Characterization of Low Earth Orbit
dynamics by frequency analysis
Candidata:
Annalisa Mazzuca
Relatori: Dott.ssa Giulia Schettino Prof. Giacomo Tommei Controrelatore: Dott. Giulio Baù
Introduction 1
1 The issue of Space Debris 4
2 The perturbed problem for Earth satellites 10
2.1 Earth’s Gravitational Potential . . . 13
2.1.1 Effect due to J2 . . . 17
2.1.2 Effect due to odd high-degree zonal harmonics . . . . 18
2.2 Lunisolar gravitational perturbations . . . 20
2.2.1 The solar disturbing function,RSun . . . 20
2.2.2 The lunar disturbing function,RM oon . . . 21
2.3 Solar radiation pressure . . . 23
2.4 Resonances . . . 24
2.4.1 Effect on the eccentricity . . . 30
3 Frequency analysis 32 3.1 Numerical simulations . . . 32
3.1.1 The Fast Orbit Propagator (FOP) . . . 33
3.1.2 Simulation scenario . . . 34
3.2 Mathematical method . . . 35
3.2.1 Fast Fourier Transform . . . 35
3.2.2 Analysis in the frequency domain . . . 37
4.1.1 An exemplificative case in the time domain . . . 42 4.1.2 An exemplificative case in the frequency domain . . . 43 4.1.3 The role of SRP . . . 47 4.2 Characterization of the orbital dynamics: low area-to-mass ratio 50 4.2.1 An exemplificative case in the time domain . . . 51 4.2.2 An exemplificative case in the frequency domain . . . 53 4.3 Frequency "Cartography" . . . 56 4.4 A possible exploitation of the frequency charts . . . 61 4.5 Final remarks and future perspectives . . . 64
Acknowledgements 67
Bibliography 68
Space debris have been generated in the last 60 years as the by-product of the exploration and exploitation of the circumterrestrial space. Due to the dramatic growth of the number of debris in space, the active intervention by the scientific community has become necessary. Indeed, an uncontrolled increase of debris in the circumterrestrial space can provoke both environ-mental damages and possible risks of collision for active missions. In 2007, the United Nations General Assembly has developed a set of guidelines to limit the amount of debris generated by space launches and to minimize the possibility of later fragmentations.
In this context, the Revolutionary Design of Spacecraft through Holis-tic Integration of Future Technologies project (ReDSHIFT) was funded by the European Union. The main idea is to approach the debris mitigation issues from different perspectives, prone to suggest innovative solutions to this pressing problem, having always in mind, beyond the specific mission requirements, the minimization of the environmental impact of the space-craft.
The first step toward this direction is to understand if the dynamical perturbations in the circumterrestrial space can drive the spacecraft toward a natural re-entry. Indeed, perturbations acting on the spacecraft can induce periodical variations in the orbital eccentricity and inclination, which can potentially be exploited. Passive solutions of this kind would be preferred if they result to be affordable for the space operators and not risky for the space environment.
In the present work, we will focus on the Low Earth Orbit (LEO) re-gion, which constitutes the most densely populated orbital environment, and on the dynamical effects caused by three different perturbations: the high-degree zonal harmonics of the geopotential, the lunisolar gravitational attraction, and the solar radiation pressure (SRP). At specific values of in-clination, these perturbations can foster the orbital decay inducing a quick increase of the eccentricity, if a well-defined resonance condition is satisfied. Following the work of [37, 39], we make a deep analysis of the role of the resonances which act on the spacecraft dynamics in LEO. By means of a numerical computation of the Fourier transform, we characterize the evolu-tion of the eccentricity of a dense set of orbits in terms of the main spectral
R⊕= 6378.1363 km is the radius of the Earth, and inclination i, from i = 0
up to i = 90◦. Moreover, we selected two different values of the area-to-mass ratio of the object, in order to understand how natural perturbations, in particular SRP, can influence the orbit dynamics, depending on the A/m value.
The aim of the spectral analysis described in this thesis is to clearly link each frequency signature found in the eccentricity spectrum to the dynam-ical effect which generates it, in order to build a frequency cartography of the LEO region. Indeed, the detailed analysis of the principal spectral com-ponents turns out to be a powerful tool to enable a better understanding of the relative importance of each gravitational and non-gravitational per-turbation in the LEO region as a function of the initial semi-major axis, eccentricity and inclination of the object. In particular, the amplitude of the spectral signature produced by a perturbation on a given orbit gives an estimate of the corresponding eccentricity variation; this quantity can be compared with the numerical results in the time domain and the analytical expressions provided by theory, in order to give a comprehensive and more robust picture of the eccentricity evolution. Together with the dynamical maps in the time domain, performed in the framework of ReDSHIFT (see, e.g., [1, 2]), our analysis can be exploited to identify the orbits where a sig-nificant growth of eccentricity, led by one or more perturbations, can assist the passive disposal of objects at their end-of-life or to optimize the design of low-cost manoeuvres, aimed at re-entering. Indeed, for some orbits, the frequency cartography is capable to disclose effects which do not show up as clearly in the temporal maps.
This thesis is organized as follows: in Chapter 1 we briefly present the issue of space debris and the recommended mitigation strategies, focusing, in particular, on the aims of the ReDSHIFT project. In Chapter 2 we de-scribe the dynamical model adopted for the numerical propagation in the time domain. We recall the analytical developments of the natural pertur-bations acting on objects in LEO, and, by means of the Lagrange planetary equations, we describe the main effects on the orbital elements due to each perturbation. Furthermore, after having defined what we mean by reso-nant condition, we analyse different kinds of resonances, due to one or more perturbations. In particular, we focus on their effect on the eccentricity evolution, in order to compare, later in Chapter 4, our results with the ana-lytical findings. In Chapter 3 we extensively explain the simulation scenario and the mathematical tools exploited in our study. We describe the Fourier transform and the method we used to identify the main frequency signa-tures, which characterize the eccentricity evolution of the orbits. Finally, in Chapter 4, we outline the numerical results of our analysis. This will be
Finally, we draw some conclusions on possible practical applications of the frequency cartography.
The issue of Space Debris
Since the first space mission was launched in 1957, the number of objects orbiting around the Earth has drastically increased year after year. Actu-ally, only 6% of the objects in orbit are active satellites (see, e.g., [15] for details), while the remaining objects, which represent, in fact, most of the objects orbiting around the Earth, are both spacecraft after their end-of-life or residual parts of spacecraft or space experiments.
The term space debris is used to designate all the human-made objects of all size and all chemical compositions which orbit around the Earth at differ-ent altitudes, such as artificial material resulting from space missions but no longer serving any function (abandoned satellites, launch vehicle upper stage) and fragments from objects breakup (paint flakes, solid fuel fragments).
These objects have been catalogued based on their size by ESA’s Space Debris Office1. As of February 2020, the catalogue shows the presence in the circumterrestrial space of nearly 34,000 artificial objects larger than 10 cm, 900,000 objects between 1 cm and 10 cm and more than 128 million of pieces smaller than 1 cm.
Due to accidental collisions and to the fact that, until few years ago, the issue of handling a satellite after its lifetime was not considered during the mission design phase, the space debris population has increased continuously, therefore representing a risk of impact for future missions. This issue was investigated for the first time in 1990 by Donald Kessler [22]. He described the consequences of a self-sustained growth of the space debris population, initially triggered by collisions between intact objects and ultimately sus-tained by collisions between the resulting orbiting fragments (this was later known as the Kessler syndrome). Such a cascade process, which cannot be stopped in its advanced stage, can act in such a way that certain altitude shells in the region around the Earth could become not accessible for a long time, and, thus, not anymore safe for orbit insertion of new satellites.
Recently, several measures have been taken to preserve current space
1https://www.esa.int/Safety_Security/Space_Debris/Space_debris_by_the_numbers
missions from collisions: manoeuvres are performed regularly by the active satellites to avoid collisions with the tracked debris and special equipments and armour plating protections are scheduled for the new spacecraft, in order to limit the possible damages caused by objects below the centimetre size, which cannot always be tracked due to their small size. Examples of possible solutions are shown in Table 1.1, depending on the size of the debris. How-ever, all these expedients increase the cost of the space mission and they are often inefficient to protect from the effects of debris between 1-10 cm, which are difficult to be detected but could cause significant damages.
