Investigation on a doubly-averaged model for the Molniya satellites orbits

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Universit`

a degli Studi di Universit`

a di Pisa

FACOLT `

A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea Magistrale in Matematica

Investigation on a Doubly-Averaged Model for the Molniya

Satellites Orbits

Candidato:

Tiziana Talu

Relatori:

Giacomo Tommei

Elisa Maria Alessi

Correlatore:

Giulio Ba´

u

Anno Accademico 2019-2020

Controrelatore: Relatori: Candidato:

Università di Pisa

Corso di Laurea Magistrale in Matematica

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Introduction

On April 23, 1965 the first Molniya-1 spacecraft was launched by former Soviet Union, [1]. After that many others were launched until 2004. These satellites were initially designed for Russian communication networks and their orbits form a class of special orbits around Earth: the Molniya orbits. Let us call generically Molniya satellite a object orbiting along a Molniya type orbit.

The main dynamical features of Molniya orbits are: • period of approximatively 12 hours

• eccentricity e ≥ 0.7 • inclination i ≈ 63.43 deg

• argument of perigee ω = 270 deg

Figure 1: Sub satellite ground track of a Molniya satellite.

Why did they choose such orbital elements? The territory to cover was, and still is, enormous and is located at a high latitude, thus a high eccentricity and a quite stable apogee above the re-gion of interest are needed. A constellation of at least three Molniya satellites provide a continuous

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2 CONTENTS coverage. The value of the inclination used is close to the critical inclination value; in this way the oblateness of our planet does not induce a precession of the line of apsides, therefore apogee and perigee are almost frozen because ω = 270 deg is a stable position. Finally the time interval between two passages through the ascending node must be half of sidereal day to ensure that the ground track repeats every 24 hours.

These orbits are particularly interesting from a dynamical point of view. The orbital period of a Molniya satellite is commensurable with the Earth’s rotation period: each day a Molniya satellite revolves around the Earth two times. The consequence is a 2 : 1 tesseral resonance1 _{whose effects}

couple with the critical inclination resonance effects. Moreover, a Molniya satellite undergoes sev-eral perturbations. The low value of the altitude of the perigee gives a non-negligible atmospheric drag, which deeply affects the evolution of the semi-major axis. Besides, the satellite spends the most of the time at high altitudes, thus the lunisolar effects play a fundamental role on timescale larger than a satellite orbital period.

Some Molniya satellites launched before 1974 experienced a quick decay, but the satellites launched after 1974 did not. Molniya orbits are considered quite chaotic, that is, the dynamical evolution strongly depends on the initial conditions. Sometimes the chaotic growth of the eccen-tricity leads to quite low value of the perigee altitude, and, if the perigee altitude decrease below a certain value, the satellite decays. Moreover, the satellite operative lifetime is heavily influenced by the available propellant reserves: a Molniya satellite needs frequent station-keeping maneuvers [5]. The propellant typically exhausted in two years; sometimes it was possible to extend the lifetime of a few years.

The purpose of this thesis is to investigate the long-term effects caused by the lunisolar pertur-bation on a Molniya satellite dynamics. The work will be structured as follows. Ch. 1 is a short presentation of the mathematical theoretical tools that are useful to deal with the specific problem of a Molniya satellite. Ch. 2 contains an overview of the Molniya orbits. Some previous results found in the literature are also presented. In the second part of Ch. 2 the analytical doubly-averaged model for a Molniya satellite is built, up to the third order expansions of the lunisolar disturbing functions. Ch. 3 exhibits some numerical investigations on the doubly-averaged lunisolar model in order to identify the dominant perturbing terms. The main resonant terms, selected in Ch. 3, are studied as isolated resonances in Ch. 4. The aim of Ch. 4 is to identify a possible resonances overlapping region in the proximity of the Molniya orbital environment. Finally, Ch. 5 provides an overview of the main results obtained so far, and offers some further investigations which may be the topic of future works. All the numerical results are achieved by using Mathematica.

1_{It is called mean motion resonance in [24], but it does not arise from a commensurability between mean motions.}

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Chapter 1

Theoretical Background

1.1 The Two-Body Problem and the Perturbed Two-Body

Problem

The restricted two-body problem (2BP), or Kepler problem, can be stated in the following way: “Let us consider two isolated point masses, m2 and m1, such that m2<< m1. The only force

acting on the system is the mutual gravitational attraction between the two bodies. The attraction of m1 by m2 is negligible because of the big inequality between the two masses, thus m1 is at rest.

How does the point mass m2 behave following the gravitational force exerted by m1?”

m1 m2 r Z Y X

Figure 1.1: The masses m1and m2in an inertial reference frame {X, Y, Z} centered in m1.

Let us fix an inertial reference frame {X, Y, Z} centered where m1 is located. The equation of

motion of the point mass m2, in an inertial reference frame {X, Y, Z} centered in m1, is:

d2_r

dt2 = −

µ1

r2ˆr (1.1)

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4 CHAPTER 1. THEORETICAL BACKGROUND where:

• µ1= Gm1 is the gravitational parameter of the mass m1,

• G = 6.67 × 10−11 m3

s2_kg the Newtonian constant of gravitation,

• ˆr= r

r is the radial unit vector and r is the distance between the masses.

The (2BP) admits as first integrals1_:

• the specific angular momentum vector h:

h= r ×dr

dt, (1.2) • the eccentricity vector e:

e= 1 µ1

−µ1

r r+ ˙r × h, (1.3) • the total energy of the system E:

E = −µ1

2a (1.4)

Vectors h and e are not independent because

h · e= 0. (1.5) Anyway, we get 6 integrals independent from each other and this fact allows to solve the dynamical system. The general solution of (2BP) is a conic section lying in a two-dimensional space, the orbital plane. Depending on the initial conditions, m2 moves along a circular, elliptic or hyperbolic

trajectory and the Keplerian elements are particularly useful to catch the geometry of the problem. We will focus on the elliptic case.

The Keplerian elements are:

• a: semi-major axis of the conic section; • e: eccentricity of the conic section;

• i: inclination of the orbital plane with respect to a reference plane, in Figure 1.2 the reference plane is the equatorial plane;

• Ω: longitude of the ascending node identifies the position of the ascending node lying along the nodal line2 _{(the dark green line in Figure 1.2), with respect the X-axis of the inertial}

reference system;

• ω: argument of pericenter identifies the position of the nearest point to the mass m2 along

the orbit;

1_{A first integral is a quantity conserved along the solutions of the dynamical system, that is along the phase flow.} 2_{The line of nodes, or nodal line, is the intersection between the orbital plane and the reference one. The nodal}

line intersects the trajectory of m2 in two different points: the ascending node and the descending node. The mass

m2 passes from below to above the reference plane through the ascending node and from above to below through

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1.1. THE TWO-BODY PROBLEM AND THE PERTURBED TWO-BODY PROBLEM 5 m1 Z X Ω Y i

Equatorial Reference Plane Pericenter e ω m2 f h i

Figure 1.2: The mass m2orbits counterclockwise along an elliptic trajectory in an inertial reference

frame centered in m1. The reference plane is represented by the equatorial plane. All the angles

are taken counterclockwise.

• an angle to identify the position of m2along the orbit.

The Keplerian angle identifying the position of m2 along the orbit can be choosen in different

ways, depending on the particular issue. We use:

• f : true anomaly, angular coordinate identifying the real position of m2;

• M : mean anomaly the angle between the eccentricity vector and the radius vector, that is a linear function of the mean motion n:

M = nt , n=r µ1

a3. (1.6)

The (2BP) gives the simplest model to handle analytically the real satellite problem; the role of the point masses m1and m2is played by the Earth and a satellite respectively. The mass of the

Earth m⊕ is several orders larger than the mass of every artificial satellite m, thus the hypothesis

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6 CHAPTER 1. THEORETICAL BACKGROUND hence the Earth gravitational field is not perfectly symmetric. Moreover the hypothesis of a com-pletely isolated system can never be realized in practice. The gravitational attractions by other bodies in the solar system, the atmospheric drag and the solar radiation pressure are just some examples of additional sources of external forces. However, the acceleration on the satellite caused by these effects is some order of magnitude weaker than the acceleration caused by the monopole term of the Earth’s gravitational attraction. Therefore the real satellite problem is mathematically modelled as a two-body problem with an extra weak effect, that is as a two-body perturbed problem (2BPP).

In the general case of the (2BPP), the equation of motion of a point mass m2, moving in a

gravitational field generated by a point mass, or an extended body, m1in a non-isolated system is:

d2_r

dt2 = −

µ1

r2ˆr+ F. (1.7)

The term F includes all the perturbing contributions. F is commonly called perturbing force or perturbation but, dimensionally, it is an acceleration because of the simplification of the masses thanks to the equivalence principle. Because of F, the resulting motion is no longer purely Keple-rian and a general solution cannot be found; depending on the perturbation, the (2BPP) defines a non-integrable dynamical system. We talk off osculating orbit as the set of orbital elements com-puted at precise instant of time.

