Source Extent and Errors

The apparent sizes and associated errors of sources reported in the Chandra Source Catalog are determined using a Mexican-Hat optimization method, based on the extent of a source before subtraction of overlapping source regions. This is a refinement of the source extent results produced by wavdetect, the source detection algorithm which identifies source candidates in each observation in catalog processing. Only point sources and compact sources with extents < ~30 arcseconds are included in the catalog. Highly extended sources, and sources located in selected fields containing bright, highly extended sources, have been excluded from the first catalog release.

Source Region | Modified Source Region | Source Extent | Point Spread Function Extent | Deconvolved Source Extent

Source Region

Source Observations Table:
ra_aper, dec_aper, mjr_axis_aper, mnr_axis_aper, pos_angle_aper, mjr_axis1_aperbkg, mnr_axis1_aperbkg, mjr_axis2_aperbkg, mnr_axis2_aperbkg, pos_angle_aperbkg

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

Note: In the first catalog release, the source region is defined on a tangent plane projection. The 0 deg position angle reference is defined on that tangent plane to be parallel to the true North direction at the location of the tangent plane reference (refer to the tangent plane reference right ascension (ra_nom), declination (dec_nom), and roll angle (roll_nom)).

Modified Source Region

Source Observations Table:
area_aper, area_aperbkg

The modified source region and modified background region for each source are defined as the areas of intersection of the source region and background region for that source with the field of view, excluding any overlapping source regions.

Source Extent

Source Observations Table:
mjr_axis_raw, mnr_axis_raw, pos_angle_raw, mjr_axis_raw_err, mnr_axis_raw_err, pos_angle_raw_err

The source extent is a rotated elliptical Gaussian parameterization of the raw extent of a source, i.e., the extent of a source before subtraction of overlapping source regions, computed from a wavelet transform analysis of the counts in the raw source region in a given science energy band. The parameter values defining the source extent are the 1σ radii along the major and minor axes, the position angle of the major axis of the source ellipse, and the associated errors. The source extent has the following form:

$s(x,y;a_{1},a_{2},\phi) = \frac{s_{0}}{a_{1}a_{2}} \exp[-\pi(Ax)^{2}] .$

Here, φ (pos_angle_raw) is the clockwise angle between the positive x-axis and the ellipse major axis; a₁ and a₂ are the 1σ radii along the major and minor axes of the source ellipse (mjr_axis_raw, mnr_axis_raw,); s₀ is the amplitude of the source elliptical Gaussian distribution, and

$\mathbf{A} = \left(\begin{array}{cc} \frac{1}{a_{1}} \quad 0 \\ 0 \quad \frac{1}{a_{2}} \end{array} \right) \left(\begin{array}{cc} \cos\phi \quad \sin\phi \\ -\sin\phi \quad \cos\phi \end{array} \right) , \mathbf{x} = \left(\begin{array}{cc} x \\ y \end{array} \right) .$

Note: In the first catalog release, the source extent is defined on a tangent plane projection. The 0 deg position angle reference is defined on that tangent plane to be parallel to the true North direction at the location of the tangent plane reference (refer to the tangent plane reference right ascension (ra_nom), declination (dec_nom), and roll angle (roll_nom)).

Point Spread Function Extent

Source Observations Table:
psf_mjr_axis_raw, psf_mnr_axis_raw, psf_pos_angle_raw, psf_mjr_axis_raw_err, psf_mnr_axis_raw_err, psf_pos_angle_raw_err

The point spread function extent is a rotated elliptical Gaussian parameterization of the raw extent of the point spread function (PSF) at the location of the source. The parameterization of the PSF is computed from a wavelet transform analysis of the PSF counts in the source region in a given science energy band, and can be compared with the parameterization of the detected source to determine whether the latter is consistent with a point source. The point spread function extent is defined by the values and associated errors of the 1σ radii along the major and minor axes, and position angle of the major axis of the point spread function ellipse that the detection process would assign to a monochromatic PSF at the location of the source, and whose energy is the effective energy of the given energy band. The point spread function has the following form:

