The Post Flight Study of Micro Accelerations On-Board of Russian

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                  The Post Flight Study of Micro
                Accelerations On-Board of Russian
                      Spacecraft FOTON-12




                   European Space Agency Contract Report




Prepared by:

Dr. Valentina Shevtsova
Mr. Denis Melnikov
Prof. J.C. Legros



All communications should be addressed to:

Dr. Valentina Shevtsova

MRC
Chem.Phys.E.P.Dept.,
ULB, CP 165/62
Av.F.D.Rooselvelt 50
B-1050 Bruxelles BELGIUM

Phone: + 32 2 650 30 24
Fax:     +32 2 650 31 26
e-mail: vshev@ulb.ac.be
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                             Document Change Record

Issue/Revision           Description          Date        Authors
     1 -0                First original     08.01.01   Shevtsova V.M.
                                                        Melnikov D.E
                                                        Legros J.C.
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                        of Microacceleration



                                                   Table of Contents

1.      APPLICABLE AND REFERENCE DOCUMENTS .................................................... 6

2.      ABBREVIATIONS USED IN THIS DOCUMENT ...................................................... 8

3.      INTRODUCTION ............................................................................................................ 9

4.      ASSIGNMENT AND BASIC CHARACTERISTICS OF S/C FOTON-12 .............. 10

5.      PRECAUTIONS IN THE PROCESSING OF THE RESULTS................................ 13

6.      CENTER OF MASS AND CENTER OF GRAVITY ................................................. 15

7.      MICRO ACCELERATION AT THE CENTER OF MASS ...................................... 20

8. THEORETICAL CONSIDERATION OF FOTON-12 ATTITUDE (POST
FLIGHT TREATMENT) ...................................................................................................... 33
     8.1   DETERMINATION OF THE S/C FOTON-12 MOTION BASED ON MEASUREMENTS OF THE
     EARTH MAGNETIC FIELD ........................................................................................................ 35
     8.2   DETERMINATION OF THE S/C FOTON-12 MOTION USING OF QSAM DATA .............. 47
9.      MEASUREMENTS OF MICRO ACCELERATIONS BY TAS INSTRUMENT... 51

10.  MEASUREMENTS OF MICRO ACCELERATIONS BY SINUS
INSTRUMENT ....................................................................................................................... 56

11. ANALYSIS OF DATA FROM QSAM AND COMPARISON WITH THE
OTHER INSTRUMENTS ..................................................................................................... 71
     11.1 MEASUREMENTS BY THE HIGH FREQUENCY ACCELEROMETER ................................... 71
     11.2 MEASUREMENTS BY QSAM GYROSCOPE .................................................................. 82
     11.3 ROTATION OF THE S/C FOTON-12 ........................................................................... 94
     11.4 ANALYSIS OF DC COMPONENT AND HIGH FREQUENCIES ........................................... 97
       11.4.1 Comparison of the QSAM, TAS, SINUS power spectrum signals in frequency
       range 0.2- 10Hz ................................................................................................................ 97
       11.4.2 On the DC and high frequency components of micro acceleration ................. 98
       11.4.3 Influence of high frequency components of angular rate on acceleration ..... 100
12.         THEORETICAL MODELS DESCRIBING FOTON-12 MOTION. .................. 101
     12.1 FIRST THEORETICAL MODEL .................................................................................... 103
       12.1.1 Empirical formula for acceleration due to own rotation and precession ...... 103
       12.1.2 Relation between angular velocities of precession and own rotation ............ 107
       12.1.3 A formula of acceleration on board of FOTON-12 due to its own rotation,
       precession and oscillations............................................................................................. 108
     12.2 SECOND THEORETICAL MODEL ................................................................................ 116
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13.      VALIDATION OF THE MODELS OF THE FOTON-12 S/C MOTION .......... 119
  13.1      RESULTS OF CALCULATIONS ACCORDING TO THE MODEL 2...................................... 122
  13.2      RESULTS OF CALCULATIONS ACCORDING TO THE MODEL 1...................................... 126
14.      CONCLUSIONS....................................................................................................... 134
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1. APPLICABLE AND REFERENCE DOCUMENTS


RD1 “PF-C/D-VERH-PR-609”, Issue 2, Rev. 0 (28/2/2001)

RD2 "The processing of the data measured by QSAM during Foton-12 mission for
    the determination of its motion." Technical report of KBOM N259-034/00
    issued in 2001.

RD3 Senchenkov A.S, A.V.Egorov A.V., Barmin I.V. Dynamic situation on board of
    the ISS and problem of crystal growth. IAA-01-IAA.12.1.06 52-nd IAF
    Congress, 1-5 Oct. 2001.Toulouse, France

RD4   V.A. Sarychev, V.V. Sazonov, M.Yu. Belyaev, N.I. Efimov, I.L. Lapshina.
      Determining passive attitude motion of the Mir orbital station from
      measurements geomagnetic field intensity. Cosmic research, 1995, vol. 33,
      No. 1, pp. 10 - 16.

RD5 V.V. Sazonov, S.Yu. Chebukov,V.I. Abrashkin, A.E. Kazakova, A.S. Zaitsev.
    Analysis of low frequency microgravity environment on board the Foton-11.
    Cosmic research, No. 4, vol. 39, 2001, p. 419-435 (in Russian).

RD6 GOST (State standard) 22721-77. Model of the upper atmosphere for ballistic
    calculations, Moscow, 1978.

RD7 Hannan E.J. Multiple time series. John Wiley and Sons, Inc. 1970

RD8 V.I.Abrashkin, V.L.Balakin, I.V.Belokonov, K.E.Voronov, V.V. Ivanov, A.S.
    Zaitzev, A.E.Kazakova, A.S.Zaitsev, V.V.Sazonov. Determination of the
    spacecraft Foton-12 motion on the measurements of the Earth magnetic
    field. Preprint No.60 of the Keldysh Institute of applied mathematics, 2000 (in
    Russian)

RD9 V.I.Abrashkin, M.V.Volkov, A.V.Egorov, A.E.Kazakova, A.S.Zaitsev,
    V.V.Sazonov. Analysis of low frequency component in measurements of the
    angular rate and the acceleration done by the system QSAM on board the
    spacecraft Foton-12 Preprint of the Keldysh Institute of applied mathematics,
    2001 (in Russian)

RD10 G. Anshakov, O.Mumin, V.Peshekhonov, Features of acceleration
     measurement system SINUS, Proceedings of 18th International Microgravity
     Group Meeting, Cocoa Beach, Fl, USA, June 1999.
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RD11 B.Juergens, P.Sickinger, A.Egorov, H.Richter, H.Hamacher, Microgravity
     Characterization of the FOTON-12 Mission. Findings of the measurement
     assembly QSAM. Post Flight Review Meeting, ESTEC, March 16th, 2000

RD12 Results of processing of data describing the motion of S/C Foton-11. KBOM
     Technical report N253-034/99, 1999

RD13 H. Hamacher, H.–E. Richter, S. Drees, A.V. Egorov, A.S.Senchenkov,
     P.Sickinger, Microgravity Characterization of the FOTON-11 Mission. IAF-99-
     J.3.05. 50th International Astronautical Congress, 4-8 Oct, 1999,Amsterdam,
     The Netherlands.

RD14 C. Albanese, F.Peluso, D.Castagnolo , Thermal radiation forces in
     Microgravity: The TRUE and TRAMP experiments: Results and future
     perspectives. Proc. Of 1st Int. Symp. "Microgravity Research and Application
     in Physical Science and Biotechnology", Sorrento, Italy, 2000.
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2. ABBREVIATIONS USED IN THIS DOCUMENT


CNES         Centre National d'Etudes Spatiales
c.o.g.       center of gravity
c.o.m.       center of mass
c.o.p.       center of pressure
DLR          German Space Agency
ESTA         Russian accelerometer inside Sinus
HFA          High Frequency Accelerometer
LFA          Low Frequency Accelerometer
MIRAGE       facility to measure magnetic field (Samara, Russia)
MSTA         Russian accelerometer inside Sinus
PSD          Power Spectral Density
S/C          Spacecraft
SINUS        Russian accelerometer package
TAS          Three Axes Servo accelerometer package
TRAMP        Experiment in FluidPac
QSAM         DLR accelerometer package
mg           10-3g
µg           10-6g
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3. INTRODUCTION

The effects of gravity can be undesirable if experiments are conducted want to
understand certain physical or biological phenomena or to study the complex
interactions of different forces involved in a process.

ESA, DLR, CNES and Rosaviakosmos provided the different scientific payloads on
FOTON-12. Besides the facilities with scientific experiments, different instruments for
measuring the micro accelerations have been arranged in the re-entry module.
Scientists are interested in micro acceleration data in order to correlate them with
experiment results and properties of materials generated in orbit.

FOTON-12 carried European Microgravity experiments including the FluidPac
module with experiments in Fluid Physics. Although the steady residual acceleration
provided by the satellite FOTON is one of the best available, the related induced
convection is not necessarily negligible for some classes of investigations, especially
when the characteristic time is large as in diffusion controlled phenomena. In these
experiments, any convective flow in the experimental cell is deleterious to obtain the
desirable results.

A residual gravity with low amplitude and small frequency variation exists due to the
atmospheric drag, the stabilizing rotation and complex motion of the platform along
the trajectory. They are causing accelerations for payloads out of the center of
gravity.

The motion of the S/C causes the low-frequency accelerations. Unlike to the general
opinion, this type of g-jitter is dangerous for a wide class of processes. For example,
in growth crystal process it can induce motions in the melted liquid in the ampoule.
As result the final crystal will have a stripped structure [1].

The different facilities working on board and the vibration of the S/C itself induce the
high-frequency accelerations. The impact of this type of g-jitter is intensively studied
due to the ISS utilization program.

During the FOTON-12 mission, September 1999, the German system QSAM, the
Russian equipment SINUS-12KU and the ESA instrument TAS have monitored the
Microgravity environment during different time periods. Moreover, the Institute of
Applied Math (Moscow) has calculated the possible motion of the S/C and the values
of micro acceleration using the measurements of the level of magnetic field (Mirage
equipment) during the mission.

Besides the fact that the different accelerometers have worked at different time
intervals, each instrument has positive and negative points. The results from different
instruments and comparative analysis of them will be described below.
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4. ASSIGNMENT AND BASIC CHARACTERISTICS OF S/C FOTON-12

The FOTON satellite is derived from the Vostok family of manned capsules as used
by Yuri Gagarin. FOTON -12 Space Craft (S/C) was launched on September 9, 1999
at 18:01 according Universal Coordinated Time, UTC (20:00:01 according Central
European time) by a LV Soyuz rocket from the Plesetsk launch site.

The time used in all Russian documents corresponds to winter Moscow time, one
should add +3 hours to the UTC time. The orbit parameters were the following:

Perigee: 225km.
Apogee: 405 km.
Inclination: 62.8 deg.
Duration: 14.64 days.

The mass of space craft (S/C):
       The mass of S/C completely assembled, with fuel – 6410,8 kg (real mass).
       The mass of S/C to the moment of switching on the brake system – 5745,01
kg (calculated value).

      Geometrical dimensions of the S/C:
      Length – 6.2 m;
      Diameter of service module is 2.50 m, and diameter of re-entry capsule is
2.30m.

The first "switch on" of the Control System of Flight of S/C "CSF" was performed at
18:08:45 and "Switch off" CSF– 19:38:45. After the switching off the control system
the spacecraft slowly spins with angular rate 0.03 degree/s. While orbiting, the control
system has not been used.

The sketch of the of S/C FOTON and location of different co-ordinate systems is
shown in fig.4.1

The origin of the S/C basic coordinate system (in fig.4.1 subscript Б) is located at the
cross point of the longitudinal axis of S/C with the contact plane of the service module
and the retrorocket. The positive direction of the axis +ОХb is from the service
module to the return capsule. The axis +ОYb is directed towards to the plane I, the
axis +ОZb is directed towards to the plane II.

The location of the mass center of S/C (completely assembled, with fuel) with respect
to the basic coordinate system S/C:

                            Х = + 1724 mm;
                            Y = -2 mm;
                            Z = - 6 mm.
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a. Return capsule                            b. Service module




              Figure 4.1. The locations of different coordinate systems.


The coordinate system with origin in the mass center has a subscript c in Figure 4.1:
       The axis +OXc is parallel to the ОХБ and directed from return module to the
service module;
       The axis +OYc is parallel to the +ОYБ and directed towards to the plane III;
       The axis +OZc is parallel to the OZБ and directed towards to the plane II.

Another co-ordinate system with its centre, located at the centre of the re-entry
module has been used for different technical reasons. This co-ordinate system has
subscript CA in Figure 4.1.

The calculated values of the moments of inertia are:

                                       Jx = 3124 kg m2;
                                      Jy = 13548 kg m2;
                                      Jz = 13540 kg m2.

The values are given in co-ordinate system of centre of mass. Accepted dispersion of
values is ± 10 %.

The co-ordinates of the mass centers of different modules working on-board in the
co-ordinate system fixed with a mass center of S/C are given in Table 4.1.
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 Table 4.1 Coordinates of mass centers of instruments with respect to the S/C
                               center of mass.



      System             Device (sensors)           Хс, мм   Ус, мм    Zс,мм
                         Module 1А                  -369      -651       -84

       QSAM              Module 1В                  -1288     -292      -518
                         Module 2     (LFA, gyro)   -1736    +218       +55
    instruments
                         Sensor      (HFA)           -18      -24       +300
                         INUK.402131.010
                         Experimental module        -735      -315       -46
                         Electronic module          -1218     -488       -21
     FLUIDPAC            ТAS (MAGIA)                -655      -291      +103
                         ТAS (BAMBI)                -655      -587      -120
                         ТAS (TRAMP)                -655      -613      +86
                         Device МSТА(1)             +109     +108       -254
      System             Device МSТА(2)             -1216     -552      -528
   SINUS -12KY           Device МSТА(3)             -424     +730       +92
                         Device МSТА(4)             -426      -462      -242
                         Device КХ97-016(I)         -932     +522       -396
     BDUS-НХ             Device КХ97-016(II)        -932     +544       +376
                         Sensor ВУС                 +1544     -962       -39
Sensor ДУС-Б-2Б          Sensor РУС                 +1544     -892       -14
                         Sensor ТУС                 +1544     -907      +46
                         Sensor КХ.000.001(I)       +125      -32       -254
     Instrument          Sensor КХ.000.001(II)      -1161     -72       -484
     "MIRAGE"            Sensor КХ.000.001(III)     -452      -315      -271
                         Sensor КХ.000.001(IV)      -507     +190       -314
                         Sensor КХ.000.001(V)       -1754     -397       -54
  POLIZON facility       П310.00.00.000             -126      +70        -12
    AGAT facility        АG.00.00                   -258       -2       -486
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5. PRECAUTIONS IN THE PROCESSING OF THE RESULTS

Analyzing the data for the micro accelerations from different instruments during the
flight of FOTON -12 one should remember, that there are a few time scales. One of
them is linked with the motion of the satellite along the orbit, another is related to the
motion of S/C around its center of mass and some scales (the shortest ones) will
appear due to the work of internal equipment.

The largest time is the orbital period, which is 90.53 min at the beginning of the flight.
To detect in Fourier spectrum the frequency corresponding to this rotation any
accelerometer should record at least 10-15 periods of rotation. It means that in case
when instrument has worked perfectly, it should work continuously during minimum
10-15 hours. Among the available instruments on board only SINUS and MIRAGE
could detect it. The longest time interval of QSAM was 5 hours. Looking carefully at
the power spectrum related to this run, one can find the tiny peaks near the
frequency f ≈ (3-4)⋅10-3Hz, but the accuracy due to the short time interval is not
enough to determine clearly this frequency.

The next characteristic time is related to the rotation of FOTON around its own axis
and precession due to atmospheric drag. The periods of these motions are varying
between 7 and 12 min. It means that minimal time of this run should be around 1.5
hours. The typical run of QSAM system was 1.7 hours; therefore the processed data
can reveal this frequency.

One should remember that the amplitude of a signal in Fourier spectrum also
depends on the amount of points, for which the analysis is done (2048 or 4096).
Therefore performing a comparative analysis of the amplitudes of different
frequencies this must be taken into account.

And of course, the quality and sensitivity of the instruments are important. For
example, the QSAM system should be able to detect the frequency range from 0Hz
up to 70Hz. The TAS instrument has worked in two different regimes: a) it can detect
the signals in the range from 1Hz up to 12Hz in the case of 50Hz sampling
frequency, b) the low frequencies in the range from 2.0 •10-3 Hz up to 1.2 Hz can be
detected in the case of 5Hz sampling frequency.

Besides the fact that the different accelerometers have worked at different time
intervals, all instruments had failures. The low frequency data from TAS instrument
are not reliable due to the strong dependency of the offset upon the temperature.
Due to unknown reason the low frequency accelerometer of the QSAM system cut-off
the amplitude of recorded time signals above 10-4g. Although QSAM gives the most
valuable and reliable data, they are received by filtering of the data from high
frequency accelerometer.
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It seems that the problem of non-zero offset has arisen within the operation of all
accelerometers. The process allowing to distinguish the offset value from the DC
value on the recorded time signals by the holders of the different instruments remains
enigma for the authors of this report.

The frequencies caused by the motions of the satellite are rather low, about f ≈ (0.3-
2.2) •10-3 Hz. But for weak flows this time-varying acceleration can change the
trajectory of the experimental tracers like in the TRAMP experiments. And the post
flight analysis of some experimental data has shown that the knowledge of the gravity
level is really important for the complete explanations of physical phenomena.

The present report consists of 13 Chapters, which are listed in the table of content.
The plots, corresponding to the chapters, are given at the end of each chapter.

Throughout the report two co-ordinate systems will be used: orbital and system
coordinate fixed with the axes of the inertia of FOTON. They will be described in text.

Some particular plots and calculations will be done at the point, corresponding to the
location of the center of mass of the TRAMP experiment, FluidPac, ESA.
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6. CENTER OF MASS AND CENTER OF GRAVITY

Going away from the Earth the gravitational acceleration is reducing as

   g=G/r2.                                                                        (6.1)

Here, the parameter G is the gravity constant. The radius of the Earth is 6.371x106m.
Then gravitational acceleration in the c.o.m. of FOTON will be

g=g0 (6.371/6.596)2=0.933 g0 in perigee (225km)

g=g0 (6.371/6.776)2=0.884 g0 in apogee (405km)

where g0 =9.81 m/s2

Let us introduce the orbital right-hand Cartesian coordinate system OX1X2X3, shown
in fig.6.1. The Points C and O are located respectively in the centers of mass of the
Earth and of spacecraft. The axis OX3 is directed along the vector R=R(CO) and is
called "local vertical". The axis OX2 is directed perpendicularly to the orbital plane.




                           Fig.6.1 Orbital co-ordinate system


For the rigid body the position of the center of mass does not depend on the
gravitational condition, but the position of the center of gravity does. By the definition:
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                                                           r                                             r
                                 r
                                              ∑m r
                                              j =1, n
                                                          j j
                                                                       r
                                                                                     ∑m
                                                                                     j =1, n
                                                                                                 j   g j rj
                                 rc.o.m. =                       and   rc.o. g . =
                                                ∑m
                                                j =1, n
                                                           j                           ∑m
                                                                                       j =1, n
                                                                                                     j   gj


The gravity inside S/C on the distance r from the mass center is defined as:
                                                  v
      r              r        r                ∂g(R c.o.m. ) r        r      r
   g( R c .o . m . + r ) ≈ g( R c .o . m . ) +      r        r , as | r |<<| R |c.o.m.                        (6.2)
                                                  ∂R
        r
Here R c.o.m. is a radius vector of the center of mass of the spacecraft. The variation
of the acceleration on the length r can be evaluated as:

   g( R c .o . m . + r ) − g( R c .o . m . )         r
                                             ≈ −2                                                             (6.3)
                g( R c .o . m . )                 R c .o . m .

