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Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star:

$$u=\frac el\cos\theta+\frac 1l$$

where $u$ is the reciprocal of the radial distance between the planet and the star, $e$ is the eccentricity of the orbit, $l$ is the semi latus rectum, $h$ denotes the angular momentum per unit mass, and $\theta$ is the angular coordinate.

$e$,$h$ and $l$ turn out to be independent from one another, and they are independent from $t$ and $\theta$. At time $t=0$, we let the radial speed vanish, and we also let the angular coordinate vanish. To find the relationship between time and angular speed $\omega$, we assume that $u$ is a smooth fuction of $t$, and differentiate $u$ w.r.t. $t$, and use $\omega=hu^2$ to find out an expression for $\omega$. To do this we can differentiate $u$ w.r.t. $\theta$, then multiply it by $\omega$, which equals to $hu^2$. Now we differentiate the first derivative of $u$ w.r.t $\theta$, then multiply the result by $hu^2$ and so on.

Since the whole process involves the differentiation w.r.t. $\theta$ only, we can assume that $e=0.5, l=h=1$. We set $e=0.5$ only because we wish to study bounded orbits so that we can apply Kepler's law and verify our result. However, the whole process is time consuming since the formula for the derivatives of $u$ becomes complicated very quickly, even if we assume explicit values for $e, l, h$. The only effective way, therefore, is to design an algorithm for this process. But I do not have any knowlege about computer science. Notice here that I'm not interested in finding out time, and I want to expand the anuglar speed at t=0 in terms of a polynomial at least 10 degree.

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At the moment you are on the physics part of this site.

As I'm sure you know, some computer languages are written especially for solving maths problems and nothing else.

If you also posted your question on this site https://scicomp.stackexchange.com/ I think you will find someone who can help you.

There is probably code already written that might solve your equations or needs only slight alterations to solve it.

Can I suggest that you edit your question into sections? It's quite difficult to follow as it stands now

"If anyone know about some other ways to find out ω in terms of t, please share."

I don't know myself but someone better at the maths than me might be able to help you if you leave the post here as well.

But do try to make it a good bit more readable. More people will notice it that way, I think.

hope this is of some help to you

Best of luck

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  • $\begingroup$ By that I mean approximate $\omega$ by a polynomial as a function of time. $\endgroup$
    – user43796
    Mar 15, 2015 at 22:07
  • $\begingroup$ if you want an algo then you will need to go to the link I gave you. If they can't help you get the algo directly, they will know someone who can. This part is not a physics question, so for sure ask the computer people $\endgroup$
    – user74893
    Mar 15, 2015 at 22:28

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