THE STELLAR KINEMATICS OF VOID DWARF GALAXIES USING KCWI
5.3 Analysis
5.3.1 Binning and covariance correction
5.2.2 Observations and data reduction
IFU data were obtained for the sample over 2.5 nights using KCWI, an optical integral field spectrograph on the Nasmyth platform of the 10 m Keck II telescope (Morrissey et al. 2018). KCWI has multiple configurations; in order to match the typical angular size of the dwarf galaxies in my sample, I used the medium slicer and blue BL grating centered atλ= 4500 Å. This combination yields a 2000×16.500 field of view, nominal spectral resolution of ∼ 2.5 Å (σ ∼ 71 km s−1) at 4500 Å, and a usable wavelength range of 3500−5100 Å.
Table 5.2 describes the observations of each galaxy. For galaxies with multiple exposures, I rotated the position angles by ±10° for each exposure in order to minimize spatial covariance during stacking. For each object exposure, I observed a patch of nearby sky with the same exposure time and position angle to perform sky subtraction. I processed all object exposures using the most recent version of the KCWI data reduction pipeline2, which produces flux-calibrated data cubes. Data cubes of the same object were then aligned and stacked using a drizzling algorithm (O’Sullivan and Chen 2020)3.
like galaxy outskirts or faint dwarf galaxies. It is therefore important to spatially bin spaxels, averaging multiple adjacent spaxels to increase the S/N. To do this, I use vorbin, an adaptive spatial binning algorithm that produces Voronoi tessellations (Cappellari and Copin 2003).
I first define the nominal S/N of an individual spaxel, assuming that each spaxel is independent. Because I am primarily interested in measuring information from the stellar continuum in each spaxel, I use the formula for the detrendedcontinuumS/N defined by Rosales-Ortega et al. (2012):
S N
!
c
= µc
σc
, (5.1)
where µc is the mean of the flux in the continuum band f(λ)c, and σc is the detrended standard deviation (i.e., the standard deviation in the difference between f(λ)c and a linear fit to f(λ)c). I take f(λ)c to be the flux across the continuum range 4700−4750 Å, which lacks strong emission features.
The S/N values from Equation 5.1 are likely overestimates, since stacking data cubes introduces covariance between adjacent spaxels. To account for this, rather than computing full covariance matrices for each spaxel in every data cube, I use an empirical formula to estimate the ratio between the “true” noisetrue and the noise assuming no covarianceno covar. This ratio, denotedη, is assumed to be a function of the bin size, with the form suggested by Husemann et al. (2013):
η =true/no covar=
β(1+αlogn) N ≤ Nthreshold
β(1+αlogNthreshold) N > Nthreshold. (5.2) Here, N is the number of spaxels in each bin, α describes the strength of the dependence of η on bin size, and β is a normalization factor. Above a certain bin sizeNthreshold, additional spaxels are assumed to be far enough apart that they do not add any extra covariance, so the ratioη is capped at a constant.
I estimate the value of the free parameters{α, β,Nthreshold}following the procedure of Law et al. (2016) as follows. First, I create mock data cubes in which all pixels have fluxes independently drawn from a normal distributionN ∼ (1,1)with mean and variance both unity. These mock cubes are stacked following the same drizzling procedure as the actual data cubes from individual exposures, producing mock in- tensity and variance cubes. For a stacked cube, the spaxels are binned using a simple boxcar of size n2 where n varies. The standard deviation of each bin in the mock
0 50 100 150 200
Bin sizeN
1.6 1.8 2.0 2.2 2.4 2.6
η
Fit: α=0.110,β=1.641,Nthreshold=75.339
Figure 5.1: The ratioη =true/no covaras a function of bin size for an example with 4 stacked exposures. Black points represent empirical estimates from mock data cubes, as described in the text. The red line indicates the best-fit empirical curve of the form Equation 5.2.
intensity cube is an estimate of the “true” noisetrue, since it accounts for the effects of stacking. The stacked mock variance cube, on the other hand, is used to compute a separate noise estimate no covar using simple error propagation rules, assuming that each spaxel is independent. The ratio of these two estimates can be plotted as a function of bin size Nand fit with a curve of the functional form described in Equa- tion 5.2 to determine the best-fit values of{α, β,Nthreshold}. Figure 5.1 demonstrates an example of this fitting for four stacked exposures, showing that Equation 5.2 is a good representation ofη.
By multiplying this empirical estimate for η by the noise estimate (i.e., dividing Equation 5.1 byη), the S/N within a bin can be corrected for the effects of spatial covariance. Using covariance-corrected S/N values, the vorbin algorithm then creates bins with a target S/N while optimally preserving spatial resolution. The target S/N is at least 10 for all galaxies; some galaxies with longer exposures have higher overall S/N, so the target S/N per bin for these galaxies is higher (for reference, the maximum target S/N is 50 for galaxy 955106). Figure 5.2 illustrates the effect of binning without and with this covariance correction (bottom left and right panels, respectively); the unbinned white light image is shown for comparison in the top panel.
11h36m42.5s 42.0s 41.5s 26◦4304000
3500
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2500
KCWI RA
KCWIDEC
Original white-light image
0.0 0.1 0.2 0.3 0.4 0.5
11h36m42.5s 42.0s 41.5s
26◦4304000
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KCWI RA
KCWIDEC
Binned image (no covar correction)
0.0 0.1 0.2 0.3 0.4 0.5
11h36m42.5s 42.0s 41.5s
26◦4304000
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KCWI RA
KCWIDEC
Binned image (with covar correction)
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Figure 5.2: The effect of covariance correction on spatial binning of a stacked IFU data cube. Top: White-light image from stacked data cube of reines65. Bottom:
Same image, but spatially binned usingvorbinalgorithm to reach a target S/N = 15 without (left) and with (right) accounting for spatial covariance. All colorbars represent the average flux (erg s−1cm−2) per pixel (for top figure) or per bin (bottom figures).