Phase space description of the dynamics due to the coupled effect of the planetary oblateness and the solar radiation pressure perturbations
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Abstract
The aim of this work is to provide an analytical model to characterize the equilibrium points and the phase space associated with the singly averaged dynamics caused by the planetary oblateness coupled with the solar radiation pressure perturbations. A twodimensional differential system is derived by considering the classical theory, supported by the existence of an integral of motion comprising semimajor axis, eccentricity and inclination. Under the single resonance hypothesis, the analytical expressions for the equilibrium points in the eccentricityresonant angle space are provided, together with the corresponding linear stability. The Hamiltonian formulation is also given. The model is applied considering, as example, the Earth as major oblate body, and a simple tool to visualize the structure of the phase space is presented. Finally, some considerations on the possible use and development of the proposed model are drawn.
Keywords
Solar radiation pressure Planetary oblateness Singly averaged dynamics Phase space Central and hyperbolic manifolds1 Introduction
The main objective of this work is to derive an analytical model in the threedimensional case of the equilibrium points associated with the singly averaged dynamics induced by the coupled effect of the solar radiation pressure (SRP) and the planetary oblateness. As far as we know, such derivation does not exist at the moment and it can represent a fundamental tool in different fields. In particular, the stationary configurations, the invariant curves in the libration regions corresponding to the elliptic points and the hyperbolic invariant manifolds associated with the hyperbolic points can have several applications in planetary dynamics (e.g., Mignard 1982; Krivov et al. 1996) and in mission design (e.g., Colombo and McInnes 2012b; Wittig et al. 2017).
The literature shows that the pioneers in the identification of the role of the SRP effect coupled with the zonal harmonics \(J_2\) of a primary body in the orbital evolution of a small body were Musen (1960) and Cook (1962). They developed the corresponding singly averaged equations of motion, for a satellite orbiting the Earth, in terms of Keplerian orbital elements, and remarked the existence of six orbital resonances involving the rate of precession of the longitude of the ascending node, of the argument of pericenter and the apparent mean motion of the Sun. Later on, Hughes (1977) expanded the disturbing function associated with SRP using the Kaula’s method up to highorder terms, and provided some examples on whether they can be relevant for satellites in Low Earth Orbit (LEO). Recall also that Breiter (1999, 2001) addressed the analytical treatment of lunisolar perturbations in canonical coordinates, considering not only the resonant condition, but more in general the critical points associated with the dynamical system, together with their stability.^{1}
In terms of applications, in the vast literature on the dynamics of dust in planetary systems, we can mention Krivov et al. (1996) and Hamilton and Krivov (1996), who studied the eccentricity oscillation associated with the singly averaged disturbing potential of SRP and \(J_2\) for a dust particle orbiting a planet. Krivov and Getino (1997) applied the same model, but in the planar case and only considering one of the resonant terms, also for obtaining a phase space representation of the orbit evolution of lowinclination Earth orbits, in the perspective of highaltitude balloons.
Colombo et al. (2012) performed a parametric analysis of the SRP\(J_2\) phase space for different values of semimajor axis and areatomass ratio, identifying the equilibrium points corresponding to either heliotropic (apogee pointing toward the Sun) or antiheliotropic (perigee pointing toward the Sun) orbits. The equilibrium conditions computed analytically for the planar case were numerically extended to nonzero inclinations. For such frozen orbits, different applications were proposed: the use of heliotropic orbits for Earth observation in the visible wavelength (Colombo et al. 2012; Colombo and McInnes 2012a), or the use of heliotropic orbits at an oblate asteroid (Lantukh et al. 2015), or also the use of antiheliotropic orbits for geomagnetic tail exploration missions (McInnes et al. 2001; Oyama et al. 2008; Luo et al. 2018). Following (Krivov and Getino 1997) a particular application was proposed by Lücking et al. (2012, 2013) for passive deorbiting of spacecraft at the end of life. Given the mass parameter of the main body, the dynamics under consideration depends on three constants, the semimajor axis of the orbit, a given integral of motion and the areatomass ratio of the small body. In Lücking et al. (2012, 2013) and Colombo and de Bras de Fer (2016), the areatomass required to obtain a given eccentricity increase was computed by means of an optimization procedure, considering also the associated time.
With the same aim of looking for natural perturbations that can support a reentry from LEO at the end of life, Alessi et al. (2018) performed an extensive numerical mapping of the region, and identified deorbiting natural corridors, defined in terms of inclination and eccentricity for given semimajor axis. In Schettino et al. (2019), a numerical frequency portrait of the LEO region was presented, highlighting also the role that highorder resonances associated with SRP (Hughes 1977) might have.
The other typical example of mission that can be designed on the basis of the coupled SRP\(J_2\) dynamics is a mission to an asteroid or a comet (e.g., Russell 2012; Scheeres 2012).
In this work, the threedimensional equations of motion which describe the variation in eccentricity, inclination, and a given angle accounting for the motion of the longitude of the ascending node and the argument of pericenter are analyzed, they are reduced to a twodimensional case by means of a wellknown integral of motion, and the analytical expressions for resonant conditions and equilibrium points are provided. The corresponding stability is given for the six main resonances, and the possible different phase space portraits for orbits at the Earth are described. Finally, the tools presented are analyzed in the perspective of a practical exploitation.
Throughout the work, it is always assumed that the major body is the Earth, but the analytical treatment developed is general and it can be applied to any oblate body. In our formulation, we adopted the Earth equatorial plane as reference plane, because our major interest is a possible exploitation for deorbiting. We notice, however, that for other applications, e.g., Scheeres (2012), the ecliptic reference system can be more suitable.
Moreover, the concept proposed will hold certainly in the cases described above, where the two perturbations—oblateness and solar radiation pressure—play a major role, but it will be of high interest to see how the solutions presented may persist under the effect of an additional effect and under which conditions they remain the skeleton of the longterm dynamics. For orbits at the Earth, the key case will be the dynamics due to lunisolar gravitational perturbations for Highly Elliptical Orbits (Colombo 2019) and in the geosynchronous region (e.g., Casanova et al. 2015; Gkolias and Colombo 2019).
2 Dynamical model
Let us assume that a small body (e.g., a spacecraft) moves under the effect of the Earth’s gravitational monopole, the Earth’s oblateness and the solar radiation pressure. In particular, for the SRP it is assumed in the cannonball model that the orbit of the spacecraft is entirely in sunlight and that the effect of the Earth’s albedo is negligible (e.g., Krivov et al. 1996).
Argument \(\psi _j=n_1{\varOmega }+n_2\omega + n_3 \lambda _S\) of the periodic component in terms of \(n_1, n_2, n_3\)
j  \(n_1\)  \(n_2\)  \(n_3\)  Cook 

