One Class of Motions for a Satellite Carrying a Strong Magnet

Cosmic Research, Vol. 40, No. 2, 2002, pp. 133–141. Translated from Kosmicheskie Issledovaniya, Vol. 40, No. 2, 2002, pp. 147–155. Original Russian Text Copyright © 2002 by Sidorenko. One Class of Motions for a Satellite Carrying a Strong Magnet V. V. Sidorenko Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl., 4, Moscow, 125047 Russia Received October 30, 2000 Abstract—The attitude motion of an artificial satellite carrying a strong magnet is studied. The approximate first integrals of the problem, i.e., adiabatic invariants, are indicated. The basic properties of the satellite motions close to the regular precessions with slowly varying parameters are established via the analysis of the adiabatic invariants. 1. EQUATIONS OF MOTION The motion about the center of mass of a dynamically symmetric satellite carrying a strong magnet is considered. The satellite’s center of mass moves along a circular polar orbit. In the geomagnetic field, the satellite is under the action of the following mechanical moment [1] Mm = mm × H. The origin of the satellite-fixed frame of reference Oxyz is also at the center of mass O, and the Oz axis of this system coincides with the longitudinal axis of the body. In this frame of reference Here, mm is the satellite’s magnetic moment and H is the field strength vector. The magnitude and direction of vector H depend on the location of the satellite in the orbit, which is determined by the value of the anomaly ν = ω0(t – t0), where ω0 is the angular velocity of the orbital motion and t0 is the instant of passage through the orbit ascending node. The typical value of Mm is O(µemm /R3), where µe is the Earth’s magnetic moment, R is the radius of the orbit, and mm = |mm |. Let us assume that the magnet placed on the satellite is sufficiently strong so that |Mg |/|Mm | ~ Aµ/µemm ! 1, where Mg is the gravitational moment, A is the satellite’s transverse moment of inertia, and µ is the Earth’s gravitational parameter. In the following, the gravitational moment is not taken into account. Let us introduce two Cartesian frames of reference: magnetic and satellite-fixed. The origin of the magnetic frame of reference OXYZ is the satellite’s center of mass O, the OZ axis is directed along the vector of strength of the geomagnetic field H, and the OX axis is directed along the normal to the plane of the orbit. Due to variation of the direction of vector H when the satellite moves along the orbit, the magnetic frame of reference rotates around the OX axis with an angular velocity m m = ( m m, e x ), y Ω ( ν ) = 3 ( 1 + sin ν )/ ( 1 + 3 sin ν ). 2 The geomagnetic field is represented by the field of a dipole with the axis coinciding with the Earth’s rotation axis. z where ex, ey, and ez are the unit vectors of the axes Ox, Oy, and Oz and m m = ( m m, e y ), x m m = ( m m, e z ). y z Let us choose the axes Ox and Oy in such a manner that x m m = 0, m m = m m sin ε m , m m = m m cos ε m . y z Let us assume in the following that the magnet is located approximately along the satellite’s longitudinal axis (εm ! 1). We determine the orientation of the satellite-fixed frame of reference with respect to the magnetic coordinate system by the Eulerian angles ψ, ϑ, and ϕ (Fig. 1). Z y ϑ z Y O x ϕ X ω m ( ν ) = Ω ( ν )ω 0 , 2 x mm = mm e x + mm e y + mm ez , ψ Fig. 1. Frames of reference: OXYZ is the magnetic frame of reference and Oxyz is the satellite-fixed frame of reference. 