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Solution to the Main Problem of the Artificial Satellite by Reverse

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Solution to the Main Problem of the Artificial Satellite by Reverse Nonlinear Dynamics manuscript No. (will be inserted by the editor) Solution to the main problem of the artificial satellite by reverse normalization Martin Lara draft of April 16, 2020 Abstract The non-linearities of the dynamics of Earth arti- tion of the zonal harmonic of the second degree, whose co- ficial satellites are encapsulated by two formal integrals that efficient is usually denoted J2, is traditionally known as the are customarily computed by perturbation methods. Stan- main problem of artificial satellite theory. The J2-truncation dard procedures begin with a Hamiltonian simplification that of the gravitational potential is known to give rise to noninte- removes non-essential short-period terms from the Geopo- grable dynamics [12,26,5,8] that comprise short- and long- tential, and follow with the removal of both short- and long- period effects, as well as secular terms [36,31]. However, period terms by means of two different canonical transfor- due to the smallness of the J2 coefficient of the Earth, the mations that can be carried out in either order. We depart full system can be replaced by a separable approximation, from the tradition and proceed by standard normalization to which is customarily obtained by removing the periodic ef- show that the Hamiltonian simplification part is dispensable. fects by means of perturbation methods [53]. Decoupling first the motion of the orbital plane from the in- When written in the action-angle variables of the Kepler plane motion reveals as a feasible strategy to reach higher problem, also called Delaunay variables, the main problem orders of the perturbation solution, which, besides, permits Hamiltonian immediately shows that the right ascension of an efficient evaluation of the long series that comprise the the ascending node is a cyclic variable. In consequence, its analytical solution. conjugate momentum, the projection of the angular momen- tum vector along the Earth’s rotation axis, is an integral of Keywords main problem · Hamiltonian mechanics · the main problem, which, therefore, is just of two degrees normalization · canonical perturbation theory · floating- of freedom. Then, following Brouwer [6], the main prob- point arithmetic lem Hamiltonian is normalized in two steps. First, the short period effects are removed by means of a canonical transfor- mation that, after truncation to some order of the perturba- 1 Introduction tion approach, turns the conjugate momentum to the mean anomaly (the Delaunay action) into a formal integral. Next, The dynamics of close Earth satellites under gravitational a new canonical transformation converts the total angular effects are mostly driven by perturbations of the Keplerian momentum into another formal integral. The main problem arXiv:2007.00058v1 [math.DS] 30 Jun 2020 motion induced by the Earth oblateness. For this reason, the Hamiltonian is in this way completely reduced to a function approximation obtained when truncating the Legendre poly- of only the momenta in the new variables, whose Hamilton nomials expansion of the Geopotential to the only contribu- equations are trivially integrable. M. Lara In spite of the normalization procedure is standard from Scientific Computing Group–GRUCACI, University of La Rioja, the point of view of perturbation theory, it happens that not Madre de Dios 53, 26006 Logrono,˜ La Rioja, Spain Tel.: +34-941-299440 all the action-angle variables of the Kepler problem appear Fax: +34-941-299460 explicitly in the Geopotential, as is usually advisable in the E-mail: [email protected] construction of perturbation solutions [32,43,44,49]. In par- M. Lara ticular, the mean anomaly remains implicit in the gravita- Space Dynamics Group, Polytechnic University of Madrid–UPM, tional potential through its dependence on the polar angle. Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Unavoidably, Kepler’s equation must be solved to show the 2 Martin Lara mean anomaly explicit, a fact that entails expanding the el- could seem to be unavoidably attached. On the other hand, liptic motion in powers of the eccentricity [23,7]. This stan- when extended to higher orders, it removes more terms than dard procedure is quite successful when dealing with orbits those needed in a partial normalization. Last, the fact that the of low eccentricities, like is typical in a variety of astronomy technique was originally devised in the canonical set of po- problems [14,57]. However, it involves handling very long lar variables, to which the argument of the perigee does not Fourier series in the case of orbits with moderate eccentric- pertain, neither helps in grasping the essence of the trans- ities [19,33], and hence the application of this method to formation. Reimplementation of the procedure in the usual different problems of interest in astrodynamics is de facto set of action-angle variables makes the process of convert- prevented. ing the argument of the