The effect of modelling assumptions on predictions of the space debris environmentR. Blake and H.G. Lewis

Astronautics Research Group, Faculty of Engineering & the Environment, University of Southampton, UK

IAC-14-A6.2.4

• Evolutionary models are used to guide technical solutions to the space debris problem– These tools incorporate simplified models for

estimating orbital motion and collision probability, for example

– Simulations using these models make assumptions to reduce the many degrees of freedom that exist

• Some research has already been done to understand the influence of assumptions made about external drivers (e.g. solar activity)

• Little research has been done to understand the influence of the model simplifications/assumptions

Introduction

Focus of this presentation is the Cube approach

External driversSolar activity Launch traffic

ExplosionsCompliance with

mitigation measures

• Evaluates collision probabilities between orbiting objects using a “sampling in time” approach:

Number of collisions:

Collision rate:

Spatial density:

The Cube Approach

𝑁𝑡𝑜𝑡= ∫𝑠=0

𝑠=𝐿

[𝑡 𝑠+1− 𝑡𝑠 ] 𝑃 𝑖 , 𝑗 ( 𝑠) 𝑑𝑠

𝑃 𝑖 , 𝑗=𝑠𝑖𝑠 𝑗𝑉 𝑖𝑚𝑝 𝜎𝑈

𝑈=𝑑3

Ud𝑠𝑖=𝑃𝑟𝑒𝑠

𝑈=

𝑃𝑟𝑒𝑠

𝑑3

• Two identical objects i and j in circular, polar orbits of a = 7000 km and intersecting at 90:

Idealised case

𝑉 𝑖𝑚𝑝=( 2𝜇𝑎 )

12

𝑃𝑟𝑒𝑠≈𝑑2𝜋 𝑎

Relative velocity:

Residential probability:

• Collision rate for this case:

Idealised case𝑃 𝑖 , 𝑗=

12𝜋 2𝑑 ( 2𝜇

𝑎5 )12 𝜎

𝜎=4 𝐴

Combined collision cross-sectional

area:

• Space Debris Environment Tool Kit:– Orbit propagator & Cube approach implemented in

Python

Implementation in SDETK

Parameter Value

(year) 2009

(year) 3009

ts+1 - ts (days) 0.5, 0.05 and 0.005

d (km) 1, 10 and 100

• Comparison of collision rates:

Theory v Implementation in SDETK

• Collision rate is inversely proportional to the cube size:

• Increasing time-interval or decreasing cube size reduces the consistency of collision rate estimates– Cube sizes ≥ 10 km, and

– Time-intervals 0.05 days, are preferred

• Increasing the number of Monte Carlo runs also enables good sampling of the space

Findings

𝑃 𝑖 , 𝑗=1

2𝜋 2𝑑 ( 2𝜇𝑎5 )

12 𝜎

Computational cost

• DAMAGE: full LEO-to-GEO evolutionary model– Uses target-centred version of Cube:

Cube Implementation in DAMAGE

Identifies all cases where a debris object resides within a bounding sphere centred on the target

Size of volume element is proportional to the size of the cube element

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LEO 10 cm Population (May 2009)

ESA MASTER 2009 population seen in

DAMAGE

29,370 objects ≥ 10 cm

• Overall collision rate estimates:

DAMAGE Results

• For the idealised, two-object case the number of co-occurring pairs in the cube remains constant but the volume increases (A): collision rate decreases

• In DAMAGE simulation, the number of unique co-occurring pairs in each cube increases as volume increases (B) or (C): overall collision rate appears ~constant

DAMAGE Results

A B C

• Collision rate between two orbiting objects is inversely proportional to the cube size:– Shown in theory

– Observed in SDETK implementation

• Increasing number of Monte Carlo runs and cube size, or decreasing time-interval improves the consistency of collision rate estimates– Default parameters in DAMAGE (and other

evolutionary models using Cube) likely to be sub-optimal

– Collision rates appear ~constant for changing cube size

– Difficult to address due to computational cost

• Further research is required to understand implications

Conclusions

Thank you for your attention

Contact: hglewis@soton.ac.uk

Thanks to Holger Krag (ESA Space Debris Office) for permission to use the MASTER reference population, and Aleksander Lidtke (University of Southampton) for valuable discussions about the work