[QED]

Below is a link to a dissertation which describes a numerical integration technique used for propagating the orbits of satellites.

Below that is a description of a numerical package along with embedded links for initial conditions for describing orbits and trajectories. The model is displayed for view.

This piece serves as illustration that Newtonian dynamics are not only solvable for explaining and modeling orbits, but also used to position artificial satellites around objects in our solar system.

Let me know if you have any questions as these numerical techniques would prove useful for FET pursuits.

https://www.colorado.edu/ccar/sites/default/files/attached-files/numerical_algorithms_for_preci.pdf

http://www.maia.ub.edu/dsg/wsem/documents/validated01.pdf

[sandokhan]

The manner in which you formulate your own sentences about these matters betrays your lack of expertise. It may be convincing to the uninitiated, but to a trained physicist it becomes immediately obvious that you are not really understanding what you write.

You have just presented two papers which describe UNDERGRADUATE LEVEL orbital mechanics: the application of the Runge-Kutta method and ephemeris calculations.

Is this your current level of understanding of the orbital equations of motion of a satellite?

This piece serves as illustration that Newtonian dynamics are not only solvable for explaining and modeling orbits, but also used to position artificial satellites around objects in our solar system.

You really need to update your knowledge on the subject.

Here is the equation of motion describing the librational motion of an arbitrarily shaped satellite in a planar, elliptical orbit:

(1 + εμcosθ)ψ" - 2εμsinθ(ψ' + 1) + 3K_isinψcosψ = 0

ψ' = δψ/δθ

K_i = (I_xx - I_zz)/I_yy

εμ = eccentricity of the orbit

For small ε, and using 1/(1 + εμcosθ) = 1 - εμcosθ + O(ε²), we obtain


ψ" + 3K_isinψcosψ = ε[2μsinθ(ψ' + 1) + 3μK_isinψcosψcosθ] + O(ε²)

This is a fully nonlinear ordinary differential equation (initial conditions). For weakly nonlinear ODE, we can use methods such as multiple scaling and averaging.

For a fully nonlinear ODE, we need very advanced perturbation techniques: the Melnikov method.

Even for a simpler version of this fully nonlinear differential equation, the orbit of a tethered satellite system, we will get chaotical motions for realistic/real flight parameters:

http://www.uni-magdeburg.de/ifme/zeitschrift_tm/1996_Heft4/Peng.pdf

In theory, time delay feedback control methods are used to try to minimize the chaotical motion; however, in real time flight, parameters values can and will exceed the data used in the theorized version.

The undergraduate papers you presented amount to nothing at all: they ASSUME that the orbital equations motion can be integrated without having to take into consideration the THEORETICAL aspects of a system of nonlinear ordinary differential equations.

[QED]

The manner in which you formulate your own sentences about these matters betrays your lack of expertise. It may be convincing to the uninitiated, but to a trained physicist it becomes immediately obvious that you are not really understanding what you write.

You have just presented two papers which describe UNDERGRADUATE LEVEL orbital mechanics: the application of the Runge-Kutta method and ephemeris calculations.

Is this your current level of understanding of the orbital equations of motion of a satellite?

This piece serves as illustration that Newtonian dynamics are not only solvable for explaining and modeling orbits, but also used to position artificial satellites around objects in our solar system.

You really need to update your knowledge on the subject.

Here is the equation of motion describing the librational motion of an arbitrarily shaped satellite in a planar, elliptical orbit:

(1 + εμcosθ)ψ" - 2εμsinθ(ψ' + 1) + 3K_isinψcosψ = 0

ψ' = δψ/δθ

K_i = (I_xx - I_zz)/I_yy

εμ = eccentricity of the orbit

For small ε, and using 1/(1 + εμcosθ) = 1 - εμcosθ + O(ε²), we obtain


ψ" + 3K_isinψcosψ = ε[2μsinθ(ψ' + 1) + 3μK_isinψcosψcosθ] + O(ε²)

This is a fully nonlinear ordinary differential equation (initial conditions). For weakly nonlinear ODE, we can use methods such as multiple scaling and averaging.

For a fully nonlinear ODE, we need very advanced perturbation techniques: the Melnikov method.

Even for a simpler version of this fully nonlinear differential equation, the orbit of a tethered satellite system, we will get chaotical motions for realistic/real flight parameters:

http://www.uni-magdeburg.de/ifme/zeitschrift_tm/1996_Heft4/Peng.pdf

In theory, time delay feedback control methods are used to try to minimize the chaotical motion; however, in real time flight, parameters values can and will exceed the data used in the theorized version.

The undergraduate papers you presented amount to nothing at all: they ASSUME that the orbital equations motion can be integrated without having to take into consideration the THEORETICAL aspects of a system of nonlinear ordinary differential equations.

You are quite mistaken, my friend. Integratability has not been assumed. I recommend you read these a bit more carefully. One does not obtain a tethered satellite as a result of an approximation. Rather, the ODEs are APPLIED to a certain case, and that is the reference you have cited (but not understood).

Also, the last time you blasted equations at me you didn’t even know what they were - recall our maxwell equation conversation? Would you like to take one more look at the equations you posted here before I reply?

I find it hilarious that you refer to a thesis as an undergraduate paper.

To answer your question, I deliberately found more simple treatments to appease Tom, who gets confused when the analysis becomes advanced.