Spent rocket bodies in geostationary transfer orbit (GTO) pose impact risks to the Earth’s surface when they reenter the Earth’s atmosphere. To mitigate these risks, reentry prediction of GTO rocket bodies is required. In this paper, the reentry prediction of rocket bodies in eccentric orbits based on only TwoLine Element (TLE) data and using only ballistic coefficient (BC) estimation is assessed. The TLEs are preprocessed to filter out outliers and the BC is estimated using only semimajor axis data. The BC estimation and reentry prediction accuracy are analyzed by performing predictions for 101 rocket bodies initially in GTO and comparing with the actual reentry epoch at different times before reentry. Predictions using a single and multiple BC estimates and using state estimation by orbit determination are quantitatively compared with each other for the 101 upper stages.
Rocket bodies in geostationary transfer orbits (GTOs) have their apogee near geosynchronous altitude and their perigee within the Earth’s atmosphere. The atmospheric drag reduces the orbital energy of the rocket bodies and lowers the orbit until reentry occurs. Lunisolar perturbations speed up or slow down this process by changing the eccentricity of the orbit and raising or lowering the perigee altitude, which in extreme cases results in direct reentry without draginduced decay. The reentry of spent rocket bodies is desirable because the deorbiting of these uncontrolled bodies prevents collisions with functional spacecraft and potential generation of new space debris. However, the reentry poses a risk to the Earth’s population because rocket bodies consist of components likely to survive the reentry and impact the Earth’s surface (such as propellant tanks) [
The major source of error in orbit prediction is the computation of the atmospheric drag [
For stateoftheart reentry prediction, the accuracy of atmospheric density calculations can be improved by calibrating the density models using near realtime satellite tracking data [
When density correction models and 6DoF propagation techniques are not available (e.g., because the object details are unknown or the measurements necessary for density corrections are unavailable), the drag coefficient
The application of highly accurate models and orbital data is required for accurately predicting the impact point of reentering objects. Sufficiently accurate orbital data is, however, often not available and TwoLine Element sets (TLEs) provided by the United States Strategic Command are the only available data to perform reentry prediction. The accuracy of TLE data is, however, limited due to the application of simplified perturbation models (SGP4 and SDP4) [
In this paper, the reentry prediction of rocket bodies in eccentric orbits based on only TLE data is assessed. Because attitude and density correction data are not directly available from TLEs, the predictions are carried out using 3DOF propagation and a standard empirical atmospheric density model. Different methods have been developed in the past to improve TLEbased reentry prediction by preprocessing TLE data and by estimating the BC, solar radiation pressure coefficient (SRPC), object state vector, or a combination of these. In this paper, reentry predictions using only an estimate for the BC are investigated. This approach is straightforward and can be used to obtain a firstorder guess of the reentry date several weeks or months before reentry when accurate prediction of the impact point is not feasible due to uncertainties in future space weather predictions. In addition, reentry predictions using only BC estimates can easily be automated to perform daily predictions for many objects. Within this assumption (only BC estimation), the goal of this paper is to provide guidelines on how to estimate the BC to obtain the most accurate reentry predictions.
It should be noted that all these methods estimate a single, and thus fixed, ballistic coefficient. In reality, the BC, however, varies over time due to, for example, rotation of the object or changes in
The estimation of the BC is tailored for reentry predictions by comparing the decay of the mean semimajor axis according to TLE data and according to a highfidelity propagator considering all perturbations.
The impact of the initial state used for BC estimation on the reentry prediction is shown.
The performance of the method is assessed and improved based on predicting the reentry dates of 101 upper stages in highly eccentric orbits (all initially in GTO) and the sources of inaccurate predictions are analyzed.
The good performance of using a single BC estimate versus the use of a median BC estimate and versus BC and state estimation is shown.
Because the considered rocket bodies are in highly eccentric orbits, all relevant perturbations (geopotential, lunisolar, drag, and SRP) are always considered during orbit propagation.
The methods used in this approach are discussed in the following section. After that the BC estimation and reentry prediction results using a single and multiple BC estimates are discussed.
The orbital propagator and BC and state estimation and TLE preprocessing methods used for TLEbased reentry prediction are discussed in the following.
The orbital propagator used in this study is the Accurate Integrator for Debris Analysis (AIDA), a highprecision numerical propagator tailored for the analysis of space debris dynamics using uptodate perturbation models. AIDA includes the following force models [
NASA’s SPICE toolbox (
The approach used for the estimation of the BC is based on the method for deriving accurate satellite BCs from TLEs proposed by Saunders et al. [
Compute the change in semimajor axis between the two TLEs,
Take guess for value of the BC.
