# Tiangong-1’s accelerated self-spin before reentry

**Part of the following topical collections:**

## Abstract

## Keywords

Tiangong-1 Rotational state estimation Space debris SLR## Introduction

When the fourth artificial satellite in human history, Vanguard I, was launched in 1958, its rotational speed was found to decay exponentially with time, which was suggested to be the effect of eddy current torque (Wilson 1959; LaPaz and Wilson 1960). After decades of research, eddy current torque has been verified as the main dissipation factor that causes a decrease in spacecraft rotational speed (Smith 1962; Williams and Meadows 1978; Praly et al. 2012; Lin and Zhao 2015). However, some observations of individual rockets show that they have increasing rotational speed (Meeus 1971), and some satellites begin to spin after losing attitude control. The causes of accelerated spin have not been clearly explained. Whether these phenomena are due to satellite fuel leaks (Boehnhardt et al. 1989), accidental collisions, or the continuous action of certain factors in space cannot be confirmed from scattered observations.

In recent years, the task of active debris removal has spurred the need for the detection of and research on the rotational state of space debris (Liou and Johnson 2009; Liou 2011; Bonnal et al. 2013; Deluca et al. 2013; Shan et al. 2016). Estimating the rotational state of space debris is very difficult when using ground-based observations because of the limited detection capability and the small amount of acquired data. The existing detection and research results are limited to objects at orbital altitudes of approximately 500–1500 km or higher, and such objects are relatively easy to detect (Kucharski et al. 2013; Koshkin et al. 2016; Lin and Zhao 2018; Pittet et al. 2018). In the extremely low-orbit region (altitudes lower than 300 km), the small number of detectable objects coupled with their high velocity, short pass duration, and short orbital lifetime require better observation conditions and higher observation capability. To date, comprehensive data on the rotational evolution of extremely low-orbit objects are unavailable, and the high-profile Tiangong-1 reentry event provided an excellent opportunity for data collection.

## Methods

### Processing of SLR data

### Estimation of the orientation of angular momentum

*H*and

*E*are the angular momentum magnitude and the kinetic energy of self-spin, respectively. Then,

*L*can indicate the position of the trajectory of the angular velocity and the angular momentum in the inertia ellipsoid. The polhode when \({L = I_y / I_z}\) (the brown line in Fig. 3) divides the inertia ellipsoid into four regions (Markeyev 2006), in which the polhode surrounds \(\pm\,x\) or \(\pm\,z\). When

*L*is equal to 1 and \({I_x / I_z}\), the axes of rotation coincide with the

*Oz*- and

*Ox*-axes, respectively. Because the magnitude of angular velocity is not constant in a polhode, when comparing the rotational speeds at different times, the equivalent rotational speed \(H / I_z\) can be used to characterize the change in rotational speed due to external torque.

*H*, \(\psi _H\), \(\theta _H\), \(\varphi _H\), \(\theta '\), and \(\varphi '\) (Lin et al. 2016; Crenshaw and Fitzpatrick 1968) shown in Fig. 4, we can establish the transformation between the body-fixed coordinate system and the orbital plane coordinate system. Define \({\varvec{R}}_{\varvec{\xi }}(\eta )\) as a rotation matrix around axis \({\varvec{\xi}}\) with angle \(\eta\). Then, the transformation matrix from the orbital plane coordinate system to the body-fixed system can be written as

*L*and conforming the change in \(\psi _H\) to the law. This process eventually yields a certain orientation of the angular momentum (red region in Fig. 5).

### Solutions of angular momentum magnitude

Determining the exact angular momentum magnitude *H* is another challenge. Directly solving the rotational speed is extremely difficult because the valid observation duration in a single pass is much smaller than the rotation period of Tiangong-1, the time interval between adjacent passes is much larger than this rotation period, and the large dimension of the algorithm makes \(\Delta P\) not sensitive to the changes in *H*.

*i*is the orbit inclination, \(\dot{\Omega }\) is the orbital precession caused by the oblateness of the Earth,

*G*is the gravitational constant,

*M*is the mass of the Earth,

*R*is the geocentric radius, and \(\bar{\theta }_H\), \(\bar{\theta }'\), and \(\bar{\varphi }'\) are the mean values of \(\theta _H\), \(\theta '\), and \(\varphi '\), respectively (Lin et al. 2016). The first-order linear change rate of \(\psi _H\) is approximately linear with 1/

*H*. This correspondence is easy to understand because the gravity gradient torque is independent of the angular momentum, and under the same gravity gradient torque effects, the smaller the magnitude of the angular momentum is, the more easily the direction is changed, which results in a faster change rate.

