Orbital Dynamics and Impact Probability Analysis

Orbital Dynamics and Impact Probability Analysis (ISAS/JAXA) 1

Overview This presentation mainly focuses on a following point regarding planetary protection. - How to prove that a mission satisfies the requirement from planetary protection policy quantitatively? And what should we care in the mission design? This is presented referring to our actual activity in Hayabusa-2 mission, 2nd Japanese sample return mission to small body. JAXA launched Hayabusa-2 in 2014, and it will explore C-type asteroid Ryugu, and return back to the Earth in 2020. Mission Schedule Earth Departure: Nov-Dec, 2014 Earth Swing-by: Dec, 2015 Ryugu Arrival: Jul, 2018 Ryugu Dep.: Dec, 2019 Earth Reentry: Dec, 2020 Credit: JAXA/Hayabusa-2 Credit: JAXA/Hayabusa-2 2

Planetary protection policy When designing interplanetary missions, we must consider and obey planetary protection policy - not to contaminate planets where the origin of life may exist (forward contamination). - not to endanger the Earth by bringing extraterrestrial organisms, if such exist (backward contamination). COSPAR develops recommendations for avoiding interplanetary contamination. In particular, Mars, Europa, Enceladus are targets that should be taken care of. 3

Requirements from COSPAR According to the destination and spacecraft type (e.g. orbiter, flyby, sample return, ), the mission is categorized into one (or more than one) Category defined in COSPAR Planetary Protection Policy. Hayabusa-2 mission was considered as Category-II* during the outbound journey with particular attention needed to avoid impact with Mars under all mission scenarios, and as Category-V** during the inbound journey, corresponding to unrestricted Earth return. Orbit of Asteroid Ryugu (cf. Mars) Perihelion 0.96AU (1.38AU) Aphelion 1.42AU (1.67AU) Eccentricity 0.19 (0.093) Inclination 5.88deg (1.85deg) (w.r.t. Ecliptic plane) Ryugu itself is far enough (i.e. has zero collision probability) both from the Earth and Mars. *Category II: Flyby, Orbiter, Lander: Comets; Carbonaceous Chondrite Asteroids; Jupiter; Saturn; Uranus; Neptune; Pluto/Charon; Kuiper-Belt Objects; others TBD. **Category V: Any Earth-return mission. Restricted Earth return : Mars; Europa; others TBD; Unrestricted Earth return : Moon; others TBD. 4

Process of proof Required probability value differs depending on missions. In case of the Hayabusa-2 mission, the requirement was that the probability of impact of the spacecraft with Mars under all possible mission scenarios must be less than 10-4 in the 50-year period after its launch. How to prove it quantitatively? Failure probability - System failure rate - Meteoroid kill rate Impact probability after failure This differs depending on missions. Simple analysis? High-fidelity Monte-Carlo simulation necessary? 5

Failure probability Spacecraft system failure rate When estimating spacecraft system failure, reliability of subsystems is referred to. - What is the most critical component? - Which component is the dominant factor? - How to estimate failure probability? Design philosophy of the spacecraft is important. In Hayabusa-2, the design philosophy was set such that any bus subsystem (i.e. data handling, communication, power management, thermal control, attitude, etc.) of the spacecraft have higher reliability than the IES (Ion Engine System). -> IES is considered to be the critical component. -> Failure of IES is taken into consideration. 6

Failure probability Spacecraft system failure rate Relationship between reliability R and failure rate λ is usually represented as R = exp λt In Hayabusa-2, the IES (critical component) was designed such that 3 out of 4 thrusters (75%) must be in good condition for 6 years. -> R = 0.75, t = 6 yr -> λ = 1.3 10 4 1 day R Reliability decreases as time passes. t Even if system failure occurs, an impact with Mars can be avoided as long as at least one of the four IES thrusters is operative. R MI t = 1 1 R 4 f bus R MI t = 1 1 exp λt 4 f bus R MI t 1 1 exp λt 4 Reliability related to Mars impact Failure probability due to bus subsystem malfunction 7