0010-9525/02/4002-0133$27.00 © 2002 MAIK “Nauka /Interperiodica” 134 SIDORENKO Let us present the satellite equations of motion in the form of Hamilton equations for the canonically conjugate variables pψ, pϑ, pϕ, ψ, ϑ, ϕ, d pψ ∂H d p ∂H d p ∂H --------- = – -------, --------ϑ- = – -------, --------ϕ- = – -------, ∂ψ dτ ∂ϑ dτ ∂ϕ dτ (1.1) dϑ ∂H dϕ ∂H ∂H dψ ------- = ---------, ------- = --------- , ------ = --------- . dτ ∂ p ϑ dτ ∂ pϕ ∂ pψ dτ In (1.1), the dimensionless quantity τ = (µemm /AR3)1/2(t – t0) = ε–1ν is used as an independent variable, where ε = (Aµ/mmµe)1/2 ! 1. The anomaly ν appears in (1.1) as a slowly varying parameter. Let us assume that ε and εm are commensurable small quantities, εm = κmε, κm = O(1). In this case H = H 0 + εH 1 + R H , dϑ ------- = p ϑ , dτ pϕ cos ϑ dϕ - ( p ψ – p ϕ cos ϑ ) + -----. ------ = – -----------2 λ dτ sin ϑ Up to designations, Eqs. (2.1) coincide with the equations of motion of a Lagrange gyroscope, integrable in quadratures [4]. System (2.1) admits a two-parameter family of partial solutions p ψ ≡ p ψ0 , ϑ ≡ θ, p ϕ ≡ p ϕ0 , p ϑ ≡ 0, (2.2) ϕ = ω ϕ0 τ + ϕ 0 . ψ = Wτ + ψ 0 , The constants ψ0 and ϕ0 in (2.2) are arbitrary, and pψ0, pϕ0, ωϕ0, W, and θ are linked by the relations χ ( ν ) sin θ p ψ0 cos θ – p ϕ0 = ------------------------, W 2 1 ( p ψ – p ϕ cos ϑ ) 2 H 0 = --- ------------------------------------- + p ϑ + ----- – χ ( ν ) cos ϑ, 2 2 λ sin ϑ 2 2 pϕ sin ψ H 1 = Ω ( ν ) ------------ ( p ψ cos ϑ – p ϕ ) – p ϑ cos ψ sin ϑ – κ m χ ( ν ) sin ϑ cos ϕ, 2 p ϕ0 -. ω ϕ0 = – W cos θ + ------λ R H = O ( ε ). 2 The function χ(ν) = (1 + 3sin2ν)1/2 specifies the variation of the geomagnetic field strength along the orbit, and parameter λ = C/A is the ratio of the axial and transverse moments of inertia. The current status of the theory of attitude motion of a magnetized satellite is discussed in [1, 2]. In [3], using the methods of the theory of adiabatic invariants, the conditions under which the satellite motions in the plane of the orbit can be transformed from rotational to oscillating motions and vice versa were determined. In this work, the methods of the theory of adiabatic invariants are used to investigate another class of motions, namely, motions close to regular precessions. 2. SATELLITE MOTIONS CLOSE TO REGULAR PRECESSIONS 2.1. The Case ε = 0, ν = const This is the case of the motion of an axially symmetric magnetized body in a constant field. When ε = 0 and ν = const, Eqs. (1.1) take the form dp ---------ψ = 0, dτ Solutions (2.2) correspond to the motions where the satellite symmetry axis moves with a constant angular velocity W at a constant angular distance θ around the magnetic field line, and the satellite itself rotates around the symmetry axis with a constant angular velocity ωϕ0. Such motions are called regular precessions. Let us take the precession velocity W and the angle of nutation θ as parameters in family (2.2). We can pass to variables “action–angle” I = (I1, I2, I3), w = (w1, w2, w3) in integrable system (2.1) [5]. The relations determining the canonical transformation ( p ψ, p ϑ, p ϕ, ψ, ϑ, ϕ ) ( I 1, I 2, I 3, w 1, w 2, w 3 ) (2.3) in the vicinity of trajectories corresponding to regular precessions (I3 ≡ 0) can be written in the following form: pϕ = I 1 , pψ = I 2 , ∞ pϑ = ∑I k/2 3 P ϑk ( I 1, I 2, w 3, ν ), k=1 ∞ ϕ = w1 + ∑I k/2 3 ϕ k ( I 1, I 2, w 3, ν ), k/2 3 Ψ k ( I 1, I 2, w 3, ν ), k=1 ∞ ( p ψ – p ϕ cos ϑ ) ( p ϕ – p ψ cos ϑ ) d pϑ - – χ ( ν ) sin ϑ, --------- = – ----------------------------------------------------------------------3 dτ sin ϑ dp --------ϕ- = 0, dτ p ψ0 – p ϕ0 cos θ = W sin θ, ψ = w2 + ∑I k=1 ∞ dψ 1 ------- = ------------ ( p ψ – p ϕ cos ϑ ), 2 dτ sin ϑ (2.1) ϑ = θ ( I 1, I 2, ν ) + ∑I k/2 3 ϑ k ( I 1, I 2, w 3, ν ). k=1 COSMIC RESEARCH Vol. 40 No. 2 2002 ONE CLASS OF MOTIONS FOR A SATELLITE CARRYING A STRONG MAGNET cosθe 1 cosθp 1 0 0 –1 0 1 2 3 –1 0 1 2 We 135 3 Wp Fig. 2. Variation of the parameters of the reference regular precession during the motion of a satellite along the orbit. The series converge if the value of I3 is sufficiently small. The expressions for the coefficients of the series will not be needed in the following. Let us consider the solution to system (2.1) close to the regular precession with parameters W and θ. This solution and precession lie at the same common level of integrals pϕ and pψ. Let c be the amplitude of nutation oscillations in this solution. It was suggested in [6, 7] to use the quantities W, θ, and c as evolution variables in the study of the perturbed motion. The evolution variables W, θ, and c are linked to the “action” variables I1, I2, and I3 by the relations χ ( ν ) cos θ I 1 = W – ------------------------ , W χ( ν) I 2 = W cos θ – -----------, W c ω ϑ0 ( W , θ, ν ) 3 I 3 = ----------------------------------- + O ( c ), 2 2 (2.4) where ωϑ0 = √ ( W + 2W χ ( ν ) cos θ + χ ( ν ) )/W is the frequency of small nutation oscillations. We call the regular precession with parameters W and θ the reference precession for the solution with parameters W, θ, and c. 4 2 2 2 2.2. The Case ε ≠ 0, ν ≠ const The Hamiltonian of the considered system in the variables I and w takes the form K = K 0 + εK 1 + R K . (2.5) Here, K0(I1, I2, I3, ν) is the Hamiltonian of the Lagrange gyroscope, K 1 ( I 1, I 2, I 3, w 1, w 2, w 3, ν ) +∞ = ∑ ( 10 ) K n ( I 1, I 2, I 3, ν )e inw 3 n = –∞ COSMIC RESEARCH Vol. 40 No. 2 ∑ 2002 2 + ∑∑K ( 1k ) m, n ( I 1, I 2, I 3, ν )e i ( mw k + nw 3 ) , k = 1 m = –1 R K ( I 1, I 2, I 3, w 1, w 2, w 3, ν ) +∞ = ∑ ∑R K m, n ( I 1, I 2, I 3, ν )e i ( mw 1 + nw 3 ) = O ( ε ). 2 m = – 1.1 n = – ∞ It was shown in [8] that ∂ ( ω 1, ω 2, ω 3 ) -------------------------------≠ 0, ∂ ( I 1, I 2, I 3 ) where ωk = ∂K0 /∂Ik, k = 1, 2, 3. Consequently, system (1.1) is nondegenerate, and the variables I1, I2, and I3 are its almost adiabatic invariants [9]. It follows from the analysis of the Fourier-series expansion of the components of Hamiltonian K that the destruction of adiabatic invariants is connected with the passage through the resonances ω k ( I 1, I 2, I 3, ν ) + nω 3 ( I 1, I 2, I 3, ν ) = 0, (2.6) n = 0, ± 1, …, k = 1, 2. The destruction of adiabatic invariants at resonances will be studied in more detail in Sections 3 and 4 by means of a general method (suggested in [10]) of investigating resonance effects in multifrequency Hamiltonian systems depending on a slowly varying parameter. Let us consider a motion close to a regular precession in which the condition of resonance (2.6) does not hold at any value of the anomaly ν. Since the adiabatic invariants I1, I2, and I3 conserve their values in this case (to an accuracy O(ε)), relations (2.4) implicitly determine the variation of the evolution variables W, θ, and c, when the satellite moves along the orbit. Let We, θe and Wp, θp be the values of the parameters of the reference precession when the satellite moves above the equator and the pole, respectively. Figure 2 demonstrates what occurs when the mapping (We, θe) (Wp, θp) of the set of possible values of 136 SIDORENKO I1 2.0 1.8 1 1.6 2 1.4 3 1.2 –90 –45 South pole 0 45 90 135 North pole 180 ν 270 South pole 225 Fig. 3. Variation of I1 along the solutions with different initial data. The dashed line is the plot of the function Q(I2, ν) in the case I2 = 2. 