Open Astron. 2020; 29: 210–219 Research Article Zhang Wei*, Cui Wen, Wang Xiuhong, Wei Dong, and Liu Xing Re-entry prediction of objects with low-eccentricity orbits based on mean ballistic coeflcients https://doi.org/10.1515/astro-2020-0006 Received Feb 17, 2020; accepted May 05, 2020 Abstract: During re-entry objects with low-eccentricity orbits traverse a large portion of the dense atmospheric region almost every orbital revolution. Their perigee decays slowly, but the apogee decays rapidly. Because ballistic coefficients change with altitude, re-entry predictions of objects in low-eccentricity orbits are more difficult than objects in nearly circular orbits. Problems in orbit determination, such as large residuals and non-convergence, arise for this class of objects, especially in the case of sparse observations. In addition, it might be difficult to select suitable initial ballistic coefficient for re-entry prediction. We present a new re-entry prediction method based on mean ballistic coefficients for objects with low-eccentricity orbits. The mean ballistic coefficient reflects the average effect of atmospheric drag during one orbital revolution, and the coefficient is estimated using a semi-numerical method with a step size ofone period. The method is tested using Iridium-52 which uses sparse observations as the data source, and ten other objects with low-eccentricity orbits which use TLEs as the data source. We also discuss the performance of the mean ballistic coefficient when used in the evolution of drag characteristics and orbit propagation. The results show thatthemean ballistic coefficient is ideal for re-entry prediction and orbit propagation of objects with low-eccentricity orbits. Keywords: re-entry prediction, low eccentricity orbit, ballistic coefficient, orbit propagation 1 Introduction predicting the re-entry time. The drag coefficient depends on many parameters, such as the object’s shape, the surface material, and the composition and the temperature of the Massive space objects cannot completely burn up during atmosphere. An object’s drag coefficient may vary under a re-entry; 10% to 40% of the mass may survive (Ailor et different altitudes and solar activity conditions (Moe and al. 2005), and the surviving components may pose a threat Moe 2005). Usually the space object’s exact shape, mass, to humans, buildings, and the environment (Choi et al. attitude, and surface material are unknown; therefore, it 2017). Usually we call these objects risky re-entry objects. is difficult to estimate the drag coefficient, cross-sectional To mitigate these risks we need predictions of re-entry of area, and mass. Hence, it is reasonable to combine these pa- massive space objects. Atmospheric drag is the dominant rameters into the ballistic coefficient B to scale how much non-gravitational perturbation acting on re-entering ob- the space object suffers from atmospheric drag (Bowman jects. Precise area-to-mass ratio of the re-entry object and 2002). B is defined by Equation (1): proper modeling of the drag characteristic are the key issues A in calculating the acceleration due to atmospheric drag and B C = D × m (1) where CD, A and m are the drag coefficient, cross-sectional Corresponding Author: Zhang Wei: Xian Satellite Control Center, area, and mass, respectively. Xian, China; Email: [email protected] Cui Wen: Xian Satellite Control Center, Xian, China; Atmospheric drag decelerates the re-entry object, thus Email: [email protected] causing a gradual decrease in eccentricity and semi-major Wang Xiuhong: Xian Satellite Control Center, Xian, China; axis. In this paper we classify re-entry objects according to Email: [email protected] their initial apogee altitude, Ha_ini, as follows (here initial Wei Dong: Purple Mountain Observatory, Chinese Academy of Sci- means ten days before re-entry, which is also the limitation ences, Nanjing, China; Email: [email protected] to the perigee altitude): Liu Xing: Xian Satellite Control Center, Xian, China; Email: [email protected] 1. Nearly circular orbit: Ha_ini < 500 km; Open Access. © 2020 Z. Wei et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Z. Wei et al., Re-entry prediction based on mean ballistic coeflcient Ë 211 2. Low-eccentricity orbit: 500 km ≤ Ha_ini < 5000 km; axis due to the drag computed by propagation using an ini- 3. High-eccentricity orbit: Ha_ini ≥ 5000 km. tial state from TLEs (Saunders et al. 2012; Sang et al. 2013; Mutyalarao and Sharma 2011; Lu and Hu 2017). Agueda et al. According to this classification, the statistics for large derived pseudo-observations from TLEs and then estimated uncontrolled objects that have re-entered from 2012 to 2018 the ballistic coefficient, solar radiation pressure coefficient, are shown in Figure 1. The information of the re-entered ob- state vector, or a combination of these (Agueda et al. 2013; jects