Orbit Determination of Chinese Rocket Bodies from the Picosecond Full-Rate Laser Measurements

The problem of space debris population is a significant challenge in today's world. Over the past few years, the number of space objects has been increasing rapidly. According to the latest data from the SPACETRACK service (SPACETRACK, 2023), there are currently 26,991 known objects larger than 10 cm. However, not all objects are being tracked and cataloged. The Space Debris Office of the European Space Agency estimates that there are around 36,500 debris objects larger than 10 cm based on statistical models (Space Debris User Portal, 2022). These numbers are influenced by various factors such as the deployment of commercial satellite operators and large constellations in low earth orbit (LEO) (Space Debris Office, 2022), as well as the anti-satellite tests conducted by several countries including China, USA, Russia, and India (Czajkowski, 2021, Harrison et al., 2021). According to the latest predictions from Canadian scientists, there is a 10% chance of a human casualty occurring in the next decade due to space debris falling (Byers et al., 2022). Due to the threats posed by space debris, it is necessary to continuously monitor this environment using passive and active techniques such as radars, optical telescopes, and laser acquisition. One of these techniques is satellite laser ranging (SLR), which is coordinated by the International Laser Ranging Service (ILRS) (Pearlman, Noll et al., 2019). SLR is a high-accuracy observational technique that allows for tracking satellites with millimeter accuracy. The results of SLR are widely used in geodesy and geophysics. They include determining Earth's polar motion and length of day, determining coordinates for laser stations and the geocenter, determining coefficients of Earth's gravity field and tidal parameters, and determining satellite ephemerides, fundamental physical constants, and others (Pearlman, Arnold et al., 2019). SLR has been successfully used by multiple teams for many years to track space debris as well (Greene et al., 2002, Smith, 2006, Sang and Smith, 2012, Zhang et al., 2012, Kirchner et al., 2013, Voelker et al., 2013, Zhongping et al., 2017, Steindorfer et al., 2020, Zhang, 2021). Accurate information about the orbits of space debris is needed to determine their attitude or spin dynamics (Kucharski et al., 2014, 2016, 2017, Silha et al., 2016, Schildknecht and Silha, 2017, Rodriguez-Villamizar and Schildknecht, 2022, Phipps et al., 2012, Scharring et al., 2021). Laser ranging activities in Poland are aimed at joining the countries involved in space debris tracking (Lejba et al., 2018).

The problem of determining the orbits of space debris has been extensively studied and analyzed in several papers. These studies include angular measurements right ascension/declination (RA/DEC) and laser ranging data. The results obtained depend on various factors, such as the type of data used (angular data, laser ranging data, or both), the length of the orbital arc, the number of observations for the given arc, the geometry of the observations, and the distribution of the observations along the orbit (Bennett et al., 2015, Cordelli et al., 2016, 2020, Zeitlhofler et al., 2023).

The studies in this paper used laser measurements from the Borowiec laser station (BORL) sensor in full-rate format. The paper demonstrates how the orbital solution depends on the length of the orbital arc and the number of measurements per arc. It also determines an alert time (AT), indicating when a laser measurement of an analyzed object should be repeated (using the three-sigma criterion). In addition, the paper determines the semi-major axis (a) and eccentricity (e) parameters of the analyzed objects, as well as the mean errors in the radial (R), along-track (S), and cross-track (W) systems. The orbital computation made the following criteria: measurements come from one sensor (BORL), measurements are in full-rate format, and successive measurements occur at intervals no longer than 24 h.

In mid-2016, the Polish laser sensor located at Borowiec and owned by the Space Research Centre of the Polish Academy of Sciences (CBK PAN) began regularly tracking space debris objects. These objects include inactive satellites with retroreflectors (cooperative targets) such as ENVISAT, ERS-1, ERS-2, SEASAT-1, or TOPEX/Poseidon, as well as uncooperative targets like European Space Agency (ESA)'s ARIANE, USA's DELTA and TITAN, China's CZ, or Russian's SL rocket bodies (RBs) from the LEO regime. From 2016 to 2020, the sensor recorded 1971 successful passes of space debris, with 23,612 normal points and 1,107,829 single measurements (full rate). The average precision was approximately 0.4 m root mean square (RMS), including both cooperative and uncooperative targets (Smagło et al., 2021). Currently, BORL regularly tracks about 100 different space debris objects. In the past few years, the number of successfully tracked objects has doubled or even tripled (Fig. 1).