682057. precession parameters on itself is carried out (symmetry with respect to the straight line W = 0 allows one to present only the region of positive values of the precession rate). The most essential variation of the precession parameters takes place when |We | ≈ 1 and θe ≈ π. When the parameters of the reference precession belong to the area shaded in Fig. 2, the amplitude of small nutation oscillations of the satellite above the equator exceeds the amplitude of similar oscillations above the pole. In the case |We | = 1 the variation of the evolution variables is described by the following formulas: W ( ν ) = ( sgn W e ) χ ( ν ), θ ( ν ) = arccos [ 1 + ( cos θ e – 1 )/ χ ( ν ) ], ω ϑ0 ( 1, θ e, 0 ) c ( ν ) ≈ c e -----------------------------------------------. ω ϑ0 ( W ( ν ), θ ( ν ), ν ) Here, ce is the value of the amplitude of nutation oscillations when the satellite moves above the equator. 3. MOTION OF A SATELLITE AT RESONANCE ω1 = 0 3.1. Numerical Results Numerical integration of Eqs. (1.1) demonstrates that some solutions halt in the vicinity of the resonance surface ω1(I1, I2, I3) = 0 for a prolonged time interval. This leads to the destruction of the adiabatic invariant I1. The examples are presented in Figs. 3 and 4 where the variation of I1 along the solutions with different initial data is plotted on the time interval corresponding to a single rotation of the satellite around the Earth. The following values of parameters were taken λ = 0.75, κ m = 1.31, ε = 0.001. Curve 1 in Fig. 3 corresponds to the motion in which, when the satellite moves above the South pole (ν = –90°), p ψ = 2.682057, p ϑ = 0.035575, p ϕ = 1.558129, ψ = 0.610334, ϑ = 0.836650, ϕ = 0.030491. (3.1) At ν ≈ 45°, the solution was captured into a resonance; at ν ≈ 135°, the solution left the vicinity of the resonance surface. The jumps of the invariant I1 at ν ≈ –45° and ν ≈ 225° are due to the passage through the resonance without capture. Curves 2 and 3 represent the behavior of the invariant I1 along the solutions with initial conditions differing only in the value of the angle of proper rotation, ϕ = 1.570796 and ϕ = –1.570796, respectively. No capture of the solution into a resonance is observed. COSMIC RESEARCH Vol. 40 No. 2 2002 ONE CLASS OF MOTIONS FOR A SATELLITE CARRYING A STRONG MAGNET 137 I1 2.0 1.8 1.6 1.4 1.2 1.0 0.8 –90 –45 South pole 0 45 90 135 North pole 180 225 ν 270 South pole Fig. 4. Variation of I1 along the solution that does not escape the neighborhood of the resonance surface ω1 = 0. The dashed line is the plot of the function Q(I2, ν) in the case I2 = 2.681933. Figure 4 corresponds to a solution with the following initial data p ψ = 2.681933, d I 1, 2, 3 /dτ ∼ ε, p ϑ = – 0.081161, p ϕ = 1.828881, In the ε -neighborhood of the resonance surface ω1(I1, I2, I2, ν) = 0, the following estimates hold: dw 1 /dτ ∼ ε, dw 2, 3 /dτ ∼ 1. This solution does not escape the resonance surface ω1 = 0. Following [10], let us begin to analyze the behavior of the solutions at a resonance from a canonical transformation (I, w) (J, v). This transformation is close to identical and has the generating function S1(J, w, ν) = w1J1 + w2J2 + w3J3 + O(ε). After this, the terms of the order of ε in the new Hamiltonian do not depend on the fast variables: The motion of the satellite in presented examples was close to the regular precession. At a resonance ω1 = 0, there is no rotation around the symmetry axis in such motions. ∂S Λ = K + ε --------1 ∂ν = Λ 0 ( J 1, J 2, ν ) + εΛ 1 ( J 1, J 2, v 1, ν ) + R Λ , (3.2) ψ = 2.134383, ϑ = 0.722098, ϕ = 0.106525. where In the space (I1, I2, I3) the surface I3 = 0 intersects the resonance surface ω1(I1, I2, I2, ν) = 0 along the curves determined by the equation ωϕ0(I1, I2, ν) = 0. When ν varies, these curves move slowly and sweep some regions on the plane I3 = 0. If for the solution to system (1.1) corresponding to a motion close to the regular precession (I3(τ) ! 