Failure probability Meteoroid kill rate Hypervelocity impacts by meteoroids are unavoidable. - What are scenarios of fatal meteoroid impact consequences? - What is the minimum mass of meteoroid that realizes fatal impact? - How to estimate the meteoroid kill rate? Analysis of Hayabusa-2 (i) (ii) (iii) Scenario Component dimension Disruption of all the 4 IES grids 150 mm diameter Penetration of 1 of the 12 RCS thrusters Dark current by impact intruding to spacecraft circuits through the honeycomb panel 64 64 mm 1.6 1.0 1.25 m -> Probability of meteoroid impact Minimum mass of meteoroid 8 10-6 g 1 10-4 g 1 10-3 g <- Structural strength Kill rate 2.5 10-7 [yr -1 ] 2.1 10-4 [yr -1 ] 2.6 10-2 [yr -1 ] <- Meteoroid flux 8

Failure probability Meteoroid kill rate Dark current scenario overwhelms other effects so that the total meteoroid impact kill rate is approximated as φ total 0.026 yr 1. Thus, the total failure probability in a period between t 1 and t 2 is calculated as q t 1, t 2 = R MI t 1 R MI t 2 + φ total t 2 t 1 Important notes are - Design philosophy of the spacecraft - Critical component for system failure - Reliability function - Critical component for meteoroid kill - Kill rate analysis referring to component properties 9

Impact probability after failure Overview Total probability P total of Mars impact is represented as Mars P total = pqdt p: Mars impact probability after spacecraft failure. -> Probability that the spacecraft out of control -> reaches Mars on a ballistic trajectory Original trajectory Mars impact? How does the spacecraft enters the ballistic trajectory? -> Orbital dynamics Mars orbiter - Gravity of Mars - Air drag Interplanetary probe - Gravity of the sun - Possibly gravity of the Earth 10

Impact probability after failure Rough estimation It should begin with rough estimation by looking at minimum distance to Mars. The minimum distance between the Hayabusa-2 spacecraft and Mars is 14 10 6 km for trajectory of the nominal window. For two backups, it is 14 10 6 km and 5 10 6 km, respectively. -> Since the trajectory guidance accuracy is a few hundred kilometers at worst, -> impact of the spacecraft with Mars due to uncertainties in trajectory -> determination is unlikely. Hayabusa2 Mars Earth Minimum Dist.=0.091AU (=14M km) Minimum Dist.=0.091AU (=14M km) Minimum Dist.=0.032AU (=5M km) End-to-end mission trajectory for Nominal Window[N] End-to-end mission trajectory for Backup Window1[BU1] End-to-end mission trajectory for Backup Window2[BU2] 11

Impact probability after failure Guidance error Next, guidance error should be evaluated. Operation frequency is important. In Hayabusa-2, - Trajectory navigation and guidance is performed using one-week cycles and OD (orbit determination) is updated no later than one month. - Maximum acceleration produced by the IES is 100 m/s per month. - The attitude to operate the IES is constrained to within 10 from the sun direction. -> In the worst case, the maximum guidance error for one month is* 100 Maximum ΔV by the IES m s 10 deg. π 180 20 Maximum angle error *ΔV by the RCS is negligible. m s Worst ΔV to undesired direction 10 deg. 20 m/s 100 m/s 12

Impact probability after failure Making transfer trajectory to Mars by Lambert s problem Whether the worst ΔV may realize Mars impact or not is investigated. Lambert s problem Supposing a spacecraft under the influence of a central gravitational force travels from point P 1 to a point P 2 in a time T, transfer ballistic trajectory can be solved. Supposing Kepler orbit of two-body problem. -> If gravity of the sun is dominant, it is useful. r 2 P 2 v 2 r 1, r 2, T: given -> Initial velocity v 1 is solved. T Impulse ΔV to reach P 2 can be solved numerically. r 1 P 1 v 1 13

Impact probability after failure Impact analysis ignoring Mars orbital position The minimum departure ΔV to intercept the Mars orbit (ignoring Mars orbital position) is evaluated based on the Lambert s problem. -> Conservative estimation. Phase-free analysis for Launch Window [N] Leg1 Leg2 Leg3 Leg4 Mars orbit 20m/s Threshold [E] [M] Phase-free analysis for Launch Window [BU1] Leg1 Leg2 Leg3 Leg4 In Hayabusa-2, The minimum ΔV is 250 m/s. -> The probability of Mars direct impact is zero. Earth orbit intercept ΔV is almost zero. -> The probability to impact Mars via Earth gravity assist should be discussed. [M], [E] denote the minimum ΔV for Mars and Earth orbit intercept, respectively. The highest Mars orbit intercept risk occurs here (Jan.1,2018) 20m/s Threshold [M] Phase-free analysis for Launch Window [BU2] Leg1 Leg2 Leg3 Leg4 20m/s Threshold [E] [E] [M] 14

