Find out the angular speed in terms of time.

Question

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, and let $h$ denote the angular momentum per unit mass, $\theta$ is the angular coordinate. $e$,$h$,$l$ 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. $\theta$ first then multiply it by $\omega$, which equals to $hu^2$. Then differentiate the first derivate of $u$ w.r.t $\theta$ first, 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, if anyone knows how to design algorithms or know about some other ways to find out $\omega $ in terms of $t$, please share.

Answer 1

Kepler actually had three laws. One of them is the "equal areas" law. Draw a line from the star to the planet; as time passes and the planet moves, the line between the star and the planet sweeps out an area of the ellipse. The total area swept between time $t_i = 0$ and time $t_f$ is proportional to the elapsed time $t_f$.

To the time at any point along the orbit, simply determine how much area was swept out up to that point and divide by the constant rate at which the swept area increases. Determine the angular velocity at that same point (again using the equal area law) and you have a formula relating angular velocity and time.

Once you have that formula, solve for angular speed, and you will have angular speed as a function of time, as you wanted.

This is all possible to do with pencil and paper without involving a computer.