1), the point (I1(τ), I2(τ)) is located in such a region, then the solution will be located in √ ε -neighborhood of the resonance surface for some value of the anomaly ν. Let us demonstrate that, in this case, the capture into a resonance observed in the calculations is actually possible under some conditions. COSMIC RESEARCH Vol. 40 No. 2 1 ( J 2 – J 1 cos θ ) J 1 Λ 0 = --- ---------------------------------- + ----- – χ ( ν ) cos θ, 2 2 λ sin θ 2 3.2. Analytical Investigation 2002 2 Λ 1 = – κ m χ ( ν ) sin θ cos v 1 , R Λ = O ( J 3 + ε ), 1/2 2 θ = θ ( J 1, J 2, ν ). This canonical transformation can be constructed by the method of Zeipel [11]. The behavior of the solutions to system (1.1) close to regular precessions at the resonance ω1 = 0 depends upon the properties of the system with a single degree of freedom with the Hamiltonian Λ res = Λ 0 ( J 1, J 2, ν ) + εΛ 1 ( J 1, J 2, v 1, ν ). (3.3) 138 SIDORENKO Let us now pass to the slow time ζ = cosθe 1 ∂ dp ∂Γ ------ = – ------- = L c /L a – ------- ( L b + R Γ ), ∂Φ dζ ∂Φ ∂R dΦ ∂Γ ------- = ------ = L a + --------Γ-, ∂p dζ ∂p 0 –1 1 (b) 1 2 Γ = --- L a p + L b – ( L c /L a )Φ + R Γ , 2 –1 R Γ ( p, J 2, Φ, ν, ε ) 1 = ε R N ( ε p, J 2, Φ, ν, ε ) = O ( ε ). –1 (c) System (3.5) is a “pendulum” system [2, 10]; i.e., when ν = const, Eqs. (3.5) resemble the equations of motion of a pendulum with a constant torque moment Lc /La. 0 –2 –1 0 We –1 (3.5) where 0 –3 ε τ in sys- tem (3.4). Introducing p = P/ ε , we arrive at (a) 1 2 3 Fig. 5. The sets of values of parameters of the reference regular precession at the instant of the satellite passage above the equator, at which in the motion close to the regular precession the resonances occur at a definite value of the anomaly: (a) ω1 = 0, (b) ω1 ± ω3 = 0, and (c) ω2 ± ω3 = 0. The ratio of the satellite’s longitudinal and transverse moments of inertia is λ = 0.75. The quantity J2 enters in (3.3) as a parameter. In order to put system (3.3) into the “pendulum” form [2, 10], let us make the canonical transformation (P, Φ) specified by the generating function (J1, v1) S2 = v1[P + Q(J2, ν)], where the function Q(J2, ν) is such that ∂Λ0 /∂J1(Q(J2, ν), J2, ν) = 0. In the new variables, the Hamiltonian gets the following form: ∂S 2 N = Λ res + ε --------2 = L 0 + 1/2L a P ∂ν + ε [ L b – ( L c /L a )Φ ] + R N . Here L 0 ( J 2, ν ) = Λ 0 ( Q ( J 2, ν ), J 2, ν ), ∂ Λ L a ( J 2, ν ) = ----------2-0 ( Q ( J 2, ν ), J 2, ν ), ∂J 1 2 L b ( J 2, Φ, ν ) = Λ 1 ( Q ( J 2, ν ), J 2, Φ, ν ), ∂ Λ0 - ( Q ( J 2, ν ), J 2, ν ), L c ( J 2, ν ) = -------------∂J 1 ∂ν 2 R N ( P, J 2, Φ, ν, ε ) = O ( ε ε ). (3.4) ∂L ∂L Let l– = min --------b and l+ = max--------b. If Lc /La ∈ (l−, l+), Φ ∂Φ Φ ∂Φ then, at ε = 0 and ν = const, there exist regions on the phase portrait of system (3.5) filled with closed trajectories corresponding to oscillatory solutions. This means that initial system (1.1) has solutions along which ω1 ≈ 0. If Lc /La ∉ (l–, l+), the pendulum system has no oscillatory solutions and a capture into a resonance in system (1.1) is impossible. When ν varies slowly, some phase trajectories of system (3.5) intersect the separatrix enveloping the