The full gravitational 2-body problem (FG2BP) accounts for the dynamics of two rigid bodies ℬ1 and ℬ2, with masses
The FG2BP is far from being well understood and requires physical and mathematical simplifications to be tackled, [27]. Therefore, only partial approaches are possible by now. The strongest streamlining of the FG2BP is carried out by assuming two point masses, then leading to the Kepler system by setting the Jacobi coordinates. A possible generalisation of this situation keeping the integrability is assuming two spherical masses. Thus, obtaining Keplerian motion for the orbital part, which is independent of the uniform rotation of the two spheres. Next level in the generalisation of the ideal situation is to consider an asymmetric body and a sphere of homogeneous density. Nevertheless, this situation still leads to a wide set of scenarios depending on the distances and masses involved. For instance, truncation of the gravitational potential associated to this problem up to the second order, the so called MacCullagh’s approximation, is possible when the orbiting body is small compared with the distance between the mass centres. In our work, we precisely consider this situation and discard terms on the gravitational potential higher that two. As such, from now on we will refer indistinctly to this approximation as the MacCullagh’s truncation and the full model as well. A different story would be if we considered two generic bodies, then the fourth order of the gravitational potential includes terms of the body-body interaction and not only spin-orbit interaction, as it happens with the second order term. Therefore, the truncation of the potential in this case is recommended at least to the fourth order, see [4].
The study of the attitude dynamics of a generic triaxial spacecraft in a central gravitational field permeates along the space era, from [3], [7], [8] and [22], up to the very recent research including [12], [28] and [31]. This problem covers aspects such as determination, propagation and control, that continue to be areas of research, see [20, 29, 33] for further details. Common to these studies has been the assumption of fast rotation. Hence, distinguishing between fast and slow variables, rotational motion has been approached by perturbative techniques [24].
We approach the gravity-gradient problem making use of the concept of
where the intermediary 𝓗0 defines a non-degenerate and simplified model of the problem at hand, which includes the Kepler and free rigid-body as particular cases and 𝓗1 is usually dubbed as the perturbation. A special realization of an intermediary occurs for the case in which it is an integrable 1-DOF system. The work of Hill on the Moon motion [32] is, perhaps, the best known example. The interest on an intermediary is twofold. On the one hand, it allows us to identify special solutions that could become nominal trajectories in missions design whereas it alleviates usual heavy computations. On the other hand, it can be used to build a perturbation theory based on a new unperturbed part avoiding the degenerate character inherent to the classical superintegrable models (Kepler or free rigid body systems in astrodynamics). In other words, a first order perturbed solution based on intermediaries might be accurate enough for tracking purposes. In astrodynamics, when dealing with orbital dynamics applied to artificial satellites, some lines of research on intermediaries arose during the seventies by Garfinkel, Aksnes, Cid, Sterne, etc. (see review in [14]), whose benefits are now seen in areas such as the relative motion in formation flights, an example is given in [25]. Nevertheless, less work has been done when dealing with attitude dynamics, where the proposal of intermediaries is more recent [2, 17] and, to our knowledge, no systematic study has been done on them.
In this paper, we continue our work on the intermediary model proposed in previous works [11, 19, 30], where an integrable 1-DOF intermediary was obtained by an uncontrolled truncation of the MacCullagh approximation [26] and assuming the secondary body to be in a Keplerian orbit. The accuracy of this model was tested by comparing with the MacCullagh’s truncation and showing a good performance in the numerical experiments. In our preceding work, we restrict to the study of the axis-symmetric case. The present manuscript complements this analysis by extending it to the case of a triaxial body and keeping all the remaining assumptions made previously. This change leads us to a 2-DOF intermediary model, which is expressed with variables referred to the total angular momentum, see [15, 17].
The validity and applicability of the model is assessed numerically. For that purpose, we study the accuracy of simulation in three different scenarios. First, we consider the triaxiality parameter introduced in [10],
We also include the study of the relative equilibria, finding constant radius solutions filling a 4-D torus. In addition, equilibria leading to lower dimensional torus are identified and conditions for periodic orbits are detected. The intermediary model is endowed with several distinguishing and physical parameters. That is to say, physical constants are related to the bodies’ features and integrals are characterized by their initial configuration. Having a high multidimensional parametric space introduces complexity in the study of bifurcations. With the aim of simplifying this scenario, we consider cos
This paper is organized as follows. Section 2 is devoted to introduce the triaxial intermediary into the Hamiltonian formalism and to describe the canonical variables in which it is expressed. In Section 3, we present the numerical simulations bounding the applicability and validity of the model. In addition, conditions leading to families of relative equilibria are identified in Section 4. We establish the connection of our families of relative equilibria with the classical ones reported in the related literature in Section 5. Finally, we present conclusions and include an Appendix section.
