I'm having a hard time trying to propagate errors for my current work. I have two known positions $X$ and $Y$ that depend on several orbital parameters $r, i, \Omega, \omega, f$ ($r$ is a distance and $i, \Omega, \omega, f$ are angles).
I know the errors $\Delta X$, $\Delta Y$ associated to the positions $X$ and $Y$ and I'd like to determine the errors associated to the other parameters $r, i, \Omega, \omega, f$ except that the relation between them is non linear.
$X$ and $Y$ can be written as follows :
\begin{equation} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} \cos{\Omega} & -\sin{\Omega} \\ \sin{\Omega} & \cos{\Omega} \\ \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & \cos{i} \\ \end{pmatrix} \begin{pmatrix} \cos{\omega} & -\sin{\omega} \\ \sin{\omega} & \cos{\omega} \\ \end{pmatrix} \begin{pmatrix} r\cos f \\ r\sin f \end{pmatrix} \\ \\\\ \end{equation}
which also writes :
\begin{equation} \begin{matrix} X = r (\cos\Omega\cos(\omega +f)-\sin(\Omega)\sin(\omega +f)\cos{i}) \\ Y = r (\sin\Omega\cos(\omega +f)+\cos(\Omega)\sin(\omega +f)\cos{i}) \end{matrix} \end{equation}
As $X$ and $Y$ are not correlated, I've tried to determine the error with this equation :
\begin{equation} \sigma^2_{\mu} \approx \left| \frac{\partial f_{\mu}}{\partial X}\right| ^2\sigma^2_X+\left| \frac{\partial f_{\mu}}{\partial Y}\right|^2\sigma^2_Y \end{equation}
where $\mu$ is one of my orbital parameters. But for that, I have to establish for each of them the function $f_{\mu}$ that depends on $X$ and $Y$.
I tried with $i$ for example but it looks odd... here is what I have for the moment :
\begin{equation} \cos i\Bigg|_{\omega+f \ne 0} = \left\{ \begin{array}{ll} \dfrac{X+Y - r\cos(\omega +f) \left( \cos\Omega+\sin\Omega \right)}{r\sin(\omega + f)(\cos\Omega-\sin\Omega)} & \mbox{if } \Omega \ne \dfrac{\pi}{4}(4k+1) \mbox{, } k \in \mathbb{Z} \\ & \\ \dfrac{Y-X}{2 r \cos\Omega \sin(\omega+f)} & \mbox{ else.} \end{array} \right. \end{equation}
\begin{equation} f_{i}(X,Y) = \left\{ \begin{array}{ll} \arccos\left( \dfrac{X+Y - r\cos(\omega +f) \left( \cos\Omega+\sin\Omega \right)}{r\sin(\omega + f)(\cos\Omega-\sin\Omega)} \right) & \mbox{if } \Omega \ne \dfrac{\pi}{4}(4k+1) \mbox{, } k \in \mathbb{Z} \\ & \\ \arccos\left( \dfrac{Y-X}{2 r \cos\Omega \sin(\omega+f)} \right) & \mbox{ else.} \end{array} \right. \end{equation}
But near $\Omega=\pi/4$ and $\Omega=5\pi/4$, my expression of $\cos i$ explodes and the arccos function is only defined between $[-1,1]$ so I'm not sure if this is the right way to propagate the errors. Plus, establishing a differentiable function that depends on $X$ and $Y$ will not be possible for all of my orbital parameters. So I'd like to know if there is another way, especially with the matrice formulation of the problem, I searched for a while but nothing seems to really answer my problem...
Thank you very much to those who'll answer, and have a nice day !