**Distribution authorized to U.S. Government Agencies and their
contractors for Administrative or Operational Use, 21 July 2006. Other
requests for this document shall be referred to Detachment 15, Air Force
Research Laboratory, 535 Lipoa Parkway, Suite 200, Kihei HI 96753.**

AMOS Observations of NASA's IMAGE Satellite

Doyle Hall^1,2^, John Africano^1,2^, David Archambeault^2^, Brian
Birge^2^,

David Witte^3^ and Paul Kervin^4^

*^1^The Boeing Company, Colorado Springs, CO*

*^2^Boeing LTS / AMOS, Kihei, HI*

*^3^ Pantera Consulting, Albuquerque, NM*

*^4^ Air Force Research Laboratory / Detachment 15, Kihei, HI*

# SUMMARY

NASA's Imager for Magnetopause-to-Aurora Global Exploration (IMAGE)
satellite stopped transmitting telemetry to ground stations in December
2005, after functioning for more than 5 years on Earth orbit. Before
this loss of telemetry, the IMAGE satellite actively maintained a
spin-stabilized attitude with spin axis perpendicular to the orbital
plane and a nominal rotation rate of about 0.5 rpm. The spinning action
served to both stabilize the satellite and keep the 250 m-long radial
wire antennas of the satellite's Radio Plasma Imager under tension
perpendicular to the satellite spin axis. After loss of telemetry, it
was unclear whether the spacecraft remained in this spin-stabilized
configuration, or whether it could continue to receive and execute
up-linked commands. In late January and early February of 2006 the AMOS
3.6m Advanced Electro Optical System (AEOS) conducted an initial set of
observations in an effort to help diagnose the state of the unresponsive
spacecraft. The AEOS observations employed the Visible Imager (VisIm)
instrument in the photometric I-band as well as the long-wavelength
infrared (LWIR) imager. The wide field-of-view VisIm images clearly show
the long radial wire antennas glinting in reflected sunlight during each
revolution of the spinning spacecraft, creating a photometric signature
characterized by large amplitude periodic variations. Analysis of
concurrent AEOS LWIR observations indicates radiometric temperatures
ranging from 250 to 310 Kelvin, with the higher temperatures occurring
when more of the continuously-sunlit portions of the spacecraft were
observable from AMOS. A detailed periodic analysis of the VisIm
photometric signatures acquired on 2006 day-of-year (DOY) 028, 031 and
034 indicates a spin axis orientation consistent with that reported in
the last telemetry down-linked from the satellite approximately seven
weeks earlier. However, the periodic variations indicate a satellite
spin rate of 0.4741 ± 0.0005 rpm, measurably slower than the last known
spin rate from down-linked telemetry. Shortly after these initial AEOS
observations were conducted, the NASA IMAGE satellite team up-linked
commands to the spacecraft to increase the spin rate up to 0.52 rpm in
order to test if the spacecraft could receive and execute such commands.
Subsequent AMOS observations conducted 2006 DOY 150, however, did not
show evidence of an increased spin rate, but instead indicated a further
reduction down to 0.4709 ± 0.0004 rpm. The AEOS observations therefore
confirm that the IMAGE spacecraft has lost the ability to receive and/or
execute up-linked commands, and indicates that, between 2005-DEC-12 and
2006-MAY-30, the spacecraft's spin axis orientation remained stable to
within detection limits but the spin rate declined at a rate of (3.1±
0.3) × 10^-5^ rpm/day, a deceleration most likely caused by
magnetically-induced environmental torques.

# INTRODUCTION

The IMAGE satellite (SSN 26113, 2000-17A) was launched in March 2000,
the first of NASA's Medium-class Explorer spacecraft to fly, and was
equipped with a payload designed to reveal the workings of Earth's
magnetosphere/ionosphere system (see \[1, 2\] and references therein for
a detailed description of the spacecraft and mission). Figure 1 shows a
schematic illustration of the spacecraft. Soon after being inserted into
an eccentric 1,640km × 45,230km × 90° orbit, the satellite began five
highly successful years of mission activity during which it actively
maintained a spin-stabilized attitude with the spin axis perpendicular
to the orbital plane and a nominal rotation rate of about 0.5 rpm. In
addition to stabilizing the spacecraft, this spinning action kept the
long radial wire antennas of the satellite's Radio Plasma Imager (RPI)
under tension and perpendicular to the satellite spin axis. These four
wire antennas were originally 250 m long and gave the newly-deployed
spacecraft a remarkable end-to-end equatorial width of 0.5 km. Although
some of these 0.3 mm-diameter antennas suffered loss in length during
the extended mission, the RPI experiment and the other components of the
satellite's scientific payload returned a tremendous scientific bounty
of data via down-linked telemetry.

During December of 2005, after exceeding its original two year design
lifetime by more than three years, ground-stations stopped receiving
telemetry from the IMAGE spacecraft. The last telemetry from the
attitude control system \[3\] included attitude quaternion vectors
spanning the period 2005-Dec-12 06:00-07:38 UT indicating a spin rate of
0.47594 ± 0.00012 rpm and a spin axis orientation with right ascension
(RA) of (291.8±1.4)° and declination (DEC) of (--0.7±1.4)°, within
statistical uncertainty of the orbit-normal vector. After loss of
telemetry, it was unclear whether the spacecraft remained in this
spin-stabilized configuration, whether it had entered a "safe"
operational mode, or whether it could continue to receive and execute
up-linked commands. In order to ascertain these unknowns and further
characterize the anomaly, NASA and AMOS agreed to use the AEOS 3.6m
telescope to perform observations designed to characterize the status of
the spacecraft as much as possible through remote observations.

