Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2 Текст научной статьи по специальности «Физика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 651-660. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220803

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 74H10, 74H40, 74H55

Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2

This paper considers the rotational motion of a satellite equipped with flexible viscoelastic rods in an elliptic orbit. The satellite is modeled as a symmetric rigid body with a pair of flexible viscoelastic rods rigidly attached to it along the axis of symmetry. A planar case is studied, i.e., it is assumed that the satellite's center of mass moves in a Keplerian elliptic orbit lying in a stationary plane and the satellite's axis of rotation is orthogonal to this plane. When the rods are not deformed, the satellite's principal central moments of inertia are equal to each other. The linear bending theory for thin inextensible rods is used to describe the deformations. The functionals of elastic and dissipative forces are introduced according to this model. The asymptotic method of motions separation is used to derive the equations of rotational motion reflecting the influence of the fluctuations, caused by the deformations of the rods. The method of motion separation is based on the assumption that the period of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time of these oscillations' decay, which, in its turn, is much smaller than the characteristic time of the system's motion as a whole. That is why only the oscillations induced by the external and inertial forces are taken into account when deriving the equations of the rotational motion. The perturbed equations are described by a third-order system of ordinary differential equations in the dimensionless variable equal to the ratio of the satellite's absolute value of angular velocity to the mean motion of the satellite's center of mass, the angle between the satellite's axis of symmetry and a fixed axis and the mean anomaly. The right-hand sides of the equation depend on the mean anomaly implicitly through the true anomaly. A new slow angular variable is introduced in order to perform the averaging for the perturbed system near the 3:2 resonance, and the averaging is

Received March 14, 2022 Accepted July 08, 2022

Albina V. Shatina shatina_av@mail.ru Maria I. Djioeva mdjioeva@yandex.ru

MIREA - Russian Technological University, pr. Vernadskogo 78, Moscow, 119454 Russia

Lyubov S. Osipova l.s.shatina@gmail.com

Lomonosov Moscow State University

ul. Vorobjevy gory 1, Moscow, 119992 Russia

A. V. Shatina, M. I. Djioeva, L. S. Osipova

performed over the mean anomaly of the satellite's center of mass orbit. In doing so the well-known expansions of the true anomaly and its sine and cosine in powers of the mean anomaly are used. The steady-state solutions of the resulting system of equations are found and their stability is studied. It is shown that, if certain conditions are fulfilled, then asymptotically stable solutions exist. Therefore, the 3:2 spin-orbital resonance capture is explained.

Keywords: Keplerian elliptical orbit, satellite, spin-orbit resonance, dissipation

1. Introduction

The 3:2 spin-orbital resonance is a type of satellite's motion, where its angular velocity with respect to its center of mass is 1.5 times larger than the mean orbital motion of its center of mass. Until the mid-twentieth century it was assumed that Mercury's orbital period around the Sun is equal to its rotation period. But the data obtained through radar observation [1] suggested that in fact Mercury is locked in a 3:2 spin-orbital resonance. This discovery leads to the numerous studies of the spin-orbital resonance dynamics [2]. The studies of resonances occurring in a planar motion of a satellite (a rigid body) in an elliptic orbit are described in [3-7].

Dissipation is the most important factor causing an arbitrary motion of a system to become resonant. In [8] numerical analysis is used to obtain the resonant types of motion for one model of a satellite with internal dissipation.

In this paper we use the asymptotic separation of motion and averaging methods for mechanical systems with infinite degrees of freedom [9] to obtain a system of second-order differential equations describing the evolution of the rotational motion of a satellite in the proximity of the 3:2 spin-orbital resonance. The satellite is modeled by a dynamically symmetric rigid body with viscoelastic rods rigidly attached to it along the axis of symmetry. The central tensor of inertia of the satellite in the nondeformed state of the rods is spherical. Previously in [10] the described model was used to explain the 1:1 spin-orbital resonance capture.

2. Problem statement. The equations of perturbed motion

Let us consider the motion of a satellite in the central Newtonian gravitational force field. The satellite is modeled by a dynamically symmetric rigid body with viscoelastic rods of length d rigidly attached to it along the axis of symmetry. The linear density of the rods p is assumed to be constant. We consider the case where the satellite's principal moments of inertia are equal to each other if the rods are not deformed.

