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ISSN 0010 9525, Cosmic Research, 2014, Vol. 52, No. 5, pp. 353­364. © Pleiades Publishing, Ltd., 2014. Original Russian Text © G.S. Zaslavskiy, V.A. Stepan'yants, A.G. Tuchin, A.V. Pogodin, E.N. Filippova, A.I. Sheikhet, 2014, published in Kosmicheskie Issledovaniya, 2014, Vol. 52, No. 5, pp. 387­398.

Trajectory Correction of the Spektr R Spacecraft Motion
G. S. Zaslavskiya, V. A. Stepan'yantsa, A. G. Tuchina, A. V. Pogodinb, E. N. Filippovab, and A. I. Sheikhetb
a

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia e mail: zaslav@kiam1.rssi.ru b Lavochkin Research and Production Association, Khimki, Moscow oblast, 141400 Russia e mail: flen@laspace.ru
Received December 16, 2013

Abstract--The results of refining the parameters of the Spektr R spacecraft (RadioAstron project) motion after it was launched into the orbit of the Earth's artificial satellite in July 2011 showed that, at the beginning of 2013, the condition of staying in the Earth's shadow was violated. The duration of shading of the spacecraft exceeds the acceptable value (about 2 h). At the end of 2013 to the beginning of 2014, the ballistic lifetime of the spacecraft completed. Therefore, the question arose of how to correct the trajectory of the motion of the Spektr R satellite using its onboard propulsion system. In this paper, the ballistic parameters that define the operation of onboard propulsion system when implementing the correction, and the ballistic characteristics of the orbital spacecraft motion before and after correction are presented. DOI: 10.1134/S001095251405013X

1. INTRODUCTION In the standard case, the trajectory of the Spektr R spacecraft is corrected in order to change the charac teristics of its future flight into operational orbit of the Earth's artificial satellite, i.e., eliminating unwanted spacecraft setting in the shadow of the Earth or Moon (the light source is the Sun) and increasing the ballistic lifetime of the spacecraft. By definition, the ballistic lifetime of the spacecraft at current time instant t is provided when it flies above the Earth's surface at no less than a given altitude h l . Each time the spacecraft falls in the shadow of the Earth or Moon is characterized by the duration of its stay in full or partial shadow (half shadow). The stan dard correction of the spacecraft motion into the operational orbit of the Earth's artificial satellite is executed by carrying out special correction sessions, in which the necessary spatial orientation of the thrust vector of the onboard propulsion system (PS) and the switching of the PS on and off are provided at given time instants. The cyclogram of the standard correc tion (correction scheme) of the operational orbit of the Spektr R spacecraft is chosen in view of the tech nical spacecraft features, as well as the accuracy of our knowledge of the parameters of its motion and the technology of implementing the correction sessions. In this case, the total cost of the working body when PS operating should be close to the minimum.

2. CHARACTERISTICS OF PS AND THE SPACECRAFT CORRECTION SESSION The calculation of the ballistic parameters neces sary to choose the scheme for spacecraft trajectory correction, to implement and to analyze the correc tion execution is performed under the following assumptions on the technical spacecraft characteristics, the structure and the logic of corresponding session. (1) At every time instant, the total thrust vector of operating PS (PS thrust) belongs to a line passing through the center of mass (CM) of the spacecraft. (2) During the session, the spacecraft's motion in is correction by the continuous operation of the PS dur ing the time interval, i.e., from the time instant tthn, designated as the time instant when the PS is switched on, to the time instant tthe, designated as the time instant when PS is switched off. (3) During the time interval [tthn, tthe] of PS opera tion, the thrust retains its direction in the inertial space. The unit vector of the thrust eth is considered as the vector collinear to a given or specially calculated vector e in the coordinate system (CS) of J2000 [1]. Hereafter, for convenience, it is assumed that, in the case, when vectors eth and e coincide in the direction, the value Vch of the increment of characteristic velocity at the cost of the PS operation has nonnegative value. Otherwise, the value Vch is taken with a minus sign. Thus, Vch 0, if eth = +e, and Vch < 0, if eth = ­e. (4) The values of the thrust P and the specific PS pulse Isp are constant for the entire time interval of its