The first AEOS observations, conducted in late January and early
February of 2006, were used to determine the spin state of the
spacecraft using time-resolved photometry from the VisIm instrument
\[5\], as well as to measure the figure-averaged temperature using
two-band thermal-IR radiometry from the LWIR instrument \[6\]. The LWIR
observations indicated radiometric temperatures in the 250 K to 310 K
range, with higher temperatures sensibly corresponding to times when
more of the continuously-sunlit portions of the spacecraft were
observable from AMOS. However, the LWIR measurements proved difficult
for the spacecraft engineering team to use for characterization because
of the relatively small expected difference in the thermal signature of
the spacecraft in nominal operational mode versus safe mode.
Fortunately, the VisIm observations yielded measurements of the
spacecraft spin rate with sufficient accuracy to allow a further test if
the spacecraft could still execute commands to adjust its spin rate.
Soon after the initial AEOS observations, NASA up-linked commands to the
spacecraft to increase its spin rate by a detectable amount. Subsequent
AEOS VisIm observations conducted in late May 2006 indicated no such
spin-up, yielding the conclusion that the IMAGE spacecraft cannot
receive and/or execute up-linked commands and is currently passively
spinning down on orbit.

# AEOS VISIM OBSERVATIONS

The data presented in this paper were acquired using the AMOS AEOS 3.6m
telescope Visible Imager (VisIm) instrument \[7, 8\]. The VisIm detector
is a Si-based 512 × 512 pixel cooled Pentamax CCD array, sensitive to
light in the 700−1100 nm wavelength range. Although the instrument is
equipped with several spectral band-pass filters, all observations for
this study employ the Bessel I-band filter which spans the 740−880 nm
spectral range, creating a peak photometric response near 790 nm.
Observing in the standard astronomical spectral I-band provides the
means of establishing an accurate photometric calibration by observing
cataloged photometric calibration stars \[5, 9\]. All observations
employed the VisIm 300 μrad × 300 μrad wide field-of-view with exposure
times in the 0.25-1.0 second range, but did not use the adaptive optics
image compensation capability of the AEOS telescope.

AEOS conducted four observations of the IMAGE satellite on 2006-028,
2006-031, 2006-034, and 2006-150 (dates specified as YEAR-DOY). Many of
the wide field-of-view VisIm images clearly show the long radial wire
antennas glinting in reflected sunlight as the spacecraft rotates.
Figure 2 shows eight selected images from the first AEOS observation (on
2006-028), when the observatory-to-satellite range was between 2,500 and
3,050 km. The eight images show almost two complete rotations of the
satellite, each separated by approximately ¼ of a rotation period (≈30
s). The images clearly show that each radial wire antenna glints in
sunlight along its entire length twice per rotation, as expected for
specular cylindrical structures extending outward from a stably spinning
body. This leads to the conclusion that the IMAGE satellite has
maintained a stable spinning attitude (as opposed to going into a
chaotic tumble, for instance) throughout the observation period after
loss of telemetry. If not, the glinting antennas would not show the
regular temporal and spatial structure apparent in the AEOS images. The
imagery in Figure 2 also shows that two of the antennas have been
truncated in length since their original deployment and that one is
completely missing. This confirms through remote observation the
long-known status of these RPI antennas to IMAGE satellite flight
engineers: on 2000-Oct-03 approximately 130 m of the antenna along the
spacecraft --X axis was lost probably to micro-meteoroid or orbital
debris impact; on 2001-Sep-18 roughly 25 m of the +X axis was lost; and
on 2002-Aug-11 most or all the +Y antenna was lost \[1\]. Remarkably,
despite these losses the RPI instrument remained scientifically
productive throughout the lifetime of the IMAGE satellite.

The glinting antennas shown in Figure 1 create large-amplitude periodic
variations in the measured flux of the satellite. Figures 3 through 6
show plots of the calibrated I-band stellar magnitudes for the four AEOS
observations, along with illustrations of the satellite ground track as
the satellite passed over Maui. The I-band magnitudes plotted in Figures
3−6 have been range normalized to account for the 1/R^2^ effects of
observatory-to-object range variations as well as Sun-to-object range
variations. All magnitudes reported here have been normalized to an
object range of 1000 km and a solar range of one astronomical unit (AU).
Each of the four photometric signatures clearly shows periodic
variations with amplitudes often exceeding 1 magnitude superimposed on
longer term upward, downward, or curved trends.

# PHOTOMETRIC SIGNATURE PERIODIC ANALYSIS

Periodic analysis of the photometric variations using methods commonly
applied to astronomical data \[10, 11\] indicates characteristic
full-cycle repetition periods close to (but not exactly equal to) the
rotation period of (126.066 ± 0.003) seconds indicated in the last
down-linked telemetry from the satellite. A more detailed analysis of
the AEOS data, however, indicates that the measured periods vary
significantly during each of the four individual observations, and that
these variations reflect the distinction between "synodic" (or apparent)
periodicities versus "sidereal" (or actual) rotation rates. The
distinction between synodic and sidereal periods has been discussed in
detail by the asteroid observational community (see \[12, 13\] and
references therein). Synodic periods are measured quantities, and can be
influenced by the relative motion of the satellite, observer, and
illumination source. For instance, if a spinning body moves quickly with
respect to an observer, the apparent rotation rate *as seen from the
observer's perspective* can deviate significantly from the actual
rotation rate of the body. An extreme example in this respect is the
Moon, which always presents the same face to the Earth showing no
*apparent* rotation even though it is actually rotating once every 28
days. In order to evaluate the actual spin rates of the IMAGE satellite
using the AEOS time-resolved photometry, two analyses are required.
First, the photometric data must be analyzed to determine accurate
synodic modulation periods. Second, the synodic periods must be
converted to sidereal rotation periods by accounting for the effects of
the relative motion of the satellite, observatory, and illumination
source during the observations.

