## Source Position Errors in the Master Sources Table

### Summary

The position and 2-dimensional positional uncertainty of a
source listed in the Master Sources Table represents the
best estimate of the source position based on several
independent measures, where the master sources table entry is
the merged
result of multiple stacked observation detections of the same
source. To determine the best estimate of the position of a
source from previous independent estimates of its position, we
employ a 2-dimensional optimal weighting
formalism to statistically average the detection
positions resulting from the set of individual
stacked observations of the source. We decided to
use this technique because it offers an improved estimate of
source position where simple averaging fails, *e.g.*,
where the area defining the detection position varies
significantly from measure to measure. We express the
uncertainties of the estimates in the form of error ellipses
centered upon the estimated detection positions.

### Input Ellipses and the Combined Ellipse

The Stacked Observation Detections Table error ellipses that are combined to produce the best estimate error ellipse for the Master Sources Table entry are determined from the best-fitting ellipse to the position-uncertainty–fit-statistic surface computed from the MLE's Markov chain Monte Carlo (MCMC) draws, as described in the page "Source Position Errors in the Stacked Observation Detections Table".

The use of the multivariate optimal weighting formalism used in the CSC, described below, is discussed in John Davis' memo "Combining Error Ellipses".

### Error Ellipses

#### Multivariate Optimal Weighting

The multivariate optimal weighting formalism used to combine error ellipses can be distilled to the following formula

\[ X = \sigma \sum_{a} \sigma_{a}^{-1} X_{a} \ ,\ \]where \(X_{a}\) represents the \(a\mathrm{th}\) estimate of the mean of the 2-dimensional source position, \(\sigma_{a}\) denotes the \(2 \times 2\) covariance matrix associated with this estimate, and

\[ \sigma = \left[\sum_{a} \sigma_{a}^{-1}\right]^{-1} \ .\ \]#### Tangent Plane Projection

Before the covariance matrix, \(\sigma\), may be computed—permitting the ellipses to be combined via the multivariate weighted sum—the error ellipses must be mapped from the celestial sphere onto a common tangent plane. The \(a\mathrm{th}\) estimate of the source position is specified as a confidence-ellipse centered upon the celestial coordinate, \((\alpha_{a},\delta_{a})\), with the major axis making an angle \(\theta\) \((-\pi \leq; \theta < \pi)\) with respect to the local line of declination at the center of the ellipse. The tangent plane coordinates \((x_{a},y_{a})\) of the center of the \(a\mathrm{th}\) ellipse are

\[ x_{a} = \frac{\left(\hat{p}_{a} \cdot \hat{e}_{x}\right)} {\left(\hat{p}_{a} \cdot \hat{p}_{0}\right)} \ , \ y_{a} = \frac{\left(\hat{p}_{a} \cdot \hat{e}_{y}\right)} {\left(\hat{p}_{a} \cdot \hat{p}_{0}\right)} \ , \ \]where \(\hat{p}_{a}\) is a unit-vector on the celestial sphere corresponding to the \(a\mathrm{th}\) estimate of the celestial coordinate \((\alpha_{a},\delta_{a})\) defining the source position, given by

\[ \hat{p}_{a} = \hat{x} \cos{\alpha_{a}} \cos{\delta_{a}} + \hat{y} \sin{\alpha_{a}} \cos{\delta_{a}} + \hat{z} \sin{\delta_{a}} \ . \ \]Similar equations give the end-point positions \(\hat{p}_{a}^{minor}\) and \(\hat{p}_{a}^{major}\) of the semi-minor and semi-major axes of each ellipse:

\[ \hat{p}_{a}^{minor} = \hat{p}_{a} \cos{\phi_{a}^{minor}} + \hat{\alpha}_{a} \sin{\phi_{a}^{minor}} \cos{\theta} - \hat{\delta}_{a} \sin{\phi_{a}^{minor}} \sin{\theta} \]and

\[ \hat{p}_{a}^{major} = \hat{p}_{a} \cos{\phi_{a}^{major}} + \hat{\alpha}_{a} \sin{\phi_{a}^{major}} \sin{\theta} - \hat{\delta}_{a} \sin{\phi_{a}^{major}} \cos{\theta} \ , \ \]where \(\phi_{a}^{minor}\) is the arc-length of the semi-minor axis and \(\phi_{a}^{major}\) is that of the semi-major axis.

\(\hat{p}_{0}\) denotes the
position on the celestial sphere where a tangent plane
is to be erected; \(\hat{p}_{0}\)
is taken to be the arithmetic mean of the ellipse
centers \(\hat{p}_{a}\), *i.e.*,

A coordinate system may be given to the tangent plane
with the origin
at \(\hat{p}_{0}\) and
orthonormal basis
vectors \(\hat{e}_{x}\)
and \(\hat{e}_{y}\) parallel to
the local lines of right ascension and declination
at \(\hat{p}_{0}\), *i.e.*,

and

\[ \hat{e}_{y} = \hat{\delta}_{0} = - \hat{x} \sin{\delta_{0}} \cos{\alpha_{0}} - \hat{y} \sin{\delta_{0}} \sin{\alpha_{0}} + \hat{z} \cos{\delta_{0}} \ , \ \]where \((\alpha_{0},\delta_{0})\) are the celestial coordinates that correspond to \(\hat{p}_{0}\).