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continuous operation. Outside this interval, the PS thrust is absent, i.e., P = 0. (5) In the correction session, the PS thrust is switched off (the increment of characteristic velocity of the spacecraft at the expense of PS is finished) after the tth duration of its continuous operation, tth = tthe tthn, is achieved. (6) To implement the session for correcting the tar get spacecraft, it is sufficient to determine the neces sary values tthn and tth, as well as vector eth of the direction of the PS thrust in the J2000 CS, designated as the parameters of the spacecraft correction session. 3. DEFINITIONS AND ASSUMPTIONS IN THE PROBLEMS OF THE SPACECRAFT CORRECTION The ballistic problems required for choosing the parameters of forthcoming correction of the space craft operational orbit or an analysis of the results of the performed spacecraft correction are considered under the following definitions and assumptions rela tive to the parameters that characterize the motion, as well as the light and shade situation for the spacecraft. (1) When solving to the problem of ballistic support (BS) of the spacecraft flight control, the mathematical simulation of the motion the spacecraft CM is per formed taking into account the attractive forces of the Sun, Moon, planets of the solar system considered as material points. Moreover, it is necessary to take into account forces caused by the noncentrality of the Earth's gravitational field [2], the aerodynamic resis tance of the spacecraft motion in the Earth's atmo sphere (the dynamic model of the atmosphere is used [3]), and light pressure on the spacecraft. The acceleration wa of the spacecraft caused by the atmospheric influence is calculated by the formula w a = s Va Va, where is the atmospheric density in the vicinity of the spacecraft, Va is the spacecraft velocity relative to the atmospheric flow, and s is the so called ballistic coefficient. The value of this coeffi cient depends on the dimensionless aerodynamic coefficient cx, the midsection area S relative to the atmospheric flow and mass m of the spacecraft, s = (c x 2) (S m) . Taking into account that the space craft flight in operational orbit passes outside the dense layers of the Earth's atmosphere and that it rarely approaches these layers, s is taken as constant in ballistic calculations. The air density is calculated in full accordance with the dynamic model of the Earth's atmosphere. In this case, the input parameters of the model (the current level of the intensity of solar radia tion, etc.) are overestimated with respect to the aver age, air density. Generally speaking, the light pressure force is char acterized by a dimensionless variable, i.e., value Sd of the ratio of the absolute value of indicated force to the attractive force of the spacecraft by the Sun. However,

in the ballistic calculations of the spacecraft flight tra jectories, for each specific trajectory, it is taken as con stant and refined by the trajectory measurements and the telemetry (TM) information. It is considered to be a matching parameter that generally allows one to take into account forces that are small in the magnitude not simulated that act on the spacecraft when predicting the motion of the spacecraft CM. (2) The trajectory of passive (without PS switching) spacecraft flight at each current time instant t is char acterized by six dimensional vector (x, y, z,V x ,V y ,V z ) of kinematic parameters of motion. The first three com ponents of this vector are the coordinates of the posi tion vector r() = (x(), y(), z()) and the last three com t ttt ponents are the coordinates of the vector V() = (V x (),V y (),V z ()) of the spacecraft velocity in t t t t the J2000 CS. The collection of nine values {, r(t), V(t), s, S d } are designated as the initial condi t tions (IC) of spacecraft motion at time instant t. It is used the decreed Moscow time (DMT), which is 3 h earlier than the corresponding Coordinated Universal Time UTC. It is assumed that the current flight space craft trajectory before PS switching is given by IC at the time instant t0 before time instant tthn of PS switch ing on as follows: (t 0, x0, y0, z 0,V x0,V y0,V z0, s, S d ). (3) The value of the spacecraft mass m at the time instant tthn of PS switching is known as follows: m(tthn) = m0. (4) The time of continuous PS operation can be determined by three ways, i.e., explicitly based on the values tthn and tthe; based on values tthn and Vch of the increment of the characteristic velocity as a result of the correction execution; and by the average time instant t* of time segment of the PS operation, t* ­ tthn = tthe ­ t*, and the value of increment of characteristic velocity Vch as a result of the correction session execution. In the third case, the time instant t* is usually found implic itly. It is determined as the first time instant (after a given time tg) at which a certain condition on the kine matic parameters of the spacecraft motion is fulfilled. That time instant can be, e.g., the time instant of reaching the minimum (or maximum) distance between the satellite and the Earth's CM, assuming its passive flight. In this case, the trajectory is corrected at the pericenter (or apocenter) of the spacecraft orbit. (5) The dependence between the duration of the PS operation and the corresponding increment of charac teristic velocity is set by the Tsiolkovskii formula

t th (V ch ) = (I

sp

P )g 0 m0 (1 - e x p [- V

ch

I

sp

g 0 ] ), (1)

where the thrust P and specific pulse Isp are given by the PS parameters. Acceleration due to the force of gravity is taken equal to be g0 = 9.80665 m/s2. The parameters in formula (1) have the following dimen sionalities: [s] for tth, [s] for Isp, [N] for P, and [kg] for m0.
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(6) The light and shade situation on the spacecraft at any fixed time instant is characterized by the dimen sionless coefficient KT of the Sun shading by the Earth (Moon), if looking from the spacecraft CM toward the Sun. The shading coefficient KT is determined as fol lows based on the ratio of the area ST of the shaded (hidden by the Earth or Moon) part of the Sun to the area SS of the entire Sun visible from the spacecraft, assuming the absence of the Sun shading: KT = ST/SS. It is assumed that the shapes of the Sun, Earth, and Moon are spheres with given radii. The geometric center of each sphere coincides with CM of the corresponding luminary. Obviously, the shading coefficient KT can take the values that remain within the limits of the segment [0, 1] of the number axis. At KT = 0, the spacecraft is in the light (there is no shading the Sun, Earth, or Moon). At KT = 1, the spacecraft is in the shadow (there is a total solar eclipse, if to look from the space craft). At a numeric value of the coefficient of shading, which belongs to the set of interior points of the above segment, it can be assumed that the spacecraft is in the half shadow (the partial solar eclipse occurs if one is looking from the spacecraft). (7) By definition, the entire time segment of shad ing [tshn, tshe] on which the coefficient KT is more than zero at each time instant t is characterized as follows by the coefficient KT max of the degree of shading:

nated as the time segment of verifying the spacecraft lifetime as follows:
M {t1} = {t1 min , t1 min + ht1, t1 min + 2ht1, . . . , t1 min + q1ht1} . (3) Here, htl > 0 is a given step of verifying the spacecraft lifetime (fulfilling condition (2)) and forming set (3), while ql is determined by the time instant tl max, namely, the condition tl min + qlhtl tl max < tl min + (ql + 1)htl. For definiteness, it is assumed that, if condition (2) is not fulfilled at the first point of set (3), then tle = tl min.