region of oscillatory solutions of the system with a fixed value of this parameter. Let S(J2, ν) be the area of the region of oscillatory solutions. If dS/dζ = ε (dS/dν) > 0, the trajectories intersecting the separatrix enter in this region. Such trajectories correspond to the solutions to system (1.1) captured in the resonance. In the case dS/dζ = √ ε( dS/dν) < 0, the separatrix is intersected by the trajectories escaping the region of oscillatory solutions. In system (1.1) this corresponds to the exit of solutions from the resonance. The action–angle variables Σ and σ can be introduced in the region of oscillatory solutions to system (3.5) at ε = 0 and ν = const. The quantity Σ(p, Φ, J2, ν) is an adiabatic invariant of system (3.5) [10]. If a solution intersects the separatrix and begins to oscillate at ν = νin, then along this solution the adiabatic invariant Σ∗ ≈ S(J2, νin)/2π. In the opposite direction the solution will intersect the separatrix at ν = νout such that S(J2, νout ) ≈ 2πΣ∗. Let us write the conditions of the capture of a solution in a resonance in a more descriptive form using the evolution variables W, θ, and c introduced in Section 2.1. COSMIC RESEARCH Vol. 40 No. 2 2002 ONE CLASS OF MOTIONS FOR A SATELLITE CARRYING A STRONG MAGNET Instead of a set of the “resonance” values of adiabatic invariants I1 and I2 (Section 3.1), we consider a set of “resonance” values of the parameters of the reference regular precession when the satellite moves above the equator. If We and θe belong to this set, then the condition of the resonance ω1 ≈ 0 will hold at a later time in the motion close to the regular precession with some value of the anomaly ν. “Resonance” sets on the plane (I1, I2) and on the plane (We, θe) transform into each other under the oneto-one mapping cos θ e I 1 = W e – -------------, We 1 I 2 = W e cos θ e – ------- . We Example. For the solution with initial data (3.1), the parameters of the reference precession with passage of the equator We = 2.90499 and θe = 0.87338 (cosθe = 0.64225). These values of the parameters belong to the “resonance” set pictured in Fig. 5a. Let the value of the anomaly be ν = νres and the value of the angle of nutation be θ = θres at the instant of the entrance of the solution into the ε -neighborhood of the resonance surface (note that 0 < θres < π/2 for the satellite with a prolate ellipsoid of inertia (0 ≤ λ < 1) and π/2 < θres < π for the satellite with an oblate ellipsoid of inertia (1 < λ ≤ 2)). The quantities νres and θres are related to the adiabatic invariants I1 and I2 by the formulas χ ( ν res ) cos θ res I 1 ≈ λ --------------------------------, 1–λ χ ( ν res ) 2 I 2 ≈ [ 1 + ( λ – 1 ) cos θ res ] ---------------------------------. ( 1 – λ ) cos θ res The following relations hold: ∂ Λ ∂θ 2 2 1 2 L a = ----------2-0 = --- + cot θ res – ω ϑ0  -------- ,  ∂J 2 λ ∂J 1 2 l + = – l – = κ m χ ( ν res ) sin θ res , ∂ Λ0 dχ ∂θ L c = -------------- = – sin θ res ------ -------- , dν ∂J 2 ∂J 1 ∂ν 2 where 1 + ( 1 – λ ) cos θ res χ ( ν res ) ∂θ - ---------------------------------------- = – --------------------------------------------, 2 ∂J 2 ( 1 – λ ) cos θ res ω ϑ0 sin θ res 2 2 ω ϑ0 = 2 ω 3 ( I 1 ( θ res, ν res ), I 2 ( θ res, ν res ), 0, ν res ) χ ( ν res ) 2 - [ 1 + ( λ – 3 ) ( λ – 1 ) cos θ res ]. = --------------------------------( 1 – λ ) cos θ res COSMIC RESEARCH Vol. 40 No. 2 2002 139 θres π/2 π/4 νres π/4 0 π/2 3π/4 π 1 2 Fig. 6. The conditions of capture into the resonance ω1 = 0 and the conditions of escape from this resonance: (1) is the region of the values of