Propagate the orbit with the full dynamical model between the two TLE epochs and simultaneously compute
where
Integrate
Update the BC estimate value using the Secant method:
where
Repeat the procedure from step 3 until convergence is reached.
The first guess,
Several changes were made to the original method by Saunders. First, during the BC estimation process, it may happen that the object unexpectedly reenters during propagation. Such a reentry is generally the result of a toohigh estimate for the BC. Therefore, the propagation is then repeated assuming a smaller value for BC, namely, 90% of the initial value. This prevents failure of BC estimation due to reentry but may require several iterations to sufficiently reduce the BC value.
By default forward propagation is applied for BC estimation, that is, taking the state at the earliest TLE and propagating it until the epoch of the latest TLE. In addition, also backward propagation was implemented starting from the latest TLE and propagating backward until the prior one. By propagating backward one prevents reentry occurring during propagation. This is especially useful when estimating the BC close to reentry where an inaccurate BC guess can easily cause unexpected reentry.
Furthermore, the change in semimajor axis due to drag (see (
Finally, the average semimajor axis is computed from osculating data from AIDA to compare the change in semimajor axis with TLE data. This improves the estimation because the osculating data includes shortperiodic variations whereas the mean TLE data does not [
Besides estimating the BC also the SRPC can be estimated. DoladoPerez et al. [
The state estimation performed in this work is carried out by fitting accurate orbit propagation states to pseudoobservations derived from TLEs using nonlinear leastsquares. This is a consolidated method widely used for offline (groundbased) orbit determination (OD) [
The TLEs have to be filtered because incorrect, outlying TLEs and entire sequences thereof could be present in the data from SpaceTrack, and using such aberrant TLEs in subsequent analyses would deteriorate the accuracy of the results. Filtering out aberrant, or incorrect, TLEs consists of a number of stages [
filter out TLEs that were published but subsequently corrected;
find large time gaps between TLEs because they hinder proper checking of TLE consistency;
identify single TLEs with inconsistent mean motion, as well as entire sequences thereof, using a sliding window approach;
filter out TLEs outlying in perigee radius;
filter out TLEs outlying in inclination;
filter out TLEs with negative
TLEs with negative
To determine the quality of the BC estimates, the estimates were compared with BC values derived from
To test the reentry prediction performance, a set of 101 rocket bodies that reentered in the past 50 years was selected. This makes it possible to compare the predicted reentry date with the real one. The reentry dates were taken from satellite decay messages from the SpaceTrack.org website (
In addition, all objects have been used to predict the reentry 10, 20, 30, 60, 90, and 180 days before the actual reentry date. Some of the 101 objects were not suitable for several reentry prediction tests, because they had no TLEs within a specific number of days before the reentry (e.g., last TLE is 90 days before reentry).
In real reentry prediction cases, the actual reentry date of the object is, of course, not known. Analyzing the results has therefore not only the goal to examine the quality of the reentry predictions but also the goal to define guidelines for real reentry prediction scenarios.
Figure
BC estimates and BC from
Besides, there is a clear relation between outliers in TLE perigee radius and estimated BC; an outlier in perigee radius results in an outlier in the BC estimates. More precisely, of the two TLEs that are used for BC estimation, the outlying TLE that is used to obtain the initial state for propagation results in an outlier in BC estimate. The other TLE is only used to compute the change in semimajor axis according to the TLEs and does not have such a strong effect. Therefore, it can be concluded that the BC estimate strongly depends on the initial state used in the estimation. Because the atmospheric drag depends largely on altitude, an incorrect value of the initial state that translates in an aberrant perigee height results in a poor BC estimate. The BC estimate compensates for the incorrect initial state such that the state and BC together give the correct decay in the estimation period.
Figures
To have a closer look at the dependency of the BC estimate on the perigee radius, the BC estimates are plotted against perigee radius according to TLE data for object 27808 in Figure
BC estimates (a), the osculating perigee radius according to TLE data (b), and BC estimates against perigee radius (c) for object 27808 in the 180 days before reentry.
This proves that to obtain a good single BC estimate the TLEs should be filtered on perigee radius, or on both semimajor axis and eccentricity. Another option to reduce the impact of outliers on the estimate is to compute multiple BC estimates and take the median of the estimates as the final BC estimate. The reentry prediction results using a single and a median BC estimate are discussed in the next two sections.
