Application Possibility of Light Curve Inversion Method Regarding Space Debris Hideaki Hinagawa1) and Toshiya Hanada1), 2)
1) Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Japan
2) International Centre for Space Weather Science and Education, Kyushu University, Fukuoka, Japan
Abstract
Space debris has the possibility to collide with other spacecraft at about 10 km/s relative velocity. In the
filed of debris removal, electric tether attached to space debris is suggested to decay an object using its
Lorenz force. However, it will be difficult to approach those space debris without the knowledge of their
rotational motion, and acquisition of this knowledge on ground is required in terms of debris removal.
The purpose of this research is to apply the light curve inversion method often used for asteroid to the
space debris using ground optical observation, and investigate the possibility of pole determination and
shape estimation.
ライトカーブを用いたスペースデブリの回転運動の推定 概要
スペースデブリは約 10km/sの相対速度で他の軌道上物体に衝突する可能性がある.スペースデ
ブリ除去の分野において,導電性テザーをスペースデブリに取り付けることでローレンツ力を
用いた軌道高度低下を促す手法が主として考案されているが,そのようなデブリを除去するた
めの装置を取り付ける場合,そもそもの対象とするスペースデブリの回転運動を把握していな
い限り,その回収は難しくなる.ゆえに地上からのスペースデブリの回転運動を把握すること
が求められる.本研究は,この地上からの光学観測によるライトカーブを用いたスペースデブ
リの回転運動を調査することを目的としている.
1. Introduction
The existence of space debris in orbit has been
becoming a serious problem and a huge thread
to the rapidly developing space utilization
society. Space debris has the possibility to
collide with other spacecraft at about 10 km/s
relative velocity. In the filed of debris removal,
low density material is under development to
capture or decelerate space debris[1], as it is said
that we cannot stop their chain reaction of the
collision as known as “Kessler Syndrome[2]”
unless we effectively remove the relatively
large debris such as dead satellite and rocket
upper body. Electric tether attached to space
debris is suggested to decay an object using its
Lorenz force[3]. However, it will be difficult to
approach these space debris without the
knowledge of their rotational motion, and
acquisition of this knowledge on ground is
required in terms of debris removal. This study
is becoming one of serious topics[4][5], since
there are about 500 objects that have been
confirmed as potential threads to trigger big
effect to space debris environment. The
purpose of this research is to apply the light
curve inversion method often used for asteroid
to the space debris using ground optical
observation, and investigate the possibility of
pole determination and shape estimation.
2. Pole Orientation Estimation
2.1. Strategy
The Attitude motion prediction starts with a
pole orientation estimation that is the simplest
case of attitude motion. In the case, the object
attitude is assumed that the object rotates
around the shortest axis, and its rotational axis
is fixed in inertia frame.
However, this assumption might be
insufficient when the real attitude motion is
considered. Then, the analysis and
understanding of the real attitude motion is
needed for further study. We present possibility
of the concern that a rocket body would not act
on that way described above.
With the fully analyzed attitude motion, we
will conduct the Euler angle estimation by
future optical observation that is supposed to be
conducted on November 2012.
For the advanced research, we will perform
the estimation of the object shape by the optical
observation.
These study will contributes to the activities
for space debris removal and mitigation study.
2.2. Coordinate System
The three-axes ellipsoid is taken as a model of
rocket body in this paper. The X-axis is the
observer direction. The Z-axis is parallel with
the Earth’s north polar direction. Y-axis is
defined that it is normal to both X and Y axis
by right handed system.
Fig. 1 ecliptic coordinate system
2.3. Measurements and States
The measurements are the object’s brightness,
sun direction, line-of-sight from JAXA’s 35 cm
LEO-object -observation with time. We don’t
possess the measurement data so we used the
measurement data[4] that provided by Dr.
Yanagisawa.
The states are the object’s rotational pole
direction in the object frame that is parallel to
the inertia frame.
2.4. Estimation Filter
We accept three assumptions for this analysis.
First, the object rotates about its shortest axis.
Next, the shape of the object is a three-axes
ellipsoid. Third, rotational axis is fixed on the
celestial sphere.
Intensities at maximum 𝑆!"# and minimum
𝑆!"# brightness could respectively happen at
the maximum projected area and the minimum
one described by
𝑆!"# = 𝜋𝑎𝑏𝑐𝑠𝑖𝑛!𝐴𝑏!
+𝑐𝑜𝑠!𝐴𝑐!
!! (2-1)
𝑆!"# = 𝜋𝑎𝑏𝑐𝑠𝑖𝑛!𝐴𝑎!
