Snappshot: Suite for the Numerical Analysis of Planetary Protection

SNAPPSHOT: SUITE FOR THE NUMERICAL ANALYSIS OF PLANETARY PROTECTION Francesca Letizia∗, Camilla Colomboy, Rudiger¨ Jehn Jeroen P.J.P. Van den Eyndez, Roberto Armellinx. Mission Analysis University of Southampton ESOC Astronautics Research Group Darmstadt, 64293 Southampton, SO17 1BJ Germany United Kingdom ABSTRACT [3]. These requirements are in place to limit the risk of con- tamination of celestial bodies due to space missions and they For interplanetary missions and missions to the Lagrangian prescribe a maximum allowed impact probability that should points, the compliance with planetary protection requirements be verified over a time window of 50-100 years, considering should be verified as spacecraft and launchers used for these possible inaccuracies of the launcher or failures of the propul- applications may be inserted in a trajectory that will impact sion system. the Earth or other planets. A new tool, SNAPPshot, was For this purpose, a new tool, called SNAPPSHOT (Suite developed for this purpose. A Monte Carlo analysis is per- for the Numerical Analysis of Planetary Protection), was de- formed considering the dispersion of the initial condition and veloped by the University of Southampton under a study for of other parameters such as the area-to-mass ratio. Each run is the European Space Agency. The tool is based on a Monte characterised by studying the close approaches through the b- Carlo approach to study the dispersion of the initial conditions plane representation to detect conditions of impacts and reso- of launchers or spacecraft. Each trajectory is then analysed nances. The application of the tool to two missions (Solo and with the b-plane representation to identify impacts and reso- BepiColombo) is presented. nances with celestial bodies. In the rest of the paper, Section Index Terms— planetary protection, Monte Carlo analy- 2 will give an overview of the method and Section 3 will show sis, b-plane the application to the launcher of Solo and to BepiColombo. 1. INTRODUCTION 2. SNAPPSHOT Spacecraft and launchers used for interplanetary missions and mission to the Lagrangian points may be inserted into or- SNAPPSHOT is a code written in modern Fortran for the bits that can come back to the Earth or impact other planets. analysis of a mission compliance with planetary protection This has already happened as in the case of the third stage of requirements, based on a Monte Carlo analysis. Figure 1 Apollo 12 that was discovered to be in an orbit that gets tem- shows a block diagram of the tool. The first main block is the porary captured by the Earth every 40 years [1]. Recently, the initialisation of the Monte Carlo (MC) simulation. The user object WT1190F, which is also supposed to belong to a lunar specifies the maximum allowed probability of impact with a mission, re-entered in the atmosphere [2]. specific body and the desired confidence level, together with The probability of these events to happen should be es- providing the covariance matrix associated to the initial state timated during the phase of the design of a mission to en- and, if necessary, additional distributions such as the one of sure it is compliant with planetary protection requirements area-to-mass ratio (A=M). The tool computes the required number of Monte Carlo runs and samples the covariance ma- ∗The authors acknowledge the use of the IRIDIS High Performance trix to obtain the initial conditions. Each initial condition is Computing Facility, and associated support services at the University of propagated and then analysed with the representation on the Southampton, in the completion of this work. yThe present study was funded by the European Space Agency through b-plane of the relevant bodies. The number of detected im- the contract 4000115299/15/D/MB, Planetary Protection Compliance Verifi- pacts is computed: if the number is not compatible with the cation Software. requirements, the number of MC runs is increased until the z Currently Research Fellow, European Space Agency, ESTEC, Noord- requirements are fulfilled or the maximum number of runs is wijk, 2200AG, The Netherlands. xCurrently Marie Curie Research Fellow at Scientific Computing Group reached. At the end of the runs, the output is produced and its (GRUCACI), Universidad de La Rioja, ES-26004 Logrono,˜ Spain post-processing is performed in MATLAB. max impact probability confidence MC runs level Monte Carlo Trajectory trajectories B-plane initial conditions covariance initialisation propagation analysis matrix other distributions Are the Increase No Yes Generate requirements MC runs output satisfied? Fig. 1: SNAPPSHOT building blocks. 