Introducing piXedfit -- a Spectral Energy Distribution Fitting Code Designed for Resolved Sources
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Introducing piXedfit - a Spectral Energy Distribution Fitting Code Designed for Resolved Sources
Abdurro’uf , Yen-Ting Lin , Po-Feng Wu , and Masayuki Akiyama Institute of Astronomy and Astrophysics, Academia Sinica,11F of AS/NTU Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C. National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Astronomical Institute, Tohoku University, Aramaki, Aoba, Sendai 980-8578, Japan (Received ; Revised ; Accepted )
Submitted to ApJSABSTRACTWe present piXedfit , pixelized spectral energy distribution (SED) fitting, a Python package thatprovides tools for analyzing spatially resolved properties of galaxies using multiband imaging dataalone or in combination with integral field spectroscopy (IFS) data. piXedfit has six modulesthat can handle all tasks in the spatially resolved SED fitting. The SED fitting module uses theBayesian inference technique with two kinds of posteriors sampling methods: Markov Chain MonteCarlo (MCMC) and random densely-sampling of parameter space (RDSPS). We test the performanceof the SED fitting module using mock SEDs of simulated galaxies from IllustrisTNG. The SED fittingwith both posteriors sampling methods can recover physical properties and star formation histories ofthe IllustrisTNG galaxies well. We further test the performance of piXedfit modules by analyzing 20galaxies observed by the CALIFA and MaNGA surveys. The data comprises of 12-band imaging datafrom GALEX, SDSS, 2MASS, and WISE, and the IFS data from CALIFA or MaNGA. piXedfit can spatially match (in resolution and sampling) of the imaging and IFS data. By fitting only thephotometric SEDs, piXedfit can predict the spectral continuum, D n H α , and H β well. Thestar formation rate (SFR) derived by piXedfit is consistent with that derived from H α emission.The RDSPS method gives equally good fitting results as the MCMC and it is much faster than theMCMC. piXedfit is a versatile tool equipped with a parallel computing module for efficient analysisof large datasets, and will be made publicly available a) . Keywords: methods: data analysis – methods: statistical – galaxies: evolution – galaxies: fundamentalparameters INTRODUCTIONThe accumulated multiwavelength photometric andspectroscopic observations over the past decades haveplayed a crucial role in our current understanding ofgalaxy formation and evolution. To interpret the mul-tiwavelength data, modeling of the galaxy spectral en-ergy distribution (SED) is required. Motivated by suchneeds, stellar population synthesis modeling has beensystematically developed since the pioneering work by
Corresponding author: Abdurro’[email protected] a) https://github.com/aabdurrouf/piXedfit Tinsley (1972) and Searle et al. (1973). Since then,numerous efforts from various groups have been madeto improve the methods (Buzzoni 1989; Bruzual A. &Charlot 1993; Bruzual & Charlot 2003; Maraston 1998,2005; Conroy et al. 2009; Eldridge & Stanway 2009).Recently, extensive developments have been made to in-clude more physical components into the SED modeling,to account for the complexity of the physics underlyingthe SED of a galaxy. These components include nebularemission (e.g., Ferland et al. 1998, 2013), dust emission(Burgarella et al. 2005; Draine & Li 2007; da Cunhaet al. 2008; Groves et al. 2008; Noll et al. 2009; Lejaet al. 2017), dusty torus emission from an active galac-tic nucleus (AGN; e.g., Nenkova et al. 2008a; Stalevski a r X i v : . [ a s t r o - ph . GA ] M a r Abdurro’uf et al. et al. 2012), and synchrotron radio emission (e.g., Bo-quien et al. 2019).In parallel with the development of the SED modeling,the statistical method for comparison between the ob-served SED and model SED, the so-called SED fitting,has been extensively developed over the past few decades(see reviews by Walcher et al. 2011; Conroy 2013). Tra-ditionally, SED fitting was considered as an optimiza-tion problem, where χ minimization technique is usedto find a model that best reproduce the observed SED(e.g., Sawicki & Yee 1998; Arnouts et al. 1999; Cid Fer-nandes et al. 2005; Kriek et al. 2009; Sawicki 2012). Asthe number of parameters in the SED modeling becomeshigher (due to the incorporation of various physical com-ponents, as described above) which introducing more op-portunities of degeneracy among the parameters, we seethe emergence of the Bayesian inference technique. Thistechnique infers the parameters from posterior proba-bility distributions produced by taking into account thelikelihoods of all models. Pioneered by Kauffmann et al.