The total length of FOTON-12 is 6.2m. In the case of orientation of S/C along the
local vertical (OX3), the variations of the acceleration on its half length r =± 3m in
apogee and perigee are:

                                   (g - gc.o.m.) / gc.o.m = ±0.919x10-6 in perigee

                                 (g - gc.o.m. ) / gc.o.m = ±0.885x10-6 in apogee

Despite the small variation of the gravity g(r) upon the distance inside S/C the
position of the center of gravity (c.o.g.) will differ from the position of the mass center
(c.o.m.). The difference between two projections of the positions of c.o.g. and c.o.m.
on the local vertical (OX3) will be maximal in the case, when the longitudinal axis of
Foton is directed towards the center of the Earth, see fig.6.2c. The center of gravity
lies below the mass center (closer to the Earth).

The estimations above allow to explain the change of the initial attitude, which is
shown in fig.6.2a. At this initial point the c. o. g. coincides with the c.o.m, there is no
gravitational torque. This orientation is not stable. As soon as small disturbances
apply to the initial attitude, the S/C will turn and the c.o.g. will move away from the
c.o.m. Assume, that from each side of c.o.m. one chooses a "local' center of mass on
the distance r from the c.o.m. of the S/C. A tiny rotational momentum will arise, see
fig.6.2b. From eqs.(6.1)-(6.2) it follows:

           r r r             r             r r          m ⋅ r2            r r r
    M g = mr × ( g − g ' ) = e M mr∆g sin( r , g ) = −4G 3           sin( r , g ) ⋅ e M (6.4)
                                                        R c .o . m .
                                         r
The direction of this momentum e M is perpendicular to the orbital plane, parallel to
the OX2. This momentum will cause clockwise (or opposite) rotation of the spacecraft
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                              Fig.6.2a. Initial orientation




                    Fig.6.2b. Appearance of gravitational torque




                         Fig.6.2c. Gravity stable orientation
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in the orbital plane, OX1X3,. Further disturbances (it can be due to traversing apogee
and perigee) will increase the slope of S/C with respect to the previous orientation,
and as a result the torque will increase. It will continue up to the moment when the
longitudinal axis of S/C will coincide with the local vertical (OX3). At that point the
gravitational moment will be equal zero M g = 0 . This attitude is called "the gravity
stable orientation". Due to inertia the S/C will perform some oscillations around the
equilibrium position. These oscillations under impact of small external forces can give
rise to rotations, precession or nutations.

The gravitational torque written by eq.(6.4) represents a simplified version of the task.
This form has been chosen for the visual physical explanation. Strictly speaking, one
should write integral equations. In this case the gravitational torque with respect to
the spacecraft centre of mass is

              v r
    M g = ∫ [ r × g(R c.o.m. + r )] dm .                                                                            (6.5)

Using the relations from eqs.(6.1)-(6.2) it can be written as
                                     r
        r      r                 r ∂g(R c.o.m. ) r           G                          r r                r  r
M g = ∫ r dm × g(R c.o.m. ) + ∫ r ×              r  dm = −2 3                        ∫ [er × e R ] ⋅ r sin(r ,R c.o.m. )dm =
                                                                                                        2

                                      ∂r                   R c .o . m .

               G              r          r              r r                 r
    − 2⋅
           R   3
               c .o . m .
                            ∫ [e   r   × e R ] r 2 sin( r , R c.o.m. ) ⋅ρ ( r ) ⋅ dV                                (6.6)


Here, the first integral is equal to zero by the definition of the mass center. With a
good accuracy FOTON-12 can be considered as a symmetrical body, but the mass is
not uniformly distributed along the longitudinal axis of S/C. Therefore the value of the
gravitational torque M g according eq.(6.6) will be different from the one in eq.(6.4).
One can take integral in eq.(6.6) and show, that S/C should turn its heavier part
(service module) closer to the Earth.

Two external torques have an influence upon the spacecraft attitude motion. The role
of the gravitational momentum is considered above. The second one is an
aerodynamic momentum. Comparison of fig.6.2a and fig.6.2b shows that the surface
of a contact of the S/C with the incoming stream of air is changing (increasing) with
the attitude of S/C.

Designers call the geometrical center of this surface of the cross section as the
center of the pressure of S/C. At the initial orientation, fig.6.2a, the center of mass
coincides with a center of pressure. With inclination of the S/C these two centers will
be shifted, being located both on the axis OX3 due to symmetry of FOTON-12.

Due to the fact that FOTON does not contain solar panels the distance between the
c.o.m. and center of aerodynamic pressure is not really large. Therefore the arising
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aerodynamic momentum should not be large. This is one of the reasons why the
Microgravity conditions on-board of the satellite FOTON are favorably distinguished
from those on the ISS.

Mathematically aerodynamic torque will be written as

                                     M a = ρ | v | ( v × d) .

Here ρ is the atmosphere density, v is the S/C velocity with respect to the
atmosphere, d is a vector describing the plane of the cross section.

The attitude motion, in which the spacecraft longitudinal axis coincides with the
normal to the orbit plane (parallel to the OX2) and the spacecraft rotates around it, is
an exact solution of the motion equations in the case M a = 0 . It means, that such
position corresponds to the stable aerodynamic orientation.

Analyzing the role of the both torques, one can see that their stable positions are
different.
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7. MICRO ACCELERATION AT THE CENTER OF MASS

A quasi-steady (low-frequency) acceleration component exists due to satellite attitude
motion, the gradient of the Earth gravitational field and the atmosphere drag. This
component can be calculated from information about satellite motion.

A satellite is considered as a rigid body. Let us choose a point P on its frame. The
difference between the gravitational field strength at the point P and the absolute
acceleration is called a residual acceleration at this point P. Taking into account
gravitational and aerodynamic momentums the residual acceleration, a, can be
defined as [2]


              dω         r        µ  3( R ⋅ r)R     
    a= r×        + ( ω × r ) × ω + e3       2
                                                 − r  + cρ v v                   (7.1)
              dt                  R      R          
                                                    

Here r is the radius vector of the point P with respect to the satellite center of mass
O, R is the geocentric radius vector of the point O, t is the time, ω is the absolute
angular rate of the satellite. v is the velocity of the point O with respect to the Earth's
surface (see fig.6.2), µe is the gravitational parameter of the Earth, ρ is the density of
the aerodynamic flow incoming on the satellite, c is the satellite ballistic coefficient.
This coefficient depends on the cross section of the S/C with incoming flow.

The center of mass of FOTON-12 moves around the Earth along the orbit, close to
Keplerian elliptic orbit. The only aerodynamic acceleration component acts on the
spacecraft center of mass. From eq.(7.1), when r = 0, follows that acceleration in the
c.o.m. is

                                      a = cρ v v
Assuming that |v| ≈ const, c ≈ const, and ρ is proportional to the density of
atmosphere in center of mass, ρ ∼ K(t), we will get

   | a | = cρ |v|2 ∼ K(t)                                                         (7.2)

The density of the atmosphere changes between the apogee and the perigee about
40 times. Equation (7.2) shows that the residual acceleration in c.o.m. should
oscillate with a period of the changing of the density of atmosphere, e.g. close to the
orbital one.
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Another right-hand Cartesian coordinate system should be introduced for describing
of the S/C motion: the system Oxyz formed by the satellite central principal axes of
inertia. The origin of the system, point O, is located at the mass center of S/C. The
axis Ox is directed along the axis of the satellite, which is the axis of a minimal
momentum of inertia. Due to axial symmetry of S/C the choice of two other axes Oy
and Oz is not so important. Therefore they are chosen in agreement with designers
co-ordinate system, see fig.4.1




          Fig.7.1 Coordinate system formed by S/C principal axis of inertia.

The absolute value of acceleration in the c.o.m. of FOTON-12 at different days of the
mission is shown in figs.7.2-7.6. The values in plots are given according theoretical
calculations, when the Keplerian orbit has been correlated according the
measurements of magnetic field by the system Mirage. See description of the system
in chapter 8.1

The module of acceleration in the c.o.m. |a| performs almost perfect sinusoidal
oscillations throughout the mission, although the shape of signal is slightly changing
with time. The maximal amplitude is increasing from |a|max ≈10.67⋅10-6 m/s2 at the
beginning of the flight to the |a|max ≈13.55⋅10-6 m/s2 at the end of the flight. The
difference is about 26%.

The minimal amplitude, achieved in the apogee, slowly varies between |a|min ≈
0.27⋅10-6m/s2 to |a|min ≈0.36⋅10-6m/s2 during the mission.

The projections of the acceleration on the axes, shown by fig.7.2 reveal more
complex behavior. In the beginning three acceleration components perform
oscillations with a period about Π ≈ 90 min. To the 7th day, the component ax
continues to oscillate approximately with the same period, but the additional
oscillations with a period about 7min are established for the perpendicular directions
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ay and az, see figs.7.2-7.5. To the last days this period reduces till ≈6min (5.94 min),
see fig.7.6.

The acceleration in the c.o.m. except the non-direct calculations considered above,
has been obtained from producers of the system SINUS (headed by Prof.
O.L.Mumin, St.Petersburg, Russia). The absolute values of accelerations in the
center of gravity and 1.5m away at different days throughout the mission are shown
in the figs.7.7-7.16, where | a | = a x + a y + a z . These experimental data are
                                     2   2    2    2


averaged on intervals of 300s.

The absolute values of the residual acceleration in plots 7.7 - 7.16 do not exceed
1*10-5 m/s2. The minimal values of |a|, around ∼4*10-6 m/s2, have been recorded from
15.09.00 until 20.09.99, which increases to the end of the mission. It is difficult to
estimate the period of oscillations due to the averaging data on each 300s. With a
large tolerance it may be accepted around 10min.

According the time schedule fig.7.2 corresponds to fig.7.7; fig.7.5 corresponds to
fig.7.10; fig.7.6 corresponds to fig.7.16. Although these figures match in time, direct
comparison is not reasonable. One can compare only the value of maximal
amplitudes from the both sources they increase to the end of mission, and do not
considerably exceed ∼1⋅10-5 m/s2.

Regarding the data presented by SINUS there is one non-clear point: are the
accelerations in the c.o.m. and 1.5 m away really so similar?

Looking ahead, one may estimate the µg-level only due to the rotation of FOTON
around own axis. For example, on 15.09.99 (4th run) the angular velocity was about
0.84 deg/s or at another units Ω=0.01471/s. (These values will be defined later.)
Then, a = Ω2R ≈ 3 ⋅10-4m/s2. The evaluated µg-level of the acceleration is two order
of value larger than indicated in Fig.7.10. But they were another motions of the S/C
except spinning.
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                                          time, min
Fig.7.2. Residual accelerations at the spacecraft center of mass at the beginning of
the mission (scaled by 10-6 m/s2). The instant t = 0 corresponds to 20:40:00 UTC
09.09.1999, vector components regard to the coordinate system of S/C.
| a |2 = a x + a 2 + a z2
           2
                 y
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   ax




   ay




   az




   |a|



                                         time, min
Fig.7.3. Residual accelerations at the spacecraft center of mass at the beginning of
the mission (scaled by 10-6 m/s2). The instant t = 0 corresponds to 06:00:00 UTC
10.09.1999, vector components regard to the coordinate system of S/C.
| a |2 = a x + a 2 + a z2
           2
                 y
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   ax




   ay




   az




   |a|



                                         time, min
Fig.7.4. Residual accelerations at the spacecraft center of mass on the 3rd day of the
mission (scaled by 10-6 m/s2). The instant t = 0 corresponds to 07:00:00 UTC
11.09.1999, vector components regard to the coordinate system of S/C.
| a |2 = a x + a 2 + a z2
           2
                 y
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   ax




   ay




   az




   |a|



                                         time, min
Fig.7.5. Residual accelerations at the spacecraft center of mass on the 7th day of the
mission (scaled by 10-6 m/s2). The instant t = 0 corresponds to 03:00:00 UTC
15.09.1999, vector components regard to the coordinate system of S/C.
| a |2 = a x + a 2 + a z2
           2
                 y
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   ax




   ay




   az




 |a|




                                         time, min
Fig.7.6. Residual accelerations at the spacecraft center of mass at the last day of the
mission (scaled by 10-6 m/s2). The instant t = 0 corresponds to 11:50:00 UTC
22.09.1999, vector components regard to the coordinate system of S/C.
| a |2 = a x + a 2 + a z2
           2
                 y
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                                               set 1

   1,00E-05
   8,00E-06
   6,00E-06
   4,00E-06
   2,00E-06
  0,00E+00
              1               13               25             37          49



                               Time, one unit corresponds to 300 s


Fig.7.7. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
at the first day of the mission (scaled by m/s2). The lines connect points distant by
300s. The instant t = 0 corresponds to 19:39:26 UTC 09.09.1999,



                                               set 2

  1,20E-05
  1,00E-05
  8,00E-06
  6,00E-06
  4,00E-06
  2,00E-06
  0,00E+00
              1                           13                         25



                               Time, one unit corresponds to 300 s


Fig.7.8. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 13:09:41 UTC 13.09.1999,
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                                              set 3

   1,00E-05
   8,00E-06
   6,00E-06
   4,00E-06
   2,00E-06
  0,00E+00
              1               13              25         37          49




                             Time, one unit corresponds to 300


Fig.7.9. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 16:59:38 UTC 13.09.1999,



                                              set 4

   1,00E-05
   8,00E-06
   6,00E-06
   4,00E-06
   2,00E-06
  0,00E+00
              1                          13                   25




                            Time, one unit corresponds to 300 s


Fig.7.10. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 18:40:00 UTC 15.09.1999,
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                                              set 5

 1,00E-05
 8,00E-06
 6,00E-06
 4,00E-06
 2,00E-06
 0,00E+00
              1                          13                       25




                               Time, one unit corresponds to 300 s


Fig.7.11. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 04:10:00 UTC 16.09.1999,




                                               set 7

   1,00E-05
   8,00E-06
   6,00E-06
   4,00E-06
   2,00E-06
  0,00E+00
              1               13               25            37        49




                               Time, one unit corresponds to 300 s


Fig.7.12. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 18:40:00 UTC 19.09.1999,
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                                                  set 8

   1,00E-05
   8,00E-06
   6,00E-06
   4,00E-06
   2,00E-06
  0,00E+00
                  1                                            13




                                Time, one unit corresponds to 300 s


Fig.7.13. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 04:40:00 UTC 20.09.1999,



                                                  set 9

   1,00E-05
   8,00E-06
   6,00E-06
   4,00E-06
   2,00E-06
  0,00E+00
              1                13            25           37        49   61




                                Time, one unit corresponds to 300 s


Fig.7.14. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 20:30:00 UTC 21.09.1999,
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                                          set 10

 1,00E-05
 8,00E-06
 6,00E-06
 4,00E-06
 2,00E-06
 0,00E+00
              1                                      13



                            Time, one unit corresponds to 300 s


Fig.7.15. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 19:50:00 UTC 21.09.1999,




                                          set 11

  1,20E-05
  1,00E-05
  8,00E-06
  6,00E-06
  4,00E-06
  2,00E-06
  0,00E+00
             1                13           25             37         49



                            Time, one unit corresponds to 300 s


Fig.7.16. Residual accelerations |a| at the spacecraft center of mass and 1.5 m away
(scaled by m/s2). The lines connect points distant by 300s. The instant t = 0
corresponds to 11:10:00 UTC 22.09.1999,
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8. THEORETICAL CONSIDERATION OF FOTON-12 ATTITUDE (POST
   FLIGHT TREATMENT)

The level of micro accelerations at different points of S/C can be calculated precisely
from information about a satellite motion, see eq.7.1.

The behavior of S/C during the mission has been theoretically studied at the Keldysh
Institute of Applied Mathematics (Moscow) under some assumptions.

1. The satellite is a rigid body.
2. The center of mass of S/C moves along a Keplerian elliptic orbit.
3. The shape of S/C is a sphere with a center displaced from the satellite center of
   mass. (It affects only equations for the aerodynamic torque acting on the satellite.)

Two right-hand Cartesian coordinate systems have been introduced for describing of
the S/C motion: the orbital one OX1X2X3 shown in fig.6.1 and the system Oxyz
formed by the satellite central principal axes of inertia. Point O is located at the mass
center of the S/C. The axis OX3 is directed along the vectors R and the axes OX2 is
directed perpendicular to the orbital plane and along the vector R×dR/dt respectively.
Later on one can find also "the structural coordinate system" similar to the Oxyz. In
this system one axis is parallel to the satellite longitudinal axis, and one of others is
fixed with the position on-board sensor or accelerometer.

To have a compact form of the governing equations the indices 1,2,3 will be used for
notations of the axis x,y,z. With admissible errors, the axes Oxi are parallel to the
axes of the satellite structural coordinate system where axis Ox1 = Ox is parallel to
the satellite longitudinal axis. The transition matrix from the system Oxyz to the orbital
system OX1X2X3 is denoted by (aij)3i,j=1.. Below in equations the components of
vectors are referred to the system Oxyz .

Taking into account the gravitational and restoring aerodynamic torques the
governing equations of a satellite attitude motion will be written as [RD4-RD5]

    dω i
         = µ i (ω jω k − νa 3 j a 3k ) + λi ρv ( v j p k − v k p j ),            (8.2)
     dt
    da1i
          = a1 jω k − a1k ω j − ω 0 a 3i ,                                       (8.3)
     dt
    da 3i
          = a 3 j ω k − a 3k ω j + ω 0 a1i ,                                     (8.4)
     dt
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                  I j − Ik            I1
           µi =              , λi =      , ν = 3µ3 , v =
                                                 e
                                                             v12 + v2 + v3 .
                                                                    2    2
                     Ii               Ii       R

The indices i, j, k run for numbers 1, 2, 3, resulting in 3 equations for angular rates ωi
and 6 equations for a1i, a2i. Relations a 2i = a3 j a1k − a 3k a1 j will give the missing
elements of the transition matrix ||aij||.

Here ωi and vi are the components of the vectors ω and v, I i are the moments of
inertia of the satellite relative to the axes Oxi, ω0 is the module of the absolute
angular rate of the orbital coordinate system, pi are the parameters characterizing the
aerodynamic torque. R is the geocentric radius vector of the point O, ρ is the density
of the aerodynamic flow incoming on the satellite, µe is the gravitational parameter of
the Earth.

The parameters µi and λi are known with accuracy ±10% from designers of S/C, and
the density ρ can be calculated according to some model [RD6]. To solve the
problem (8.2)-(8.4) one should define 9 parameters: 6 initial values of a solution of
the differential equations of the satellite attitude motion and 3 aerodynamic
parameters pi.

The rough solution of the eqs.(8.2)-(8.4) can be found without using any real data
from S/C instruments. For this reason the mathematical formulation should be
simplified at least in 2 lower levels of complexity:
a) S/C is moving on a circular orbit in motionless atmosphere, the density of the
   atmosphere is constant;
b) The most simple one; in addition to the previous assumptions, the external
   moments and forces are dropped, e.g. aerodynamic is not taken into account at
   all. This is valid in the case, when the angular rate of S/C is much larger then the
   orbital frequency (ω1 >>2π/T, where T is about 90 min)

The solution of the simplest model (b) is used as initial guess for solving the next
level problem (a). At the next step the solution of the model (a) can be used as initial
guess for solving eqs.(8.2)-(8.3). The three aerodynamic parameters pi are
considered as parameters of the problem. Of course, the final solution will not
describe the actual motion of the satellite FOTON-12.

To incorporate the real motion of S/C in the theoretical model the initial values for the
satellite attitude and the parameters pi are estimated from the processing of the
experimental data (Mirage or QSAM).

Processing the measurement data related to a sufficiently great number of intervals it
is possible to find typical features of a satellite attitude motion. Then, with some
accuracy, one can estimate a level of a quasi-steady acceleration component during
the flight.
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This problem is a usual one of estimating some parameters by processing indirect
measurements e.g. measurements of other parameters or functions. There are some
basic techniques for solving this problem [RD7]. The most popular one is the least
square method which has been used at the Keldysh Institute of Applied Mathematics
(Prof. Sazonov). The whole procedure is rather complicated and it is based on
modern statistical analysis. The detailed description can be found in the issues [RD5,
RD8, RD9]. The developed methods allow not only to find unknown values pi, but
also to correct the values of other numerous parameters, such as λ i, µ i, etc.


