SURVVIVABILITY ANALYSIS IN THE SHADOW OF APOLLO: PART II – SPACECRAFT CHARGING VULNERABILITY NEAR THE STABLE EARTH-MOON LAGRANGE POINTS
By Maj. Robert Bettinger, Nathan Boone, Maj. Nicolas Hamilton, and Lt. Col. Bryan Little
Though the reality of routine travel and sustained operations 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. 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.
In Part I of this two-part series— published in the fall 2021 issue of Aircraft Survivability —the threats to spacecraft due to kinetic impacts resulting from catastrophic explosions were examined, as was the likelihood of survivability assessed for a variety of different locations in cislunar space. Now, in this second part of the series, the spacecraft threats shift away from kinetics to that of electrical charging for spacecraft operating in the dust-enveloped environment of the stable L4 and L5 Lagrange points.
As more spacecraft begin to operate in cislunar space, an accurate understanding of the threats to spacecraft due to debris and the general natural space environment is needed. Certain orbits in cislunar space may be especially vulnerable to environmental effects, particularly near the L4 and L5 Earth-Moon Lagrange points, which are stable in the restricted three-body problem. These points could be useful for space operations due to this very stability, thus allowing a spacecraft to remain relatively stationary in a rotating reference frame with minimal station-keeping. However, this stability leads to an accumulation of dust grains and meteoroids.
(Note that, in many sources, dust grains are considered as a subset of meteoroids, and no distinction between the two types of natural debris is observed. For this article, a dust grain refers to a grouping of atoms ranging in size from tens of atoms to 1,012 or more atoms in micrometer-scale particles . By comparison, meteoroids are larger rocky or metallic objects smaller than asteroids and ranging in size from several centimeters to approximately 1 m in diameter.)
Several previous studies have investigated the existence of “Kordylewski clouds,” which are accumulations of lunar and interplanetary dust and debris observed near the Earth-Moon L4 and L5 points. In their study of the Kordylewski clouds, Slíz-Balogh, Barta, and Horváth noted that dust grains and rock-sized particles should be able to circulate in the vicinity of the L4 and L5 Lagrange points for weeks or even months . The potential risk to spacecraft operations from these natural debris objects has been largely unstudied.
The research described herein examines the risks to spacecraft due to dust grain accumulations at the Earth-Moon L4 and L5 points from the perspective of spacecraft charging rather than particle kinetic impact. For an assessment of kinetic impact risk, see Boone and Bettinger . Spacecraft charging is a phenomenon that can produce effects such as localized arc-discharging and a transient broadband electromagnetic pulse. These effects can cause onboard electronic components to fail or suffer degraded performance. If the charging is great enough on the exterior and/or internal spacecraft structure, then localized component heating and material sputtering may also occur .
Using a trajectory model that incorporates the gravitational influences of the Earth, Moon, and Sun, this article simulates the motion of dust particles trapped near L4 and L5 to assess the risks of spacecraft charging to nearby spacecraft. Research into the motion and resulting charge effects of dust grains near the stable Earth-Moon Lagrange points will enable an improved understanding of the potential risks posed by the Kordylewski clouds and ultimately enhance operational planning for potential Lagrange point missions in the future.
NATURAL DEBRIS ENVIRONMENT
The natural debris environment in cislunar space has been a subject of research since the mid-20th century. In a 1961 study, Singer  analytically predicted the existence of a dust shell around the Earth with a concentration a few times higher than the concentration of dust in interplanetary space. In a 1965 NASA report, Burbank, Cour-Palais, and McAllum  developed expressions for the flux of meteoroid particles in cislunar space as a function of both physical size and mass. This report noted that the size of meteoroids in the Earth-Moon system is greater than 0.3 μm for metallic particles because smaller particles are swept out of the Solar System by solar radiation pressure.
More recently, Altobelli, Grün, and Landgraf  studied data collected from the Helios spacecraft to analyze the density, composition, and interaction of interplanetary dust with solar radiation pressure near Earth. In this article, the ratio of solar radiation pressure to the gravitational force was calculated as a function of particle mass. This ratio is extremely small for particles heavier than approximately 10-10 kg, thus indicating that gravity imparts a more significant influence on the motion of larger particles.