(2003), the Bayesian framework for SED fitting has beenapplied widely in the literature (e.g., Burgarella et al.2005; Salim et al. 2007; da Cunha et al. 2008; Noll et al.2009; Boquien et al. 2019). Currently, a Bayesian infer-ence with state of the art posteriors sampling technique,such as the Markov Chain Monte Carlo (MCMC) andthe nested sampling techniques, has become a standardpractice in the SED fitting (e.g., Acquaviva et al. 2011;Serra et al. 2011; Johnson et al. 2013; Han & Han 2014;Chevallard & Charlot 2016; Calistro Rivera et al. 2016;Leja et al. 2017; Carnall et al. 2018; Zhou et al. 2020).Despite the fact that galaxies are extended objects,the majority of the studies over the past decades haveonly utilized their integrated light, particularly for SEDfitting; in the case of spectroscopic studies, the inte-grated spectrum of a galaxy is obtained with the single-fiber spectroscopy over a small diameter of the galaxy’scenter (e.g., Sloan Digital Sky Survey, SDSS, Galaxy andMass Assembly survey, GAMA, York et al. 2000; Driveret al. 2009, respectively). These observations have re-vealed many important evolutionary trends and correla-tions among physical properties of galaxies that shapedour current understanding of galaxy evolution.Despite the huge amount of information obtained fromthe above surveys, we have not made the full use of theavailable information, namely the omission of spatiallyresolved SED with which physical properties of spatialregions in the galaxy can be derived. As spatially ex-tended objects, galaxies have properties that vary acrosstheir bodies. The advent of the integral field spec-troscopy (IFS) surveys has revolutionized the studies ofgalaxy formation and evolution: in the local universe, we have SAURON (de Zeeuw et al. 2002), ATLAS (Cap-pellari et al. 2011), CALIFA (S´anchez et al. 2012), SAMI(Croom et al. 2012), and MaNGA (Bundy et al. 2015); inthe high redshifts, KMOS D (Wisnioski et al. 2015) andSINS/zC-SINF (F¨orster Schreiber et al. 2018). Thanksto these surveys, spatially resolved properties of galaxiesare recently being studied, allowing for a better under-standing of the galaxy evolution.While the SED fitting technique has been widely ap-plied to the integrated SEDs of galaxies over a widerange of redshifts, its potential for applications to thespatially resolved SEDs has only been explored by alimited number of studies. Abraham et al. (1999) didfitting of spectral synthesis models to spatially resolvedmulticolor photometry of 32 galaxies at 0 . < z < z < ∗ ) of 9 nearbygalaxies. Wuyts et al. (2012, 2013) applied the standardSED fitting technique to the spatially resolved broad-band SEDs (from the Hubble Space Telescope, HST)of 0 . < z < . ∗ , age, and dust attenuation as a functionof galactocentric radius, and measure structural param-eters of the galaxies.Recently, Sorba & Sawicki (2015, 2018) used multi-band images covering rest-frame ultraviolet (UV)–optical to conduct pixel-by-pixel SED fitting of 67nearby galaxies and 1222 galaxies in high redshifts (upto z ∼ .
5) to study the systematic effect introducedby the integrated SED fitting on the total stellar mass( M ∗ ) estimate. By comparing the total M ∗ from sum-ming up the spatially resolved mass estimates with thatobtained from the integrated SED fitting (i.e., spatially-unresolved M ∗ ), they found that the M ∗ can be severelyunderestimated using the integrated SED, especially onstar-forming galaxies. They argue that this systematiceffect is caused by the outshining effect by young stars,i.e., young stars (which have low M / L) are so brightsuch that their light dominates the galaxy’s SED in the patially resolved SED fitting with
P I X E D F I T / L) .In our previous studies (Abdurro’uf & Akiyama 2017,2018), we conducted spatially resolved SED fitting of93 local (0 . < z < .
02) and 152 high redshifts(0 . < z < .
