## Position Errors in the Stacked-Observation Detections Table

## Detection Position Error Estimates

In the Stacked-Observation Detections
Table, compact detection positions (ra, dec)
are determined from the best fitting Maximum Likelihood Estimator (MLE)
model. MLE uses *Sherpa* to perform a simultaneous fit to the
set of individual observations that comprise the stack,
using the local PSF model for each observation convolved
with a rotated elliptical Gaussian. For the point source
model, the Gaussian sigmas are set to approximate a
single-pixel delta function, whereas for the extended
compact source model the sigmas are allowed to vary.

After the *Sherpa* model fit is performed, the confidence
intervals for the position parameters are calculated. The
*Sherpa* implementation of the pyBLoCXS
algorithm (Van
Dyk et al. 2001, Siemiginowska
et al. 2011) is used to obtain a MCMC sample from the posterior
probability density for all the fitted parameters. The
starting point for the MCMC run is the best fit (maximum
likelihood) model parameter values, and the assumed priors
for the position parameters are flat within the boundary of
the fitted regions.

A sample of 5,000 draws from the posterior probability distribution for a point source model, and 15,000 draws for an extended compact source model. The convergence and correlations within the draws for each parameter are calculated using the \(\hat{R}\) diagnostic parameter (Gelman et al. 2013). In calculations of \(\hat{R}\) the full MCMC chain is split into 10 sub-sections and the variance within each section and between the sections is used as input to \(\hat{R}\) calculation. We verify that the \(\hat{R}\) value for each parameter is close to 1.0 for detections classified as TRUE based on their likelihood. In marginal detections this value can be larger, and this can be due to an incorrect model being used, potential multi-source regions fit with single source, additional emission components not taken into account in the model etc.

Provided that \(\hat{R} < 1.2\) and the acceptance fraction \(> 0.2\), the 90% confidence best-fitting error ellipse is calculated using the MCMC posterior probability draws. If these criteria are not met, then an estimate of the 90% confidence error circle is estimates from a set of position uncertainty equations calibrated from the equivalent circular radii of a large sample of good-quality best-fitting error ellipses. The circular error position uncertainty equation takes the form:

\[ \log_{10}\left(error\ radius \right) = C_{0} + C_{1} \theta + C_{2} \log_{10}\left(net\ counts \right) + C_{3} \theta \log_{10}\left(net\ counts \right) \]where the constants \(C_{n}\) are listed in the table below.

Energy Band | \(C_{0}\) | \(C_{1}\) | \(C_{2}\) | \(C_{3}\) |
---|---|---|---|---|

b | -0.031 | 0.173 | -0.526 | -0.023 |

h | 0.054 | 0.176 | -0.607 | -0.021 |

m | 0.056 | 0.154 | -0.578 | -0.018 |

s | 0.088 | 0.153 | -0.582 | -0.018 |

u | 0.242 | 0.134 | -0.714 | 0.001 |

w | 0.075 | 0.206 | -0.544 | -0.026 |

Position errors in the Stacked-Observation Detections
Table are internal errors (i.e., calculated from
the model fit to the data) only. They do not include
uncertainties in the absolute position of the
*Chandra* data relative to the ICRS astrometric reference
frame. Such uncertainties are included in
the source position error estimates included in the
Master Sources
Table.