## Fourier-series Method to Determine Synodic Periods

Following the method suggested in ref. \[12\], we have developed and
applied a Fourier-series-based analysis to determine synodic periods
from the AEOS time-resolved photometric data. This entails modeling the
fluxes observed during a specified time interval with a series of the
form:

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$F^{'}(t) = \left\lbrack \sum_{m = 0}^{M}{a_{m}\ t^{m}} \right\rbrack + \left\lbrack \sum_{n = 1}^{N}{b_{n}\text{cos}(n\omega\ t) + \ }c_{n}\text{sin}(n\omega\ t) \right\rbrack$$   ()

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------

The first polynomial series allows the reproduction of long term
non-periodic trends in flux and the second Fourier series allows
reproduction of superimposed periodic variations, both of which are
observed in the AEOS VisIm observations (see the middle panels of
Figures 3−6). Equation () contains a total of *M+2N+2* unknown
parameters: *M*+1 polynomial series coefficients (*a~m~*), a total of
*2N* Fourier series coefficients (*b~n~*, *c~n~*), and one Fourier
frequency, *ω*. As explained below, a least-squares analysis provides
the means to determine these unknown parameters and generate a best-fit
to the data.

A model of the *i*^th^ measured flux can be expressed by averaging the
flux given in equation () over the CCD exposure time:

  -------------------------------------------------------------------------------------------------- --------
  $$F_{i}^{'} = \frac{1}{\tau_{i}}\ \int_{}^{}{\ \left\lbrack F^{'}(t) \right\rbrack}\ \text{dt}$$   ()

  -------------------------------------------------------------------------------------------------- --------

where *t~i~* denotes the time at the beginning of the exposure and
*τ~i~* denotes the exposure time. This model flux provides the means to
write the least-squares chi-squared metric:

  ------------------------------------------------------------------------------------------- --------
  $$\chi_{F}^{2} = \sum_{i}^{}\left( \frac{F_{i} - F_{i}^{'}}{{\Delta F}_{i}} \right)^{2}$$   ()

  ------------------------------------------------------------------------------------------- --------

where *F~i~* denotes the observed fluxes and Δ*F~i~* the associated
measurement uncertainties. Minimizing this function with respect to the
*M+2N+2* unknown parameters yields the best-fit to the data.
Fortunately, the best-fit *M+2N+1* series coefficients (i.e., *a~m~*,
*b~n~*, *c~n~*) and can be found efficiently using linear matrix methods
\[14\], leaving only the frequency parameter *ω* to be found using
non-linear methods.

This Fourier-series based least-squares procedure provides a robust
estimate of the characteristic frequency of modulation even in the
presence of significant long-term trends and it has the capability to
model variations in flux occurring on timescales shorter than the
observation exposure times, given observations of sufficient quality.
The best-fit frequency *ω* calculated using this procedure can be used
to derive the corresponding synodic period, *T~SYN~* *=* 2π */ω.*

## Measured Synodic Periods for the IMAGE Satellite

The Fourier-series periodic analysis method provides a means to measure
variations in synodic period during the AEOS VisIm photometric
observations of the IMAGE satellite shown in Figures 3−6. The adopted
method employs a "sliding-window" technique, where multiple
least-squares analyses were performed sequentially for a series of time
"windows" spanning the observations. For the IMAGE data, the procedure
employed a quadratic polynomial series (*M=2*), a total of 30 Fourier
terms (*N=15*) and a window width of *w* = 252s (approximately two
rotation periods). The analysis proceeded as follows:

> 1\) For a given time, *t*, find all photometric measurement
> measurements within the time-window interval *t-w/2* to *t+w/2*;
>
> 2\) perform a least-squares Fourier-series analysis to determine the
> best-fit synodic period for that interval; and
>
> 3\) increment the time and repeat the procedure "sliding" the window
> across the entire set of observations.

The bottom panels in Figures 3−6 show the best-fit synodic periods
generated using this procedure, plotted on the same time-axis scale as
the photometric data shown directly above. The analysis indicates
significant variations in synodic periods between about 121 s and 127 s
during the four observations.

# SPIN-STATE ANALYSIS USING TIME-RESOLVED PHOTOMETRY

As discussed earlier, converting synodic periods (or frequencies) into
sidereal values requires accounting for the relative motion of the
observatory, the satellite, and the Sun, and has been discussed in
detail by the asteroid observation community \[12, 13\]. The
synodic/sidereal difference depends on the spin axis orientation of the
spinning body. In fact, using an analysis procedure called the "epoch
method" \[13\], well-observed variations in synodic frequencies can be
used to derive complete spin-state solutions, yielding both spin axis
orientation and sidereal rotation rates of the observed object. Although
developed for asteroid characterization, the method applies equally well
to artificial satellite observations because it is independent of the
shape and reflectivity characteristics of the observed object. (An
alternate method applied to asteroids for spin-state determination
called the "amplitude method" is not as well suited for artificial
satellite observations because it relies more on the nearly ellipsoidal
shapes and diffuse reflectivity characteristics of asteroids).