+𝑐𝑜𝑠!𝐴𝑐!
!! (2-2)
𝐴𝑀𝑃 = 2.5𝑙𝑜𝑔𝑆!"#𝑆!"#
1 + 𝛽𝛼 (2-3)
where A denotes the aspect angle. a, b, c (a > b
> c) are the length of each axis (x, y, z). β is
the Sun phase angle coefficient. α is the Sun
phase angle.
Then we define the unit rotational axis vector
by 𝜆!,𝛽! . 𝜆! is the right ascension, and 𝛽!
is the declination. Line-of-site direction is also
able to be defined by 𝜆,𝛽 in ecliptic frame.
Finally, you can obtain the aspect angle A from
the dot product of 𝜆!,𝛽! and 𝜆,𝛽 by
𝑐𝑜𝑠𝐴 = 𝑐𝑜𝑠𝛽𝑐𝑜𝑠𝛽!𝑐𝑜𝑠 𝜆 − 𝜆!
+ 𝑠𝑖𝑛𝛽𝑠𝑖𝑛𝛽! (2-4)
With this measurement model, the unknown
state can be estimated from least squares fitting.
2.5. Result of Pole Orientation
From the provided measurements data, pole
orientation estimation was conducted. The final
estimate from the least squares filter is ended
shown below.
The rotational axis could be reckoned as
𝜆!,𝛽! = 289.43 ± 9.10 −3.98 ± 9.52
The estimated axis ratio of the three axes
ellipsoid and the Sun’s phase coefficient are
𝑏, 𝑐 !!! = 0.38 ± 0.74 0.38 ± 0.77
𝛽 = 0.21 ± 1.42
Figure 2 depicts the error distribution of the
estimation filter with respect to the rotational
axis. You can see the circular areas that possess
e small error against the measurement
amplitude. We might be able to say that the
rotational axis contains precession (figure 3).
The two circle means that the both side of
rotational axis existence separated by 180 [deg]
of the phase angle.
Fig. 2 error distribution of rotational axis
Fig. 3 precession motion as a possible
explanation of figure 1
3. Orbit Determination
3.1. Strategy
The previous section explains that orbit
determination has an important role for more
precise comprehension of the attitude motion.
To begin with, orbit determination for a GEO
object was adopted. For this purpose of this
study, orbit determination can be composed by
the initial orbit determination (IOD) and the
precise orbit determination (POD).
The IOD is assumed that the orbit is a circular
orbit and essentially required for the POD’s
initial estimate. Then, the POD process goes
with the IOD initial state by a given estimation
filter[6]. We consider five scenarios presented in
Table 1 and each scenario has different
perturbation forces in the orbit model.
Table 1 Scenario Conditions
GEOID SRP MOON SUN
Scenario1 × × × ×
Scenario2 ○ × × ×
Scenario3 ○ ○ × ×
Scenario4 ○ ○ ○ ×
Scenario5 ○ ○ ○ ○
SAKURA 2A (ID: 13782U) was set as the
target object in GEO whose inclination is 9
[deg], and the measurement data history is
illustrated in figure 4.
3.2. Coordinate System
𝒓 , 𝒓 is the object’s position and velocity
vector respectively in inertia frame. 𝜆, 𝛽 are
the object’s right ascension and declination
respectively in inertia frame.
3.3. Observation Model
The dynamic model that provides the object’s
position and velocity with major perturbations
was constructed. The position and velocity
vectors vary with time and need to be
integrated regarding the position and velocity at
an initial epoch. In this paper, considered
perturbation accelerations 𝒂𝒑 are the
gravitational forces including geo-potential
effect and the acceleration of the Sun and the
Moon, and solar radiation pressure. 𝑦(𝜆,𝛽) is
used as the measurements of optical
observation. The state model are given by
𝑿 = 𝒓, 𝒓 𝑻 (3-1)
𝑿 = 𝑭 𝑿, 𝑡 (3-2)
𝒓 = −𝜇𝒓𝒓 𝟑 + 𝒂𝒑 (3-3)
where 𝑭 𝑿, 𝑡 is the nonlinear differential
equation of orbit motion.
The measurement model are given by
𝒀𝒊 = 𝜆, 𝛽 ! = 𝑮 𝑿𝒊, 𝑡 + 𝝐 (3-4)
𝜆 = 𝑎𝑡𝑎𝑛 𝑟! 𝑟! (3-5)
𝛽 = 𝑎𝑠𝑖𝑛 𝑟! 𝒓 (3-6)
where 𝑿𝒊 is the true state at time 𝑡𝒌, 𝒀 is a
two dimensional measurement vector at time
𝑡𝒌. You might get linearized equation of motion
by assuming that the unknown true state 𝑿 is
sufficiently close to some reference state during
a given time interval by
𝒙 𝑡 = 𝑨 𝑡 𝒙 𝑡 (3-7)
𝒚𝒊 = 𝑯𝒊𝒙𝒊 + 𝝐 (3-8)
where 𝑨 𝑡 is the Jacobian of the state.
