A note on the use of generalized sundman anomalies in the numerical integration of the elliptical orbital motion

The orbital motion around a central body is an interesting problem that involves the theory of artificial satellites and the planetary theories in the solar system. Nevertheless some difficult situations appear while studying this apparently simple problem, depending on each particular case. The real problem consists of searching the perturbed solution from a basic two-body motion problem. In addition, the perturbed problem must be solved using a numerical method and its efficiency depends on the selected coordinate system and the corresponding time. In fact, local and global errors are not necessarily homogeneously distributed over the orbit. In other words, there is a strong relationship between the spatial distribution of the selected points and the temporal independent variable. This is particularly dramatic in specially difficult cases. This issue leads us to consider different anomalies as temporal variables, searching for both precision and efficiency. Therefore, we are interested in the study of techniques to integrate the orbital motion equations using different anomalies as temporal variables which are functions of one or more parameters. The final aim of this paper is the minimization of the integration errors using an appropriate choice of the parameter depending on the eccentricity value in the family of the generalized Sundman anomalies.This research has been partially supported by Grant P1-06I455.01/1 from Bancaja.Lopez Orti, JA.; Marco Castillo, FJ.; Martínez Uso, MJ. (2014). A note on the use of generalized sundman anomalies in the numerical integration of the elliptical orbital motion. Abstract and Applied Analysis. 2014:1-8. https://doi.org/10.1155/2014/691926S182014Velez, C. E., & Hilinski, S. (1978). Time transformations and Cowell’s method. Celestial Mechanics, 17(1), 83-99. doi:10.1007/bf01261054Nacozy, P. (1977). The intermediate anomaly. Celestial Mechanics, 16(3), 309-313. doi:10.1007/bf01232657Ferr�ndiz, J. M., Ferrer, S., & Sein-Echaluce, M. L. (1987). Generalized elliptic anomalies. Celestial Mechanics, 40(3-4), 315-328. doi:10.1007/bf01235849Brumberg, E. V. (1992). Length of arc as independent argument for highly eccentric orbits. Celestial Mechanics and Dynamical Astronomy, 53(4), 323-328. doi:10.1007/bf00051814López Ortí, J. A., Gómez, V. A., & Rochera, M. B. (2012). A note on the use of the generalized Sundman transformations as temporal variables in celestial mechanics. International Journal of Computer Mathematics, 89(3), 433-442. doi:10.1080/00207160.2011.611502Janin, G. (1974). Accurate computation of highly eccentric satellite orbits. Celestial Mechanics, 10(4), 451-467. doi:10.1007/bf01229121Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics. doi:10.1007/978-1-4757-2063-

A Study about the integration of the elliptical orbital motion based on a special one-parametric family of anomalies

This paper aimed to address the study of a new family of anomalies, called natural anomalies, defined as a one-parameter convex linear combination of the true and secondary anomalies, measured from the primary and the secondary focus of the ellipse, and its use in the study of analytical and numerical solutions of perturbed two-body problem. We take two approaches: first, the study of the analytical development of the basic quantities of the two-body problem to be used in the analytical theories of the planetary motion and second, the study of the minimization of the errors in the numerical integration by an appropriate choice of parameters in our family for each value of the eccentricity. The use of an appropriate value of the parameter can improve the length of the developments in the analytical theories and reduce the errors in the case of the numerical integration.This research has been partially supported by Grant P1.1B2012-47 from University Jaume I of Castellon.Lopez Orti, JA.; Marco Castillo, FJ.; Martínez Uso, MJ. (2014). A Study about the integration of the elliptical orbital motion based on a special one-parametric family of anomalies. Abstract and Applied Analysis. 2014:1-11. https://doi.org/10.1155/2014/162060S1112014Simon, J.-L., Francou, G., Fienga, A., & Manche, H. (2013). New analytical planetary theories VSOP2013 and TOP2013. Astronomy & Astrophysics, 557, A49. doi:10.1051/0004-6361/201321843Brumberg, V. A. (1995). Analytical Techniques of Celestial Mechanics. doi:10.1007/978-3-642-79454-4Nacozy, P. E. (1969). Hansen’s Method of Partial Anomalies: an Application. The Astronomical Journal, 74, 544. doi:10.1086/110833Brumberg, E., & Fukushima, T. (1994). Expansions of elliptic motion based on elliptic function theory. Celestial Mechanics & Dynamical Astronomy, 60(1), 69-89. doi:10.1007/bf00693093López, J. A., & Barreda, M. (2007). A formulation to obtain semi-analytical planetary theories using true anomalies as temporal variables. Journal of Computational and Applied Mathematics, 204(1), 77-83. doi:10.1016/j.cam.2006.04.029López Ortí, J. A., Martínez Usó, M. J., & Marco Castillo, F. J. (2008). Semi-analytical integration algorithms based on the use of several kinds of anomalies as temporal variable. Planetary and Space Science, 56(14), 1862-1868. doi:10.1016/j.pss.2008.02.035Sundman, K. F. (1913). Mémoire sur le problème des trois corps. Acta Mathematica, 36(0), 105-179. doi:10.1007/bf02422379Nacozy, P. (1977). The intermediate anomaly. Celestial Mechanics, 16(3), 309-313. doi:10.1007/bf01232657Janin, G. (1974). Accurate computation of highly eccentric satellite orbits. Celestial Mechanics, 10(4), 451-467. doi:10.1007/bf01229121Velez, C. E., & Hilinski, S. (1978). Time transformations and Cowell’s method. Celestial Mechanics, 17(1), 83-99. doi:10.1007/bf01261054Ferr�ndiz, J. M., Ferrer, S., & Sein-Echaluce, M. L. (1987). Generalized elliptic anomalies. Celestial Mechanics, 40(3-4), 315-328. doi:10.1007/bf01235849Brumberg, E. V. (1992). Length of arc as independent argument for highly eccentric orbits. Celestial Mechanics and Dynamical Astronomy, 53(4), 323-328. doi:10.1007/bf00051814López Ortí, J. A., Gómez, V. A., & Rochera, M. B. (2012). A note on the use of the generalized Sundman transformations as temporal variables in celestial mechanics. International Journal of Computer Mathematics, 89(3), 433-442. doi:10.1080/00207160.2011.611502Deprit, A. (1979). Note on Lagrange’s inversion formula. Celestial Mechanics, 20(4), 325-327. doi:10.1007/bf01230401Chapront, J., Bretagnon, P., & Mehl, M. (1975). Un formulaire pour le calcul des perturbations d’ordres �lev�s dans les probl�mes plan�taires. Celestial Mechanics, 11(3), 379-399. doi:10.1007/bf0122881