## The Epoch Method of Spin-state Determination

Asteroid researchers developed the epoch method for determining the
spin-state of non-resolved objects observed in reflected sunlight using
time-resolved photometry (see \[12, 13\] and references therein). The
epoch method formulation employs a set of measured synodic frequencies
to determine a rotating object's sidereal angular spin rate, Ω, and spin
axis orientation. The spin axis can be specified by RA/DEC angles (*α*,
*δ*) or, alternatively, Euler angles \[4\] given by *φ* = 90°+ *α* and
*θ* = 90°− *δ*. The sidereal spin rate is typically reported in terms of
frequency of revolution, *ν* = Ω / 2π (often in units of rpm). The
sidereal period is given by *T~SID~* = 2π / Ω. Generally, researchers
have applied the epoch method to observations acquired over time scales
sufficiently short so that temporal changes in the three parameters that
specify the spin-state (Ω*,φ*, *θ,*) can be neglected, although this
should be justified *post facto*.

The epoch method relies on the two fundamental assumptions: 1) that the
observed body is in a stable spin rotational state, and 2) that each
full-cycle synodic period measured by the observer corresponds to the
time interval required for the body to rotate such that the "phase angle
bisector" makes one complete revolution in azimuthal angle around the
spin axis, as discussed in more detail below \[13\]. Most asteroids
satisfy the first assumption rigorously because their rotational motions
have relaxed over very long time scales into stable, principal-axis spin
states \[12\]. Spin-stabilized spacecraft also satisfy the first
assumption by design. However, if attitude control is lost, such
spacecraft can potentially evolve into unstable or chaotic rotational
states depending on the characteristics of their inertia tensor and the
presence of environmental torques \[4\]. The phase angle bisector (PAB)
is the direction midway between the body-to-observer and body-to-Sun
direction vectors. Over short time scales, corresponding to small
changes in observation and solar illumination angles, the single-cycle
reflected-light signature of a spinning body is controlled to a large
extent (but not entirely) by the position of the PAB with respect to the
body's spin axis, providing the basis of the second assumption employed
in the epoch method.

## The Phase-Angle Bisector and the Relation Between Synodic and Sidereal Frequency

In the Earth-centered inertial (ECI) reference frame, denoted here by
the "\*" superscript, the PAB unit vector can be expressed as

  ------------------------------------------------------------------------------------------------------------ --------
  $${\hat{b}}^{}(t) = \frac{\ {\hat{o}}^{} + {\hat{s}}^{}}{\ \left| \ {\hat{o}}^{} + {\hat{s}}^{} \right|}$$   ()

  ------------------------------------------------------------------------------------------------------------ --------

where ${\hat{o}}^{}$and ${\hat{s}}^{}$denote the ECI-frame
body-to-observer and body-to-Sun direction unit vectors. In general, the
PAB unit vector varies as a function of time. This vector can be
transformed into the "spin axis reference frame" which has its z-axis
directed along the spin axis, and its x- and y-axes fixed in the ECI
frame (see \[4\]):

  ----------------------------------------------------------------------------------------------- --------
  $$\hat{b}(t,\varphi,\theta) = \left\lbrack R(\varphi,\theta) \right\rbrack\ {\hat{b}}^{}(t)$$   ()

  ----------------------------------------------------------------------------------------------- --------

where the 3×3 rotation matrix, **R**, depends on the spin axis Euler
angles as follows:

  -------------------------------------------------------------- --------
  $$R(\varphi,\theta) = \begin{bmatrix}                          ()
  1 & 0 & 0 \\                                                   
  0 & \text{cos}\theta & \text{sin}\theta \\                     
  0 & - \text{sin}\theta & \text{cos}\theta                      
  \end{bmatrix}\begin{bmatrix}                                   
  \text{cos}\varphi & \text{sin}\varphi & 0 \\                   
   - \text{sin}\varphi & \text{cos}\varphi & 0 \\                
  0 & 0 & 1                                                      
  \end{bmatrix}$$                                                

  -------------------------------------------------------------- --------

The PAB azimuthal and axial angles (Ψ, Θ) relate to the unit vector
given in equation () as follows:

  -------------------------------------------------------------- --------
  $$\hat{b} = \begin{bmatrix}                                    ()
  \text{cos}\Psi\ \text{sin}\Theta \\                            
  \text{sin}\Psi\ \text{sin}\Theta \\                            
  \text{cos}\Theta                                               
  \end{bmatrix}$$                                                

  -------------------------------------------------------------- --------

Equations ()−() provide the means to derive the PAB azimuthal angle as a
function of time and spin axis orientation, Ψ = Ψ(*t*, *φ* , *θ*).