3.4. Estimation Filter
This paper provides initial orbit determination
that based on the assumption of circular orbit,
and a conventional weighted batch least squares
filter. The state transition matrix 𝚽 , is the
conventional solution to the linear differential
equation in equation (3-7),
𝒙 𝑡 = 𝚽 𝑡, 𝑡! 𝒙𝒌 (3-9)
Finally, weight batch least squares estimator
can be given by
𝒙𝒌 = 𝐇𝑻𝐇!𝟏𝐇 !𝟏𝐇𝑻𝐑!𝟏𝒚 (3-10)
where 𝐇 = 𝐇𝚽. 𝐑 is the weighting matrix..
By adding the calculated 𝒙𝒌 to initial state,
you might obtain the updated state. However,
only one iteration is not enough and need to
iterate until 𝒙𝒌 satisfy a given tolerance state.
3.5. Result of Orbit Determination
This section describes orbit determination
result of SAKURA, whose measurement data
set is provided by Dr. Yanagisawa. Figure 4
and figure 5 illustrate the position and the
velocity respectively.
Fig. 4 R.A. and Dec. Measurements Data
The IOD’s result ended in
𝑠𝑒𝑚𝑖 𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 = 41741.300 𝑘𝑚
𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛 = 13.001 [𝑑𝑒𝑔]
𝑟𝑖𝑔ℎ𝑡 𝑎𝑠𝑐𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑎𝑠𝑐𝑒𝑛𝑑𝑖𝑛𝑔 𝑛𝑜𝑑𝑒
= 356.076 [𝑑𝑒𝑔]
𝑚𝑒𝑎𝑛 𝑚𝑜𝑡𝑖𝑜𝑛 = 1.018 [𝑟𝑒𝑣/𝑑𝑎𝑦]
Figure 5 and figure 6 respectively depict the
estimated radius with error bar defined by
𝜎!"#$%& = 𝜎!! + 𝜎!! + 𝜎!! and 𝜎!!"#$%&' =
𝜎!!! + 𝜎!!
! + 𝜎!!! .
Fig. 5 radius with error bar
Fig. 6 velocity with error bar
Scenario 1 is a simple ecliptic orbit with no
perturbation. This case has the minimum error
of position compared with other perturbation
cases. On the other hand, Scenario 3 has the
minimum error of velocity. We can say that
adding perturbations might increase the error of
position, but decrease the error of velocity.
the Moon’s gravity force’s effect improved the
error of position by 20 km , and the error of
velocity by 0.3 km/sec. Adding the Sun’s
gravity force and solar radiation pressure might
not affect greatly compared with scenario 2.
In GEO, the solar radiation pressure has the
most important role on the orbit motion.
However, it is very hard to give the reflectivity
that is required for the orbit propagation
considering solar radiation pressure. Moreover,
the measurement data is sampled from the
object in GEO, and it always limited mean
anomaly, which is not for an object in LEO
because the observation site and the object in
LEO orbit with different rotational speed. This
could make hard to estimate the full orbit.
4. Conclusions
For pole orientation from light curve
measurement, we showed that it is quite
possible to estimate the object rotational axis
and the size ratio. Moreover, the motion of
precession can be seen from the error
distribution between the amplitude and the
measurement model. However, in order to
provide more precise estimation, the orbit
determination is also required because the orbit
information is also needed to the pole
orientation estimation.
For orbit determination from optical
observation in GEO, the initial orbit
determination and precise orbit determination
were developed. The initial orbit determination
is assumed as a circular orbit. The designed
precise orbit determination considers the
Earth’s geo-potential effect, the solar radiation
pressure, the Sun and the Moon’s gravity forces.
These estimation filters were evaluated with the
measurement observed in GEO, but might be
hard to determine the full orbit from the limited
measurement data because the site and the
object in GEO rotates with the same speed.
Acknowledgement
The authors grateful acknowledge the support
and of Dr. Yanagisawa from JAXA, who
provided us with the measurement data of GEO
optical observation.
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[4] T. Yanagisawa, "Shape and motion estimate of LEO debris using light curves."Advances in Space Research . 50, 2012, 136-145.
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