Let us introduce an inertial coordinate frame OXYZ with its origin at the gravitational center, a coordinate frame Cxlx2x3 rigidly attached to the satellite, and the Koenig coordinate frame C^^^. Here C is the satellite's center of mass in the case of nondeformed rods. In this case the rods are straight and are aligned to the Cxl axis. The axes of the coordinate frame are parallel to the corresponding inertial axes at all times. Since the linear

dimensions of the satellite are small relative to the characteristic dimensions of the orbit, the motion of the satellite relative to its center of mass does not influence the motion of the center of mass itself [3].

Let us consider the planar case, when the satellite's center of mass C moves in a Keplerian elliptical orbit lying in the OXY plane, and the axis of the satellite's rotation is orthogonal to this plane. We choose the OX axis so that it contains the pericenter and the Cxl axis is

orthogonal to the orbital plane. The points belonging to the rod are moving in the Cx1x2 plane coinciding with the orbital plane (Fig. 1).

Fig. 1. Problem statement

The radius vector R of point C in the coordinate frame OXYZ is as follows:

R = R(cos tf; sin tf; 0),

where R = |R|, tf is the true anomaly. The values of R, tf are the given functions of time:

R =

a (1 - e2) - dtf • (1 + ecos tf)2

—;-tf = —I =-zrrih n

1 + ecos tf dl (i - e2)3/2

7 \ 1/2

-I J , l = n(t-t0).

Here a is the major semiaxis of the radius vector R orbit, e is the eccentricity, l is the mean anomaly, n is the mean motion of point C, y is the gravitational constant, t is the current instant in time, and t0 is the instant when point C passes the pericenter (a, e and n being constant).

The radius vector of an arbitrary point of the rod in the coordinate frame CX1X2X3 can be written as follows [11]:

r rod

= r + u, u = u(s,t)e2, r = sei,

s e K = [-b - d < s < -b] U [b < s < b + d],

where u(s, t) is the deviation of the rod's section with coordinate s when the rod is bent, and ei is the unit vector aligned with the Cxi (i = 1, 2, 3) axis. Let us denote the transformation operator from the frame Cx1x2x3 to the frame C(1(2(3 as r, and the angle between axes Cx1 and C(1 as p (Fig. 1). Then

r

cos y — sin (f

ü\

V

sin p cos p 0

0 0 iy

Let us denote the area of three-dimensional Euclidean space occupied by the satellite with nondeformed rods as V. The radius vector of a point M belonging to the satellite can be written in the inertial frame OXYZ as

R

■m

R + r(r + u), r e V.

(2.1)

For the points belonging to the rigid part of the satellite we assume u = 0. The expression for the velocity of point M can be derived from (2.1):

vM = R + r[pe3 x (r + u) + U].

The kinetic energy of the satellite is given by the functional

11

/2

M

V V

r=2/vM^=2 / r-'R + ipea x (r + u) + u dp,

where i is the measure function on area V. The potential energy is given by

V

Let us transform the right-hand side of (2.2) taking into account that |r + u| ^ R. Keeping quadratic terms with respect to and linear terms with respect to we get

n = + ^ /«. u)Pds ~ § /«. u)Pds- (2-3)

K K

Here

r

£ = T"1— = (cos(i? - <p), sm(§ - <p), 0). R

While deriving formula (2.3) it was also considered that

J(xl + x3) di = J(xl + x|) di = J(x\ + x3) di = A,

where A is the central moment of inertia of the satellite with nondeformed rods with respect to the axis Cx^, i = 1, 2, 3.

We will use the linear bending theory for thin inextensible rods. The functionals of potential energy for elastic deformations and dissipation are as follows:

£ =

2 J \ds2 J ~ 2

KK

where N is the bending stiffness of the rods and % (x > 0) is the constant characteristic of the dissipation of energy when the rod is bent. Let us define the generalized impulse I:

I = T = pj [e3 x (r + u)]2 di + J (u x r_lR + (r + u) x U, e3) pds.

VK

Let us introduce the dimensionless variable k = In the absence of deformations this variable is equal to the ratio of the angular velocity of the satellite to the mean orbital motion of its center of mass.

2

In [10] a system of equations for the rotational motion of a satellite in an elliptical orbit was obtained via the asymptotic method of motion separation [9, 12], taking into account perturbations caused by elasticity and dissipation. The method of motion separation is based on the assumption that the period T1 of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time T2 of these oscillations' decay, which, in its turn, is much smaller than the characteristic time T3 of the system's motion as a whole:

Tl « T2 « T3. (2.4)

Therefore, when studying the evolution of the system during intervals T3 or larger it is possible to ignore the autonomous oscillations of the points of viscoelastic material and consider only the oscillations induced by the external and inertial forces.