Size (r) Characteristics Protection Number (object) r < 0.1 cm cumulative effects
sur-face erosion
not necessary 0.1 < r < 1 cm significant damages
perforation
armour plating 128 000 000
1 < r < 10 cm important damage no solution 900 000 r > 10 cm catastrophic events manoeuvres 34 000 Table 1.1: Characteristics and solutions provided for each type of space debris depending on the size (first column). The number of catalogued objects for each range of size is also shown (fourth column).
It is well known that the atmospheric drag can help some objects to re-enter to the Earth but in most cases it is unlikely that drag alone can lead to the re-entry. Indeed, the drag is effective only for orbits with low altitudes: for example, the lifetime of a typical satellite is estimated to be 1 month for an altitude of 300 km over the Earth surface, 1 year for 400 km, 10 years for 500 km, decades for 700 km, centuries for 900 km and millennial for 1200 km (see, e.g., [26]).
In order to support the success of the mission and at the same time to minimize the environment impact of the spacecraft, in 2007 the United Nations General Assembly provided a set of guidelines 2. They stressed the importance of avoiding potential breakups during and after the active phase of a space mission and they defined the well-known limit of 25 years as the maximum desirable remaining time for a spacecraft in orbit after its end-of-life. This limit has been set by estimating the trade-off between the complete removal of the space structure and the risk of collision with other objects. How to manage and dispose these objects is a study task that must be accomplished as soon as possible to prevent the proliferation of potential targets and projectiles, therefore avoiding the theoretical scenario pictured by Kessler.
In order to develop suitable mitigation measurements, it is crucial to sup-port technological advances with a better understanding of the evolution of
2https://www.unoosa.org/oosa/oosadoc/data/documents/2010/stspace/stspace49_0.html
the long-term dynamics of debris population. In this context, the Revolution-ary Design of Spacecraft through Holistic Integration of Future Technologies (ReDSHIFT) project was funded by the European Union in the framework of the PROTECT Call of Horizon 2020. The project started in 2016 and ended in 2019. It involved 13 European partners with the common scope to resolve space debris related issues by means of either theoretical and experimental studies.
One of the main challenge of ReDSHIFT was to introduce a new paradigm in the space debris mitigation, showing how modern celestial mechanics and astrodynamics can help in finding de-orbiting solutions for every orbital zone, thus minimizing the accumulation of objects in any regime. For objects al-ready in orbit, one must not only make the most out of the payload and propulsion systems, but also take advantage of the natural perturbations acting on satellites, which could lead to very beneficial, sometimes unex-pected or not considered, end-of-life orbital configurations. In general, two kinds of de-orbiting strategies can be identified: active and passive disposal. The former involves external systems, which apply forces on the objects lead-ing them to suitably re-enter. It could be a very expensive approach in terms of costs and required propellant. Conversely, the latter is based on the ex-ploitation of natural perturbations acting on satellites, such as atmospheric drag or solar radiation pressure, which can help the natural re-entry or in manoeuvres towards advantageous orbits in terms of lifetime. The main goal of this thesis is to explore in details the second approach, in order to identify suitable strategies for disposal, depending on the altitude.
Moreover, one of the purposes of the ReDSHIFT project was also to ap-proach the challenge of the complexity of spacecraft design through a holistic method. This means that, for future missions, the type and number of avail-able end-of-life options have to be clearly identified from the early phases of the mission design, in order to properly evaluate the trade-off among op-erational requirements and costs, spacecraft capabilities and environment protection. An approach of this kind focuses on the sustainability of the "space project" as a whole.
The described goals can be achieved by accurately mapping the circum-terrestrial phase space identifying stable and unstable regions, resonances (from simple mean motion resonances to complex interaction with lunisolar perturbations and/or non-gravitational perturbations) and chaotic islands. Once this mapping is performed, the manoeuvres needed to reach the optimal disposal orbit can be estimated identifying the optimal de-orbiting device to be used for any particular mission. After all, it is the synergy between the detailed dynamical mapping, the knowledge about the dynamics in differ-ent orbits around the Earth and the technical solutions that can really push forward the exploitation of the disposal techniques which, left alone, cannot guarantee the de-orbiting from high orbits at the end-of-life.
In the framework of the ReDSHIFT project, it has been performed a
comprehensive “cartography” of the phase space in the Earth vicinity span-ning a dense grid of initial conditions. Depending on the altitude over the Earth surface, we can typically distinguish between three main regions:
• Low Earth Orbits (LEO): orbits with altitude between 900 and 2000 km;
• Medium Earth Orbits (MEO): orbits lying at altitudes between 2000 and 30000 km;
• Geosynchronous Equatorial Orbits (GEO): orbits in a thin shell around the circular geosynchronous orbit at an altitude of 35786 km.
The density of objects (in three different regimes of size) as a function of the altitude above the Earth surface in shown in Figure 1.1 (see [33]). As it can be seen, the most populated areas in the LEO, MEO and GEO regions are clearly detectable. In the case of ReDSHIFT, a detailed analysis of the most populated orbits in each region was carried out. The results concerning the MEO and GEO regions can be found, respectively, in, e.g., [40] and [9].
Figure 1.1: Density of objects as a function of altitude for three different size thresholds objects with diameter larger than 1 mm, 1 cm and 10 cm [33].
In the following, we will focus only on the LEO region, which constitutes the most densely populated orbital environment, having more than half of the total amount of space debris. Due to its strategical importance for remote sensing/weather and Earth observation missions, the LEO region has been defined by international agreements (see [41] for further details) as one of the "Protected Regions", together with the GEO region.
As it can be seen from Figure 1.2 (cfr. [2]), according to the MASTER model3, the most critical altitudes in LEO, i.e., the most populated orbits, are located in the ranges of altitudes [700:1000] km and [1300:1600] km, with
3
The MASTER (Meteoroid and Space debris Terrestrial Environment Reference) model is distributed by the ESA’s Space Debris Office https://sdup.esoc.esa.int.
eccentricities lower than 0.02 (almost-circular orbits). The same figure shows the mean altitude corresponding to a re-entry in 25 years for circular orbits (horizontal line), thanks to the perturbation exerted by the atmospheric drag. It is clear from the figure that for most of the objects an impulsive de-orbiting strategy must be applied to comply with the guidelines and re-enter within 25 years.
Figure 1.2: Orbital distribution in terms of inclination i and semi-major axis a of the objects in LEO with a mass greater than 100 kg. The colour bar reports the eccentricity of a given object [2].
A comprehensive description of the dynamical behaviour in LEO was performed in [2], in the framework of the ReDSHIFT project. The aim of that work was to identify stable and unstable regions in LEO. The results are collected in a series of colour maps, which describe when a re-entry is feasible, and, if it is the case, how much time it is required 4. Moreover, in order to study the effectiveness of passive disposal from LEO orbits, in [3] a general analysis on the role of resonances induced by solar radiation pressure (SRP) was provided.
The authors pointed out the connection between dynamical resonances and the growth of orbital eccentricity, which can favour the re-entry of an object. In practical cases, if a small amount of propellant is available for an object at its end-of-life, a suitable disposal strategy can be to perform a manoeuvre to move the object to a nearby orbit, where perturbations can induce a significant growth of eccentricity, i.e., a significant lowering of the perigee, in order to reach a region where drag becomes effective and re-enter (see, e.g., [38]). To properly investigate if the growth of the orbital eccentricity, led by one or more perturbations, can assist or not the passive disposal of spacecraft at the end-of-life, [39] provided a deeper analysis of
4
A complete set of the colour maps can be found at: http://redshift-h2020.eu/results/leo/.
the resonances which act in the LEO region, by performing an analysis of the evolution of the eccentricity in terms of frequency.
The final aim, according to the holistic approach adopted in ReDSHIFT, is to identify, since the early stages of the mission design, suitable post-mission disposal orbits, re-entry strategies of the debris in the lower atmo-sphere or, alternatively, the possibility of a transfer to a graveyard orbit5, in order to help the decrease of space debris population.