An improved algorithm to develop semi-analytical planetary theories using Sundman generalized variables

One of the main problems in celestial mechanics is the construction of analytical theories of planetary motion. The most common solution of this problem is arranged by means of Poisson series developments. These developments depend on the selection of the anomaly to be used as temporal variable. In this paper we develop an improved algorithm in order to use arbitrary anomalies included in the family of the generalized Sundman anomalies as temporal variables.This research has been partially supported by Grant P1-1 B2012-47 from Universidad Jaume I of Castellón

Geometrical definition of a continuous family of time transformations generalizing and including the classic anomalies of the elliptic two-body problem

[EN] This paper is aimed to address the study of techniques focused on the use of a family of anomalies based on a family of geometric transformations that includes the true anomaly f, the eccentric anomaly g and the secondary anomaly f' defined as the polar angle with respect to the secondary focus of the ellipse. This family is constructed using a natural generalization of the eccentric anomaly. The use of this family allows closed equations for the classical quantities of the two body problem that extends the classic, which are referred to eccentric, true and secondary anomalies. In this paper we obtain the exact analytical development of the basic quantities of the two body problem in order to be used in the analytical theories of the planetary motion. In addition, this paper includes the study of the minimization of the errors in the numerical integration by an appropriate choice of parameters in our selected family of anomalies for each value of the eccentricity. (C) 2016 Elsevier B.V. All rights reserved.This research has been partially supported by Grant P1.1B2012-47 from University Jaume I of Castellón and Grant AICO/2015/037 from Generalitat Valenciana.López Ortí, J.; Marco Castillo, FJ.; Martínez Uso, MJ. (2017). Geometrical definition of a continuous family of time transformations generalizing and including the classic anomalies of the elliptic two-body problem. Journal of Computational and Applied Mathematics. 309:482-492. https://doi.org/10.1016/j.cam.2016.02.041S48249230

A new bi-parametric family of temporal transformations to improve the integration algorithms in the study of the orbital motion

One of the fundamental problems in celestial mechanics is the study of the orbital motion of the bodies in the solar system. This study can be performed through analytical and numerical methods. Analytical methods are based on the well-known two-body problem; it is an integrable problem and its solution can be related to six constants called orbital elements. To obtain the solution of the perturbed problem, we can replace the constants of the two-body problem with the osculating elements given by the Lagrange planetary equations. Numerical methods are based on the direct integration of the motion equations. To test these methods we use the model of the two-body problem with high eccentricity. In this paper we define a new family of anomalies depending on two param- eters that includes the most common anomalies. This family allows to obtain more compact developments to be used in analytical series. This family can be also used to improve the efficiency of the numerical methods because defines a more suitable point distribution with the dynamics of the two-body problem.This research has been partially supported by Grant P1-1B2012-47 from Universidad Jaume I of Castell ́on and Grant AICO/2015/037 of Generalitat Valenciana

Study of errors in the integration of the two-body problem using generalized Sundman's anomalies