When the perturbation is conservative, there exists a perturbing potential R such that: F=∂R

∂r. (1.8)

and the Lagrange planetary equations [20] depict the dynamical system, in terms of Keplerian elements, whose solution is the solution of (1.7) :

                                           da dt = 2 na ∂R ∂M de dt = 1 − e2 na2_e R ∂M − √ 1 − e2 na2_e ∂R ∂ω di dt = − 1 na2√_{1 − e}2_{sin i} ∂R ∂Ω − cos i ∂R ∂ω dΩ dt = 1 na2√_{1 − e}2_{sin i} ∂R ∂i dω dt = √ 1 − e2 na2_e ∂R ∂e − cos i na2√_{1 − e}2_{sin i} ∂R ∂i dM dt = n − 2 na ∂R ∂a − 1 − e2 na2_e ∂R ∂e (1.9)

The system (1.9) leads to the following considerations :

• The variations of the semi-major axis occur if the perturbing potential R depends on the mean anomaly of the satellite.

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1.2. HAMILTONIAN FORMALISM AND DELAUNAY VARIABLES 7 • The variation of the eccentricity is particularly linked to the semi-major axis variation, it is caused by the presence of the mean anomaly in R. The eccentricity may undergo long-term and secular variations, caused by the term ∂R

∂ω. These effects are periodic (long-term effects),

with a period larger than the orbital period of the satellite, or accumulating over time (secular drifts)3_.

• The harmonics of the disturbing potential containing the longitude of the ascending node and the argument of the perigee of the satellite produce the variations of the inclination of the orbital plane.

1.2 Hamiltonian Formalism and Delaunay Variables

An autonomous N-degree of freedom dynamical system in R2N _{is called Hamiltonian system if there}

exist q = (q1, ..., qN) ∈ RN coordinates, p = (p1, ..., pN) ∈ RN conjugate momenta and a properly

regular function defined on the phase space:

H = H(p, q) (1.10) such that the autonomous dynamical system can be written as follows:

˙p ˙q = J∇(p,q)H where: J = 0N −IN IN 0N ∈ R2N ×2N_, _(1.11) where:

• H is called Hamiltonian of the system, • (p, q) are called canonical variables.

Writing equations (1.11) in a more explicit way we get the so called Hamilton equations: (

˙pi= −∂H_∂q

i for: i = 1, ..., N

˙qi= ∂H_∂p_i for: i = 1, ..., N

(1.12) Non-autonomous dynamical system are given by a non-autonomous Hamiltonian, that is an explic-itly time dependent Hamiltonian

H = H(p, q, t) (1.13) fulfilling Hamilton equations (1.12). Anyway a non-autonomous dynamical system can always be converted into an autonomous one by adding one dimension to the phase space. For example: we can consider the new momenta (T , p), new coordinates (t, q) and the new Hamiltonian H0_:

H0_{(T , p, t, q) = T + H(p, q, t) −→}              ˙ T = −∂H ∂t ˙pi= −∂H_∂q_i for: i = 1, ..., N ˙t = 1 ˙qi= ∂H_∂p i for: i = 1, ..., N (1.14)

3_{A more exhaustive description of these effect will be shown further on. Now, it is important to underline how}

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8 CHAPTER 1. THEORETICAL BACKGROUND It defines the same dynamics of H = H(p, q, t) for p and q but without an explicit formal de-pendence on time. Therefore we can show all the general results just for autonomous Hamiltonian system.

We say that a N-degree of freedom generic dynamical system is integrable if it is possible to integrate, by quadrature, the differential equations defining the dynamics. The Liouville-Arnold-Yost theoremgives the integrability criteria to establish whenever an Hamiltonian dynamical system is integrable or not. We just focus on the statements due to Liouville and Arnold [19]:

• an integrable Hamiltonian system admits N independent first integrals (I1, ..., IN), such that

the Poisson’s bracket {Ik, Ih} = 0 ∀k 6= h, where:

{Ik, Ih} = N X i=1 ∂Ik ∂qi ∂Ih ∂pi −∂Ik ∂pi ∂Ih ∂qi ;

• if the N-dimensional surface Σ, defined by the constant of motions (I1, ..., IN), is compact,

then an integrable Hamiltonian system admits a special set of coordinates, called action-angle variables(P, Q) ∈ Σ×TN 4_{. The new Hamiltonian, written in action-angle variables, depends}

only on the actions, namely the new momenta, P:

H = H(P). (1.15)

1.2.1 Hamiltonian Form of the Two-Body Problem

The (2BP) is an autonomous dynamical system but the Keplerian elements are not a set of action-angle variables. The Keplerian elements are not even canonical variables. A special action-action-angle variables are the so-called Delaunay variables, related to the Keplerian elements through a coordi-nate change transformation:

(a, e, i, ω, Ω, M ) 7−→ (L, G, H, `, g, h) ;                      L=p(µtot)a G= L√1 − e2 H = G cos i `= M g= ω h= Ω (1.16)

where µtot is the gravitational parameter referred to the total mass of the system:

µtot= G(m1+ m2) ≈ µ1

The Hamiltonian of the (2BP) is

Hkep= − µ2 tot 2L2 ≈ − µ2 1 2L2 (1.17) 4

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1.2. HAMILTONIAN FORMALISM AND DELAUNAY VARIABLES 9 and defines the Hamiltonian system:

       ˙ L= 0 ˙` = µ21 L3 = n ˙ G= 0 ˙g = 0 ˙ H = 0 ˙h = 0 (1.18)

Although the resulting motion is more understandable in terms of Keplerian elements, for any initial conditions (L0, G0, H0, `0, g0, h0) it is easy to integrate the system (1.18):

                     L(t) = L0 G(t) = G0 H(t) = H0 `(t) = `0+ nt g(t) = g0 h(t) = h0

−→ It means that semi-major axis, eccentricity and inclination are constant in time.

−→ It means that mean anomaly evolves with constant velocity, while the pericenter direction and the longitude of the ascending node stay constant in time.

(1.19)

Now, let us consider a (2BPP) with a conservative perturbation F =∂R

∂r: it is an Hamiltonian

dynamical system. Its Hamiltonian in Delaunay variables is of the form:

H(L, G, H, `, g, h) = H0(L, G, H) + εH1(L, G, H, `, g, h) (1.20)

where:

• H0 = Hkep(L) is the unperturbed term or the Keplerian term; it is the integrable part, that

is, H would be an integrable system if ε = 0; • εH1= −R is the perturbative term;

• ε is the small parameter, ε << 1. It is useful to highlight the fact that the unperturbed term is some orders of magnitude larger than the perturbing terms.

The unperturbed term is highly degenerate5_{, because the Eq. (1.17) only depends on the action}

L. For this reason, even a small perturbation of this problem may give a complicated dynamics [19].

The system defined by Eq. (1.20) belongs to a special class of non-integrable dynamical systems: the nearly integrable Hamiltonian system.

The corresponding Hamilton equations are:        ˙ L= −εH1 ∂` ˙` = ∂H0 ∂L + ε ∂H1 ∂L = n + ε ∂H1 ∂L ˙ G= −εH1 ∂g ˙g = ∂H0 ∂G + ε ∂H1 ∂G = ε ∂H1 ∂G ˙ H = −εH1 ∂h ˙h = ∂H0 ∂H + ε ∂H1 ∂H = ε ∂H1 ∂H (1.21)

Action variables are no longer constant in time but evolve slowly because of the small parameter. Conversely the angles, usually, that is in a non degenerate problem, evolve faster. Our problem

5_{An Hamiltonian H}

0(P), written in action-angle variables, is called non-degenerate if det ∂

2_H 0

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10 CHAPTER 1. THEORETICAL BACKGROUND is degenerate, thus the only fast variable is the mean anomaly `. To make clear the distinction between slow and fast variables we can use the example of a Molniya satellite case6_:

       ˙` ≈ n = 1.459 × 10−4 _[rad s ] −→ T`= 11.9622 [h] ˙h ≈ −2.63614 × 10−8 _[rad s ] −→ Tg = 7.5527 [yr] ˙g ≈ 1.938 × 10−10 _[rad s ] −→ Tg= 1027.35 [yr] (1.22)

• A Molniya satellite takes approximately half a day to make a full orbit around the Earth, thus the period of the mean anomaly T` is calculated in term of hours.

• The angular variables related to the perigee, g, and the longitude of the ascending node, h, have much smaller frequencies. The corresponding periods, Tg and Th respectively, are

calculated in a time scale of the order of the years.

