# Error propagation methods for orbit parameters

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?

• For a good statistical analysis answer, you might consider posting this in crossvalidated stackexchange instead. It's certainly possible you'll get a good answer here, but you might get it a lot faster there. They even have a tag specifically for error-propagation! – uhoh Dec 31 '17 at 17:29
• Analytical error propagation is usually possible when you know the underlying distribution of the measurments. As in all generality for all astrophysical sources this distribution is unknown and will also never be the same it is common practice to 'smash' this problem with the 'MC' Hammer. – AtmosphericPrisonEscape Dec 31 '17 at 17:52
• Unless you can linearise the model, there is no analytic error estimate. – Rob Jeffries Dec 31 '17 at 20:34