K

T max

= max K T(t).
t[t
shn ,t she

]

(8) The time segment [tashn, tashe] in which the equality KT = 1 is designated as the time segment of the total solar eclipse at each time instant t. In this case, the segment (if it exists) is a unique subset of the cor responding segment of the time of shading [tshn, tshe]. (9) The prediction of the light and shade situation on the spacecraft after calculating the correction of its motion trajectory is reduced to the determination of a set of time segments of shading and the corresponding time segments of the total solar eclipse if they exist. The desired set of segments is caused by the parame ters of the spacecraft trajectory correction and given time interval tsh of the prediction of the light and shade situation on the spacecraft. The initial time instant tshn for each desired segment of the time of shading should satisfy the inequality tthe tshn toff + tsh. (10) The lifetime of the spacecraft operational orbit is determined as the last time instant tle that belongs to a given set of M {t1} of sequential time instants before which the following condition is ful filled: the altitude h(t1) of the current spacecraft orbit exceeds given below the acceptable value h1 of the altitude of the spacecraft flight, i.e., (2) h(t1) h1. The indicated set represents a collection of individual time instants (ql pieces) that belong to the given seg ment [t1m in , t1m ax ] of the number axis, which is desig
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3. SPACECRAFT ORIENTATION FOR IMPLEMENTING THE CORRECTION SESSION This chapter is devoted to the ballistic analysis of the possibility of constructing the acceptable orienta tion for the spacecraft as a rigid body in order to imple ment the forthcoming correction session. In this case, it is assumed that the unit vector eth of the PS thrust is a vector collinear to the vector V* of the spacecraft velocity at the time instant t* in the passive spacecraft flight V * = V (t*) . The corresponding ballistic problem is presented in the chapter, as a result of which, when it is possible to construct the indicated spacecraft orientation, we obtain a solution containing values of its basic (orien tation) parameters. We introduce the following right rectangular coordinate systems bound with the space craft: BCS O x b y bz b, the center of which, point O, coincides with the spacecraft CM, and the O x b axis is directed along the vector of the PS thrust and BCS0, which, at time instant t*, coincides with BCS provided that its O x b axis is directed along the vector eth, the Oxbzb coordinate plane contains the unit vector eS of the direction from the spacecraft CM (the point O) to the Sun CM, and the positive direction of the Ozb axis is an acute angle with the vector eS. Before the session of correction spacecraft motion, the possibility of constructing the acceptable BCS ori entation in the J2000 CS is verified, which is condi tioned by the implementation of the restriction pre sented below. Restriction. The angle between the direction of the Oxb axis of BCS and the direction from the space craft to the Sun CM belongs to the given range: (4) min max , where the boundaries of the range can be specified during flight and construction spacecraft tests and deviate from the values min = 90° and max = 165° within a few degrees, respectively. In connection with the foregoing, the ballistic analysis of the possibility of constructing the space craft orientation in order to implement a correction session for its motion can be reduced to solving the fol lowing problem, which we will call the orientation analysis problem.


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Orientation Analysis Problem

t 0, x 0, y0, z 0,V x0,V y0,V z 0, s, S d are initial conditions of the spacecraft motion in the J2000 CS; t* is middle of the interval of the PS operation of the correction ses sion; I v is the indicator of the PS thrust direction when the spacecraft is corrected (in the case of I v 0, the thrust is directed along the velocity V (t*) and, in the case of I v < 0, the thrust is directed against the indicated velocity); min, max are the segment bound aries of acceptable angles between the PS thrust when making corrections and the direction from the spacecraft toward the Sun CM (see (4)). Set: {r(t*) , V (t*) } are the kinematic parameters of spacecraft motion in the J2000 CS at the time instant when the spacecraft achieves (in passive flight) the point at which it is supposed to implement the session of the target correction of spacecraft motion; the sequence of the parameters (rS is the distance from the spacecraft to the Sun CM, eS is the unit vec tor of the direction from the spacecraft to the Sun CM, rE is the distance from the spacecraft to the Earth's CM, eE is the unit vector of the direction from the spacecraft to the Earth's CM) that characterize the position of the Sun and the Earth relative to the space craft at the time instant t*; I o is an indicator of the possibility of constructing the orientation of the spacecraft as a rigid body to implement the session of correcting its motion; at I o 0 , it is possible to construct the necessary space craft orientation (inequalities (4) are fulfilled); at I o < 0, it is considered to be impossible to construct this orientation (inequalities (4) are not fulfilled) and the procedure for solving to the problem is completed; is the angle between the Oxb axis of BCS directed along the PS thrust and the direction from the space craft to the Sun CM at the time instant t*; a sequence of unit vectors in the J2000 CS that cor responds to the directions of the BCS axis in its refer ence position of BCS0 (see above), i.e., e Ox along the direction of the Oxb axes, eOy along the direction of the Oyb, and eOz along the direction of the Ozb axis.
4. CHOOSING THE PARAMETERS FOR THE CORRECTION SESSION FOR THE TRAJECTORY OF THE SPACECRAFT The characteristics of the correction session depend significantly on the time instant t* (see above). At given values of mass m0 before PS switching and the value Vch of increment of characteristic velocity, it uniquely determines (using formula (1)) the time instants of switching the PS on (tthn) and off (tthe) dur ing the session of correction the trajectory of the spacecraft. An algorithm for choosing the parameters of the correction session is based on searching for acceptable values of the time instant t*.