8.1   Determination of the S/C FOTON-12 motion based on measurements of
      the Earth magnetic field

To determine non-oriented motion of S/C FOTON-12 the instrument Mirage was
employed. It was designed in the Laboratory of Space Instrument Engineering (The
Volga Branch of the Russian Academy of Cosmonautics).

The equipment Mirage on-board FOTON-12 has measured the magnetic field inside
the re-entry capsule during the space mission. It consisted of five three-axis
magnetometers. Records of all five magnetometers digitised at the same instants
with steps of 5s. Also, there were some short time intervals (of length less than 10
min) in which the steps of sampling were 1s. The analysis of the measurements
showed that the magnetic field inside the space vehicle was basically determined by
the Earth magnetic field.

Processing the magnetic field measurement data was made as follows. Two of the
five magnetometers had recorded non-expectable data. Either it was some failure or
they were measuring an additional magnetic field. The operating technology
equipment may cause that variations of magnetic field, e.g. the sensors located near
the setup Polizon displayed the highest values. The measurements from those
sensors were omitted. Measurements of the other magnetometers were transformed
to the structural coordinate system and then they were averaged component-wise
with equal weights. Averaging was applied only to the measurements related to the
same instant. The averaged values were considered to be measured values of the
components hi (i=1,2,3) of the Earth magnetic field strength at that instant.

The components of the Earth magnetic field strength Hi(t) in the orbital coordinate
system at certain time were constructed using the Kepler approximation for the
satellite orbital motion and analytical model of the Earth's magnetic field IGRF1995.
(This document may be requested from Russian Space Agency.) Functional
describing the difference between hi and Hi(t) in the same co-ordinate system has to
be minimized over initial conditions at the instant t0 and the parameters of the
problem written above (Section 8). The final solution should determine the satellite
attitude motion and define more precisely the parameters of the task.
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The results obtained in a way clearly demonstrate the spinning of Foton-12 along the
longitudinal axis. The spacecraft angular rate ωx at the very beginning of the flight
was slow down to 0.03 degree/s. The increase of this rate during the mission is
shown in fig.8.1-.8.3. At the beginning of the mission it was varied in a chaotic way.
The most interesting behavior of ωx is observed in fig.8.2-a at the 3d day of the
mission. The angular rate, being oscillating, increases by small jump each 90min,
passing the perigee. After 5 day of the FOTON-12 mission (see fig.8.2-b) the growth
of ωx was slowing down achieving the value about 1 deg./s at the end of mission.

Two others projections of the angular rate, ωy and ωz, were oscillating around zero.
The amplitudes of these oscillations were slowly increasing during the flight arriving
to the values about 0.15 deg./s to the end of the mission. Non-zero amplitude points
out to the fact that some other rotations or oscillations exist in the system.

According to the conclusions, drawn at the Keldysh Institute of Applied Mathematics
(Prof. Sazonov), the S/C performs motion, close to an Euler's regular precession of
an axially symmetric rigid body at the last days of the flight.

The last plot on each fig.8.1-.8.3 shows the function K(t), which is the ratio of the
atmosphere density at the point O and the instant t to the minimal value of this
density on the interval under processing. During one orbital period the magnitude of
K(t) changes by 40 times, and achieves the maximum near perigee and minimum
near an apogee. As it was mentioned above, (see figs.7.2-7.6) the time dependence
of the micro acceleration at the center of mass of satellite is similar to behavior the
function K(t).

The time dependence of micro acceleration at the location of the ESA equipment
FluidPac (particularly at the centre of mass of TRAMP) calculated by Keldish Institute
(Moscow) through the magnetic field (Mirage) is shown in fig.8.4-8.6 at different days
throughout the mission. The following co-ordinates have been used (point T = -655
mm, -613 mm, 86 mm) in the S/C system of co-ordinate OcXcYcZc, shown in fig.5.1.


The absolute value of the residual acceleration | a |=        a x + a 2 + a z2 at point T
                                                                2
                                                                      y

continuously increases during the mission. At the last days |a| was oscillating near
the average value of about |aav|∼2.24⋅10-4 m/s2, see fig.8.6. Despite the complex
structure of the oscillations, their amplitude is quite small, ∼1.4⋅10-5 m/s2. It is about
6% with respect to |aav|.

All three projections of accelerations grow from the beginning of the flight to the end
having different magnitudes at the same time. The projection ax in the longitudinal
direction of S/C oscillates around zero with an amplitude ∼6.0⋅10-5 m/s2. Two others
are oscillating around some average, non-zero value. The projection ay, has a
maximal average value, as in y-direction the point T is most remote from the center of
mass YC. Of course, this projection gives the main income to the absolute value |aav|.
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The time dependencies on figs.8.4-8.6 have been carefully analysed and on their
bases an empirical formula has been elaborated at the time interval during the
operation of TRAMP.

                  n
   ai = ai 0 + ∑ (bi j cos 2πf i j t + cij sin 2πf i j t ) .                             (8.5)
                 j =1


Where the coefficients for n=5 are determined as

                  Table 8.1. Coefficients in eq. 8.5 for the i=1, (x-direction)

                         j          a x0             b xj          c xj          f xj

                         1        -3.6615       -2.4896∗101    3.0786∗101     1.645209


                                              i=2, (y-direction)

                         j          a y0             b yj          c yj          f yj

                         1     -1.1665∗102          1.5392     4.2332∗10-1    1.157556
                         2          "               3.3976       -7.5033      2.051535
                         3          "               4.1744        1.4193      1.872925
                         4          "              -4.7073       -2.6550      2.237846
                         5          "           -9.6563∗10-1     -1.4247      3.281500



                                              i=3, (z-direction)

                         j          a z0             bzj           c zj          f zj

                         1      1.6508∗101          5.5583        2.8422      1.643534
                         2           "          -4.5562∗10-1   -7.0390∗10-1   1.870828
                         3           "             -1.5791     -9.4072∗10-2   2.073682
                         4           "             -2.8356        1.0229      3.290399
                         5           "          -9.1597∗10-1   -2.7967∗10-1   4.149568



The time dependencies of the projections of the micro acceleration according to the
full calculations of eqs.(8.2)-(8.4) and according the empirical formula (8.5) are
shown in figs.8.7-8.9 at the upper plots, and their differences are shown at the lower
plots. The formula reproduces time dependency and the levels of the micro
accelerations pretty well, for the ay and az the difference is below 10% of the average
value. In the longitudinal direction the difference a little bit higher than 10%.
 Reference:   ESA Contract Report
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              of Microacceleration




                        a                                       b

ωx




ωy




ωz



K




                   time, min                                time, min



Fig.8.1. Components ωi of spacecraft angular rates measured in deg./s at successive
days of mission. The function K(t) is the ratio of the atmosphere density at the point
O and the instant t to the minimal value of this density on the interval under
processing. (a) The instant t = 0 corresponds 20:40:00 UTC 09.09.1999;
(b) The instant t = 0 corresponds 06:00:00 UTC 10.09.1999
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              of Microacceleration



                           a                                  b


ωx




ωy




ωz




K



                        time, min                           time, min


Fig.8.2. Components ωi of spacecraft angular rates measured in deg./s at different
days of the mission. The function K(t) is the ratio of the atmosphere density at the
point O and the instant t to the minimal value of this density on the interval under
processing. (a) The instant t = 0 corresponds 07:00:00 UTC 11.09.1999;
(b) The instant t = 0 corresponds 23:40:00 UTC 14.09.1999
 Reference:   ESA Contract Report
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 Title:       The Post Flight Study
              of Microacceleration



                               a                                 b


ωx




ωy




ωz




 K



                        time, min                            time, min


Fig.8.3. Components ωi of spacecraft angular rates measured in deg./s at second
part of the mission. The function K(t) is the ratio of the atmosphere density at the
point O and the instant t to the minimal value of this density on the interval under
processing. (a) The instant t = 0 corresponds 03:00:00 UTC 15.09.1999;
(b) The instant t = 0 corresponds 11:50:00 UTC 22.09.1999
 Reference:   ESA Contract Report
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              of Microacceleration




                               a                                    b



 ax




 ay




 az




 |a|



                      time, min                                time, min

Fig.8.4. Vector components of the micro accelerations ai and the absolute value of
| a |2 = a x + a 2 + a z2 at the FluidPAC (particularly TRAMP) location at the successive
           2
                 y
days of the mission (scaled by 10-6 m/s2) (scaled by 10-6 m/s2). The S/C attitude was
calculated using the measurements of magnetic field. (a) The instant t = 0
corresponds 20:40:00 UTC 09.09.1999; (b) The instant t = 0 corresponds 06:00:00
UTC 10.09.1999
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              of Microacceleration



                           a                                        b


 ax




 ay



 az




 |a|



                      time, min                                 time, min


Fig.8.5. Vector components of the micro accelerations ai and the absolute value of
| a |2 = a x + a 2 + a z2 at the FluidPAC (particularly TRAMP) location at the different
           2
                 y
days of the mission (scaled by 10-6 m/s2) (scaled by 10-6 m/s2). The S/C attitude was
calculated using the measurements of magnetic field. (a) The instant t = 0
corresponds 07:00:00 UTC 11.09.1999; (b) The instant t = 0 corresponds 23:40:00
UTC 14.09.1999
 Reference:   ESA Contract Report
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 Title:       The Post Flight Study
              of Microacceleration


                           a                                     b


 ax




 ay



 az




 |a|



                      time, min                                  time, min

Fig.8.6. Vector components of the micro accelerations ai and the absolute value of
| a |2 = a x + a 2 + a z2 at the FluidPAC (particularly TRAMP) location at second part of
           2
                 y
the mission (scaled by 10-6 m/s2). The S/C attitude was calculated using the
measurements of magnetic field. (a) The instant t = 0 corresponds 03:00:00 UTC
15.09.1999; (b) The instant t = 0 corresponds 11:50:00 UTC 22.09.1999
     Reference:   ESA Contract Report
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     Title:       The Post Flight Study
                  of Microacceleration




  a x,
  ax,emp
                                                                                (a)




                                                                                 (b)
ax - ax,emp




                                         time , t (103s)


    Fig.8.7. ax component of the micro acceleration at the point T (FluidPAC, TRAMP)
    scaled by 10-6 m/s2. The duration of the step 5 (TRAMP run) corresponds to 0.738 < t
    < 3.278 and step 6 corresponds to the interval 3.358 < t < 5.968.
     (a) ax - is calculated according eqs.(8.2)-(8.4) using the measurements of magnetic
         field, ax,emp - according the empirical formula (8.5).
     (b) The difference between theoretical and empirical dependencies, ax - ax,emp.
     Reference:   ESA Contract Report
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                  of Microacceleration




  ay,
  ay,emp
                                                                                (a)




                                                                                 (b)
ay - ay,emp




                                         time , t (103s)


    Fig.8.8. ay component of the micro acceleration at the point T (FluidPAC, TRAMP)
    scaled by 10-6 m/s2. The duration of the step 5 (TRAMP run) corresponds to 0.738 < t
    < 3.278 and step 6 corresponds to the interval 3.358 < t < 5.968.
    (a) ay - is calculated according eqs.(8.2)-(8.4) using the measurements of magnetic
    field, ay,emp - according the empirical formula (8.5).
    (b) The difference between theoretical and empirical dependencies, ay - ay,emp.
     Reference:   ESA Contract Report
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                  of Microacceleration




    az ,                                                                        (a)
    az,emp




                                                                                 (b)
az - az,emp




                                         time , t (103s)


    Fig.8.9. az component of the micro acceleration at the point T (FluidPAC, TRAMP)
    scaled by 10-6 m/s2. The duration of the step 5 (TRAMP run) corresponds to 0.738 < t
    < 3.278 and step 6 corresponds to the interval 3.358 < t < 5.968.
    (a) az - is calculated according eqs.(8.2)-(8.4) using the measurements of magnetic
    field, az,emp - according the empirical formula (8.5).
    (b) The difference between theoretical and empirical dependencies, az - az,emp.
 Reference:   ESA Contract Report
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 Title:       The Post Flight Study
              of Microacceleration



8.2   Determination of the S/C FOTON-12 motion using of QSAM data

As it was mentioned above [RD2] the QSAM low frequency accelerometer had a
failure. Therefore the data from the high frequency accelerometer has been used for
determination of the S/C motion. The third instrument of QSAM facility, a gyroscope
has measured angular velocities.

The mean value of records of high frequency accelerometer is considered to be not
exact value ‘by definition’. Therefore low frequency data were extracted in two steps
by a method, developed at the Keldysh Institute. After such processing the mean
value were dropped down, only the oscillatory components around zero value was
further analyzed. The oscillatory signals have been used for the spectral analysis.

The deviation of the components of the micro accelerations a ′ = a − a        from the
                                             -6    2
mean values shown in fig.8.10 scaled by 10 m/s . There are 2 curves on each plot.
One of them, with a filled small rectangular corresponds to the filtrated data of QSAM
measurements. The other one, without labels corresponds to calculated micro
acceleration using measurements of magnetic field. Both dependencies are in a good
agreement.

A similar procedure was applied for processing of the signals from the gyroscope.
The low frequency signals for angular rates estimated through the magnetic field and
QSAM are in a good agreement.

Using complicated spectral analysis of a ′(t ) and ω ′(t ) a few fundamental frequencies
have been found for each components of angular rate and for micro accelerations.
The frequencies and their amplitude obtained both using the result of QSAM during
run 18 and magnetic fields are given in the Table 8.2 and Table 8.3. All frequencies
in these Tables are enumerated, the number is written in the first column. By different
means 10 different frequencies were identified during the run 18: (0.18; 0.33; 0.397;
0.76; 2.235; 2.449; 2.629; 2.916; 3.00; 3.18)•10-3Hz. The amplitudes are given in the
next column after the frequency.

A similar table coming from a Fourier analysis performed at MRC, ULB (Brussels) is
shown in Table 8.4. The spectral analysis has been applied directly to the available
2.5*106 points on big computer without preliminary processing. Thus all frequencies
can be plotted. Only small frequencies are currently analysed.

As QSAM run 18 lasted during 5h, the filtrated data of the gyroscope had recorded
the low frequency, f ~0.18·10-3Hz, corresponding to the orbital period, see Table 8.2.
From the filtered data of the high frequency QSAM accelerometer this frequency is
not determined. For the higher frequencies, f >1.0·10-3Hz, gyroscope data of QSAM
and those of magnetic field are in a good agreement.
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                    of Microacceleration

Table 8.2. Frequencies fi (0.001Hz) and amplitudes Ai of components of
angular rates ωi and micro accelerations ai according to QSAM filtrated data.
Run 18: 22.09.99


          Data according to the QSAM gyroscope                 Data according to the QSAM high frequency
                      measurements                                   accelerometer measurements

           ωx
            ′                 ω′
                               y                 ωz
                                                  ′                   a′
                                                                       x                 a ′y                   a′
                                                                                                                 z


      f         A         f         A        f         A        f           A        f           A        f           A
1    0.181    0.107
2    0.331    0.137
3    0.397    0.280
4    0.761    0.108
5    2.242    0.014    2.235       2.004   2.234      2.485   2.234    21.465
6                                                             2.423        1.249   2.452        1.685   2.443        1.164
7    2.636    0.025    2.629       1.470   2.634      1.033   2.619        1.875   2.633        5.302   2.632        5.743
8    2.817    0.016                                                                2.816        5.938   2.815        5.561
9                      2.999       0.585   2.990      0.436                        3.002        3.025   2.997        3.297
10                                                                                 3.180        1.487   3.191        1.291
11                                                            3.384        1.191




Table 8.3. Frequencies fi(0.001Hz) and amplitudes Ai of components of angular
rates ωi and micro accelerations ai according calculated data through the
measurements of magnetic field. Data correspond to run 18.


     Results of frequency analysis of angular rates                 Results of frequency analysis of micro
                                                                                 accelerations

           ωx                 ωy                 ωz                   ax                 ay                     az
      f         A         f         A       f          A       f            A       f            A       f            A
5    2.233    0.189    2.234       2.240   2.234   2.441      2.234    23.009      2.232        1.381   2.234        1.249
6                                                             2.436        0.746   2.450        0.153
7    2.642    0.035    2.649       0.145   2.649   0.163      2.656        1.707   2.638        0.478   2.664        0.091
8    2.816    0.041                                                                2.816        0.661
9    2.995    0.029                                                                2.998        0.440
10                                                                                 3.182        0.131
11                                                            3.397        1.265
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                   of Microacceleration

Fourier analysis made in the MRC has found this frequency from the high frequency
QSAM accelerometer (Table 8.4) with rather large errors. Eventually, the values
0.191⋅10-3 Hz in angular rate and in acceleration 0.143⋅10-3 Hz were found. The
accuracy of Fourier spectrum is, ∆f ≈ 9.536⋅10-5 Hz. (For FOTON-11 the processing
QSAM data had been done with ∆f ≈ 7.5⋅10-5 Hz). Therefore the values 0.191⋅10-3 Hz
in angular rate and in accelerations 0.143⋅10-3 Hz may belong to the same
theoretically expected frequency f≈0.180⋅10-3 Hz.


Table 8.4. Frequencies fI (0.001Hz) and amplitudes Ai (0.001) of components of
angular rates ωi and micro accelerations ai according to QSAM run 18. (Fourier
analysis MRC).


           ωx                 ωy                 ωz                   ax                   ay                   az
№      f         A        f         A        f         A        f           A        f           A        f           A
     0.191    0.107                                           0.143                                     0.143±
 1     ±                                                        ±                                        0.045
     0.045                                                    0.045

 2   0.381      0.137

 3

 4

 5                      2.235      1.803   2.235      2.820                        2.235        0.077   2.235        1.005

 6                                                            2.646        0.105   2.646        0.107   2.646        0.077

 7                      2.670      0.724   2.670      0.809

 8                                                                                 2.789        0.077

 9                      3.051      0.159   3.051      0.157   3.004        0.040   3.004        0.028

10

11                                                                                                      0.429        0.143

12                                                                                                      0.572        0.150

Precautions! The amplitude of angular rates Aω can not be compared with an
amplitude of accelerations Aa. They are calculated using a different number of
points.
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   a'x




   a'y




   a'z




                                      time, t(103s)

Fig.8.10. Deviation of the components of the micro accelerations from the mean
values scaled by 10-6 m/s2. The lines with labels correspond to the filtrated data of
QSAM measurements. The solid lines without labels correspond to calculated micro
acceleration using measurements of magnetic field.
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9. Measurements of micro accelerations by TAS instrument

One of the accelerometers, the TAS instrument (Three-Axes Servo accelerometer
package) was placed inside FluidPac. The TAS contains three accelerometers
developed by the Swiss company CSEM. They allow the measurement of the
residual acceleration level in three orthogonal axes.

The TAS was programmed according to the requirements of the Principal
Investigators. For Magia, the PI (Prof. Schwabe) selected sequential acquisitions at
50Hz and 5Hz. For Tramp measurements the only sampling rate 5Hz was selected
(Prof. Gaeta). The TAS has worked from the first day of the mission 09.09.99 until
16.09.99, when FluidPac was shut down.

The description of the results obtained from onboard FluidPac sensors and analysis
of failures are discussed in detail in the Verhaert Company Report [RD1]. The overall
conclusion formulated in this report is that the technology of the CSEM sensor is
adequate for evaluating high frequency g-jitter but not for measuring the steady and
quasi steady residual accelerations in the micro-g range.

It means that one can trust only the data for frequencies larger than 0.2 Hz. At this
point it is reasonable to compare results of TAS with the Russian instrument Sinus-
12K.

To sum up, one can trust only to the values of the frequencies, data concerning the
amplitudes do not deserve much confidence.

Although, as it was presented in this report, the average residual acceleration
amplitude is of about |a| ≈ 4mg. It was determined by the averaging of the signals.
Possibly, for this case DC component was somehow subtracted.

These values of |a| cannot be confirmed or rejected. The present study shows that
the acceleration amplitude strongly depends upon the day of the mission, the
frequency range and the point, at which acceleration is measured. It is proved by
different means, S/C was turning around longitudinal axis and on 5th day of the flight
(14.09.99) the angular rate velocity was ωx ∼ 0.7deg/s. Although the FluidPac was
distant from centre of mass, the value |a| ≈ 4 mg is rather large, but it is close to the
values of the accelerations revealed by SINUS for the frequency range of 10-200Hz.

