Determining & Evaluating High Risk Conjunction Events

Improving Space Operations Workshop

Boulder, CO5 – 6 April 2011

Introduction

• Conjunction Assessment Process• Collision Probability• Sigma Level Analysis• Alignment of Radial Vectors• Collision Probability Sensitivity• Maximum Collision Probability• High Risk Events

Conjunction Assessment Process (1/2)

• Conjunction assessment is performed based on state and state uncertainty data generated and disseminated by the JSpOC.

• Close approach predictions are generated based on a 5-day screening span.

• Results are sent out daily, or more frequently for high risk conjunction events.

• Predictions are made using data from the JSpOC high-accuracy space object catalog.

Conjunction Assessment Process (2/2)

• The typical conjunction assessment process for a satellite program is to receive Orbital Conjunction Messages (OCMs) for secondary objects that are predicted to violate a designated screening volume around the primary satellite.

• OCMs contain sufficient data to calculate a Pc value; i.e., state vector and state uncertainty data.

• The screening volume must be large enough to capture close approaches with objects with a wide range of covariance sizes—this can result in large amounts of data and many conjunction events that are not a threat.

Collision Probability (1/2)

• Collision probability (Pc) is a measure of the overlap of the error distribution of the two objects, where the error distribution is given by the covariance matrix.

• When the covariances of the objects are combined, Pc can be thought of as the relationship of the miss vector to the combined covariance.

Primary & Secondary object with covariance ellipsoids

Combined covariance with keep-out region positioned by miss vector

Collision Probability (2/2)

• Pc is used as the primary measure of risk since it captures the miss distance, the relative geometry, and the associated uncertainty of the close approach.

• A conjunction assessment process based on miss distance alone does not account for the uncertainty inherent in the problem.

• Miss distance is used as the basis of the screening process, leading to the receipt of many OCMs for conjunction events with zero Pc values.

• This approach can lead to large amounts of data; especially for satellites in LEO.

Sigma Level Analysis (1/2)

• The miss vector and the state uncertainty of the two objects can be used for a streamlined screening process.

• The sigma level can be calculated using the Radial, Intrack, and Crosstrack (RIC) directions of the primary.

where and

• OCMs are not required for these calculations.

𝑅𝑎𝑑𝑖𝑎𝑙 𝑠𝑖𝑔𝑚𝑎𝑙𝑒𝑣𝑒𝑙=𝜌𝑅

𝜎𝑅𝑝+𝜎 𝑅𝑠

𝐼𝑛−𝑝𝑙𝑎𝑛𝑒𝑠𝑖𝑔𝑚𝑎𝑙𝑒𝑣𝑒𝑙=√𝜌 𝐼❑2+𝜌𝐶

❑2

𝜎 𝐼 𝑝+𝜎 𝐼 𝑠=

𝜌 𝐼𝐶

𝜎 𝐼𝑝+𝜎 𝐼𝑠

Sigma Level Analysis (2/2)

�̂�𝑝

𝐼𝑝�̂�𝑝

𝜌𝜌𝑅

• The two semi-major axes equal the largest possible in-plane uncertainty component, ; and the semi-minor axis equals the sum of the radial uncertainties, .

• A high sigma level from either calculation should result in a near zero Pc value, but a low sigma level does not necessarily lead to a high Pc value.

• A sigma level > 4 is recommended.

𝜌 𝐼𝐶𝜌𝐶

𝜌𝐶

Sigma Level Analysis Results

• Sample report of data required:

• Sample results with corresponding collision probability:

Predicted Miss Distances Primary Error at TCA Secondary Error at TCA

Total Radial Intrack Cross-track

Radial Intrack Cross-track

Radial Intrack Cross-track

Case Radial Sigma Level

In-plane Sigma Level

Collision Probability

1 2.29 0.35 7.78e-5

2 1.13 3.42 5.18e-9

3 -9.29 0.05 0

4 0.84 0.36 1.18e-3

5 4.17 0.09 4.74e-012

Alignment of Radial Vectors

• The Radial unit vectors of the two objects nearly align for conjunction events, therefore, the Radial direction can be decoupled from the Intrack and Crosstrack directions.

𝜑

�̂�𝑝

�̂�𝑝 𝐼𝑝𝜌 𝐼𝐶

𝑟𝑝 𝑟 𝑠

Earth center

�̂�𝑠

𝐼 𝑠

�̂� 𝑠

Since for tangible conjunction events, is small and the two Radial unit vectors can be considered collinear for the purposes of this analysis.

𝜌𝑅

𝜑

𝜌 𝐼𝐶

Collision Probability Sensitivity

• Covariance size can be scaled to determine the sensitivity of the Pc value.

• Since covariance is propagated from OD epoch to Time of Close Approach (TCA), a reduction in covariance size gives an indication of how the Pc will evolve.

• This calculation is performed with a static miss distance.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-9

-8

-7

-6

-5

-4

Normalized Covariance Size

log

Pc

Collision Probability Sensitivity

Maximum Collision Probability

• The Pc sensitivity curve allows evaluation of the Pc max condition.

• Pc max tends to occur when the miss vector lies on the 1-sigma uncertainty ellipsoid.

• Most conjunction events evolve to the left of the Pc max condition.

• Those that don’t tend to be of concern …

• Notice that in this example the covariance was enlarged to show Pc max.

0 0.5 1 1.5 2 2.5 3 3.5 4-10

-9

-8

-7

-6

-5

-4

-3

-2

Normalized Covariance Size

log

Pc

Pc .

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

12

Sig

ma

Leve

l

Collision Probability Sensitivity

Radial Sigma LevelIn-plane Sigma Level

High Risk Events

• Some conjunction events show little change in Pc as the covariance is contracted.

• This occurs when the miss vector is within the 1 or 2-sigma of the combined covariance ellipsoid.

• This condition can be a sign that the conjunction event will remain a threat as the TCA approaches.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3.5

-3

-2.5

Normalized Covariance Size

log

Pc

Pc .

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Sig

ma

Leve

l

Collision Probability Sensitivity

Radial Sigma LevelIn-plane Sigma Level