When searching for each fixed acceptable time t*, the calculations are performed for the ballistic lifetime of the spacecraft and the light and shade situation onboard the spacecraft (during indicated lifetime) for a finite set M{Vch} of values of the increment of the characteristic velocity Vch. Set M{Vch} is considered to be a set of all possible separate points (q pieces) that belong to a given segment [Vchmin, Vchmax] of the num ber axis and is determined by given step hV ch > 0 as fol lows:

M {Vch } = {Vch min ,Vch min + hV ch , Vch min + 2hV ch , ...,Vch min + q hV ch } .

(5)

In this case, it should be remembered that the quan tity Vch can take both positive and negative values and, hence, the values Vch min and Vch max can also be positive or negative values. In connection with this, the solution to the prob lem of choosing the parameters of forthcoming cor rection can be reduced to solving the partial problem of choosing the correction parameters at which the increment of characteristic velocity Vch is fixed value from set (5). In this case, the vector e is calculated from the fact that it is directed along the spacecraft velocity vector at the time instant t*, assuming the pas sive spacecraft flight in orbit of the Earth's artificial satellite. Problem of Selecting the Correction Parameters

t 0, x 0, y0, z 0,V x0,V y0,V z 0, s, S d are the initial condi tions of the spacecraft motion in the J2000 CS; m0 is the value of the spacecraft mass m at time instant tthe of PS switching; Vch is the increment of characteristic velocity as a result of the PS operation; t* is the middle of the interval of the continuous PS operation in the correction session; hS is acceptable flight altitude below the spacecraft; htS is a test step for the spacecraft lifetime (the fulfillment of condition (2)); and tg is the time instant until which the verification of the ballistic spacecraft lifetime is implemented (in this case, it is taken to be tl min = tthe and tl max = tg). Set: tthn, tth is the time instant when the PS is switched on to implement the correction of spacecraft motion and the duration of its operation; eth is unit vector (in the J2000 CS) of the direction of the PS thrust when correcting the spacecraft motion; {tthn, r(tthn, V(tthn)} are the kinematic parameters of the spacecraft motion in the J2000 CS at the time instant of finishing the PS operation; me = m(tthe) is the spacecraft mass at the time instant when PS operation is finished, (only spacecraft mass losses are me = m0 - P I sp g 0t th taken into account because of fuel consumption when correcting);
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tle is time of the ballistic lifetime of the operational spacecraft orbit (in this case, if the ballistic spacecraft lifetime is provided up to the time instant tg, then it is accepted that tle = tg); tshnj, tshej, tashnj, tashej, K T max j is the ordered (over the time of the beginning of shading time segments, tshn1 < tshn2 < ... < tthnN) is a sequence of five numbers, each of which characterizes (after corrections): the beginning and end of the shading time segment ([tshnj, tshej]), the beginning and end of the segment of total solar eclipse ([tashnj, tashej]) and the degree of spacecraft shading (the coefficient K T max j ) on the jth shading time segment (all of the considered segments belong to the interval [tthe, tle]); N is the number of five numbers indicated above, i.e., shading intervals, that belong to the interval [tthe, tle]; and an array of the following parameters of the osculating at the time instant tthe spacecraft orbit after correction in the J 2000 CS, where h is altitude of pericenter above the Earth's surface, h is apocenter altitude above the Earth's surface, is argument of latitude pericenter, i is inclination, is longitude of the ascending node, Po is orbit period, t is the time instant when the pericenter of the orbit is passed by the spacecraft in the previous orbit, and t is the time instant when the beginning of the current flight orbit is passed by the spacecraft. Here and below, it is accepted that, when calculat ing the altitudes of the perigee and apogee of the spacecraft orbit, because the Earth's shape is consid ered to be a sphere with an average radius of RE = 6378.2 km; the values and take the values from the half interval [0, 2), and the value i is taken from the interval [0, ]. The number of the orbit is riced by one at the time instant when the spacecraft passes the ascending node, i.e., when the spacecraft crosses the reference plane of the J2000 CS and the applicate changes its sign from negative to positive. 5. SCHEME OF CORRECTING SPACECRAFT TRAJECTORY Over time, when refining the parameters of space craft motion in the operational orbit formed after launching the spacecraft in November 2011, it became necessary to correct the spacecraft's trajectory in 2013. At the initial (before correction) spacecraft trajectory, its ballistic lifetime was restricted by the time instant that occurred at the end of 2013 to the beginning of 2014. This time instant was defined as the time when the spacecraft is first found at the altitude less than 400 km above the spherical Earth's surface. Moreover, at the indicated trajectory of spacecraft flight in the beginning of 2013 the spacecraft set into the Earth's shadow occurs, which essentially in the duration
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exceeds the maximum acceptable set (about 2 h) and is approximately 5.7 h. Since the middle of November 2011, corrections were made on the future orbit of the spacecraft that included refining the initial conditions of the space craft's motion according the trajectory measurements and the TM data based on the laborious calculations of the parameters. For the target orbit (after corrections), we considered an orbit for which the time instant tl occurs no earlier than in the middle of 2018 (at the expiration of 7 years after spacecraft launching into the operational orbit), and the spacecraft set into half shadow does not exceed 2 h in duration by more than 10 min for 5 years after launching the spacecraft into the operational orbit of the Earth's artificial satellite. In this case, indicated conditions should be fulfilled taking into account errors of initial conditions of the spacecraft motion and the coefficient Sd of solar pres sure, possible errors in the orientation and the value of PS thrust when implementing each of its switching. Preliminary calculations showed that the effect of errors in the thrust orientation on the further motion of the spacecraft CM is negligible compared with the influence of errors in the value of thrust, the current knowledge of the initial conditions, and the prediction of the value of coefficient Sd. All subsequent calcula tions of the correction parameters were performed taking into account the value of limit error of the PS thrust, which is equivalent to the relative error in the implementation of the increment of characteristic velocity Vch equal to 9% of the value of increment and was previously agreed upon with the Main Operational Control Group (MOCK). In this case, there was a solution to the problem of the calculating correction parameters, which provides the above requirements for the spacecraft trajectory after correction for three values of the coefficient of light pressure, i.e., (1) Sd equal to the current (before correction) value Sd0, (2) Sd = 0, and (3) Sd = 2Sd0. When solving the correction problems, the correc tion parameters are the time instants when the PS is switched on. Problems when the PS is switched on once and twice are considered. In the case when the PS is switched on twice, it should be possible to refine the spacecraft trajectory parameters before the second PS switching according the trajectory measurements and the TM data. Calculations have shown that this requirement is satisfied when the time instant at which the PS is switched on are spaced by no less than approximately the period of the satellite orbit with the practically possible intensity of the trajectory mea surements. The possible direction of the PS thrust is selected to be almost uniquely based on the condition of its parallelism to the spacecraft velocity vector in view of restriction (4) by the angle. The rejection of the correction with one PS switch ing occurs when the absolute value of the increment of characteristic velocity is so high that, after corrections,