Besides, different epoch separations between the two TLEs used for BC estimation have been tested, namely, 2, 5, 10, and 20 days. A TLE separation of 10 days was found to be least sensitive to outliers and shortperiod effects, because the difference between mean and median of the estimates was the smallest and the dispersion in terms of standard deviation and median absolute deviation was small as well. Therefore, 10day separation is used for BC estimation, which is in agreement with Saunders et al. [
Finally, BCs were estimated for the 101 test objects in the 180 days before reentry. It was found that 80% of the medians of the BC estimates were within the range of possible areatomass ratio (assuming
Median of the BC estimates and the minimum and maximum BC according to object data for all 101 objects. Median BC estimates outside the BC range according to data are indicated with an orange dot. (Objects are sorted on increasing average areatomass ratio.)
The objective of this section is to show that, for reentry prediction using only a BC estimate, it is of fundamental importance to run the reentry predictions using the same state that is used for BC estimation.
As described in Section
Reentry predictions 30 days before reentry using an older or newer TLE for BC estimation and the same or a different state for BC estimation and reentry prediction. All reentry predictions start from the TLE at 30 days before reentry. BC estimation starts from the same TLE (orange and blue lines) or ends there and starts at a different TLE (yellow and green lines). The other TLE used of BC estimation is either an older or a newer TLE with respect to the TLE at 30 days. (The colors of the plots in (a) and the arrows in (b) correspond.)
Cumulative distributions and 90%confidence regions of reentry prediction errors using only an estimate for BC for 91 objects 30 days before reentry
Schematic diagram of BC estimation
Using the same state for BC estimation and reentry prediction gives better results, because the BC estimate is computed such that together with the state it gives the correct decay rate of the semimajor axis in the estimation period. Using that BC estimate with another state will generally not result in the correct decay rate and the reentry prediction is thus more likely to be less accurate. Therefore, the same initial state for BC estimation and reentry prediction should be applied.
The reentry predictions using a single BC estimate that are presented in the following sections are computed using the “older TLE, same state” approach such that the latest available TLE is used for the initial state.
Instead of using a single estimate, one can compute multiple estimates and take the mean or median of the set that may better represent the average BC behavior. This approach was tested by estimating the BC for every TLE between 90 and 30 days and from 180 to 60 before reentry and use the median of the estimates for reentry prediction at 30 and 60 days before reentry, respectively. The prediction errors are shown in Figure
Cumulative distribution and 90%confidence region of reentry prediction error using a single BC estimate (orange) or the median BC (blue) for (a) 91 objects 30 days before reentry and (b) 93 objects 60 days before reentry.
30 days before reentry; median taken from BC estimates between 90 and 30 days before reentry
60 days before reentry; median taken from BC estimates between 120 and 60 days before reentry
It was found that especially for orbits with a high eccentricity and low inclination the predictions with median BC are less accurate. Figure
Reentry prediction error 60 days before reentry using a single BC (orange) or median BC (blue) plotted against eccentricity with different markers for different inclination ranges.
Median absolute deviation (MAD) of detrended mean perigee radius data in 180 days before reentry against eccentricity at 60 days before reentry.
These results suggest that estimation of the perigee altitude or eccentricity is required in order to improve the perigee data and thus the BC estimation and reentry prediction. Indeed, Sharma et al. [
The reentry predictions using only BC estimates are compared with those after full state estimation using OD. Figure
Cumulative distributions and 90%confidence regions of reentry prediction error of 91 objects 30 days before reentry using only an estimate for BC and (a) after OD to estimate state and BC and (b) subsequently reestimate the BC.
Prediction errors using only BC estimate and after OD
Prediction errors using only BC estimate and after OD with subsequent BC reestimation
To assess whether an accurate state and BC estimate result in an accurate reentry prediction, the six objects with the lowest position residuals after state and BC estimation using OD at 30 days before reentry were analyzed. Table
Mean position residuals and reentry prediction errors before OD (only BC estimation) and after OD (see Section
NORAD 

Mean position residual [km]  Prediction error [%]  

Before OD  After OD  Before OD  After OD  
19332  0.153  660.0  9.9  2.3  1.4 
7252  0.070  662.3  7.8  2.2  4.8 
7794  0.050  105.5  3.0  6.3  6.1 
9017  0.084  513.2  7.3  7.7  6.4 
25240  0.087  422.6  6.7  8.2  9.7 
25372  0.046  303.3  7.9  11.9  16.5 
This outcome may be the consequence of taking a fixed BC for prediction. Figures
Finally, the reentry prediction results for 10, 20, 30, 60, 90, and 180 days before reentry using single BC estimates are shown in Figure
Cumulative distributions of reentry prediction error 10, 20, 30, 60, 90, and 180 days before reentry and all prediction errors together with 90%confidence region using only an estimate for BC.