perigee into a cyclic variable much On the contrary, the integrals appearing in the solution more evident [45], but it still bears the same differences with of the artificial satellite problem can be solved in closed respect to a classical normalization procedure. form of the eccentricity [6,36]. It only requires the help of We disregard the claimed benefits of Hamiltonian sim- the standard relation between the differentials of the true plification procedures and compute the solution to the main and mean anomalies that is derived from the preservation problem of the artificial satellite theory by standard normal- of the total angular momentum of the Keplerian motion. ization. It is called reverse normalization because we ex- Regrettably, the closed form approach soon finds difficul- change the order in which the formal integrals are tradition- ties in achieving higher orders of the short-period elimi- ally introduced when solving the artificial satellite problem. nation, which stem from the impossibility of obtaining the More precisely, the total angular momentum is transformed antiderivative of the equation of the center (the difference into a formal integral in the first place, in this way decou- between the true and mean anomalies) in closed form of pling the motion of the orbital plane from the satellite’s mo- the eccentricity in the realm of trigonometric functions [28, tion on that plane.1 Then, a second canonical transformation 54]. Nonetheless, the difficulties are overcome by the arti- converts the mean anomaly into a cyclic variable, in this way fact of grouping the equation of the center with other func- achieving the total reduction of the main problem Hamilto- tions appearing in the procedure previously to approaching nian. their integration [37,1,17]. Alternatively, the application of The procedure for making the argument of the perigee a preliminary Hamiltonian simplification, the elimination of cyclic in the first place follows an analogous strategy to the the parallax [16,46], eases the consequent removal of short- one devised in the classical elimination of the perigee trans- period effects to some extent [9,25]. formation [2]. However, in our approach it is applied directly From the point of view of the perturbation approach, re- to the original main problem Hamiltonian, and differs from moving the short-period effects in the first place seems the the original technique, as well as from an analogous proce- more natural in view of the degeneracy of the Kepler prob- dure carried out in [56], in which the parallactic terms (in- lem. Indeed, the Kepler Hamiltonian in action-angle vari- verse powers of the radius with exponents higher than 2) are ables, on which the perturbation approach hinges on, only not removed from the new, partially normalized Hamilto- depends on the Delaunay action [22,41]. However, the order nian. In spite of that, we did not find trouble in dealing with in which the formal integrals are sequentially introduced in the equation of the center in closed form in the subsequent the perturbation solution is not relevant in a total normaliza- Delaunay normalization [17], a convenience that might had tion procedure, whose result is unique [3]. In fact, it hap- been anticipated from the discussions in [50]. pens that relegating the transformation of the Delaunay ac- The Hamiltonian reduction has been approached in De- tion into a formal integral to the last step of the perturbation launay variables. Unfortunately, these variables share the de- approach provides clear simplifications in dealing with the ficiencies of their partner Keplerian elements, which are sin- equation of the center [2]. In this way the task of extending gular for circular orbits and for equatorial orbits. Because the solution of the main problem to higher orders is notably of that, the secular terms are reformulated in a set of non- simplified. singular variables that replaces the mean anomaly, the argu- Converting the total angular momentum into a formal in- ment of the perigee, and the total angular momentum, by tegral of the main problem requires making cyclic the argu- the mean argument of the latitude and the projections of ment of the perigee, up to some truncation order of the per- the eccentricity vector in the nodal frame, which are some- turbation approach, in the transformed Hamiltonian. How- times denoted semi-equinoctial variables [34]. For the pe- ever, as it appeared in the literature, the transformation called riodic corrections, we find convenience in using polar vari- by their authors the elimination of the perigee [2] is not the ables, which in the particular case of the main problem are typical normalization procedure, although it operates anal- 1 ogous results. Indeed, on the one hand, the elimination of The advantages of decoupling the motion of the instantaneous or- bital plane from the in-plane motion are well known, and are com- the perigee is only applied to a Hamiltonian obtained af- monly pursued in the search for efficient numerical integration meth- ter the elimination of the parallax, to which simplification it ods, q.v.
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