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the trajectory is implemented with unacceptable errors. The correction problem is a difficult mathematical programming problem, the functional of which is the sum of the absolute values of the increments of the characteristic velocities when the PS is switched on while making corrections, the value of which is pro portional to the consumption of the working body for implementing the target correction. Searching for its solution is performed with human participation using the developed algorithms to solve the above basic problems for an orientation analysis and choosing the correction parameters. When searching for the scheme of making correc tion to the Spektr R spacecraft, schemes in which the correction sessions, as well as preliminary and final operations, are performed within visible zones for at least one of the two ground stations (in Medvezhyi Ozera and Ussuriisk), are preferable. In the period from November 2011 to January 2012, at the ballistic center of the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, the solution to more than 700 of the indicated prob lems of mathematical programming were performed in order to choose the parameters of spacecraft motion correction in 2012. Corresponding information for the forthcoming correction of spacecraft motion trajec tory and the epy scheme for its implementation was presented by the Main Operational Group for Space craft Flight Control (MOCG). The indicated scheme has been designed in view of the need to implement a PS burn (a series of technological operations with the PS switching) before the first PS switching in order to implement the targeted correction of the operational spacecraft orbit. In this case, it was taken into account that the PS burn leads to an increment of the charac teristic spacecraft velocity within 2 cm/s. As a result, the following decisions were made: (1) Corrections are performed in order to provide (when maintaining the model of forces acting on the spacecraft) a ballistic spacecraft lifetime until the mid dle of 2018 (the spacecraft altitude above the Earth's surface is not less than 640 km); the absence of contin uous intervals of shadow on the spacecraft from the Earth with an unacceptable shading coefficient for a duration of more than 2.2 h before the beginning of 2017; and the conservation (for carrying out effective researches) of the evolution of spacecraft orbit, which is achieved upon small variations of the spacecraft orbit parameters. (2) Correction is implemented according the scheme proposed by KIAM: burn + first pulse on Feb ruary 21, 2012; second pulse on March 1, 2012. Reserve versions: (1) burn and the first correction pulse on February 21, 2012; the second correction pulse on March 10, 2012; (2) burn and the first correc tion pulse on March 1, 2012; the second correction pulse on March 10, 2012. In all cases, the ballistic parameters necessary for the implementation of the

