## 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 source entry is the merged result of multiple observations 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 source positions resulting from the set of individual 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 source position varies significantly from measure to measure. We express the uncertainties of the estimates in the form of error ellipses centered upon the estimated source positions.

*An example of input ellipses (green) and the combined
ellipse (red). The x and y values represent
tangent plane
coordinates.*

The error ellipses that are combined to produce the "best
estimate" error ellipse for the merged source entry result from
the Chandra Multiwavelength Project (*ChaMP*) positional
uncertainty relations, as
described in the page "Source
Position Errors in the Source
Observations Table."

The use of the multivariate optimal weighting formalism in the Chandra Source Catalog, described below, represents its first application to astrophysical data, as it is based on an analysis of weapons targeting from a Master's thesis from the Naval Postgraduate School. For more information on this analysis and its use in the Chandra Source Catalog, see Joseph R. Orechovesky's Master's thesis "Single Source Error Ellipse Combination" and John Davis' CSC document "Combining Error Ellipses," respectively.

### Error Ellipses

#### Multivariate Optimal Weighting

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

where *X _{a}* represnts the

*a*th estimate of the mean of the 2-dimensional source position,

*σ*denotes the 2 x 2 covariance matrix associated with this estimate, and

_{a}#### Tangent Plane Projection

Before the covariance matrix *σ* 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*th
estimate of the source position is specified as a
confidence-ellipse centered upon the celestial coordinate
(α_{a}, δ_{a}), with the major axis making an
angle *θ* (-pi <= *θ* < pi) with
respect to the local line of declination at the center of the
ellipse. The tangent plane coordinates
(*x _{a}*,

*y*) of the center of the

_{a}*a*th ellipse are

where * p_{a}* is a unit-vector
on the celestial sphere corresponding to the

*a*th estimate of the celestial coordinate (α

_{a}, δ

_{a}) defining the source position, given by

Similar equations give the end-point positions
* p_{a}^{minor}* and

*of the semi-minor and semi-major axes of each ellipse*

**p**_{a}^{major}
where *φ _{a}^{minor}* is the arc-length of
the semi-minor axis and

*φ*is that of the semi-major axis.

_{a}^{major}
* p_{0}* denotes the position on
the celestial
sphere where a tangent plane is to be erected; it is taken to be
the arithmetic mean of the ellipse centers

*, i.e.,*

**p**_{a}
A coordinate system may be given to the tangent plane with the
origin at * p_{0}* and
orthonormal basis vectors

*and*

**e**_{x}*parallel to the local lines of right ascension and declination at*

**e**_{y}*, i.e.,*

**p**_{0}
where (α_{0}, δ_{0}) are the celestial
coordinates that correspond to * p_{0}*.

The tangent plane coordinates that correspond to the end-point
positions * p_{a}^{major}* and

*of the semi-major and semi-minor axes of the ellipse are denoted by (*

**p**_{a}^{minor}*x*,

_{a}^{major}*y*) and (

_{a}^{major}*x*,

_{a}^{minor}*y*). The lengths of the semi-major and semi-minor axes in the tangent plane are given by

_{a}^{minor}respectively. Finally, the angle that the semi-major axis makes with respect to the local line of declination is

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 *θ* 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*σ* 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, * σ' _{1}* is the 1

*σ*confidence value along the minor axis of the ellipse, and

*σ'*is that along the major axis (

_{2}*σ'*>=

_{2}*σ'*). The form of the covariance matrix

_{1}*σ*in the unrotated system follows from

where *R* is a rotation matrix that transforms a vector
*X* to *X'* = *RX*. Here, *R* is defined as

which yields

At this point, the lengths of the semi-major and semi-minor axes
of the soure position error ellipses
in the tangent plane, * σ' _{1}* and

*σ'*defined at the end of the previous section, may be input to the covariance matrix

_{2}*σ*above, permitting the error ellipses to be combined via the multivariate weighted sum.

^{ —> \longrightarrow }

^{ = σa = \sigma_a —> \longrightarrow }

This process produces the geometric
parameters of a combined 2-D error ellipse on the tangent plane
(*X*), which are recorded in the
"Position and Position
Errors" fields *err_ellipse_r0*,
*err_ellipse_r1*, and *err_ellipse_ang* in the Master
Chandra Source Table.