2.3 Additional Data
2.3.4 V Max Correction for Selection Effects
DA(z) = DM 1 +z, (2.3b) DL(z) = (1 +z)DM, (2.3c) DM = 5log(DL) + 25 (2.3d)
using the notation given in Hogg (1999). The angular diameter distance is reported in the DERT in units of kpc per arcsec to allow for easy conversion between apparent and physical sizes of galaxies.
2.3.4 VMax Correction for Selection Effects
A VMax term is also calculated for each galaxy. This represents the observable volume
given the constraints on magnitude, surface brightness, and redshift used to construct the catalog. Volume corrections are calculated similar to that of Shen et al. (2003) (i. e., we account for all of the catalog cuts when calculating the observed volume for any galaxy in the catalog). As discussed in Chapter 2.2, the SDSS spectroscopic survey is complete down to a galactic extinction corrected Petrosian r-band magnitude of 17.77 and an extinction
2. CATALOG SELECTION
corrected Petrosianr-band surface brightness of 23.0 mag/arcsec2. Additionally, a minimum redshift cut of zmin >0.005 and a maximum redshift cut of zmin <1.0 were imposed. The
upper redshift cut removes essentially no galaxies and changes the VMax value of less than
10 galaxies. In addition, a minimum magnitude cut was included to exclude bright galaxies. Therefore, the volume in which any particular galaxy is observable must account for these effects.
The lower and upper bounds of redshift that limit this volume are determined by the conditions imposed on the data. The magnitude limits in the r-band, mr,min ≤ mr,obs ≤ mr,max correspond to a lower and upper limit on the redshift at which the galaxy remains
observable through the luminosity distance.
DL(zmax,m) =DL(z)10−0.2(mr−mr,max), (2.4a)
DL(zmin,m) =DL(z)10−0.2(mr−mr,min) (2.4b)
The surface brightness limit constrains VMax mainly through the (1 +z)4 cosmological
dimming effect. The maximum redshift from which a galaxy with mean surface brightness
µ50,r at redshift z can be detected at the surface brightness limit of µ50,r = 23 is given by
zmax,µ= (1 +z)10(23.0−µ50,r)/10−1 (2.5)
As is mentioned by S11, our selection criteria do not include a cut on size of the target galaxy, so the maximum redshifts may differ from that of Shen et al. (2003). The maximum and minimum redshifts for a galaxy in this work are then
zmin = max(zmin,m,0.005), (2.6a)
zmax= min(zmax,m, zmax,µ,1.0) (2.6b)
With the above limits on redshift, the comoving volume in which a target galaxy satisfies the sample selection criteria is:
VMax= Z dΩf(θ, φ) Z zmax(θ,φ) zmin(θ,φ) (1 +z)2DA2 H(z) cdz (2.7)
2.3 Additional Data
whereH(z) is the Hubble parameter at redshiftzand cis the speed of light. f(θ, φ) is the sampling fraction of point (θ, φ) on the sky, and Ω is the solid angle. Following S11,f(θ, φ) is treated as constant. In the case of a flat universe, Equation 2.7 reduces to
VMax=
4πfsky
3 (DM(zmax)
3
−DM(zmin)3) (2.8)
wherefsky= 0.195 is the fraction of the full sky covered by SDSS (8032 square degrees cov-
erage for the SDSS DR7 Legacy survey out of 41253 square degrees in the sky). Calculated values of VMax are provided as part of the data released with this work.
Chapter 3
Fitting Process
This chapter describes the fitting process. Section 3.1 describes the choice of cutout size and the data used for fitting. Section 3.2 describes the profiles used during fitting. Section 3.3 briefly describes thePyMorphpipeline and its components used to fit the catalog. Finally, Section 3.4 describes the masking and neighbor identification process used in this work.
3.1
Preprocessing of SDSS Images
FpC images and psField files are the primary data used in the fitting procedure. The data were downloaded from the SDSS Data Archive Server as described in Chapter 2.2. Postage stamp images of each source were extracted from the fpC image such that the stamp was 40 Petrosian half-light radii on a side (20×petroR50 rfrom the center of the image to the edge) and centered on the target source.
The decision to cut at 20 half-light radii is justified in chapter 4.2.5 based upon sim- ulations and provides a large number of background pixels (about 30 000-40 000 pixels for an average-sized image). In addition, a minimum size of 80 pixels on each side was set to ensure that enough pixels were retained to properly determine the background. In reality, with such a large postage stamp size, this minimum of 80 pixels is rarely required. With
3.1 Preprocessing of SDSS Images
these settings, the estimate of background sky brightness is accurate to∼0.1% with a slight bias toward underestimating the background level using this stamp size. This sky bias does not noticeably bias the other fitted parameters.
A smaller cutout size could potentially be used when fitting the galaxy and would reduce the time needed to fit each galaxy. Figure 4.18 and the related discussion show that the effect of further reducing the cutout size is insignificant to estimates of the total magnitude, half-light radius, and sky brightness. However, the original 20 half-light radii cutout sizes are used throughout the rest of this work.
The PSF is extracted from the PsField files using the readAtlasImages-v5 4 11
program distributed on the SDSS website2. The PSF provided by SDSS has a standard
image size of 51 pixels on each side.
In addition, prior to fitting the 1000 DN soft bias is removed from the images and PSF. Sigma images are created from the SDSS image cutout following the standard deviation calculation3
Wi,j =
s
Fi,j
gain+ dark variance (3.1)
whereWi,j is the pixel sigma in DN,Fi,j is the pixel flux (again in DN),gain for the image
as specified in SDSS CasJobs and used to account for the fact that the photo electrons (rather than the DN) obey Poisson statistics, and ‘dark variance’ is the term used by SDSS to describe the contribution of the read noise and dark current to the image noise. Finally, the postage stamp and sigma images are normalized to a 1 s exposure prior to fitting. A simple diagram of this process is shown in Figure 3.1.
2The use of readAtlasImages-v5 4 11 for PSF extraction is described at
http://www.sdss.org/dr7/products/images/read_psf.html
3See the SDSS DR7 online documentation athttp://www.sdss.org/dr7/algorithms/fluxcal.htmlfor further discussion.
3. FITTING PROCESS Weight Image Final Weight Postage Stamp psField file Final PSF fpC Image de-biased Image SDSS DAS PSF Image Final Stamp download
calculate sigma for each pixel
extract using readAtlasImages-v5 4 11
subtract 1000DN bias download
extract 20*PetroR50rimage
subtract 1000DN bias
normalize to 1s. exposure
normalize to 1s. exposure