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, were excluded from the first catalog release; they will be added as a supplementary list in the second release.
Source Region
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.
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
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
mjr_axis_raw, mjr_axis_raw_lolim, mjr_axis_raw_hilim, mnr_axis_raw, mnr_axis_raw_lolim, mnr_axis_raw_hilim, pos_angle_raw, pos_angle_raw_lolim, pos_angle_raw_hilim
In order to estimate the intrinsic extent of a source in the sky, one first needs to realize that the measured extent of the source on the detector is the result of a convolution between the source itself and the PSF corresponding to that particular observation. It is therefore necessary to estimate the convolved extent of the source and of the PSF, and then perform a deconvolution.
Convolved Source Extent
The extent of the convolved source is estimated in a given science energy band with a rotated elliptical Gaussian parametrization of the raw extent of a source, i.e., the extent of a source before subtraction of overlapping regions. The corresponding ellipse has the following form:
where
Here, φ (pos_angle_raw) is the clockwise angle between the positive x-axis and the ellipse major axis; a_{1} and a_{2} are the 1σ radii along the major and minor axes of the source ellipse (mjr_axis_raw, mnr_axis_raw); s_{0} is the amplitude of the source elliptical Gaussian distribution.
For source extent purposes, the parameters of the ellipse are estimated by performing a spatial transform with a Mexican-Hat wavelet (also known as Ricker wavelet) directly on the counts in the raw source region.
The idea is simple: the two-dimensional correlation integral (i.e., the transform) between the wavelet function W and the ellipse function S is defined as:
where α = (a_{1},a_{2},φ) are the semi-major axis, semi-minor axis, and rotational angle of the Mexican-Hat wavelet. The quantity ψ(x,y;α) = C(x,y;α)/(a_{1}a_{2})^{1/2} is maximized if the dimensions of the ellipse and the Mexican-Hat wavelength are related as: a_{i} = 3^{1/2}σ_{i} and φ = φ_{0}. We can therefore estimate the parematers of the source extent ellipse by maximizing φ(x,y;α). Note that this assumes that sources can always be described as elliptical Gaussians. In practice, the maximization is evaluated as a discrete version of the equations above on the pixels of the image.
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)).
PSF Extent
The same approach as for the convolved source extent is used to estimate the elliptical parameters that best represent the instrumental point spread function (PSF) in each science band at the location of the source. The inputs are the the PSF counts in the source region. The parameterization of the PSF can be compared with the parameterization of the detected source to determine whether the latter is consistent with a point source (see below).
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
Determination of the extended nature of sources
An isolated source with a significant number of counts above the background is likely to be extended if min(σ_{i}) > σ_{t}, where:
In the equation above, f_{psf} ≡ Δσ_{psf}/σ_{psf} is the fractional uncertainty in the PSF size and f_{mho} is the fractional uncertainty in the source region size which can be approximated:
N_{σ} is the number of counts above background inside σ_{i}. A minimum of 15 counts are required for this estimate to be meaningful.
The deconvolved source extent is (well-described below).
Point Spread Function Extent
psf_mjr_axis_raw, psf_mjr_axis_raw_lolim, psf_mjr_axis_raw_hilim, psf_mnr_axis_raw, psf_mnr_axis_raw_lolim, psf_mnr_axis_raw_hilim, psf_pos_angle_raw, psf_pos_angle_raw_lolim, psf_pos_angle_raw_hilim
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:
Here, ψ (psf_pos_angle_raw) is the clockwise angle between the positive x-axis and the ellipse major axis; b_{1} and b_{2} are the 1σ radii along the major and minor axes of the PSF ellipse (psf_mjr_axis_raw, psf_mnr_axis_raw); p_{0} is the amplitude of the PSF elliptical Gaussian distribution, and
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
major_axis, major_axis_lolim, major_axis_hilim, minor_axis, minor_axis_lolim, minor_axis_hilim, pos_angle, pos_angle_lolim, pos_angle_hilim
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.
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.
major_axis, major_axis_lolim, major_axis_hilim, minor_axis, minor_axis_lolim, minor_axis_hilim, pos_angle, pos_angle_lolim, pos_angle_hilim
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,
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 σ_{1} and σ_{2} (minor_axis, major_axis), and has the form:
Here, δ (pos_angle) is the clockwise angle between the positive x-axis and the ellipse major axis, s_{0} and p_{0} are the respective amplitudes of the source and PSF elliptical Gaussian distributions, and
In the Chandra Source Catalog, the deconvolved source extent is defined by the 1σ radius along the major axis σ_{2}, the 1σ radius along the minor axis σ_{1}, the position angle δ of the major axis of the elliptical Gaussian defining the PSF-convolved source, and the associated errors.
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)).
Changes with Respect to Earlier Versions
With respect to earlier versions of the catalog, a number of improvements have been included in Release 2 of the catalog to improve the source extent estimate. The main changes were:
- We use the results from wavdetect to set the initial parameter guess for the source size. The correlation integral is maximized using the Nelder-Mead Simplex optimization method, but only the scale and orientation of the Mexican-Hat wavelet are free parameters. The centroid position is estimated prior to the fit by maximizing a simplified version of the wavelet that uses the initial guesses for a_{i} (from wavdetect), and φ=0. Therefore, the position of the pixel where the maximum occurs is found first, and then the orientation and size of the ellipse are optimized for.
- Adding the effect of aspect blur to the PSF. Both the aspect solution and detector effects add an aspect blur to the instrumental PSF that effectively increase its extent. We have added an estimated blur in quadrature to the PSF extent in order to improve our estimate of the deconvolved source extent.
- Improving the PSF image fitting by adjusting image centering and size, and using sub-pixelated PSFs where appropriate.