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.

References [1] H. Hirayama, “Orbital Debris Removal Using

Special Density Materials,” 2010 Beijing Orbital Debris Mitigation Workshop, October 18, 20.

[2] Kessler, D. J, “Collisional Cascading: The Limits of Population Growth in Low Earth Orbit,” Advances in Space Research, V0l. 11, Issue 12, pp. 63-66, 1991

[3] C. Pardini, T. Hanada, P. H. Krisko, “Potential benefits and risks of using electrodynamics tethers for end-of-life de-orbit of LEO spacecraft,” IADC Action Item 19.1, 2006

[4] T. Yanagisawa, "Shape and motion estimate of LEO debris using light curves."Advances in Space Research . 50, 2012, 136-145.

[5] A. Pospieszalska-Surdej, "Determination of the Pole Orientation of an Asteroid."Astron. Astrophys.149, 1985,186-194.

[6] A. Gelb, “Applied Optimal Estimation,” The analytic Sciences Corporation, 1974.