The Applicability Research of Nonlinear Collision Probability

Transcription

1 The Applicability Research of Nonlinear Collision Probability XIONG Yongqing, XU Xiaoli Purple Mountain Observatory Chinese Academy of Sciences Nanjing, Jiangsu , China

2 XU Xiaoli

3 Outline Introduction The integral limit along the relative path The scope of application of nonlinear method Conclusion

4 Chan and Patera proposed the concept of collison probability for nonlinear relatve motion, and programmed different methods to calculate this nonlinear collision probability. INTRODUCTION But the linear relative motion is not always valid: Most encounters are essentially instantaneous at high relative velocities. Therefore, the assumption of linear relative motion is valid for calculating collision probability, and the calculation methods are well defined by researchers. In this case, collision probability calculations are compressed to the encounter plane which is perpendicular to the direction of relative velocity. If the relative velocity between two encounter objects is quite small, the combined hard body will move within the combined covariance ellipsoid which also varies with time.

5 Chan s method: an analytical expression Patera s method: a numerical scheme The main rule of nonlinear method: The nonlinear trajectory is divided into several small approximate linear pieces. All the linear pieces yield the sectional probabilities, and summed to derive a total collision probability under a user's integral limit is reached. We mainly discuss : How the integral limits along the relative path affect collision probability and integral time. In order to compare the linear probability with the nonlinear, a special limit threshold is suggested. At last, test cases are given to analyze the applicability research of nonlinear collision probability.

6 The integral limit along the relative path For nonlinear encounters, a 3-dimensional integration should be reconsidered, and the corresponding integral limits along the relative path must be set to ensure the calculation stable. However, it is difficult to set the limit threshold in practical application.

7 The sectional probability gradually decays, and is accompanied by periodic changes simultaneously. Fig.1 The variation of sectional probability in every integration step For example, if 10-8 is taken as cutoff, integration along the relative path stop soon, even no more than a quarter of orbital period, leading to the subsequent many periods which sectional value is above the cutoff not be included. If the cutoff is 10-10, there are two periods to be excluded yet, it will have immediate impact on the integration.

8 To show how the integral limits affect the integral range and probability, an encounter between two objects in circular orbits is discussed. Table 1: the second object is just above the first 20 meters at TCA, the angle i between the two orbital planes is adjusted to change the relative velocity, a constant combined spherical position error of 0.5 km is used, and the size of combined hard body is set to 50 m

9 Table 2. Integral time and collision probability for different threshold limits limits P P P 10 8 Angle ( ) Time ( T ) Collision probability Time ( T ) It is found that collision probability derived from nonlinear method is nearly close to that from linear method,when the angle is 0.1, and both are still equal even when the angle larger than integral time is different and less than Different threshold limits are adopted half of orbital period; Collision probability Time ( T ) Collision probability collision probability is nearly the

10 When the angle is lower than 0.02, eg : ⅰ If ΔP is used, collision probability from nonlinear method reaches to which is about 20 times greater than that from linear method, and integral time has overtaken 50 orbital periods. ⅱ IfΔP is used, integral time reduces to 30 orbital periods and collision probability is just like the result of ΔP ⅲ If the limit is ΔP 10-8, collision probability is only , which is smaller than the linear result. The reason is that the integration along the relative path is quickly cut off and no more than half period. At last, if the angle is 0.0, the two orbits is coplanar, we notice the result is similar to 0.01 angle, except for limits ΔP So consecutive limits bring about discontinuous integral range and probability, and the probability is very sensitive to the limits. In fact, the sensitivity arises from the relative distance between two objects in the integration along the relative path.

11 There are two intersections in one orbital period, so the relative distance increases gradually overall, but there is also a small cyclic variation; Because the relative distance is smaller near the intersection point, the corresponding sectional probability is larger. That means the collision probability increases faster near the intersection point, whereas increases slower in other area. Fig. 2 shows the accumulated probability and relative distance variation with forward integration from TCA. ( The angle is 0.01, and the threshold limit is ΔP ) At last, if the relative distance greater than 3.5 km (about 5 covariance shell), the limit is satisfied, and the integration is stopped.