The epoch method assumes that azimuthal motion of the PAB (i.e.,
temporal changes in Ψ) creates the difference between synodic and
sidereal frequencies \[13\], leading to the following expression for the
expected (or model) synodic frequency at a particular time given the
three spin-state parameters (Ω, *φ*, *θ*):

  ------------------------------------------------------------------------------------------------------ --------
  $$\omega^{'}(t,\Omega,\varphi,\theta) = \Omega - \frac{\partial\Psi(t,\varphi,\theta)}{\partial t}$$   ()

  ------------------------------------------------------------------------------------------------------ --------

To model a specific measurement of the synodic frequency *ω~i~* averaged
over a window *t~i~ − w/2* ≤ *t* ≤ *t~i~+w/2* (as described in section
4.2 above) equation () can be integrated in time

  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----
  $$\omega_{i}^{'}(\Omega,\varphi,\theta) = \Omega - \left\lbrack \int_{}^{}{\ \frac{\partial\Psi(t,\varphi,\theta)}{\partial t}\text{dt}} \right\rbrack\Omega - \left\lbrack \frac{\Psi(t_{i}\frac{+ w}{2},\varphi,\theta) - \Psi(t_{i}\frac{- w}{2},\varphi,\theta)}{w} \right\rbrack$$   ()

  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----

where the second expression is applicable over time windows short enough
to ensure that the two terms in the square brackets are approximately
equal (i.e., their difference must be small compared to Δ*ω~i~*, the
measurement uncertainty on *ω~i~*).

## Least-squares Spin-state Determination

The epoch method of determining an object's spin state from measured
synodic frequencies can be formulated as a least-squares problem. Given
a time-series of measured synodic frequencies (*t~i~*, *ω~i~*), such as
those plotted in the bottom panels of Figures 3−6, the epoch method
goodness-of-fit indicator may be defined as follows:

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$\chi_{\omega}^{2} = \chi_{\omega}^{2}(\Omega,\varphi,\theta) = \sum_{i}^{}\left( \frac{\omega_{i} - \omega_{i}^{'}(\Omega,\varphi,\theta)}{{\Delta\omega}_{i}} \right)^{2}$$   ()

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------

where Δ*ω~i~* denotes the uncertainty on the measured frequency and
*ω~i~*′ and denotes the model synodic frequency given by equation ().
Determining the best-fit spin state requires minimizing this function
with respect to the three parameters (Ω, *φ*, *θ*). Fortunately, the
best-fit sidereal period can be found analytically by solving the
equation

  ---------------------------------------------------------------------------------------------------------------------------------------- --------
  $$\left( \frac{\partial\left\lbrack \chi_{\omega}^{2}(\Omega,\varphi,\theta) \right\rbrack}{\partial\Omega} \right)_{\Omega_{0}} = 0$$   ()

  ---------------------------------------------------------------------------------------------------------------------------------------- --------

which yields the following expression

  -------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$\Omega_{0} = \Omega_{0}(\varphi,\theta) = \frac{\sum_{i}^{}{W_{i}\left( \omega_{i} + {\overset{˙}{\Psi}}_{i}(\varphi,\theta) \right)}}{\sum_{i}^{}W_{i}}$$   ()

  -------------------------------------------------------------------------------------------------------------------------------------------------------------- --------

where
$W_{i} \equiv {\Delta\omega}_{i}^{- 2}$and${\overset{˙}{\Psi}}_{i}(\varphi,\theta)$is
the quantity given in square brackets in equation (). This best-fit
frequency can be inserted into the goodness-of-fit indicator from
equation () to define a new indicator as follows:

  --------------------------------------------------------------------------------- --------
  $$\chi_{0}^{2}(\varphi,\theta) = \chi_{\omega}^{2}(\Omega_{0},\varphi,\theta)$$   ()

  --------------------------------------------------------------------------------- --------

The best-fit spin axis can be found by minimizing this quantity with
respect to (*φ*, *θ*). Because of the highly non-linear nature of this
2-dimensional function, this procedure does not lend itself to a
straightforward analytical solution but can be calculated numerically
relatively efficiently even for large data sets. Our analysis employs a
grid-search at fine increments over the entire (*φ*, *θ*) domain,
followed by a simplex non-linear numerical minimization procedure to
home-in on the best-fit angles \[14\].

# IMAGE SATELLITE SPIN-STATE ANALYSIS

The last telemetry from the IMAGE satellite attitude control system on
2005-DEC-12 \[3\] indicated a sidereal spin rate of *ν* =
(0.47594±0.00012) rpm and spin axis angles *φ* = (21.8±1.4)° and *θ* =
(90.7±1.4)°. Applying the epoch method analysis procedure to the four
AEOS measurements yields best-fit spin axis angles consistent with these
telemetry values to within statistical uncertainty. However, best-fit
spin rates show a statistically significant declining trend during the
170-day period following loss of telemetry, despite intervening attempts
by ground-controllers to command an increase in spin rate.

## IMAGE Spin Axis Determination from AEOS Photometric Data

Table 1 lists spin state parameters for the image satellite. The first
entry in Table 1 shows parameters taken from the last down-linked
quaternion telemetry. The second entry shows the best-fit parameters
from the epoch-method analysis applied to the combined data from AEOS
observations conducted 2006-028, -031 and -034; Figure 7 shows the
corresponding gray-scale plot of the goodness-of-fit indicator from
equation () as a function of the spin axis angles *φ* and *θ*. The
analysis yields best-fit axis angles of *φ* = (17.9±4.4)° and *θ* =
(88.7±3.9)°, within statistical uncertainty of those last down-linked
from the satellite. Table 1 also shows the best-fit spin-state
parameters derived from these three AEOS data sets analyzed individually
(entries 3−5), as well as the best-fit parameters from the AEOS data
from 2006-150 (entry 6). Notably, even the 2006-150 observation
indicates the same axis orientation to within detection limits, implying
that the spin axis remained stable to within detection limits during the
170-day period after loss of telemetry.