Eventually, the raw QSAM data from HFA show that the projections of the
acceleration vector have DC values a little bit higher than 2mg, see chapter 11. But
the values of the offsets, unknown for the authors of this study, were hidden in this
DC.
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Concerning results from Verhaert company Report [DR1] it has to be noticed, that
according to the authors of the report:

 Residual acceleration DC components cannot              be   evaluated   from   the
measurements.
 Precision of the results is rather poor: up to 10-4g.
 Only one (XTAS) sensor is reliable.

The direction of XTAS coincided with Z-Foton during TRAMP experiment. During
MAGIA the angle between XTAS and Z-Foton was equal to 135°.




  Fig.9.1. Power spectral density versus frequency on 12.09.99. (Courtesy VDD).


For the higher frequencies (f>0.2Hz) the results of the Fourier analysis of the TAS
data are quite reliable. These frequencies are caused by on-board operating
equipment. Positive results were obtained from the signal’s analysis: known
excitation frequencies of some payloads are clearly observed and can be correlated
to the operational phase of these instruments.

The main results of the report RD1, including frequency analysis, are given as
waterfalls. Typical waterfall presenting power spectral density versus frequency on
12.09.99 is shown in fig.9.1.
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 Title:        The Post Flight Study
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The spectral power density (G2/Hz) is shown on the plots of waterfalls. In order to
calculate the amplitude in G, it is necessary to know the width of the Kaiser-Bessel
sampling window. Tom Beuselinck (one of the authors of RD1) explained such
choice of results presentation in private discussions. The Kaiser Bessel window was
selected because literature identifies it as the best choice for spectral separation.
This means that a harmonic signal does not "leak" too much to nearby frequency
bands. The choice of a window does influence the amplitude; therefore only results
obtained using the same windowing-technique should be compared. It is not possible
to use no window. This corresponds to a rectangular flat window, which is
demonstrated in literature to be a very bad choice for this type of measurements.

Fourier transform generates a discrete spectrum, not a continuous one (although
often the result is shown as a continuous plot). The distance between two adjacent
points (spectral lines) on the frequency axis is the spectral resolution. It is identified
(by TAS, QSAM and Sinus) that IBIS centrifuge has caused vibrations with a
frequency 1.14Hz. Probably the frequency of this pure harmonic source will fall in
between two spectral lines. Therefore a 2µg signal at 1.14Hz might show up as 1.3µg
at 1Hz and 0.7µg at 1.5Hz (this assume a spectral resolution of 0.5Hz). When slightly
changing the spectral resolution (this is an immediate consequence of changing the
total acquisition duration), one might have a spectral line at 1.14Hz, in this case the
complete amplitude would show up at this frequency.

In order to avoid this artefact of accidental coinciding of sources with spectral line,
the authors have chosen to show the power spectral density (PSD). This is the
square of the amplitude divides by the spectral resolution. It is a measurement for the
energy in a single spectral line.

From the point of view of scientist this way to present the result is rather confusing.
For example, the set of fundamental frequencies was obtained from careful analysis
of waterfalls. The data is presented in Table.9.1 for the different axes of FOTON
centre of mass.


Table 9.1. The set of determined frequencies according the measurements by
TAS instrument in co-ordinate system of FOTON:


X-axis        2.0 •10-3 Hz 1.14Hz         3.14Hz                       9.8Hz 10.0Hz

Y-axis        2.0 •10-3 Hz 1.14Hz         3.14Hz         4.7Hz         9.8Hz

Z-axis        2.0 •10-3 Hz 1.14Hz         3.14Hz         4.7Hz         9.8Hz


To follow the variation of the value of the amplitude of these frequencies during the
flight is not so simple.
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Title:       The Post Flight Study
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      Table 9.2. The profile of frequencies from Table 9.1 and PSD of the
        X- projection in co-ordinate system of FOTON center of mass:


  Frequency, Hz   PSD, 10-8 • G2/Hz      Time of observation, [DD/MM;H:M:S]
  ≈ 1.14          ≈ 0.0                  [09/09; 22:36:27] - [14/09;15:43:27]
  ≈ 3.19          ≈ 1.9                  [11/09; 04:04:17] - [14/09;15:43:27]
  ≈ 9.52          ≈ 4.8                  [11/09; 04:04:17] - [12/09;22:39;36]
                  ≈ 1.9                  [12/09; 22:48:54] - [14/09;15:43:27]
  ≈ 10.0          ≈ 6.8                  [11/09; 04:04:17] - [12/09;20:07:12]
                  ≈10.2                  [12/09; 20:29:09] - [13/09;12:40:32]
                  ≈ 7.8                  [13/09; 12:51:22] - [14/09;10:12:59]
                  ≈ 10.9                 [14/09; 10:42:03] - [14/09;15:43:27]
  0.002           ≈ 90.3                 [11/09; 03:45:53] - [14/09;08:03:04]
                  ≈ 13.0                 [14/09; 09:25:22] - [14/09;15:11:43]



      Table 9.3. The profile of frequencies from Table 9.1 and PSD of the
        Y- projection in co-ordinate system of FOTON center of mass:


 Frequency, Hz    PSD, 10-8 • G2/Hz      Time of observation, [DD/MM;H:M:S]
 1.14             ≈ 1.0                  [09/09;22:36:27] - [11/09;04:04:17]
                  ≈ 4.8                  [11/09;14:38:55] - [11/09;22:07:58]
                  ≈ 0.6                  [11/09;22:17:36] - [12/09;13:24:41]
                  ≈ 4.8                  [12/09;13:44:13] - [12/09;22:39:36]
                  ≈ 7.8 - 9.6            [12/09;22:48:54] - [13/09;09:52:26]
                  ≈ 2.0                  [13/09;10:07:55] - [14/09;04:17:04]
                                         [14/09;14:33:52] - [14/09;15:43:27]
                  ≈ 3.2 - 9.6
                                         [14/09;22:07:36] - [16/09;08:57:31]
                  ≈ 4.0 - 32.5
                                         [16/09;10:24:42] - [16/09;20:04:12]
                  ≈ 0.0
 3.14             ≈ 5.3 - 7.8            [09/09;22:36:27] - [14/09;15:43:27]
 4.7              ≈ 5.2 - 9.6            [09/09;22:36:27] - [14/09;15:43:27]
 9.8              ≈ 1.0 - 1.4            [09/09;22:36:27] - [14/09;15:43:27]
 0.002            ≈ 0.0                  [09/09;22:20:51] - [12/09;07:05:59]
                  ≈ 0 - 9.6              [12/09;07:25:30] - [13/09;06:44:29]
                  ≈ 0.0                  [13/09;06:57:55] - [14/09;15:11:43]
                  ≈ 0 - 4.0              [14/09;20:24:28] - [16/09:20:04:12]
 Reference:    ESA Contract Report
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 Date:
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 Title:        The Post Flight Study
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       Table 9.4. The profile of frequencies from Table 9.1 and PSD of the
         Z- projection in co-ordinate system of FOTON center of mass:

   Frequency, Hz       PSD, 10-8 • G2/Hz     Time of observation, [DD/MM;H:M:S]
       1.14                   ≈ 4.0           [09/09;22:36:27] - [10/09;03:00:29]
                          ≈ 16.0 - 19.4       [11/09;04:04:17] - [11/09;21:44:42]
                          ≈ 9.0 - 11.6        [11/09;22:36:51] - [12/09;11:47:14]
                             ≈ 10.9           [12/09;12:02:43] - [12/09;22:28:27]
                          ≈ 16.0 - 19.4       [12/09;22:48:54] - [13/09;07:08:23]
                          ≈ 37.2 - 56.2       [13/09;07:27:46] - [13/09;23:59:25]
                                              [14/09;00:18:25] - [14/09;10:57:31]
                          ≈ 16.0 - 19.4
                                              [14/09;11:27:07] - [14/09;15:43:27]
                          ≈ 1.0 - 23.04
                                              [14/09;20:24:28] - [15/09;03:18:44]
                              ≈ 2.0           [15/09;04:32:42] - [16/09;06:45:10]
                              ≈ 1.4
       3.14               ≈ 5.8 - 13.7         [09/09;22:36:27] - [14/09;15:43:27]
        4.7               ≈ 5.8 - 13.7         [09/09;22:36:27] - [14/09;15:43:27]
        9.8                ≈ 2.0 - 4.0         [09/09;22:36:27] - [14/09;15:43:27]
       0.002                  ≈ 0.0            [09/09;22:20:51] - [12/09;05:07:09]
                           ≈ 1.0 - 9.6         [12/09;05:20:34] - [16/09;20:04:12]

In the Tables 9.2, 9.3, 9.4 the Power Spectrum Density and distribution of defined
above frequencies by the date and by projections on co-ordinate system of FOTON
center of mass are listed. All the data presented above were subtracted from the
waterfall plots of the Report RD1.

The value of the power spectrum density (PSD) depends upon sampling and the
amount of points, which is expressed as coefficient measured in units Hz. As a rule
in the Tables 9.2, 9.3, 9.4 the amplitude of two successive values of the same
frequency, e.g. 1,14Hz, cannot be compared without complicated processing, as they
are multiplied by 2 different coefficients.

Summarising the discussion written above one may draw a conclusion that different
ways of presentation of the spectral results has negative and positive points. The
power spectral density gives more precise results, but for non-specialist it is not
simple to substrate the level of micro accelerations. Or PSD has to be calculated for
the same sampling and the same amount of points. The power spectrum, usually
used by scientist, is more visual, but not so accurate.

As the main goal of present study is to clarify the level of quasi-steady micro
acceleration, e.g. low frequency accelerations, the data of TAS will not be used for
                                       r
the elaboration of empirical formula a (t ) and for discussion of the S/C orbital
position.
 Reference:    ESA Contract Report
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 Date:
 Page:         Page 56 of 135
 Title:        The Post Flight Study
               of Microacceleration



10.    Measurements of micro accelerations by SINUS instrument

The CSRI Elektropribor, St. Petersburg, designs Russian accelerometers named
Sinus. According to the opinion of independent Russian specialists, SINUS is
comparable with the instruments, which are known to be the leaders: SAMS
designed by Glenn Research Centre (NASA), BETA designed by CNES, QSAM
designed by DLR.

It is assessed by the large potential of the accelerometers MSTA and ESTA, (basic
parts of SINUS) developed in the CSRI Elektropribor exclusively for measurements
of space acceleration. Accelerometers do not belong to the main business of this
company, which deals with the design of wide class of super accurate devices.

The accelerometer MSTA (magnetic spherical triaxial accelerometer) is an
instrument with a contactless magnetic suspension of a levitating spherical rotor. The
characteristics of the accelerometer may vary in a wide range due to the possibility to
change standard electronic components only. The accelerometer ESTA
(electrostatic spherical triaxial accelerometer) presents a spherical rotor that levitates
in the electrical field of the suspension. ESTA has capabilities to reach a sensitivity of
up to 10-10g.

The system Sinus-6K was first used on board of FOTON-11. The next modification
Sinus-12 was designed for FOTON-12. The main peculiarities of SINUS-12K as
compared with SINUS-6 are the increase of the number of measuring channels from
6 to 12, improvement of the sensitivity to 10-7g and extension of the time period over
which it is possible to measure the micro acceleration from 32 to 400 hours [RD10]

The drawback of the available results is that all of them are averaged during a period
of 300s. The results for low frequency are missing.

Another negative point is that the contact with CSRI Elektropribor is extremely
difficult. In reality it was possible to receive some detailed data only after an official
request by TsKB signed by Deputy General Director G. Anshakov.

The designers declared that measurements of accelerations have been done during
the 12 runs, which are listed in Table10.1, in the three frequency ranges of 0 -0.1 Hz,
0.2-10 Hz and 10-300 Hz.

Unfortunately it is impossible to receive results in the low frequency range, except
those shown in fig.10.1, fig.10.2. The projections of quasi-steady acceleration on
different axes are given for the 7th run (19.09.99-20.09.99) for the frequency range 0-
0.1Hz at two different points. The averaged amplitude of the projections of
accelerations were plotted versus Moscow time and they do not exceed 5⋅10-6m/s 2.
These amplitudes look a bit low, as it corresponds to the value of |a| acceleration in
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 Title:         The Post Flight Study
                of Microacceleration

the centre of mass. Comparison of the accelerations in the centre of mass and 1.5m
away according to the SINUS-12K results, shown in figs .7.8-7.16, reveals that they
are similar. It can be the truth only in the case when this remote point is located on
the symmetry axis. Unfortunately, the authors of present report did not succeed to
get information allowing to identify which points belong to the c.o.m. and which ones
belong to the remote point. Without notations it is difficult to say which one of the 2
curves belongs to the |a| in the centre of mass.


          Table 10.1 Time schedule of SINUS-12K on-board FOTON-12


   Operation         Moscow winter time                  UTC             Duration

                     Days           Hours      Days         Hours
    Run 1 on        09.09.99       22:39:26   09.09.99     18:39:26
    Run 1 off       10.09.99       03:40:40   10.09.99     22:40:40      5 hours
    Run 2 on                       17:09:41                12:09:41
    Run 2 off       13.09.99       20:11:01   13.09.99     14:11:01      3 hours
    Run 3 on        13.09.99       20:59:38                15:59:38
    Run 3 off       14.09.99       01:59:49   13.09.99     20:59:49      5 hours
    Run 4 on                       12:11:33                07:00:00
    Run 4 off       15.09.99       15:12:53   15.09.99     10:00:00      3 hours
    Run 5 on                       07:59:49                03:59:49
    Run 5 off       16.09.99       11:00:37   16.09.99     06:00:37      3 hours
    Run 6 on                       11:20:00                06:20:00
    Run 6 off       19.09.99       14:20:00   19.09.99     10:20:00      3 hours
    Run 7 on        19.09.99       22:40:00                17:40:00
    Run 7 off       20.09.99       03:40:00   19.09.99     22:40:00      5 hours
    Run 8 on                       08:40:00                03:40:00
    Run 8 off       20.09.99       14:40:00   20.09.99     09:40:00      3 hours
    Run 9 on                       00:30:00                19:30:00
    Run 9 off       21.09.99       02:30:00   20.09.99     21:30:00      2 hours
   Run 10 on        21.09.99       23:50:00                18:50:00
   Run 10 off       22.09.99       01:50:00   21.09.99     20:50:00      2 hours
   Run 11 on                       15:10:00                10:10:00
   Run 11 off       22.09.99       20:10:00   22.09.99     15:10:00      5 hours
   Run 12 on        23.09.99       08:40:00   23.09.99     03:40:00
   Run 12 off       24.09.99       11:10:00   24.09.99     06:10:00      27 hours
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 Title:        The Post Flight Study
               of Microacceleration


The results of the measurements of micro acceleration made by SINUS-12K are
presented in a different way in comparison with the results from TAS instrument.
Basically the amplitudes are supplied by CSRI Elektropribor as the mean square root
values for a fixed frequency interval. The physical meaning of this value can be
described in the following way.

The mean power of a periodic signal of the type
                                         N −1
                                                                                           1
                              A(t ) = ∑ B ( k ) exp( 2πkf 0 t ) , here f0=
                                         k =0                                              T
over the time interval T can be written as
                                                                T
                                                             1
                                                             T∫
                                                A2 ( t ) =       A2 ( t ) dt ,
                                                               0

Here A2 ( t ) is equivalent to the mean power of a signal A (t). The Parseval theorem
says, that the mean power value is equal to the sum of the power of its harmonics
                                           1 N −1
                                 A2 ( t ) = ∑ B 2 ( k )
                                           2 k =0

The square root of mean power calls mean square root value of А(t) and it is a
measure of the amplitude of complex type oscillations. Below the simple notation
−
A(t ) will be used for this parameter:
                                          −                               1 N −1 2
                                         A(t ) =        A 2 (t ) =          ∑ B (k )
                                                                          2 k =0

The output signal of the system at each used channel includes the following
components:
                        50                                           60
                A(t) = ∑B(k1 ⋅ f01,t) exp(2π ⋅ k1 ⋅ f01 ⋅ t) + ∑B(k2 ⋅ f02,t) exp(2π ⋅ k2 ⋅ f02 ⋅ t) ,
                       k1=1                                         k 2=2



where f 01 = 0 ,2 Hz, f 02 = 5 Hz.

The first term is a component of a signal, defined by the spectral components in the
interval of frequencies 0.2-10Hz, the second one is corresponding to the spectral
components in the interval of frequencies from 10 Hz to 300 Hz.

Then mean square root value of the acceleration defined by the spectral components
in the frequency interval 0.2-300 Hz is calculated as,


                                              50                               60
                              −                     1                                  1
                                           ∑ 2 B (0,2k 1, t )                 ∑ 2 B (5k 2, t )
                                                                      2                          2
                              A( t ) =                                    +                          ,
                                           k 1= 1                             k 2= 2
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 Title:       The Post Flight Study
              of Microacceleration


And the mean square root value of acceleration defined by the different spectral
components in the frequency interval 0.2-10 Hz is equal

                                 −           50   1                2
                               A1 (t ) =     ∑      B ( 0 ,2k 1,t ) ,
                                            k 1=1 2


And respectively for the frequency range of 10 – 300 Hz it is:

                                                         60
                                              −                 1
                                                        ∑ 2 B(5k 2, t )
                                                                          2
                                            A2 (t ) =                         .
                                                        k 2=2




   Table 10.2. Mean square root values А(t) for different frequency intervals

                                                  −                       −
                                                A1 (t )                   A 2 (t )
                                           f ∼0.2 – 10 Hz           f ∼10 – 300 Hz

                 Run 1                1.3• 10-5 g0                  2.5• 10-5 g0

                 Run 2                1.4• 10-5 g0                  2.0• 10-5 g0

                 Run 3                1.6• 10-5 g0                  2.0• 10-5 g0

                 Run 4                1.3• 10-5 g0                  3.0• 10-5 g0

                 Run 5                1.4• 10-5 g0                  1.9• 10-5 g0

                 Run 7                1.4• 10-5 g0                  2.3• 10-5 g0

                 Run 8                1.4• 10-5 g0                  2.5• 10-5 g0

                 Run 9                1.4• 10-5 g0                  3.8• 10-5 g0

                 Run 10               1.5• 10-5 g0                  3.8• 10-5 g0

                 Run 11               1.6• 10-5 g0                  3.0• 10-5 g0

                 Run 12               1.7• 10-5 g0                  2.5• 10-5 g0
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               of Microacceleration

                −          −
The values of A1 (t ) and A2 ( t ) for different runs are shown in Table10. 2.

According to the results summarised in Table 10.2 the mean spectral power was
about 1.4⋅10-5 g0 throughout the flight. As for the amplitudes of projections of
acceleration, they were for the frequency range 0-0.2Hz around 5.0⋅10-5 g0 and one
order of the magnitude higher for the frequency range 10-300Hz, 2.0⋅10-4 g0

To compare the results by SINUS-12K with the results by TAS and QSAM the values
of the projections of acceleration were requested at different days of the flight,
13.09.99 and 22.09.99. Waterfalls for the frequency range of 0.2-10 Hz have been
drawn on the basement of the obtained results. They are shown in fig. 10.3-10.8.
Amplitudes are given at the same units as in table 10.2, e.g. in g0.

     Table10.3. Amplitudes of the projections of the acceleration vector for
                          frequency range 0.2-10Hz

              run          f            ax              ay              az

                          Hz           10-5g           10-5g          10-5g
               1       0.2 - 6         4-8            1–5             2–5
                       6.2 -10         6 - 10         6–8             6 - 10
               2       0.2 - 6        6 – 10          1–4             2–5
                       6.2 -10        10 - 14         5 - 10          5 - 10
               3       0.2 - 6         4–8            1–5             2–5
                       6.2 -10        6 - 10          6-8             6 - 10
               4       0.2 - 6         1–8            1–5             1–6
                       6.2 -10        8 - 10          6-8             6 - 10
               5       0.2 - 6         1–6            1–8              1 –8
                       6.2 -10        6 - 12          1-8             1-8
               7       0.2 - 6        2 - 10           2-8             1-8
               8       0.2 - 8         2–5             1 -8           1–5
                       8.2 -10        6 - 10                          6 - 10
               9       0.2 - 10        2–9            1 - 14          1 - 10
              10       0.2 - 8         2 –8            2 –8            2 –6
                       8.2 -10         8 -10          8 -10           6 -10
              11       0.2 - 8         5-8            1-10             1-8
                       8.2 -10        8 - 12

The amplitudes of the projections of the acceleration vector for frequency range 0.2-
10Hz are presented in Table 10.3. As it follows from this table, the µg-level for this
frequency range has order of magnitude 10-5g for the f∼0.2-6Hz, and for the
frequencies 6-10Hz it is higher ∼10-4g.
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 Title:          The Post Flight Study
                 of Microacceleration


     Table10.4. Amplitudes of the projections of the acceleration vector for
     frequency range 10-300Hz. The high peaks are placed in separate line.