The purpose of this thesis is to complement and deepen such works in order to support the cartography of LEO with a detailed frequency analysis, and to take a deeper look into some aspects left behind, such as the arise of resonances due to solar radiation pressure disturbing function or the role of high-degree zonal harmonics of geopotential in the variation of eccentricity. The challenge of managing the space debris issue, which has now reached a critical point, has offered the opportunity to adapting, testing and devel-oping known techniques and methods, in a new environment. Mathematical tools, such as Fourier transform, and well-known theories, as the pertur-bation theory, are at the basis of the analysis that will be carried out in this thesis and they could be the key to achieve new contributions to our knowledge of the space debris dynamics.
5
With the term “graveyard orbit” we refer to stable suitable orbits, lying away from common operational orbits, which can be used for disposal in the case that re-entry is not feasible.
The perturbed problem for
Earth satellites
In this chapter we describe the perturbed problem for an object orbiting around the Earth. We will focus on the analytical development of the main forces acting on the debris population in the LEO region and we will analyse the long period effects on the dynamics.
Referring to several studies devoted to the modelling of space debris (see, e.g., [2, 26, 39]), in order to provide an analytical expression for the long term dynamics of an object in the LEO region, we must take into account the following perturbations:
• the Earth’s gravitational potential, including high-degree (up to 5) zonal harmonics;
• the gravitational influence of the Sun and the Moon; • the solar radiation pressure (SRP).
Looking at Figure 2.1 (cfr.[26]), we can observe that the main perturbations acting on satellites orbiting up to an altitude of 20000 km are the Earth’s monopole and the second zonal harmonic of the geopotential, followed by the third zonal harmonic and the lunisolar perturbations. Conversely, the effect due to the solar radiation pressure strongly depends on the area-to-mass ratio of the considered object.
Concerning the atmospheric drag, as we already mentioned in Chapter 1, it turns out that this perturbation is effective in driving a re-entry, within a relatively short time, only up to an altitude of about 1000 km [2]. The purpose of this thesis is to describe the long-term evolution of a LEO object in order to identify which perturbations, other than atmospheric drag, can drive passive disposal, depending on the initial orbit. Thus, we decided to remove the drag from the dynamical model adopted in the following study; in this way, it will be easier to characterize the specific effect of the other
Figure 2.1: Contribution of the major perturbing accelerations acting on satellites around the Earth as a function of the distance from the Earth’s centre [26]. The effect due to the atmospheric drag is not shown.
perturbations (geopotential, solar radiation pressure and lunisolar perturba-tions) on the dynamics.
The contribution of each perturbation on the orbital evolution is modelled in terms of Keplerian elements: semi-major axis a, eccentricity e, inclina-tion i, argument of perigee ω, longitude of the ascending node Ω and mean anomaly M . In the following, we will sometimes refer also to the perigee alti-tude, hp, which depends on a and e in the following way: hp = a(1 − e) − R⊕,
where R⊕ = 6378.1363 km is the radius of the Earth. Then, the effect on
the orbital elements due to each perturbation is studied by means of the Lagrange planetary equations. Since the gravitational force exerted by the Earth and the third-body perturbation are conservative forces, they can be written as the gradient of a scalar function, usually called disturbing func-tion or perturbing funcfunc-tion (see, e.g., [35]). Notice that this approach could not be suitable in the case of solar radiation pressure. However, if the light aberration is neglected, SRP can be considered as a conservative force, thus, in this case, we can define the SRP disturbing potential [16, 20].
The Lagrange planetary equations in terms of a generic disturbing func-tion,R, can be written as (see, e.g., [35])
da dt = 2 na ∂R ∂M, de dt = 1 − e2 na2e ∂R ∂M − √ 1 − e2 na2e ∂R ∂ω, di dt = − 1 na2√1 − e2sin i ∂R ∂Ω − cos i ∂R ∂ω , dω dt = √ 1 − e2 na2e ∂R ∂e − cos i na2√1 − e2sin i ∂R ∂i, dΩ dt = 1 na2√1 − e2sin i ∂R ∂i, dM dt = n − 2 na ∂R ∂a − 1 − e2 na2e ∂R ∂e. (2.1)
Consider a small body (i.e., a spacecraft, a space debris) subject to the gravitational attraction of the Earth and to the perturbations due to the Sun and the Moon. The equations of motion can be written as
d2r
dt2 = ∇U + ∇RSun+ ∇RM oon− ∇RSRP, (2.2)
where r is the position of the object in the inertial reference system with origin in the centre of mass of the Earth, U is the Earth’s gravitational potential, RSun and RM oon are the gravitational disturbing functions due to the Sun and the Moon, and RSRP is the SRP perturbing function and the symbol ∇ stands for the gradient of a function in the inertial reference frame.
The expressions of the above disturbing functions can be expanded in se-ries involving an infinite number of combinations of the argument of perigee, the longitude of ascending node and the mean anomaly of either the object (ω, Ω, M ) and the third-body perturber, (ω∗, Ω∗, M∗). When the
third-body perturber is the Sun, the symbol will be adopted, while $ will be used in case of the Moon. In the following, we will show which terms of the expansion of the disturbing functions will be necessary to keep for our purposes and which terms can be neglected.
We remark that the orbital elements of the Sun and the Moon are known functions of time. The changes of the Sun’s orbital elements with respect to the celestial equator is well approximated by linear functions of time; thus, Ω and ω may be considered constant, while M changes with a constant slow rates. Concerning the Moon, we refer to its orbital elements with respect to the ecliptic (and not to the celestial equator), since they vary linearly in time: M$ changes with constant moderate rates, while Ω$ and ω$ undergo slow variations (see, e.g., [20] for further details).
Thus, in our problem we can identify three types of angular variables: “fast angles”, such as M ; “semi–fast angles”, such as M$ and M ; “slow
angles”, such as ω, Ω, ω$ and Ω$.
We can characterize each term of the expansion of the disturbing func-tions by means of the frequency (or period) associated with the corresponding sinusoidal component in the expansion. Therefore, each term of the expan-sion may be classified as follows:
• secular terms, in which the argument do not depend on the mean anomalies M , M , M$;
• semi-secular terms, in which the argument is independent of M , but depends on the semi-fast angles;
• short periodic terms, in which the argument involves fast and semi-fast angles;
• resonant terms, in which the argument involves all angles and in which there is a commensurability between ˙ω, ˙Ω, M , ˙˙ ω∗, ˙Ω∗, M˙∗.
Reso-nances involving commensurabilities between the Earth’s rotational period and the orbital period of the space debris are called tesseral resonances (or gravitational), while the resonances involving the rates of the Sun and Moon’s angles are called lunisolar resonances. They can arise from the gravitational potential due to the Sun and the Moon, but also from the disturbing function of SRP (see [17] for further details). The degree of influence of the terms in the series expansions depends on the given space region we are considering (i.e., mainly on the semi-major axis of the orbit) and on the time interval over which the dynamics is studied. Since our purpose is to model the long-term dynamics, we can perform an average over the faster variables. Indeed, according to the average principle (see, e.g., [35]), taking the mean value of the disturbing functions with re-spect to either M and M∗, the short-period terms average out over a long
time-scale, while only the long-period perturbations matter. Hence, short-period terms can be dropped from the expansions and we can focus only on the long-period terms, which are the ones we are interested in. Moreover, we do not discard, of course, the semi-secular terms, since they might have a relevant role in the LEO region. For example, the semi-secular terms of the geopotential (i.e., zonal harmonics of degree l ≥ 2) as well as the lunisolar semi-secular resonances may directly affect the variation of the eccentricity [3, 31].
2.1
Earth’s Gravitational Potential
In this section we derive the expression of the Earth’s gravitational potential U to be used in Equation 2.2. We recall that the gravitational potential must
satisfy the Laplace equation in any region outside the attracting mass (see, e.g., [21]), namely
∇2U = 0. (2.3)
For the Earth, it is useful to solve the above Equation in spherical coor-dinates (r, φ, λ), related to the Cartesian coorcoor-dinates by:
x = r sin φ cos λ, y = r cos φ cos λ, z = r sin λ,
where r is the distance from the object to the centre of the Earth, φ ∈ [−π/2 π/2] is the latitude and λ ∈ [0, 2π] is the longitude.