second correction pulse are calculated using the tra jectory measurements after implementing the first correction pulse taking into account the possible implementation of the second pulse with error with respect to modulus not exceeding 9% of its absolute value. All versions provide the conditions for correct ing the operational spacecraft orbit. The first pulse is about 1.49 m/s. The burn imparts a total pulse of about 0.01 m/s to the spacecraft. The second pulse is about 2 m/s. The beginning of PS operation for the pulse implementation occurs on February 21, 2012 at 21.00.00. Switching PS to implement the second pulse occurs in the region of the apocenter of the current orbit. All necessary ballistic data for the real implementa tion of sessions for correcting operational spacecraft orbit were calculated in accordance with the above scheme. The total duration of the PS operation at burning was determined to be equal to 2 s. The time instants of PS switching were taken to be no more than a few min utes before the PS was switched on in the first session of the target correction of the spacecraft orbit. In bal listic calculations, the burn was simulated as one inter val of the PS operation for 2 s. In this case, the average time t* of the PS operational interval coincided with the middle of the interval between the beginning of the first and the end of the last from expected PS switching when implementing the burn and the direction of the PS thrust coincided with the direction of the thrust in the first session of the target correction. For simulating burn, t* = 21.02.2012 at 20.56.47,1 was agreed with MOCG. As a result of burn, the esti mated value of the absolute increment of characteris tic velocity is about 0.01 m/s. In the first session of the target correction of the operational satellite orbit, PS should be switched on February 21, 2012 at 21.00.00,0 and operate during 300 s. At the cost of this PS operation, the calculated value of the absolute increment of the characteristic spacecraft velocity is about 1.75 m/s. In the second session of the target correction of the operational sat ellite orbit, PS should be switched on March 1, 2012 at 14.45.00,0 and operate during 332 s. At the cost of this PS operation, the calculated value of the absolute increment of the characteristic spacecraft velocity is about 1.86 m/s. The above mentioned burn and two sessions of the target spacecraft correction were per formed. 6. ANALYSIS OF THE RESULTS OF SPACECRAFT TRAJECTORY CORRECTION An analysis of the results of executing the target correction of the Spektr R spacecraft is performed by comparing the characteristics of sequences of three trajectories of its passive flight, i.e., before PS burning, refined after corrections, and current (refined on November 20, 2013) trajectory. The indicated trajec
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TRAJECTORY CORRECTION OF THE SPEKTR R SPACECRAFT MOTION Table 1. Ballistic lifetime and intervals of spacecraft shading Number of trajectory 0 22.XII.2013 08.04.29 8.I.2013 22.04.19 20.492 8.I.2013 23.05.35 1 14.I.2020 21.45.22 11.I.2018 12.29.27 22.129 11.I.2018 13.50.06 12.118 2 >18.VII.2021 12.00.00 21.I.2017 11.36.36 24.062 21.I.2017 13.20.55 11.688

359

t t

she shn

t sh, hours t
ashn
ash

t

, hours

13.062

tories are assigned to corresponding numbers of 0, 1, and 2. Table 1 shows the calculated characteristics of bal listic lifetime and the first unacceptable spacecraft shading during flight with the coefficient for each of the three trajectories. The following additional desig nations are used in the table, except for previously pre sented designations: tsh = tshe ­ tshn is the duration of the shading interval and tash = tashe ­ tashn is the dura tion of the total solar eclipse. The data in Table 1 show that the assigned prob lems of correcting the operational orbit are success fully solved with regard to the support (when conserv ing the model of forces acting on the satellite) of long term ballistic spacecraft lifetime and acceptable light and shade situation onboard the spacecraft to about the beginning of 2017. In order to get information on the difference between trajectories 0, 1, and 2, for each of them, at certain intervals of the spacecraft flight, the parame ters of osculating orbits were calculated in the sequence (by orbits) of time instants in order to reach the minimum distance of the spacecraft from the Earth's CM. The following previously introduced parameters are considered to be the parameters of the osculating orbits: h, h, , i, , Po and two parameters that are interesting from the point of view of imple menting the scientific program of the Spektr R space craft, namely, the right ascension e and the declina
, deg 115 114 113 112 111 110 109 108 0 8 16 24 32 40 48 j 0, j 1 56 64 72

tion e of apocenter in the rectangular right ecliptic coordinate system (ECS), the beginning of which coincides with the beginning of the J2000. CS. The directions of the abscissa axes of these CS coincide. The applicate axis of ECS is orthogonal to the plane of the ecliptic and directed towards the Earth's North Pole. The calculation results are shown in Figs. 1­16 in the form of graphs with broken lines. The number axis is considered to be the abscissa axis. Abscissa values of the vertices of broken lines belong to a finite set jk(D1, D 2 ) of natural numbers. Elements of this set are serial numbers of points in the course of the satellite flight from the beginning of a date (D1) up to the begin ning of a date (D2 ), on the kth trajectory with minimal (in orbit) distance from the Earth's CM. The ordinate axis of the vertex of the broken line is equal to the calcu lated value of the parameter of the osculating orbit indi cated in the figure. The number (k) of the trajectory is specified directly next to the broken lines. Instead of the designations h and h, the designations hmin and hmax are used, respectively. Figures 1­8 show the dependences of the parame ters of osculating orbits for trajectories 0 and 1 on the interval, the beginning of which is approached to the time instant of finishing the last session of correction, where D1 = 04.03.2012 and D2 = 22.12.2013. These values reflect the changes in the characteristics of the
, deg 16 8

1 0

0 ­8 ­16 ­24 ­32 0 1

80

0

8

16

24

32

40 48 j 0, j 1

56

64

72

80

Fig. 1. COSMIC RESEARCH Vol. 52 No. 5 2014

Fig. 2.