The bending rigidity N of the rods is assumed to be "large", which allows us to introduce a small parameter e inversely proportional to the bending rigidity:

e = pw0d4 N-1 « 1,

where w0 is the constant limiting the absolute value of the satellite's initial angular velocity. Conditions (2.4) lead to the following relations between the problem's parameters:

0 < e « xn « 1.

The perturbed system of equations is as follows:

Here

3

k = -tin^i, fi = kn — £1 nF2, l = n.

_ pdn2 (6362 + 91 bd, + 33d2) t_1 = 21ÖÄw2

Fi = Ft(k, fi, 0(l)), i = 1, 2, F1 = Gf cos ß{ (1 + 3xnG3) sin ß — 2xn(k — G2) cos ß},

F2 = gA Ig3 sin ß — (k — G2) cos ß-

(2.5)

xn

3 2

-G4 sin ß + 2G3 (3k - 4G2) cos ß + 2(k - G2)2 sin ß

(1 + ecostf)3 (1 + ecostf)2 esintf(l + ecostf)

1= (1 — e2)3 ' G2= (1-e2)3/2 ' 3 = (1 - e2)3/2 '

e(1 + ecos tf)2 [cos $(1 + ecos tf) — 4esin2tfl

G4 = ---i-=---P = 2ip-2ti.

4 (1-e2)3

The right-hand sides of Eqs. (2.5) depend on the mean anomaly l through the true anomaly tf. Leaving only no higher than second-order terms with respect to eccentricity e,

we get [2]

ti = l + 2e sin l + 1.25e2 sin 2l + o (e2),

e2(9sin3l — 7sinl) ,

sin v = sin I + e sin 21 H-----h o (e j, (2 6)

9e2(cos3l — cos l) , 2n

cos § = cos Z + e(cos 21 - 1) H------- + 0 (e2).

8

Next, we assume that

e2 < xn. (2.7)

3. The evolution of the satellite's motion near the 3:2 spin-orbital resonance

Let us introduce a new angular variable

In the proximity of the resonance in question ^ is a "slow" angular variable. Let us transform the right-hand sides of the perturbed system (2.5), using expansions (2.6) and keeping only the

terms of no higher than second order with respect to eccentricity e. As a result we get

3

k = -exnQiik, 0, I),

ip=(k-^jn-e1nQ2(k, 0, I), i3-1)

3 2

l = n.

Here

Qi(k, 0, l) = h0 — xn[hik — hi], Q2(k, 0, l) = b0 — b±k — xn [62 + 63k + bAk2], hi = ht(^,l), i = 0, 1, 2; bj = bj(0, l), j = 0, 1, ..., 4,

1e M'0, 1) = 2 s[n('21 + + 2 t7sin^ + " + +

e2

+—[117 sin 40 - 34 sin(2Z + 40) + sin(4Z + 40)], 8

h1 (0, l) = 1 + cos(2l + 40) + e[6cosl + 7cos(l + 40) — cos(3l + 40)] + e2

+—[30 + 54 cos 21 + 117 cos 40 + cos(4Z + 40) - 34 sin(2Z + 40)],

e

/?,2(0, I) = 1 +cos(2Z + 40) + -[32cos / + 35cos(Z + 40) - 3cos(3Z + 40)] + e2

+—[108 + 176 cos 21 + 351 cos 40 + cos(4Z + 40) - 68 cos(21 + 40)], 8

e e2

6q (0, I) = cos (I + 20) + -[21 cos 20 - cos(2/ + 20)] + y [34 cos (I - 20) - 5 cos (I + 20)],

e e2

6^0, /) = cos(Z + 20) + -[7cos20 -cos(2/ + 20)] + — [17cos(Z - 20) - 5cos(Z + 20)],

g

62(0, I) = 2sin(Z + 20) + -[63sin20 - 17sin(2/ + 20)]-

—g2[4sin(3l + 20) + 48sin(l - 20) + 9sin(l + 20)], b3(0, l) = —4sin(l + 20) + e[9sin(2l + 20) - 21 sin 20] + g2[68sin(l - 20) + 10sin(l + 20)], b4(0, l) = 2sin(l + 20) + e[7sin20 — sin(2l + 20)] — e2[17sin(l — 20) + 5sin(l + 20)].