The complete solution of Equation 2.3 is given by:
U(r) = µ⊕ r " 1 + ∞ X l=2 l X m=0 Ulm # ,
where µ⊕ is the Earth gravitational parameter, defined as µ⊕ = GM⊕ with
G the gravitational constant and M⊕ the mass of the Earth, andUlm is the
spherical harmonic component of degree l and order m given by (see [21] for a comprehensive discussion):
Ulm(r, φ, λ) = µ⊕
R⊕
r l+1
Plm(sin φ) [Clmcos(mλ) + Slmsin(mλ)] ,
where the functions Plm are defined in terms of the Legendre polynomials Pl(x), which are shown in Table 2.1 for l = 2, ..., 5, as
Plm(x) = (1 − x2)m/2 dm dxmPl(x). l Pl(x) 2 1 2(3x 2− 1) 3 1 2(5x 3− 3x) 4 1 8(35x 4− 30x2+ 2) 5 1 8(63x 5− 70x3+ 15x)
Table 2.1: Legendre polynomials of degree l ≤ 5.
Lastly, the quantities Clm and Slm in Equation 2.6 are given coefficients,
In order to describe the effect of the geopotential on the orbital elements of an object in LEO, we adopt the Kaula’s formulation of the geopotential (see [21] for details), introducing the explicit dependency from the Keplerian elements. In this case, a given harmonic, Ulm, of the Earth’s gravitational potential can be written as:
Ulm = µ⊕ Rl⊕ al+1 l X p=0 Flmp(i) ∞ X q=−∞ Glpq(e)Slmpq(ω, M, Ω, θ), (2.4)
where θ is the sidereal time, in order to account for the rotation of the Earth. The functions Flmpare called Kaula’s inclinations functions and are defined by Flmp(i) = min(p,k) X t (2l − 2t)! t!(l − t)!(l − m − 2t)!22l−2t sin l−m−2ti m X s=0 m s cossi ×X c l − m − 2t + s c m − s p − t − c (−1)c−k, (2.5) where k is the integer part of (l −m)/2 and c is summed over the values mak-ing the binomial coefficients non-zero. The functions Glpq, called eccentricity
functions, are given by
Glpq(e) = (−1)|q|(1 + β2)lβ|q| ∞ X k=0 PlpqkQlpqkβ2k, (2.6) where β = e 1 +√1 − e2, Plpqk = h X r=0 2p0− 2l h − r (−1)r r! (l − 2p0+ q0)e 2β r ,
with h = k + q0 when q0 > 0 and h = k when q0 < 0, and
Qlpqk = h X r=0 −2p0 h − r 1 r! (l − 2p0+ q0)e 2β r ,
where h = k when q0 > 0 and h = k − q0 when q0 < 0, p0 = p, q = q0 for p ≤ l/2 and p0= l − p, q0 = −q for p > l/2.
Finally, the terms Slmpq are given by
Slmpq =
(
− Jlmcos ψlmpq, l − m even,
where the quantities Jlm and λlm are defined as Jlm = q C2 lm+ Slm2 (m 6= 0), Jl0 ≡ Jl= −Cl0, Clm= −Jlmcos(mλlm), Slm = −Jlmsin(mλlm).
The angle ψlmpq is called Kaula gravitational argument and is given by ψlmpq = (l − 2p)ω + (l − 2p + q)M + m(Ω − θ) − mλlm.
Depending on the values of l and m, we can identify three types of har-monics:
• zonal harmonics: l 6= 0 and m = 0; • sectorial harmonics: m = l 6= 0; • tesseral harmonics: m 6= 0 and m 6= l.
A list of the value of the main zonal coefficients of the Earth is given in Table 2.2 and we can visualize the behaviour of spherical harmonics up to degree and order 5 in Figure 2.2.
Zonal Coefficient Approximated Value
J2 1.08 x 10−3
J3 −2.53 x 10−6
J4 −2.27 x 10−7
J5 5.40 x 10−7
Table 2.2: A list of the zonal coefficients Jl= −Cl0. The values are computed from [21].
Once the development of the Earth’s potential has been provided, we can describe the effects on the orbital elements of the spacecraft due to the secular and semi-secular terms associated to each harmonics of the geopotential. Of course, the main secular effect on the rate of variation of the longitude of ascending node and the argument of perigee is due to the J2 zonal harmonic (this can be easily seen by comparing the amplitude of zonal harmonics shown in Table 2.2). Anyway, following the analysis performed in [2, 39], it turns out that we can not omit in the analysis the long-term effects of odd high-degree zonal harmonics, namely J3 and J5. Indeed, as it will be shown
in Chapter 4, these semi-secular terms are significant in the description of the evolution of eccentricity and inclination in time. In the following, we will detail the contribution to the dynamics due to each zonal harmonics of the geopotential.
Figure 2.2: Behaviour of spherical harmonics up to degree and order 5.
2.1.1 Effect due to J2
The perturbation due to second zonal harmonic, also called Earth’s oblate-ness, plays an important role in estimating the secular effects upon the space debris orbits, since J2= −C20 is, at least, 100 times greater than any other
Cl0 (see Table 2.2). From Equation 2.4, the contribution due to J2 is given
by: U20= µ⊕J2 a R⊕ a 2 X p,q
F20p(i)G2pq(e) cos[(2−2p)ω +(2−2p+q)M ]. (2.7)
The above expression can be rearranged in order to account only for the long-term effects. This is done by taking the mean value over the orbital period of the spacecraft, namely by integrating over the mean anomaly M and dividing by 2π. In this way the short terms, which depend on M , are discarded.
Actually, the desired expression corresponds to the suitable combination of p and q such that Equation 2.7 does not depend anymore on the mean
anomaly (i.e., it corresponds to the case: 2 − 2p + q = 0). Since it holds that G20−2(e) = G222(e) = 0 [21], the only term ofU20such that the dependency
on M disappears is given by p = 1, q = 0. Finally, taking the values of F201(i) = 14(3 sin2i − 2) and G210(e) = (1 − e2)−3/2 from Equations 2.5 and
2.6, the secular component is:
¯ U20=U2010 = − µ⊕J2 a R⊕ a 2 F201(i)G210(e) = = −µ⊕J2 4 R2⊕ a3(1 − e2)3/2(3 sin 2i − 2).
The Lagrange planetary equations applied to ¯U20 provide
da dt = de dt = di dt = 0,
while for the argument of perigee ω the variation in time is given by: dω dt = p(1 − e2) na2e ∂ ¯U20 ∂e − cos i na√1 − e2sin i ∂ ¯U20 ∂i = 3 4 J2R2⊕n a2(1 − e2)2(5 cos 2i − 1), (2.8)
for the longitude of ascending node Ω it is dΩ dt = 1 na2√1 − e2sin i ∂ ¯U20 ∂i = −3 2 J2R2⊕n a2(1 − e2)2cos i, (2.9)
and for the mean anomaly M it is dM dt = n − 2 na ∂ ¯U20 ∂a − 1 − e2 na2e ∂ ¯U20 ∂e = n + 3 2 J2R2⊕n a2(1 − e2)2 1 −3 2sin 2i p 1 − e2. (2.10)
Equations 2.8 - 2.10 shows that the main effects of J2 are a slow change
in the rate of the mean anomaly, a precession of the perigee and a secular regression of the orbital node.
2.1.2 Effect due to odd high-degree zonal harmonics
The spherical harmonic corresponding to J3 = −C30 is called "pear-shape"
term and it provides a long-term periodic perturbation on the orbit of the satellite.
In this case, the suitable combinations of the coefficients which make the third zonal harmonicU30 to be independent from the mean anomaly M correspond to the cases p = 1, q = −1 and p = 2, q = 1. Referring to Tables 1 - 2 in [21] (Chapter 3, p. 34 and p. 38) we can compute the corresponding Kaula’s inclination functions and eccentricity functions.
Thus, the perturbing function associated with the long-period effects due to J3 turns out to be:
¯ U30=U301−1+U3021 = 3 2µ⊕J3 R3⊕ a4 e (1 − e2)5/2sin i 1 −5 4sin 2i sin ω.