                        ax                         ay                         az


   run         f, Hz         ax ⋅10-4g    f, Hz         ay ⋅10-4g    f, Hz         az ⋅10-4g

               10-300        0.5-1.5     10-300          0.5-2      10-300         0.6-0.8
    1            75           30-32                                   75              4
              200-220          15        200-245         10-30       200             15

               10-300        0.5-1.5      10-300          2-5        10-300          1-2
    2         200-220         30-40      200-240         10-30      200-235          10
                240            10

               10-300          1-3       10-300           2-5        10-300           1-3
    3            75           1-16.5                                   75            2-43
              200-235         10-40      200-235         10-30      200-235           10

               10-300          1–3        10-300          1-3        10-300         1–3
                 75           1-16.5        75                         75           1-45
    4         200-235         10-38      200-235         10-28      200-235          10
                                                                    265-270          15
               10-300         1–5         10-300          1-5        10-300         1 –4
    5         200-235         10-35      200-240         10-25      200-235          10


               10-300          1-5        10-300          1-6        10-300          1–6
    7         200-235         10-45      200-235         10-32      200-235         10 -14


               10-300          1-5        10-300          1-5        10-300         1–5
    8         200-235         10-28      200-235         10-28      200-235         8-13


               10-300          1-5        10-300          1-5        10-300          1–5
    9         200-235         10-45      200-235         10-30      200-235         10 -30


               10-300          1-5        10-300          1-5        10-300         1–5
    10        200-235          5-90      200-235         10-68      200-235         5-25


              10-300          36-38      10-300            1-4      10-300            1-5
    11          75                       130-135          8-40        75            0.5-18
                                         200-210            7



The relative values of amplitudes of micro accelerations for very large frequency
range are written in Table 10.4 for all available runs of SINUS-12K. In average, the
order of magnitude of µg-level is about 10-4g. But for a few discrete frequencies it
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 Title:       The Post Flight Study
              of Microacceleration

achieve 10-3g and even higher. This dependency during one run is shown in the
fig.10.9. One may see that frequency f∼200Hz is the most pronounced one.




Fig. 10.1. Results of measurements of quasi-steady accelerations at point P1 of
FOTON- 12 by SINIUS-12K in 7th run (19.09.99-20.09.99)
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 Title:       The Post Flight Study
              of Microacceleration




Fig. 10.2. Results of measurements of quasi-steady accelerations at point P2 of
FOTON-12 by SINIUS-12K in 7th run (19.09.99-20.09.99)
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                of Microacceleration




 22:04:38




                                                                         10.0
            21:04:33
                       0.0


Fig. 10.3. FOTON-12 mission, 13.09.99: Waterfall plots for x-projection of
acceleration according to the SINUS-12K accelerometer data. The plot begins from
frequency 0.2Hz.
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                 of Microacceleration




   22:04:38




                                                                         10.0
              21:04:33
                         0.0



Fig. 10.4. FOTON-12 mission, 13.09.99: Waterfall plots for y-projection of
acceleration according to the SINUS-12K accelerometer data. The plot begins from
frequency 0.2Hz.
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     22:04:3




                                                                          10.0
        21:04:33
                    0.0



Fig. 10.5. FOTON-12 mission, 13.09.99: Waterfall plots for z-projection of
acceleration according to the SINUS-12K accelerometer data. The plot begins from
frequency 0.2Hz.
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 Title:        The Post Flight Study
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    16:19:41




                                                                           10.0
              15:14:34
                          0.0



Fig. 10.6. FOTON-12 mission, 22.09.99: Waterfall plots for x-projection of
acceleration according to the SINUS-12K accelerometer data. The plot begins from
frequency 0.2Hz.
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                 of Microacceleration




    16:19:41




                                                                          10.0
              15:14:34
                            0.0



Fig. 10.7. FOTON-12 mission, 22.09.99: Waterfall plots for y-projection of
acceleration according to the SINUS-12K accelerometer data. The plot begins from
frequency 0.2Hz.
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                 of Microacceleration




      16:19:41




                                                                           10.0
              15:14:34
                            0.0



Fig. 10.8. FOTON-12 mission, 22.09.99: Waterfall plots for z-projection of
acceleration according to the SINUS-12K accelerometer data. The plot begins from
frequency 0.2Hz.
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Title:       The Post Flight Study
             of Microacceleration




Fig. 10.9. Dependence of the amplitudes of micro acceleration on frequency
measured by SINUS-12K during FOTON-12 mission. Courtesy Agarkov V.F.TsSKB,
Samara.
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 Title:       The Post Flight Study
              of Microacceleration



11. Analysis of data from QSAM and comparison with the other
   instruments

The QSAM (Quasi-Steady Acceleration Measurement System) facility has measured
Microgravity quality and disturbances during several flight periods. DLR specialists
developed the facility. Installation and flight of QSAM were managed by Kayser-
Threde in co-operation with KBOM, Moscow.

The QSAM instrument includes 3 accelerometers: two for the low frequencies and
one for the high frequency and a gyroscope for the measurements of angular
velocities. The co-ordinates of the low frequency accelerometers and gyroscope
inside FOTON-12 correspond to the Module 2 in Table 4.1. The co-ordinates of the
high frequency accelerometer are corresponding to the co-ordinates of sensor INUK
in Table 4.1.

The authors of this report would like to thank the colleagues from KBOM (Moscow),
and personally Mr. A. Egorov, who gave them an opportunity to use the raw data of
QSAM facility recorded on CD-ROM.


11.1 Measurements by the high frequency accelerometer

The data from the low frequency accelerometers were useless for processing. QSAM
sensors were out of their working range (10-4g) after 500 -1000s from the switching
on due to the drift of DC component. A typical set of signals from the low frequency
accelerometer is shown in fig.11.1.

To subtract information about the S/C motions all research works were done with
high frequency and gyroscope data, using different methods of processing. The time
dependence of the projections of the acceleration on FOTON axes within run 10,
when TRAMP was working, and at the end of the flight, run 18, are presented in
figs.11.2 - 11.7.

For comparison the projections of the acceleration vector on x-axis for the runs 10
and 18 are given on the same page. Due to the shortage of information about offset
magnitude, the mean value of a x ∼ 2.15mg in the plot cannot be considered as
realistic level of micro acceleration, e.g. as DC components. Actual DC value could
be smaller or larger. In the present case only the amplitudes of accelerations around
the mean value can be taken into account.
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Title:         The Post Flight Study
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             Table 11.1 Time schedule of QSAM on-board of FOTON-12


              Operation                      UTC             MET            Duration


                                      Days         Hours    Hours

    Launch                      09:09:99       18:00:00
    Separation of capsule                      18:09:00     0:00:00
    Stabilization off                          19:38:45     1:29:45
    Run 1 on                    09.09.99       19:39:35     1:30:00
    Run 1 off                   10.09.99       00:39:35     6:30:00    5 hours
    Run 2 on                                   18:31:00    96:22:00
    Run 2 off                   13.09.99       20:11:0     98:02:00    1h 40min
    Run 3 on                                   21:31:00    99:22:00
    Run 3 off                   13.09.99       23:11:00    101:02:00   1h 40min
    Run 4 on                    13.09.99       23:51:00    101:42:00
    Run 4 off                   14.09.99       01:31:00    103:22:00   1h 40min
    Run 5 on                                   05:01:00    106:52:00
    Run 5 off                   14.09.99       06:41:00    108:32:00   1h 40min
    Run 6 on                                   07:21:00    109:12:00
    Run 6 off                   14.09.99       09:01:00    110:52:00   1h 40min
    Run 7 on                                   13:31:00    115:22:00
    Run 7 off                   14.09.99       15:11:00    117:02:00   1h 40min
    Run 8 on                                   17:01:00    118:52:00
    Run 8 off                   14.09.99       18:41:00    120:32:00   1h 40min
    Run 9 on                                   19:31:00    121:22:00
    Run 9 off                   14.09.99       21:11:00    123:02:00   1h 40min
    Run 10 on                   14.09.99       23:31:00    125:22:00
    Run 10 off                  15.09.99       01:11:00    127:02:00   1h 40min
    Run 11 on                                  02:31:00    128:22:00
    Run 11 off                  15.09.99       04:11:00    130:02:00   1h 40min
    Run 12 on                                  05:31:00    131:22:00
    Run 12 off                  15.09.99       07:11:00    133:02:00   1h 40min
    Run 13 on                                  10:31:00    136:22:00
    Run 13 off                  15.09.99       12:11:00    138:02:00   1h 40min
    Run 14 on                                  02:35:00    152:26:00
    Run 14 off                  16.09.99       04:05:00    154:06:00   1h 40min
    Run 15 on                                  18:35:00    192:26:00
    Run 15 off                  17.09.99       20:15:00    194:06:00   1h 40min
    Run 16 on                                  09:05:00    230:56:00
    Run 16 off                  19.09.99       10:45:00    232:36:00   1h 40min
    Run 17 on                                  17:04:00    238:55:00
    Run 17 off                  19.09.99       18:44:00    240:35:00   1h 40min
    Run 18 on                                  11:50:20    305:41:34
    Run 18 off                  22.09.99       16:50:20    310:41:34   5h
    Run 19 on                                  09:04:30    326:55:44
    Run 19 off                  23.09.99       14:04:30    331:55:44   5h

    landing                     24:09:00       09:18:00    351:09:14
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As a general tendency one may notify that for raw signals the mean value a i and the
amplitudes of oscillations Ai are similar for all runs. They have magnitude about a x ∼
2.15mg, a y ∼ -1.22mg, a z ∼ 2.26mg and Ax ∼ 0.4mg, A y ∼ 0.5mg, Az ∼ 0.9mg
(according to the run 18). On the fine time scale the difference in signals structure at
different runs will be visible.

The most indicative difference between runs 10 and 18 is the existence of discrete
peaks of the amplitude, existing in run 10, but not in run 18, (compare the fig10.2-
10.7at the same pages). The amplitudes of these peaks for x- and z- projections
exceed the amplitude of the oscillations, ∆Ax ∼ 0.5mg, ∆Ay ∼ 0.4mg, ∆Az ∼ 0.9mg.

Possibly, the operation of some on-board equipment caused the splashes of
amplitude. It is known that the IBIS facility induced the frequency about 1.14Hz.
Pronounced influence on the micro acceleration level was caused by Polizon facility.
But during run 10, POLIZON did not yet begin experiments. Some other on-board
equipment has to be responsible of these disturbances.

The strongest influence of Polizon operation was exhibited in run 14 and 17. The
switch-on of this facility results in splashes and in the increase of the amplitude of the
oscillations. The time dependencies of |ax| and |az| during run 14 are shown in
fig.10.8-10.9. The increase of the amplitude of the oscillation was caused by the
displacement of the Polizon body. The fundamental frequency of the induced
vibrations was 7.7 Hz. During the run 17 the change of the amplitude had even
stronger behaviour. During this run a fundamental frequency 12 Hz was induced by
displacement of the Polizon body. It is interesting to note, that the experimental
container moved with low velocity of V=15mm/h. The details of operation of Polizon
facility one may find in [RD2, RD3, RD12].

Figs.11.10-11.12 give other views of micro accelerations from figs.11.2-11.7. Unlike
to disorderly looking time signals in fig.11.2-11.7 if the plots are in the form ai vs. aj (i,
j = {x, y, z}) they have a well-ordered structure, although all plots consist of an
enormous amount of points grouped on a finite surface. If we would connect them
successively, the pictures would look as some shaded surfaces. The transition from
one point to the next one in time does not follow lines parallel to either x or y-axes.
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             of Microacceleration




                                                                   Time [s]




                                     a [µg]
Fig.11.1. The data from QSAM low frequency accelerometer during 11th run. Due to
      the DC drift after 500s the measurements are out of its working range.
                             Courtesy KBOM, Moscow
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 Title:       The Post Flight Study
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 ax [mg]




Fig.11.2. Projection of the acceleration on x-axis of FOTON vs. time. The signal was
recorded by high frequency QSAM sensor during 10th run. The instant t = 0
corresponds to 23:31:00 UTC 14.09.1999. (The time interval corresponds to the
carrying out of TRAMP experiment)
 ax [mg]




Fig.11.3. Projection of the acceleration on x-axis of FOTON vs. time. The signal was
recorded by high frequency QSAM sensor during 18th run. The instant t = 0
corresponds to 11:50:00 UTC 22.09.1999. (This run has one of the longest duration –
5 hours)
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                of Microacceleration

 ay [mg]




Fig.11.4. Projection of the acceleration on y-axis of FOTON vs. time. The signal was
recorded by high frequency QSAM sensor during 10th run. The instant t = 0
corresponds to 23:31:00 UTC 14.09.1999. (The time interval corresponds to the
carrying out of TRAMP experiment)
ay [mg]




Fig.11.5. Projection of the acceleration on y-axis of FOTON vs. time. The signal was
recorded by high frequency QSAM sensor during 18th run. The instant t = 0
corresponds to 11:50:00 UTC 22.09.1999. (This run has one of the longest duration –
5 hours)
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   Title:       The Post Flight Study
                of Microacceleration
az [mg]




Fig.11.6. Projection of the acceleration on z-axis of FOTON vs. time. The signal was
recorded by high frequency QSAM sensor during 10th run. The instant t = 0
corresponds to 23:31:00 UTC 14.09.1999. (The time interval corresponds to the
carrying out of TRAMP experiment)
     az [mg]




Fig.11.7. Projection of the acceleration on z-axis of FOTON vs. time. The signal was
recorded by high frequency QSAM sensor during 18th run. The instant t = 0
corresponds to 11:50:00 UTC 22.09.1999. (This run has the longest duration – 5
hours)
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                  of Microacceleration
ax [mg]




  Fig.11.8. Projection of the acceleration on x-axis of FOTON vs. time. The signal was
  recorded by high frequency QSAM sensor during 14th run. The instant t = 0
  corresponds to 02:35:00 UTC 16.09.1999. The increase of the amplitude of |ax| is
  caused by operation of the facility Polizon.
   az [mg]




  Fig.11.9. Projection of the acceleration on z-axis of FOTON vs. time. The signal was
  recorded by high frequency QSAM sensor during 14th run. The instant t = 0
  corresponds to 02:35:00 UTC 16.09.1999. The disturbance of the amplitude of |az| is
  caused by operation of the facility Polizon.
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   Date:
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                of Microacceleration
ay [mg]




                                        az [mg]


 Fig.11.10. Projection of ay vs. az; another view of the micro accelerations, shown in
 fig.11.5 and 11.7. The signal was recorded by high frequency QSAM sensor during
 18th run. The instant t = 0 corresponds to 11:50:00 UTC 22.09.1999. (This run has
 the longest duration – 5 hours)
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    ax [mg]




                                             az [mg]

Fig.11.11. Projection of ax vs. az; another view of the micro accelerations, shown in
fig.11.3 and fig.11.7. The signal was recorded by high frequency QSAM sensor
during 18th run. The instant t = 0 corresponds to 11:50:00 UTC 22.09.1999. (This
run has the longest duration – 5 hours)
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       ax [mg]




                                             ay [mg]

Fig.11.12. Projection of ax vs. ay; another view of the micro accelerations, shown in
fig.11.3 and 11.5. The signal was recorded by high frequency QSAM sensor during
18th run. The instant t = 0 corresponds to 11:50:00 UTC 22.09.1999. (This run has
the longest duration – 5 hours)
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11.2 Measurements by QSAM gyroscope

As it was shown in previous chapter, the high frequency accelerometers signals are
rather complicated and difficult to analyse. The gyroscopes' signals are more simple
and regular; therefore they are mainly used for the different studies of FOTON-12
motion. Besides, the QSAM gyroscope recorded more or less similar signals during
both the FOTON-11 and FOTON-12 missions. It gave the idea that one may trust to
the gyroscopes' data.

The projections of the angular rates on the axes of FOTON-12 versus time are
shown in figs.11.13-11.18 for the runs 10 and 18. The angular velocities oscillate
around some non-zero value, which may be connected with its real mean value. As
offsets of ωi are unknown it is impossible to draw a conclusion from the data on the
plots either the mean value of angular rates changes with time and what is their
orders of the magnitude. Fortunately, the magnitude of these mean values can be
evaluated by non-direct methods.

Some information about the frequencies describing the motion of the S/C can be
obtained directly from the plots. Except for some high frequency oscillations non-
distinguishable in the plots, the angular rate ωx in longitudinal direction of S/C, see
fig.11.13, performs some oscillations with a period Π10∼51min during run 10. At the
run 18 this large-period (large-scale) oscillations became non-regular, see fig.11.14,
but still the rough estimation of the period gives the close value Π18∼43min. The
orders of magnitude of Π10 and of Π18 are close to half of the orbital period.

The angular rates in perpendicular directions ωy and ωz, shown for run 10 in fig.11.15
and 11.16, perform the oscillations with a period around 10 min besides the large-
period oscillations. The large-scale period oscillations between ωy and ωz are shifted
by value about of π. Comparing fig.11.15 and 11.16 one can see that the maximum
of ωz corresponds to the minimum of ωy.

The phase shift between ωy and ωz is accurately visible on longer time interval during
run 18, see fig.11.17 and 11.18. For this last day of the mission the period of large-
scale oscillations is about of Π18 ∼ 42-43 min, and the period of small-scale
oscillations is about 7min. The values of the frequencies are not used intentionally at
these rough estimations, as they may be found more precisely by Fourier analysis.

The angular rates in fig.11.19-11.21 are plotted in the form ωi vs. ωj (i, j = {x, y, z}).
The plots, including component ωi are similar to those for accelerations. They have
well-ordered structure. But the dependency ωy upon ωz looks chaotic, see fig.11.20.

QSAM specialists, ref. [RD11], have made Fourier analysis of the gyroscope data
shown above to display the low fundamental frequencies describing the motions of
FOTON-12. Results of their processing's are shown in fig. 11.22-11.24 with accuracy
∆f = 0.56⋅10-4Hz. The fundamental low frequency f0 was increasing from f0 ≈ 1.38⋅10-3
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Hz at the 4th day of the flight up to f0 ≈ 2.25⋅10-3Hz at the end of the mission. There is
no one frequency with distinguishable amplitude in the x-direction of FOTON (axis of
symmetry). Theoretical calculations using the magnetic field data displayed a few
frequencies of relatively small amplitudes, where the fundamental frequency f0 has
the largest amplitude, see Table 8.3.

The first point to note is that during a few runs, beginning from run 4, the second
frequency, close to f0, exists in spectrum. It is seen on y- and z- projections in fig.
11.23 and 11.24.That new frequency cannot be harmonic of f0, it means that there
are at least two different types of motions: vibrations or/and oscillations. Further, in
run 8 on the z-projection, a third frequency appeared in spectrum.

The second point to remark is the appearance of 2 small peaks at the beginning of 1st
and 18th runs. As they are the only runs, which were lasted 5 hours, more then 3
orbital turns, the low frequency f ∼10-4Hz, which corresponds to the orbital period T≈
90 min, has appeared in spectrum. Moreover, there are 2 frequencies with the order
of magnitude ∼ 10-4Hz.

The independent spectral analysis of the QSAM gyroscope and of the high frequency
accelerometer (HFA) data has been performed at the Keldysh Institute of Applied
Math (Moscow) using another method, ref. [RD9]. Results below concern only the
low frequencies. The data for the different runs: 9, 15,16,18 are presented in the
Tables 11.2; 11.3; 11.4; and 11.5, respectively.

