Harvesting Electromagnetic Energy from Hypervelocity Impacts for Solar System Exploration

URSI GASS 2020, Rome, Italy, 29 August - 5 September 2020

Sean A. Q. Young*(1), Nicolas Lee(1), and Sigrid Close(1) (1) Department of Aeronautics & Astronautics, Stanford University, Stanford, California, USA, 94305

Abstract

CubeSats have opened new opportunities for solar system exploration by decreasing mission development and launch costs. While they have proven successful in the inner solar system, providing sustained power to these spacecraft Figure 1. Graphical representation of the HVI energy har- at large distances from the sun requires novel power gener- vesting concept. ation methods. Solar power is dramatically diminished at these distances and radioisotope thermoelectric generators do not scale well to spacecraft with small dimensions. The CubeSats as free-flying sensors can produce plasma mea- space environment could provide a solution; by harnessing surements spread over hundreds of kilometers — a signif- energetic hazards in the environments under investigation, icant improvement over the tens of meters that a boom- the spacecraft could forgo traditional power sources. Radio mounted sensor would provide. This concept has heritage frequency emissions are abundant in the space environment in the Cluster [1] mission for example. with sources originating from planetary aurora, particle dy- namics in the magnetosphere, and electromagnetic pulses Power presents a challenge to small spacecraft in the outer (EMPs) associated with hypervelocity impacts between mi- solar system. The extreme distance from the sun precludes crometeoroids and spacecraft. A case study of a mission the use of solar panels for Cubesats, and radioisotope ther- to the rings of Saturn is presented and analyzed for feasi- moelectric generators are expensive, difficult to handle, and bility. Upper bounds on the power emitted by the hyper- bulky relative to the amount of power delivered. Another velocity impact EMPs are produced by combining results approach under consideration is to use the power within the from ground-based experiments and Saturnian environment space environment to power these spacecraft. Radio fre- models. We find that although the kinetic energy of hyper- quency (RF) sources within the space environment present velocity impacts is significant, the conversion efficiency to a tempting target considering their abundance. EMPs is too low for spacecraft to effectively use. Hypervelocity impacts (HVIs) are one such source of radio 1 Introduction and Motivation frequency emissions. HVIs on spacecraft surfaces and air- less bodies occur frequently in the space environment due Exploration of the outer solar system is an expensive en- to micrometeoroids, dust, and orbital debris (referred to col- deavor and maximizing the science return of the few mis- lectively as impactors throughout this paper). As described sions that will make the journey is of paramount importance in detail in [2], HVIs on metallic targets emit an electro- to mission designers. Nanosatellites (spacecraft between 1- magnetic pulse (EMP) which can interfere with spacecraft 10 kg in mass) such as CubeSats (spacecraft made up of cu- electronics, resulting in mission anomalies. Given that the bic units 10 cm to a side) succeeded in driving down costs emitted energy is significant enough to be destructive, we for missions to Earth orbit and provided platforms for ex- aim to characterize the amount of energy carried by the perimentation with mission concepts, hardware, and pay- EMP and to determine whether it can be harnessed to power loads. While solo flights to the outer planets may be cur- nanosatellites. Figure 1 illustrates this concept using a rently infeasible for them, Cubesats traveling with larger surface-mounted patch antenna. motherships have the potential to advance science while controlling costs. This study analyzes the amount of expected energy available to spacecraft traversing Saturn’s rings from impact For example, nanosatellites can enable distributed sensing EMPs. First, we present general expressions for the up- missions to the outer planets. Magnetospheres are large- per bound of energy available to a spacecraft given an im- scale, complex structures whose properties are constantly in pactor flux model and EMP measurements from laboratory flux. Understanding physical processes that drive the mag- experiments. Characteristics of the Saturnian space envi- netosphere requires resolution of spatial gradients in plasma ronment are derived from spacecraft data while our own properties. Distributed sensing missions using swarms of measurements provide the expected EMP energy. Synthe- sizing these elements resutls in an estimate of the power where A is the impact area and available to these spacecraft. 