*H*can be obtained from the change in \(\psi _H\) from two passes based on the relationship between

*H*and the precession rate of the angular momentum. In actual operation, the resulting error will be excessive if the two passes are too close, and if their time interval is too long, the large difference in orbital altitudes will affect the \(\psi _H\)–

*H*relationship. Therefore, we select the interval from the observations to be more than half a day and less than 10 days to calculate the change in \(\psi _H\). After eliminating the incorrect solutions for the difference angles of \(\psi _H\) in the same quadrant, the magnitude of the angular momentum can be obtained by numerically calculating the \(\psi _H\)–

*H*relationship (Fig. 7).

## Results and discussion

### Tiangong-1’s rotational state

*z*-axis (i.e., the maximum principal axis of inertia) with maximum angles of approximately \(1^\circ\) and \(8.5^\circ\) in the

*x*and

*y*directions, respectively (Fig. 3), while in the inertial space, the angular momentum \({\varvec{H}}\) precesses around the normal direction \({\varvec{N}}\) of the orbital plane due to gravity gradient torque (Lin et al. 2016) at an angle \(\theta _H = 23.1^\circ \pm\,2.5^\circ\) (Fig. 8). During the 5-month observation period, the rotation mode of Tiangong-1 was relatively stable, indicating that the satellite was not subjected to a strong sudden effect (such as a collision). Any dissipating effect in space will cause the angular momentum to migrate to the axis of maximum inertia (the rotation with the lowest energy), but Tiangong-1’s angular momentum remained at a certain angle to the

*z*-axis for a long duration. A numerical test showed that gravity gradient torque produces only periodic deviations of up to \(0.01^\circ\) (along the

*x*-axis) and \(0.1^\circ\) (along the

*y*-axis). Therefore, we speculate that Tiangong-1 was also subject to other effects, which may be additional stable driving torques. Combined with the attitude of Tiangong-1 before losing attitude control (Fig. 8), these effects may have caused the

*Oz*-axis to be deflected by approximately \(90^\circ\) in the normal direction of the orbit plane, finally reaching a balanced state.

*H*can be derived through the relationship of

*H*with the precession rate of the angular momentum due to the gravity gradient torque, thus enabling the variation in rotational speed with time to be obtained (Fig. 9). The rotational speed of Tiangong-1 increased in the 5 months before reentry. This acceleration phenomenon has been verified by a very small amount of imaging and radar data. Although a complete period evolution cannot be obtained from these additional data, the detected rotational speed in early 2018 is clearly faster than that at the end of 2017. As mentioned in Introduction, this difference is an unexpected phenomenon. Within the known major external torques in space, the gravity gradient torque generally acts as a conservative torque and does not have a secular effect on the angular momentum (Lin et al. 2016; Crenshaw and Fitzpatrick 1968; Holland and Sperling 1969; Hitzl and Breakwell 1971). The eddy current torque and magnetic torque are dissipative torques that decrease the rotational speed (Smith 1962; Williams and Meadows 1978; Praly et al. 2012; Lin and Zhao 2015). Light pressure torque may theoretically have a slight acceleration effect under specific orbit and attitude configurations (Kucharski et al. 2016), but no secular effect was found in our numerical test on Tiangong-1. Classic aerodynamic torque, which affects objects only at very low orbital altitudes (Williams and Meadows 1978; Lyle and Stabekis 1971; Hart et al. 2014), will also decrease rotational speed (blue line in Fig. 9).

### Atmospheric density gradient torque

However, we conjecture that the upper atmosphere is the factor causing the increase in the rotational speed because the acceleration of the spin increases significantly with decreasing orbital altitude, which is consistent with the increase in atmospheric density with decreasing orbital altitude. In this regard, we propose an atmospheric density gradient torque (ADGT) model (Fig. 8), which takes into account the torque generated by the change in atmospheric density with orbital altitude at the satellite scale. For the rotational state of Tiangong-1, the ADGT is nearly in the same direction as the angular momentum, resulting in an acceleration effect. The atmospheric density gradient at Tiangong-1’s position after February 2018 also increases faster (upper panel of Fig. 9), which is consistent with the increasing accelerated rate of the rotational speed.

Normally, \(\delta\) is set to 1. The acceleration effect of the ADGT, which is independent of rotational speed, overcomes the deceleration effect of classic aerodynamic torque (the brown and cyan lines in Fig. 9). However, a gap exists between the simulation results and the measured data. If we multiply the force from the molecule–surface interaction model by a coefficient \(\delta = 3\) in the numerical simulation, we can obtain a result that is similar to the measured data (red line in Fig. 9). Hence, the acceleration effect may have a strong correlation with the atmospheric density gradient, and the ADGT model can be inferred to qualitatively conform to the actual situation. The atmospheric density model used in Fig. 9 is MSIS 90. We have also compared other models such as Ciral1972, DTM1994, and JB2008, and their results are not significantly different.