is obtained from the space-track website. We see that Lidtke et al. 2016). Gondelach et al. studied the re-entry objects with low-eccentricity orbits are not in the minority, predictions of GTO rocket bodies, and proposed that TLEs exceeding one-sixth in the past seven years, and even 40% should be preprocessed to filter out outliers (Gondelach et in 2018. al. 2017; Lemmens and Krag 2014). When compared with objects in nearly circular or- 100 Nearly Circular Orbit bits, we find that re-entry predictions of objects in low- Low−Eccentricity Orbit 80 High−Eccentricity Orbit eccentricity orbits are more difficult. Because objects with low-eccentricity orbits traverse a large portion of the dense 60 atmospheric region almost every orbital revolution, the Number 40 drag coefficients change significantly in one revolution. In 20 a practical test it was found that it might be very difficult to obtain small residuals for each pass, particularly if we 0 2012 2013 2014 2015 2016 2017 2018 adopted the strategy that estimated a single ballistic coeffi- Year cient from multiple passes of observations. If we adopted Figure 1. Number of large uncontrolled re-entered objects from 2012 the strategy that estimated multiple ballistic coefficients to 2018 from observations, it might be difficult to select a suitable estimated result as the initial value for re-entry prediction, even though residuals for each pass could be small. The main difficulties in re-entry prediction are de- Almost all of the earlier studies assume the ballistic termining the orbit and modeling the atmosphere drag coefficient is constant, which might not be a reasonable et al. (Gondelach 2016). The optimal method for re-entry choice for objects with eccentric orbits. This paper presents prediction is to determine the precise orbit based on abun- a new method for re-entry prediction of objects with low- dant observations, estimate the ballistic coefficient as a fit- eccentricity orbits. This new method is based on mean bal- ting parameter, and then propagate the orbit until re-entry. listic coefficients, which reflect the average effect of atmo- et al. Xiong carry out the re-entry predictions by mainly spheric drag during one orbital revolution. The method is employing a numerical integration method based on ob- developed in Section 2. Then in Section 3 the method is servations acquired from the optical surveillance network. tested using the Iridium-52 satellite which uses sparse ob- They obtain the same re-entry time from two precise orbits servations as the data source, and ten other objects with by adjusting the ballistic coefficient to improve the predic- low-eccentricity orbits which use TLEs as the data source. tion accuracy, and they propose that the change in the drag Evolution of drag characteristics and orbit propagation us- coefficient with altitude should be considered whenthe ing the mean ballistic coefficient are also discussed in Sec- et al. mean altitude is below 180 km (Xiong 2009). tion 3. Finally, some conclusions are drawn in Section 4. Only a few countries, however, have the ability to ac- quire observations of the re-entry objects. Most re-entry pre- dictions are performed based on TLE (Two Line Element) sets provided by United States Strategic Command. Pardini 2 Method et al. used Satellite Re-entry Analysis Program (SATRAP) software, which uses TLEs as the input, to study the pre- The mean ballistic coefficient estimation method and the re- diction errors under different solar activities and different entry prediction method are laid out in this section. In this atmospheric model conditions (Pardini and Anselmo 2001, paper we use classical orbital elements σ(a, e, i, Ω, ω, M) 2008; Anselmo and Pardini 2013). Different methods have to describe the orbital motion, where (a, e, i, Ω, ω, M) are been developed in the earlier studies to improve TLE-based semi-major axis, eccentricity, inclination, right ascension re-entry prediction. Saunders et al. estimated the ballis- of the ascending node, argument of perigee, and mean tic coefficient by comparing the change in the semi-major anomaly, respectively. axis according to TLE data with the change in semi-major 212 Ë Z. Wei et al., Re-entry prediction based on mean ballistic coeflcient 2.1 State Estimation 2011): 2 (︁ 2)︁ 1 (︁ 2)︁ CD = p exp −s + 1 + 2s erf(s) (3) State estimation can be carried out based on observations πs s2 or TLEs. When multiple passes of observations are used, s 1 p Tk r problems such as large residuals and non-convergence may + π , s Ti arise for the re-entering objects with low-eccentricity orbits, especially when there are sparse observations. Therefore, where s is the speed ratio, defined as the ratio of the rela- we perform state estimation based on a single pass of ob- tive velocity of the re-entry object with respect to the atmo- servations using a numerical method.