> Table 1. IMAGE Satellite Spin-state Parameters

  ------- ------ --------------- ------------ ------------ -----------------
  Entry   Year   DOY Range       *φ*          *Θ*          ν (rpm)
                                 (degrees)    (degrees)    

  1       2005   346.25 − 346.32 21.8 ±       90.7 ±       0.47594 ±
                                 1.4^\*^      1.4^\*^      0.00012^\*^

  2       2006   028.66 − 034.61 17.9 ±       88.7 ±       0.47452 ±
                                 4.4^\#^      3.9^\#^      0.00041^\#^

  3       2006   028.66 − 028.68 16.9 ± 5.4   90.4 ± 8.4   0.47472 ± 0.00051

  4       2006   031.62 − 031.65 17.1 ± 4.5   89.8 ± 2.4   0.47462 ± 0.00051

  5       2006   034.59 − 034.61 20.4 ± 4.5   81.7 ± 2.5   0.47457 ± 0.00029

  6       2006   150.23 − 150.25 23.1 ± 4.4   89.2 ± 3.9   0.47138 ± 0.00035

  7       2006   028.66 − 028.68 21.8 ±       90.7 ±       0.47435 ± 0.00059
                                 1.4^\*^      1.4^\*^      

  8       2006   031.62 − 031.65 21.8 ±       90.7 ±       0.47412 ± 0.00059
                                 1.4^\*^      1.4^\*^      

  9       2006   034.59 − 034.61 21.8 ±       90.7 ±       0.47413 ± 0.00040
                                 1.4^\*^      1.4^\*^      

  10      2006   150.23 − 150.25 21.8 ±       90.7 ±       0.47094 ± 0.00042
                                 1.4^\*^      1.4^\*^      
  ------- ------ --------------- ------------ ------------ -----------------

> ^\*^Parameters taken from last down-linked IMAGE satellite quaternion
> telemetry \[3\].
>
> ^\#^Parameters determined by combining three AEOS observations
> conducted on 2006-028, -031, and -034.

## IMAGE Spin Rate Determination from AEOS Photometric Data

Because of the implied axis orientation stability, entries 7−10 of Table
1 list best-fit spin rates derived using the last down-linked spin axis
parameters directly in equation (), instead of finding best-fit values
for (*φ*, *θ*). Figure 8 shows the corresponding synodic periods,
indicating that the models reproduce the observations exceedingly well.
The derived sidereal spin rates listed in Table 1 clearly show a
declining trend. Notably, AEOS measured a significant decline between
the observations conducted on 2006-034 and 2006-150, which is especially
significant because IMAGE flight engineers attempted to up-link commands
to the spacecraft during this period to increase the satellite's spin
rate. Specifically, on 2006-047 (Feb. 16) and again on 2006-061 (Mar. 2)
commands were issued to increase to a target rate of 0.52 rpm \[3\]. The
AEOS data clearly indicate that the spacecraft failed to receive and/or
execute these up-linked commands.

Figure 9 shows a plot of the steadily declining spin rate of the IMAGE
satellite. The five plotted points comprise the last down-linked spin
rate (Table 1, entry 1) and the four AEOS-measured spin rates (Table 1,
entries 7−10). These data can be fit using an exponentially decaying
model expected for a satellite experiencing a dampening torque, such as
that arising from magnetically induced currents (c.f., \[15\]):

  -------------------------------------------------------------- --------
  $$\nu(t) = \nu(t_{0})\ e^{- (t - t_{0}\frac{)}{\tau}}\ $$      ()

  -------------------------------------------------------------- --------

The best-fit, shown as the dotted line in Figure 9, has a decay time
constant of *τ* = 41 ± 3 years corresponding to a spin-down rate of
(3.1±0.3)×10^-5^ rpm/day during the 170-day span of the data. This decay
rate seems to exceed by a factor of ≈3 the spin-down rate experienced by
the spacecraft during its last two years of operation \[3\]  an as-yet
unexplained discrepancy.

# CONCLUSIONS

The following conclusions can be drawn from the analysis of AEOS
telescope observations of NASA's IMAGE satellite:

1.  The IMAGE satellite maintained a stable spin axis for at least 170
    days after loss of telemetry. If this were not the case, the
    glinting radial antennas would not show the regular temporal and
    spatial structure apparent in the AEOS VisIm images, and the
    photometric spin-state determinations would not consistently yield
    best-fit spin axis angles within statistical uncertainty of those
    last down-linked from the spacecraft.

2.  The remote AEOS imagery confirms the known status of the IMAGE
    satellite RPI radial wire antennas: that ≈130 m of --X antenna, ≈25
    m of the +X antenna, and most of the +Y antenna has been lost since
    the original deployment.

3.  The spacecraft spin rate declined steadily during the 170 days after
    loss of telemetry at a rate of (3.1±0.3)×10^-5^ rpm/day,
    approximately three times faster than the decay rate observed during
    the last two years of active satellite operation.

4.  The attempts by IMAGE flight controllers to command an increase in
    spin rate up to 0.52 rpm up-linked on 2006 Feb 16 and again Mar 2
    did not have the intended effect, implying that the spacecraft has
    lost the ability to receive and/or execute such commands.

5.  AEOS LWIR measurements of figure-averaged radiometric temperatures
    proved difficult for the IMAGE spacecraft engineering team to use
    for remote characterization because of the relatively small expected
    difference in the thermal signature of the spacecraft in nominal
    operational mode versus safe mode.