If we assume that the ADGT model is valid, the atmospheric force acting on Tiangong-1 is larger than that in the molecule–surface interaction model, which indicates that other effects may be acting on the satellite. Traditionally, the atmosphere on satellite orbits has been treated as being under free molecular flow (Hart et al. 2014), and the density has been regarded as constant at the satellite scale. Here, the density gradient is taken into consideration; thus, the accuracy of the molecular–surface interaction model may not match the accuracy required for modeling weak aerodynamic torques. Further refinements are needed to establish an in-depth atmosphere–surface interaction model. Other minor effects, such as multiple reflection or hydrodynamic effects, that were omitted in classic models should be considered in further research.

## Conclusions

By processing Tiangong-1’s laser ranging data, we obtained the rotational state in the 5 months before reentry and detected an unexpected increase in rotational speed. Then, we proposed a new torque model ADGT to explain this spin acceleration. Because we are unable to determine the status of solar panels, we performed our simulations for two extreme cases, i.e., the panels are parallel (\(\theta _{{\rm p}}=0^\circ\)) and perpendicular (\(\theta _{{\rm p}}=90^\circ\)) to the *xy* plane, respectively. Although several times weaker than the measured data, the acceleration effect due to the ADGT model is still a non-negligible factor in the increasing angular velocity of Tiangong-1. This effect provides a new explanation for the mechanism of the observed rotation acceleration of extremely low-orbit objects. The data on the rotational evolution of Tiangong-1 obtained during the 5-month-long joint observations will help to improve the relevant models and to promote research on the rotational evolution of extremely low-orbit objects, which has an important impact on the prediction of reentry, satellite attitude control, and the overall active debris removal strategy. More observations of extremely low-altitude debris objects are needed to better understand the cause of the rotational acceleration of Tiangong-1 prior to reentry.

## Notes

### Authors' contributions

HYL and CYZ conceived and designed the study. TLZ and ZPL performed the processing of raw data. DW, WZ, JNX, and JWZ performed orbit determination and scheduled the observation plans. XWH, HFZ, ZBW, YQL, QLS, HTZ, and HRD performed satellite laser ranging and contributed data. CZ and YDP processed the photometry and imaging data for verification. HYL and TLZ wrote the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

We thank Hong-Bo Wang and Xin Wang for their discussions. We wish to thank the anonymous reviewers for their valuable comments.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

The dataset supporting the conclusions of this article is available in the Zenodo repository, https://doi.org/10.5281/zenodo.1452298.