1  1  1  \(\)1  8 
2  1  \(\)1  \(\)1  7 
3  0  1  \(\)1  15 
4  0  1  1  14 
5  1  1  1  6 
6  1  \(\)1  1  9 
2.1 Resonances and equilibrium points

for \(j=1\), if Eq. (8) is verified at prograde orbits, then the longitude of the periapsis is sunsynchronous;

for \(j=2\), if Eq. (8) is verified at retrograde orbits, then the longitude of the periapsis is sunsynchronous;

for \(j=3\), the argument of periapsis is sunsynchronous;

for \(j=4\), the argument of periapsis is sunantisynchronous;

for \(j=5\), if Eq. (8) is verified at prograde orbits, then the longitude of the periapsis is sunantisynchronous;

for \(j=6\), if Eq. (8) is verified at retrograde orbits, then the longitude of the periapsis is sunantisynchronous.
Coefficients of the quadratic equation \(c_1\cos ^2{i}+c_2\cos {i}+c_3=0\) associated with the equilibrium point at \(\psi _j=0\) for \(j=1,2,5,6\)
j  \(c_1\)  \(c_2\)  \(c_3\) 

1  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta +2C_\mathrm{{SRP}}a^4\beta ^4\gamma \)  \(~~2C_\mathrm{{SRP}}a^4\beta ^4\gamma (12e^2)e\beta \left( 3C_{\oplus }+4a^5\beta ^4nn_S\right) \) 
2  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta +2C_\mathrm{{SRP}}a^4\beta ^4\gamma \)  \(2C_\mathrm{{SRP}}a^4\beta ^4\gamma (12e^2)+e\beta \left( 3C_{\oplus }4a^5\beta ^4nn_S\right) \) 
5  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta +2C_\mathrm{{SRP}}a^4\beta ^4\rho \)  \(~~2C_\mathrm{{SRP}}a^4\beta ^4\rho (12e^2)e\beta \left( 3C_{\oplus }4a^5\beta ^4nn_S\right) \) 
6  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta +2C_\mathrm{{SRP}}a^4\beta ^4\rho \)  \(2C_\mathrm{{SRP}}a^4\beta ^4\rho (12e^2)+e\beta \left( 3C_{\oplus }+4a^5\beta ^4nn_S\right) \) 
Coefficients of the quadratic equation \(c_1\cos ^2{i}+c_2\cos {i}+c_3=0\) associated with the equilibrium point at \(\psi _j=\pi \) for \(j=1,2,5,6\)
j  \(c_1\)  \(c_2\)  \(c_3\) 