To simplify the comparison between the data obtained from gyroscope and HFA, the
numbers of non-zero frequencies are written in the first column. The next column
after the value of frequency shows the value of the amplitudes. All the data are
obtained by the same method, allowing the comparison of the amplitudes of the
different frequencies.

For the fundamental frequency f0 with the largest amplitude, the spectral data from
the gyroscope are in good agreement with those of DLR, e.g. on run 18 the value
from Table 11.5 is f0 ≈ 2.25⋅10-3Hz to be compared with f0 ≈ 2.25⋅10-3Hz for DLR
data. Again, in a good agreement both gyroscope and DLR did not find this
frequency with pronounced amplitude in spectrum of Ωx.

But, this frequency f0 is always present in any spectrum of ax measured from HFA,
where it has the largest amplitude and f0 do not so pronounced in the spectrum of ay
and az. Thus, there is a discrepancy between spectral results of gyroscope and HFA.
Moreover, the spectral results based on the measurements of magnetic field are in a
good agreement with the spectral results of gyroscope; this is visible by comparing
the Table 8.2 and 8.3. It means, that the results of HFA are not really reliable for the
determination of the quasi-steady residual accelerations in the micro-g range.

Unlike to HFA, QSAM gyroscope data look reliable and they will be used further for
the development of the theoretical model.
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    Table 11.2. Frequencies fi (0.001Hz) and amplitudes Ai of components of
    angular rates ωi and micro accelerations ai according to QSAM filtrated data.
    Run 9: 14.09.99; 19:31:00-21:11:00


        Data according to the QSAM gyroscope                 Data according to the QSAM high frequency
                   measurements                                    accelerometer measurements

            Ω ′x               Ω ′y               Ω ′z               a′
                                                                      x               a ′y           a′
                                                                                                      z

№       f          A       f          A       f          A     f           A      f          A   f         A
1    0.227 0.237
2                                                            0.351        3.375
3    0.612 0.134
4                                                     1.160 5.181 1.284                                   4.039
5                1.612 1.828 1.616 2.620 1.619 15.122 1.637 6.117 1.747                                   7.609
6    1.859 0.045 1.838 1.864 1.830 0.874 1.841 3.809 1.864 7.073
7                                                                 1.973                                   8.671
8                2.204 0.545 2.235 0.279                          2.257                                   3.373
9                                        2.433 2.010 2.389 4.276 2.397                                    2.415




    Table 11.3. Frequencies fi (0.001Hz) and amplitudes Ai of components of
    angular rates ωi and micro accelerations ai according to QSAM filtrated data.
    Run 15: 17.09.99; 18:35:00-20:15:00



        Data according to the QSAM gyroscope                 Data according to the QSAM high frequency
                   measurements                                    accelerometer measurements

            Ω ′x               Ω ′y               Ω ′z               a′
                                                                      x               a ′y           a′
                                                                                                      z

№   λ     A     λ     A     λ     A     λ     A      λ     A     λ     A
1 0.383 0.271
2 0.705 0.154
4 2.058 0.022 2.026 2.439 2.028 2.844 2.030 23.138
5             2.385 1.690 2.388 1.045 2.354 2.691 2.360 6.196 2.309 4.426
6                                                  2.563 6.569 2.558 6.872
7             2.756 0.596 2.748 0.390 2.751 1.139 2.776 3.237 2.781 3.021
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Table 11.4. Frequencies fi (0.001Hz) and amplitudes Ai of components of
angular rates ωi and micro accelerations ai according to QSAM filtrated data.
Run 16: 19.09.99; 09:05:00-10:45:00


      Data according to the QSAM gyroscope                  Data according to the QSAM high frequency
                  measurements                                    accelerometer measurements

           Ω ′x                Ω ′y              Ω ′z               a′
                                                                     x                   a ′y                   a′
                                                                                                                 z

№      f          A        f          A      f          A     f            A       f             A        f           A
1    0.337    0.208
2    0.708    0.085
3                        2.184    1.792   2.167     1.897   2.172    16.541
4    2.510    0.029      2.499    1.286   2.510     0.844   2.503    2.120       2.496          4.634   2.531        4.873
5    2.700    0.015                                                              2.677          4.824   2.677        5.474
6                        2.871    0.484   2.860     0.337                        2.819          3.753   2.872        3.707
7                                                           3.097        1.575




Table 11.5. Frequencies fi (0.001Hz) and amplitudes Ai of components of
angular rates ωi and micro accelerations ai according to QSAM filtrated data.
Run 18: 29.09.99; 11:50:20-16:50:20



      Data according to the QSAM gyroscope                  Data according to the QSAM high frequency
                  measurements                                    accelerometer measurements

           Ω ′x                Ω ′y              Ω ′z               a′
                                                                     x                 a ′y                     a′
                                                                                                                 z

      f           A        f          A      f          A    f            A       f              A       f            A
1    0.181    0.107
2    0.331    0.137
3    0.397    0.280
4    0.761    0.108
5    2.242    0.014      2.235    2.004   2.234     2.485   2.234    21.465
6                                                           2.423     1.249      2.452          1.685   2.443        1.164
7    2.636    0.025      2.629    1.470   2.634     1.033   2.619     1.875      2.633          5.302   2.632        5.743
8    2.817    0.016                                                              2.816          5.938   2.815        5.561
9                        2.999    0.585   2.990     0.436                        3.002          3.025   2.997        3.297
10                                                                               3.180          1.487   3.191        1.291
11                                                          3.384        1.191
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    ωx [degrees/s]




Fig.11.13. Projection of the angular rate on x-axis of FOTON ωx vs. time. The signal
was recorded by QSAM gyroscope during 10th run. The instant t = 0 corresponds to
23:31:00 UTC 14.09.1999. (During this time interval TRAMP experiment was carried
out)
     ωx [degrees/s]




Fig.11.14. Projection of the angular rate on x-axis of FOTON ωx vs. time. The signal
was recorded by QSAM gyroscope during 18th run. The instant t = 0 corresponds to
11:50:00 UTC 22.09.1999. (This run has the longest duration - 5h)
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        ωy [degrees/s]




Fig.11.15. Projection of the angular rate on y-axis of FOTON ωy vs. time. The signal
was recorded by QSAM gyroscope during 10th run. The instant t = 0 corresponds to
23:31:00 UTC 14.09.1999. (During this time interval TRAMP experiment was carried
out)
     ωz [degrees/s]




Fig.11.16. Projection of the angular rate on z-axis of FOTON ωz vs. time. The signal
was recorded by QSAM gyroscope during 10th run. The instant t = 0 corresponds to
23:31:00 UTC 14.09.1999. (During this time interval TRAMP experiment was carried
out)
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                                     of Microacceleration




                    ωy [degrees/s]




Fig.11.17. Projection of the angular rate on y-axis of FOTON ωy vs. time. The signal
was recorded by QSAM gyroscope during 18th run. The instant t = 0 corresponds to
11:50:00 UTC 22.09.1999. (This run has the longest duration - 5h)
   ωz [degrees/s]




Fig.11.18. Projection of the angular rate on z-axis of FOTON ωz vs. time. The signal
was recorded by QSAM gyroscope during 18th run. The instant t = 0 corresponds to
11:50:00 UTC 22.09.1999. (This run has the longest duration - 5h)
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                     ωx [degrees/s]




                                                 ωy [degrees/s]

Fig.11.19. Projections ωx vs. ωy;. The signal was recorded by QSAM gyroscope, 18th
run.
              ωy [degrees/s]




                                                  ωz [degrees/s]
Fig.11.20 Projections of ωy vs. ωz; The signal was recorded by QSAM gyroscope,
18th run.
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 ωx [degrees/s]




                                         ωz [degrees/s]

Fig.11.21 Projections ωx vs. ωz; another view of the angular rate, shown in fig.11.13
and 11.18. The signal was recorded by QSAM gyroscope, 18th run.
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Fig. 11.22. Spectrums of the signals measured by QSAM Z-gyroscope at different
                    runs. It corresponds to X-axis of FOTON.
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Fig. 11.23. Spectrums of the signals measured by QSAM Y-gyroscope at different
                    runs. It corresponds to Y-axis of FOTON.
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Fig. 11.24. Spectrums of the signals measured by QSAM X-gyroscope at different
                    runs. It corresponds to Z-axis of FOTON.
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11.3 Rotation of the S/C FOTON-12

The angular rate ωx of the rotation of FOTON-12 around its symmetry axis at the
beginning of the mission was as slow as 0.03 degree/s. The theoretical results of
Keldysh Institute of Applied Mathematics (Moscow, Prof. Sazonov) obtained on the
basis of the measurements of the magnetic field have demonstrated the spinning of
Foton-12 along the longitudinal axis, see chapter 8.1. The increase of this rate during
the mission was shown in fig.8.1-.8.3. At the beginning of the flight, the mean value
of the ωx angular rate increases by small jumps each 90min, passing the perigee.
The most obvious proof of this jumping behaviour can be seen in fig.8.2-a. After a
few days these jumps became too small to be easily distinguished. Except that, the
angular velocity ωx performs almost regular oscillations in time, but their amplitude is
rather small in comparison with the mean value.

The change of spinning velocity ωx of the S/C with time throughout the mission is
shown in fig.11.25 and fig.11.26. Due to oscillatory behaviour of the ωx these plots
present the variation with time of the mean value of the angular velocity during some
time interval, ω x (t ) .

The curve in fig.11.25 was obtained via engineering method to process the QSAM
gyroscope data in KBOM, Moscow. Both solid line and diamantes in fig.11.26 were
obtained via powerful numerical calculations at the Keldysh Institute of Applied
Mathematics. The solid line was calculated using the results of Mirage facility
(measurements of the magnetic field) and the diamantes co-ordinate were calculated
using QSAM gyroscope data. The co-ordinates of the diamantes are written in Table
11.6.

The good agreement between solid line and diamantes in the plot 11.26, underlines
again the fact that the results recorded by Mirage and QSAM gyroscope are reliable.
Therefore, the gyroscope data will be used for elaboration of the empirical formula by
the authors of this report.

The behavior of the curves in fig.11.25 and fig.11.26 exhibits sufficiently strong
increasing of the S/C rotation around the symmetry axis during the first 4 days.
Eventually, something happened on-board of Foton-12 on the 5th day, when the
angular velocity jumped up, see fig.11.25. A tiny jump exists also in fig.11.26 at the
same time. It is visible with a scale magnification. It is difficult to say, on which plot
the value of the jump is correct. The two solid curves present the results of averaging
during some short time period with respect to the duration of the mission. The
magnitude of the value ω x (t ) depends on the number of the perigees, which have
been passed on the time interval under processing. For fig.11.26 the typical time
period is 1h40 min (run duration), one or two perigees can be passed.
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The stars in the plots correspond to the theoretical model-1, which will be described
further.

All results merge to the fact that to the velocity of the S/C rotation around the
symmetry axis increases from almost zero value up to 1 degree/s at the end of the
mission.




         Table 11.6. Variations of angular velocity ωx during the mission

              Date           Time         MET               ωx
              (IX.99)       (UTC)       (QSAM)

                 13            18:21     96.22             0.654
                 13            21:21     99.22             0.669
                 13            23:41    101.42             0.672
                 14            04:51    106.52             0.696
                 14            07:11    109.12             0.694
                 14            13:21    115.22             0.729
                 14            17:25    118.52             0.731
                 14            19:21    121.22             0.743
                 14            23:21    125.22             0.759
                 15            03:21    12.8.22            0.780
                 15            05:21    131.22             0.782
                 15            10:21    136.22             0.798
                 16            02:25    152.26             0.865
                 17            18:25    192.26             0.938
                 19            08:55    320.56             1.008
                 19            16:54    238.55             1.008
                 22            11:50    305.41             1.004
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ωx [degrees/s]




Fig.11.25. The angular velocity of the S/C rotation around own axis throughout the
mission FOTON-12. The solid line is obtained by the processing QSAM gyroscope
data. (Courtesy of KBOM, Moscow)
       ωx [degrees/s]




Fig.11.26. The angular velocity of the S/C rotation around own axis throughout the
mission FOTON-12. The solid line is theoretically obtained using the measurements
of magnetic field, the diamonds correspond to the theoretical results using QSAM
gyroscope data. (Courtesy of Prof. Sazonov, Moscow)
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11.4 Analysis of DC component and high frequencies


11.4.1 Comparison of the QSAM, TAS, SINUS power spectrum signals in
       frequency range 0.2- 10Hz

The results, described above in section 8, deal with a quasi-steady motion of S/C. At
the Keldysh Institute of Applied Mathematics (Prof. Sazonov) the input signals have
been filtered below the level f < 0. 2Hz. Due to their method the higher frequencies
were cut off and they cannot appear in the final results. As the Inst. of Appl. Math is
the official partner of the FOTON Design Bureau for this question, this information
appeared in the official reports issued by TsSKB. But it does not mean that these
high frequencies did not exist during the FOTON-12 flight.

Moreover, no one of the accelerometers present on-board of FOTON-12 (TAS,
SINUS and QSAM) did not provided reliable data for the measurements of steady
and quasi-steady micro acceleration. As a result, the high frequencies were indeed
measured more exactly than low ones.

Table 11.7. Comparison of the results of spectral analysis in the frequency
range 0.2-10Hz from different accelerometers available on-board of FOTON-12


          Num             TAS, 12-14.09.99        SINUS, run 3       QSAM, run 3

        Spectral          X,        Y,    Z,     X,     Y,   Z,    X,     Y,     Z,
       maximums           Hz        Hz    Hz     Hz     Hz   Hz    Hz     Hz     Hz

              1          1.14      1.14   1.14   1.2   1.2   1.2   1.14   1.14   1.14

              2                                        2.0   2.0

              3          3.14      3.14   3.14                     2.93   2.85   2.95

              4                    4.7    4.7    4.2   4.6   4.6   4.37

              5                                        7.0   7.0   7.70   7.70   7.39

              6          9.8       9.8    9.8    9.8   9.6   9.8          9.89   9.18




The current study has been done to compare the performance of these instruments
in higher frequency range. There is only one interval of time, during which the data
from all accelerometers are available and be compared. TAS data were subtracted
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from the VDD report [RD1], SINUS data were received from CSRI Elektropribor, St.
Petersburg and the power spectrums of QSAM data were calculated at MRC. As the
three devices recorded the data more or less at the same time, one expects the
coincidence of their power spectrums. The Table 11.7 illustrates this fact with several
exceptions.

SINUS accelerometer reveals the signal with main frequency 2Hz and the two other
accelerometers do not. Also the TAS data do not reveal frequency around 7Hz. It
seems that 3 payloads were operating on-board at that time generated 3 distinct
frequencies: 1.14-1.2 Hz, 4.2-4.7 Hz and 9.6-9.9 Hz. All of them were discovered by
the three accelerometers. The origin of the first of them is known: the operation of the
IBIS facility.

The slight difference between the three accelerometers data (1.14 vs. 2 Hz) may be
explained via the different number of points taken to perform the Fourier analyses
and as a consequence of that the different spectral resolution. For the Fourier
analysis of QSAM data of the 3rd run we took 524288 points. To compare, for TAS
data frequency analysis usually 8192 points were taken. There is no information
about processing the SINUS data

Comparison of the measured amplitudes even for high frequencies is not so nice. For
frequency f=1.14 Hz SINUS and QSAM displayed 25 µg and 31 µg, while TAS gave
2µg.


11.4.2 On the DC and high frequency components of micro acceleration

Concerning the DC component of a signal, it is worth mentioning that it was almost
always discarded in data processing. It is treated as an offset, although it is not clear.
The authors of this report did not succeed to obtain the true values of the offsets for
any of instruments on-board of FOTON-12. The only information, published currently
on this subject is that the data of the BETA accelerometer (CNES) from FOTON-11
mission reveal a very large DC component in the sensors' signals. The raw data from
the BETA accelerometer were shown together with the low frequency approximating
signal in the report on the FOTON-11 mission [RD5]. It was also mentioned in the
report of KBOM [RD12] that QSAM instruments have rather large values of offset on
FOTON-11.

The present treatment on the raw QSAM data from CD-ROM for FOTON-12 mission
discovers the large DC values for all components of acceleration vector. One can
clearly see (fig.11.2-11.9) that DC components of the signal, i.e. the averaged in time
raw data, is of the order of 2⋅10-3 m/s2, while the DC of the filtrated oscillations (low
pass filter f=10Hz) is ≈ 0.6⋅10-3 m/s2, see [RD11]. At the same time, presented in the
figures amplitude, filtered on the low frequencies (low pass filter f=0.1Hz), is ≈3⋅10-5
m/s2, where DC component is discarded.
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Of course, we are not sure whether it is the offset or what its value is, since it is not
MRC who developed the accelerometers. But modeling of the real satellite motions
result in the values of residual acceleration 10-100 times larger than the ones
reported before. Just to estimate, it is known that own angular rate of FOTON-12
during the 18th QSAM run was about ω x≈ 1.0 degrees/s. The accelerometer was
placed on the distance of about r≈300mm from the symmetry axis. A simple
estimation of the acceleration in this point gives the value of micro acceleration due
to the own S/C rotation:

a= ω2r ≈ 9.1⋅10-5 m/s2 ≈ 10-5g0

At the same time, QSAM accelerometer at its 18th run gives just a ≈ 2.7⋅10-5 m/s2.
The values have the same order of magnitude, but the real motion of FOTON-12 was
much more complicated than just its own rotation. This fact is confirmed by the data
of the three QSAM gyroscopes. It means, that after taking into account all the factors,
one will receive larger value of acceleration. But QSAM data are suffering of an offset
that authors of the report cannot evaluate.

The motion of the satellite itself is characterized with just low frequencies (90 minutes
orbital motion and approximately 6 minutes rotation around its symmetry axis). After
filtering the signal of accelerometer at low frequencies, it allows the researchers
having dealt with the raw data to conclude that the real satellite motion generates
rather small residual acceleration on board. It is true, but it is not the complete picture
of what really might take place. As there are a lot of facilities on FOTON satellite
generating high frequencies vibrations with noticeable amplitudes and even low
frequencies. To complete the describing of what was going on during the FOTON-12
flight it is better not to drop these components out.


   Table 11.8 Mean spectral acceleration on board of FOTON-12 obtained via
                    QSAM measurements for z-component

       Frequency, HZ            1.14    7.5    18.5    33.0   37.0    51.0    66.0
       Amplitude, µg            31.0    2.0     1.6     4.0   17.0    52.0     5.0

Table 11.8 shows high frequencies observed in some runs and their amplitudes. The
data were taken from the materials of the presentation made by DLR [RD11].

Actually, it is of importance for those who intend to carry out experiments on board of
a spacecraft to know the actual level of residual gravity and not only low frequencies
components. Even if the latter have "non-regular" nature as due to on board
payloads and their locations may be unique for each flight.
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11.4.3 Influence of high frequency components of angular rate on acceleration
High frequency acceleration ranges from ≈0.01 through ≈300 Hz. It is associated with
the operation of the satellite (on-board disturbers such as machinery, devices,
ventilators, experiment racks etc.).

Due to time dependence of angular rates (precession and oscillations) the resulting
formula for acceleration has their time derivatives (12.11)-(12.13), and high
frequency oscillations may have remarkable influence on oscillations even if their
amplitudes are relatively small. Suppose, there are two peaks in the spectrum of
angular rate:

Ω = A1sin(ω1t) + A2sin(ω2t)
dΩ/dt = A1ω1cos(ω1t) + A2ω2cos(ω2t).




 Fig. 12.27: Power spectrum of Z'-projection of angular rate in the 18th run of QSAM.

So, even if A1 >> A2 the second signal cannot be neglected as products A1ω1 and
A2ω2 may be of the same order of magnitude. To be more precise, let us have a look
at the power spectrum of angular rate for the RUN-18 (fig. 12.9).

Consider two frequencies in the signal of Z'-projection of the angular rate: the main
frequency ω1 = 0.0138, A1 ≈ 2⋅10-3, and the second one ω2 = 18.165, A2 ≈ 8⋅10-7.
Then A1ω1/A2ω2 ≈ 1.9. As one can see the input of the high frequency component of
angular rate is just approximately 2 times smaller than of the main frequency,
although A1/A2 ≈ 2500. Possible responses of the system to the oscillations with
different frequencies may be completely different but in the considered case the
second frequency generates high frequency noise in the acceleration signal with
relatively large amplitude (just 2 times smaller).
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12.   THEORETICAL MODELS DESCRIBING FOTON-12 MOTION.