# FIGURES

![](media/image1.pct){width="3.2243055555555555in"
height="3.8333333333333335in"}

Fig 1. A schematic illustration of the IMAGE spacecraft \[3\] including
the octagonal central bus, the triangular axial antennas extending
parallel to the spin axis, and portions of the four radial wire antennas
that originally extended 250 meters outward from the spacecraft equator.
The ≈0.5-rpm spin of the spacecraft keeps these four wire-thin radial
antennas extended outward under tension. During operation, the
spacecraft's attitude control system maintained the spin axis in an
orientation perpendicular to the orbital plane.

![](media/image2.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image3.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image4.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image5.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image6.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image7.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image8.jpeg){width="1.46875in"
height="1.1027777777777779in"}![](media/image9.jpeg){width="1.46875in"
height="1.1027777777777779in"}

Fig. 2. Eight AEOS VISIM wide field-of-view images of NASA's IMAGE
satellite showing the radial wire antennas glinting in reflected
sunlight. These false-color images were acquired 2006-028 when the
observatory-to-spacecraft range was between 2,500 and 3,050 km. The
images show almost two complete satellite rotations (from top-left to
top-right and bottom-left to bottom-right) each separated by one-quarter
of a rotation period (approximately 30 seconds). The first (top-left)
image shows the +X antenna extending to the lower left of the central
body, and the --X antenna extending to the upper right, confirming that
the -X antenna has lost about 130 m of its original 250 m length. The
second image acquired approximately ¼ of a rotation later shows the
slightly truncated --Y antenna extending to the left of the central
body, but with no antenna extending on the opposite side of the central
body, confirming that the +Y antenna is completely missing. The third
image shows the ±X antennas again, but on opposite sides of the central
body than in the first image, as expected after ½ of a satellite
rotation. Similarly the fourth image shows the --Y antenna again, but on
the opposite side of the central body than in the second image. The
second row shows that this same pattern is repeated in the
immediately-following satellite rotation. The zenith (+EL) direction is
upward in all of these images, explaining the slightly changing aspect
angle of the antennas from frame to frame.

![](media/image10.png){width="2.579861111111111in"
height="2.579861111111111in"}

![](media/image11.png){width="3.925in" height="4.7125in"}

Fig. 3. AEOS VisIm photometric observation of the IMAGE satellite,
conducted on 2006-028 (Jan 28). The top panel shows the ground-track of
the satellite as it passed over Maui with the red circles indicating the
signal acquisition and loss points. The middle panel shows the I-band
stellar magnitudes as a function of time normalized to remove object and
solar range variations (upward-pointing triangles indicate saturated
measurements and diamonds indicate unsaturated data). Periodic
variations from the spinning satellite are clearly evident in the middle
panel, superimposed on a brightening trend. The bottom panel shows
synodic periods derived from the time-resolved photometric data using a
sliding-window Fourier analysis, indicating that full-cycle modulation
periods decrease from 126 s to 122 s during the observation.

![](media/image12.png){width="2.5597222222222222in"
height="2.5597222222222222in"}

![](media/image13.png){width="3.932638888888889in" height="4.7125in"}

Fig. 4. AEOS VisIm I-band photometric observation of the IMAGE satellite
conducted on 2006-031 (Jan 31). The top panel shows the ground-track of
the satellite with the red circles indicating the signal acquisition and
loss points and the arrow indicating the direction of motion. The middle
panel shows the range-normalized I-band stellar magnitudes measured as a
function of time, with clearly evident periodic variations from the
spinning satellite superimposed on longer term temporal variations. The
bottom panel shows synodic periods derived from the time-resolved
brightness data using a sliding-window Fourier analysis, indicating that
full-cycle modulation periods vary between 121.6 s and 126.5 s during
the observation.

![](media/image14.png){width="2.5597222222222222in"
height="2.5597222222222222in"}

![](media/image15.png){width="3.932638888888889in" height="4.7125in"}

Fig. 5. AEOS VisIm I-band photometric observation of the IMAGE satellite
conducted on 2006-034 (Feb 03). The top panel shows the ground-track of
the satellite with the red circles indicating the signal acquisition and
loss points and the arrow indicating the direction of motion. The middle
panel shows the range-normalized I-band stellar magnitudes measured as a
function of time, with clearly evident periodic variations from the
spinning satellite superimposed on longer term temporal variations. The
bottom panel shows synodic periods derived from the time-resolved
brightness data using a sliding-window Fourier analysis, indicating that
full-cycle modulation periods vary between 121.6 s and 125.8 s during
the observation.

![](media/image16.png){width="2.5597222222222222in"
height="2.5597222222222222in"}

![](media/image17.png){width="3.932638888888889in" height="4.7125in"}

Fig. 6. AEOS VisIm I-band photometric observation of the IMAGE satellite
conducted on 2006-150 (May 30). The top panel shows the ground-track of
the satellite with the red circles indicating the signal acquisition and
loss points and the arrow indicating the direction of motion. The middle
panel shows the range-normalized I-band stellar magnitudes measured as a
function of time, with clearly evident periodic variations from the
spinning satellite superimposed on a slight dimming trend. The bottom
panel shows synodic periods derived from the time-resolved brightness
data using a sliding-window Fourier analysis, indicating that full-cycle
modulation periods vary between 122.7 s and 126.5 s during the
observation.

