Recently I've encountered an article in our local astronomical popularization magazine. It is about a well known task of Sgr A*-black hole's mass estimation. The article is written as a step by step guide for young researchers. For example, the orbital parameters of the S2 star should be found using a drawing on a sheet of paper.
That is why I believed there is no need for very precise calculations for this task.
However, in addition to the $ x $, $ y $ coordinates of the S2 star, I was also given the corresponding uncertainty values $ \Delta x $, $ \Delta y $ (19 coordinate measurements for the time period from 1992 to 2003):
\begin{array}{|c|c|c|c|} \hline time & x & \Delta x & y & \Delta y \\ \hline 1992.226 & 0.104 & 0.003 & -0.166 & 0.004 \\ 1994.321 & 0.097 & 0.003 & -0.189 & 0.004 \\ ... & .. & .. & .. & \\ 2003.353 & 0.077 & 0.002 & -0.030 & 0.002 \\ 2003.454 & 0.081 & 0.002 & -0.036 & 0.002\\ \hline \end{array}
Using direct least squares method for S2 orbital fitting by an ellipse I've found parameters $a,\ b,\ c,\ d,\ e,\ f$ of the conic equation: $$ax^2+bxy+cy^2+ dx+ey+f=0$$ I've also decided to simulate the error propagation in a Monte Carlo fashion to find the uncertainty in the estimates of $a,\ b,\ c,\ d,\ e,\ f$. And I succeeded in that.
But, here is the question: is there any other appropriate, "analytical" way (not a Monte Carlo method) to find the uncertainty in the estimates of $a,\ b,\ c,\ d,\ e,\ f$ which may be used in usual astronomers' practice?
error-propagation
! $\endgroup$ – uhoh Dec 31 '17 at 17:29