Figure 1.

In the record year of 2020, there were 698 successful passes of space debris acquired, accounting for 34% of all passes obtained that year. On average, around 27% of all passes registered by the BORL sensor each year are space debris. The BORL sensor tracks typical RB objects that do not have retroreflectors and provide reflections. Measurements are taken using a nanosecond laser, with an average precision of approximately 0.7 m RMS for objects with a radar cross section greater than 1 m². In the years 2016–2021, our laser sensor registered 53 different objects and their 393 passes. For picosecond measurements, the average precision is approximately 0.02 m RMS for objects with a radar cross section greater than 1 m². In the years 2016–2021, our laser sensor registered 11 different objects and their 325 passes.

The BORL sensor operates independently using two neodymium-doped yttrium aluminum garnet Nd:YAG pulse laser modules (picosecond and nanosecond). It has a Cassegrain telescope with an aperture of 65 cm and an effective area diameter of 60 cm (Lejba et al., 2016). The mount is equipped with a 20-cm guiding Maksutov telescope for visual control. The sensor's measuring equipment includes a high-speed start Si photodiode FDS025, a STANFORD SR620 time interval counter, and a HAMAMATSU H5023 detector. The laser sensor's reference time scale is Universal Time Coordinated Astrogeodynamic Observatory at Borowiec UTC(AOS), which is a local realization of UTC based on an active hydrogen maser CH1-75A that provides a 10 MHz and 1 pulse per second (PPS) time signal to the laser sensor. Table 1 shows the basic configuration of the laser modules used for space debris tracking.

Both lasers are mounted on an optical table and operate at a repetition rate of 10 Hz. They have significant differences in pulse width and pulse energy, resulting in the nanosecond module having an average power that is 9 times higher. This module is preferred for laser ranging to space debris.

For this study, two Chinese RBs were chosen. They have North American Aerospace Defense (NORAD) numbers 28480 and 31114. These objects were selected because they have a relatively large number of laser measurements performed by the BORL sensor. The BORL sensor successfully obtained its first measurements of object 28480 on December 30, 2016 and of object 31114 on November 9, 2016. Both objects are orbiting in the LEO regime, which has the highest spatial density of objects with a diameter greater than 10 cm. Table 2 provides a brief description of these objects based on information from the Database and Information System Characterizing Objects in Space (DISCOS) of ESA.

Table 3 presents the detailed observational statistics of objects 28480 and 31114 from the years 2016 to 2022, as provided by the BORL sensor. The table includes data on successful passes, registered returns, pass RMS, range bias (RB), and time bias (TB).

Both objects are orbiting below 1000 km. The orbit of object 28480 is slightly flattened, with an eccentricity of 0.014. The orbit of object 31114 is close to a circular orbit, with an eccentricity of approximately 0.006. Their physical parameters, including mass and height, are identical.

The most productive years for laser measurements of these objects, as recorded by the BORL sensor, were 2017–2018. During this time, there were a total of 75 successful passes of object 28480 and 172 successful passes of object 31114. The detailed observational statistics can be found in Table 4.

The higher average RMS value in 2017 for both objects is due to the initial use of a nanosecond module for laser measurements. It was discovered that both objects emit a significant amount of reflections, likely due to having reflectors. As a result, laser measurements switched to using a picosecond module, which is several dozen times more accurate than the nanosecond module. The average pass RMS in this case was several centimeters. These objects still produced a significant number of reflections when measured, with an average of 403 reflections per pass for object 28480 and 718 reflections per pass for object 31114. These values are typical for cooperative targets equipped with retroreflectors. Based on this, it was presumed that objects 28480 and 31114 are equipped with retroreflectors or other components (reflective coatings/optical elements) that can easily reflect the laser beam (Lejba et al., 2018). However, this information could not be officially confirmed in any available publication, technical report, or conference paper. The subsequent measurements of those objects made in the following years using a picosecond laser continued to yield many returns from them. This suggests that these registered returns come from the retroreflectors. Figures 2 and 3 provide convincing evidence of this, as they both show numerous returns similar to those observed for the cooperative targets tracked regularly by the BORL sensor.