1 Z a A = r∆Ωdr (3) eff 2 2 Methodology 0 is the effective power collection area. In the small space- A = a2/ = A/ The physics governing the generation of the EMPs are not craft limit, eff π 4 4. well understood, nor is the Saturnian impactor mass and velocity distribution completely known. In addition, the de- 3 Physical models sign space of harvesting circuit topologies is too vast to enu- merate completely or optimize for the harmonic structure Models for the electromagnetic field energy EEMP and the of the EMP. A number of simplifying assumptions, there- meteoroid flux density F(m,u) are needed to proceed fur- fore, are required to obtain a usable power estimate. En- ther. The former can be estimated using detections of EMPs ergy conservation allows us to constrain the EMP energy to from terrestrial impact experiments while the latter is con- a fraction of the impactor kinetic energy, resulting in an up- structed using data from spacecraft measurements. per bound on available energy. Using spacecraft geometry and physical models for the EMP energy as a function of 3.1 Electromagnetic pulse energy impactor properties and the ambient impactor environment, we can compute the expected RF energy flux to a rectify- An empirical power law relation between the equivalent ing antenna (rectenna). This analysis assumes that EMPs isotropic radiated power, the mass m, and velocity u of α β result from every HVI event, that the EMP radiation pattern the impactor is PEMP ∝ m u where α ≈ 1 and β ≈ 3.5 is isotropic, and that HVI measurements from lower veloc- [3]. This relationship agrees with similar empirical laws ity ground-based experiments can be extrapolated to orbital describing the amount of charge produced in HVI exper- speed HVI events on spacecraft. Model-specific assump- iments [4]. We use this proportionality to estimate EEMP tions are presented in their respective sections. over a range of masses and velocities. α β EEMP = Cm u (4) The amount of energy in the electromagnetic fields EEMP is less than the combined impactor kinetic energy and en- Data from experiments can be used to determine the con- ergy stored in the electrostatic field due to spacecraft charg- stant of proportionality C. We recently conducted a set of ing. Since the charging state is variable, we use the kinetic experiments using the light gas gun at the NASA Ames Ver- energy alone as a weak upper limit. Experimental measure- tical Gun Range (AVGR). In these tests, small, 1.6 mm di- ments of EMPs can be used to compute a tighter bound. The ameter aluminum spheres were fired in partial vacuum (0.5 Torr) at speeds between 4-6 km/s. Targets were made of amount of energy Ec stored by a rectenna with frequency- various materials and biased to different voltages mimick- dependent efficiency η0( f ) is ing spacecraft charging conditions. A suite of RF sensors Z ∞ ∆Ω — including patch antennas, dipole antennas, and a 0.9- Ec = η0( f )ESD( f )d f < η0,max EEMP (1) 0 4π 1.8 GHz log-periodic monopole antenna (LPMA) — was placed inside the chamber about 1 m from the target. We where ESD( f ) is the energy spectral density of the incident detected EMPs on two of the thirteen shots (m = 5.9 mg, signal, ∆Ω is the solid angle subtended by the antenna el- u = 5.4 km/s) on aluminum targets biased to -300 V. ements, and η0,max = max f η0( f ) is the peak efficiency of the antenna and rectifying electronics. In the far-field, ∆Ω Given its broadband characteristics, the LPMA was used to would simply exhibit the usual inverse square attenuation estimate the EMP energy. The signal energy within a short with distance. For small spacecraft, however, the rectenna temporal window around the EMP was computed, account- is more likely to be in the near-field. As an example, for an ing for amplifier gain, cable losses, and space losses. For antenna element with height ` and diameter d protruding at ◦ these shots, EEMP ≈ 70 nJ. Using Equation (4), this results 90 from the spacecraft, ∆Ω ≈ π sr for nearby impacts and −15 2 in a proportionality constant C ≈ 1 × 10 when m and u asymptotically approaches 2`d/r as the distance r → ∞. are measured in SI units. Orbital impactors can travel at speeds up to about tens of km/s, bringing with them com- When more than one impactor impinges upon the surface, paratively higher EMP energies. In addition, the flux of the power collected by the rectenna is computed using a impactors can be quite high, especially in the rings of Sat- model of the impactor flux density F(m,u)dudm (number urn. of particles with mass m and u impacting per area per second). Deployable structures can increase the number of im- 3.2 Saturn environment characteristics pacts on the spacecraft and thus the harvested energy. As- suming that the deployable surface is circular with radius Equation (5) gives the in-plane density profile of im- a with the antenna at its center, the power per unit impact pactors within Saturn’s rings derived from Cassini Radio area can be derived from (1) as and Plasma Wave Science data [5]. ∞ ∞ Aeff Z Z ρ p ΓEMP < η0,max dm duF(m,u)EEMP(m,u) (2) n(ρ) = n0 (5) A 0 0 3.95 −3 where n0 = 0.035m , ρ is the orbit radius measured in 0 Saturn radii, and p = 16±2 for ρ < 3.95 and p = −10±1.6 10 for ρ > 3.95. The peak density at ρ = 3.95 coincides with the orbit of Enceladus, indicating a promising destination 10-5 for harvesting energy. Above and below the ring plane, the