Objects orbiting the L4 and L5 points are in stable equilibrium and will tend to return to those points if perturbed. Any object orbiting at one of the collinear Lagrange points—that is, L1, L2, or L3—is in unstable equilibrium and will tend to diverge from these points if perturbed. Therefore, natural satellites at the L4 and L5 points are common.
For example, astronomers have observed more than 7,000 asteroids at the Sun-Jupiter L4 and L5 points . Due to the stability of these points, Polish astronomer Kazimierz Kordylewski first began searching for large objects near the Earth-Moon L4 and L5 points with a telescope in 1951. After the initial search proved unsuccessful, Kordylewski instead began looking for a cloud of small dust particles collectively visible on dark and clear nights. He first observed large patches of dust near the L5 point in 1956 and then succeeded in photographing the dust patches in 1961. There have since been numerous attempts, both successful and unsuccessful, to observe these clouds, which are now known as the Kordylewski clouds.
Several papers have sought to explain the dynamics of the hypothesized Kordylewski clouds and demonstrate their existence. In 1964, Pohle  concluded that a three-body dynamical model allows for the motion of dust cloud shapes like those observed by Kordylewski. Salnikova, Stepanov, et al. have extensively studied the Kordylewski clouds and showed that dust is stable at the L4 and L5 points even with the dynamical inclusion of gravitational perturbations from the Sun as a fourth body [9, 11–13]. These studies included the optimal times for maximum visibility of the clouds as seen from Earth , the motion of charged dust grains in the clouds , the development of a probabilistic model for the distribution of dust , and the possibility of multiple dust clouds .
In 2018, a detailed study by SlízBalogh, Barta, and Horváth [4, 14] was reported to have confirmed the existence of the Kordylewski clouds. In the first phase of their study , the authors sought to determine if lunar or interplanetary dust would become trapped near the L5 point by simulating the movement of 1,860,000 particles using four-body dynamics. Computer simulations reveal that the cloud structure and density are not constant as a function of time. The resulting distribution of particle positions matched well with optical analysis conducted using ground-based imaging polarimetry in the second part of the study . From this optical analysis, polarization patterns indicate an excess of silicate or limonite rather than ice or metallic particles in the clouds. By comparison, the number density of particles comprising the Kordylewski clouds is greater than the density of the background zodiacal dust.
Dust Grain Characteristics
The Kordylewski clouds likely comprise dust grains and micrometeoroids originating from interplanetary and lunar sources. From Misra and Mishra (2013) , interplanetary dust clouds are typically fine-grain mixtures of thousands of millions of mineral grains and amorphous components. The dust is primarily composed of graphite and silicate particles, with sizes ranging from several angstroms to a few millimeters in approximate diameter [2, 15]. Graphite grains are characteristically identified as being “metallic,” while silicate grains are “dielectric” . Lunar dust and soil, on the other hand, are primarily composed of silicates . For lunar dust and regolith studies, Marshall, Richard, and Davis  indicates that silica fume and limonite dust (Fe2O3) are sometimes used as experimental surrogates due to their dielectric and ultrafine properties, while basalt (a silicate, mainly SiO3) is considered to be a “reasonable sample” for the optical properties of the lunar regolith .
The two main components of the spacecraft charging simulation are the trajectory generation and spacecraft charging models. The governing equations, mathematical relationships, and analytical assumptions used for each component of the simulation are discussed in this section.
Bi-Circular Restricted Four-Body Problem (BCR4BP)
Although the Circular Restricted Three-Body Problem (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. An extension of the CR3BP, the BCR4BP 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 1. The equations of motion are again derived in the Earth-Moon barycentric rotating reference frame shown in the figure, and these equations form the main trajectory model for dust grain motion in this article.
