Because the two ATLAS survey regions are small (Ωă4 deg2) we can make small angle approx-
imations for the calculation of the comoving volumes enclosed by the ATLAS survey; that is, we simply scale the spherical volume equation by Ω{4π. This results in a simple conical volume for a comoving distanceDM˚:
V“ 1
3DM
3Ω. (3.1)
To correct for the under-counting of radio sources inherent to any flux-limited survey, we as- sume homogeneity of source distributions within the ATLAS volume. We considerVto be the volume calculated at the detection distance of a source, andVmaxto be the corresponding vol- ume for the distance at which that source would fall below the detection limit of the survey. Fol- lowing the adaptation of theV{Vmaxcorrection (Schmidt 1968) established byBest & Heckman (2012) to allow for a sliding redshift window (i.e., the specification of a minimum redshift of in- terest corresponding to the minimum volume element,Vmin), the contribution to the comoving volume density of each source count is scaled by the volume weighting:
ωV“ V
´Vmin
Vmax´Vmin
. (3.2)
Because the sources contributing to the radio-luminosity function require an optically ob- tained redshift as well as a 1.4 GHz radio flux-density,ωVneeds to account for both radio and optical survey limitations; that is, a source will only be detectable within the most limiting vol- ume. So we can take
Vmax “minimum
!
Vmax,o,Vmax,r,Vmax,z , (3.3) whereVmax,o,Vmax,r,andVmax,zare the optical, radio and redshift maximum volumes respec- tively.Vmax,zis included for cases where a manual redshift-cut is applied to the data that results in a smaller maximum comoving volume than either the optical or radio limitations (e.g., Fig- ure 3.1). Going forward, all calculations adoptΛ-CDM cosmological parameters of Ωm “ 0.3, ΩΛ “ 0.7, andH0 “ 70 km s´1Mpc´1(Spergel et al. 2003) to maintain consistency with
literature.
˚In practice, volumes calculated in this chapter useastropy.cosmology.comoving_volumeand
astropy.cosmology.comoving_distanceto correctly integrate over a non-spherical cosmology
3.2.2 Maximum optical volume
Calculation ofVmax,osimply requires the calculation of an absolute magnitude for each source via the magnitude-distance relationship:
M“m´5rlog10pDLq ´1s. (3.4)
followed by the inversion of this relationship to solve for a maximum luminosity distanceDL,max using the calculated absolute magnitudes and a specified limiting apparent magnitude,mlim.
3.2.3 Maximum radio volume
Calculations of the maximum radio volumes are more complicated than optical volumes due to the non-uniform sensitivity of the survey area (see Figure 3, Page 10). Instead of calculating a maximum volume for an averaged flux-density sensitivity limit, I break the survey area into discrete sensitivity bins (similar toMao et al. 2012). The final maximum radio volume is the sum of the binned volumes for which the source would have been visible; that is, an individual bin only contributes to the summedVmaxif the flux density limit of that bin is less than the flux- density of the source (see Figure 3.2).
Figure 3.1:Diagram demonstra ng the effect of slice-wise volume calcula ons for a redshi volume limited source. Each er
represents the detectable volume of a source for a hypothe cal RMS limit slice. The red lined region shows the maximum extent of an arbitrary redshi volume, and the stepped outline is theVmax,rover all bins for a source. The blue lined region
is the finalVmax,ronce accoun ng for the excluded black do ed region and redshi regions.
The k-corrected 1.4 GHz luminosity of each source is calculated from the observed 1.4 GHz flux,S1.4, via:
L1.4“ 4πS1.4DM 2
Figure 3.2:Diagram demonstra ng the effect of slice-wise volume calcula ons for a radio flux-density limited source. Each er represents the detectable volume of a source for a hypothe cal RMS limit slice. The blue lined region is the finalVmax,r
for this source. The black do ed regions are excluded as the source flux-density is less than five mes the RMS of these bins.
where the spectral indexα “ ´0.7 (Condon et al. 2002;Sadler et al. 2002) is chosen to main- tain consistency with the literature used for comparison later in this chapter.
For a survey area separated intoiregions, with corresponding sky-areas Ωiand flux-density limit,Slim,i:the maximum co-moving distance for each source within this slice is calculated via a rearrangement of Equation (3.5), and the correspondingVmax,rfor each source is then the sum of all contributing volume slices:
Vmax,r“ 1 3 n ÿ i“1 DM,maxi3Ωi. (3.6)
3.2.4 Comoving Source Density
To assemble the radio-luminosity function sources are binned by 1.4 GHz luminosity in loga- rithmic space, and the comoving density of sources within these luminosity bins is calculated according to: Φbin“ 1 Δbin ÿ sources ωV,source Vsource ˆ 1 Csource, (3.7) :For alli S
lim,iis taken to be five times the median root-mean-square (RMS) flux-density in that bin. Because the two ATLAS fields, CDFS and ELAIS-S1, have different RMS profiles these values are calculated independently. The binned median RMS values for ELAIS-S1 are
r17.7,25.1,35.6,51.5,73.8,106sμJy beam´1and for CDFSr15.9,21.7,33.4,50.3,74.7,114sμJy beam´1 coveringr1.40,0.76,0.51,0.36,0.31,0.27sdeg2andr1.46,1.37,0.72,0.54,0.41,0.37sdeg2respectively.
in this equationωV,sourceis calculated from Equation (3.2) according to the most limiting vol- ume (see Equation (3.3)). Regardless of the limiting volume,Csourceis the spectroscopic com- pleteness at the corresponding magnitude of each source (see§3.4.1). The density in each bin is normalised by the (logarithmic) width of the bin, Δbin.