Figure 2.

Figure 3.

Even though CZ's RBs have retroreflectors mounted, it is unknown how many have been installed, where they are located, and in what configuration (single or group, ring or panel, etc.). This means that the computed coordinates of the objects are referenced to the position of the reflective element, not the center of mass of the objects. The center of mass correction for the studied objects is unknown and was not applied in the orbital computations.

The best time for observing these objects using the laser sensor is when they pass close to the day/night terminator (object is visible) due to the use of two-line-elements (TLE) as input ephemeride. However, blind tracking was also successful, and the laser returns have been received multiple times.

The laser measurements in full-rate format that are considered in the presented research were collected in 2018. This year, the BORL sensor was most effective in terms of laser ranging to objects 28480 and 31114. The selected measurements were divided into two groups (slots) for each object, with one set of measurements following another within a short period of approximately 24–25 h. Each slot was composed of three to five observations, as shown in Table 5. This table provides information on the number of full-rate measurements and their duration. The average measurement time from one pass for object 28480 was 195.5 s (3.26 min) and for object 31114, it was 309.9 s (5.16 min). The details about each observation can be found in Table 5, including the date of measurements, the start epoch of laser measurements (in UTC time), the time slot during which the returns were registered (time of measurements), the number of accepted returns (full-rate measurements), TB, RB, and pass RMS. Finally, object 28480 creates five observations in slot number 1 and three observations in slot number 2. Object 31114 creates four observations in slot number 3 and three observations in slot number 4.

The RMS pass values given in Table 5 deserve particular attention. The average values for objects 28480 and 31114 are 2.70 and 3.25 cm, respectively. The maximum value does not exceed 4.64 cm, and the minimum value is 1.80 cm. All values are comparable for both objects, indicating that the registered returns most likely come from dedicated optical elements that easily reflect the laser beam, such as retroreflectors. A pass RMS greater than 4 cm may suggest that the returns come from different parts of the objects, possibly indicating the presence of multiple retroreflectors mounted in different places.

The orbital calculations were conducted using NASA Goddard's GEODYN-II program (Beall et al., 2015). The parameters and models used in the GEODYN-II calculations are listed in Table 6. The laser measurements in full-rate format were adjusted for atmospheric refraction using the Mendes and Pavlis model (Mendes et al., 2002, Mendes and Pavlis, 2004), with an elevation cut-off angle of 10°. The orbit integration step size was 1 s. The size and mass of the objects were obtained from ESA's DISCOS database (DISCOS, 2022). The input orbital elements were Cartesian state vectors that were determined from the TLE set that was closest to the epoch of the beginning of the laser observations. The TLE sets were obtained from the US Space Surveillance Network. For many years, TLE element sets have been regularly used in many applications related to determining and predicting the orbit of space objects.

Unfortunately, they are not published with an associated estimate of uncertainty. Laser ranging to objects 28480 and 31114 based on TLE sets generally does not pose a problem for tracking them. However, it should be noted that, whenever possible, the most recent TLE data should be downloaded before making observations. The accuracy of TLE sets is limited and depends on factors such as the type of object and its orbit, the mathematical formulas used to generate the sets, the number and quality of observations, etc. (Kelso 2007, Vallado and Cefola, 2012).

For this study, the author selected observations of objects 28480 and 31114 that met the following assumptions: −

the measurements are from a single station,

Based on the assumptions mentioned above, four different data slots were chosen, with two slots assigned to each object (Table 5). The slots were not merged together, and each one represented a separate range of data (with no overlapping). For object 28480, the first slot consisted of five observations, while the second slot had three observations. As for object 31114, the first slot contained four observations and the second slot had three observations. There was no overlap between the slots.

Orbital calculations were performed in three different scenarios for both objects. In scenario No. 1, each orbital arc was 24 h long. The initial satellite's state vector was taken from TLE. Then, a new initial state vector was determined by incorporating laser measurements into the calculations. In the next step, the orbital calculations were repeated, this time considering the corrected initial state vector. Scenario No. 2 was very similar to Scenario No. 1, that is, each orbital arc was formed by only one observation and each orbital arc lasted for 24 h. However, in addition, the input data consisted of deviations of the object coordinates determined in Scenario No. 1 for individual orbital arcs. In Scenario No. 3, the observations from each slot that followed were combined to create a longer orbital arc. The length of the orbital arcs was increased by 1–4 days (depending on the slot). The orbital calculations were performed in the same manner as in Scenario No. 1. Each scenario produced eight different solutions for object 28480 and seven different solutions for object 31114.