1.3 The Averaging Procedure

The averaging procedure is an important tool deeply used to handle problems of celestial mechanics and astrodynamics. Let us consider a generic nearly integrable system, not necessarily Hamiltonian:

( _{˙I = εf(I, ϕ)} ˙

ϕ= ω(I) + εg(I, ϕ) (I, ϕ) ∈ W × T

3_,

W ⊂ R3. (1.23) The corresponding averaged system is defined as follow [2]:

˙J = ε ¯f(J), where: ¯f(J) = 1 (2π)3

Z

T3

f(J, ϕ)dφ (1.24) From [2], we say that the system (1.23) verifies the averaging principle if there exists ¯t >0 such that for every initial condition I(0) = J(0) and ϕ(0), the following relation is satisfied

max

|t|<¯t ε

||I(t) − J(t)||ε→0−→ 0 (1.25) This means that the solution J(t) of the averaged system well approximate the evolution of the actions I(t) given by the system (1.23), at least in a time interval depending on the small parameter: |t| < ¯t

ε. Not all the systems like (1.23) satisfy the averaging principle and, for a better

understanding, it is convenient to use a simple example. Let us consider the nearly integrable Hamiltonian

H(I, ϕ) = ω · I + εX

k∈K

ˆ

fkcos(k · ϕ) (1.26)

where:

6_{The values of the slow frequencies in (1.22), will be more clear later. The regression rate of the longitude of}

the ascending node is given by the oblateness effect, while the value of ˙g is given by the precession rate caused by oblateness effect and lunisolar second order mean terms.

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1.3. THE AVERAGING PROCEDURE 11 • W ⊂ R3 _{open and (I, ϕ) ∈ W × T}3_{action-angle variables for the integrable part;}

• K ⊂ Z3_{\ {0} is finite set;}

• the frequencies are constant, ω ∈ R3_{, and not all equal to zero, thus the system is isochronous;}

• the coefficients ˆfk are constants for all the multi-indices k = (k1, k2, k3) ∈ K.

The Hamiltonian (1.26) leads to the following Hamilton equations ( _˙

Ih= εPk∈Kkhfˆksin(k · ϕ)

˙ ϕh= ωh

h= 1, 2, 3. (1.27) Choosing a set of initial conditions (I(0), ϕ(0)) at t = 0, then the angles evolve linearly in time: ϕh(t) = ϕh(0) + ωht h= 1, 2, 3 (1.28)

and the dynamics of the action variables can be easily integrated: I(t) = I(0) + ε Rt 0 P k∈Kk ˆfksin[k · (ϕ(0) + ωt)]dt = = I(0) − εt P k∈Kresk ˆfkcos(k · ϕ(0)) + −ε P k∈K\Kresk ˆ fk (k·ω)cos(k · ϕ(0) + k · ωt) − cos(k · ϕ(0)) (1.29) where: Kres= {k ∈ K t.c. k · ω= 0} (1.30)

Kres is called resonant set, because it is the set of the multi-indices k satisfying the resonance

conditions k · ω= 0 (k 6= 0). Integer combinations of angles for which there exists a multi-index ¯k fulfilling the resonance conditions are called critical angles or critical arguments.

The dynamics of actions, in Eq. (1.29), can be decomposed into:

• secular evolution or secular drift, accumulating over time. It is given by resonant terms, that is all the terms for which k ∈ Kres;

• periodic oscillations, given by non-resonant terms, where time dependency is found within a trigonometric function. The amplitudes of these oscillations strongly depend on the ratio

1

|k · ω| (1.31) If |k · ω| is small, we get a near resonant condition. It may provide very large oscillations and numerical instability.

The corresponding averaged system is: ˙ Jh= ε (2π)3 Z T3 X k∈K

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12 CHAPTER 1. THEORETICAL BACKGROUND Does the system satisfy the averaging principle?

To answer this question we have to check the condition (1.25). For h = 1, 2, 3 we get: |Jh(t)−Ih(t)| ≤ ε|t|| X k∈Kres khfˆkcos(k·ϕ(0))|+ε| X k∈K\Kres khfˆk k · ωcos(k·ϕ(0)+k·ωt)−cos(k·ϕ(0))| (1.33) • If no resonance occurs, Kres = ∅, and neither small divisors appear, in the equation (1.33)

only the second term remains. Therefore the time dependency is within cosine argument and we can always take the estimate:

max |t|<1 ε ||J(t) − I(t)|| ≤ max |t|<1 ε X k∈K\Kres ε| ˆfk| |k · ω|||k|| ε→0 −→ 0 (1.34)

• If some resonances occur then for all t ≈ ˜t

ε and h = 1, 2, 3 the first term in (1.33):

ε|t|| X k∈Kres khfˆkcos(k · ϕ(0))| ≈ ˜t| X k∈Kres khfˆkcos(k · ϕ(0))| ε→0 6−→ 0 (1.35) and the averaging principle does not hold.

The averaging principle is based on the assumption that the dynamics defined by nearly inte-grable system (1.23) can be split into secular drift and periodic small oscillations [2]. Consequently, it is possible to well approximate the (1.23) with the averaged system.

Let us extract from (1.23) the system of four differential equations, with only one frequency, the faster frequency:

( _{˙I = εf(I, ϕ)} ˙

ϕ= ω(I) + εg(I, ϕ) (I, ϕ) ∈ W × S

1_,

W ⊂ R3. (1.36) with f and g periodic function of ϕ. If :

1. f , g, ω are well defined and bounded on W × S1_{with their derivatives up to the second order}

included,

2. there exists a positive constant C0such that in W

ω(I) > C0>0 (1.37)

3. J(t) ∈ W and B(J(t), d) ⊂ W 7_{, where J(t) is the solution of the singly-averaged system:}

˙J = ε ¯f(J), f¯(J) = 1 2π

Z 2π

0

f(J, ϕ)dϕ (1.38)

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1.4. RESONANT DYNAMICS 13 then for small enough value of the small parameter (0 < ε < ε0):

||I(t) − J(t)|| < C1ε ∀t ∈ h 0,1 ε i (1.39) where the constant C1>0 does not depend on ε.

We can consider the averaging procedure as the iterative procedure of extracting, at every step, a new system, averaged over one given period, which should approximate the one obtained in the previous step. We should notice, as pointed out by [2], that at every step we obtain a simpler dynamical system but its solution may be less and less significant if the goal is to approximate the dynamics of the initial system, and that resonant terms or near resonant terms may obstruct the averaging procedure.

Usually in the (2BPP), thanks to these results, all the short-period non-resonant terms depend-ing on the fast angle, namely the mean anomaly, are averaged out and the sdepend-ingly-averaged dynamics gives a good approximation of the initial problem. It is often useful to study the doubly-averaged model, that is, the model obtained by averaging twice. In the case we will study, we average over the mean anomalies of the satellites and of the perturbing bodies.

1.4 Resonant Dynamics

In this section we show how to study the dynamics associated to a resonance. As a concrete example, useful in Ch. 4, let us consider the Hamiltonian

H = H0(I) + εH1(I, ϕ) where:

      

H1(I, ϕ) =P_k∈Kfˆk(I) cos(k · ϕ)

K ⊂ Z3_{\ {0} is a finite set}

(I, ϕ) ∈ W × T3

, W ∈ R3

(1.40)

and the corresponding Hamilton equations ( _{˙I = −ε}∂H1(I,ϕ) ∂ϕ ˙ ϕ= ∂H0(I) ∂I + ε ∂H1(I,ϕ) ∂I (1.41) We call main frequencies the components of the vector:

ω0(I) =

∂H0

∂I (I). (1.42) A resonance happens if there exists a index ¯k ∈ K such that:

¯

k · ˙ϕ= 0 → ¯k · ω0+ ε¯k ·

∂H1(I, ϕ)

∂I = 0 (1.43) Let us study the resonant dynamics associated to a precise multi-index: be ¯k= (¯k1, ¯k2, ¯k3) ∈ K

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14 CHAPTER 1. THEORETICAL BACKGROUND is a single resonance8 _{and isolated. Consequently, in a neighborhood of the resonant region, the}

dynamics is dominated by the unperturbed term and by the resonant term9

Hres= H0(I) + ε ˆf¯_k(I) cos(¯k · ϕ).

  

˙I = ε¯k ˆf¯_k(I) sin(¯k · ϕ)

˙ ϕ=∂H0

∂I (I) + ε ∂ ˆfk¯

∂I (I) cos(¯k · ϕ)

(1.44) There always exists a canonical transformation10 _{of the following type:}

(I, ϕ) 7→ (J, φ) ( φ= U ϕ J= U−T_I where: U =   ¯ k1 ¯k2 ¯k3 u21 u22 u23 u31 u32 u33   (1.45)

and its inverse transposed: U−T =   v11 v12 v13 v21 v22 v23 v31 v32 v33  . After this coordinate change the Hamiltonian in the (J, φ) variables is:

H0_res(J1, φ1; J2, J3) = H00(J) + ε ˆf 0 ¯

k(J) cos(φ1) (1.46)

The angles φ2, φ3 are cyclic11 variables in (1.46) and therefore the conjugate actions variables J2

and J3 are constants of motion. The motion takes place along the J2= const. and Js= const. in

the (J1, φ1) plane, where φ1= ¯k · ϕis the critical angle. In the standard resonance model [19] the

Hamiltonian (1.46) gives a pendulum-like dynamics in the proximity of the resonant region. Let I∗_{∈ W the value of the exact resonance:}

¯ k · ω0(I∗) = 0 and J∗= U−TI∗. (1.47) Then up to order ε: ˙ φ1(J∗) = ¯k · ˙ϕ(I∗) = ε ∂ ˆf0 ¯ k(J ∗₎ ∂J1 cos(φ1) ≈ 0 (1.48)

Let us fix the constants of motion:

(

J2= J2∗

J3= J3∗

(1.49) In order to focus our attention in the proximity of the resonant region

| ˙φ1(J∗)| = |¯k · ϕ(I∗)| ≈ 0 (1.50)

we expand the resonant Hamiltonian (1.46) in a Taylor series of J1 around J1∗ up to the second

order: H00 res = H0res(J1∗) + ∂H0_res ∂J1 |J ∗ 1(J1− J ∗ 1) + 1 2 ∂2H0 res ∂2J1 |J ∗ 1(J1− J ∗ 1)2+ ... (1.51)

8_{In general if ¯}_{k · ˙}_{ϕ then, for every integer value C, the index k = C¯}_{k is also a resonant index. In the case we will}

study in next chapters we do not deal with multiple resonances.