1  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta 2C_\mathrm{{SRP}}a^4\beta ^4\gamma \)  \(2C_\mathrm{{SRP}}a^4\beta ^4\gamma (12e^2)e\beta \left( 3C_{\oplus }+4a^5\beta ^4nn_S\right) \) 
2  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta 2C_\mathrm{{SRP}}a^4\beta ^4\gamma \)  \(~~2C_\mathrm{{SRP}}a^4\beta ^4\gamma (12e^2)+e\beta \left( 3C_{\oplus }4a^5\beta ^4nn_S\right) \) 
5  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta 2C_\mathrm{{SRP}}a^4\beta ^4\rho \)  \(2C_\mathrm{{SRP}}a^4\beta ^4\rho (12e^2)e\beta \left( 3C_{\oplus }4a^5\beta ^4nn_S\right) \) 
6  \(15C_{\oplus }e\beta \)  \(6C_{\oplus }e\beta 2C_\mathrm{{SRP}}a^4\beta ^4\rho \)  \(~~2C_\mathrm{{SRP}}a^4\beta ^4\rho (12e^2)+e\beta \left( 3C_{\oplus }+4a^5\beta ^4nn_S\right) \) 
Coefficients of the cubic equation \(\sin ^3{i}+s_1\sin ^2{i}+s_2\sin {i}+s_3=0\) associated with the equilibrium point at \(\psi _j=0\) for \(j=3\) and \(j=4\)
j  \(s_1\)  \(s_2\)  \(s_3\) 

3  \(\frac{2C_\mathrm{{SRP}}}{15C_{\oplus }e}a^4\beta ^3\sin {\epsilon }\)  \(\frac{4}{15C_{\oplus }}\left( 3C_{\oplus }a^5\beta ^4n n_s\right) \)  \(\frac{2C_\mathrm{{SRP}}}{15C_{\oplus }}ea^4\beta ^3\sin {\epsilon }\) 
4  \( \frac{2C_\mathrm{{SRP}}}{15C_{\oplus }e}a^4\beta ^3\sin {\epsilon }\)  \(\frac{4}{15C_{\oplus }}\left( 3C_{\oplus }+a^5\beta ^4n n_s\right) \)  \(\frac{2C_\mathrm{{SRP}}}{15C_{\oplus }}ea^4\beta ^3\sin {\epsilon }\) 
Coefficients of the cubic equation \(\sin ^3{i}+s_1\sin ^2{i}+s_2\sin {i}+s_3=0\) associated with the equilibrium point at \(\psi _j=\pi \) for \(j=3\) and \(j=4\)
j  \(s_1\)  \(s_2\)  \(s_3\) 