Notice that in the case of J3 the disturbing function depends on the argument of perigee ω, so that de
dt 6= 0, di
dt 6= 0. In particular, the variation in eccentricity is given by:
de dt = − 3 2 √ 1 − e2 na2e µJ3 R3⊕ a4 e (1 − e2)5/2sin i 1 −5 4sin 2i cos ω = −3 2nJ3 R3⊕ a3(1 − e2)2 sin i 1 −5 4sin 2i cos ω, (2.11)
while the rate of the inclination is: di dt = 3 2 cos i na2√1 − e2sin iµJ3 R⊕3 a4 e (1 − e2)5/2 sin i 1 −5 4sin 2i cos ω = 3 2eJ3 R3 ⊕ a3(1 − e2)3 cos i cos ω 1 −5 4sin 2i . (2.12)
According to the classical results shown, e.g., in [21, 35], in order to achieve the final expression for the long-term effect of J3, we need to integrate
Equations 2.11 - 2.12 with respect to the argument of perigee. In terms of the finite increment of e and i, we finally obtain, respectively:
∆eJ3 = − 1 2 J3 J2 R⊕ a sin i sin ω, (2.13) and: ∆iJ3 = 1 2 J3 J2 R⊕ a e 1 − e2cos i sin ω.
Remark 1. For the previous computations we have assumed that the total secular variation of the argument of perigee is given by Equation 2.8, neglect-ing the variation induced by the third-zonal harmonic. This is justified by the order of magnitude of J3 with respect to J2, which is 10−3J2 (see Table 2.2).
Therefore, we can assume that J3 does not provide a significant contribution
Analogously, we can derive the disturbing potential due to the fifth zonal harmonic, J5= −C50, and analyse its effects on the orbital elements.
We have already pointed out that we are interested in removing the dependency from the mean anomaly. In this case, it corresponds to the condition: 5 − 2p + q = 0. Recalling Equations 2.5 and 2.6, we can compute the coefficients corresponding to the combinations of p and q that satisfy the above condition, that are the following: (1, −3), (2, −1), (3, 1), (4, 3). By averaging the Lagrange planetary equations over the period of the argument of perigee, we obtain the long-term effect on the eccentricity
∆eJ5 = 5 2 J5 J2 R⊕ a 3 1 (1 − e2)2 sin i
(3 + 5 cos 2i)(f (e, i) sin ω + g(e, i) sin 3ω) , (2.14) with f (e, i) = 1 −7 2sin 2i + 21 8 sin 4i 1 +3 4e 2 , g(e, i) = 7 24e 2sin2i 1 −9 8sin 2i .
Remark 2. This result is consistent with the one obtained in [2].
2.2
Lunisolar gravitational perturbations
Let us consider the gravitational perturbation due to the presence of a third body, which can be either the Sun or the Moon in the case of a satellite around the Earth.
Following the Kaula development shown in [20], we provide the expan-sion of the solar and lunar disturbing functions in terms of the orbital el-ements of both perturbed and perturbing body, denoted, respectively, by (a, e, i, ω, Ω, M ) for the perturbed body and (a∗, e∗, i∗, ω∗, Ω∗, M∗) for the
perturbing body. The symbol is adopted in case of the Sun and$ in case of the Moon.
2.2.1 The solar disturbing function, RSun
The variation of the Sun’s orbital elements with respect to the celestial equa-tor are well approximated by linear functions of time, therefore, the expan-sion can be successfully described by the original Kaula expresexpan-sion in terms of solar equatorial elements.
Suppose that the Sun moves on a Keplerian orbit with semi-major axis a = 1 au, eccentricity e = 0.0167, inclination i = 23◦26021.40600, argu-ment of perigee ω = 282.94◦and longitude of the ascending node Ω = 0◦.
The mean anomaly changes as ˙M ' 1◦/day. According to [20], the expan-sion of the gravitational solar potential is given by
RSun = µ ∞ X l=2 l X m=0 l X p=0 l X h=0 +∞ X q=−∞ +∞ X j=−∞ al al+1 m (l − m)! (l + m)! × Flmp(i)Flmh(i )Hlpq(e)Glhj(e ) cos(ψlmphqj),
(2.15)
with µ = Gm , m denoting the mass of the Sun. The angle ψlmphqj is
given by
ψlmphqj = (l−2p)ω+(l−2p+q)M −(l−2h)ω −(l−2h+j)M +m(Ω−Ω ),
while the quantities m are defined as
m =
(
1 m = 0,
2 m ∈ Z \ {0}. (2.16)
The functions Hlpq and Glhj are the Hansen coefficients Xl−2p+ql,l−2p (e),
Xl−2p+j−(l+1),l−2h(e ) (see [18, 25] for explicit expressions). The functions Flmp(i),
Flmh(i ) are the Kaula inclination functions already defined in Equation 2.5.
2.2.2 The lunar disturbing function, RM oon
We remark that an expansion similar to Equation 2.15 holds for the Moon (provided the solar elements are replaced by the lunar ones), when the Moon’s orbital elements are referred to the equatorial plane.
As it has been noted in various works (see, e.g., [10, 20]), the Moon’s inclination, node, and argument of perigee are not simple functions of time. In particular, the longitude of the ascending node varies between −13◦ and +13◦ with a period of 18.8 years. Within the same interval, the inclination of the lunar orbit with respect to the celestial equator oscillates between 18.4◦ and 28.6◦. Also the change in the argument of perigee is nonlinear. Thus, we can adopt a different approach with respect to the case of the Sun. Indeed, following [13, 17, 23], it is convenient to introduce a rotation of the spherical harmonics for the Moon, so that its orbital elements are referred to the ecliptic plane, while the orbital elements of the satellite (or space debris) remain unchanged, that is, they are referred to the equatorial plane.
In this framework the inclination i$ is approximately constant, while the variation of the longitude of the ascending node and the argument of perigee are approximately linear, with rates of change, respectively, equal to
˙
Ω$ ' −0.053◦/day, ˙ω$ ' 0.164◦/day, while the mean anomaly changes as ˙
M$ ' 13.06◦/day.
To express the potential due to the Moon, we can assume that the tra-jectory of the Moon is a Keplerian ellipse with semi-major axis a$ = 384748
km, eccentricity e$ = 0.0549 and inclination i$ = 5◦150. Thus, the expan-sion of the lunar disturbing function is given as follows [8, 23]
RM oon= µ$ ∞ X l=2 l X m=0 l X p=0 l X s=0 l X q=0 +∞ X j=−∞ ∞ X r=−∞ (−1)m+s(−1)k1ms 2a$ (l − s)! (l + m)! a a$ l
× Flmp(i)Flsq(i$)Hlpj(e)Glqr(e$)
× {(−1)k2Um,−s l cos( ¯ψlmpj+ ¯ψ0lsqr− ysπ) + (−1)k3Um,s l cos( ¯ψlmpj− ¯ψ 0 lsqr− ysπ)}, (2.17) where µ$ = Gm$, m$ denoting the mass of the Moon, ys = 0 for s even
and ys = 1/2 when s is odd, k1 = dm/2e, k2= t(m + s − 1) + 1, k3 = t(m + s)
with t = (l − 1) mod 2. The quantities ¯ψlmpj, ¯ψ0lsqr are given by
¯
ψlmpj = (l − 2p)ω + (l − 2p + j)M + mΩ,
¯
ψ0lsqr= (l − 2q)ω$ + (l − 2q + r)M$ + s(Ω$ − π/2), while the functions Ulm,s are defined as
Ulm,s= min(l−s,l−m) X r=max(0,−(m+s)) {(−1)l−m−r l + m m + s + r l − m r × cosm+s+2r(/2) sin−m−s+2(l−r)(/2)},
where = 23◦26021.4500 denotes the obliquity of the ecliptic. The functions Flmp(i) and Flmh(i$) have been introduced in Equation 2.5, while Hlpj(e)
andGlqr(e$) are the Hansen coefficients Xl−2p+jl,l−2p (e), Xl−2q+r−(l+1),l−2q(e$).
Remark 3. Since our purpose is to analyse the long-term effects on the spacecraft orbit, it is convenient to provide the main secular terms of the potentials due to the Sun and the Moon.
Considering the expansion in Equations 2.15 and 2.17 up to the degree 2 in the ratio of semi-major axes and averaging over the mean anomalies M and M∗ (the asterisked quantities refer to the third-body perturber, either the
Sun or the Moon), one obtains:
¯ RSun = µ 2 X m=0 2 X p=0 a2 a3 1 (1 − e )3/2m (2 − m)! (2 + m)! × X02,2−2p(e) cos((2 − 2p)ω + m(Ω − Ω ), (2.18)
¯ RM oon= 1 2µ$ 2 X m=0 2 X p=0 2 X s=0 a2 a3 $ 1 (1 − e$)3/2(−1) [m/2] ms (2 − s)! (2 + m)! × F2mp(i)F2s1(i$)X02,2−2p(e)
× {U2m,−scos((2 − 2p)ω + mΩ + sΩ$ − sπ
2 − ysπ) + U2m,scos((2 − 2p)ω + mΩ − sΩ$ + sπ
2 − ysπ)}.