In all orbital calculations, three main parameters were determined, that is, arc RMS, RMS of position (RMSPOS) and RMS of velocity (RMSVEL), and AT. The arc RMS is the root mean square of the post-fit residuals and is defined as follows: (1) RMS=Σi=1nOi−Ci2n−1 {\rm{RMS}} = \sqrt {{{\Sigma _{i = 1}^n{{\left( {{O_i} - {C_i}} \right)}^2}} \over {n - 1}}} where O_i − C_i is the difference between the observed and calculated distance values (residuals) and i is the number of laser measurements.

RMSPOS and RMSVEL are defined as follows: (2) RMSPOS=σX2+σY2+σZ2 {\rm{RMSPOS}} = \sqrt {\sigma _X^2 + \sigma _Y^2 + \sigma _Z^2} where σ_X is the standard deviation of the determined component X, σ_Y is the standard deviation of the determined component Y, and σ_Z is the standard deviation of the determined component Z. (3) RMSVEL=σVx2+σVy2+σVz2 {\rm{RMSVEL}} = \sqrt {\sigma _{Vx}^2 + \sigma _{Vy}^2 + \sigma _{Vz}^2} where σ_Vx is the standard deviation of the determined component Vx, σ_Vy is the standard deviation of the determined component Vy, and σ_Vx is the standard deviation of the determined component Vz.

The positions and velocities of the analyzed objects were determined using two processes: orbit propagation (OP) and orbit determination (OD). In the OP process, the data was obtained directly from the TLE ephemeride, while in the OD process, the data was obtained directly from laser measurements. The input state vector, taken from the TLE set, was the same for both processes. The computed cartesian coordinates from both processes were converted to the radial-along-track-cross-track (RSW) coordinate system and compared with each other. In the RSW system, which moves with the object, the radial axis (R) always points from the Earth's center along the radius vector toward the object. The along-track axis (S) points in the direction of the velocity vector and is perpendicular to the radius vector. This axis is not always aligned with the velocity vector. The cross-track axis (W) is normal to the orbital plane of the object (Vallado 2003). In the next step, the differences between the compared coordinates and their sigmas have been determined. In the last step, AT has been determined at which the computed difference for one of the components, R, S, or W, is greater than or equal to 3 sigma. The AT parameter indicates the interval at which the laser measurement of the object should be repeated.

In addition, the semi-major axis and eccentricity of the analyzed objects have also been calculated. These parameters were consistently determined at the start of the orbital arc. The results are provided in the following section.

The results of the orbital calculations are shown in Table 7 (object 28480) and Table 8 (object 31114). Both tables present the values of RMSPOS and RMSVEL, the number of used full-rate measurements for a given arc, and RMS of determined orbital arcs (arc RMS).

Furthermore, the results of the calculated RMSPOS parameter are displayed in Figures 4 and 5 for objects 28480 and 31114, respectively. The conclusions drawn from the obtained results are very similar for both objects. In Scenario No. 1, it is evident that when using orbital arcs with a length of 24 h, the value of the calculated RMSPOS parameter is approximately 500 m, regardless of the used slot of laser measurements. The arc RMS for Scenario No. 1 ranges from 2.28 to 6.29 cm for object 28480 and from 2.49 to 12.85 cm for object 31114. In Scenario No. 2, it is clearly visible that the values of RMSPOS parameter for subsequent slots are significantly lower compared to the results obtained in Scenario No. 1. These values are lower by 100–300 m. However, the arc RMS values show only slight changes. In 11 cases, the differences are below 1 mm, in one case the difference is 1.7 mm, in three cases it ranges between 7 and 9 mm, and in only one case it is 38 mm. Therefore, determining the position of an object based on the previous solution's results is a justified approach for improving the object's orbit. The results for Scenario No. 3 indicate that extending the orbital arc greatly reduces the RMSPOS values. However, this also leads to significantly higher arc RMS values, ranging from approximately 100 cm to over 300 cm. The drawback of Scenario No. 3 is that it relies on laser measurements from only one sensor. To enhance the orbital solution, it would be advisable to strengthen the measurement geometry by adding measurements from a different sensor distributed over a different part of the orbital arc. This would help reduce the arc RMS.