The tangent plane coordinates that correspond to the end-point positions \(\hat{p}_{a}^{major}\) and \(\hat{p}_{a}^{minor}\) of the semi-major and semi-minor axes of the ellipse are denoted by \((x_{a}^{major},y_{a}^{major})\) and \((x_{a}^{minor},y_{a}^{minor})\). The lengths of the semi-major and semi-minor axes in the tangent plane are given by

\[ \sigma_{2}^{\prime} = \sqrt{ \left(x_{a}^{major} - x_{a}\right)^{2} + \left(y_{a}^{major} - y_{a}\right)^2 } \]and

\[ \sigma_{1}^{\prime} = \sqrt{ \left(x_{a}^{minor} - x_{a}\right)^{2} + \left(y_{a}^{minor} - y_{a}\right)^{2} } \ , \ \]respectively. Finally, the angle that the semi-major axis makes with respect to the local line of declination is

\[ \theta_{a}^{\prime} = \tan^{-1}{ \left(\frac{x_{a}^{major} - x_{a}} {y_{a}^{major} - y_{a}}\right) } \]Armed with these relations, it is easy to compute the tangent plane projections of the error ellipses.

### Computing Covariance Matrices

Three of the parameters specifying the geometry of each
projected error ellipse are the semi-major and semi-minor
axis lengths, and the position angle \(\theta\) that
the major axis of the ellipse makes with respect to the
tangent plane *y*-axis. The semi-major and
semi-minor axis lengths correspond to the \(1\sigma\)
confidence intervals along these axes. More specifically,
in a basis whose origin is at the center of the ellipse, and
whose *y*-axis is along the major axis of the
ellipse, the correlation matrix is

Here, \(\sigma^{\prime}_{1}\) is the \(1\sigma\) confidence value along the minor axis of the ellipse, and \(\sigma^{\prime}_{2}\) is that along the major axis \((\sigma^{\prime}_{2} \geq \sigma^{\prime}_{1})\). The form of the covariance matrix \(\sigma\) in the unrotated system follows from

\[ \mathbf{\sigma}^{\prime} = \mathcal{R} \mathbf{\sigma} \mathcal{R}^{-1} \ , \ \]where \(\mathcal{R}\) is a rotation matrix that transforms a vector \(X\) to \(X^{\prime} = \mathcal{R}X\). Here, \(\mathcal{R}\) is defined as

\[ \mathcal{R} = \left[\begin{array}{cc} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{array} \right] \ , \ \]which yields

\[ \mathbf{\sigma} = \left[\begin{array}{cc} {\sigma_{1}^{\prime}}^2 \cos^{2}\theta + {\sigma_{2}^{\prime}}^{2} \sin^{2}\theta & \left({\sigma_{2}^{\prime}}^{2} - {\sigma_{1}^{\prime}}^{2} \right) \cos{\theta}\sin{\theta} \\ \left({\sigma_{2}^{\prime}}^{2} - {\sigma_{1}^{\prime}}^{2}\right) \cos{\theta}\sin{\theta} & {\sigma_{1}^{\prime}}^{2} \sin^{2}\theta + {\sigma_{2}^{\prime}}^{2} \cos^{2}\theta \end{array} \right] \ . \ \]At this point, the lengths of the semi-major and semi-minor axes of the source position error ellipses in the tangent plane, \(\sigma^{\prime}_{1}\) and \(\sigma^{\prime}_{2}\) defined at the end of the previous section, may be input to the covariance matrix, \(\sigma\) above, permitting the error ellipses to be combined via the multivariate weighted sum:

\[ \begin{array}{c} \begin{array}{r} \sigma_{2}^{\prime} = \sqrt{ \left(x_{a}^{major} - x_{a}\right)^{2} + \left(y_{a}^{major} - y_{a}\right)^2 } \\ \sigma_{1}^{\prime} = \sqrt{ \left(x_{a}^{minor} - x_{a}\right)^{2} + \left(y_{a}^{minor} - y_{a}\right)^{2} } \end{array} \\ \longrightarrow \mathbf{\sigma} = \left[\begin{array}{cc} {\sigma_{1}^{\prime}}^2 \cos^{2}\theta + {\sigma_{2}^{\prime}}^{2} \sin^{2}\theta & \left({\sigma_{2}^{\prime}}^{2} - {\sigma_{1}^{\prime}}^{2} \right) \cos{\theta}\sin{\theta} \\ \left({\sigma_{2}^{\prime}}^{2} - {\sigma_{1}^{\prime}}^{2}\right) \cos{\theta}\sin{\theta} & {\sigma_{1}^{\prime}}^{2} \sin^{2}\theta + {\sigma_{2}^{\prime}}^{2} \cos^{2}\theta \end{array} \right] = \mathbf{\sigma_{a}} \\ \longrightarrow X = \sigma \sum_{a} \sigma_{a}^{-1} X_{a} \ . \ \end{array} \]
This process produces the geometric parameters of a combined
2-D error ellipse on the tangent plane \((X)\), which
are recorded in
the "Source Position and
Position Errors"
fields *err_ellipse_r0*, *err_ellipse_r1*,
and *err_ellipse_ang* in the Master Chandra Sources
Table.