The second-order nondimensional equations of motion of the BCR4BP in the Earth-Moon rotating frame are
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
Spacecraft and Dust Grain Initial Conditions
Dust grains are assumed to be initially moving extremely slowly relative to a generic spacecraft positioned at the L4 and L5 Lagrange points. The spacecraft is represented as a sphere with a radius of 10 m to account for solar arrays and/or other hardware extending from the central bus structure. The dust grain initial conditions are similar to those used by Slíz-Balogh, Barta, and Horváth  in their study of the Kordylewski clouds. The grains are distributed randomly in position and velocity using random numbers generated from a standard normal distribution, as shown in Table 1, with the parameter Zi representing a random number generated from the standard normal distribution. All generated random numbers are scaled by 10-7 for position and by 10-6 for velocity.
The dust clouds are assumed to be homogeneous in composition, containing only spherical silicate grains with a uniform particle mass density of 2 x 10-18 kg⁄m3 and a radial size of 0.1 μm [20, 21]. The Kordylewski cloud temperature is assumed to be t ≈ 250 K, which is equivalent to the approximate temperature of background interplanetary dust particles . Only dust particles that could induce charging effects in the event of a kinetic impact are considered in this article; therefore, all particles larger than 0.1 μm are ignored to isolate potential charging-derived damage effects. The simulation assumes that all dust grains, similar to lunar dust, will adhere to the spacecraft following impact and not feature any specular motion with respect to the spacecraft surface . Lastly, the simulation will not model dust mass flux into and out of the Earth-Moon system.
In terms of electrical characteristics, silicate grains may feature a maximum positive potential of approximately +14 V when exposed to the maximum phase of solar wind at 1 astronomical unit (AU), or the mean distance of the Earth from the Sun. Mukai  notes a potential range of +4.2 to +6.5 V for an average solar wind phase at 1 AU, while Mann  indicates an average potential of +3.2 (+0.5/–0.05) V. In the maximum solar wind phase, the electric potential for silicate grains becomes nearly independent of grain radius. In the interplanetary medium, the dominant dust charging is process photoelectron emission, followed by secondary electron emission, and electron/ion accretion on the dust particle surface [15, 20]. Despite size variations, dust grains of the same material in complex plasma exhibit a uniform electric potential, and the charge on a particle is generally proportional to its grain radius .
In the Earth-Moon system, silicate grains are assumed to exhibit a maximum positive potential of approximately +14 V when exposed to the maximum phase of solar wind; for average solar wind phase, this potential reduces to between +4.2 and +6.5 V . The dependence of grain electric potential (Φ) on solar radiation incidence angle (α) is approximated the following cosine function:
where Φd is the approximate maximum potential of +14 V for silicate dust grains and α ∈ [0,π⁄2] rad.
According to , the surface charge of silicate grains is approximated by
where ∈0 is the electrical permittivity of free space and rd is the grain radius. For grains with a radius of 0.1 μm, q = –240e and e = 1.60217653 x 10-19 J⁄eV, or a single elementary charge. When silicate grains impact the spacecraft surface, the collected electric current is given by 
where kB= 1.3806505 x 10-23 J⁄K (Boltzmann’s constant), n is the number density, v is the particle speed, T is the particle temperature, q is the charge, and rSC corresponds to the spacecraft radius.
For comparison, the current induced by the ambient solar plasma on a spacecraft at L4 and L5 is also computed using equation (9). In total, the plasma contains electrons, ions, and neutral atoms. The average plasma number density is assumed to be 5 x 106 m-3, with a temperature of 2 x 105 K and a speed of 500 km/s based on an average speed range of 300–800 km/s expected for solar wind in the Earth-Moon system [15, 20, 24]. The electric potential of the solar plasma is estimated with the following expression :
where the denominator is again representing a single elementary charge and Tplasma is the plasma temperature. The interactions from plasma particles originating from the interstellar medium are assumed to be negligible .
ANALYSIS AND RESULTS
The following sections provide the results of the spacecraft charging study for silicate dust grains simulated using the BCR4BP trajectory model. Dust grains were simulated for the five initial start times of cases t0 = [0.0,0.2,0.4,0.6,0.8] TU with Δt = 0.2 TU. For the first case where t0 = 0.0 TU, the simulation begins with the Sun, Earth, and Moon initially positioned in a line (i.e., ω1t and ω2t in Figure 1 equal to zero). In total, the motion and estimated charging effects of 100,000 grains were simulated relative to a spacecraft station-keeping at either the L4 or L5 point for each time case. As an example, Figure 2 illustrates the initial position of 100,000 dust grains about the L4 point.