Figure 4.

Figure 5.

The values of the RMSVEL parameter do not change significantly. This is because this parameter generally has low values, around 1 m/s. The highest values of this parameter were obtained in Scenario No. 1 for both object 28480 and object 31114, which were 1.3812 and 1.2512 m/s, respectively. In Scenario No. 1 and Scenario No. 2, the RMSVEL parameter remained stable and fluctuated around the average values of 1.0691 and 0.9125 m/s, respectively, for object 28480 and 0.9359 and 0.8339 m/s, respectively, for object 31114. In Scenario No. 3, the value of the RMSVEL parameter decreased significantly for both objects, especially as the orbital arc lengthened. For object 28480, with the longest orbital arc covering five observations, RMSVEL was 0.0026 m/s. For object 31114, with the longest orbital arc covering four observations, RMSVEL was 0.0041 m/s.

One of the most important parameters that was determined is AT, which is the time after which the observation (measurements) of a given object should be repeated. Tables 9 and 10 list the determined ATs for all scenarios for both objects, as well as the triple sigma range error for each R, S, and W component. It is clear that the largest triple sigma values are for the along-track component, and they increase rapidly over time. This fact is well illustrated as an example in Figure 6. The largest errors are found in the along-track component, which is not surprising. However, an interesting result was obtained for the cross-track component. This is likely because that orbital solution relies only on laser measurements and it is not possible to accurately determine the orientation of the orbital plane of the analyzed objects. The use of angular observations, as shown in Cordelli et al. (2020), would be helpful in determining the orientation of the orbital plane.

Figure 6.

All values of ATs are shown in Figure 7. Regardless of which scenario is used, the laser measurements of objects 28480 and 31114 should be repeated within the next 24 h to maintain the 3-sigma criterion. The average ATs for both objects were approximately 70,000 s each (19.44 h). However, there is an exception for object 28480 in orbital solution nr 16, where AT was as high as 148,200 s (41.17 h). In this case, two sets of laser measurements of object 28480 were considered from April 18 and 19 (see Table 5 for more details). This is likely due to the accurate fit of the ephemeris to the laser measurements during this period. Despite a time difference of 23.5 h between these two sets of measurements, the arc RMS in this case was only 2.98 cm. Generally, the average ATs for Scenario No. 1 and Scenario No. 2 are similar for both objects, approximately 75,000 s (20.83 h). The results presented in this paper indicate that computational scenarios No. 1 and No. 2 should be used for similar computations using laser measurements for other RBs in the LEO regime, such as Russian SL's or other Chinese CZ's objects.

Figure 7.

However, the computational strategy presented as Scenario No. 3 in this work suggests that this approach is not the best option. Specifically, the RMSPOS parameter decreases significantly, but this leads to a poorer alignment between the object's ephemeris and its laser measurements. This misalignment is reflected in the high value of the arc RMS parameter, which can be as high as several meters. It is important to note that this scenario should be analyzed under the assumption that the correction for the center of mass of the objects studied in this work is known. Currently, this parameter is unknown and considering the size of the objects, it could be up to several meters. Taking into account the center of mass correction and reusing picosecond laser measurements may yield different results.

Furthermore, the semi-major axis and eccentricity were also determined for all analyzed scenarios. The laser measurements are suitable for determining these geometrical parameters because they provide direct ranges. The semi-major axis and eccentricity were determined at the beginning of each orbital arc and compared with values directly obtained from TLE ephemeris. The results of this comparison are shown in Figures 8 and 9. For object 28480, the mean absolute differences in the semi-major axis are 203, 920, and 718 m for Scenarios No. 1, No. 2, and No. 3, respectively. For object 31114, the mean absolute differences in the semi-major axis are 805, 654, and 4635 m for Scenarios No. 1, No. 2, and No. 3, respectively. On the other hand, in case of eccentricity, for object 28480, the mean absolute difference is 1.64E-05, 3.80E-05, and 1.24E-04 for Scenarios No. 1, No. 2, and No. 3, respectively. For object 31114, the mean absolute difference in eccentricity is 6.72E-05, 5.36E-05, and 5.08E-04 for Scenarios No. 1, No. 2, and No. 3, respectively.