8) massive disk galaxies to study the evo-lution of the spatially resolved star formation main se-quence (SFMS) and the radial trends of disk growth andquenching. Overall, we found that massive disk galaxiestend to build their stellar masses and quench their starformation activities in the inside-out fashion.Until recently, the wide area IFS surveys (mentionedpreviously) have been mostly targeting local galaxies be-cause such large surveys for high redshifts galaxies areprohibitively expensive. The spatially resolved SED fit-ting method can serve as a powerful alternative to study-ing the spatially resolved stellar population propertiesof galaxies across a wide range of redshifts, as shownby previous studies mentioned above. Some advantagesof this method over the IFS surveys are the following:(1) the current and future abundance of high spatialresolution and deep multiband imaging data, particu-larly those from space missions such as
Euclid , JWST ,and
Roman Space telescope , which allow us to performthis method to a large number of galaxies across widerange of redshifts, (2) the recent developments in SEDmodeling and fitting methods enable a robust and rapidestimation of galaxy properties, (3) the usage of a sin-gle method to study galaxies over a wide range of red-shift can reduce systematic biases (which would arisewhen different methods are used for different redshift)in the study of evolutionary trends of the galaxy prop-erties. Motivated by these, in this study, we develop piXedfit , pixelized SED fitting, a Python packagethat provides a self-contained set of tools for analyzingspatially resolved properties of galaxies from imagingdata as well as the combination of imaging data andIFS data.The structure of this paper is as follows. We de-scribe the data sets used for the analysis of this paperin Section 2. In Section 3, we explain the piXedfit design, including descriptions of 4 out of 6 modules.The description of the SED fitting approach and the2 modules associated with it is given in Section 4.In Section 5, we test the SED fitting performance of However, the discrepancy between the both total M ∗ estimatesis not observed by Wuyts et al. (2012) and Smith & Hayward(2018). Smith & Hayward (2018) used synthetic galaxy imagescovering FUV–FIR that are constructed by performing dust ra-diative transfer on a 3D hydrodynamical simulation of an isolateddisk galaxy. piXedfit using mock SEDs of the simulated galaxiesfrom the IllustrisTNG. In Section 6, we empirically test piXedfit modules using spatially resolved spectropho-tometric data of local galaxies. Finally, we summarizethe analysis of this paper in Section 7. As sections 2 to 4are primarily technical and describing the architectureof piXedfit , readers who are more interested in theperformance can start from section 5 while referring toTable 1.Throughout this paper, the cosmological parametersof Ω m = 0 .
3, Ω Λ = 0 .
7, and H = 70kms − Mpc − , theAB magnitude system, and the Chabrier (2003) initialmass function (IMF) are assumed. DATAIn the analysis throughout this paper, two kinds ofdata sets are used: imaging data set ranging from far-ultraviolet (FUV) to near-infrared (NIR) and the IFSdata. Each of the data sets is briefly described in thefollowing. 2.1.
Broad-band Imaging Data
GALEX
The Galaxy Evolution Explorer (GALEX; Martinet al. 2005) is a space mission equipped with a 0 . .
13 deg , a pixel res-olution of 1 . (cid:48)(cid:48) , and a point spread function (PSF) fullwidth at half-maximum (FWHM) of 4 . (cid:48)(cid:48) and 5 . (cid:48)(cid:48) inthe FUV and near-ultraviolet (NUV) bands (effectivewavelengths: 1538 . . σ limiting magnitudes in FUV (NUV)of those three survey modes are 19 . . . . . . SDSS
The SDSS (York et al. 2000) and its following surveysare providing the largest dataset combining imaging andspectroscopic data, using a dedicated 2 . u , g , r , i , and z ) with central wavelengths rangingfrom 3551 to 8932˚A and pixel resolution of 0 . (cid:48)(cid:48) . TheSDSS imaging is 95% complete to u = 22 . g =22 . r = 22 . i = 21 . z = 20 . . (cid:48)(cid:48) in the r -band (see Ross et al.2011). Abdurro’uf et al.
The Two Micron All Sky Survey (2MASS; Skrutskieet al. 2006) is an imaging survey of the whole sky in theNIR. The survey uses two 1 . J (1.24 µ m), H (1.66 µ m), and K s (2.16 µ m) bands. The imageproduct is resampled to 1 . (cid:48)(cid:48) pixel − . The point-sourcesensitivities at signal-to-noise ratio of S/N=10 are: 15 . .
1, and 14 . J , H , and K s , respectively. Theseeing is ∼ . − . (cid:48)(cid:48) (Skrutskie et al. 2006).2.1.4. WISE
The Wide-field Infrared Survey Explorer (WISE;Wright et al. 2010) mapped the whole sky in four in-frared bands: 3 .