360 hmin, thous. km 72 64 56 48 40 32 24 16 8 0 8 16 24 32 40 48 j 0, j 1
Fig. 3.

ZASLAVSKIY et al. h
max,

thous. km

336 320 0 1 288 272 56 64 72 80 0 8 16 24 32 40 48 j 0, j 1 56 64 72 80 304 1 0

Fig. 4.

, deg 320 240 160 80 0 8 16 24

1 0

32

40 48 j 0, j 1

56

64

72

80

i, deg 88 80 72 64 56 48 40 32 24 0

0 1

8

16

24

32

40 48 j 0, j 1
Fig. 6.

56

64

72

80

Fig. 5.

, deg 300 296 292 288 284 280 278 0 8 16 24 32 40 48 j 0, j 1 56 64

Po, days 8.80 8.72 8.64 8.56 8.48 8.40 8.32 8.24 72 80 0 8 16 24 32 40 48 j 0, j 1 56 1

0 1

0

64

72

80

Fig. 7.

Fig. 8.

operation spacecraft orbit as a result of burning and correcting the trajectory of its motion. Figures 9­16 show the dependences of the same parameters of osculating orbits for trajectories 1 and 2 as in Figs. 1­8 over an interval of about 5 years. The beginning of this interval is more than 1.5 year away from the time instant of the completion of the last ses sion of correction and coincides with the end of the interval in Figs. 1­8, where D1 = 22.12.2013 and D2 = 20.07.2018.

Tables 2 and 3 set up a correspondence between numbers of jk points on the kth trajectory, where the spacecraft reaches minimum (in orbit) distance from the Earth's CM and calendar time. The dependences shown in Figs. 9­16 allow us to make a qualitative estimate of the model of the Spektr R spacecraft motion used in ballistic calculations. The motion model is adequate for real satellite motion from the point of view of scheduling the flight control of the spacecraft over a few years.
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TRAJECTORY CORRECTION OF THE SPEKTR R SPACECRAFT MOTION e, deg 128 120 112 104 0 16 32 48 64 80 96 112 128 144 160 176 192 j1, j 2 2 1

361

e, deg 24 16 8 0 ­8 2 ­16 1 ­24 ­32 0 16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 10.

Fig. 9.

hmin, thous. km 72 64 56 48 40 32 24 16 8 0

2 1

hmax, thous. km 360 344 328 312 296 280 0

2 1

16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 11.

16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 12.

, deg 320 240 160 80

1

i, deg 72 56 40 24 2 1

2 0 16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 13.

8 0 16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 14.

CONCLUSIONS The trajectory of the spacecraft's flight is corrected in accordance with the proposed basic scheme. The calculated time of switching on the PS in the first cor rection session is on February 21, 2012 at 21.00.00,0 and in the second correction session on March 1, 2012 at 14.45.00,0. In this case, in agreement with the tech nical administration of the spacecraft flight, the burn ing of the spacecraft orbit in ballistic calculations was simulated by switching on the PS for 2 s of continuous
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operation at the time instant of February 21, 2012 at 20.56.46,1. The calculated increments of the charac teristic spacecraft velocity during PS burning and the implementation of the first and second correction ses sions were 0.01, 1.75, and 1.86 m/s, respectively. After finishing the correction, subsequent calcula tions showed (see Tables 1­3 and Figs. 1­16) that the correction was implemented successfully, i.e., the basic requirements for flight trajectory after correc tions are performed, when keeping the mathematical


362 , deg 320 240 160 80 0 2

ZASLAVSKIY et al. Po, days 10.25 9.75 9.25 8.75 1 16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 15.

21

8.25 0 16 32 48 64 80 96 112 128 144 160 176 192 j 1, j 2
Fig. 16.