Figure 8.

Figure 9.

Object 28480 had an elliptical orbit, while object 31114 had a nearly circular orbit (see Table 2 for more details). The average time between the start of laser measurements and the epoch of TLE elements was much longer for object 28480, at 8.24 h. For object 31114, this average time was 4.91 h. Both parameters were determined in OP mode from TLE and in OD mode from laser measurements for each scenario and compared. The largest differences in semi-major axis and eccentricity were found for object 31114 in Scenario No. 3 (with increasingly longer orbital arcs). In case of object 28480, the differences were smaller, but the time between the start of laser measurements and the epoch of TLE elements was about 1 h shorter for object 28480 compared to object 31114. This means that object 28480 had more recent TLE elements. For Scenarios No. 1 and No. 2 (with 1-day orbital arcs), the differences in semi-major axis and eccentricity for both objects were much smaller, even though the time between the start of laser measurements and the epoch of TLE elements ranged from several to several dozen hours (with a maximum of 15 h for object 28480). The conclusion here is that short orbital arcs, no longer than 24 h, are a beneficial solution for determining semi-major axis and eccentricity parameters, even when we have laser measurements from only one sensor.

It is sensible to determine the orbits of RB objects using full-rate laser measurements, even if these measurements come from only one sensor, and it yields accurate results. It is crucial to perform these measurements regularly, at a specific time interval. This paper presents a modern approach to determining the orbit of Chinese RBs using single picosecond full-rate laser measurements collected by a single sensor (BORL station). The computation strategy was carried out in three different scenarios, where several key parameters were determined, that is, arc RMS, RMSPOS, RMSVEL, and AT. The results obtained confirm the potential and usefulness of two computational strategies, 1 and 2, which involve short orbital arcs no longer than 24 h. For Scenario No. 1, the determined RMSPOS parameter is approximately 500 m and in Scenario No. 2, RMSPOS is around 100–300 m lower. The RMSVEL parameter remains stable and fluctuates around average values of 1 m/s for both Scenarios No. 1 and No. 2. In Scenario No. 3, when subsequent passes are combined into longer orbital arcs, it does not improve the OD process described in this paper. In this case, RMSPOS and RMSVEL decrease to single centimeters and single millimeters, respectively. However, the arc RMS increases significantly to decimetres and more when the length of the orbital arc is 2 days or more. The results obtained in Scenario No. 2 are the most optimal. The reduction of the RMSPOS parameter is clearly visible, with an arc RMS of several centimeters. This means that if we consider the deviations of individual coordinate components obtained from the previous orbital solution in the current orbital arc, we can see the importance of regular observations of space objects like the ones analyzed in this study. This fact is well illustrated in this paper by the determined AT parameter, which is approximately 19.5 h.

It would be beneficial to conduct similar studies using laser measurements from other sensors, but these measurements must be performed regularly at specific time intervals. The geometry of the orbital solution would be significantly improved, which would affect the quality of the orbital solution.

The average duration of the laser measurements used in these studies was 195 and 310 s for objects 28480 and 31114, respectively. For instance, within 10 h, only one laser sensor could measure up to 120 passes of RBs, assuming good weather conditions (cloudless) and 5 min of time per pass. Including more laser sensors, such approach would allow optimization of the tracking campaign of many RB’s targets. This could benefit greatly programs like Space Surveillance and Tracking (SST) and Space Traffic Management (STM).

This paper presents another interesting result: the Chinese RBs 28480 and 31114 are likely equipped with retroreflectors. However, there is no available information on the location and method of mounting the retroreflectors on these objects, which have a height of several meters and a mass of a few tons. Nevertheless, it is very easy to obtain returns using a low-energy picosecond laser. It would be worthwhile to verify and track other Chinese RBs to see if they also have retroreflectors. The BORL sensor has already planned an appropriate observing campaign for this purpose. Regular laser tracking of targets with retroreflectors is very useful for topics such as rendezvous proximity operations, attitude motion, collision avoidance, characterization, and maneuvering.