3  \(\frac{2C_\mathrm{{SRP}}}{15C_{\oplus }e}a^4\beta ^3\sin {\epsilon }\)  \(\frac{4}{15C_{\oplus }}\left( 3C_{\oplus }a^5\beta ^4n n_s\right) \)  \(\frac{2C_\mathrm{{SRP}}}{15C_{\oplus }}ea^4\beta ^3\sin {\epsilon }\) 
4  \( \frac{2C_\mathrm{{SRP}}}{15C_{\oplus }e}a^4\beta ^3\sin {\epsilon }\)  \(\frac{4}{15C_{\oplus }}\left( 3C_{\oplus }+a^5\beta ^4n n_s\right) \)  \(\frac{2C_\mathrm{{SRP}}}{15C_{\oplus }}ea^4\beta ^3\sin {\epsilon }\) 
2.2 Hamiltonian formulation
3 Dynamical description
The solutions corresponding to inclination equal to zero, for \(j=1\) and \(j=2\), are equivalent to the ones described in Colombo et al. (2012). For \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\) and \(j=1\) or \(j=2\), the solutions presented in Colombo and McInnes (2012a), for the inclined case when considering only \(J_2\), are equivalent to the ones shown here, because for low A / m values the effect of the solar radiation pressure on \(({\varOmega }, \omega )\) is negligible.
Notice how the inclination changes as the semimajor axis increases, increasing or decreasing depending on j and on whether the equilibrium is at \(\psi _j=0\) or \(\psi _j=\pi \). Note also that, by increasing the areatomass ratio, the main effect is to enlarge the range of semimajor axis where the effect can be detected. In general, for quasicircular orbits, the dynamics is coupled up to about \(a=15000\, {\hbox {km}}\), except for the cases \(\psi _1=0\) and \(\psi _2=\pi \) of the debris fragment, for which the effect can be considerable up to Geosynchronous Earth Orbits (GEO) altitudes. In Figs. 5, 6, we show the inclination values corresponding to the equilibrium points for \(j=3\) and \(j=4\), as a function of (a, e) for the same three values of areatomass ratio. The inclination shown is computed by considering one of the solutions of the cubic equations displayed in Tables 4 and 5. The other solutions correspond to the opposite value shown and to the singularity \(i=0.\)
Notice that Fig. 8 can give also the information on the range of inclination within which we can have a given number of equilibrium points with the corresponding stability, and how the inclination changes according to the eccentricity, for given semimajor axis and \({\tilde{{\varLambda }}}\). In other words, the inclination corresponding to a given phase space portrait, such as the ones in Fig. 9, can be inferred by looking into Fig. 8. A preliminary definition of this domain was considered in Schettino et al. (2017), taking into account only the strictly resonant behaviors for almost zero eccentricities.
For \(j=1\), \(A/m=1 \, {\hbox {m}}^2/\hbox {kg}\), by increasing the semimajor axis, the number of possible equilibrium points can increase. Spanning a range of a up to the GEO ring, the maximum number is 5, and it is found already at \(a=8178\, {\hbox {km}}\). In Fig. 10, we show the corresponding behavior of the \((e,{\tilde{{\varLambda }}})\) curves and the phase space portrait associated with 5 equilibrium points for \(a=12078\, {\hbox {km}}\). Moreover, we note that \((e,{\tilde{{\varLambda }}})\) curves cannot be continuous, because Eq. (7) sets specific constraints on the range of \({\tilde{{\varLambda }}}\) (\(\cos {i}\in [1,1]\)).
Looking into the \((e,{\tilde{{\varLambda }}})\) curves for \(j=1\), always considering the solution for \(\cos {i}\) on the first column of Fig. 1, for \(A/m=20 \, {\hbox {m}}^2/{\hbox {kg}}\) it is noticed a similar behavior, while for \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\) the number of equilibrium points can be 4 also in nondegenerate configurations. Taking as reference the phase space portrait depicted in Fig. 10 on the right, for \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\) and given \({\tilde{{\varLambda }}}\), the equilibrium point existing at the lowest eccentricity can disappear, and there persist one unstable equilibrium at \(\psi _1=0\) and one stable equilibrium at \(\psi _1=\pi \) corresponding to a same eccentricity, and one stable equilibrium at \(\psi _1=0\) and one unstable equilibrium at \(\psi _1=\pi \) corresponding to a different eccentricity.
With respect to the other resonant behaviors, the following is stressed. Generally speaking, it appears that the dynamics associated with the resonant terms #1, #2 and #4 is richer than for the other three resonant terms, in terms of maximum possible number of equilibrium points. Also, the value of areatomass ratio can change the value of semimajor axis at which a bifurcation can occur.