density falls, following roughly a Lorenztian distribution in ) the plane normal coordinate. 2 10-10 (W/m The density given in Equation (5) is properly an integration over the impactor mass and velocity distribution. Gurnett 10-15 KE EMP max( ) max( ) [6] approximates the mass distribution using Voyager 2 data KE EMP from a ring crossing at 2.88 Saturn radii. At peak intensity KE,lim EMP,lim during the crossing, several hundred impacts occurred ev- 10-20 ery second. This distribution takes the form of an inverse 0 0.2 0.4 0.6 0.8 1 i/ cube law. 2 dn 2m0 Figure 2. Orbit averaged (solid), maximum (dashed), and = n 3 (6) dm (m0 + m) upper bound (dotted) estimates of power per unit area incident on the spacecraft due to the impactor kinetic energy where m is the impactor mass, m0 is the mean impactor (blue) and EMP energy (red) incident on the spacecraft. mass and n is the density at the orbit radius of inter- −13 est. Though m0 is quoted as 1.44 × 10 g by Gurnett [6], a more recent estimate [7] places the mean mass at 4 Results and Discussion 7.7 × 10−11 g, though the discrepancy could be due to the location of the ring crossings used in the estimation. Com- Equipped with these physical models, we can estimate the bining these two models results in an estimate of the im- utility of using impact EMPs as power for spacecraft for pactor distribution at the orbit of Enceladus. several orbits around Saturn. As a threshold for usefulness, we use 77 mW: the power used by the sprite spacecraft de- Finally, the velocity distribution of the impactor must be es- ployed from Kicksat [8]. timated. For a particle traveling at ~uimp, the impact speed is given by ∆u = k~usc −~uimpk where ~usc is the velocity of the Substituting the models from Equations (4) and (7) into spacecraft. Even for impactors above and below the ring Equation (2), the expression for the power per unit area plane, the angular momentum vector averaged over the dis- ΓEMP satisfies tribution should be oriented along the normal to the ring A plane. Therefore, as an approximation, the mean velocity Γ < eff η nm C(∆u)9/2. (8) EMP A 0,max 0 vector should lie in this plane as well. We estimate the mean speed to be the circular orbit speed with radius equal The remaining parameters Aeff/A, and η0,max still require to the projected radius in the ring plane. In this model, the specification. For small spacecraft, the ratio of areas can impactors orbit on average like concentric cylinders, rotat- be roughly estimated as 1/4 in the a → 0 limit from Equa- p 3 ing at ω = µ/ρ where µ is the Saturnian gravitational tion (3). The efficiency η0,max is assumed to be 10%, but parameter and ρ is the projected radius in the ring plane. in general depends upon the precise antenna and rectifier Although this model tends to overestimate the velocity in topology. polar orbits, the densities in polar regions are sufficiently tenuous that the estimates are not significantly affected. The density and velocity difference are functions of the orbit used by the spacecraft. For simplicity, consider a cir- An upper limit on the impact speed is found when the im- cular orbit with radius equal to that of the semi-major axis ◦ ◦ pactor and spacecraft are traveling in opposite directions. of Enceladus and inclination varying between 0 and 180 . Further, the speed of any impactor in the ring popula- With the approximations used, the peak power for the equation is bounded from above by the escape speed p2µ/r. torial, prograde orbit is zero, then increases steadily with in- u Therefore, the upper limit on ∆u is given by ∆ulim = usc + clination as ∆ grows. Using the concentric cylinder model, p2µ/r. ∆u is found to be

s 3 r µ r r 2 Using either model for ∆u, the velocity dependence can be ∆u = 1 + − 2 cosi (9) approximated as a delta function, peaked at the speed ∆u. r ρ ρ With this approximation, the impactor flux distribution can p 2 2 be written as where r is the orbit radius, ρ = r 1 − sin isin ν, i is the inclination and ν is the true anomaly. Most of the en- 2 2m0 ergy harvesting will occur as the spacecraft crosses the ring F(m,u) = nu 3 δ (u − ∆u). (7) (m0 + m) plane where the debris is most dense. The absolute upper energy fluxes to the spacecraft surface on the order of the 10-9 solar constant at Saturn. However, we have shown that the conversion rate from kinetic to electromagnetic energy of the EMP generated in HVIs is far too low to power even 10-10 minimal electronics on small, free flying spacecraft. Other

KE sources of power in the environment — such as other natu- ral radiators or spacecraft charging — may prove more ef- / 10-11 fective in realizing the distributed sensing architecture with EMP nanosatellites.