9_{It is always possible to push the non-resonant terms to a higher order in ε using a canonical transformation. In}

our example the perturbation is periodic in ϕ thus only the unperturbed term and the resonant term remain in the Hamiltonian up to the first order in ε.

10_{A canonical transformation is a coordinate change preserving Hamiltonian structure of the dynamical system.} 11_{A variable is cyclic if the Hamiltonian does not depend on it.}

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1.4. RESONANT DYNAMICS 15 Neglecting the constant term, which does not affect the dynamics, and the perturbative terms of the first order in (J1− J1∗), the Hamiltonian becomes:

H00 res = β 2(J1− J ∗ 1) 2_{+ c cos(φ} 1) β = ∂2_H0 0 ∂J2 1 (J∗ 1), c= ε ˆf¯_k0(J1∗). (1.52)

In conclusion the dynamics is pendulum-like because the form of the Hamiltonian (1.52) is of the same type of the well-known pendulum Hamiltonian. There are two equilibria in φ∗

1= 2nπ, π +2nπ:

(J1∗,2nπ) (J1∗, π+ 2nπ) for n ∈ Z (1.53)

whose stability depends on the sign of the coefficients c and β. The linearized matrix is

A(J1, φ1) = ∂ ∂J1 dJ1 dt ∂ ∂φ1 dJ1 dt ∂ ∂J1 dφ1 dt ∂ ∂φ1 dφ1 dt ! (1.54) If the eigenvalues of A(J∗

1, φ∗1) are complex conjugate then (J1∗, φ∗1) is an elliptic fixed point and

A(J∗

1, φ∗1) gives a rotation matrix. Taking a set of initial conditions in an appropriate neighborhood

of this equilibrium, the corresponding orbits will remain confined in a neighborhood of the elliptic fixed point, that is, the point is a stable equilibrium.

If the eigenvalues of A(J∗

1, φ∗1) are real with opposite sign then (J1∗, φ∗1) is a hyperbolic fixed point.

In this case, (J∗

1, φ∗1) is the unstable equilibrium because of the presence of the stable separatrix and

of the unstable separatrix.

Figure 1.3: Contour plot of the Hamiltonian of the pendulum with c and β both positive, in the x-axis the critical angle φ1 ∈ [0 deg, 360 deg] and in the y-axis the values of (J1− J1∗). The stable

equilibrium is at φ∗

1= 180 deg and the unstable equilibrium is at φ∗1= 0 deg, 360 deg.

The resonant region is the libration region around the stable equilibria because for all initial condition the near resonance condition | ˙φ1| ≈ 0 is fulfilled [19]. Hence the amplitudes of the libration

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16 CHAPTER 1. THEORETICAL BACKGROUND regions measured at the apex of the separatrices give us the maximum half-width of the resonant region: |J1− J1∗| ≤ 2 r c β = 2 v u u tε| ˆ f_¯_k0(J∗ 1) ∂2_H0 0 ∂J2 1 (J ∗ 1) | (1.55) The resonance half-width is:

• proportional to the amplitudes of the oscillating resonant term, • inversely proportional to the torsion of the resonance [18], that is

∂2_H0 0 ∂J2 1 (J1∗) = ∂ ∂J1 ∂ H00 ∂J1 (J1∗) (1.50) = ∂(¯k · ω0) ∂[v11I1+ v12I2+ v13I3] . (1.56) If there are two or more resonances close together, then we can separately study the dynam-ics relatively to each one making the assumption that they are isolated. The resulting motion is pendulum-like with appropriate coordinate change for every single resonance and the pendulum-like model gives a well approximation as long as the corresponding libration regions stay apart.

If there exist some overlapping resonant region then the pendulum-like model breaks down. The Chirikov resonance-overlapping criterion states that when two or more critical arguments librate in the same region of phase space then a large scale chaos may be expected [8]. It happens that the separatrices of different resonances are connected and the chaotic regions may extend over all the resonant regions. Therefore an initial condition in this region may pass from one libration region to an other one showing chaotic diffusion.

There is one other case in which the pendulum-like approximation would not seem to give a good approximation of the real resonant dynamics. In the standard resonance model the exact resonance is used as the libration center but the real equilibria of the resonant Hamiltonian (1.46) are solutions of: _

  ˙ J1= ε ¯fk¯0 sin(φ1) = 0 ˙ φ1= ∂H0₀(J) ∂J + ε ∂ ˆf¯k0(J) ∂J cos(φ1) = 0 (1.57) Therefore, the first equation of the above system gives:

˙

J1= 0 → φ∗1= 2nπ, π + 2nπ, n ∈ Z (1.58)

By replacing the values of φ∗

1in the second equation of (1.57), then we get two different equations:

˙ φ1= 0 →    ∂H0₀(J) ∂J + ε ∂ ˆf¯k0(J) ∂J = 0 ∂H0 0(J) ∂J − ε ∂ ˆf¯k0(J) ∂J = 0 (1.59) Hence, solutions of (1.59) are not necessarily the same although they are close. It brakes down the symmetry found the phase space of the pendulum-like approximation given by the fact that the stable equilibria and the unstable equilibria are the same value of J1.

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1.4. RESONANT DYNAMICS 17

Figure 1.4: On the left: the phase portrait of the pendulum-approximation of the resonance 2 ˙g in the (G0_{, g}0_{) plane. On the right: the phase portrait of the resonant Hamiltonian, not developed in}

Taylor series, in the same plane. g0 _{= 2g is the critical argument, G}0₌ 1

2Gis the conjugate action.

For a better understanding of such loss of symmetry we need a concrete example. Let us con-sider the case of the resonance 2 ˙g that we are going to study in the Ch. 4.

Figure 1.4 displays the contour lines of the resonant Hamiltonian suitable for the resonance 2 ˙g (on the right) and of its Taylor development (on the left) around the stable equilibria given by the exact resonance. The suitable reference frame is given by (G0₌1

2G, g

0 _{= 2g), as shown in Ch. 4 in more}

exhaustive way. On the left the stable and unstable equilibria lies on the same line G0 _{= 35815.4,}

consequently the phase portrait is symmetric with respect to this line. On the contrary, in the phase portrait on the right, the libration region stretches upwards because the unstable equilibria lies above the stable equilibria. In such deep asymmetrical case it is better to use a different ap-proach to get the libration width: the method adopted in [24].

Let us suppose that (Js, φs) is the stable equilibrium and (Ju, φu) is the unstable one of the

dynamics defined by the resonant Hamiltonian H0

res, such that Js 6= Ju. The maximum and the

minimum value of J1, respectively Jmax and Jmin, in the libration region12 are solutions of the

following relation:

H0res(J1, φs) = H0res(Ju, φu) (1.60)

It means that Jmax and Jmin are respectively the largest and the lowest value at the edge

be-tween the libration region and the separatrices. In fact, the relation (1.60) means that Jmax and

Jmin are the intersections between φ = φs and the contour line, on the phase space, at the level

H0

res(Ju, φu), corresponding to the separatrices.

Actually, the width of the libration region in the standard model, given in the formula (1.55), is obtained following the same idea, but for a symmetric situation produced by a Hamiltonian developed in Taylor series. Hence, it well describe a quite symmetric case. Conversely, the

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18 CHAPTER 1. THEORETICAL BACKGROUND tion (1.60) gives more reliable range [Jmin, Jmax] of the resonant region in a deep asymmetrical case.

In Ch. 4, the resonance 2 ˙g will be treated with more detail, and more concrete example about the symmetric and asymmetric cases will be shown.

1.5 Gravitational and Non Gravitational Perturbations

In this section we describe two of the most important perturbations acting on a Earth’s satellite, that are at the basis of this work.

Gravitational perturbations are generated by a mass distribution and are conservative, otherwise perturbations are non-gravitational and they are not necessarily conservative. The analytic expres-sions of the perturbing forces can be easily found in literature, but in the majority of the cases the developments are given in terms of Keplerian elements.

1.5.1 Earth Gravitational Potential

Considering the Earth as an extended body, it is pretty far from a perfect sphere. The distribution of the internal mass is not radially symmetric. For example centrifugal forces arising from the rotation around the Earth’s spin axis causes a flattering at the poles and an equatorial bulge giving an oblate shape to our planet. For instance, oceans regions are at a lower level than the lands.