Resonant term #2 For \(A/m=1 \, {\hbox {m}}^2/{\hbox {kg}}\), considering the solution for \(\cos {i}\) on the second column of Fig. 2, the curve corresponding to \(\psi _2=0\) is higher than the curve corresponding to \(\psi _2=\pi \), contrary to what happens for \(j=1\). For the lower values of semimajor axis, the maximum possible number of equilibrium points is 3: one stable at \(\psi _2=\pi \) and one stable and one unstable at \(\psi _2=0\), that is, as in Fig. 9 on the bottom right, but the curves are shifted by \(\pi \) in \(\psi \). At \(a=9878\, {\hbox {km}}\), it can appear also one unstable equilibrium at \(\psi _2=\pi \) and the 3 equilibrium points can be as before or all associated with \(\psi _2=\pi \), as in Fig. 9 on the top right, but again shifted by \(\pi \) in \(\psi \). From about \(a=10078\, {\hbox {km}}\), the phase space can be structured around 5 equilibrium points, as in Fig. 10 on the right, but also shifted by \(\pi \) in \(\psi \). For \(A/m=20 \, {\hbox {m}}^2/{\hbox {kg}}\), the same behavior is found, while for \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\), the only difference is that, analogously to what happens for the resonant term #1, it is possible to have 4 equilibrium points, as the one associated with the lowest eccentricity drifts as much toward \(e=0\) as to disappear.
Resonant term #5 The dynamics is analogous to what described for \(j=1\) except that we do not ever notice 5 equilibrium points and that there exist large regions where the number is only 2: one stable at \(\psi _5=0\) and one unstable at \(\psi _5=\pi \), or vice versa. In particular, for \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\) and \(A/m=1 \, {\hbox {m}}^2/{\hbox {kg}}\), this is always the case from about \(a=19000\, {\hbox {km}}\).
Resonant term #6 The dynamics is analogous to what described for \(j=2\), but, analogously to the case \(j=5\), the maximum number of equilibria is 3 and in many cases we can see only 2. In particular, for the three values of areatomass explored this is always the case from about \(a=16500\, {\hbox {km}}\).
Resonant term #3 For \(A/m=1 \, {\hbox {m}}^2/{\hbox {kg}}\), there are 3 equilibrium points: one stable equilibrium point at \(\psi _3=0\) at a low eccentricity, and one unstable at \(\psi _3=0\) and one stable at \(\psi _3=\pi \). This occurs until about \(a=15500\, {\hbox {km}}\) (about \(a=15000\, {\hbox {km}}\) for \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\) and about \(a=17000\, {\hbox {km}}\) for \(A/m=20 \, {\hbox {m}}^2/{\hbox {kg}}\)), where the equilibrium at the lowest eccentricity disappears. From that value of semimajor axis on, the 2 equilibrium points drift toward higher values of eccentricity, analogously to what happens for the other resonant terms.
Resonant term #4 In this case, the dynamics can be as rich as in the case of resonant terms #1 and #2, in the sense that there can exist up to 5 equilibrium points: one stable at \(\psi _4=0\) at a low eccentricity, one unstable at \(\psi _4=0\) and one stable at \(\psi _4=\pi \), and then one stable at \(\psi _4=0\) and one unstable at \(\psi _4=\pi \) at increasing eccentricity. This occurs for \(A/m=0.012 \, {\hbox {m}}^2/{\hbox {kg}}\) until about \(a=10000\, {\hbox {km}}\), for \(A/m=1 \, {\hbox {m}}^2/{\hbox {kg}}\) until about \(a=11600\, {\hbox {km}}\), for \(A/m=20 \, {\hbox {m}}^2/{\hbox {kg}}\) until about \(a=19000\, {\hbox {km}}\). From that value on, there persist 4 equilibrium points, which are associated with increasing values of eccentricity for increasing values of semimajor axis.
4 Discussion and future directions
The analysis proposed in this paper provides the fundamental ingredients to describe the dynamics of a body with a high areatomass ratio, which orbits a given major oblate body, and which is subject to solar radiation pressure effects.
In particular, it is possible to apply all the dynamical systems theory tools (e.g., Parker and Chua 1989; Wiggins 2003) which can characterize the elliptic (stable) and hyperbolic (unstable) behaviors associated with the equilibrium points of the system. The information given by the eigenvalues and eigenvectors of the Jacobian matrix, Eq. (11), computed at the corresponding equilibrium point, can be used to compute the invariant librating curves in the neighborhood of the elliptic points and the hyperbolic invariant manifolds in the neighborhood of the hyperbolic points. Concerning the time required to move along a curve, at an elliptic point the two eigenvalues are conjugate pure imaginary, say \(\pm i\nu \). The closer the libration curve to the elliptic point, the closer its period to \(T=2\pi /\nu \). In Fig. 11, we show the value of the period T in years for the cases shown in Figs. 8, 9 and 10. The asymptotic behavior that can be seen in the figure corresponds to the line of transition between different behaviors—the point of bifurcation described before. As the distance with respect to the elliptic point increases, the time needed to cover the curve also increases, in the limit to reach the hyperbolic manifold, which tends, by definition, asymptotically to the unstable point. Note that the resonant curves corresponding to \(\psi _j=\pi /2\) and \(\psi _j=3\pi /2\), Eqs. (9)–(10), always act as separatrices between libration and circulation motion, when an equilibrium point at low eccentricity exists.