The formulation of the triaxial intermediary model follows the same derivation as the one made in [11], which is based in six simplifying assumptions. More specifically, the following set of simplifications are assumed in order to define our modelization:
The Hamiltonian of the roto-orbital model is obtained from the mechanic energy function. Thus, denoting
in other words, the potential is usually split in two parts: a term which depends only on 1/
where
where
The variables in which the problem is posed may have a significant impact on its treatment. Our choice is the use of the total angular momentum as the key object to define them, which application for the roto-translatory problem was first introduced in [17] as a result of the application of the elimination of the nodes in the
The complete set of canonical variables is (
In addition, the conjugate momenta of the variables read as follows
where
Finally, we gather below the formulas for the relative positions of the different planes, as functions of the canonical momenta, which are given by
and the following relations between angles and momenta hold
where
The direction cosines appearing in (3) may be expressed in the body frame by means of the following composition of rotations:
where
where
which is independent of
and the notation has been abbreviated by writing
Facing a non-integrable Hamiltonian system requires the development of a perturbation theory. A usual way to proceed is to expand the Hamiltonian function in power series and truncate it at a certain order. This procedure is in general an uncontrolled approximation since, for most cases, we are no longer sure that the solutions of the truncated model stay close to the full model. In the case of the FG2BP, it is customary to consider the orbital and rotational kinetic energies and choose the first term in the expansion of the gravitational potential, see (1). Thus, we end up with the free rigid body Hamiltonian (the rotational kinetic energy) and the Kepler system Hamiltonian. That is, the orbital kinetic energy plus the first non-zero term of the gravitational potential. However, there is no any theorem claiming that this is the only way it can be done. The novelty of an intermediary model is that it allows to consider truncation of partial order in the power series expansion. In other words, we pick some “entire” terms of the power series plus a “piece” of one term. The only requirement that must be fulfilled in this procedure is that the intermediary model should be searchable.
In [11], the authors proposed an axis-symmetric integrable intermediary model, whose accuracy was tested by comparing with the MacCullagh’s truncation and showing a good performance in the numerical experiments. Here, we complete this previous study by investigating the triaxial case. One of our aims is to analyze the physical-parametric bifurcations of relative equilibria due to the elimination of the axial symmetry of the body. Keeping this motivation in mind, we propose our intermediary following exactly the same procedure than in [11], except for the triaxial parameter. That is to say, we only take into account the first line of
which leads us to the final expression of the intermediary Hamiltonian
where
Nevertheless, for the sake of simplicity, we keep the original notation without primes on the variables. Then, the 2-DOF Hamiltonian system of differential equations associated with (13) is given by the following expressions:
together with the integrals Θ̇ =
Note that, in general, a 2-DOF system is not integrable. Thus, in the triaxial case, the analytical integration is not provided and the integrability of the system remains as an open question, which is not in the scope of the present paper.
In this section we present numerical experiment comparing the intermediary versus the full model (MacCullagh’s approximation). For this purpose, we sweep three parameters that allow us to define the usability of the triaxial intermediary model. In the first scenario, we recover the triaxiality parameter
and allows us to assess the impact of the triaxiality in the performance of the model. Next, we analyze the role of the eccentricity and height (distance between the bodies surfaces) by carrying out simulations with these parameters ranging from
Before we get into the numerical evaluations, it is important to pay some attention to the way in which we proceeded. With the aim of keeping track of the geometry, the initial conditions for the variables are not given directly. Namely, we introduce some of the canonical momenta by means of the set (
Additionally, in order to reproduce the numerical experiment, it is very important to keep in mind throughout this section that we have introduced some simplifications. Namely, we have considered the Hamiltonian per unit of mass and the canonical and inertia momenta have been scaled, see (14). Furthermore, we have changed internally the units for longitudes by choosing the new one as the radius of the spherical body
In this part we analyze the impact of the triaxiality parameter
Before we present our experiments in the general case, we study in detail the transition from
Now we consider the case of higher triaxialities. Despite of what have been said, these experiments still show a very high accuracy for moderate values of
Initial conditions for simulations in Figure 2 and Figure 3.
making use of the formulas in the Appendix, we are led to the following initial values for the canonical variables
In the formulation of the triaxial model in Section 2 we restrict the validity of the model to small eccentricities. That is to say, we consider eccentricity up to 0.5, since for higher values even the full model (MacCullagh’s approximation) does not provide a valuable estimation. In Figure 4
Next, we specify the initial conditions and parameters to assess the role of the eccentricity. Firstly, we fix the dimensions of the secondary body to be 60, 55 and 45 Km. Then, we obtain the following associated moments of inertia
The remaining variables, inclinations and momenta are also fixes with the exception of the eccectricity
making use of the formulas in the Appendix, we are led to the corresponding initial values for each value of the eccentricity. Namely, for
and for
This section present a representative selection for a low range of altitudes. In Figure 5 we compare the results obtained for 1000 and 2000 Km, with fixed
Initial conditions and parameters are set as follows
Then, we obtain the following initial conditions for heights
for height = 1000Km and
for height = 2000Km.