Polar radius

Equatorial radius

Figure 1.5: Simple scheme of an oblate Earth, in black, with respect to a perfectly symmetrical Earth with radius equal to R⊕, dashed in grey. The figure is in scale: R⊕= 2 cm.

The perturbation given by such asymmetric distribution of a mass can be written as a convergent series in terms of the geogentric distance r from the center of the mass distribution, in our case the Earth’s center of mass. The Earth gravitational potential, also called geopotential, is:

U⊕(r) = µ⊕ r ( 1 − ∞ X l=2 JlPl(sin φ) R⊕ r l + ∞ X l=2 l X m=1 JlmPlm(sin φ) R⊕ r l cos(mλ − mλlm) ) (1.61)

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1.5. GRAVITATIONAL AND NON GRAVITATIONAL PERTURBATIONS 19 • r is the geocentric radius vector.

• R⊕ is the mean equatorial radius of the Earth.

• φ is the geocentric latitude. • λ is the longitude.

• Plm(x) is the associated Legendre polynomials of degree l and order m

• Jlm=pClm2 + Slm2 with Clm and Slm are coefficient determined experimentally, Jl= Jl0.

• λlm = ( 1 mtan −1₍Slm Clm) l − m even 1 mtan −1₍ Clm −Slm) l − m odd

U⊕is a harmonic function, that is ∇2U⊕= 0, and the index l represents the order of the harmonics

appearing in (1.61). To simplify the discussion we use the following notation: U⊕=

µ⊕

r + R⊕ where: R⊕ = Rzonal+ Rtesseral. (1.62) The first term of U⊕ is the monopole term of the gravitational potential and the second term gives

the deviation from a perfect spherical planet, the disturbing effect of the geopotential. There exists a useful expansion due to Kaula for both the zonal and tesseral harmonics, [11]. For the zonal part:

Rzonal= ∞ X l=2 l X p=0 ∞ X q=−∞ µ⊕ a R⊕ a l

JlFl0p(i)Glpq(e) cos((l − 2p + q)M + (l − 2p)ω + θl) (1.63)

θl=        0 Cl0>0, l even π 2 Cl0>0, l odd π Cl0<0, l even 3π 2 Cl0<0, l odd (1.64)

The second order J2 coefficient is three orders of magnitude larger than the next harmonic, it

takes into account that the equatorial radius of the Earth is larger than the polar one: this fact means that the oblateness of the Earth is the dominant perturbation. Therefore J2harmonic refers

to the oblateness effect.

We are interested on an averaged dynamics and to average Rzonal on the mean anomaly of the

satellite M is equivalent to retain the terms in Eq. (1.61) such that:

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20 CHAPTER 1. THEORETICAL BACKGROUND thus, the averaged zonal disturbing potential is:

¯ Rzonal= ∞ X l=2 l X p=0 ∞ X q=−∞ µ⊕ a R⊕ a l

JlFl0p(i)Glpq(e) cos((l − 2p)ω + θl) =

=1 4 J2µ⊕R2⊕ a3_{(1 − e}2₎32 1 − 3 cos2i + ∞ X l=3 l X p=0 ∞ X q=−∞ µ⊕ a R⊕ a l

JlFl0p(i)Glpq(e) cos((l − 2p)ω + θl)

(1.66) The collection of these terms gives the long-term and the secular effects of zonal geopotential perturbation to the dynamics of a satellite orbiting around the Earth.

In Eq. (1.66) the first term (l = 2) is the secular oblateness term and higher order terms (l ≥ 3) are O(J2 2), therefore: ¯ RJ2= 1 4J2 µ⊕R2⊕ a3_{(1 − e}2₎3₂ 1 − 3 cos2_i (1.67) is usually the most important effect in the secular dynamics of the slow frequencies of the satellite. One of the first problems studied in astrodynamics is the long-term evolution of a satellite in a gravitational field generated by an oblate planet. From the Lagrange planetary equations (1.9) we get the dynamics:

             ˙a = 0, M˙ = n + J2 3nR2 E

4a2_(1−e2₎32(3 cos2_i−1)

˙e = 0, ˙ω = J2 3nR2 E 4a2_(1−e2₎2(5 cos 2_{i −}₁₎ ˙i = 0, _{Ω = −J}˙ ₂ 3nR2 E 2a2_(1−e2₎2cos i (1.68)

Thus the oblateness main effect does not affect the semi-major axis, the eccentricity and the inclination of the orbit but yields a precession or regression, depending on the sign, of the argument of pericenter and of the longitude of the ascending node.

In system (1.68), the values of inclination leading to ˙ω = 0 for all values of a and e are called critical inclinations. These values correspond to i = 63.43 deg for prograde orbits and i = 116.57 deg for retrograde ones.

In the Molniya case the inclination is close to the critical inclination value, thus the oblateness effect on perigee is cancelled out, at least for a first order approximation. Choosing ω = 270 deg leads to stable equilibrium thus the argument of pericenter of Molniya orbits should librate around this equilibrium.

The Kaula’s expansion of the tesseral part is: Rtesseral= ∞ X l=2 l X m,p=0 ∞ X q=−∞ µ⊕ a R⊕ a l

Flmp(i)Glpq(e)Jlmcos((l−2p+q−

m so )(M +ω)+m(θ−λlm)−qω) (1.69) where: θ= 1 so (M + ω) − (θ⊕− Ω) (1.70)

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1.5. GRAVITATIONAL AND NON GRAVITATIONAL PERTURBATIONS 21 • θ denotes the stroboscopic node13_,

• θ⊕ the mean Greenwich sidereal angle,

• sois the integer nearest ratio between the satellites mean motion and the Earth rotation rate.

The Eq. (1.69) is particularly convenient when the period of the mean anomaly of the satellite is commensurable with the rotation period of the planet, that is a tesseral resonance. Resonant tesseral harmonics verify:

l −2p + q = m so

(1.71) As stated before, a Molniya satellite completes two orbits every day, therefore Molniya orbits shows a 2 : 1 tesseral resonances. Therefore: so= 2 and resonant tesseral harmonics are those such that

m= 2(l − 2p + q).

1.5.2 Third Body Perturbation

m1 Y Z X m mk rk r ρk

Figure 1.6: Relative positions of the three bodies in an inertial reference system.

The gravitational action of a third body generates a conservative perturbation. The perturbing function is called disturbing function:

Rk= µk rk −ρk r3 k (rk· r) (1.72)

where the subscript k stands for a generic third body, in our case k = $ stands for the Moon and k= stands for the Sun, and:

• ρk means the distance between the satellite and the third body,

13_{The geographic position of the sub-satellite point at the time of intersection of the Earth’s equator by a satellite}

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22 CHAPTER 1. THEORETICAL BACKGROUND • rk means the distance between the central body (Earth) and the third body.

The development as a series of Legendre polynomials, when r << rk, was originally done by

Kozai in 1959 but a more convenient form is usually needed [6].

A further expansion involving Keplerian elements referred to the celestial equator for both the per-turbing and the perturbed body was developed by Kaula in 1962. The big advantage of Kaula’s expansion of the disturbing function is the opportunity of figuring out which harmonic is responsible of short/intermediate/long-periodic or secular motion.

The orbital elements of the Sun with respect to the celestial equator are well approximated by linear functions of time, thus the solar disturbing function can be written as:

R=P∞l=2 Pl m,p,q=0 P∞ j,r=−∞µ al al+1 m (l−m)! (l+m)!Flmp(i)Flmq(i)Hlpj(e)Glqr(e)× × cos(l − 2p + j)M − (l − 2q + r)M+ (l − 2p)ω − (l − 2q)ω+ m(Ω − Ω) (1.73) where: m= 1 se m = 0 2 se m ∈ Z \ {0} (1.74) • Flmp are Kaula’s inclination functions, the analytical expressions for various indices is given

in Appendix A. In [20], the estimate about the order of magnitude of the inclination functions are given: Flmp(i) = O sin i 2 |m−l+2p| (1.75) • Hlpjand Glqrare Kaula’s eccentricity functions, computed by the Hansen’s coefficients whose

analytical expressions are given in Appendix A. From [20]: Hlpj(e) = X l,l−2p l−2p+q(e) = O(e |q|_), _G lqr(e) = X −(l+1),l−2q l−2q+r (e) = O(e |r|₎ _(1.76)

The expressions of the eccentricity functions needed are given in Appendix A. A simple precessing ellipse was used in [24] to model the solar dynamics:

       a= 1.49597870 × 108 km e= 0.016709 i= 23.4393 deg ω= 282.9372 + 0.32 T deg (1.77)

where T is the Julian centuries date from J2000.0.

The ecliptic plane is fixed with respect to the equatorial reference plane because ˙Ω = 0, thus to

choose the value of Ω is equivalent to fix a reference frame. Usually Ω= 0.

The gravitational influence of the Moon is pretty much laborious. The motion of our moon is quite disturbed by our star and its orbital elements are nonlinear function of time in a reference system with the celestial equator as the reference plane. If, instead of the celestial equator, the

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1.5. GRAVITATIONAL AND NON GRAVITATIONAL PERTURBATIONS 23 ecliptic is adopted as reference plane, then i$ is approximately constant and ω$ and Ω$ are ap-proximately linear function of time.