4, 4 .
6, 12, and 22 µ m ( W W W W
4, respectively). In this paper, we use the imag-ing data product from the AllWISE data release. Thefour wavelength bands ( W W W
3, and W
4) havespatial resolutions PSF FWHM of 6 . (cid:48)(cid:48) , 6 . (cid:48)(cid:48) , 6 . (cid:48)(cid:48) , and12 . (cid:48)(cid:48) , respectively. The spatial sampling of the imagingproduct in the four wavelength bands is 1 . (cid:48)(cid:48) pixel − .WISE achieved 5 σ point source sensitivites better than0 .
08, 0 .
11, 1, and 6 . W W Integral Field Spectroscopy (IFS) Data
CALIFA
The Calar Alto Legacy Integral Field Area (CAL-IFA) survey (S´anchez et al. 2012) is an IFS survey de-signed to obtain spatially resolved spectra of around 600galaxies in the local universe (0 . < z < . . R ∼ − R ∼ − σ surface brightness limits of ∼ . − and ∼ . − , respec-tively (S´anchez et al. 2012). In the analysis of this paper,we use the combined data product so-called COMB datacubes from the DR3 release (S´anchez et al. 2016a). TheCOMB data product is a collection of data cubes thatcombines the spectra from the two observation setups.The COMB spectra cover 3701 − . ∼ . (cid:48)(cid:48) , with aspatial sampling of 1 . (cid:48)(cid:48) spaxel − .2.2.2. MaNGA
Mapping nearby Galaxies at Apache Point Observa-tory (MaNGA; Bundy et al. 2015), a part of SDSS IV(Blanton et al. 2017), is a wide area IFS survey target-ing ∼ ,
000 local galaxies at 0 . < z < .
15. TheMaNGA hexagonal fiber bundles make use of the BOSSspectrographs (Smee et al. 2013). The observed spec-tra cover 3600 − , R ∼ − . (cid:48)(cid:48) (Lawet al. 2015) and spatial sampling of 0 . (cid:48)(cid:48) spaxel − . Inthe analysis of this paper, we use the LOGCUBE datacubes from the data reduction pipeline (DRP; Law et al.2016). The data cubes reach a typical 10 σ limiting con-tinuum surface brightness of 23 . − in a five-arcsecond-diameter aperture in the g band (Law et al.2016). Detailed descriptions on the survey design andobserving strategy are given in Law et al. (2015), Yanet al. (2016), and Wake et al. (2017). PIXEDFIT
DESIGN piXedfit is designed to be modular, and each mod-ule can be run independent of each other. Due toits modularity, users can use a particular module in piXedfit without the need of using other modules.For instance, it is possible to use the SED fitting mod-ule to fit integrated SED of a galaxy (not limited tospatially resolved SED) without the need of using theimage processing module. This way piXedfit canbe beneficial for various applications. Figure 1 showsthe design of piXedfit . piXedfit has six mod-ules: (1) piXedfit_images is for image processing,(2) piXedfit_spectrophotometric is for spatiallymatching multiband imaging data with IFS data to ob-tain spatially resolved spectrophotometric SEDs of agalaxy, (3) piXedfit_bin is for pixel binning to maxi-mize S / N ratio, (4) piXedfit_model is for generatingmodel SEDs, (5) piXedfit_fitting is for perform-ing the SED fitting, and (6) piXedfit_analysis isfor visualization of fitting results. In this section we de-scribe the first four modules, leaving the last two mod-ules to Section 4.3.1. piXedfit images : Image Processing
In the pixel-by-pixel SED fitting process, it is very im-portant to make sure that the multiband images are allmatched to the same spatial resolution and spatial sam-pling, so that a given pixel represents the same region onthe sky in all the images used. Such an image processing patially resolved SED fitting with
P I X E D F I T Figure 1.
The piXedfit design. piXedfit has six modules: (1) piXedfit images is for image processing, (2) piXedfit spectrophotometric is for spatially matching of multiband imaging data and IFS data to obtain spatially re-solved spectrophotometric SEDs of a galaxy, (3) piXedfit bin is for pixel binning, (4) piXedfit model is for generatingmodel SED, (5) piXedfit fitting is for performing SED fitting, and (6) piXedfit analysis is for visualizing SED fittingresults.