νres, θres, which make the capture in the resonance possible, and (2) is the region of the values of νres, θres, at which only the escape of solutions from the resonance is observed. The quantities La, Lc, l–, and l+ are π-periodic functions of νres. The condition Lc /La ∈ (l–, l+) is satisfied in some region of the plane (ν, θ). As an example, this region is constructed in Fig. 6 for a satellite with the parameters λ = 0.75 and κm = 1.31. The adiabatic invariant I2 is conserved at the resonance. The relation I2(ϑ, ν) ≈ I2(θres, νres) implicitly determines the variation of the angle of nutation in the solutions that are captured in a resonance. Figure 6 presents the curves specified by this relation at different values of θres and νres. The variation of I1 along the solutions captured in a resonance looks like oscillations with an amplitude O( ε) with respect to I *1 = Q(I2, ν) + RQ, RQ(I2, I3, ν) = O(I3) (Figs. 3 and 4). The area S(θres, νres) = S(I2(θres, νres), νres) of the region of oscillatory solutions on the phase portrait of the system Γ = Lap2/2 + Lb – (Lc /La)Φ is also a π-periodic function of νres, and S ( θ res, ν res ) = S ( θ res, π – ν res ), ∂S ------ ( I 2 ( θ res, ν res ), ν res ) ∂ν ∂S = – ------ ( I 2 ( θ res, π – ν res ), π – ν res ). ∂ν The regions in Fig. 6 where ∂S/∂ν > 0 and the capture of solutions in the resonance is possible and the regions where ∂S/∂ν < 0 and capture is impossible, differ by the type of hatching. If νin = νres is the value of the anomaly at the instant of capture, the value of the anomaly at the instant of exit from the resonance is νout ≈ π – νin. The 140 SIDORENKO value of the angle of nutation at the instant of exit and the value of the angle of nutation immediately before the capture coincide within the error O( ε |lnε| + I3). The probability of capture Pcap(θres, νres), whose formal definition is given in [12], specifies a fraction of solutions captured in a resonance at θ = θres and ν = νres in some totality of the solutions to system (1.1) with close initial conditions. Taking into account the results of [12], we arrive at P cap ( θ res, ν res ) = 1 L a ∂S -----ε ------ ---2π L-c ∂ν with the proviso that L c /L a ∈ ( l –, l + ), ∂S/∂ν > 0. 4. MOTION OF A SATELLITE AT THE RESONANCES ωk + nω3 = 0 (n = ±1, …; k = 1, 2) Let us consider the behavior of adiabatic invariants in the ε -neighborhood of the resonance surface ω1 + nω3 = 0 (n = ±1, …). The case ω2 + nω3 = 0 (n = ±1, …) can be studied in a similar manner. As a first step towards our goal, let us make the change of variables (I, w) ( Ĩ , w̃ ), which takes into account the topological variations of the adiabatic invariants at a resonance. Let us introduce the “resonance” variables by a canonical transformation with the generating function S 3 = w 1 Ĩ 1 + w 2 Ĩ 2 + ( w 1 + nw 3 )Ĩ 3 . In this case, where Λ̃ 0 = Λ 0 ( J̃ 1 + J̃ 3, J̃ 2, ν ) + O ( J̃ 3 ), ( 10 ) Λ 1 = K 0 ( J̃ 1 + J̃ 3, J̃ 2, nJ̃ 3, ν ) ( 11 ) + K –1, –n ( J̃ 1 + J̃ 3, J̃ 2, nJ̃ 3, ν )e ( 11 ) + K 1, n ( J̃ 1 + J̃ 3, J̃ 2, nJ̃ 3, ν )e – i ṽ 3 i ṽ 3 , R Λ̃ = O ( ε ). 