Detection and Conjunction Assessment of Long Encounters

Transcription

1 Detection and Conjunction Assessment of Long Encounters F. B a b i k e r V. A b b a s i M D o y o n Detection and Conjunction Assessment of Long Encounters F. B a b i k e r V. A b b a s i M D o y o n

2 Introduction long encounters are thought to be rare occurrences typically in the geostationary orbit and attributed to low relative velocities in case of relative motion at very small approach angles. The Canadian space Agency Collision Risk Assessment and Mitigation System (CRAMS) recently enhanced with rectilinear motion approximation validity test has been used to process Conjunction Summary messages for several close approaches for LEO and GEO satellites. The test will be described the test results will be reviewed. the challenges presented will be discussed together with suggested responses to deal with them. In particular the need to compute non-linear probability and the availability of sufficient data to do that will be addressed. 2

3 Collision Risk Assessment The probability of collision is assessed using the formula below : P = 1 (2π) 3 C V e 1 2 rt C 1r dxdydz (1) evaluated in the encounter system xyz. If the encounter is short the curvilinear motion can be approximated by rectilinear motion. The velocities and the covariances are considered constant. The volume of integration becomes a cylinder of infinite length and circular cross-section of combined hard body radius. 3

4 Collision Risk Assessment Since the value of the integral in the relative velocity direction (the y axis) is unity, the probability integral reduces to the following 2-D integral in the xz plane-the encounter plane: P = 1 2πσ x σ z 1 ρ xz 2 A e x σx 2 2ρ xz x σx 2(1 ρxz 2 ) z σz + z σz 2 dxdz To test the validity of rectilinear motion approximation, a test is devised based on the analysis in Spacecraft collision Probability by Kenneth Chan. 4

5 The Path Length of Integration The requisite path length is estimated by varying the finite limits of the integral I defined by I = 1 y σ 2π න e y2 /2σ 2 dy y F0r Y=3σ, I=0.997 For Y = 8.5σ, I differs from unity in the seventeenth decimal place.so the path lengths for 2-digit and 15- digit accuracies are 6σ and 17σ respectively. 5

6 Rectilinear Motion Approximation The relative motion path should be sufficiently linear over a 17 σ length. Sufficient linearity if the deflection angle is small is about 3.4 deg. That corresponds to a transit-time orbital period ratio of 2%. The opposite figure ( based on Figure 3.2 of the reference above) shows two objects approaching each other in circular orbits of radius R and their straight line approximations L. The objects are separated by a distance S and have approach angle φ. C is the arc length of each of arcs AU and BU. Q is the length of line OV. They are related by the equations below: 6

7 Rectilinear Motion Approximation L = S 2 sin(φ/2) C = Rtan 1 ( L R (4) (5) The radial and in-track errors are D and E respectively, D = Q R (6) E = L C (7) Q 2 = R 2 + L 2 (8) L = (D 2 +2RD (9) t = S Vr (10) t is transit time and Vr is inertial speed. The deflection angle α = tan 1 ( D ) is used as an L indication of the degree of linearity. 7

8 Rectilinear Motion Approximation Validity Test The CRAMS implementation of the rectilinear motion test is based on the following: If the time to traverse the encounter frame expressed as a percentage of the orbital period is less than or equal to 2 percent, then it is a short encounter and the assumption of rectilinear motion is valid. Otherwise it is a long encounter and equation 1- the 3-D probability formula should be used to compute non-linear probability. 8

9 Rectilinear Motion Approximation Validity Test (t/t c )*100 <= 2 Where t = 17* σ /V r t is the transit time. It is a function of both σ and V r Tc = 2π (r 3 /µ) is the corresponding circular orbit period. V r is the inertial relative speed σ is the relative position error given by the expression below: σ = ( σ 2 x + σ 2 y + σ 2 z ) Where σ 2 x, σ 2 y and σ 2 z are the diagonal entries of the combined covariance matrix in the encounter system. 9

10 Rectilinear Motion Test Results-Regular LEO There have been 52 long encounters reported from November 5, 2015 to October 20, 2016 for LEO satellites with JSpOC regular screening. They have been reported in 106 out of a total of 5621 alerts. Some linearity test results are shown below: R primary Vrel (m/s) Transit (s) Period (s) Ratio Validity INVALID INVALID INVALID Sigma ( m) Data Qual. Probab Good 1.9E Good 1.3E Bad 2.1E-05 10

11 Rectilinear Motion Test Results-Regular LEO The first row shows a long encounter due an error of 1472 m and a relative velocity of m/s. The second row shows typical long encounter due to a low relative velocity of 7.26 m/s. The data quality is good for both encounters. The long encounter in the third row is due the high error of km despite the high relative velocity of km/s. 11

12 Rectilinear Motion Test Results-LEO(Advanced) 7061 alerts have been received over the period from November 5,2015 to October 10, 2016 for 5 satellites on advanced screening. 61alerts were long encounters due to large errors. Test results are similar to the bad data case reported in the test results for LEO satellites with regular screening. 12

13 Rectilinear Motion Test Results-GEO 932 alerts out of a total of 1401 ones have reported long encounters for 19 GEO satellites from January 14, 2016 to October 13, of them required action either on high linear probability or small miss distance. A significant number of these encounters were intra-constellation ones. 36 of them were with known objects and defunct satellites. 13

14 Rectilinear Motion Test Results-GEO Some test results are shown below: R primary Vrel (m/s ) Transit (s) Period (s) Ratio Validity Sigma ( m) Data Qual. Probab Bad 1.9E E INVALID 4.2E INVALID Bad 19.7E E INVALID 3859 Good 7.2E-06 14

15 Rectilinear Motion Test Results-GEO GEO Long encounters are mainly due to low relative velocities and exasperated by large errors. The test results in the Ratio column show a high degree of non-linearity. The GEO test results emphasize the need to compute non-linear probability. 15

16 Conclusion It has been shown that: long encounters have a high occurrence in GEO orbit but also occur in LEO as well but not as often. They are mainly attributed to low relative velocities in GEO and large errors in LEO. There is a pressing need to have sufficient data to enable the computation of non-linear probability. The rectilinear motion approximation validity test is an important component of any conjunction analysis tool. 16

17 Conclusion The orbit propagation parameters in the updated CDM provide the potential for providing the necessary data for the computation of the non-linear probability.the validation of their adequacy and quality is still work in progress. Operator ephemeris files with realistic covariances are another source of data for the computation of non-linear probability. 17

18