Abdurro’uf et al. task in piXedfit is done by the piXedfit_images module. The piXedfit_images module is a Pythonscripting module that combines together various imageprocessing functions in
Astropy (Astropy Collabora-tion et al. 2013), Photutils (Bradley et al. 2019), and SExtractor (Bertin & Arnouts 1996) such that an im-age processing task for any combination of imaging datacan be done automatically. The user only need to spec-ify a set of photometric bands, the names of input FITSfile for the science image associated with each band, thenames of input FITS file for the variance image (whichis the square root of an uncertainty image) associatedwith each band, and coordinate (right ascension, RA,and declination, DEC) of the target galaxy. Using aspecific function in piXedfit_images , the varianceimage is calculated for each band . The current ver-sion of piXedfit can perform image processing to thefollowing list of imaging data: GALEX, SDSS, 2MASS,WISE, Spitzer , Herschel , and Hubble Space Telescope(HST). The workflow of image processing is shown inFigure 1. In the following, each of the image processingtasks will be described.3.1.1.
Background Subtraction In piXedfit_images , the background estimationis done using the Background2D function from
Photutils . The
Background2D function estimatesthe background by first dividing an image into certainnumber of grids and then, for each grid, backgroundlevel is estimated using the sigma-clipping method. In piXedfit_images , grid size is required as an input.The background subtraction is only applied to the sci-ence images. After the background subtraction process,the background and RMS images are stored into FITSfiles. 3.1.2.
PSF Matching
In order to obtain accurate multiwavelength photo-metric SED from a set of multiband images, it is im-portant that all the images are brought to the samePSF size. Commonly, PSF matching between two im-ages is done by convolving the higher resolution image(i.e., smaller PSF size) with a pre-calculated kernel. Thematching kernel between the two PSFs is derived fromthe ratio of Fourier transforms (see e.g., Gordon et al.2008; Aniano et al. 2011). https://photutils.readthedocs.io/en/stable/ The description on how to estimate the uncertainty of pixel valueand derive the variance image are described at https://pixedfit.readthedocs.io/en/latest/list imaging data.html
Previous studies have constructed convolution kernelsfor matching the PSFs of imaging data from varioustelescopes including both space-based and ground-basedones. Gordon et al. (2008) constructed convolution ker-nels for matching the PSFs of the
Spitzer /IRAC and
Spitzer /MIPS images . Aniano et al. (2011) constructedconvolution kernels for matching the PSFs of imagingdata from various space-based and ground-based tele-scopes that includes GALEX, Spitzer , WISE, and
Her-schel . Besides that, Aniano et al. (2011) also con-structed convolution kernels for some analytical PSFsthat includes Gaussian, sum of Gaussians, and Moffat .The analytical PSF forms are expected to represent thenet (i.e., effective) PSFs of ground-based telescopes.We use convolution kernels from Aniano et al.(2011) for the PSF matching process in the piXedfit_images module. Since the PSFs of SDSSand 2MASS are not explicitly covered in the list ofPSFs analyzed by Aniano et al. (2011), to find the an-alytical PSFs representative of those imaging data, weconstruct empirical PSFs of the 5 SDSS bands and 32MASS bands, then compare them with the analyticalPSFs of Aniano et al. (2011). We present this analysisin Appendix A. In short, we find that the empiricalPSFs of SDSS u , g , and r bands are best represented bydouble Gaussian with FWHM of 1 . (cid:48)(cid:48) , while the otherbands (i.e., i and z ) are best represented by doubleGaussian with FWHM of 1 . (cid:48)(cid:48) . The two Gaussian com-ponents have a fix center, the relative weights of 0 . .