12 Fig. 3 shows the comparison of linear method and nonlinear method. ( The integration of nonlinear method is done forward and backward for more than 16 orbital periods with a start at TCA) Considering encounter is a process including two sides, the total collision probability for this encounter derived from nonlinear method is If we use linear method to calculate the probability, there are 32 orbital periods including 64 encounters, corresponding to the range of integration for nonlinear method, so 64 linear probabilities exist.

13 The scope of application of nonlinear method In order to analyze the scope of application of nonlinear method, it is necessary to compare the results of the linear and nonlinear methods. As mentioned above, the encounter defined by linear method is very short with several seconds or minutes, while the encounter defined by nonlinear method maybe covers many orbital periods.obviously they aren t comparable. A new additional integral threshold limit along the relative path is suggested for nonlinear method.because there are two encounters in one orbital period, the threshold limit of half orbital period is adopted. Namely the integration is done forward and backward no more than a quarter of orbital period with start at TCA, so the two methods describe one and the same encounter.

14 Fig. 4 shows the nonlinear probabilities for five different angles. Due to the symmetry of circular motion and constant combined covariance, the integral results forward and backward is symmetrical also, and centered on TCA. Therefore, the two symmetrical probabilities are summed to derive a total nonlinear collision probability. The accumulated probabilities for 0.1 angle quickly increase in just 200 seconds forward and backward, and then keeps stable basically. But for the 0.0 case, the accumulated probabilities keep increasing slowly in half orbital period. According to the geometric, the collision probability calculated by linear method is , which is consistent with the result of the 0.1 case, so nonlinear method is still valid in this case. However, the result from nonlinear method is much less than that from linear method in the 0.0 case.

15 Whether nonlinear method is valid or not is determined by the duration of the encounter, which mainly depends on the relative velocity. Because most satellites move at extremely high velocities ( about 7~8 km/s for low orbits and 3 km/s for geostationary orbits ), the relative velocity between the two objects is mainly affected by the angle between the two orbital planes, larger angle corresponds to larger relative velocity. Therefore the angle can be used as a criterion to determine the scope of application of nonlinear method. Additionally, the duration of the encounter generally depends on orbital altitude. We will discuss how it affects the scope of application of nonlinear method.

16 The encounter just like in table 1, only element a changed, is still used to analyze the scope of application of nonlinear method. Three types of orbital altitudes corresponding to LEO, MEO and GEO are applied to the calculation of collision probability respectively. The calculation results are shown in table 3. The linear probabilities obtained are all , ratio is defined by the nonlinear probability over the linear probability here.

17 Table 3, the ratio of probability values based on the two methods is completely close to 1 if the i angles are greater than 0.02, 0.01 and corresponding to LEO, MEO and GEO. Therefore we can get a preliminary conclusion that linear method is valid if the angles are above 0.02, 0.01 and for LEO, MEO and GEO respectively, and can be used to calculate collision probability fully. When the angle less than this values, it will bring out larger errors if linear method is used. In this case, nonlinear method should be used for the calculation of collision probability.

18 Fig. 5 The ratio of the nonlinear result to the linear result are presented in, three curves depict the ratios for LEO, MEO and GEO respectively. They all increase with the increasing of included angle between the two orbital planes, if the included angles are bigger enough, the ratios are all close to 1. That means, for LEO, MEO and GEO orbits, if the included angle exceeds 0.02, the nonlinear collision probability is similar to the result of linear method, so we calculate the collision probability based on linear method trusting. And more, for the same included angle, the ratio also varies with orbital altitudes, the higher the orbital altitude is, the closer to 1 the ratio is. In other words, the application scope of nonlinear method for LEO is larger than that for GEO.

19 CONCLUSION Nonlinear method of collision probability calculation is an exceptionality for linear case, it happens only in some extreme encounters with very small included angles. The probability is very sensitive to the threshold limit along the relative path for nonlinear method; consecutive limits bring about discontinuous integral range and probability. A new additional integral threshold limit is suggested for nonlinear method. Half orbital period is suggested, namely the integration is done forward and backward not more than a quarter of orbital period with start at TCA. Under this integral threshold limit,linear and nonlinear methods describe the same encounter, and can be compared so as to analyze the application of nonlinear method. Study shows, if the included angle exceeds 0.02, the nonlinear collision probability is consistent with the result of linear method for LEO, MEO and GEO, so we calculate the collision probability based on linear method trustingly. And more, the higher the orbital altitude is, the smaller the application scope of nonlinear method is.

20 Thanks