Now let us perform the averaging for the perturbed system (3.1). Near the 3:2 resonance the angular variable 0 becomes a slow one. So, we shall average over the fast angular variable l. We shall use the same notations for the averaged variables as for the initial ones:

2n 2n

k = J ^nQ^k, 0, I) dl, 0 = ^k J £i nQ2(k, 1) dl- (3-2)

0 0

After calculating the integrals on the right-hand sides of (3.2), we arrive at the system of second-order differential equations describing the evolution of the rotational motion of the satellite in the elliptical orbit:

3 f117 2

k = 1 <J ^r—e2 sin 40 — xn(k — 1) + O (xne2) >,

J (3-3)

0 = n ( k - ^ ) <j 1 + exe

^ cos 20 + 7xn (k — ^ ) sin 20

4. Steady-state solutions of the evolutionary system and their stability

Equating the right-hand sides of (3.3) to zero, we can find the steady-state solutions of the system:

»4

2 (4.1)

. „ , 4xn

sm40 = q, Q =

If q > 1, then equation (4.1) has no solutions. If q = 1, then

n nk ,

0 = - + —, ke Z.

If q < 1, then (4.1) has two series of solutions:

, a nk n — a nk

0! = - + —, 02 = —— + —, k G Z, a = arcsm q.

It is important to point out that the condition q < 1 imposes certain constraints on the orbit's eccentricity:

2> fx« 117"

Taking into consideration condition (2.7), we get the following estimate for the eccentricity:

< e2 < xn. (4.2)

Let us study the stability of the steady-state solution obtained above. First let us consider the solution k = §, tp = Setting k = f + y1, tp = + y2 in (3.3), we get the following first-order approximation system of equations:

y1 = a1lVl + al2V2^ y2 = a21 Vl- (4-3)

3 2 351elne2 du = ~2SlXn K a'12 =-4-cosa >

f 7el e fa , \ 1

a2i = " H—2~ cos v"2 + J | >

The characteristic equation of system (4.3) is as follows:

l11 - A a12 a21 -A

= 0 ^ A2 - a11A - a12a21 = 0. (4.4)

The discriminant of Eq. (4.4) is positive: D = a21 + 4a12a21 > 0, therefore the characteristic polynomial has two distinct real roots A1 and A2. According to Viete's theorem, A1A2 = = — a21a12 < 0. Therefore, the roots A1 and A2 have different signs, and so the steady state solution k = tp = is unstable.

In the case of the steady-state solution k = tp = tp2 the first-order approximation system of equations takes the form (4.3) with the following coefficients:

3 2 351e1ne2 an = ~2eiXn < 0, a12 =----cos a < 0,

a-2i = n j1 + -y- sin - ^k) | > 0.

All coefficients in the characteristic equation (4.4) are positive, therefore, the steady-state solu-t.ion k = |, tp = tp2 is asymptotically stable [13].

Figure 2 presents the integral curve for the system of differential equations (3.3) in the (0, k) plane for the following values of the problem's parameters: e1 = 0.005, xn = 0.05, e = = 0.17. In this case q = 0.05915 < 1, f = 0.0148, ^ = 0.7706. For these values the condition (4.2) holds. The initial values are taken in the proximity of the unstable steady state solution k = |, tp = j: k(0) = 1.501, 0(0) = 0.015. The integration time is equal to the 100 orbital rotations of the satellite. The integral curve is an intertwining spiral. The attraction point is the asymptotically stable steady-state solution k = ■§, tp =

5. Conclusion

The model problem considered above explains the 3:2 spin orbital resonance capture of a celestial body with dissipation.

In this model problem the 1:1 spin orbital resonance capture can happen for an arbitrary small (including zero) value of the eccentricity of the satellite's center of mass orbit [10]. In this paper it is demonstrated that the existence of a stable steady-state motion near the 3:2 resonance requires condition (4.2), e.g., the value of the eccentricity squared must be limited not only from above, but also from below.

In the Solar system the motion of Mercury is the most well-known example of a 3:2 spinorbital resonance. It should be pointed out that the Mercury's orbit has the eccentricity of 0.2056, which is one of the largest among the eccentricities of the planetary orbits in the Solar system.

Conflict of interest

The authors declare that they have no conflict of interest.

References

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