(2.19)
2.3
Solar radiation pressure
We provide now the analytical expansion of the solar radiation pressure disturbing function, by following the development supplied by Hughes in [16]. The purpose is to suitably express the quantityRSRP in Equation 2.2
in terms of the orbital elements of both the satellite and the Sun, in order to have an expression to be compared with the lunisolar gravity disturbing potentials, shown in Equations 2.15, 2.17.
The classical model to study the effect of SRP on artificial satellites with mass m is the "cannonball model" [16]. In this case, the solar radiation flux is assumed to be constant and directed along the line connecting the centre of the Sun and the Earth. We also suppose that the cross-sectional area of the satellite exposed to the Sun’s radiation, A, and the fraction of the incident radiation absorbed by the satellite, σ, are constant, that is a suitable approximation for a sphere of uniform surface texture. Finally, we assume that the orbit of the object is entirely in the sunlight and that the effect of the Earth’s albedo is negligible, so that the force due to the solar radiation pressure can be considered as a conservative one. In this way, its effect can be represented by the gradient of the disturbing function RSRP.
Let P be the solar radiation pressure at the distance of the Earth (1 au). The development of the disturbing potential in terms of the spherical coordinates of the object is given by:
RSRP = − P RCR A m ∞ X n=1 Pn(cos δ) r R n , (2.20)
where CR is the reflectivity coefficient, r is the distance of the satellite from
the Earth’s centre, R is the Earth-Sun distance and Pn(cos δ) are the Leg-endre polynomials of argument cos δ, where δ is the angle subtended at the Earth’s centre by the satellite and the Sun. The n = 0 term has been omitted from the summation because it is independent of the satellite’s coordinates, and will, therefore, produce no change in its orbital elements.
By applying the Kaula’s method to Equation 2.20 (see [16] for details), we get:
RSRP = −P CR A m ∞ X l=1 l X m=0 l X p=0 l X h=0 ∞ X q=−∞ ∞ X j=−∞ al (a )l−1m (l − m)! (l + m)! × Flmp∗ (i)Flmh∗ (i )Hlpq(e)Glhj(e ) cos(ψlmphqj),
(2.21)
where
ψlmphqj = (l − 2p)ω + (l − 2p + q)M − (l − 2h)ω − (l − 2h + j)M + mΩ,
in which Ω = 0, by definition, for the Sun. The coefficient m is given by
Equation 2.16. The functions Flmp∗ (i) and Flmh∗ (i ) are the modified Kaula’s inclination functions, also called Allan inclinations functions [16]. Finally, the functions Hlpj(e) and Glqr(e ) are the Hansen coefficients Xl−2p+jl,l−2p (e),
Xl−2q+r−(l+1),l−2q(e$).
Notice that the usual expression for the solar radiation pressure disturb-ing function (see, e.g., [28]) can be obtained if only the long-period terms (corresponding to the condition: l − 2p + q = 0) with l = 1 and j = 0 are considered, that is we get:
R0 SRP = 3 2P CR A mae × {2csc s [cos(ω − ω − M ) − cos(ω + ω + M )] + c2[c2 cos(ω − ω − M + Ω) + s2 cos(ω + ω + M + Ω)] + s2[c2 cos(−ω − ω − M + Ω) + s2 cos(−ω + ω + M + Ω)]}, (2.22) where c = cos(i/2), s = sin(i/2), c = cos(i /2), s = sin(i /2).
Remark 4. It is worth noting that the most important long-period terms are those for which l = 1 and j = ±1, i.e.:
R1 SRP = 3P CR A maee × {2csc s [cos(ω − ω − 2M ) − cos(ω + ω + 2M )] + c2[c2 cos(ω − ω − 2M + Ω) + s2 cos(ω + ω + Ω)] + s2[c2 cos(−ω − ω − 2M + Ω) + s2 cos(−ω + ω + 2M + Ω)]}. (2.23)
2.4
Resonances
The aim of this thesis is to analyse the dynamical evolution of a set of orbits in the LEO region in order to identify if some orbits are affected in a
significant way by one or more of the perturbations described in the previous sections. The basic idea is the following: if one or more perturbations are capable to induce a significant increase of the initial eccentricity of the orbit, this can be exploited in order to facilitate passive disposal of the object. Indeed, the more the orbit becomes eccentric, the more the perigee lowers, the more the atmospheric drag becomes effective making feasible the re-entry. We have seen that the disturbing function associated to each perturbation acting on the satellite can be expanded in a series of periodic terms, each one characterized by its own period, given by the argument of the cosine function. Each periodic term has its own amplitude, which depends from the orbit we are considering. Thus, given an orbit, we can rank the effect of each contribution, depending on the relative amplitude. The chance to identify the individual effect on the orbit given by each periodic term is based on the assumption that, in fact, each periodic term appearing in the disturbing functions is independent from the others (see [3]).
The perturbing forces of the Moon and the Sun acting on Earth satel-lites cause secular, semi-secular and fast periodic variations to the orbital elements, as we have seen from Equations 2.15, 2.17, 2.21. In the particular case of satellites orbiting in the LEO region, due to the vicinity of the Earth, the effects of the other perturbing bodies are often negligible, in comparison to that of the Earth’s oblateness.
However, for satellite orbits that satisfy a lunar or solar resonant condi-tion (i.e., in the case of commensurability between the rate of mean anoma-lies, M and M∗, the arguments of perigee, ω, ω∗ and the longitudes of the
ascending node, Ω, Ω∗, where the asterisked quantities refer to the third-body
perturber, either the Sun and the Moon) these perturbations can modify the elements of the orbit to a measurable extent, even on longer timescales than the Earth’s oblateness. Indeed, recent works suggest that the coupled res-onant effects of the Earth’s oblateness and lunisolar perturbations (gravity and solar radiation pressure) are quite important mechanisms for delivering high-altitude Earth orbiting satellites into regions where atmospheric drag can lead to orbital decay [2].
A formal definition of resonances can be given as follows.
Definition 2.4.1. A resonance is the locus of points in the phase space for which a specific linear combination of frequencies becomes null.
In [17] it has been shown that there are 15 types of third-body (lunar and solar) resonances. This classification accounts for all possible resonances involving a third-body perturber. The types of resonance are the following. • Secular resonance: it occurs whenever there is a commensurability between the slow frequencies of orbital precession of the satellite and the perturbing body. That is, whenever there exist (k1, k2, k3, k4) ∈ Z4
such that
k1ω + k˙ 2Ω + k˙ 3ω˙∗+ k4Ω˙∗ = 0, (2.24)
where the asterisked quantities refer to the third-body perturber, either the Sun and the Moon.
• Semi-secular resonance: it involves the mean anomalies of the Moon and Sun and it occurs whenever there exist (k1, k2, k3) ∈ Z3\ {0} such
that
k1ω + k˙ 2Ω + k˙ 3M˙∗ = 0, (2.25)
where the asterisked quantities refer to the third-body perturber, either the Sun and the Moon.
• Mean motion resonance: it occurs when the ratio of the orbital peri-ods of the perturbed and perturbing bodies equals a rational number. Since this kind of resonance does not occur in LEO (neither in MEO or GEO), but rather at a much larger distance from the centre of the Earth, the mean motion resonance is less interesting for the space de-bris dynamics.
The most important secular resonances are given by the lunar and solar gravitational perturbations. To identify them, we need to consider the aver-aged expression of the disturbing potentials shown in Equations 2.18, 2.19. Since they are independent of ω$ and ω , we find that k3 = 0 in Equation
2.24. Moreover, given the fact that ˙Ω = 0, Equation 2.24 can be re-written in the form
(2 − 2p) ˙ω + m ˙Ω = 0 m, p = 0, 1, 2, (2.26) for the Sun and
(2 − 2p) ˙ω + m ˙Ω + s ˙Ω$ = 0 m, p = 0, 1, 2, s = −2, 1, 0, 1, 2, for the Moon.