The system of differential equations defined by the Hamiltonian (13) is endowed with several distinguish and physical parameters. Thus, bifurcations occurs in several directions in the parametric space. With the aim of simplifying this scenario and provide a clear geometric interpretation of our equilibria, we organize our families of relative equilibria according to the inclinations of pairs of fundamental planes (orbital, rotational and body planes). More precisely, we consider the relative inclination between orbital and rotational planes (
However, the physical parameters do also give rise to bifurcations. Namely, the number and nature of equilibria depends on the moments of inertia (
We begin our study by looking for a possible relative equilibrium of our Hamiltonian system with constant radius, which leads to solutions living in a 4-D torus. Then, we proceed with the study of conditions leading to lower dimensional tori. We would like to remark that we are not going to carry out an exhaustive study of all the equilibria of this intermediary, since it would require the use of variables free of singularities which is beyond the scope of this paper and it is a research in progress part of [6]. Henceforth, we consider the following notation
Observe that on one hand, the cases {
Along this subsection we study the equilibria obtained under a fixed inclination of the dihedral angle
Our first approach to tackle the problem of searching for orbits of constant radius within our intermediary is by examining the subsystem ( The orbits obtained under this inclination are restricted to be into the invariant 𝕋4(
Following the classification previously indicated, we study here the equilibria obtained under a fixed inclination of the
Note that the above formula implies
As result of the appearance of the
Considering the subsystems ( which leads to an implicit relation between Using Taylor series of the previous expression of where This expression shows us how the triaxiality of Searching for conditions that lead us to find equilibria for We examine now the subsystem (
The variation of the remainder angles for these orbits are:
Under this particular body-inclined equilibria the relations between inclinations are given by:
where we introduce the following notation
These relationships lead us to the following family of curves
whose graph is studied in the Cartesian plane
Notice that the triaxiality of
A partial study of the triaxiality role is presented now in more detail due to the important effects produced on the inclinations of
Within our restriction and fixing a certain inclination
Moreover in figures 7 and 8 (left) we included a representation of the triaxiality region.
As we observe the triaxial shape of
Working on the subsystems ( This particular inclination of Particularly the variation of the remainder angles are given by: Notice that these mean motions for
Along this section we study the equilibria obtained under a fixed inclination of the
Examining the subsystems ( Observe that the radius obtained is according to [11] when the triaxial body tends to be axial-symmetric. Worth noting is that when the orbital and the rotational planes present a critical inclination, i.e.
Taking into account the subsystems ( Notice that this equilibrium has the same constant radius as the one obtained for a critical inclination, therefore we interpret that for this equilibrium The remainder angles are given by: Analyzing the subsystems ( The expression for the remainder angles are:
Consequently the mean motions of
In general, classical relative equilibria of the roto-orbital dynamics associated with a rigid body in circular orbit [16, 21, 22] are not reflected in our analysis, since they require particular inclinations of the fundamental planes involved leading to angles singularities. For instance, the solutions usually designated as
Next, we consider some special configurations. Let
These relative equilibria are similar to the types
As a final remark, we would like to point out that, a complete identification of the classical families of relative equilibria involves the use of a complete set of charts, which cover the case in which the axis
An intermediary model has been presented considering the triaxial version of the one introduced in [11]. The numerical simulations assess the validity of the model for the case of a fast rotating body. Although, more exhaustive experiments are necessary to establish bounds for its applicability. Moreover, our experiments show the influence of the triaxiality reporting a marked degradation in the precision of the fast angles, though the radial distance and slow angles remain to be approximated with high accuracy after one orbital period. Nevertheless, the evaluation of a slow rotation regime is part of our ongoing research.
We also investigate the relative equilibria of the model finding families that depart from the equilibria of the free rigid body and the classical equilibria reported in the literature. Yet, some of these equilibria are recovered in our setting and we give a detailed geometric description of how to identify them. A complete searching of classical equilibria in our setting requires the use of several charts. The role of the triaxial shape of the body is studied for some equilibria, showing partially how the triaxiality influences the inclinations which lead to equilibria. The integrability of our model and the stability of the equilibria obtained are parts of a further research.