$p(x,y;b_{1},b_{2},\psi) = \frac{p_{0}}{b_{1}b_{2}} \exp[-\pi(Bx)^{2}] .$

Here, ψ (psf_pos_angle_raw) is the clockwise angle between the positive x-axis and the ellipse major axis; b₁ and b₂ are the 1σ radii along the major and minor axes of the psf ellipse (psf_mjr_axis_raw, psf_mnr_axis_raw); p₀ is the amplitude of the psf elliptical Gaussian distribution, and

$\mathbf{B} = \left(\begin{array}{cc} \frac{1}{b_{1}} \quad 0 \\ 0 \quad \frac{1}{b_{2}} \end{array} \right) \left(\begin{array}{cc} \cos\psi \quad \sin\psi \\ -\sin\psi \quad \cos\psi \end{array} \right) , \mathbf{x} = \left(\begin{array}{cc} x \\ y \end{array} \right) .$

Note: In the first catalog release, the source region is defined on a tangent plane projection. The 0 deg position angle reference is defined on that tangent plane to be parallel to the true North direction at the location of the tangent plane reference (refer to the tangent plane reference right ascension (ra_nom), declination (dec_nom), and roll angle (roll_nom)).

Deconvolved Source Extent

Master Sources Table:
major_axis, minor_axis, pos_angle, major_axis_err, minor_axis_err, pos_angle_err

The deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-convolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization consists of the best estimate values and associated errors for the 1σ radius along the major axis, the 1σ radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that is convolved with the ray-trace local PSF at the location of the source spatial event distribution.

Note: In the first catalog release, the rotated elliptical Gaussian parameterization of the deconvolved source extent will be approximated by a circularly symmetric Gaussian parameterization. This limitation will be lifted in a future release.

Source Observations Table:
major_axis, major_axis_err, minor_axis, minor_axis_err, pos_angle, pos_angle_err

The deconvolved source extent is a parameterization of the deconvolved extent of each source, i.e., a rotated elliptical Gaussian source that is convolved with the ray-trace local PSF at the location of the spatial event distribution of the source. Using a telescope with a PSF defined by p(x,y) to observe a source represented by s(x,y), one obtains a source image, c(x,y), which is the convolution of the source and the PSF,

$c(x,y) = \int\int s(x^{'},y^{'}) p(x - x^{'}, y - y^{'})dx^{'}dy^{'} .$

If both the source and PSF are elliptical Gaussians, then the PSF-convolved source, c(x,y), is an elliptical Gaussian centered on the origin with 1σ radii along the major and minor axes σ₁ and σ₂ (minor_axis, major_axis), and has the form

$c(x,y;\sigma_{1},\sigma_{2},\delta) = \frac{s_{0}p_{0}}{\sigma_{1}\sigma_{2}} \exp[-\pi(Tx)^{2}] .$

Here, δ (pos_angle) is the clockwise angle between the positive x-axis and the ellipse major axis, s₀ and p₀ are the respective amplitudes of the source and PSF elliptical Gaussian distributions, and

$\mathbf{T} = \left(\begin{array}{cc} \frac{1}{\sigma_{1}} \quad 0 \\ 0 \quad \frac{1}{\sigma_{2}} \end{array} \right) \left(\begin{array}{cc} \cos\delta \quad -\sin\delta \\ \sin\delta \quad \cos\delta \end{array} \right) , \mathbf{x} = \left(\begin{array}{cc} x \\ y \end{array} \right) .$

In the Chandra Source Catalog, the deconvolved source extent is defined by the 1σ radius along the major axis σ₂, the 1σ radius along the minor axis σ₁, the position angle δ of the major axis of the elliptical Gaussian defining the PSF-convolved source, and the associated errors.

Notes: In the first catalog release, the rotated elliptical Gaussian parameterization of the deconvolved source extent will be approximated by a circularly symmetric Gaussian parameterization. This limitation will be lifted in a future release.

In the first catalog release, the deconvolved source extent is defined on a tangent plane projection. The 0 deg position angle reference is defined on that tangent plane to be parallel to the true North direction at the location of the tangent plane reference (refer to the tangent plane reference right ascension (ra_nom), declination (dec_nom), and roll angle (roll_nom)).