It was shown in previous chapters that the data only from two on-board instruments
could be considered as reliable: Mirage and QSAM gyroscope. They will be used for
the developments of the theoretical models.

After analysis of the numerous papers and plots the authors draw a conclusion: two
models elaborated.

I. The first one was developed having the knowledge that the motion of Foton-12 was
close to the regular precession. The complete model has been created in two steps
using QSAM gyroscope data:
    1. The angular rates in the FOTON co-ordinates system analytically calculated
        taking into account the rotation of S/C around the symmetry axis and its
        precession. The obtained projections of the angular rate were re-calculated
        onto QSAM gyroscopes co-ordinates system. The proposed motion could be
        considered as correct if the angular rate projections fit to the QSAM measured
        signals. It was not the case at the first step.
    2. In addition to the rotation and precession, the pendulum oscillations of the
        space vehicle have been introduced. The resulting attitude appears to be a
        combination of three regular motions: rotation around its symmetry axis,
        precession and oscillations. The angular rates obtained according this model
        fit quite well to the QSAM gyroscope data.

The resulting acceleration field is calculated as a result of the rotational motions and
oscillations of the S/C. The rotation of S/C around the Earth is not taken into account
in the final empirical formula for the acceleration.

II. The second one is based on the results of strict mathematical consideration. The
governing system of equations, taking into account gravitational and aerodynamic
momentum, is numerically solved. The parameters of the problem were adjusted
through the use of the results of the measurements of the magnetic field (Mirage
facility). The final model of the S/C motion have been constructed in MRC, ULB,
although in order to understand and to visualize the physical motion of the S/C
coming from the calculations performed at the Keldysh Institute of Applied
Mathematics (Prof. Sazonov).

To validate the both models, numerical simulations of the TRAMP experiment were
carried out.

Actually these two models are not very different. Looking ahead, the short physical
description of FOTON-12 will be given right now. The S/C motion in orbital co-
ordinate system according the first model is shown in fig.12.1 and for the second one
in the fig. 12.2.
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      Fig.12.1. The sketch of the Foton-12 motion according to the first model.




Fig.12.2. The sketch of the Foton-12 motion according to the second model.
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The centre of mass of FOTON, point O, is the origin of the orbital co-ordinate system
OX1X2X3. Point C is located at the centre of mass of the Earth. The temporal location
of the symmetry axis of FOTON is shown as Ox.

According to the first model, fig.12.1, the S/C rotates around Ox1, follow the
precession around a vector of the kinetic momentum K, and performs 8-shaped
oscillations within the precession radius. The direction of the vector K is constant in
time, but the angle between Ox and K is changing.

According the second model, except rotation around the longitudinal axis, Ox, the
vehicle performs a precession around kinetic momentum K. Unlike to the previous
model the vector K also performs small non-symmetrical motions, but the angle
between Ox and K is practically constant.

There is one more essential difference between these two models. The first model is
empirical one, it is developed using only system co-ordinate of FOTON and does not
have any preferable orientation in the orbital plane. The orbital motion is not taken
into account. Therefore it can perform precession near the axis OX2 or OX3.. But as
we are confident to the second model, by analogy the same axis has been chosen.

Below, justifications of both models are considered in details.


12.1 First theoretical model


12.1.1 Empirical formula for acceleration due to own rotation and precession

As a first step, the possible motions of a satellite will be modeled under assumptions
that the S/C rotates around its symmetry axis and performs regular precession.

The solid body performs regular precession only when the resulting momentum of all
external forces is equal to zero and the body has axially symmetrical shape. With
some tolerance FOTON-12 can be considered as an axially symmetrical body. It is
supposed to be known that the first condition with some accuracy is fulfilled at the
second half of FOTON-12 mission.

It is assumed that the S/C is rotating around its symmetry axis with an angular rate
denoted by Ω0 and is inclined on some angle θ with respect to the axis of precession.
To notice, if this angle is zero no precession will take place. Currently it is suggested
that angle θ is constant and non-zero, θ≠0, and also Ω0= const and Ωp= const.
Hereinafter Ωp means the angular rate of precession. A point P inside the satellite
(fig.12.3) may be associated with a position of a sensor. The goal of this section is to
write down the formula for the angular rates at the point P and then calculate
acceleration.
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At first, let us introduce two co-ordinate systems OXYZ and O'X'Y'Z' built in the
following manner (Fig. 12.3):

OXYZ: O is the center of mass of FOTON, X-axis is parallel to the vector of angular
rate of precession Ωp. YZ-plane is perpendicular to X-axis. At the very beginning of
the observations (t=0) the angle β is equal to zero.

O'X'Y'Z': O' is the cross point of the symmetry axis of FOTON and transversal plane
where the considered point P is situated, O' belongs to the symmetry axis of FOTON.

X'-axis is parallel to the vector of angular rate of the S/C rotation around its symmetry
axis Ω0. Z'-axis is directed along the radius-vector r=O'P of the P point and Y'-axis is
perpendicular to the X'Z'-plane.




                          Fig 12.3. Local co-ordinate systems
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Velocity in the P point can be written as:

          (             ) (        )
   v P = Ω 0 + Ω p × R0 + r = Ω p × R0 + Ω p × r + Ω 0 × r ,                    (12.1)
where ( Ω 0 × R0 = 0 ).

According to a definition, acceleration in the P point is


                                                       aP = d vP        .
                                                                   dt

Taking into account that FOTON satellite is a rigid and non-deformable body (R0 and
r do not depend upon time) the acceleration will be written as:

               [           ]           [       ]
a P = Ω p × Ω p × R0 + Ω p × Ω p × r + Ω p × Ω 0 × r −     [        ]             (12.2)
      [            ]           [
− r × Ω p × Ω0 + Ω0 × Ω p × r + Ω0 × Ω0 × r]       [           ]
where r and R0 are respectively the radius-vectors of the P point in O'X'Y'Z' and of
the O' one in OXYZ co-ordinate systems.

It is useful to write down the formula of acceleration namely in the O'X'Y'Z' co-
ordinates system in order to be connected to the data of accelerometers. Then, in the
O'X'Y'Z' co-ordinate system:
Ω p =Ω p ( − sin θ ⋅cos β ,sin θ ⋅sin β ,cosθ ) ,
R0 = R0 (0,0,1) ,
r = r (1,0,0) ,
Ω 0 = Ω 0 (0,0,1) .

Here β = (Ω0 - Ωp⋅cosθ)⋅ t is a linear function of time, as the angular rates Ω0, Ωp ⋅are
constant. R0 is the distance between O and O' points.

After having opened double vector products and using the relations
Ω p ⋅ R0 = Ω p R0 cos θ ,
Ω p ⋅ r = −Ω p r sin θ cos β ,
Ω p ⋅ Ω 0 = Ω p Ω 0 cos θ ,
Ω0 ⋅ r = 0 ,

one can write down the projections of the acceleration vector, eq.(12.2), at the point
P in the O'X'Y'Z' co-ordinate system. Hereinafter, the sub-indices 1, 2 and 3 read for
Z', Y' and X' projections respectively.
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       a P = (a z ' ,a y ' ,a x ' ) ,
       a z ' = − (Ω 2 R0 sin θ cos θ cos β + Ω 2 r (1 − sin 2 θ cos 2 β ) + Ω 0 r + 2Ω p Ω 0 r cos θ ) , (12.3)
                    p                          p
                                                                              2


       a y ' = Ω 2 (R0 cos θ − r sin θ cos β ) sin θ sin β ,
                 p                                                                                      (12.4)
       a z ' = − (Ω R0 sin θ + Ω r sin θ cos θ cos β + 2Ω p Ω 0 r sin θ cos β ) .
                       2
                       p
                                   2          2
                                              p                                                         (12.5)

     The angular rate Ω = (ω z ' ,ω y ' ,ω x ' ) itself in the O'X'Y'Z' co-ordinate system is:

      ω z ' = −Ω p sin θ cos β ,                                                                        (12.6)
      ω y ' = Ω p sin θ sin β ,                                                                         (12.7)
      ω x ' = Ω 0 + Ω p cos θ .                                                                         (12.8)

     For the considered regular precession Ωpsinθ is constant. It means that ωz' and ωy'
     from eqs.(12.6)-(12.7) have to be the pure periodical functions. Comparing the
     angular rates from eqs.(12.6)-(12.7) with the corresponding experimental data plots
     one can easily make a remark that besides regular precession there was something
     else that took place during the FOTON-12 mission.


                                                          (a)                                     (b)
ωy' [degrees/s]




                                        ωz' [degrees/s]                        ωz' [degrees/s]


     Fig.12.4 Theoretical (a) and experimental (b) dependencies ωy' vs. ωz'. The theoretical
     picture corresponds to the Euler regular precession and experimental one
     corresponds to the QSAM gyroscope data during 18th run.

     Theoretical dependence ωy' vs. ωz' resulting from eqs.(12.6)-(12.7) is shown in
     fig.12.4-a when Ωp = 1 degree/s. On the 12.4-b the same dependence is built on the
     basis of the QSAM gyroscope measurements. So, the real motion of FOTON-12
     could not be described only as rotation and precession. Some other motions were
     present.
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12.1.2 Relation between angular velocities of precession and own rotation

The relation between the angular rates could be easily obtained for the case of
regular precession, see [RD14].

Axially symmetrical body has the minimal momentum of inertia (I) along the
symmetry axis (x'). In the co-ordinate system, introduced in the fig.12.3, the
projection of a kinetic momentum K on the x'-axis in the case of a regular precession
can be written as
                               K x' = K cos θ = Ωx' I x'


where Ωx' is the angular velocity of rotation around own axis x’, Ωx' =Ω0.

Angular velocity of precession Ωpr can be found from other projection of a momentum
of impulse M z' = Ωp sin θ then

                      Ωz' =M z' /I z' = M sin θ /I z' and Ωp = M /Iz'

The ratio between velocity of the rotation and of the precession can be written as

                                Ω0 / Ωp = ( Iz' / I x' ) cos θ

According to the data in Chapter 4 with accuracy of 10% the momentums of inertia
are:
Iz' ≈ Iy' ≈13 548 kg⋅m2, and I x' ≈3 124 kg⋅ m2, then Iz' / I x' ≈ 4.337

                        Ω0 / Ωp = ( Iz' / I x' ) cos θ = 4.337 cos θ

The refined version of the ratio of momentums of inertia is given in [RD8]. As
FOTON-12 was not absolutely symmetrical, then it was précised that Iz' / Iy' = 1.072,
then

                       Ω0 / Ωp =     ( Iy' / I x' ) cos θ = 4.045 cos θ

This relation can work as independent test for validation of the predicted satellite
motions.
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12.1.3 A formula of acceleration on board of FOTON-12 due to its own rotation,
       precession and oscillations

Now let us assume that besides rotating with angular rate Ω0 = const around its
symmetry axis and precessing the FOTON is oscillating with respect to the axis OX
of precession. In comparison with the previously considered rotation-precession
situation there is one more angular rate vector Ωs. The vector Ωs is not constant but
rotating with Ωp rate and its absolute value is a sine function of time. In the OXYZ co-
ordinate system:

 Ω s = Ω 0 (sin(ω s t ) sin(Ω p t ),− sin(ω s t ) cos(Ω p t ),0) .
         s


In this case the point O' in will move by 8-shaped trajectory, see fig.12.3 and fig.12.1.
The vector acceleration a P = (a z ' , a y ' , a x ' ) components in the OX'Y'Z' co-ordinate
system will be written as:

             ∂Ω p            ∂Ω s
a z' =                            R0 cos β −
                      R0 sin θ sin β −
       ∂t                     ∂t
(2Ω p R0 cosθ − 2Ω p r sin θ cos β − Ω s r sin β )Ω s sin β −                              (12.9)
(Ω   p   R0 cos θ − Ω p r sin θ cos β )Ω p sin θ cos β − (Ω + Ω + Ω + 2Ω p Ω 0 cos θ )r
                                                                       2
                                                                       0
                                                                             2
                                                                             p
                                                                                  2
                                                                                  s



             ∂Ω p                                        ∂Ω s
a y' =                (R0 sin θ cos β + r cosθ ) +            R0 sin β −
                 ∂t                                       ∂t
(2Ω R cosθ − 2Ω r sin θ cos β − Ω r sin β )Ω
         p       0              p                    s            s   cos β + ,           (12.10)
(Ω R cosθ − Ω r sin θ cos β )Ω sin θ sin β
     p       0              p                p



                 ∂Ω p          ∂Ω s
a x' = −                             r cos β − (Ω 2 + Ω 2 )R0
                        r sin θ sin β +           p     s
          ∂t                    ∂t
− 2(Ω p r sin θ cos β + Ω s r sin β )Ω 0 +                    .                           (12.11)
(Ω   p   R0 cos θ − Ω p r sin θ cos β )Ω p cos θ

and the components of the vector of angular rate Ω = (ω z ' ,ω y ' ,ω x ' ) are:

ω z ' = −Ω p sin θ cos β − Ω s sin β ,                                                    (12.12)
ω y ' = Ω p sin θ sin β − Ω s cos β ,                                                     (12.13)
ω x ' = Ω 0 + Ω p cos θ .                                                                 (12.14)

Formula for Ωp is given below (eq.(12.19)).
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Here, in the above formulas, Ωs, Ωp and θ are functions of time:

Ω s (t ) = Ω 0 cos(ω s t ) , θ (t ) = θ 0 sin(ω s t ) , Ω 0 = nω sθ 0 ,
             s                                            s                          (12.15)

n is a coefficient, which is not equal to unity, as it should be. Its value will be chosen
for the best fit of the angular rates (12.12)-(12.14) to the QSAM data. This is one of
the points to be argued at.

The present goal is to work out an empirical formula, which can fit to the real data.
Also it does not take into account non-linear factors as well as high frequency
oscillations.

The value of Ω0s in eq.(12.15) is determined as following. By definition the angular
rate of oscillations Ωs = dθ / dt. Then, Ωs = θ0ωs cos ωs t and Ω0s = θ0 ωs

Looking at the QSAM 18th run angular rate data plots (fig.11.14, fig.11.17 and
fig.11.18) and compared them to the equations (12.12)-(12.14), one could make the
following conclusions:

         There are oscillations with two distinguishable frequencies present in the
angular rate plots. Oscillations of ωy and ωz with smaller frequency have some phase
shift (fig. 12.5). Namely, when amplitude is maximal for Z' projection it is minimal for
Y' projection at the same time and vice versa. It may be stated looking at the
equations (12.11)-(12.13) for angular rate that the small frequency corresponds to
                          t
β (t ) = β (t = 0) + ∫ (Ω 0 − Ω p )dt                                               (12.16)
                          0

and not to the angle θ. The latter is in the sine function in the eqs.(12.12)-(12.13) and
gives no phase shift between the two projections. Oscillations with the larger
frequency correspond to a pendulum like motion (oscillations of the angle θ).
Estimations of the QSAM 18th run data give the frequencies:

         f1≈ 3.9 ⋅10-4Hz, that corresponds to angular rate 0.1395 degrees/s; or to the
         period Π ≈ 43 min.
         f2≈ 2.22⋅10-3Hz, that corresponds to angular rate 0.802 degrees/s; or to the
         period Π ≈ 7.5 min.

Then, the difference of angular rates, which enter to the eq.(12.16) is:

Ω0 - Ω0p ≈ 360⋅f1 ≈ 0.1395 [degrees/s]                                              (12.17)

             1 / f1

Ω = f1         ∫Ω
   0
   p                  p   dt is   a characteristic, mean value of Ωp (see the of Ωp below, eq.
               0
(12.19).
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        To obtain the values of Ω0 and Ω0p it takes one more equation. Let us suggest
that relative variation of Ωp is small in comparison with its mean value. So with good
accuracy, for the QSAM 18th run (equation (12.13) and fig.11.14) the second relation
is:

Ω0 + Ω0p ≈ 2.090 [degrees/s]                                                (12.18)


Solution of the system of equations (12.16) and (12.17) gives

        Ω0 ≈ 1.115 [degrees/s],          Ω0p ≈ 0.975 [degrees/s],

As for the time dependencies in eq.(12.15), θ = θ0sin(2π⋅f2⋅t) = θ0sin(0.014t) [degrees]
and Ω0s ≈ nθ0⋅0.014 [degrees/s], where n≠1.

       For the RUN-18, θ0 = 25 [degrees] and n= 0.6 lead to the best fit of the model
to the experimental data.

       One can see on the plot fig.(11.14) for the angular rate projection on the S/C
symmetry axis that it is oscillating with the frequency 3.2 ⋅10-4Hz (own rotation minus
precession). Their amplitude may be estimated as 0.035 [degrees/s]. Keeping in
mind data from QSAM gyroscope and eq.(12.14) equation for the precession can be
written as:

      0               2π
      Ω p + 0.035 sin     (Ω 0 − Ω 0p )t  
                                           
                      360                
Ωp =                                                                        (12.19)
                     cos θ


        The angular rate's projections on the S/C cross section (Y'Z' plane) do not
oscillate around zero, as it ought to be according to the formulas for the angular rate,
see fig.12.5. This may be considered in three ways:

  There is one more rotation with constant angular rate;
  They are offsets of the gyroscopes;
• The gyroscopes were not properly oriented with respect to Y'- and Z'-axis of
FOTON. In this case Ω0 and Ωp projections on Y' and Z' have non-zero mean value.

For the current study it was decided to treat the non-zero mean values as some
offsets of gyroscopes, as for the RUN-10 they have the same values.

For the RUN-10, when the TRAMP experiment was carried out, the following
parameters have been obtained:
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       f1≈ 3.3 ⋅10-4Hz, that corresponds to angular rate 0.119 degrees/s; or to the
       period Π ≈ 51.0 min.
       f2≈ 1.72⋅10-3Hz, that corresponds to angular rate 0.619 degrees/s; or to the
       period Π ≈ 9.7 min.

Ω0 ≈ 0.927 [degrees/s], Ω0p ≈ 0.823 [degrees/s],                              (12.20)

θ = 15 sin(0.0108t) [degrees] and Ω0s ≈ 0.8 θ0⋅0.0108 [degrees/s].            (12.21)

So, based on the present model, the motion of FOTON12 in the OXYZ system of co-
ordinates of the center of mass is a combination of three independent ones:

   Ω0 is a function of time (see fig.11.13 and 11.14): for the 10th and 18th runs it
corresponds to periods equal to 6.47 and 5.38 minutes respectively.

   Precession of the S/C around X-axis with some angular rate Ωp. We call it
precession, but in the case it also might be thought to be rotation of the plane of
oscillations. Ωp is a rather complicated function of time. Also its characteristic value
Ω0p as well as Ω0 is different for each run: for the 10th and 18th runs it corresponds to
periods equal to 7.29 and 6.08 minutes respectively. The values of angular rate of
precession are close to the ones of the own rotation of the S/C FOTON12, and the
gyroscopes registered namely the difference between Ω0 and Ω0p (periods equal to
57.69 and 57.72 minutes).

  Oscillations of the S/C with respect to X-axis with Ωs angular rate. That is why the
angle of precession θ between X- and X'-axis is not constant. The period of
oscillations is about 9.69 minutes for the 10th run and 7.47 minutes for the 18th run).


