suggested that natural debris can accumulate at these points [17–18]. Thus, artificial debris accumulation at L4 and L5 could become problematic if these points become crowded in the future. Although no spacecraft currently operate at L4 and L5, these points have many potential uses due to their stability and high visibility over the rest of the cislunar region. This case study analyzed the risk that could be posed from spacecraft explosions at L4 and L5 to another spacecraft at L4 and L5. The results of this case study are summarized in Table 6. The simulation was run for 1 year.
This case study demonstrated that catastrophic spacecraft mishaps at L4 and L5 would pose some risk to other spacecraft operating at those points. L4 and L5 could be an appealing location for future spacecraft due to their stability, but much of the debris from a catastrophic spacecraft mishap at either point would be also stable at those points for at least a year following the explosion. This fact results in a much higher probability of hazard to the notional spacecraft than any prior simulation runs. Debris circulating these points would also likely be difficult to track from Earth due to the great distance, making it difficult to maneuver a spacecraft to avoid debris. However, as with the other case studies, the risk is still low, and debris would only become problematic if these points become more crowded in the future or if repeated mishaps occur.
Mishap in Lunar Orbit
The final artificial debris case study analyzes threats from debris following a catastrophic spacecraft mishap in lunar orbit. Greater numbers of spacecraft may begin to operate in lunar orbit to support lunar exploration and colonization in the coming years, thus increasing the need to study the risks resulting from a catastrophic spacecraft mishap. An improved understanding of the risks from artificial debris in lunar orbit enables an understanding of the possible importance of proper debris management techniques in the lunar environment. Depending on the risks from artificial debris in lunar orbit, disposal strategies such as those used at end-of-life phases for Earth-orbiting spacecraft may be necessary for spacecraft orbiting the Moon.
The results of the lunar orbit debris case study are summarized in Table 7. Prior to the explosion, the spacecraft that explodes is in a 110-km circular lunar orbit much like the one used by the Apollo missions. These simulations were all run for 1 day following the explosion, and the notional spacecraft is another spacecraft in the same orbit located at the Run 1 position at the start of the simulation.
A mishap in lunar orbit resulted in far greater threats to a notional spacecraft operating in the vicinity of the mishap than any of the other debris case studies examined in this research. The gravity well of the Moon causes the debris to remain within a small region following the catastrophic mishap, within which it can continue to threaten the notional spacecraft. In the other cislunar case studies, the debris expanded to fill much greater volumes immediately following the explosion. Lunar mishaps could create even more risk to a particular spacecraft than mishaps in near-Earth environments due to the smaller size of orbits around the Moon, which could give more opportunities for close approaches. The debris may also last longer in orbit due to the lack of atmospheric drag around the Moon. Future studies could use a more robust lunar trajectory model to analyze a variety of lunar catastrophic mishap scenarios over longer time periods to further quantify the risk of such events.
Each case study analyzed in this research resulted in unique risks to other cislunar spacecraft. Mishaps at the collinear Lagrange points and at the stable Lagrange points resulted in slight risks to other cislunar spacecraft, a mishap during an Apollo-like transfer would pose the greatest risk to currently operational spacecraft near Earth, and a mishap in lunar orbit would pose significant risks to another spacecraft operating at the same lunar orbital altitude. Although these risks are generally low, the longevity of debris, difficulties tracking debris in cislunar space, and the potential hazards to other spacecraft mean that care should be taken in future cislunar space missions to avoid generating significant amounts of debris in this region.
In terms of spacecraft survivability, significant debris-generating events in cislunar space could reduce the chances that a threatened spacecraft will be able to maintain mission functionality. The assessment of this ability requires analyzing the traditional, overlapping (and interdependent) aspects of survivability—namely, susceptibility, vulnerability, and recoverability (illustrated in Figure 7). Susceptibility analysis focuses on the threat and its ability to harm the space system. Vulnerability analysis relates to a spacecraft’s ability to “survive” the threats’ intended effects. And recoverability analysis focuses on the spacecraft’s (and its operators’) ability, following damage from a threat system, to take emergency action to prevent the loss of the spacecraft and/ or to regain a level of spacecraft mission capability .
Overall, the susceptibility of spacecraft to cislunar debris is assessed to be low, though it was assessed to be higher in some case studies, such as the lunar debris case study. Spacecraft would be susceptible to cislunar debris impacts if the debris cloud intersects the spacecraft, as these particles could cause significant damage to the spacecraft due to impacts at high relative speeds.
That said, vulnerability to impacts could be reduced by maneuvering the spacecraft, adding additional shielding to the spacecraft, or practicing responsible cislunar spacecraft disposal strategies that avoid generating debris that could threaten operational spacecraft. Furthermore, the ability of spacecraft to recover from a collision with a debris particle would depend on the properties of the impact, including the mass, velocity, and impact angle of the particle that strikes the spacecraft. The location of the impact on the spacecraft, of course, would also influence the spacecraft’s recoverability, as damage to certain critical components (such as antennas or the payload) could ultimately make an impact nonrecoverable.
[Editor’s Note: The analysis presented herein is part of the Cislunar Education, Research, and Technology (CERT) graduate research program at the Air Force Institute of Technology. Readers are encouraged to contact the authors for more information on this program.]
ABOUT THE AUTHORS
Mr. Nathan Boone is a doctoral student in the Department of Aeronautics and Astronautics at the Air Force Institute of Technology (AFIT) and also works on spacecraft system analysis in support of defense-related projects. His research focus includes the propagation of debris in cislunar space following a catastrophic spacecraft mishap, accumulations of debris in cislunar space, and the survivability of spacecraft. Mr. Boone holds a bachelor’s degree in mechanical engineering from the University of Cincinnati and a master’s degree in astronautical engineering from AFIT.
Maj. Robert Bettinger is an Assistant Professor of Astronautical Engineering, the Deputy Director of the Center for Space Research and Assurance, and the Curriculum Chair for the Astronautical Engineering degree program at AFIT. His research focus areas include atmospheric reentry dynamics, cislunar trajectory analysis, and spacecraft survivability. Formerly, he was the senior military analyst for the Counterspace Analysis Squadron at the National Air and Space Intelligence Center, as well as a research engineer in the Air Force Research Laboratory’s Space Vehicles Directorate. Maj. Bettinger is a graduate of the U.S. Air Force Academy and holds a master’s degree in astronautical engineering and a doctorate in astronautical engineering from AFIT.
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