Hence, in order to translate in mathematical language the presence of the Moon, it is better to use the Lane’s expansion [6] of the disturbing potential, which adopts a mixed reference frame.

The lunar disturbing function can be written as: R$ =P∞ l=2 Pl m,p,s,q=0 P+∞ j,r=−∞(−1) m+s₍₋₁₎k1µ$ms 2a_$ (l−s)! (l+m)!( a a_$) l_F

lmp(i)Flsq(i$)Hlpj(e)Glqr(e$)×

×n(−1)k2_Um,−s l cos[(l − 2p + j)M + (l − 2p + r)M$+ +(l − 2p)ω + (l − 2q)ω$ + mΩ + s(Ω$ −π 2) − ysπ] +(−1)k3_Um,s l cos[(l − 2p + j)M − (l − 2p + r)M$+ +(l − 2p)ω − (l − 2q)ω$ + mΩ − s(Ω$ −π 2) − ysπ] o . (1.78)

This expansion is obtained from Kaula’s one with some cumbersome and appropriate rotations which lead to the use of Keplerian elements of the satellite referred to the equatorial plane and the Keplerian elements of the Moon referred to the ecliptic plane.

A simple precessing ellipse can be adopted to model the lunar dynamics around the Earth, [24]:            a$ = 3.84747981 × 105 _km e$ = 0.054880 i$ = 5.1298 deg ω$ = 318.3087 + 6003.15T deg Ω$ = 125.0446 − 1934.14T deg (1.79)

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Chapter 2

Analytical Models for the Molniya

Satellites

As said before, the main features of the Molniya orbits are: an orbital period of approximately 12 hours, a high eccentricity and an inclination close to the critical inclination value.

The circumterrestrial orbital environment con-sists of some special orbital region. Particularly, the Low Earth Orbits (LEO) are characterized by an altitude between about 200 km and 2000 km, the Medium Earth Orbits (MEO) is the region surrounding the Earth above LEO region and below the Geostationary Earth Orbits (GEO) threshold, that is about 36000 km.

The altitudes, of a typical Molniya satellite, may be around 500 km for the perigee and even 40000 km for the apogee, [16] and [1]. It means that these orbits cross the entire nominal orbit space, from LEO through GEO.

The range of possible semi-major axis is approximately [23000, 27000] km. Such orbital elements provide the initial conditions for an operative mission, but they undergo more or less significant variations caused by the perturbing forces. In practice, the satellites are equipped with a propulsion system capable of correcting the route by restoring the proper orbital parameters. We will carry out the analysis assuming a passive Molniya satellite like a satellite no longer operational or, in general, like a space debris.

The main perturbative term is given by the non-central part of the gravity field of the Earth. For most of their orbital period, Molniya satellites are in MEO region and beyond, therefore they are strongly affected by the gravitational attraction of the Moon and the Sun. The main purpose of this work is to investigate on the importance of lunisolar effects on the Molniya dynamics. Hence, in our investigation, the disturbing potential will be given by the Earth disturbing potential, in Eq.

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26 CHAPTER 2. ANALYTICAL MODELS FOR THE MOLNIYA SATELLITES (1.62), and by the lunisolar disturbing function, in equations (1.78) and (1.73);

R = R⊕+ R$ + R. (2.1)

Let us call Molniya parameters the values in Keplerian elements: amoln= 26554.3 km emoln= 0.72 imoln= 63.43 deg ωmoln= 270 deg in Delaunay variables: Lmoln= 102881 km 2 s Gmoln= 71326.9 km 2 s Hmoln= 31935.2 km 2 s gmoln= 270 deg (2.2)

To have an idea of the order of magnitude of the perturbations in the Molniya region, with respect to the unperturbed Keplerian term, we can give some estimates by evaluating all the terms involved in the Molniya parameters.

The order of magnitude of the main term (1.17) is: Hkep = −

µ⊕

2a ≈ 7.50 [ km2

s2 ]

From the Kaula’s development of the zonal part, in Eq. (1.63), of R⊕, we get for the given

Molniya parameters: J2averaged-term : J2 µ4 ⊕R2⊕ 4L3_G3 1 − 3 H G ≈ 2.80 × 10 −4 J3averaged-term : 3 2J3 µ⊕R3⊕ a4 e (1 − e2₎5₂ sin i 1 −5 4sin 2_i sin ω ≈ −2.115 × 10−7 (2.3)

and the Von Zeipel correction term, from [25]: J₂2 term: J2 2 µ6 ⊕ L10 R4 ⊕ 4 A(L, G, H) cos 2ω + B(L, G, H) ≈ 2.257 × 10 −8 _(2.4)

where the expressions of A(L, G, H) and B(L, G, H) can be found in Appendix A.

The tesseral terms, associated to the 2 : 1 resonance, in R⊕are those in the Kaula’s development

(1.69) such that: (l − 2p + q) =m 2 for:        l= 2, 3, ..., +∞ p, m= 0, ..., l q= −∞, ..., +∞

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27 The resonant tesseral harmonics for l = 2, 3, not equal to 0, are shown in the table below.

l m p q 2 0 1 0 2 2 2 3 2 2 1 1 2 2 0 -1 3 2 0 -2 3 0 1 -1 3 0 2 1 3 2 3 4 3 2 1 0 3 2 2 2

The main resonant tesseral terms, according to [11] and [25], evaluated in the Molniya parame-ters are: J22− tesseral: p=0, q=-1 ≈ µ⊕ a R⊕ a 2 J22F220(i)G20−1(e) = 8.244 × 10−8 J22− tesseral: p=1, q=1 ≈ µ⊕ a R⊕ a 2 J22F221(i)G211(e) = 3.198 × 10−7 J32− tesseral: p=0, q=-2 ≈ µ⊕ a R⊕ a 3 J32F320(i)G30−2(e) = 5.638 × 10−8 J32− tesseral: p=1, q=0 ≈ µ⊕ a R⊕ a 3 J32F321(i)G310(e) = 5.752 × 10−7 J32− tesseral: p=2, q=2 ≈ µ⊕ a R⊕ a 3 J32F322(i)G322(e) = 7.601 × 10−7 (2.5)

The last two values are pretty large and tesseral harmonics affect both the semi-major axis and the eccentricity of the satellite.

From the development of the lunisolar disturbing functions (1.78) and (1.73), we get the following estimates from above:

l_{= 2 terms of R$ ∼} µ$a 2 a3 $ ≈ 6.09 × 10−5 l= 2 terms of R∼ µa2 a3 ≈ 2.79 × 10−5 l_{= 3 terms of R$ ∼} µ$a 3 a4 $ ≈ 4.20 × 10−6 l= 3 terms of R∼ µa3 a4 ≈ 4.96 × 10−9 l_{= 4 terms of R$ ∼} µ$a 4 a5 $ ≈ 2.90 × 10−7 (2.6)

The purpose of this work is to investigate on the long-term effects caused by the gravitational attraction of the Moon and the Sun. Therefore, the dominant perturbing terms that we will consider

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28 CHAPTER 2. ANALYTICAL MODELS FOR THE MOLNIYA SATELLITES are: the oblateness term, both the second and the third order lunar disturbing function l = 2, 3, and the solar disturbing function l = 2 and l = 3 for completeness. These results are consistent with the fact that as the semi-major axis increases, lunisolar perturbations become less and less negligible.

The spectrum of the frequencies involved is quite wide, that is: • the short-period frequency is the mean anomaly of the satellite:

˙` = ˙M = 1.459 × 10−4 rad

s −→ T`= 11.9622 h (2.7) • the mean anomalies of the perturbing bodies give the intermediate-period frequencies:

˙ M$ = n$ = 2.66 × 10−6 rad_s −→ TM_$= 27.39 d ˙ M= n= 1.99 × 10−7 rad s −→ TM= 365.25 d (2.8)

• the long/secular frequencies are given by the slow angle variables involved: ˙ω$ = 0.164 deg_d −→ Tω_$= 5.99 yr ˙ Ω$ = −19.341 deg_yr −→ TΩ_$= 18.61 yr ˙ω = 0.32 deg Cyr −→ Tω = 1125 Cyr ˙ Ω= 0 (2.9)

To distinguish the angular variables in terms of their frequencies, or their periods, is useful to better understand the dynamics. As a matter of fact:

• short-period oscillations are produced by harmonics containing the mean anomaly of the satellite in the argument;

• intermediate-period oscillations are generally produced by the harmonics depending on the mean anomalies of the perturbing bodies;

• long-period perturbations are produced by harmonics in which the argument depends on the argument of the perigee and do not depend on the mean anomaly of the satellite;

• secular perturbations are produced by harmonics in which the argument do not depend on the argument of perigee or on and the mean anomaly of the satellite.