[EN] As is well known, the numerical integration of the two body problem with constant step presents problems depending on the type of coordinates chosen. It is usual that errors in Runge-Lenz's vector cause an artificial and secular precession of the periaster although the form remains symplectic, theoretically, even when using symplectic methods. Provided that it is impossible to preserve the exact form and all the constants of the problem using a numerical method, a possible option is to make a change in the variable of integration, enabling the errors in the position of the periaster and in the speed in the apoaster to be minimized for any eccentricity value between 0 and 1. The present work considers this casuistry. We provide the errors in norm infinite, of different quantities such as the Energy, the module of the Angular Moment vector and the components of Runge-Lenz's vector, for a large enough number of orbital revolutions.Lopez Orti, JA.; Marco Castillo, FJ.; Martínez Uso, MJ. (2014). Study of errors in the integration of the two-body problem using generalized Sundman's anomalies. SEMA SIMAI Springer Series. 4:105-112. doi:10.1007/978-3-319-06953-1_11S1051124Brower, D., Clemence, G.M.: Celestial Mechanics. Academic, New York (1965)Brumberg, E.V.: Length of arc as independent argument for highly eccentric orbits. Celest. Mech. 53, 323–328 (1992)Fehlberg, E., Marsall, G.C.: Classical fifth, sixth, seventh and eighth Runge–Kutta formulas with stepsize control. Technical report, NASA, R-287 (1968)Ferrándiz, J.M., Ferrer, S., Sein-Echaluce, M.L.: Generalized elliptic anomalies. Celest. Mech. 40, 315–328 (1987)Gragg, W.B.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. SIAM J. Numer. Anal. 2, 384–403 (1965)Janin, G.: Accurate computation of highly eccentric satellite orbits. Celest. Mech. 10, 451–467 (1974)Janin, G., Bond, V.R.: The elliptic anomaly. Technical memorandum, NASA, n. 58228 (1980)Levallois, J.J., Kovalevsky, J.: Géodésie Générale, vol. 4. Eyrolles, Paris (1971)López, J.A., Agost, V., Barreda, M.: A note on the use of the generalized Sundman transformations as temporal variables in celestial mechanics. Int. J. Comput. Math. 89, 433–442 (2012)López, J.A., Marco, F.J., Martínez, M.J.: A study about the integration of the elliptical orbital motion based on a special one-parametric family of anomalies. Abstr. Appl. Anal. 2014, ID 162060, 1–11 (2014)Nacozy, P.: The intermediate anomaly. Celest. Mech. 16, 309–313 (1977)Sundman, K.: Memoire sur le probleme des trois corps. Acta Math. 36, 105–179 (1912)Tisserand, F.F.: Traité de Mecanique Celeste. Gauthier-Villars, Paris (1896)Velez, C.E., Hilinski, S.: Time transformation and Cowell’s method. Celest. Mech. 17, 83–99 (1978

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131?150,2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez?s method for near-circular motion under the J2 perturbation is transformed into linear.Moreover, themethod reveals to be competitive with two very popular elementmethods derived from theKustaanheimo-Stiefel and Sperling-Burdet regularizations

Time elements for enhanced performance of the Dromo orbit propagator

We propose two time elements for the orbit propagator named Dromo. One is linear and the other constant with respect to the independent variable, which coincides with the osculating true anomaly in the Keplerian motion. They are defined from a generalized Kepler’s equation written for negative values of the total energy and, unlike the few existing time elements of this kind, are free of singularities. To our knowledge it is the first time that a constant time element is associated with a second-order Sundman time transformation. Numerical tests to assess the performance of the Dromo method equipped with a time element show the remarkable improvement in accuracy for the perturbed bounded motion around the Earth compared to the case in which the physical time is a state variable. Moreover, the method is competitive with and even better than other efficient sets of elements. Finally, we also derive a time element for a null and positive total energy

An improved C++ Poisson series processor with its applications

One of major problems in celestial mechanics is the management of the long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system. In this work we will develop a software package suitable for managing these objects. This package includes the ordinary arithmetic operations with Poisson series—such as sum or product—as well as the most commonly used functions sin , cos , or exp , among others. Derivation or integration procedures with respect to time of these objects have been implemented and inversion or serialization procedures are also attainable to obtain such series. All the programming has been accomplished overloading the arithmetic and functional operators natural to C++ with the intention of allowing the programmer to work in a more friendly way and treating the series as if they were mere numbers. In this work we extend the processor in order to obtain the solution to perturbed problems which solution is in the form of a Poisson series. These algorithms have been written in C++

Minkowski Spacetime and QED from Ontology of Time

Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the necessary number of postulates. We uncover the intrinsic connection of these areas of physics and describe them using a common symplectic Hamiltonian formalism. Our approach is based on a proper distinction between variables and constants, i.e. on a basic but rigorous ontology of time. We link these concept with the obvious conditions for the possibility of measurements. The derived consequences put the measurement problem of quantum mechanics and the Copenhagen interpretation of the quantum mechanical wavefunction into perspective. According to our (onto-) logic we find that spacetime can not be fundamental. We argue that a geometric interpretation of symplectic dynamics emerges from the isomorphism between the corresponding Lie algebra and the representation of a Clifford algebra. Within this conceptional framework we derive the dimensionality of spacetime, the form of Lorentz transformations and of the Lorentz force and fundamental laws of physics as the Planck-Einstein relation, the Maxwell equations and finally the Dirac equation.Comment: 36 pages, 3 figures, several typos corrected, references with title