1, and the FWHM of the second component is twicethat of the first (Aniano et al. 2011). For 2MASS, allthe three bands ( J , H , and K s ) are best represented byGaussian with FWHM of 3 . (cid:48)(cid:48) . For consistency, we usethose analytical PSFs to represent the PSFs of SDSSand 2MASS and use the convolution kernels associatedwith them whenever needed .In piXedfit_images , the convolution of an imagewith a kernel is done using the convolve_fft functionin Astropy . Before convolving an image with a ker-nel, the kernel should be spatially resampled to matchthe spatial sampling of the image, which is done usingthe resize_psf function in
Photutils . Originally,the kernels provided by Aniano et al. (2011) are all re- Convolution kernels are available at https://irsa.ipac.caltech.edu/data/SPITZER/docs/dataanalysistools/tools/contributed/general/convkern/ ∼ ganiano/Kernels.html More information on the kernels and demonstration of theirperformaces can be seen at https://pixedfit.readthedocs.io/en/latest/list kernels psf.html patially resolved SED fitting with
P I X E D F I T . (cid:48)(cid:48) pixel − . The PSF matching process isdone to both science images and variance images.3.1.3. Spatial Resampling and Reprojection
After the PSF matching, all images are brought toa uniform spatial sampling and reprojection. The fi-nal spatial sampling is chosen to be the lowest spa-tial sampling (i.e., largest pixel size) among the imag-ing data being analyzed. The spatial resampling andreprojection task in piXedfit_images is done usingthe reproject_exact function from the reproject package (Robitaille 2018). reproject_exact repro-jects an image to a new projection using the flux-conserving spherical polygon intersection method. Be-cause the reprojection basically includes regridding andinterpolation, the pixel value of the image should be ina surface brightness unit, not in a flux unit. Thereforebefore reprojection and resampling, the images are con-verted into surface brightness whenever needed. If theoriginal unit of an image is in flux, it will be reconvertedto flux unit after the resampling process.The next step is cropping around the target galaxy.This is done using the wcs_world2pix and
Cutout2D functions available in
Astropy . The size of the finalcropped images, which retain correct WCS information,can be defined by the user. The spatial resampling, re-projection, and cropping are done to the science imagesand the variance images.3.1.4.
Image Segmentation and Defining Galaxy’s Regionof Interest In piXedfit_images , image segmentation using SExtractor is done to obtain an initial estimate forthe region of the target galaxy. The segmentation isdone in all imaging bands (only the science images), thensegmentation maps from all bands are merged (i.e., com-bined) to get a single segmentation map from which thegalaxy’s region will be determined. Due to the emer-gence of the background noise, the segmentation map ofa galaxy can have an irregular (i.e., filamentary) struc-ture at the outskirt. To remove such feature, an ellipticalaperture cropping is applied to the galaxy’s segmenta-tion region. Ellipticity, position angle, and maximumradius (along the semi-major axis) for the elliptical aper-ture cropping can be specified when providing input tothe piXedfit_images module. If those parametersare not provided by the user, elliptical isophote fittingwill be done to the final stamp image of a band around As it is often times difficult to define the boundary of a galaxy,here we refer to the region of the target galaxy to be fit simplyas the “region” of the galaxy. the middle of the rest-frame optical (such as r band)using the Ellipse class in
Photutils . In
Ellipse ,the isophotes in the galaxy’s image are measured usingan iterative method described in Jedrzejewski (1987).From the set of isophotes (as a function of radius) pro-duced by
Ellipse , the ellipse closest to the desiredmaximum radius is chosen.3.1.5.
Extracting SEDs of Pixels
The tasks described above give the final stamps ofreduced science and variance images, and the pixel co-ordinates associated with the galaxy’s region of interest.The next step is calculating fluxes and flux uncertaintiesof pixels within the galaxy’s region in the multiband im-ages. The end product of this process is the photometricSED of every pixel of interest. The conversion of pixelvalue into flux density unit of erg s − cm − ˚A − (whichis the default flux unit of data product produced by piXedfit_images ) depends on the unit of the pixelvalue in the original image. The flux uncertainty of apixel is obtained by first taking square root of the pixelvalue in the variance image then convert it into the fluxdensity unit.The pixel values of the imaging data used in our anal-ysis have a variety of units. To convert the pixel valueof an image to flux density in erg s − cm − ˚A − andestimate the uncertainty of the pixel value, we followthe relevant information from the literature and doc-umentation files from