Concerning the semi-secular resonances, according to the classification of the harmonic terms in Equations 2.15, 2.17 and 2.21, we can obtain that:
(l − 2p) ˙ω + m ˙Ω − (l − 2h + j) ˙M = 0 l ∈ N, m, p, h = 0, 1, 2, .., l, j ∈ N, (2.27) for the Sun, and
(l − 2p) ˙ω + m ˙Ω ± [(l − 2q) ˙ω$ + (l − 2q + r)M$ + s˙ Ω$] = 0˙ l ∈ N, m, p, q, s = 0, 1, 2, .., l, r ∈ N, for the Moon.
As it has been done in [5], by taking a quadrupolar approximation of Equations 2.15 and 2.17, namely considering l = 2, for the lunisolar gravita-tional perturbations and for the main semi-secular terms of the solar radia-tion pressure development, given in Equaradia-tions 2.22 and 2.23, it follows that the main semi-secular resonances have the form
α ˙ω + β ˙Ω + γ ˙M = 0 α ∈ {±1, ±2, 0}, β ∈ {±1, ±2, 0}, γ ∈ Z \ {0}, (2.28) for the Sun and
α ˙ω + β ˙Ω + α$ ˙ω$ + β$Ω$ − γ˙ M$ = 0 α, α$ ∈ {±2, 0},˙ β, β$ ∈ {±2, ±1, 0}, γ ∈ Z \ {0}, for the Moon.
As pointed out in [17] some resonances turn out to be independent on a, e, and they depend only on the inclination. In particular, this kind of res-onances occur due to lunar and solar gravitational perturbations and they belong to the general class of secular resonances. They can be identified re-ferring to Equations 2.18 and 2.19 and they are characterized by the relations given by Equations 2.24 and 2.27. The most important ones are:
• the critical inclination resonance ˙ω = 0 at 63.4◦1 and 116.6◦; • the polar resonance ˙Ω = 0 at 90◦;
• the linear combinations: ˙ω + ˙Ω = 0 at 46.4◦ and 106.9◦, − ˙ω + ˙Ω = 0 at 73◦ and 133.6◦, −2 ˙ω + ˙Ω = 0 at 69.0◦ and 123.9◦, 2 ˙ω + ˙Ω = 0 at 56.1◦ and 111.0◦.
However, we can also find resonances depending on inclination and semi-major axis as well [19]. This is the case of the six main resonances due to the SRP disturbing function, given by Equation 2.22. They are semi-secular resonances and their form is the one shown in Equation 2.28: the explicit expressions of the resonant conditions are shown in Table 2.3.
Table 2.3: List of the six main SRP resonances: argument and val-ues of coefficients α, β, γ. Argument α β γ ω + Ω − M 1 1 -1 −ω + Ω − M -1 1 -1 ω − M 1 0 -1 ω + M 1 0 1 ω + Ω + M 1 1 1 −ω + Ω + M -1 1 1 1
As we will show later, this resonant inclination also occurs due to high-degree har-monics of the geopotential.
Figure 2.3: Location of the six main SRP resonances in the (i, a) plane for the initial eccentricity condition e = 0.001.
In Figure 2.3 we show location of the six main SRP resonances as a function of the inclination and the semi-major axis, for the given value e = 0.001, assuming that, as we have already noticed, only the oblateness of the Earth is responsible of a variation in Ω and ω, with the rates of precession of the ascending node and of the argument of perigee given by Equations 2.9 and 2.8. These approximated expressions can be adopted for the case of A/m = 0.012 m2/kg, which represents a typical value of the area-to-mass
ratio of an intact object orbiting in LEO (see, e.g., [9]). Following [3], in this work we choose to show the behaviour of the eccentricity as a function of i, a, fixing a reference value of the initial eccentricity as e = 0.001 (i.e., a quasi-circular orbit, as it is the case for the most populated orbits in LEO). In such a way we can recognize that the exact value of the resonant inclinations can significantly change, depending on the choice of the initial semi-major axis. Figure 2.4 (a) shows a close-up of Figure 2.3 limited to the LEO region, up to an altitude of 3000 km. In Figure 2.4 (b) we show the location of the same resonances accounting also for the variation of Ω and ω due to SRP. Conversely, the effect of lunisolar gravitational perturbations on ˙ω and ˙Ω can be neglected, following, e.g., [28]; thus, we can write the general expression:
˙
Ω = ˙ΩJ 2+ ˙ΩSRP, ω = ˙˙ ωJ 2+ ˙ωSRP,
where ˙ΩJ 2, ˙ωJ 2 are given by Equations 2.9, 2.8, while ˙ΩSRP, ˙ωSRP are
2.22.
In this general case, the location of resonances depends also on Ω, ω e M . In particular, in the following we will consider the case of initial Ω = 0◦, ω = 0◦, M = 90◦ for A/m = 1 m2/kg, which represents a reference value for a small spacecraft equipped with a SRP enhancing device (see, e.g., [9]). Both the cases (i.e., without and with the contribution of SRP to the rate of ω and Ω) will be considered in Chapter 4 in order to compare theory with our numerical results.
Referring again to Figures 2.3-2.4, we can observe that the resonances occur at a given value of inclination and semi-major axis. Furthermore, we note that resonances of different nature can cross each other (i.e., we mean that they can occur at the same value of i for a given a). In particular, we have found:
• an overlapping between ˙ω − ˙M = 0 and ˙ω + ˙Ω + ˙M = 0 at a ≈ R⊕+ 2194 km and i ≈ 56.06◦;
• an overlapping between ˙ω + ˙M = 0 and − ˙ω + ˙Ω + ˙M = 0 at a ≈ R⊕+ 1180 km and i ≈ 69◦ (at same inclination it appears also the
well-know lunisolar gravitational resonance: −2 ˙ω + ˙Ω = 0).
Referring to the simulations performed in [3], the dynamics in the over-lapping regions does not manifest a chaotic behaviour, but rather the two resonances concur to a possible increase in eccentricity, as we will observe in Chapter 4.
(a) (b)
Figure 2.4: Location of the six main SRP resonances in the LEO region as a function of i, a for e = 0.001, ω = Ω = 0◦. In (a) the curves were computed assuming ˙Ω = ˙ΩJ 2,
˙
2.4.1 Effect on the eccentricity
In general, the analysis of the long-term stability of an orbit is based on the study of the temporal evolution of the eccentricity, since any change in e affects the perigee of the orbit, and this, in turn, influences the satellite’s lifetime.
By means of the Lagrange planetary equations, we can describe the effect of a resonance on the orbit of the satellite. In particular, we are interested in the characterization of the behaviour of the eccentricity in presence of resonances. We assume that the dynamics is driven by a single resonance; this means that we are assuming that the variation in the orbital elements is due to only one term in the expansion of the disturbing potentials of Equations 2.15, 2.17 and 2.21. Calling R the generic contribution to the disturbing potential due to the given resonance, it will have the form:
R = T(a, e, i) cos ψ,
whereT(a, e, i) is a coefficient depending on (a, e, i) and on specific constants according to the given perturbation and ψ is the resonant angle.
Thus, the resulting variation in the eccentricity given by Equation 2.1 becomes: de dt = − √ 1 − e2 na2e ∂R ∂ω = T (a, e, i) sin ψ. (2.29) where T (a, e, i) is a coefficient depending on (a, e, i).
From the above Equation, it turns out that the rate of e is proportional to sin ψ; thus, by integrating Equation 2.29 over time, we finally find the total variation of eccentricity due to a given resonance:
∆e = T (a, e, i) ˙
ψ cos ψ. (2.30)
Notice that the maximum eccentricity variation is:
∆emax= 2 T (a, e, i) ˙ ψ . (2.31)
Concerning the six main SRP resonances mentioned before, in Figure 2.5 we show the maximum variation of eccentricity achievable as a function of i, a for e = 0.001, Ω = 0◦, ω = 0◦, M = 90◦ and A/m = 0.012 m2/kg (case a) and A/m = 1 m2/kg (case b), sampling the (i, a) plane with steps of ∆i = 0.2◦ and ∆a = 20 km, respectively. The colour bar refers to the maximum eccentricity variation achievable: in the case of low A/m value, we found a maximum ∆e of 0.1 in LEO, while for the high A/m value the increment in eccentricity can be as high as 0.4, allowing in this case for a re-entry even from high orbits in LEO.
