Kepler equation solutions and orbital perturbation methods determine time dependence in elliptical orbits
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Multiple peer-reviewed sources and reference encyclopedias establish that solving Kepler's equation determines the time-of-flight and temporal position within elliptical orbits, while perturbation methods account for deviations and orbital variations.
In this study, the first kind Bessel function was used to solve Kepler equation for an elliptical orbiting satellite. It is a classical method that gives a direct solution for calculation of the eccentric anomaly. It was solved for one period from (M=0-360)° with an eccentricity of (e=0-1) and the number of terms from (N=1-10). Also, the error in the representation of the first kind Bessel function was calculated. The results indicated that for eccentricity of (0.1-0.4) and (N = 1-10), the values of eccentric anomaly gave a good result as compared with the exact solution. Besides, the obtained eccentric anomaly values were unaffected by increasing the number of terms (N = 6-10) for eccentricities (0.8 and 0.9). The Bessel function's solution appeared to be close to the exact solution for eccentricity of 1 and more than 10 number of terms. Finally, the representation of the first kind Bessel function J1(x) was closer to the exact representation only for eccentricity 0.5 and (N=1-10).
Ibrahim and Saleh Iraqi Journal of Science, 2024, Vol. 65, No. 2, pp: 1129- 1137
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1. Introduction
To solve Kepler’s equation, different analytical and graphical solutions were proposed. From
Kepler and Newton until the middle of the 20 th century, almost every mathematician
concentrated on this equation. After the beginning of the modern spaceflight era, new
algorithms were being suggested and published. Many significant developments in mathematics
were used to solve this equation, such as Fourier series, Bessel functions, and basic concepts of
complex function theory, using many numerical analysis approaches [1, 2].
The Bessel function was found to be a direct solution to Kepler ’s equation and obtained the
coefficient of the first kind, J 1(x) in sine wave form. For all values of eccentricity and the
periodic series of the mean anomaly, the coefficient was completely convergent [1]. Kepler’s
equation gives the relationship between the polar coordinates of a celestial object and the time
required for this object to move around another object. It is a fundamental equation in celestial
mechanics but cannot be directly reversed in terms of simple functions in order to determine
where the object will be at a particular time. The solution of Kepler ’s equation by the Bessel
function of the first kind depends on three parameters, which are: eccentricity, mean anomaly,
and number of terms [3, 4, 5]. For many years, many researchers have presented their solutions.
James provided a global solution by defining four M-e plane sub-domains [6]. Fouad and Anas
investigated how the sun's position affects the length of astronomical twilight [7]. Daniele et al.
used a predictor-corrector approach for orbit propagation. The value of the eccentric anomaly
was estimated under a constant time interval constraint by linear or quadratic approximations
and then corrected using a single Newton-Raphson or Halley iteration [8]. By using the Special
Trans Functions Theory, Slavica et al. discovered an analytical solution for several families of
Kepler’s transcendental equations (STFT) [9]. By using the Maclaurin expansion, Aisha and
Asrar were able to solve Kepler’s equation without the requirement to decompose the associated
nonlinearity as they would have to do with the differential transformation method and Adomian
method [10]. Mohammed and Abdul Rahman evaluated the orbital maneuvers for changing
from Low Earth Orbit to Geostationary Earth Orbit using the Newton -Raphson method [1 1].
Ibrahim and Saleh Iraqi Journal of Science, 2024, Vol. 65, No. 2, pp: 1129- 1137
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References
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[4] W.Ulrich, Astronautics:The Physics of Space Flight.Third Edition. Springer, 2019, p.105.
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The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex, involves the use of a transcendental equation (referred to as the generalized Kepler equation) that cannot be solved in closed form, and lacks smoothness near the collision point. Here, we provide some insight into the significance of this equation, along with a very simple derivation and confirmation of the inverse LS mapping. Then we use numerical methods to inves
Then we use numerical methods to investigate three applications: 1) solutions of the Kepler function, 2) calculation of orbits including time-of-flight data based on the Delaunay Hamiltonian, and 3) numerical evidence for the Birkhoff conjecture for the circular restricted 3-body problem. status released display-pdf yes is-olf no is-manuscript no is-preprint no is-journal-matter no is-scanned no is-retracted no Received 2018 Apr 30; Accepted 2018 Aug 28; Collection date 2018. Introduction The Ligon-Schaaf mapping ( LS mapping) was introduced in 1976 [ 1 ] based in part on a Diplom thesis (similar to master’s thesis) which has recently been translated into English [ 2 – 4 ].
