Toshihiro Chujo 1

Orbital Dynamics andImpact Probability Analysis

Toshihiro Chujo

(ISAS/JAXA)

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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 ScheduleEarth Departure: Nov-Dec, 2014Earth Swing-by: Dec, 2015Ryugu Arrival: Jul, 2018Ryugu Dep.: Dec, 2019Earth Reentry: Dec, 2020

Credit: JAXA/Hayabusa-2 Credit: JAXA/Hayabusa-2

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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.

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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”.

*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.

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.

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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?

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Failure probabilitySpacecraft 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 setsuch that any bus subsystem (i.e. data handling,communication, power management, thermalcontrol, attitude, etc.) of the spacecraft havehigher reliability than the IES (Ion Engine System).

-> IES is considered to be the critical component.

-> Failure of IES is taken into consideration.

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Failure probabilitySpacecraft 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

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.

𝑅𝑀𝐼 𝑡 = 1 − 1 − 𝑅 4 − 𝑓𝑏𝑢𝑠𝑅𝑀𝐼 𝑡 = 1 − 1 − exp −𝜆𝑡 4 − 𝑓𝑏𝑢𝑠𝑅𝑀𝐼 𝑡 ≈ 1 − 1 − exp −𝜆𝑡 4

R

t

Reliability decreasesas time passes.

Reliability related to Mars impact Failure probability due tobus subsystem malfunction

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Failure probabilityMeteoroid 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?

(i) (ii) (iii)

Scenario Disruption of all the 4 IES grids

Penetration of 1 of the 12 RCS thrusters

Dark current by impact intruding to spacecraft circuits through the honeycomb panel

Componentdimension

150 mm diameter

64×64 mm 1.6×1.0×1.25 m

Minimum mass of meteoroid

8×10-6 g 1×10-4 g 1×10-3 g

Kill rate 2.5×10-7 [yr-1] 2.1×10-4 [yr-1] 2.6×10-2 [yr-1]

Analysis of Hayabusa-2

-> Probability of meteoroid impact

<- Structural strength

<- Meteoroid flux

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Failure probabilityMeteoroid kill rate

Dark current scenario overwhelms other effects so that the total meteoroid impact

kill rate is approximated as 𝜙𝑡𝑜𝑡𝑎𝑙 ≈ 0.026 𝑦𝑟−1 .

Thus, the total failure probability in a period between t1 and t2 is calculated as

𝑞 𝑡1, 𝑡2 = 𝑅𝑀𝐼 𝑡1 − 𝑅𝑀𝐼 𝑡2 + 𝜙𝑡𝑜𝑡𝑎𝑙 𝑡2 − 𝑡1

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

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Impact probability after failureOverview

Total probability Ptotal of Mars impact is represented as

𝑃𝑡𝑜𝑡𝑎𝑙 = 𝑝𝑞d𝑡

p: Mars impact probability after spacecraft failure.

-> Probability that the spacecraft out of control-> reaches Mars on a ballistic trajectory

How does the spacecraft enters the ballistic trajectory?

-> Orbital dynamics

Mars orbiter Interplanetary probe

- Gravity of Mars- Air drag

- Gravity of the sun- Possibly gravity of the Earth

Original trajectory

Mars

Mars impact?

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Impact probability after failureRough 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 14×106 km for trajectory of the nominal window.

For two backups, it is 14×106 km and 5×106 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.

Mars Earth

Hayabusa2

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]

Minimum Dist.=0.091AU (=14M km) Minimum Dist.=0.091AU (=14M km) Minimum Dist.=0.032AU (=5M km)

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Impact probability after failureGuidance 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 m s × 10 deg. ×𝜋

180≈ 20 m s

Maximum ΔVby the IES

Maximumangle error

Worst ΔV toundesired direction

*ΔV by the RCS is negligible.

100 m/s

10 deg.

20 m/s

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Impact probability after failureMaking 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 P1 to a point P2 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.

r1, r2, T: given-> Initial velocity v1 is solved.

Impulse ΔV to reach P2 can be solved numerically.

P2

P1

T

v1

v2

r2

r1

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Impact probability after failureImpact 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.

20m/s

Threshold

[M], [E] denote

the minimum ΔV for

Mars and Earth orbit

intercept,

respectively.

The highest Mars orbit intercept risk occurs here(Jan.1,2018)

Phase-free analysis for Launch Window [N]

Phase-free analysis for Launch Window [BU1]

Phase-free analysis for Launch Window [BU2]

Leg1 Leg2 Leg3 Leg4

Leg1 Leg2 Leg3 Leg4

Leg1 Leg2 Leg3 Leg4

[M]

[E]

[M]

[E]

[M]

[E]

20m/s

Threshold

20m/s

Threshold

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.

Marsorbit

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Impact probability after failurePossibility 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.

Spacecraft failure

Direct impact to Mars

Encounter with the Earth

Impact to Mars

Gravityassist

Totalprobability

Probability is zero, but analyzedby Monte-Carlo simulation, too.

Analyzed by Monte-Carlo simulation.

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Impact probability after failureMonte-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

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)

The roughness of the approximation of discretization isΔ𝑅 = Δ𝑉𝑡 = 20 m s × 1 month = 51840 km

1000 cases

1000 cases

1000 cases

1 month

1 month

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Impact probability after failureSwing-by cases and no swing-by cases

If the spacecraft passes within 106 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”.

Earth B-plane

Earth

Swing-by

No swing-by

106 km

Spacecraft failure

No swing-by case

Swing-by case

Statistical algorithm ofno swing-by cases

Statistical algorithm ofswing-by cases

915 cases

85 cases

1000 cases

ex) Failure onJan. 1st 2016

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Impact probability after failureAlgorithm for no swing-by cases

Algorithm for no swing-by cases

- Every intersection point on the Mars B-planeis investigated.

- Statistical processing is conducted.

- Probability of Mars impact is calculated.

Sun Mars B-plane

Mars

z

r

Sun

Intersection positions of the spacecraft

Mars positions

Gaussian distribution

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Impact probability after failureAlgorithm 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.

Transfertrajectory

Earth

Mars

Mars

Key hole

EarthdφR

106 km

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Calculation results

Calculation results

Trajectory Leg IES ΔV/RCS ΔV Impact probability(no swing-by)∫pno_swbydt

Impact probability(swing-by)∫pswbydt

Failureprobability

Total probability∫(pno_swby+pswby)qdt

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

Nominal window

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.

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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.