Spacecraft Charging at L4
As shown in Figure 3, the first simulation case for L4 (t0 = 0.0 TU) experienced a total of 119 grain impacts on the 10-m spacecraft, which corresponds to 0.119% of the initial dust grain population. The dust grain impact geometry is illustrated by the polar histogram subplot of Figure 3, with the angle sweep of the sun vector given by the yellow triangle. Over the simulation duration of Δt = 0.2 TU, the highest induced electric current was 6 x 10-8 mA, which was a function of both the low speed of the grains relative to the spacecraft and the lower grain temperature. Compared with solar plasma featuring an assumed speed of 500 km/s, the maximum relative grain speed for this case is 0.05 m/s. Based on the assumed characteristics for solar plasma in cislunar space, equation (10) estimates the induced current as 1.8 mA—a value several orders of magnitude greater than the current induced by impacting dust grains.
Similar results are shown in Figures 4 and 5 for cases with t0 = 0.4 TU and t0 = 0.8 TU, respectively. Of note, the number of grain impacts increases from 119 (for t0 = 0.0 TU) to 151 (or 0.151%) and 141 (or 0.141%) for these cases, respectively. Changes to simulation start time also produced changes to the ingress direction or incident angle of the dust grains relative to the spacecraft. While the t0 = 0.0 TU case produces the greatest percentage of impacts in the second quadrant relative to the spacecraft, this starts to shift to the first and third quadrants for t0 = 0.4 TU and t0 = 0.8 TU cases. Table 2 lists the following impact characteristics at L4 for the five initial start time cases: the total number of grain impacts, the maximum average relative speed of the grains, the maximum electric current induced on the spacecraft, and the impact direction in terms of quadrant of arrival for the dust grains.
Spacecraft Charging at L5
To analyze charging risks at the L5 point, silicate dust grains are also distributed about L5 according to the initial conditions shown in Table 1. As with the simulation of grains about L4, dust motion is propagated for the five initial start times of cases t0 = [0.0,0.2,0.4,0.6,0.8] TU with Δt = 0.2 TU. For these simulation cases, approximately 0.12% of grains from the initial cloud impacted the spherical spacecraft. Similar relative speeds and induced electric current were observed for grains impacts with the results obtained for the L4 simulations. Grain impact characteristics for case start times of t0 = 0.0,0.4, and 0.8 TU are shown in Figures 6, 7, and 8, respectively, for L5.
When compared, noticeable differences between the dust grain behavior at the L4 and L5 points arise in terms of impact geometry and average relative speed. While the impacts at L4 were primarily focused in the first and second quadrants, the impacts at L5 shift to the second and third quadrants. For speed, the impacts at L4 were in the range of 0.05–0.06 m/s, whereas the speed increased to 0.07–0.09 m/s for L5. A combined listing of grain impact characteristics at L5 is given in Table 3.
Overall, this preliminary analysis reveals the risk to spacecraft from external charging induced by dust grain impacts at L4 or L5 to be extremely low. This low risk is due to both the slower speed and lower temperature of the dust grains in the Kordylewski clouds relative to the solar plasma. Even though the solar plasma does produce an appreciable charging risk, a similar risk is assessed to be negligible for silicate dust grains over short time periods.
In terms of internal charging, Leach and Alexander  states that electron energies on the order of 10 keV to several megaelectronvolts are required for charged particles to penetrate external spacecraft surfaces and deposit electric charge within internal components and structures. By comparison, the estimated dust grain energy is 0.0215 eV, while the plasma energy is approximated at 17.235 eV. These energies do not breach the penetration threshold of external surfaces and, as a result, internal charging due to the in situ natural environment at L4 or L5 is assessed as low.