Precession and oscillations together give an 8-shaped loop trajectory of the S/C in
the OXYZ co-ordinate system. Angular velocity projections onto the O'X'Y'Z' co-
ordinate system for the 10th and 18th runs of QSAM, calculated via the formulas
(12.12)-(12.14), (12.19) are shown in the figs. 12.6 -12.11.
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 ωz [degrees/s]
       ωy [degrees/s]




Fig. 12.5. QSAM gyroscope data for 18th run: y- and z-projections of the angular rate.
              Phase shift between oscillations with smaller frequency.
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    ωz' [degrees/s]




          Fig. 12.6: Model 1: Theoretically obtained Z'-projection of angular rate.
                                          Run-10.
    ωy' [degrees/s]




          Fig. 12.7: Model 1: Theoretically obtained Y'-projection of angular rate.
                                          Run-10.
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       ωx' [degrees/s]




                    Fig. 12.8: Model 1: Theoretically obtained X'-projection of angular rate.
                                                    Run-10.
  ωx' [degrees/s]




                    Fig. 12.9: Model 1: Theoretically obtained X'-projection of angular rate.
                                                    Run-18.
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ωz' [degrees/s]




                  Fig. 12.10: Model 1: Theoretically obtained Z'-projection of angular rate.
                                                   Run-18.
ωy' [degrees/s]




                  Fig. 12.11: Model 1: Theoretically obtained Y'-projection of angular rate.
                                                   Run-18.
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12.2   Second theoretical model

The equations corresponding to this model has been described in Section 8. Here
the only analysis of angular motions will be given.

According to this model-2 basic motions of the S/C are: rotation and precession. The
angular rate of the rotation of FOTON around own axis, ωx, was shown in fig. 11.26.
From [RD8] and [RD9] follows that angular rate of precession is


                                     ( I z − I x )( I y − I x )
         Ωp= p⋅ ωx, where p =                                     = 0.77   ⇒ Ωp = 0.77⋅ ωx,
                                              IyIz

Angular rates should be measured in the same units.

Estimation of time scales during 9th run of QSAM gives: ωx ≈ 0.758deg/s, then Ωp ≈
0.584 deg/s. The corresponding frequencies f ≈ 2.2 10-3 and f ≈ 1.62⋅10-3 can be
found in the Table 11.9 for spectral analysis. The periods of oscillations
corresponding to them are: 7.6min and 9.8min min respectively. It means the period
of oscillations, observed in experiment TRAMP, corresponds to procession.

The sketch of the FOTON-12 motion, according this model, was shown in fig.12.2.

Let us look at the attitude of FOTON in the orbital co-ordinate system. On the
fig.12.12 the dependence of angle θ, which is the angle between symmetry axis Ox
and kinetic moment K, is shown versus time. The mean value of θ is about 28.5° with
amplitude of about 3°. As the value of the angle weakly varies, this motion is not
really a regular precession; it is a nutation.

The plot below in fig.12.12 shows dependence of angle ρ versus time. ρ Is the angle
between symmetry axis and OX2, normal to the orbit. Last plot in the same figure
shows σ versus time, where σ is an angle between the axis OX 2 and the projection
of K onto the plane OX1X3.

Remains obscured yet the motion of the kinetic momentum K. The shape of the
projections of the end of the kinetic moment of the plane OX1X3 is shown in fig.12.13.
The values on the axes do not correspond to an orbital co-ordinate system. These
two pictures allow to restore the motions of the S/C.
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 ex1




                                        eX3




Fig.12.12. Projection of the trajectory of the kinetic momentum K on the plane OX1X3
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  Θ
  ρ
  σ




Fig.12.13. Dependence of the angles Θ, ρ , σ (degrees) on time in the orbital co-
                          ordinate system OX1X2X3.
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13.    Validation of the models of the FOTON-12 S/C motion

To justify the model of the FOTON-12 motion computer simulations were carried out.
The fully tested code for solving the system of 3D Navier-Stokes equations in
Cartesian co-ordinate system in Boussinesq approximation is used.

Thermocapillary convection in a fluid that is enclosed in a laterally heated, three-
dimensional rectangular cell is investigated. The system was subjected to an
acceleration field g. Two vertical isothermal sidewalls are kept at temperatures Th on
the left and Tc on the right, Th>Tc. All other boundaries are assumed to be adiabatic.
The geometry of the cell is shown in fig.13.1.

In the TRAMP experiment that took place during the 10th run of QSAM, an
observation was done through the window on the top, which occupies only one part
of rigid wall, e.g. 20×20 mm2. Here we take into account that the vector of the
residual acceleration has components in any spatial direction. In the Cartesian
coordinate system the 3D non-dimensional Navier-Stokes, the energy, and the
continuity equations in Boussinesq approximation are given in the co-ordinate system
of the TRAMP experimental cell (fig.13.1) by:

∂U          ∂U           ∂U       ∂U           ∂P
     + ΓxU       + ΓyV        +W       = − Γx     + ∆U + Grx (Θ − x + 1)       (13.1)
 ∂t          ∂x           ∂y       ∂z          ∂x
∂V          ∂V           ∂V       ∂V          ∂P
     + ΓxU       + ΓyV       +W       = − Γy      + ∆V + Gry (Θ − x + 1)       (13.2)
∂t           ∂x          ∂y       ∂z          ∂y
∂W           ∂W           ∂W       ∂W        ∂P
     + ΓxU        + ΓyV        +W       =−       + ∆W + Grz (Θ − x + 1)        (13.3)
 ∂t           ∂x           ∂y       ∂z        ∂z
∂Θ          ∂Θ              ∂Θ       ∂Θ 1
     + ΓxU       − 1 + ΓyV      +W       =      ∆Θ                           (13.4)
∂t          ∂x               ∂y      ∂z Pr
    ∂U        ∂V ∂W
Γx      + Γy       +       =0                                                  (13.5)
    ∂x        ∂y      ∂z
                          ∂2          ∂2    ∂2
where operator ∆ = Γx2          + Γy2 2 + 2 . The linear temperature profile is subtracted
                          ∂x 2       ∂y     ∂z
from the total value Θ0 to have zero boundary conditions for temperature in the
direction of applied temperature gradient Θ = Θ0 - 1 + x, where Θ0 = (T - Tc) / (Th -
Tc). The equations (12.19)-(12.23) have to be solved together with the following
boundary conditions. On the rigid walls no slip conditions are used V = (U , V , W ) = 0
and a constant temperatures are imposed on the hot and cold walls Θ(x=0) =
Θ(x=1)=0. Thermal adiabatic conditions are imposed on other walls ∂Θ = 0 . The
                                                                               ∂n
Prandtl, Grashof and aspect ratios:
     ν         g β (Th − Tc )L3          L      L
 Pr = , Gri = i                x
                                 , Γx = z , Γy = z .
     k               ν 2
                                         Lx     Ly
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Lx is the length of the cell in the direction of the temperature gradient, and Ly is the
depth of the cell. gi is acceleration along the i-axis.

The numerical results presented below correspond to physical values of mixture of
ethylene-glycol and water. It is a typical mixture with viscosity around (2-3)⋅10-6 m2/s
and it corresponds to Prandtl number Pr=20. The geometrical sizes of the considered
cell are Lx = 2⋅10-2 m, Ly =5⋅10-2 m, Lz = 5⋅10-2 m. It gives the following aspect ratios:
Γx = 2.5, Γy = 1. To solve the governing equations, written in primitive-variable
formulation, a finite volume method is applied.




         Fig. 13.1: Geometry of the TRAMP experiment and numerical domain.

Usually, the processing of the experimental results is based on the knowledge of the
trajectory of tracer particles. Due to residual gravity the tracer particles, iso-dense
with liquid, move along the typical paths of convective flow. The density of the
particles ρp has the same value as the liquid at particular temperature, i.e. ρl0 = ρp0.
The estimations show that applying high temperature gradients like ∆T= Th - Tc = 40-
60K the difference of densities δρ=ρl- ρp becomes non-negligible. The Stokes velocity
(free falling body) due to this δρ can be comparable with the velocity of the flow in the
central part of the cell:

       2 r02                     2
Vp =         (ρ l − ρ p )g = 2 r0 β∆T g = 2.852 × 103 r02 g                   (13.6)
       9η                    9ν
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For numeric the acceleration field is taken for the 10th run of QSAM, i.e. for the
TRAMP experiment (see (12.17)-(12.18)). The points of computer simulations are to
put the acceleration field based on a model of S/C motion into the 3D equations and

follow trajectories of particles. Results of the numerical analysis are presented in a
form of the trajectories of tracer particles introduced into the domain.

The numerical trajectories of tracer particles of r0 = 0.1 mm have been reproduced in
the mid-cross Y'Z'-plane. Two types of acceleration fields were taken: based on the
presented above models own rotation, precession and oscillations. Also, two different
values of the particle density (δρ= 15 and 30 kg/m3) were taken and influence of so-
called Coriolis force on the tracer trajectory was investigated.




                                                                                 Step # 5 - Trajectories
                                                                                   Cold plate
                                                                       X displacement (normal grad T)
                                         0   50            100   150    200              250     300       350      400             450   500
                                    0


                                    50
                                             Vx=-6.2e-4 cm/s

                                   100
                                                                                                                                                Trajectory 1
Y displacement (parallel grad T)




                                                                                                                                                Trajectory 2
                                   150
                                                                       Vx=-3.7e-4 cm/s                                                          Trajectory 3
                                                                                                                  Vx=-3.7e-4 cm/s
                                                                                                                                                Trajectory 4
                                   200                                                                                                          Trajectory 5
                                                                                                                                                Trajectory 6
                                                                                                                                                Trajectory 7
                                   250                                                                                                          Trajectory 8
                                                                                                                                                Trajectory 9
                                   300                                                                                                          Trajectory 10
                                                                                                                                                Trajectory 11
                                                                                                                                                Trajectory 12
                                   350


                                                                                                                 Vx=5.94e-4 cm/s
                                   400


                                   450                                               Hot plate




Fig. 13.2.Experimentally observed tracer particles in TRAMP experiment. Courtesy
the authors [RD14]
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13.1 Results of calculations according to the model 2

The formula (8.5) for the acceleration field at the point of TRAMP experiment was
derived at the Keldysh Institute of Applied Mathematics (Moscow). According to this
formula the projections of the acceleration vector during the 10th run of QSAM in the
FOTON co-ordinate system are shown in fig.13.2-fig.13.4. It is worth to notice that
the level of micro accelerations is rather high, especially in y-direction. It depends
upon the distance between Ycom and Y of location of TRAMP experiment. The
calculations of micro accelerations in fig.13.2-fig.13.4 have been done at the point
with a co-ordinate: {-655 mm, -613 mm, 86 mm}. The mean value is about 1.16⋅10-5g
and the amplitude of the oscillations is rather small, about 0.22⋅10-5g

Density difference δρ= 15kg/m3 between particle and liquid is taken into account and
Coriolis force is presented. Unlike the camera made experimental observations, the
computer simulations with such an acceleration field result in the sinusoidal type of
trajectory, see fig13.5. No loop-like trajectories of the tracer particle were observed.
It means that some points of the phenomenon are missing. Although the velocity
obtained in the domain gives values close to the experimental ones: the numerics for
this model give |vmax| ∼3.26⋅10-4 cm/s in comparison with an experimental value
|vmax| ∼ (3.7 - 6.2)⋅10-4 cm/s. The calculated value is low than the experimental one.
It might be that the real microgravity level is higher.

The displacement of particle along y-and z directions is shown in fig13.6 and 13.7.
The oscillatory trajectory is observed only in y-direction, where the µg-level is the
highest. The period of oscillatory motion is about 10 min, and it is in a good
agreement with an experimental data. According the theoretical model, this
characteristic time corresponds to the angular rate of precession, see.fig.12.2




Fig. 13.3. Component ax of the acceleration vector according formula (8.5) in the
FOTON co-ordinate system at TRAMP location. The instant t = 0 corresponds to
23:40:00 UTC on 14.09.1999. Step 5 was during 738< t <3278 and step 6 was during
3358 < t < 5968.
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Fig. 13.4. Component ay of the acceleration vector according formula (8.5) in the
FOTON co-ordinate system at TRAMP location. The instant t = 0 corresponds to
23:40:00 UTC on 14.09.1999. Step 5 was during 738< t <3278 and step 6 was during
3358 < t < 5968.




Fig. 13.5. Component az of the acceleration vector according formula (8.5) in the
FOTON co-ordinate system at TRAMP location. The instant t = 0 corresponds to
23:40:00 UTC on 14.09.1999. Step 5 was during 738< t <3278 and step 6 was during
3358 < t < 5968.
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     y [mm]




                                        x [mm]

Fig. 13.6. Model 2: Trajectory of the tracer particles corresponding to the time of
TRAMP operation. The length of displacement within one period of oscillations is
about 0.15mm. The difference of density between particles and liquid is δρ= 15kg/m3.
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       y [mm]




                                         time, s×103

Fig. 13.7. Model 2: The displacement of the tracer particles with time in y-direction.
                     The period of oscillations is about 10 min.
       x [mm]




                                        time, s×103
Fig. 13.8. Model 2: The displacement of the tracer particles with time in x-direction.
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13.2 Results of calculations according to the model 1

Knowing the angular velocities Ω0, Ωp and Ωs the components of the acceleration
vector can be calculated according the equations (12.9)-(12.11). Again, like in the
case of model 2, the calculations of micro accelerations have been done at the point
with a co-ordinate: {-655 mm, -613 mm, 86 mm}.

As it was written above, Section 12.1, the mean values of the components of the
angular rates ωx and ωz characterizing the motion of the symmetry axis of S/C were
chosen to be equal zero. Despite that fact, the mean values of the accelerations at
these directions are non-zero. The acceleration of Y-component is the largest one
and it has mean value about 5.75⋅10-5g. Comparison with a model 2, where it was
1.16⋅10-5g, shows that they have the same order of magnitude. Although the ways of
making the two models are completely different.

The time dependencies of the components of the acceleration vector are plotted in
fig.13.9, fig.13.10, fig.13.11

At the real experiment the particles and liquid has been chosen iso-dense. They
cannot stay iso-dense with increase of the temperature. As the temperature
difference in the experiment was sufficiently high, ∆T= Th - Tc = 40-60K, the velocity
due to viscous drag, see eq.(13.6) will be comparable with velocity of weak
convection.

The difference of densities, δρ, is unknown therefore it is considered as parameter.
For the simulations with an acceleration field, presented in fig.13.9-13.11 the three
different values of this parameter are investigated:

Case1 δρ= 15kg/m3, Coriolis force is presented.
Case2 δρ= 30kg/m3, Coriolis force is presented.
Case3 δρ= 15kg/m3, Coriolis force is switched off.



One can see that all the three cases considered gave almost the same trajectories of
the tracer. The particle trajectory has a loop-like form as in the real TRAMP
experiment. It indicates that the given model reflects the principal features of the real
motion of the S/C. The size of the loops is approximately 0.3-0.4 mm. In the TRAMP
experiment 1.0-1.5 mm loops were observed. Such a difference between the sizes of
the numerical and experimental loops may be explained by different parameters of
the tracer particles taken for simulations. The maximal velocity in the domain (more
or less the same for the three cases) is: |vmax| ∼5.2⋅10-4 cm/s vs. |vmax| ∼3.7 - 6.2⋅10-4
cm/s.

The fact that for the three cases trajectories and the sizes of the loops are almost the
same shows that for the given acceleration field neither density of particle nor the
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presence of the Coriolis force influence the tracer motion. The former means that in
comparison with the Stokes velocity of the particle, velocity of the liquid itself is much
larger. The latter points on the fact that loop-like trajectories of the particle are not the
results of the Coriolis force.

An analysis of the experimental results revealed the presence of a bubble in the
experimental cell. Usually, Marangoni force on gas-liquid interface produces very
strong convection in Space for so large temperature differences. As theoretical and
experimental velocities have close magnitude, it means that the effect Marangoni on
the interface bubble-liquid was sufficiently small.




Fig. 13.9. Model 1: X'-projection of acceleration vector according to eq.(12.11) at
TRAMP location.




Fig. 13.10. Model 1: Y'-projection of acceleration vector according to eq.(12.10) at
TRAMP location.
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Fig.13.11. Model 1: Z'-projection of acceleration vector according to eq.(12.09) at
TRAMP location.
        y [mm]




                                         x [mm]

     Fig.13.12. Model 1: Numerically obtained trajectory of the particle. Case 1.
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              y [mm]




                                         time, s×103
             x [mm]




                                         time, s×103
  Fig.13.13. Model 1: Displacement of the particle with time in y and x directions.
                                    Case1.
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y [mm]




                                             x [mm]
         Fig.13.14. Model 1: Numerically obtained trajectory of the particle. Case 2.
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                 y [mm]




                                              time, s×103
             x [mm]




                                              time, s×103
  Fig.13.15. Model 1: Displacement of the particle with time in y and x directions.
                                    Case2.
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  y [mm]




                                          x [mm]

       Fig.13.16. Model 1: Numerically obtained trajectory of the particle. Case 3.
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                 y [mm]




                                         time, s×103
             x [mm]




                                         time, s×103
  Fig.13.17. Model 1: Displacement of the particle with time in y and x directions.
                                    Case3.
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14.   CONCLUSIONS

Little information is available about µg-levels achievable in unmanned carriers.

A residual gravity with small amplitude and low frequency variation exists due to the
atmospheric drag, the stabilizing rotation and complex motion of the platform along
the orbital trajectory. They are causing accelerations for payloads out of the center of
gravity.

The frequencies caused by the motions of the satellite are rather low, about f ≈ (0.3-
3.1) •10-3 Hz.

All accelerometers had failures in the range of low frequencies. It is supposed that
the most reliable data were recorded by QSAM gyroscope and Mirage.

It seems that the problem of non-zero offset has arisen within the operation of all
accelerometers. The process allowing to distinguish the offset value from the DC
value on recorded time signals by the holders of different instruments remains
enigma for the authors of this report.

In the range of high frequencies the operation of all accelerometers were satisfying.

Once during the mission, on 13.09.99, three accelerometers (TAS, QSAM, SINUS)
worked together simultaneously. In the range of the frequencies 1-10Hz all of them
with some tolerance identified 3 frequencies: 1.14-1.2Hz, 4.2-4.7Hz, 9.6-9.9Hz.

Comparison of the measured amplitudes is not so nice. For frequency f=1.14 Hz
SINUS and QSAM displayed 25 µg and 31 µg, while TAS gave 2µg.

The frequency 1.14Hz was induced by IBIS, frequency 7.7Hz on 16.09.99 and 12Hz
on 19.09.99 were caused by Polizon,

The operation of some other on-board equipment caused the splashes of rather large
amplitudes, close to 1mg.

Two external torques have their influences upon the spacecraft attitude motion:
gravitational and aerodynamic. At the second part of the flight FOTON-12 performed
rather regular motions.

Gravity stable orientation corresponds to the attitude when the S/C is aimed by its
heavier part (service module) towards the Earth and lies in the orbital plane.

The attitude motion, in which the spacecraft symmetry axis coincides with the normal
to the orbit plane corresponds to the stable aerodynamic orientation.
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Under the influence of these two torques, FOTON-12 has chosen some other
orientation different from those two.

All results merge to the fact that to the velocity of the S/C rotation around the
symmetry axis increases from almost zero value up to 1 degree/s at the end of the
mission.

The µg-levels on FOTON-12 were increasing throughout the mission, although the
attitude of FOTON became more regular at the second part of the flight.

At the beginning of the flight the module of acceleration in the c.o.m. |a| was varying
from ≈0.27 10-7 g up to 1.067 10-7 g depending on the location on the orbit. To the
end of mission this values increased by 25%.

The mean value of the acceleration in FluidPac was about (1-6)⋅10-5g on 14.09.99.
The acceleration itself oscillates with a time, and the amplitude of oscillations is about
10% of the mean value.

It seems that the µg-levels reported in [RD11] are underestimated.

The amplitude of accelerations increases on going to the higher frequencies.

To describe the S/C motions two model are suggested. One of them is pure
empirical, another is the results of the strict mathematical modeling. Physically they
are not very different. The resulting motion of FOTON-12 could be decomposed as:
spinning, precession and some kind of oscillations.

The numerical simulations of 3D Navier-Stokes equations have been done to validate
the models.

The calculated maximal velocity is 5.2 ⋅10-4cm/s and 3.26 ⋅10-4cm/s for the 1st ad 2nd
models respectively. The same value from the TRAMP experiment is (3.7-6.2)⋅10-4
cm/s.

According the first model the tracer particle perform the loops, similar to the
experimental ones. The second model results in sinusoidal type motion of tracer
particle.

The size of the theoretical loops is comparable, but a little bit smaller than
experimental ones.

FOTON as unmanned carrier is well suited for the space experiments but certain
improvements of µg-levels are desirable.

				
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