Since then, the paper has been cited numerous times [ 5 – 27 ]. In particular, a few papers have provided very significant insight into the properties of the mapping and the symplectic manifolds involved in the regularization that it achieves [ 8 , 17 ]. The LS mapping is a modified stereographic projection, converting phase space in Cartesian coordinates to phase space over a 3-sphere, and maps the collision orbits to the north pole of the sphere. This makes it possible to regularize the Kepler problem by adding the poles to the original space.
However, the solutions involved two different angles, and the second one required solving a transcendental equation that cannot be solved in closed form. Then we discovered that we could modify the Moser mapping in such a way that this transcendental equation disappeared from the solutions, and both angular momentum and the Runge-Lenz vector generated simple rotations on the sphere S 3 . In other words, we had defined the LS mapping and showed that it converts the symmetry of the Kepler problem to the canonical action of SO(4).
Based on this foundation, we create a numeric solution for all mappings and use it for the following applications: 1) investigation of the “Kepler function” generated by the “generalized Kepler equation”: φ = x ∙ sin ( φ ) − y ∙ cos ( φ ), 2) calculation of time-of-flight for Kepler orbits using the Delaunay Hamiltonian and the inverse LS mapping, and 3) numerical evidence for the Birkhoff conjecture for the circular restricted 3-body problem. It has often been observed that the Kepler laws can be deduced by geometrical means, without solving differential equations, but finding the time of flight requires solving the Kepler equation.
In our second application, this time dependency is found in closed form on T * S 3 , and the Kepler equation is solved “implicitly” as part of the inverse LS mapping. The third application provides evidence for the Birkhoff conjecture within the framework of holomorphic curve theory, but all calculations in this paper are “elementary calculations” based solely on traditional algebra and trigonometry. This investigation begins with an alternative, analytical proof for the Birkhoff conjecture based on the rotating Kepler Hamiltonian[ 13 ] and then covers numerical calculations based on the circular restricted 3-body Hamiltonian.
Forward LS mapping F = F 2 ∘ F 1 The forward LS mapping is defined for H < 0, corresponding to the elliptical orbits of the Kepler Hamiltonian, and q ≠ 0, meaning that collision orbits are excluded. Definition 1 .
The denominator of ( 19 ), 1 − ξ 0 c o s ( φ ) − η 0 η s i n ( φ ) ≠ 0 , and the denominator of ( 21 ), 1 − r 0 ≠ 0, because, following the identity ( 37 ), − 2 H q = 1 − r 0 = 1 − ξ 0 c o s ( φ ) − η 0 η s i n ( φ ) , and the mapping is only defined for H < 0, corresponding to the elliptical orbits of the Kepler Hamiltonian, and q ≠ 0, meaning that collision orbits are excluded. Proposition 2. G = G 1 ∘ G 2 . G 1 = F 1 − 1 G 2 = F 2 − 1 G = F −1 . Proof.
Applications Investigate Kepler function The inverse LS mapping begins by solving a transcendental equation (Eq ( 22 )) φ = ξ 0 s i n ( φ ) − η 0 η c o s ( φ ) We refer to this as the “generalized Kepler equation”, and define what we call the “Kepler function”: (80) K F : [ − 1,1 ] 2 → R : ( x , y ) ↦ s o l u t i o n o f φ = x s i n ( φ ) − y c o s ( φ ) Since we have no solution for the Kepler function KF in closed form, we explore it using a numeric solution. Here, we numerically solve the generalized Kepler equation and calculate φ for selected values of x and y . Fig 2 provides an overview of the Kepler function. Fig 2 Kepler function.
The numerical solution, calculated on a grid from -1 to 1 with spacing .01, demonstrates the following values: m i n ( φ ) = ( − 1.2587 ( 1,1 ) ) m a x ( φ ) = ( 1.2587 ( 1 , − 1 ) ) m i n ( ∇ ( φ ) ) = ( − 4.9081 ( 1,0.01 ) − 100 ( 0.99,0 ) ) m a x ( ∇ ( φ ) ) = ( 4.9081 ( 1 , − 0.01 ) − 0.18667 ( 1 , − 1 ) ) Calculate orbits and time of flight This application can be
Here, we provide software that accepts values for q and p and calculates the Hamiltonian, the period of the orbit, the angular momentum, the Runge-Lenz vector, the values of ξ and η created by the LS mapping, the true, eccentric, and mean anomaly, the Kepler elements (semi-major axis, eccentricity, longitude of ascending node Ω , inclination, argument of pericenter ω , and time of passage), and the Delaunay variables ( l , g , h , L , G , H ). Then we can convert the coordinates using the LS mapping, calculate the orbit, including time-of-flight, by using the Delaunay Hamiltonian, and convert back to q and p using the inverse LS mapping.
excentricity of orbit , longitude of perihelion, and mass, of an assumed exterior planet,—deduced entirely from unaccounted-for perturbations of Uranus. The
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