Comparing the analytical location of SRP resonances in the (i, a) plane, shown in Figure 2.4, with the numerical findings shown in Figure 2.5, we
(a) A/m = 0.012 m2/kg. (b) A/m = 1 m2/kg.
Figure 2.5: Maximum variation in eccentricity (from 0 up to 0.1 (a) or to 0.4 (b) in the colour bar) that can be obtained along a given SRP resonance as a function of i, a for e = 0.001, Ω = 0◦, ω = 0◦, M = 90◦.
can observe that the bright “corridors” found by means of the numerical propagation exactly match the theoretical location of resonances.
The variation in eccentricity that can be obtained thanks to a SRP res-onance can allow for a re-entry and it can be exploited to design passive end-of-life solutions. Even in the case of A/m = 0.012 m2/kg, in which the variation is less significant, we can take advantage of these corridors to optimize the impulsive strategies aimed at re-entering and to support the road-map for eventually taking advantage of a drag or a solar sail. This application is faced in detail in [1].
The aim of the present work is to give an estimate of the eccentricity vari-ation due to high-order terms in the disturbing functions defined by Equa-tions 2.15, 2.17 and 2.21. Indeed, by linking the estimate of the variation with the perturbation that generates it, we can have an additional tool to ex-ploit the relative importance of each given gravitational or non-gravitational perturbation to be adopted for the design of future space missions strategies. We anticipate here that the estimate, obtained by the method described in the next chapter, will be tested, supported and compared with the analysis described in this chapter. In the following, we will make an extensive use of Equations 2.13, 2.14, 2.30, which provide the variation of eccentricity due, respectively, to high-order odd zonal harmonics of the geopotential (J3 and J5) and to a generic resonant term in the development of the solar and the
Frequency analysis
Spectral analysis considers the problem of determining the spectral content of a finite set of data, in order to detect any periodical behaviour [30]. It finds applications in many different fields; in particular, in the framework of celestial mechanics it may be used as an additional tool to reveal “hidden periodicities” in the studied data, which are to be associated with cyclic behaviour or recurring processes (see, e.g., [24, 29]).
We recall that the goal of the present work is to analyse the different role of the gravitational and non-gravitational perturbations on the orbital evolution in the LEO region. This will be done, in turn, by studying how each perturbation acts on the temporal evolution of the orbital eccentricity. The particular interest in the study of the dynamical evolution of the eccentricity lies in the fact that the final aim of such analysis is to identify strategies for passive disposal of objects orbiting in LEO. Indeed, a significant growth of orbital eccentricity can facilitate passive disposal of the spacecraft at their end of life, since in this case the lowering of the orbital perigee helps in moving the object toward a region where atmospheric drag becomes effective.
Following the work of [39], we present here a frequency characterization of a set of LEO orbits, in order to better understand their dynamical evolution. First of all, we briefly describe the numerical propagator, FOP, which provides the temporal series of eccentricity at different LEO orbits. Then, we discuss how to choose the most suitable initial orbital elements to be adopted for our simulations. Finally, we describe the standard technique to numerically compute the Fast Fourier Transform (FFT) and we show how it is implemented for our purposes.
3.1
Numerical simulations
The first step in order to perform the frequency characterization of the orbital eccentricity is to build the temporal series over a suitable time interval.
3.1.1 The Fast Orbit Propagator (FOP)
The orbital propagation is carried out over a time span of 600 years by means of the semi-analytical Fast Orbit Propagator (FOP) [12], which is an accurate, long-term predictor, based on the Long-term Orbit Propagator (LOP) [34].
FOP propagates the orbits by adopting a singly averaged formulation, by numerically integrating the Lagrange or the Gauss planetary equations applied with the gravitational and non-gravitational perturbations which act on an object orbiting the Earth. The formulation is non-singular for circular orbits and singular for equatorial orbits. The disturbing potential or the equations of motion are averaged over the mean anomaly of the object and propagated using a multi-step, variable step-size and order integrator.
The dynamical model and the disturbing potentials included in FOP have been described in Chapter 2 and they are:
• geopotential harmonics up to degree and order 5. For tesseral resonant effects, located at specific values of semi-major axis, where there ex-ists a commensurability between the satellite’s mean motion and the Earth’s rotation rate, a partial averaging procedure is applied to retain only the long-periodic perturbation associated with these harmonics; • lunisolar perturbations. The position of the Moon and the Sun, which
are held constant during the averaging process, are determined by means of accurate analytical ephemerides;
• solar radiation pressure, including shadows. We adopt the cannonball model and we set the reflectivity coefficient at CR = 1. The shadows are modelled as a solar occultation, assuming a cylindrical shape. The algorithm is based on the assumption that the Sun is a point at infinity and the spacecraft and the Sun are frozen during the occultation. Concerning the atmospheric drag, which can play a relevant role in the dynamical evolution of the LEO region below 1000 km of altitude, we ex-pressly decided to turn it off in the simulations performed for this thesis. Although this approach could seem not realistic, the idea is, rather, to char-acterize the particular effect due to SRP, geopotential and lunisolar pertur-bations and to check if a perturbation alone is capable to induce the disposal. FOP is a very suitable orbital propagator, indeed, it is set up in such a way that each dynamical perturbation in the model can be individually turned on or off. This is very useful for our purposes since, by suitably switching off some perturbations, we can link in an easier way the dynamical effect on the orbital elements to the remaining perturbations. For example, to fit our purposes, in some simulations we properly turned off the lunisolar perturbations and/or the effects of higher order terms of geopotential in order to study the effect of SRP alone on the dynamics.
3.1.2 Simulation scenario
The numerical simulations consist in propagating a number of initial con-ditions, given as orbital elements (a, e, i, Ω, ω) for 600 years from the epoch June 21, 2020, at 06:43:12, corresponding to a June new Moon solstice, where the longitude of the Sun with respect to the ecliptic plane is M ≈ 90◦. This choice has been motivated in [2] by the will of comparing the results of the numerical simulations with simpler, averaged, analytical models where the relative configuration of Sun-Earth-Moon system was expected to play a role. We refer to [2] for further details on the geometrical configuration.
We choose a suitable set of initial conditions, spanning the LEO region for different values of semi-major axis a, from a = R⊕ + 800 km up to
a = R⊕+ 1600 km, and inclination i, from 0◦ to 90◦, at steps of ∆i = 1◦. We
fixed the value of the initial eccentricity at e = 0.001, in order to study the evolution of quasi-circular orbits, i.e., orbits which typically do not facilitate natural re-entry since their perigee almost coincides with their semi-major axis. Moreover, we fixed the initial value of Ω and ω to 0◦, for simplicity. We selected two different values of the A/m ratio of the object: A/m = 0.012 m2/kg, which is a typical value for an object in LEO 1 and the augmented value A/m = 1 m2/kg, representing a typical value for a small satellite equipped with a SRP-enhancing device or a paint flakes (see, e.g., [2]). Tables 3.1 and 3.2 outline the adopted grid for the initial orbital elements in the two A/m cases, respectively.
Table 3.1: Grid of initial orbital conditions in the case of A/m = 0.012 m2/kg.
a (km) ∆a (km) e i (◦) ∆i (◦) Ω (◦) ω (◦) [800 : 1600] + R⊕ 200 0.001 [1:90] 1 0 0
Table 3.2: Grid of initial orbital conditions in the case of A/m = 1 m2/kg.
a (km) ∆a (km) e i (◦) ∆i (◦) Ω (◦) ω (◦) [800 : 1000] + R⊕ 200 0.001 [1:90] 1 0 0
[1000 : 1100] + R⊕ 20 0.001 [1:90] 1 0 0
[1100 : 1600] + R⊕ 200 0.001 [1:90] 1 0 0
As it can be seen from Table 3.2, we adopted a finer grid (∆a = 20 km instead of ∆a = 200 km) at altitudes between 1000 km and 1100 km. This has been done in order to check the evolution of some signatures identified during the frequency analysis, as will be extensively explained in Section 4.4.
1
This value was the result of a thorough analysis carried out on the MASTER catalogue (see, e.g. [9]).