Internal charging is indeed possible; however, such occurrences would likely be the result of charged particles leaking into internal spacecraft components and structures through the inadequate external shielding and/ or the introduction of highly energetic particles into the environment from interstellar origin. Although the spacecraft was assumed to be spherical in shape so as to promote system design independence, this assumption does mask potential effects due to differential charging resulting from dust and plasma impacts on complex structural geometries and different external materials.
While the risk of spacecraft charging, as shown by this analysis, is patently extremely low, the effects of prolonged spacecraft exposure to dust grains must be studied with respect to dust adhesion in order to assess long-term spacecraft electrical vulnerabilities at L4 and L5. With the Kordylewski clouds likely containing, in part, similar and/or identical dust to that which comprises lunar regolith, the likelihood of the Kordylewski cloud dust adhering to spacecraft surfaces is high. As dust coats surfaces such as payload optics or solar panels, component performance will become hindered and the dust charging effects may create a long-term accumulative charging hazard.
In future studies, the models developed in this article will need to be expanded to account for solar maximum and minimum variation in solar plasma and perturbative solar radiation effects on the dust grain trajectories. The addition of dust mass flux into and out of the Earth-Moon system will also be needed, with inbound mass originating from both interplanetary and lunar sources . The absence of this mass flux component in the simulation represents one of many limitations to the present analysis and confines all current assessments to short-duration “snapshots” of potential charging activity with respect to Kordylewski cloud grain motion at L4 and L5.
The goal of any space system is to maintain mission functionality for the planned mission lifetime, and spacecraft survivability enables the attainment of this goal. As mentioned previously , spacecraft survivability is a function of three time-separated phases—namely, susceptibility, vulnerability, and recoverability. For susceptibility, analysis focuses on the threat system or mechanism and its ability to successfully detect, be employed, intercept, and finally function as intended vis-à-vis the target space system. A spacecraft’s vulnerability relates to its ability to “survive” the threats’ intended effects. Recoverability is the ability of a spacecraft (and the spacecraft operators), 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 .
Figure 9 depicts the general survivability considerations for spacecraft, with the Venn diagram outlining the aspects of susceptibility, vulnerability, and recoverability. Overall, the spacecraft charging analysis reveals that the vulnerability risk for spacecraft at L4 and L5 is extremely low.
For susceptibility, the continued operation of spacecraft at these Lagrange points will leave them susceptible to dust impact and low-level charging due to the continuous presence of dust particles accumulating in these gravitational wells. Susceptibility to charging can be reduced by ensuring the proper shielding of electrical components, both external and internal to the spacecraft bus.
Finally, for recoverability, spacecraft operating at L4 and L5 will be able to overcome and recover from potential electric charging without considerable mission impact due to the assessed low-level risk of such charging. Recoverability could be enhanced by conducting minor re-positioning of spacecraft relative to the L4 and L5 points if space weather forecasts accumulations of dust and debris and/ or higher-than-normal levels of inbound solar radiation due to solar flares or coronal mass ejections.
[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 (AFIT). Readers are encouraged to contact the authors for more information on this program.]
ABOUT THE AUTHORS
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 (AFRL) 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.
Mr. Nathan Boone is a doctoral student in the Department of Aeronautics and Astronautics at 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. Nicolas Hamilton is an Assistant Professor of Electrical Engineering and the Program Chair for the Graduate Computer Engineering degree program at AFIT. His research focus areas include spacecraft survivability, radiation hardening spacecraft electronics through hardware and software redundancy, and field programmable gate arrays. Maj. Hamilton has also served as a mission manager for Department of Defense research, development, test, and evaluation space vehicle missions at the Space Test Program in the Advanced Systems and Development Directorate, Space and Missile Systems Center. He holds a doctorate in electrical engineering from AFIT.
Lt. Col. Bryan Little is an Assistant Professor of Astronautical Engineering at AFIT. His research focus areas include space domain awareness, cislunar dynamics, constellation design and optimization, and sensor tasking. Lt. Col. Little has also served as Assistant Director of operations at the Maui Space Surveillance Site in the AFRL Directed Energy Directorate and as a system requirements lead in the former Headquarters Air Force Space Command. He holds a doctorate in astronautical engineering from Purdue University.
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