2.1. Monte Carlo samples generation be easily inverted to find the required number of runs The number of MC runs necessary to verify the planetary pro- z2 (1 − p) n = α : (2) tection requirements is found by applying the expression of p Wilsons’s confidence interval as in Jehn [4] and Wallace [5]. Differently from the latter, the expression for the one-sided If after n runs the number of impacts is different from zero, interval is used. Equation 1 should be used again, with the new value of p^, to update the value of n. Note that when p^ = 0, Equation 1 The MC runs can be considered as a binomial process cannot be inverted analytically, so Newton’s method is used. with only two possible outcomes, generally labelled as suc- Jehn [4] proposed an analytical approximation of n = n(n ), cess and failure, which in the present application correspond I but this was not applied here because Newton’s method has to impact and no impact. In the current implementation, the a fast convergence and the time spent in the computation of user is asked to identify a reference body for this evaluation. n is negligible compared to the time used for the propagation The selection of a reference body is used only to evaluate if of the trajectories. The analysis is terminated when the com- the number of MC runs should be increased, whereas impacts pliance is demonstrated or the maximum number of runs is with all considered celestial bodies are registered. In the re- reached. This limit is currently set equal to 500000 runs. Al- sults in Section 3, Mars is set as the reference body, so only ternatevely, the user can set the desired number of MC runs. impacts with this planet will trigger the increase of the num- Once the number of runs is defined (or updated), the cor- ber required runs. This is done because Mars has more strin- responding initial conditions need to be generated. When the gent planetary protection requirements than other planets due dispersion of the initial condition is described by a covariance to possible presence of habitats (or their rest) on its surface. matrix, Cholesky factorisation [7] is used to obtained corre- The output of the MC is considered as a binomial variable lated deviations from random generated numbers. When the X ∼ B(n; p) n p , with number of trials and the success prob- case of the failure of the propulsion system is studied, the ability in each trial. The expression of the one-sided Wilson’s random generated numbers are used to define the instants of interval is failure and the initial conditions are obtained by linear inter- polation of the state vector at the closest times. Additional 0 2 q 2 1 zα 2 p^(1−p^) zα p^ + + zα + 2 distributions can be considered such as the one of the area-to- p ≤ 2n n 4n : @ z2 A (1) 1 + α mass ratio. In this case, the user provides the known values n (e.g. the minimum, the average and the maximum value) and selects a distribution; currently, the uniform and the triangular where p^ is the probability of success estimated from the sta- distribution are implemented. tistical sample (e.g. p^ = nI =n, with nI number of impacts), z is the α quantile of a standard normal distribution, α is the α 2.2. Trajectory propagation confidence level [6]. In SNAPPSHOT, the user sets the value of the maximum Once the initial state is defined, the trajectory is propagated allowed p and the one of α. The corresponding zα is obtained over 100 years, using Cartesian coordinates in the J2000 ref- from a lookup table. At the beginning of the simulation no erence frame centred in the solar system barycentre. To keep impacts are recorded, so p^ = 0. In this case, Equation 1 can the tool flexible, the initial state of the object can be provided ·10−5 1:5 Analytical Integration ESA Dynamics 1 SPICE ] s −6 [ 2:73 · 10 t Ephemeris ∆ 0:5 1:31 · 10−6 Other 0 0 20 40 60 1 2 3 4 5 Share of the computational time [%] Number of bodies Generate ephemeris Mean to true anomaly Conversion to cartesian Other Fig. 2: Timing of the call of the ephemeris routines as a func- tion of the number bodies. Fig. 3: Example of computational time break-down for the integration of a trajectory using the analytical ephemerides. by the user in a reference systems with different orientation (e.g. the ecliptic one) or a different centre (e.g. the Sun). expressed in terms of orbital parameters, so they are not di- When the results of the MC are shown in functions of the state rectly suitable for the current application. In fact, the saving vector components (as in Section 3), then the body reference in the computational time associated with the analytical for- is used because this reference frame offers a direct connection mulation vanishes because of the need of conversion to Carte- to the physics of the evolution of the trajectory. sian coordinates (Figure 3).

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