10-12 6 Acknowledgments

10-13 This work was supported by a NASA Space Technology 0 0.2 0.4 0.6 0.8 1 Research Fellowship (NASA Grant 80NSSC18K1152). i/ The authors would like to thank Jan Stupl of NASA Ames Figure 3. EMP conversion efficiency for each orbit. The Research Center for his assistance in grounding and prepar- dotted line represents the upper bound on this efficiency ing the ideas presented in this paper. In addition, the authors thank the operators at AVGR including Charles Cornelison, computed using ∆ulim. Alfredo Perez, and Jon-Pierre Wiens who made the HVI experiments possible. Those experiments were supported by √ p limit on ∆u for this orbit type is ∆ulim = (1 + 2) µ/r. AFOSR grant FA9550-14-1-0290. Inserting the spacecraft in an elliptical orbit provides a slight enhancement; the upper limit at periapsis becomes References √ √ p ∆ulim,e = ( 1 + e + 2) µ/r where 0 < e < 1 is the orbit eccentricity. Therefore, the absolute upper limit for any [1] C. Escoubet and R. Schmidt, “Cluster II: Plasma p orbit is ∆ulim = 2 2µ/r during ring plane crossings. measurements in three dimensions,” Advances in Space Research, 25, 7-8, pp. 1305–1314, 2000, doi: Figure 2 shows the power per unit area averaged over the 10.1016/S0273-1177(99)00639-0. circular orbit hΓ i from the kinetic energy of the impactor [2] S. Close et al., “Detection of electromagnetic pulses 1 3 ΓKE = 2 m0(∆u) and due to the EMP ΓEMP. ΓKE can be produced by hypervelocity micro particle impact thought of as the maximum power available to be col- plasmas,” Physics of Plasmas, 20, 9, 2013, doi: lected from HVIs in these orbits, from the EMP or through 10.1063/1.4819777. other means. The maximum power seen by the spacecraft [3] A. M. Nuttall, “Radio-Frequency Emissions from Hy- max(Γ ) at any point in the orbit is also shown, along with pervelocity Impacts on Charged Spacecraft,” PhD the- the upper limit on the power Γlim for any orbit. Figure 3 sis, Stanford University, 2018, p. 212. shows the ratio hΓEMPi/hΓKEi, which serves as an average conversion efficiency. The retrograde orbit i = π results [4] N. McBride and J. A. M. McDonnell, “Meteoroid im- in the highest available power and average conversion ef- pacts on spacecraft: sporadics, streams, and the 1999 ficiency. If kinetic energy could be harnessed in its raw Leonids,” Planetary and Space Science, 47, pp. 1005– form with perfect efficiency, the maximum power captured 1013, 1999. would be 22 mW/m2 while the amount captured from the [5] S. Y Ye, D. A. Gurnett, and W. S. Kurth, “In-situ mea- EMP is only 4.3 pW/m2. For comparison, the expected surements of Saturn’s dusty rings based on dust im- solar power at Saturn is about 4.2 mW/m2 assuming 30% pact signals detected by Cassini RPWS,” Icarus, 279, efficient solar cells facing sunward. It is clear that, to attain pp. 51–61, 2016, doi: 10.1016/j.icarus.2016.05.006. 77 mW sustained power, the spacecraft would require an [6] D. Gurnett et al., “Micron-sized particles detected infeasibly large impact area or number of rectennas. near Saturn by the Voyager plasma wave instrument,” Icarus, 53, 2, pp. 236–254, 1983, doi: 10.1016/0019- Although ΓEMP is linearly dependent on m0, the conversion 1035(83)90145-8. efficiency is dependent on ∆u alone. Since this efficiency [7] Z. Wang et al., “Characteristics of dust particles de- increases by powers of 3/2 with the relative speed, the ex- tected near Saturn’s ring plane with the Cassini Ra- pected power generation increases with more energetic ve- dio and Plasma Wave instrument,” Planetary and locity distributions, but is limited by the escape velocity. Space Science, 54, 9-10, pp. 957–966, 2006, doi: 10.1016/j.pss.2006.05.015. 5 Conclusions [8] Z. Manchester, M. Peck, and A. Filo, “KickSat : A Crowd-Funded Mission To Demonstrate The World’s Compared with other solar system environments, the large Smallest Spacecraft,” in Annual AIAA/USU Confer- density of impactors in the Saturnian rings coupled with ence on Small Satellites, Logan, UT, 2013. high relative speeds of transiting spacecraft result in kinetic