2 The qualitative behavior of the solutions to system (1.1) that are close to the regular precessions at the resonance ω1 + nω3 = 0 is dependent on the properties of the system with a single degree of freedom with the Hamiltonian Λ̃ res = Λ̃ 0 ( J̃ 1, J̃ 2, J̃ 3, ν ) + εΛ̃ 1 ( J̃ 1, J̃ 2, J̃ 3, ṽ 3, ν ). (4.1) The quantities J̃ 1 and J̃ 2 appear in expression (4.1) as parameters. The transformations differing from those considered in Section 3.2 only by designations allow one put system (4.1) in the “pendulum” form ∂ d p̃ ------ = L̃ c /L̃ a – ------- ( L̃ b + R Γ̃ ), dζ ∂Φ̃ (4.2) ∂R dΦ̃ -------- = L̃ a + --------Γ̃-. ∂ p̃ dζ Here, Φ̃ = ṽ 3 and p̃ = ( J̃ 3 – Q̃ ( J̃ 1 , J̃ 2 , ν))/ ε (the function Q̃ ( J̃ 1 , J̃ 2 , ν) is implicitly determined by the I 1 = Ĩ 1 + Ĩ 3 , w 1 = w̃ 1 , I 2 = Ĩ 2 , w 2 = w̃ 2 , I 3 = nĨ 3 , relation ∂ Λ̃ 0 /∂ J̃ 3 ( J̃ 1 , J̃ 2 , Q̃ ( J̃ 1 , J̃ 2 , ν), ν) = 0), w 1 = ( w̃ 3 – w̃ 1 )/n. ∂ Λ̃ L̃ a = ----------2-0 ( J̃ 1, J̃ 2, Q̃ ( J̃ 1, J̃ 2, ν ), ν ), ∂J̃ 1 2 The new Hamiltonian is K̃ ( Ĩ, w̃, ν, ε ) = K ( I ( Ĩ ), w ( w̃ ), ν, ε ). In the ε -neighborhood of the resonance surface ω1 + nω3 = 0 d Ĩ 1, 2, 3 /dτ ∼ ε, dw̃ 1, 2 /dτ ∼ 1, dw̃ 3 /dτ ∼ ε. Let us eliminate in the Hamiltonian the terms of the order of ε depending on the fast variables taking advantage of a canonical transformation ( Ĩ , w̃ ) ( J̃ , ṽ ) with the generating function S4, which is close to the identical transformation ∂S Λ̃ = K̃ + ε --------4 = Λ̃ 0 ( J̃ 1, J̃ 2, J̃ 3, ν ) ∂ν + εΛ̃ 1 ( J̃ 1, J̃ 2, J̃ 3, ṽ 3, ν ) + R Λ̃ , ( 11 ) L̃ b = K –1, –n( J̃ 1 + Q̃( J̃ 1, J̃ 2, ν), J̃ 2, nQ̃( J̃ 1, J̃ 2, ν), ν)e ( 11 ) – iΦ̃ + K 1, n ( J̃ 1 + Q̃ ( J̃ 1, J̃ 2, ν ), J̃ 2, nQ̃ ( J̃ 1, J̃ 2, ν ), ν )e , iΦ̃ ∂ Λ̃ 0 - ( J̃ 1, J̃ 2, Q̃ ( J̃ 1, J̃ 2, ν ), ν ), R Γ̃ = O ( ε ε ). L̃ c = -------------∂J̃ 1 ∂ν 2 The following estimates are valid L̃ a ∼ 1, L̃ c ∼ 1. Assuming that ε = 0 and ν = const, system (4.2) in the case J̃ 3 = Q̃ ( J̃ 1 , J̃ 2 , ν) ! 1 has no oscillatory solu( 11 ) ( 11 ) tions, since the Fourier coefficients K –1, –n and K 1, n appearing in the formula for L̃ b have the value O( J̃ n /2 ). This indicates the impossibility of capture of COSMIC RESEARCH Vol. 40 No. 2 2002 ONE CLASS OF MOTIONS FOR A SATELLITE CARRYING A STRONG MAGNET the solutions to system (1.1) close to the regular precessions into the resonance ω1 + nω3 = 0. The impossibility of capture of this class of solutions into the resonance ω2 + nω3 = 0 (n = ±1, …) can be proved similarly. For the case of the satellite’s motion above the equator, when the motion is close to the regular precession, Figs. 5b and 5c, respectively, represent the sets of values of the parameters of the reference regular precession for which the conditions of resonances ω1 + ω3 ≈ 0 and ω2 + ω3 ≈ 0 are satisfied for some ν. When system (1.1) passes through the resonance ωk + nω3 = 0 (n = ±1, …; k = 1, 2), the values of adiabatic invariants Ik and I3 vary by the value O( ε ). It is assumed that such “jumps” of adiabatic invariants at resonances have a statistically independent nature [10]. The diffusion of the adiabatic invariants during the multiple passage through the resonance results in violation of the condition I3 ! 1 for almost all solutions after a sufficiently long, but finite, time interval. The behavior of the system at a resonance in the general case (I3 ~ 1) calls for special investigation. 3. 4. 5. 6. 7. 8. 9. 10. ACKNOWLEDGMENTS This work was supported by the INTAS, project 00-221, and RFBR, project 00-00-00174. 11. REFERENCES 1. Beletskii, V.V. and Khentov, A.A., Vrashchatel’noe dvizhenie namagnichennogo sputnika (Rotational Motion of a Magnetized Satellite), Moscow: Nauka, 1985. 2. Sarychev, V.A. and Ovchinnikov, M.Yu., Magnetic Attitude Control Systems for Earth’s Satellites, Itogi Nauki COSMIC RESEARCH Vol. 40 No. 2 2002 12. 141 Tekh., Ser.: Issled. Kosm. Prostr., Moscow: VINITI, 1985, vol. 23, pp. 3–105. Pivovarov, M.L., Rotation of a Satellite with a Large Magnetic Moment, Kosm. Issled., 1986, vol. 24, no. 6, pp. 816–820. 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