Shape and motion estimate of LEO debris using light curves

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Abstract

We have derived a tri-axial ellipsoidal model of an LEO object, a Cosmos 2082 rocket body, including its rotational axis direction, rotation period, precession, and a compositional parameter, using only light curve data from an optical telescope. The brightness of the object was monitored for two days and least-squares fitting was used to determine these values. The derived axial ratios are 100:18:18, the coordinates of the rotational axis direction on the celestial sphere are R.A. = 305.8° and Dec. = 2.6°, and its observed average rotation period is 41 s. When precession is considered, its amplitude and precession period are 30.5° and 29.4 min. These results show that optical light curve data are sufficient to determine the shapes and the motions of LEO objects.

Introduction

Space debris is becoming a serious problem in low earth orbit (LEO). The anti-satellite experiment conducted by China and the collision between the Iridium satellite and the Russian Cosmos satellite have increased the amount of space debris in this region. It is thought that a chain reaction of cascading collisions, in which many pieces of space debris created by a collision become sources of other collisions, have begun in the 1000 km region (Liou and Johnson, 2006). Humanity must cope with this situation as soon as possible to protect its space programs.

The Japan Aerospace Exploration Agency (JAXA) is studying three aspects of the space debris problem: observation, protection and mitigation. For mitigation, a satellite is being considered that will capture large LEO objects such as damaged satellites or rocket bodies and remove them from the orbit (Kibe et al., 2003, Kawamoto et al., 2003, Ishige et al., 2002). To evaluate the potential of this method and to build such a satellite, it is necessary first to investigate the tendencies of shapes and motions of LEO objects. Radar facilities can directly observe the shapes and motions of LEO objects, but are costly to build and maintain. There is one such facility in Japan, the “Kamisaibara Spaceguard Center” in Okayama Prefecture, but it is specialized for detecting multiple targets rather than observing their shapes and motions (Taromaru et al., 2003).

It may be possible, however, to perform the same task using cheaper optical equipment. Although an optical telescope cannot observe the shapes and motions of LEO objects at an altitude of 1000–2000 km like a radar facility, this information may be extracted indirectly from light curve observations. In the field of astronomy, light curve observations of asteroids to determine their shapes and spin axis orientations have been carried out for decades (Connelly and Ostro, 1984, Zappala, 1981, Zappala and Knezevic, 1984, Taylor et al., 1988, Magnusson, 1986, Drummond et al., 1988, Kaasalainen and Torppa, 2001) and these works are very helpful to space debris researchers for extracting information on space debris from their light curves. Studies on the light curves of space objects is becoming a major activity in the space debris field (Fruh and Schildknecht, 2010, Fruh and Schildknecht, 2011, Moriba and Ronald, 2007, Jin et al., 2011, Ojakansky and Hill, 2011, Harms et al., 2011, Kurosaki et al., 2009), reflecting the degradation of the space environment by space debris.

There are two leading methods to extract source information from light curves (Magnusson et al., 1989): the “amplitude method”, which uses amplitude differences of light curves to extract the spin axis orientation and shape as will be described in Section 2; and the “epoch method” which uses the periodic difference of light curves to extract the spin axis orientation and direction of rotation, the magnitude of this difference being a function of the orientation of the spin axis of the object with respect to the observer-object-sun geometry and direction of rotation, which can thus be estimated. In the space debris field, several studies have been carried out using the epoch method (Somers, 2011, Wallace et al., 2010, Hall et al., 2006), but since it requires a timing accuracy of less than 1% of the sidereal period and is not able to extract shape information, we adopt the amplitude method in this work.

Fig. 1 shows the light curve of the Cosmos 2082 rocket body obtained by JAXA’s 35 cm telescope specialized for observing LEO objects. The x- and y- axes represent the observation time and the target’s relative brightness in ADUs (analog-to-digital units). From the periodic change in brightness in Fig. 1, it is thought that the target has an elongated shape and is rotating. Numerous observations of a single target enable us to extract more detailed information such as the axis ratio of a tri-axial ellipsoid, the direction of its rotation axis, its rotation period, a parameter related to its composition, and precession parameter. We observed the Cosmos 2082 rocket body for two days and succeeded in extracting this information.

In this paper, the observational details are described in Section 2 and the analysis and results in Sections 3 Analysis, 4 Analysis result I, 5 Analysis result II. Section 6 gives some discussions, and conclusions are presented in Section 7.

Section snippets

Observation

Before observing a target, a search was carried out for LEO objects that exhibit periodic brightness changes, as we considered that such objects might be a good starting point for this kind of study as they are relatively easy to analyze. The orbital elements of bright satellites and space debris are available on various web sites. We first searched for LEO objects observable from our location by using these orbital elements with the orbit calculation function of the STK (satellite tool kit)

Analysis

Four light curves were obtained from the target by analyzing a few thousand images from each pass with the algorithm mentioned above. The algorithm identifies a group of pixels in an image with a value higher than a threshold as the target. It then sums the value of 100 pixels around that group of pixels and subtracts 100 times the median value of the surrounding pixels to correct for the sky value. The brightness of the target in each image is measured automatically by these processes. The

Analysis result I

Fig. 7 shows the error distribution of the least-squares method for the rotational axis direction of the Cosmos 2082 rocket body. The error is determined as follows:Error=i=117(Aobs_i-Areal_i)2where Aobs_i and Areal_i are the ith observed brightness amplitudes out of 17 points and the theoretical value calculated by Eq. (3), respectively. The minimum point in the error distribution is considered to be the rotational axis direction. The x and y axes of Fig. 7 represent the coordinates on the

Analysis result II

In the previous section, the axial ratio of the tri-axial ellipsoid, the rotational direction and period, and a compositional parameter were derived assuming a tri-axial ellipsoidal body rotating about its shortest axis. However, as can be seen, these values include large amounts of error, especially in the coordinates of the rotational axis direction. This suggests that the target may have another motion such as precession. Another analysis was carried out considering precession in addition to

Discussion

The Cosmos 2082 rocket body is the second stage of a Russian Zenit rocket (http://www.russianspaceweb.com/zenit). Its diameter and length are 3.9 and 10.4 m, respectively, so its axis ratio under the tri-axial ellipsoid assumption is 100:38:38. There is a two-fold discrepancy between this and the values of b and c extracted in Section 4. The reason may be that the albedo of the target’s side is double that of its top and bottom. In this case, b and c in Sections 4 Analysis result I, 5 Analysis

Conclusion

We succeeded in determining the overall shape, rotational axis, rotation period, precession radius, precession period, and a composition parameter for a Cosmos 2082 rocket body by observing and analyzing its light curves. The axis ratio assuming a tri-axial ellipsoidal body was 100:18:18. The coordinates of the direction of the rotational axis on the celestial sphere were R.A. = 305.8° and Dec. = 2.6°, and the compositional parameter M was 0.05/degree. The rotational period was 41 s, and the

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