Impact probability after failure Possibility of gravity assist of Earth In the Hayabusa-2 mission, probability of direct impact to Mars is zero, but probability of getting gravity assist of the Earth is not zero. This happens because the original trajectory aims at the Earth swing-by. -> Original trajectory design is also important. Since the Earth swing-by makes large dispersion of probability, estimation using Monte-Carlo simulation is suitable. Direct impact to Mars Probability is zero, but analyzed by Monte-Carlo simulation, too. Spacecraft failure Encounter with the Earth Gravity assist Impact to Mars Total probability Analyzed by Monte-Carlo simulation. 15

Impact probability after failure Monte-Carlo simulation The following is the algorithm made for the Hayabusa-2 mission. Monte-Carlo to evaluate the probability quantitatively Monte-Carlo propagation -> Statistical processing 1000 cases Starting day: every 1 month Number of cases: 1000 Propagation time: 50 years after the launch Initial deviations (Gaussian random numbers) Position: σ = 100 km for x, y, z (accuracy of OD) Velocity: σ = 1 m s for x, y, z (accuracy of OD) Velocity: and 3σ = 20 m s for x, y, z (guidance error) 1 month 1 month 1000 cases 1000 cases The roughness of the approximation of discretization is ΔR = ΔVt = 20 m s 1 month = 51840 km 16

Impact probability after failure Swing-by cases and no swing-by cases If the spacecraft passes within 10 6 km (smaller enough than the roughness) from the Earth on its B-plane even once, the case is defined as a swing-by case, and otherwise, no swing-by case. No swing-by 10 6 km Earth B-plane Swing-by Earth 1000 cases Spacecraft failure ex) Failure on Jan. 1st 2016 915 cases No swing-by case 85 cases Swing-by case Statistical algorithm of no swing-by cases Statistical algorithm of swing-by cases 17

Impact probability after failure Algorithm for no swing-by cases Algorithm for no swing-by cases - Every intersection point on the Mars B-plane is investigated. - Statistical processing is conducted. - Probability of Mars impact is calculated. Sun Intersection positions of the spacecraft Mars positions z Sun Mars B-plane Gaussian distribution r Mars 18

Impact probability after failure Algorithm for swing-by cases Algorithm for swing-by cases - Transfer trajectory after gravity assist by the Earth is made by Lambert s problem. - A keyhole on the Earth B-plane that connects to the transfer trajectory is searched. - Statistical processing is conducted. - Probability of Mars impact is calculated. Mars Mars Key hole Transfer trajectory Earth R dφ Earth 10 6 km 19

Calculation results Calculation results Nominal window Trajectory Leg IES ΔV/RCS ΔV Impact probability (no swing-by) p no_swby dt Impact probability (swing-by) p swby dt Failure probability Injection No/No 1.0e-14 1.9e-12 2.1e-3 4.1e-15 Earth to Earth Yes/Yes 4.1e-13 1.9e-11 2.4e-2 4.1e-14 Earth to Asteroid Yes/Yes 0 3.7e-11 6.8e-2 8.8e-14 Asteroid Proximity No/No 0 0 4.1e-2 0 Asteroid to Earth Yes/Yes 0 3.4e-9 2.8e-2 8.0e-12 TOTAL Probability 4.1e-13 3.4e-9 1.6e-1 8.1e-12 Total probability (p no_swby +p swby )qdt Total probability for backup window 1 and 2 are 8.0e-12 and 5.6e-7, respectively. -> The Hayabusa-2 mission satisfies the COSPAR requirement that the Mars impact -> probability should be less than 10-4 for 50 years after launch for all mission -> scenarios. 20

Summary - Spacecraft failure rate and probability of impact after the failure are respectively calculated to prove the probability of Mars impact quantitatively. - Design philosophy of the spacecraft and reliability of subsystem component are important factors when considering system failure. - Making ballistic trajectory to Mars and estimating probability to enter it accidentally is the first step for estimation of impact probability. - For missions where the spacecraft may be assisted by the Earth gravity, it may be necessary to analyze the probability using Monte-Carlo simulation. 21