On the other hand, each invariant curve, as the ones depicted in Figs. 9 and 10, is characterized by a constant Hamiltonian—Eq. (14)—and this information can be considered to compute a priori the maximum and minimum eccentricity that can be achieved. For instance, departing from the unstable direction associated with a saddle point, corresponding say to \(\psi =0\), we can see if an eccentricity corresponding to \(\psi =\pi \) can be attained and, in case, its value, by inverting Eq. (14) as a function of \(\Sigma \), or \(\sqrt{1e^2}\). A detailed analysis on how this can be done will be the focus of a future investigation.
As it is well known, the single resonance hypothesis assumed in this work can fail to hold, that is, it can occur that two resonant terms can play a comparable role at the same time. This can be inferred from Figs. 1, 2, 3, 4, 5 and 6, which show that, for a same (a, e, i, A / m) combination, there exist equilibrium points associated with different resonant terms. The same situation can be also depicted as an intersection, in the (e, i) plane, between curves corresponding to equilibrium points for given (a, A / m).^{4} In Fig. 12, we show an example for \(a=10078\, {\hbox {km}}\) and \(A/m=1 \, {\hbox {m}}^2/{\hbox {kg}}\) and \(A/m=20 \, {\hbox {m}}^2/{\hbox {kg}}\). Note how the stability can change by changing the areatomass ratio and also the number of intersections. By increasing the value of semimajor axis, the curves tend to be parallel with respect to the xaxis toward higher values of eccentricity.
The behavior in the regions where two dominant terms can exist will be studied in the future by analyzing the value of the corresponding amplitude and period and also the possible occurrence of a chaotic behavior.
Future work will include the estimate of the size and location of the chaotic regions along with the corresponding Lyapunov time, not only in case of overlapping resonances but also in the neighborhood of the hyperbolic invariant manifolds associated with the hyperbolic equilibrium points. Also, specific practical applications for both bounded and escape trajectories for the six resonant terms will be investigated. For instance, as shown in Colombo et al. (2012), the equilibrium points corresponding to orbits with apogee pointing the Sun can be exploited for enhanced Earth observation in the visible wavelength, while the equilibrium point corresponding to orbits with perigee pointing the Sun can be exploited for exploration mission of the Earth magnetosphere (McInnes et al. 2001; Oyama et al. 2008). The design of disposal trajectories with a solar sail, so far performed analytically only for planar orbit (Lücking et al. 2012) and numerically for inclined orbits (Colombo and de Bras de Fer 2016), can be fully solved analytically with the method presented here. In the field of space debris, similarly to what was done in Schaus et al. (2019) for the case of the “resonant reentry corridors”, it will be of interest to analyze the space debris catalog to see whether there exists a specific case trapped in one of the resonances found. Other applications can be considered for different planetary systems, such as the frozen solutions found in Scheeres (2012).
Finally, as mentioned at the beginning, it will be our priority to see how the phase space may change when an additional perturbation comes into play. In the case of the Earth, lunisolar gravitational perturbations and the ellipticity of the Earth will be considered for highaltitude orbits (see, e.g., Casanova et al. 2015; Gkolias and Colombo 2019; Valk et al. 2009) and the role of the atmospheric drag in the final deorbiting phase (see, e.g., Vilhena de Moraes 1981).
Footnotes
 1.
Recall that the SRP effect can be written analogously to the solar gravitational perturbation, except for the amplitude of the perturbation and the first order of its expansion (Hughes 1977).
 2.
We always assume \(c_R=1\) in this work.
 3.
In our formulation, \(n_1\) and \(n_2\) are inverted with respect to the formulation in Daquin et al. (2016) and we call \(\Sigma _{1,2,3}\) which they call \({\varLambda }_{1,2,3}\).
 4.
Notes
Acknowledgements
The authors would like to acknowledge the funding received by the European Commission Horizon 2020, Framework Programme for Research and Innovation (2014–2020), under the ReDSHIFT project (Grant agreement n. 687500) and the funding received by the European Research Council (ERC) through the European Commission Horizon 2020, Framework Programme for Research and Innovation (2014–2020), under the project COMPASS (Grant agreement n. 679086). The authors are grateful to Josep Masdemont, Ioannis Gkolias and Kleomenis Tsiganis for the useful discussions.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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