![](media/image18.png){width="5.120138888888889in" height="3.375in"}

Fig. 7. Mollweide projection of the epoch method goodness-of-fit
parameter as a function of IMAGE spacecraft spin axis orientation angles
(*φ*, *θ*). This gray-scale plot of the chi-squared indicator from
equation () shows the results of the epoch method spin-state analysis
applied to the combined data from the first three AEOS observations of
the IMAGE satellite conducted on 2006-028, -031 and -034. The central
meridian of this plot corresponds to *φ* = 0 with *φ* increasing from
left to right; the top corresponds to *θ* = 0 and the bottom to *θ* =
180°. Lighter gray-scale regions show areas of higher quality fits to
the data, and the minimization analysis yields best-fit values of *φ* =
(17.9±4.4)° and *θ* = (88.7±3.9)°. These best-fit angles are within
statistical uncertainty of those last down-linked from the satellite's
attitude control system, *φ* = (21.8±1.4)° and *θ* = (90.7±1.4)°, shown
as the bold × symbol on the plot.

![](media/image19.png){width="2.9659722222222222in"
height="1.90625in"}![](media/image20.png){width="2.957638888888889in"
height="1.8944444444444444in"}![](media/image21.png){width="2.957638888888889in"
height="1.8944444444444444in"}![](media/image22.png){width="2.957638888888889in"
height="1.90625in"}

Fig. 8. Best-fit models of the measured synodic periods from AEOS
observations of the IMAGE satellite. The "+" symbols show the AEOS VisIm
measurements from 2006-028 (upper left), 2006-031 (upper right),
2006-034 (lower left) and 2006-150 (lower right). The solid lines on
each plot indicate the best-fit model synodic periods calculated using
equation () with the spin axis parameters last down-linked from the
satellite's attitude control system, *φ* = 21.8° and *θ* = 90.7°. The
error bars on the data show estimated measurement uncertainties (which
are smaller than the plotted "+" symbols in some cases).

![](media/image23.png){width="4.932638888888889in"
height="4.513888888888889in"}

Fig. 9. Sidereal spin rates for the IMAGE satellite as a function of
time (black diamond symbols) and an exponential decay model (solid
line). The first data point shows the spin rate down-linked from the
satellite attitude control system on 2005-Dec-12, just before loss of
telemetry. The last four data points show the rates measured remotely
using AEOS telescope time-resolved photometric observations. The
measured spin rates show a steady declining trend, despite intervening
efforts by IMAGE satellite controllers to command the spacecraft to
increase its spin rate.

# ACKNOWLEDGEMENTS

The authors would like to thank John Lambert for insightful explanations
of the epoch method of determining spin states, Thomas Kelecy for
discussions of the dynamics of satellite spin deceleration in the
presence of dampening environmental torques and the AEOS telescope
operations team for executing the observation plan.

# REFERENCES

1\. See the NASA Goddard Space Flight Center IMAGE mission web site
(http://image.gsfc.nasa.gov) and references therein for details on the
mission history of the spacecraft (accessed for this work Feb -- Jul
2006).

2\. Gibson, W. et al., "IMAGE, the First of the New MIDEX Missions,"
*13^th^ AIAA/USU Conference on Small Satellites*, Logan, UT, 1999.

3\. Prior, Michael, *Hubble Space Telescope Deputy Operations Manger*,
Code 440.8, Goddard Space Flight Center, Greenbelt, MD 20771,
<mprior@hst.nasa.gov>, private communication Jan.-July, 2006.

4\. Goldstein, H., "Classical Mechanics" Addison-Wesley Pub. Co.,
Reading Massachusetts, 1981.

5\. Hall D. et al., "AEOS I-band Photometry of Moving Targets," *The
2003 AMOS Technical Conference Proceedings*, Kihei, HI, 2003.

6\. Witte, D., "The AEOS LWIR Imager: A Tool for Space Object
Thermometry," *The 1999 AMOS Technical Conference Proceedings*, Kihei,
HI, 1999.

7\. Mayo, J., "AEOS 3.67 Meter Telescope Optics: Design, Fabrication,
and Performance," *The 1999 AMOS Technical Conference Proceedings*,
Kihei, HI, 1999.

8\. Berger, P., "AEOS Adaptive Optics System and Visible Imager," *The
1999 AMOS Technical Conference Proceedings*, Kihei, HI, 1999.

9\. Landolt, A., "UBVRI Photometric Standard Stars in the Magnitude
Range 11.5\<V\<16.0 Around the Celestial Equator," *The Astronomical
Journal*, Vol. 104, 340−49, 1992.

10\. Scargle, J., "Studies in Astronomical Time Series Analysis II.
Statistical Aspects of Spectral Analysis of Unevenly Spaced Data," *The
Astrophysical Journal*, Vol. 263, 834−53, 1982.

11\. Larsson, S., "Parameter Estimation in Epoch Folding Analysis,"
*Astron. Astrophys. Supp. Ser.*, Vol. 117, 197−201, 1996.

12\. Magnusson, P., "Distribution of Spin Axes and Senses of Rotation
for 20 Large Asteroids," *Icarus*, Vol. 68, 1−39, 1986.

13\. Magnusson, P. et al., "Determination of Pole Orientations and
Shapes of Asteroids," in *Asteroids II* (edited by R. Binzel et al.)
66−97, University of Arizona Press, 1989.

14\. Press, W. et al., "Numerical Recipes in FORTRAN: The Art of
Scientific Computing" (2^nd^ Edition), Cambridge University Press, New
York, NY, 1992.

15\. Bertotti, B. and Iess, L., "The Rotation of Lageos," *Journal of
Gephysical Res.*, Vol. 96, 2431−40, 1991.