The averaging principle, if verified, suggests that the secular and resonant terms dominate the dynamics. Starting from Eq. (1.78), mathematically a lunar resonance arises from the following relation:

(l − 2p + j) ˙M ±(l − 2p + r) ˙_{M$ + (l − 2p) ˙ω ± (l − 2q) ˙ω$ + m ˙Ω ± s ˙Ω$ ≈ 0} (2.10) From Eq. (1.73), we get that the solar resonances are a little bit simpler, and ˙Ω= 0:

(l − 2p + j) ˙M −(l − 2q + r) ˙M+ (l − 2p) ˙ω − (l − 2q) ˙ω+ m( ˙Ω − ˙Ω) ≈ 0 (2.11)

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29 • Semi-secular lunar resonances, that are resonances between the slow variables of the satellite

and the mean anomaly of the Moon:

±(l − 2p + r) ˙_{M$ + (l − 2p) ˙ω ± (l − 2q) ˙ω$ + m ˙Ω ± s ˙Ω$ ≈ 0;} (2.12) • Semi-secular solar resonances, that are resonances between slow variables of the satellite and

the mean anomaly of the Sun:

−(l − 2q + r) ˙M+ (l − 2p) ˙ω − (l − 2q) ˙ω+ m( ˙Ω − ˙Ω) ≈ 0. (2.13)

• Secular lunar resonances, that are resonances between slow frequencies of both the satellite and the Moon:

+(l − 2p) ˙ω ± (l − 2q) ˙ω$ + m ˙Ω ± s ˙Ω$ ≈ 0. (2.14) • Secular solar resonances, that are resonances between slow frequencies of both the satellite

and the Sun:

+(l − 2p) ˙ω − (l − 2q) ˙ω+ m( ˙Ω − ˙Ω) ≈ 0. (2.15)

The motion near a single isolated resonance is pendulum-like with an appropriate coordinate change, but sometimes more resonances overlap and this simple model breaks down.

The choice of neglecting other perturbations must be somehow justified. In particular: to neglect the atmospheric resistance is a rather subtle issue. Whenever the satellite altitude drops below 2000 km1_{the atmospheric drag exerts an acceleration on the satellite directed on the opposite direction}

with respect to the orbital velocity of the satellite. A Molniya satellite crosses periodically the LEO region: the effect due to the atmospheric drag might be quite strong, especially for the satellite lifetime [14]. The tangential perturbing acceleration due to the atmospheric drag is:

D= −1 2ρ A mv 2_C Dˆv (2.16) where:

• ρ is the atmospheric density,

• A is the area of the satellite along the direction of motion, • v = vˆvis the orbital velocity,

• CD is the drag coefficient, typically between 1.5 and 3.

The main effects concern the variation of the semi-major axis and the eccentricity causing the circulation of the orbit. Since we are not interested in accurately propagating the orbit: if we assume a small area-to-mass ratio and if we consider that the altitude of a Molniya satellite is less than 1000 km for no longer than 15 minutes within one revolution [14], then we may be somehow justified in ignoring this dynamic contribution.

Also the solar radiation pressure (SRP), due to the solar electromagnetic radiation, strongly depends on the area-to-mass ratio of the satellite.

Thus, assuming a low area-to-mass ratio is an appropriate choice but it sets a limit to the accuracy of the analytical model we are going to build.

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30 CHAPTER 2. ANALYTICAL MODELS FOR THE MOLNIYA SATELLITES

2.1 State-of-The-Art

In this section are exposed some important results concerning the gravitational perturbations of the Earth, the Moon and the Sun acting on a high eccentric and critical inclined orbit like Molniya type. From the selected past results, we can extrapolate some interesting considerations useful for this work.

Anselmo and Pardini [1] noticed that the choice of the longitude of the ascending node is one of the crucial parameters for the satellite lifetime.

In [25] and in [9] a dynamical model considering mainly the geopotential perturbation was investigated. In both works the singly-averaged Hamiltonian model is studied, and the main selected perturbing terms are:

• the oblateness term,

• the Von Zeipel correction term proportional to J2 2,

• the main tesseral resonant harmonics,

• the Coriolis term, the term resulting after the elimination of the explicit dependence on time in tesseral harmonics.

Region of the phase space associated to the mean motion resonances are found and the numer-ical simulations confirmed the presence of chaos around them. The libration center found in [25] is a ≈26554.3 km and the resonance width ∆a ≈ 38 km. About the Molniya dynamics, the resonant tesseral harmonics are more important than the J2

2-term, which indeed is more important in a

critical inclination problem without mean motion resonances. The effect of the tesseral resonances with the Earth causes an enlargement of the libration width in inclination and eccentricity for the long-period dynamics.

Such geopotential-only model is not appropriate for the Molniya case and in later works, [10] and [24] the authors introduce also the third-body effect.

Lunisolar effects are usually studied by a doubly-averaged model up to the second order. Be-cause of the expression of the doubly-averaged lunisolar potential, the argument of the perigee of both the Moon and the Sun does not appear explicitly as harmonic argument [6]. Using the as-sumption of a circular orbit for both the Moon and the Sun, the third order disturbing potential is zero thanks to the analytical expressions of the eccentricity function appearing in it [7].

The web of secular lunisolar resonances is usually studied approximating the slow frequencies of the satellite as perturbed only by the planetary oblateness, namely:

   ˙ω = 3 4J2n R⊕ a 2 1 (1−e2₎2(5 cos 2_{i −}₁₎ ˙ Ω = −3 2J2n R⊕ a 2 1 (1−e2₎2cos i (2.17)

Replacing (2.17) in equations (2.14) and (2.15) representing the lunar and solar second order secular resonances, then we get inclination-only-dependent resonances:

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2.1. STATE-OF-THE-ART 31 (2 − 2p) ˙ω + m ˙Ω + s ˙_{Ω$ ≈ 0, (2 − 2p) ˙ω + m ˙Ω ≈ 0. with:}      p= 0, 1, 2 m= 0, 1, 2 s= −2, −1, 0, 1, 2 (2.18)

Figure 2.1: On the left: lunisolar resonances for s = 0. On the right: lunisolar resonances for s = −2, −1, 0, 1, 2: the multiplets structure. Both figures are plotted by approximating the frequencies

˙g and ˙h with the precession due to the oblateness effect.

In Figure 2.1 on the left we shown the resonances occurring in the simplified case of ˙

Ω$ = ˙h$ = 0 (2.19) In reality, the motion of the Moon around the Earth is quite perturbed by the presence of the Sun and this leads to the precession of the lunar ascending node and of the lunar perigee. In other words, assuming (2.19), in a second order doubly-averaged model, is substantially equivalent to assume an unperturbed motion of the Moon.

The real perturbed motion of the Moon, namely with non negligible ˙_{Ω$, leads a multiplets structure} of the resonances [8], shown in Figure 2.1 on the right.

The lack of overlapping between the resonances usually guarantees the confinement of the motion [21], while a dense structure of multiplets creates an intricate network of resonances. In the latter case, the perturbed motion of the Moon is a particular source of chaotic phenomena due to the resonances overlapping.

Many other results concerning the dynamics associated with more complex models, including the SRP effect or the coupling of tesseral resonances and lunisolar resonances can be found in literature.

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32 CHAPTER 2. ANALYTICAL MODELS FOR THE MOLNIYA SATELLITES A coupling between tesseral and lunisolar resonances may be expected with a consequent in-crease of chaotic diffusion (and obviously obstruction to the averaging procedure). The qualitative motion of the semi-major axis is affected primary by the tesseral resonant harmonics and the inter-action between the motion in semi-axis and the motion in eccentricity is slight for orbit with period of 12 hours or lower [12].

Long period evolution caused by the SRP is negligible, with respect to the lunisolar contribu-tion, if A

m ≈ 0.1 km2

kg . For low value of the area-to-mass ratio the orbital evolution is periodic, but

for larger values may become chaotic [23].

Although the literature is rich, the problem of instability of Molniya orbits is far from be solved. We decided to focus on the lunisolar effect because it dominates the dynamics on the long-term time scale.

2.2 The Doubly-Averaged Model

Let us start from the Hamiltonian, written in Delaunay dynamical variables, including the second and the third order perturbation of the Moon and of the Sun:

H = Hkep+ HJ2+ H$ + H (2.20)

where:

• HJ2 = −R

(l=2)

⊕ is the collection of the terms in (1.63) with l = 2:

R(l=2)_⊕ = 2 X p=0 +∞ X q=−∞ µ4_⊕ L2 R⊕ L2 2 J2F20p(G, H)G2pq(G, L) cos[(2 − 2p + q)` + (2 − 2p)g + θt] (2.21) • H$ = −Rl=2

$ − Rl=3$ is the collection of the terms from (1.78) withl= 2, 3, respectively:

R(l=2)_$ =P2 p,m,q,s=0 P+∞ j,r=−∞(−1) m+s₍₋₁₎k1 µ$ 2a3 $ L2 µ⊕ 2 ms_(2+m)!(2−s)!× ×F2mp(G, H)F2sq(i$)H2pj(G, L)G2qr(e$)× ×n(−1)k2_Um,−s 2 cos[(2 − 2p + j)` + (2 − 2q + r)M$+ +(2 − 2p)g + (2 − 2q)ω$ + mh + s(Ω$ −π 2) − ysπ] +(−1)k3_Um,s 2 cos[(2 − 2p + j)` − (2 − 2q + r)M$+ +(2 − 2p)g − (2 − 2q)ω$ + mh − s(Ω$ −π 2) − ysπ] o . (2.22)