### Funding

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11533010 and 11703095), the Youth Innovation Promotion Association, CAS (Grant No. 2018353), and the Chinese Academy of Sciences Foundation (Grant No. KGFZD-135-16-012).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- Boehnhardt H, Koehnhke H, Seidel A (1989) The acceleration and the deceleration of the tumbling period of Rocket Intercosmos 11 during the first two years after launch. Astrophys Space Sci 11:297–313CrossRefGoogle Scholar
- Bonnal C, Ruault JM, Desjean MC (2013) Active debris removal: recent progress and current trends. Acta Astronaut 85:51–60. https://doi.org/10.1016/j.actaastro.2012.11.009 CrossRefGoogle Scholar
- Crenshaw JW, Fitzpatrick PM (1968) Gravity effects on the rotational motion of a uniaxial artificial satellite. AIAA J 6(11):2140–2145. https://doi.org/10.2514/3.4946 CrossRefGoogle Scholar
- Deluca LT, Bernelli F, Maggi F, Tadini P, Pardini C, Anselmo L, Grassi M, Pavarin D, Francesconi A, Branz F, Chiesa S, Viola N, Bonnal C, Trushlyakov V, Belokonov I (2013) Active space debris removal by a hybrid propulsion module. Acta Astronaut 91:20–33. https://doi.org/10.1016/j.actaastro.2013.04.025 CrossRefGoogle Scholar
- Hart KA, Dutta S, Simonis KR, Steinfeldt BA, Braun RD (2014) Analytically-derived aerodynamic force and moment coefficients of resident space objects in free-molecular flow. In: AIAA SciTech, AIAA atmospheric flight mechanics conference, National Harbor, MDGoogle Scholar
- Hitzl D, Breakwell J (1971) Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite. Celest Mech 3(3):346–383CrossRefGoogle Scholar
- Holland RL, Sperling HJ (1969) A first-order theory for the rotational motion of a triaxial rigid body orbiting an oblate primary. Astron J 74:490–496. https://doi.org/10.1086/110826 CrossRefGoogle Scholar
- Koshkin N, Korobeynikova E, Shakun L, Strakhova S, Tang ZH (2016) Remote sensing of the EnviSat and Cbers-2B satellites rotation around the centre of mass by photometry. Adv Space Res 58(3):358–371. https://doi.org/10.1016/j.asr.2016.04.024 CrossRefGoogle Scholar
- Kucharski D, Kirchner G, Lim HC, Koidl F (2013) New results on spin determination of nanosatellite BLITS from high repetition rate SLR data. Adv Space Res 51(5):912–916. https://doi.org/10.1016/j.asr.2012.10.008 CrossRefGoogle Scholar
- Kucharski D, Bennett JC, Kirchner G (2016) Laser de-spin maneuver for an active debris removal mission—a realistic scenario for Envisat. In: Proceedings of the advanced maui optical and space surveillance technologies conference, Held in Wailea, Maui, Hawaii, 20–23 Sept 2016Google Scholar
- LaPaz L, Wilson RH (1960) Magnetic damping of rotation of the Vanguard I satellite. Science 131:355–357CrossRefGoogle Scholar
- Lin H-Y, Zhao C-Y (2015) Evolution of the rotational motion of space debris acted upon by eddy current torque. Astrophys Space Sci 357(2):167. https://doi.org/10.1007/s10509-015-2396-2 CrossRefGoogle Scholar
- Lin H-Y, Zhao C-Y (2018) An estimation of Envisat’s rotational state accounting for the precession of its rotational axis caused by gravity-gradient torque. Adv Space Res 61(1):182–188. https://doi.org/10.1016/j.asr.2017.10.014 CrossRefGoogle Scholar
- Lin H-Y, Zhao C-Y, Zhang M-J (2016) Frequency analysis of the non-principal-axis rotation of uniaxial space debris in circular orbit subjected to gravity-gradient torque. Adv Space Res 57(5):1189–1196. https://doi.org/10.1016/j.asr.2015.12.036 CrossRefGoogle Scholar
- Liou J-C (2011) An active debris removal parametric study for LEO environment remediation. Adv Space Res 47(11):1865–1876. https://doi.org/10.1016/j.asr.2011.02.003 CrossRefGoogle Scholar
- Liou J-C, Johnson NL (2009) A sensitivity study of the effectiveness of active debris removal in LEO. Acta Astronaut 64(2–3):236–243. https://doi.org/10.1016/j.actaastro.2008.07.009 CrossRefGoogle Scholar
- Lyle R, Stabekis P (1971) Spacecraft aerodynamic torques. In: NASA SP-8058Google Scholar
- Markeyev AP (2006) Theoretical mechanics. Higher Education Press, BeijingGoogle Scholar
- Meeus J (1971) Satellites artificiels—Observations de périodes photométriques 1968–1971. Ciel et Terre 87:606Google Scholar
- Pearlman MR, Degnan JJ, Bosworth JM (2002) The international laser ranging service. Adv Space Res 30(2):135–143. https://doi.org/10.1016/S0273-1177(02)00277-6. arXiv:1011.1669v3 CrossRefGoogle Scholar
- Pittet JN, Šilha J, Schildknecht T (2018) Spin motion determination of the Envisat satellite through laser ranging measurements from a single pass measured by a single station. Adv Space Res 61:1121–1131. https://doi.org/10.1016/j.asr.2017.11.035 CrossRefGoogle Scholar
- Praly N, Hillion M, Bonnal C, Laurent-Varin J, Petit N (2012) Study on the eddy current damping of the spin dynamics of space debris from the Ariane launcher upper stages. Acta Astronaut 76:145–153. https://doi.org/10.1016/j.actaastro.2012.03.004 CrossRefGoogle Scholar
- Shan M, Guo J, Gill E (2016) Review and comparison of active space debris capturing and removal methods. Prog Aerosp Sci 80:18–32. https://doi.org/10.1016/j.paerosci.2015.11.001 CrossRefGoogle Scholar
- Smith G (1962) A theoretical study of the torques induced by a magnetic field on rotating cylinders and spinning thin-wall cones, cone frustums, and general body of revolution. In: NASA eTR R-129Google Scholar
- Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett. https://doi.org/10.1029/2004GL019920 CrossRefGoogle Scholar
- Williams V, Meadows AJ (1978) Eddy current torques, air torques, and the spin decay of cylindrical rocket bodies in orbit. Planet Space Sci 26:721–726CrossRefGoogle Scholar
- Wilson RH (1959) Magnetic damping of rotation of satellite 1958
*β*2. Science 130:791–793CrossRefGoogle Scholar

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