A Study about the Integration of the Elliptical Orbital Motion Based on a Special One-Parametric Family of Anomalies

Ortí, J. A. López; Marco, Francisco Javier García; Martínez, María José Martínez

doi:10.1155/2014/162060

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…From ( 24) or (25) we immediately get: On the other hand, as is well known, the coordinates of the secondary with respect to the primary (ξ, η) are given by: ξ = a(e − cos g), η = a 1 − e 2 sin g, (19) and the distances between the primary and secondary foci by (4). Many classical authors have studied the eccentric anomaly [1,2,5], among others.…”

Section: Other Symmetric Variables Not Belonging To the Biparametric ...mentioning

confidence: 93%

“…It is evident that the central anomaly Φ is not in the biparametric family of anomalies. This is not new, since the intermediate radials of Cid [17], the natural family of anomalies [19], and the geometric family of anomalies [20] are not in general in the biparametric family, but they are in the symmetrical case since they coincide, respectively, with the semifocal anomaly and with the eccentric anomaly. Therefore, we have a symmetrical anomaly that does not belong to the biparametric family.…”

Section: Other Symmetric Variables Not Belonging To the Biparametric ...mentioning

confidence: 99%

“…Notice that the anomaly Ψ α,β is symmetric to the axis of the ellipse when α − β = 1. There are some interesting anomalies not included in this biparametric family [17][18][19][20].…”

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confidence: 99%

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A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families

López Ortí,

Marco Castillo,

Martínez Usó

2024

Algorithms

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In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem is treated more adequately through temporal reparametrizations related to the mean anomaly through the partition function. The biparametric family of anomalies, with an appropriate partition function, allows a systematic study of these transformations. In the present work, we consider the elliptical orbit as a meridian section of the ellipsoid of revolution, and the partition function depends on two variables raised to specific parameters. One of the variables is the mean radius of the ellipsoid at the secondary, and the other is the distance to the primary. One parameter regulates the concentration of points in the apoapsis region, and the other produces a symmetrical displacement between the polar and equatorial regions. The three most used geodesy latitude variables are also studied, resulting in one not belonging to the biparametric family. However, it is in the one introduced now, which implies an extension of the biparametric method. The results obtained using the method presented here now allow a causal interpretation of the operation of numerous reparametrizations used in the study of orbital motion.

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Section: Other Symmetric Variables Not Belonging To the Biparametric ...mentioning

confidence: 93%

Section: Other Symmetric Variables Not Belonging To the Biparametric ...mentioning

confidence: 99%

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A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families

López Ortí,

Marco Castillo,

Martínez Usó

2024

Algorithms

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

Ortí

Marco

Martínez

2017

Journal of Computational and Applied Mathematics

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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

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A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two Body Problem with Extensions to Other Families

López Ortí,

Marco Castillo,

Martínez Usó

2024

Preprint

View full text Add to dashboard Cite

In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem is treated more adequately through temporal reparametrizations related to the mean anomaly through the partition function. The biparametric family of anomalies, with an appropriate partition function, allows a systematic study of these transformations. In the present work, we consider the elliptical orbit as a meridian section of the ellipsoid of revolution, and the partition function depends on two variables raised to specific parameters. One of the variables is the mean radius of the ellipsoid at the secondary, and the other is the distance to the primary. One parameter regulates the concentration of points in the apoapsis region, and the other produces a symmetrical displacement between the polar and equatorial regions. The three most used geodesy latitude variables are also studied, resulting in one not belonging to the biparametric family. However, it is in the one introduced now, which implies an extension of the biparametric method. The results obtained by the method presented here now allow a causal interpretation of the operation of numerous reparametrizations used in the study of orbital motion.

show abstract

A Study about the Integration of the Elliptical Orbital Motion Based on a Special One-Parametric Family of Anomalies

Cited by 4 publications

References 15 publications

A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families

A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families

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

A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two Body Problem with Extensions to Other Families

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