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2.2. THE DOUBLY-AVERAGED MODEL 33 R(l=3)_$ =P3 p,m,q,s=0 P+∞ j,r=−∞(−1) m+s₍₋₁₎k1 µ$ 2a4 $ L2 µ⊕ 3 ms_(3+m)!(3−s)!× ×F3mp(G, H)F3sq(i$)H3pj(G, L)G3qr(e$)× ×n(−1)k2_Um,−s 3 cos[(3 − 2p + j)` + (3 − 2q + r)M$+ +(3 − 2p)g + (3 − 2q)ω$ + mh + s(Ω$ −π 2) − ysπ] +(−1)k3_Um,s 3 cos[(3 − 2p + j)` − (3 − 2q + r)M$+ +(3 − 2p)g − (3 − 2q)ω$ + mh − s(Ω$ −π 2) − ysπ] o . (2.23) • H= −Rl=2 − Rl=3 is the collection of the terms from (1.73) with l = 2, 3, respectively:

Rl=2 =P 2 p,m,q=0 P+∞ j,r=−∞ µ a3 L2 µ⊕ 2 m(2−m)!_(2+m)!F2mp(G, H)F2mq(i)H2pj(G, L)G2qr(e)× × cos[(2 − 2p + j)` − (2 − 2q + r)M+ (2 − 2p)g − (2 − 2q)ω+ m(h − Ω)] (2.24) Rl=3 = P3 p,m,q=0 P+∞ j,r=−∞ µ a4 L2 µ⊕ 3 m (3−m)! (3+m)!F3mp(G, H)F3mq(i)H3pj(G, L)G3qr(e)× × cos[(3 − 2p + j)` − (3 − 2q + r)M+ (3 − 2p)g − (3 − 2q)ω+ m(h − Ω)] (2.25) Artificial satellite orbits are, usually, too low to allow mean motion resonances between the perturbed and the perturbing body, at least for low order resonances, that is for low values of the index l.

In the Molniya case, the orbital periods of the bodies of interest are:

T`≈ 12 [hours] , TM_$≈ 1 [month] , TM ≈ 1 [year]. (2.26)

Thanks to this fact there are no obstructions to averaging over the orbital period motion the spacecraft.

The singly-averaged Hamiltonian ¯H is the collections of all terms in which the mean anomaly does not appear, namely

¯

H = Hkep+ ¯HJ2+ ¯H$ + ¯H (2.27)

where:

• the oblateness term in (1.67), written in Delaunay variables, is: ¯ HJ2= J2 µ4 ⊕R2⊕ 4L3_G3 1 − 3 H G (2.28) • ¯_{H$ = − ¯}R(l=2)_$ − ¯R(l=3)_$ , that is, the collection of all the terms in (2.22) such that (2−2p+j) =

0 and all the terms in (2.23) such that (3 − 2p + j) = 0, respectively; • ¯H= − ¯R

(l=2) − ¯R

(l=3)

, that is, the collection of all the terms in (2.24) such that (2−2p+j) = 0

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34 CHAPTER 2. ANALYTICAL MODELS FOR THE MOLNIYA SATELLITES In such Hamiltonian system, the mean anomaly ` is a cyclic variable and therefore L is a first integral. In term of Keplerian elements, it is equivalent to say that the semi-major axis is constant in time.

To get the doubly-averaged Hamiltonian we have to perform one further step: we have to average over the orbital periods of the perturbing bodies. This procedure is allowed whenever no resonances occur between the slow frequencies of the satellite and the orbital periods of Moon and Sun. If we assume that no semi-secular resonances occur, at least for low order l = 2, 3, then we can extract from (2.27) the part responsible for the long-term and secular dynamics.

The doubly-averaged Hamiltonian considered in this work is: ¯ ¯ H = Hkep+ ¯HJ2+ ¯H$ +¯ ¯ ¯ H (2.29) where: • ¯_{H$ = −}¯ R¯¯(l=2)_$ − ¯R¯(l=3)_$ , • ¯H¯ = − ¯R¯ (l=2) − ¯R¯ (l=3) .

As said before, the doubly-averaged second order contribution of the Moon in ¯R¯(l=2)_$ is the collection of the terms in (2.22) such that:

(

(2 − 2p + j) = 0 (2 − 2q + r) = 0 for:

p, q= 0, 1, 2

j, r= −∞, ..., +∞ . (2.30) These relations constrain the values of the indices in the summation (2.22):

       p= 0 → j = −2 p= 1 → j = 0 p= 2 → j = 2 and        q= 0 → r = −2 q= 1 → r = 0 q= 2 → r = 2 (2.31)

The only eccentricity function G2qr not identically equal to zero is G210, as shown in Appendix

A, thus we can always assume q = 1. Consequently, the lunar argument of the perigee does not appear because the corresponding coefficient is:

(2 − 2q) = 0. Finally: ¯¯ R(l=2)_$ =P2 m,s=0(−1) m+s+[m 2] µ_$ 2a_$ L2 µ⊕ 2 ms_(2+m)!(2−s)!× n

F2m0(G, H)H20−2(G, L)F2s1(i$)G210(e$)(−1)m+s[U2m,−scos(ϕ2,0,m,s) + U2m,scos(ϕ2,0,m,−s)]+

+F2m1(G, H)H210(G, L)F2s1(i$)G210(e$)(−1)m+s[U2m,−scos(ϕ2,1,m,s) + U2m,scos(ϕ2,1,m,−s)]+

+F2m2(G, H)H222(G, L)F2s1(i$)G210(e$)(−1)m+s[U2m,−scos(ϕ2,2,m,s) + U2m,scos(ϕ2,2,m,−s)]

o (2.32)

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2.2. THE DOUBLY-AVERAGED MODEL 35 where the harmonics arguments are:

ϕ2,p,m,±s= (2 − 2p)g + mh ± s

Ω$ −π₂

− ysπ. (2.33)

It is easy to see from (2.33) that, for p = 1 and s, m = 0, we have ϕ2,1,0,0 = 0, thus the

corre-sponding harmonic term in ¯R¯(l=2)_$ depends only on the actions (L, G, H). We will call this special harmonic mean term of the doubly-averaged second order lunar potential.

Similar considerations can be done also for the doubly-averaged second order potential of the Sun. ¯R¯(l=2) is the collection of the terms fulfilling the constrain (2.30) and (2.31).

Therefore: ¯ ¯ R(l=2) =P2 m=0 µ a3 L2 µ⊕ 2 m (2−m)! (2+m)!× n F2m0(G, H)H20−2(G, L)F2m1(i)G210(e) cos(φ2,0,m)+ F2m1(G, H)H210(G, L)F2m1(i)G210(e) cos(φ2,1,m)+ F2m2(G, H)H222(G, L)F2m1(i)G210(e) cos(φ2,2,m) o (2.34)

where the harmonics arguments are:

φ2,p,m= (2 − 2p)g + m(h − Ω) p, m = 0, 1, 2 (2.35)

As in the lunar case, for p = 1 and m = 0 we have φ2,p,m = 0, and thus we will refer to the

corresponding harmonic term as the mean term of the doubly-averaged second order solar potential. The doubly-averaged third order contribution of the Moon ¯¯R(l=3)_$ is the collection of the terms in (2.23) such that ( (3 − 2p + j) = 0 (3 − 2q + r) = 0 for: p, q= 0, 1, 2 j, r= −∞, ..., +∞ (2.36) These relations constrain the values that the indices in the summation (2.23) can assume:

             p= 0 → j = −3 p= 1 → j = −1 p= 2 → j = 1 p= 3 → j = 3 and              q= 0 → r = −3 q= 1 → r = −1 q= 2 → r = 1 q= 1 → r = 3 (2.37)

Contrary to what happens for the second order harmonics, the lunar argument of the perigee does not disappear in the third order part because its coefficient ±(3 − 2q) never vanishes.

The eccentricity functions G333 and G30−2, as shown in Appendix A, are identically equal to zero,

Investigation on a doubly-averaged model for the Molniya satellites orbits

Universit`

a degli Studi di Universit`

a di Pisa

FACOLT `

A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea Magistrale in Matematica

Investigation on a Doubly-Averaged Model for the Molniya

Satellites Orbits

Tiziana Talu

Giacomo Tommei

Elisa Maria Alessi

Giulio Ba´

u

Anno Accademico 2019-2020

Università di Pisa

Corso di Laurea Magistrale in Matematica

Contents

Introduction

Chapter 1

Theoretical Background

1.1

The Two-Body Problem and the Perturbed Two-Body

Problem

1.2

Hamiltonian Formalism and Delaunay Variables

1.2.1

Hamiltonian Form of the Two-Body Problem

1.3

The Averaging Procedure

1.4

Resonant Dynamics

1.5

Gravitational and Non Gravitational Perturbations

1.5.1

Earth Gravitational Potential

1.5.2

Third Body Perturbation

Chapter 2

Analytical Models for the Molniya

Satellites

2.1

State-of-The-Art

2.2

The Doubly-Averaged Model