Tables 2. Instant times for approaching minimum distance for trajectories with numbers 0 and 1 from March 4, 2012 to December 22, 2013 j0 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 77 Date 5.III.2012 16.IV.2012 29.V.2012 11.VII.2012 22.VIII.2012 4.X.2012 16.XI.2012 28.XII.2012 9.II.2013 24.III.2013 5.V.2013 17.VI.2013 30.VII.2013 10.IX.2013 22.X.2013 5.XII.2013 13.XII.2013 Time 21.07.58,844 09.09.17,801 11.32.48,964 18.48.31,059 13.37.36,110 12.04.50,820 17.07.19,668 11.27.19,353 15.06.40,917 19.54.00,465 10.34.28,336 08.49.22,710 18.28.47,007 16.30.21,606 22.04.53,124 05.09.38,125 19.26.05,325 j1 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 77 Date 5.III.2012 16.IV.2012 29.V.2012 12.VII.2012 23.VIII.2012 5.X.2012 17.XI.2012 30.XII.2012 10.II.2013 26.III.2013 7.V.2013 18.VI.2013 1.VIII.2013 13.IX.2013 25.X.2013 6.XII.2013 15.XII.2013
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Time 21.45.26,826 12.00.28,656 15.38.36,449 11.32.12,122 12.48.21,469 01.10.09,638 22.07.01,070 02.55.54,760 13.51.14,225 08.38.35,630 14.28.15,897 14.27.05,944 05.42.36,166 08.31.48,257 02.36.23,591 22.39.45,207 14.55.32,645
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Tables 3. Instant times for approaching minimum distance for trajectories with numbers 1 and 2 from December 22, 2013 to July 20, 2018 j1 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 189 Date 24.XII.2013 5.II.2014 19.III.2014 1.V.2014 13.VI.2014 26.VII.2014 6.IX.2014 20.X.2014 2.XII.2014 13.I.2015 25.II.2015 10.IV.2015 23.V.2015 4.VII.2015 17.VIII.2015 30.IX.2015 11.XI.2015 24.XII.2015 7.II.2016 22.III.2016 5.V.2016 16.VI.2016 28.VII.2016 11.IX.2016 27.X.2016 15.XII.2016 2.II.2017 19.III.2017 29.IV.2017 13.VI.2017 1.VIII.2017 20.IX.2017 4.XI.2017 15.XII.2017 27.I.2018 17.III.2018 5.V.2018 21.VI.2018 18.VII.2018
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Time 09.40.44,974 16.31.07,614 14.26.31,051 01.37.26,802 18.09.55,794 10.55.44,955 10.27.10,341 00.57.55,052 09.15.50,866 07.59.44,944 09.38.59,431 16.33.56,866 00.42.12,985 08.08.39,040 21.14.04,220 13.56.42,976 07.25.53,389 04.32.56,136 04.22.14,270 19.57.40,505 05.52.07,501 00.51.07,116 23.55.49,689 14.22.31,448 14.51.41,697 16.23.21,437 21.38.29,864 05.20.40,978 22.09.38,920 12.56.31,166 23.32.57,999 13.23.42,452 00.24.30,866 08.07.26,097 20.22.18,898 00.16.24,622 10.26.18,450 01.11.45,195 22.45.19,348
2014

j2 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186

Date 24.XII.2013 5.II.2014 19.III.2014 1.V.2014 13.VI.2014 26.VII.2014 6.IX.2014 20.X.2014 2.XII.2014 13.I.2015 25.II.2015 10.IV.2015 22.V.2015 4.VII.2015 17.VIII.2015 30.IX.2015 11.XI.2015 24.XII.2015 7.II.2016 22.III.2016 5.V.2016 16.VI.2016 29.VII.2016 11.IX.2016 28.X.2016 17.XII.2016 2.II.2017 17.III.2017 28.IV.2017 14.VI.2017 3.VIII.2017 19.IX.2017 1.XI.2017 13.XII.2017 31.I.2018 22.III.2018 9.V.2018 28.VI.2018

Time 08.46.37,707 14.52.32,823 12.23.07,511 00.31.57,852 16.57.56,549 08.31.20,277 08.39.45,627 00.15.07,420 07.24.33,697 05.52.52,336 08.50.04,079 15.12.47,832 22.04.23,813 06.43.53,458 20.49.11,524 12.44.26,176 05.53.29,067 04.12.21,949 04.54.02,232 22.53.45,266 16.53.31,000 16.34.14,638 07.33.36,474 20.44.25,137 11.24.35,857 04.02.31,477 20.53.04,145 12.37.39,221 19.52.11,571 18.18.02,023 15.04.01,096 19.24.30,134 00.37.37,458 15.19.48,370 08.38.48,469 10.50.15,360 19.32.16,713 22.10.03,073

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model of the spacecraft motion. Moreover, the calcu lated data presented in this paper testify to the adequacy of the model for the ballistic motion of Spektr R space craft to correct its motion. ACKNOWLEDGMENTS The RadioAstron project is being performed by the Astro Space Center of Lebedev Physical Institute and Lavochkin Scientific and Production Association according to the contract with the Russian Space Agency, along with many scientific and technical organizations (including the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences) in Russia and other countries.

REFERENCES
1. Akim, E.L., Zaslavskiy, G.S., Stepan'yants, V.A., et al., Mashinostroenie. Entsiklopediya. Raketno kosmiches kaya tekhnika (Mechanical Engineering: Encyclope dia, Rocket and Space Technology), Legostaev, V.P., Ed., Moscow: Mashinostroenie, 2012. 2. Akim, E.L., Bazhinov, I.K., Zaslavskiy, G.S., et al., Navigatsionnoe obespechenie poleta orbital'nogo komple ksa Salyut 6­Soyuz­Progress (Navigation Support of Flight for the Orbital Complex Salyut 6­Soyuz­ Progress), Moscow: Nauka, 1985. 3. GOST (State Standard) R 25645.166­2004: Upper Atmosphere of the Earth: A Model of Density for Ballistic Support of Flights of the Earth's Artificial Satellites, 2004.

Translated by N. Topchiev

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No. 5

2014