All prediction errors and at 10, 20, and 30 days before reentry
All prediction errors and at 60, 90, and 180 days before reentry
Overall, with 90% confidence, 62 to 72% of the predictions is within 10% error and 85 to 95% within 20% error. Using a single BC estimate one can thus obtain a firstorder estimate of the reentry date irrespective of TLE quality and availability. More sophisticated methods, such as 6DoF propagation and density corrections, should subsequently be applied to accurately estimate the impact point of the reentering object.
The estimation of the BC is tailored for reentry predictions by comparing the decay of the mean semimajor axis according to TLE data with the decay of the average semimajor axis due to drag according to a highfidelity propagator considering all perturbations. The BC estimation results show that the estimated BC depends strongly on the initial state because TLE outliers and noise in the perigee radius result in outliers and noise in BC estimates. Therefore, filtering TLEs on eccentricity or perigee radius is important. Because of the dependency on the initial state, it is important to use the same initial state for BC estimation and reentry prediction as inaccuracy in the state is absorbed by a single BC estimate such that they provide the correct decay of the semimajor axis. Taking the median of multiple BC estimates for predicting the reentry does not give good results, because the median BC is not related to the initial state. The accuracy of reentry predictions after state and BC estimation using OD are not significantly different from using only a single BC estimate. Moreover an accurate initial state and BC do not necessarily give accurate reentry predictions. Overall, using a single BC estimate 62 to 72% of the reentry predictions is within 10% error (with 90% confidence). These conclusions are based on reentry predictions using TLE data and are thus subject to their accuracy and availability that vary largely for different objects.
Besides using more accurate orbital data, the fixedBC approach can be improved by using more accurate atmospheric density models and by applying a wind model to increase the accuracy of density and velocity calculations during both BC estimation and reentry prediction. Furthermore, if the accuracy of the orbital data is very low, estimation of the eccentricity or perigee radius could improve the predictions as they strongly affect the BC estimate and reentry prediction. However, if the drag coefficient or frontal area of the object changes over time, then the achievable accuracy using a fixed BC is limited. Knowledge of the object’s attitude and 6DoF propagation or a forecasting model for the BC could significantly reduce the reentry prediction error.
Rocket bodies with the following NORAD catalog numbers were used for reentry prediction:
625, 2609, 7252, 7794, 8479, 9017, 9787, 9859, 10983, 11072, 11718, 11719, 12562, 12810, 13025, 13087, 13098, 13136, 13294, 13447, 13599, 13684, 13940, 14130, 14168, 14287, 14332, 14369, 14423, 14787, 14989, 15157, 15165, 15679, 16600, 18352, 18923, 19218, 19332, 19877, 20042, 20123, 20254, 20778, 20920, 21057, 21141, 21654, 21766, 21895, 21990, 22118, 22254, 22906, 22928, 22932, 22997, 23315, 23416, 23572, 23797, 23916, 24314, 24666, 24770, 24799, 24847, 25051, 25129, 25154, 25240, 25313, 25372, 25496, 25776, 26560, 26576, 26579, 26641, 27514, 27719, 27808, 28185, 28239, 28253, 28418, 28452, 28623, 28703, 29497, 32764, 36829, 37211, 37239, 37257, 37482, 37764, 37805, 37949, 39499, 40142
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was partly carried out within the European Space Agency project ITT AO/18155/15/D/SR, titled “Technology for Improving ReEntry Predictions of European Upper Stages through Dedicated Observations.” The authors acknowledge Dr. Hugh G. Lewis of the University of Southampton (UoS), Dr. Camilla Colombo of Politecnico di Milano, and Dr. Tim Flohrer and Quirin Funke of the European Space Agency for their valuable contributions. In addition, the use of the IRIDIS High Performance Computing Facility and associated support services at UoS in the completion of this work are acknowledged. David J. Gondelach was funded by an EPSRC Doctoral Training Grant awarded by the Faculty of Engineering and the Environment of UoS. Aleksander A. Lidtke would like to acknowledge the funding he received from the Ministry of Education, Culture, Sports, Science and Technology of Japan. Roberto Armellin acknowledges the support received by the Marie SkłodowskaCurie Grant 627111 (HOPT, Merging Lie perturbation theory and Taylor Differential algebra to address space debris challenges).