I seem to be at a standstill on an orbital mechanics problem. My work right up until my roadblock is shown below.
Consider a satellite elliptically orbiting the Earth (the center of the Earth is located at the left foci point), which is described by the position function relative to the center of the ellipse, $$ x=a\cos(\theta) $$ $$ y=b\sin(\theta) $$
The derivatives of the position function are, $$ \dot{x}=-a\dot{\theta}\sin(\theta) $$ $$ \dot{y}=b\dot{\theta}\cos(\theta) $$
And the second derivative is, $$ \ddot{x}=-a\dot{\theta}^2\cos(\theta)-a\ddot{\theta}\sin(\theta) $$ $$ \ddot{y}=-b\dot{\theta}^2\sin(\theta)+b\ddot{\theta}\cos(\theta) $$ Now, the satellite is also influenced by the gravitational pull of the Earth. Apparently there are three parts of the acceleration: the "normal" acceleration, the "tangential" acceleration, and the "total" acceleration (which I was sure is was just the tangential and normal components added together but I don't know anymore). We are told that "total acceleration is due to gravity" and $g$ is given as.
$$ g=g_o\cdot\frac{(RE)^2}{r^2} $$
Where $g_o$ is gravity at the Earth's surface, $RE$ is the radius of the Earth, and $r$ is the position of the satellite relative to the Earth (not the center of the ellipse). A gravitational force can't really be applied to my problem because it's not in the same frame as well. I'm not sure how to remedy this part.
I need to solve for what the normal and tangential accelerations are at a specific point, θ. Given the derivatives of my position function, how am I to solve for these values? 1. I do not know the angular velocity. And 2. I do not know the angular acceleration. Are we to assume that these values are constant? For example, if $ω$ (angular velocity) is constant, that would mean that the angular acceleration is 0. However, $ω$ is always changing because the orbit is non-circular. It seems to me that the satellite orbiting Earth will have an acceleration for two reasons: because of its motion AND because of Earth's gravity.
The acceleration due to the motion of the satellite should be described by the second derivatives, which these should be the components of the centripetal acceleration. However, as I mentioned beforehand, I have unknown values for $θ'$ and $θ''$. I've also calculated the speed at a point using the vis-viva equation, although I do not know the direction of this value, only the magnitude.
I'm not sure how exactly to best approach this problem and I've been stumped for a while. Every example I try to research does not go into this much detail about this type of problem as well. Any help would be greatly appreciated.