the survey’s website from whichthe imaging data were obtained. The variance imagesassociated with the science images that are input to piXedfit_images (see Section 3.1) are constructedfollowing that information .The next step is to correct the pixel-wise SEDs forthe foreground Galactic dust extinction. For this, weestimate E ( B − V ) from the reddening ( A λ ) in theSDSS bands, obtained from the NASA/IPAC Extra-galactic Database (NED) which is based on the mapby Schlafly & Finkbeiner (2011), recalibration fromSchlegel et al. (1998). Then we use the Fitzpatrick(1999) with R V = 3 . piXedfit_images module, this The unit of pixel value in imaging data that can be analyzed withthe current version of piXedfit , and how to convert the pixelvalue into flux and estimate the flux uncertainty are described athttps://pixedfit.readthedocs.io/en/latest/list imaging data.html https://ned.ipac.caltech.edu/ Abdurro’uf et al. step is done manually using a specific function. The useronly need to input central coordinate and an estimateof the radius (in pixels) of each star.The derived maps of fluxes and flux uncertainties (inmultiple photometric bands) of the target galaxy arethen saved into one multi-extension FITS file. Figure 2shows an example of the maps of multiband fluxes pro-duced by the piXedfit_images module. The targetgalaxy for this example is NGC 309. Imaging data over12 bands ranging from the FUV to W W W
2, which are deeper than the 2MASS bands)although their spatial resolution is lower than UV andoptical bands. The inclusion of the WISE bands canprovide stronger constraint in the NIR regime.3.2. piXedfit spectrophotometric : ExtractingSpatially Resolved Spectrophotometric SEDs of aGalaxy
In the analyses of the integrated SED of a galaxy(i.e., treating the galaxy as one object), there have beenseveral attemps in combining rest-frame optical spec-tra (particularly covering 4000˚A break) and broad-bandphotometry covering wider wavelength range into a so-called spectrophotometric SED fitting (see e.g., Newmanet al. 2014; Dressler et al. 2018; Morishita et al. 2019;Abramson et al. 2020; Chen et al. 2020). By combin-ing the rest-frame optical spectrum and the broad-bandphotometry, it is expected that the constraining powerin the SED fitting can be enhanced and potentially breakthe existing degeneracies among the parameters in thefitting process.The availability of the FUV–NIR broad-band imagingand the IFS datasets for local galaxies (thanks to CAL-IFA, MaNGA, and SAMI surveys) give us opportuni-ties to conduct the spatially resolved spectrophotomet-ric SED analyses. However, for a self-consistent analysiswe need to spatially match (in spatial resolution andsampling) the broad-band imaging and IFS datasets. piXedfit provides a new capability of combining thebroad-band imaging data with the IFS data to ob-tain spatially resolved spectrophotometric SEDs of agalaxy. The tasks featuring this process is in the module piXedfit_spectrophotometric . As for the cur-rent version, the piXedfit_spectrophotometric module can only analyze the combination of broad-band imaging data from the GALEX, SDSS, 2MASS,and WISE, and the IFS data from the CALIFA/COMBand MaNGA/DRP. The final product of this module is a data cube that contains spatially-matched pixel-wisespectrophotometric SEDs of a galaxy.
Our analysis pre-sented here is the first attempt of this kind .To spatially match the three dimensional IFS datawith the broad-band imaging data, first, a two dimen-sional image (i.e., the imaging layer) of every wavelengthgrid in the IFS data is made. Before creating images outof the IFS data, the spectra are smoothed by convolv-ing them with a Gaussian kernel with a sigma valuefollowing that of the spectral resolution of the IFS data( ∼ . ∼ . W
2, the final producthas the spatial resolution of the W . (cid:48)(cid:48) FWHM) andthe spatial sampling of the FUV/NUV (1 . (cid:48)(cid:48) pixel − ).The PSF matching for an imaging layer is done by con-volving the image with a pre-calculated kernel. Since theeffective PSFs of MaNGA and CALIFA have FWHM of2 . (cid:48)(cid:48) , we use corresponding convolution kernel from Ani-ano et al. (2011). The convolution kernel was created formatching a Gaussian PSF with FWHM of 2 . (cid:48)(cid:48) to thePSF size of W
2. We have compared the reconstructedPSFs of MaNGA DRP data cube in the g , r , i , and z bands (provided in the FITS file containing the datacube of one galaxy) with the Gaussian PSF with FWHMof 2 . (cid:48)(cid:48) from Aniano et al. (2011). The MaNGA empiri-cal PSFs match well with the Gaussian PSF in all thesebands.After PSF matching, all the imaging layers are spa-tially resampled and reprojected to match the spatialsampling and projection of the broad-band imaging datacube produced by piXedfit_images . This task isdone in the same way as that for the images process-ing, described in Section 3.1.3. The next step is cor-recting the spatially resolved spectra for the foregroundGalactic dust extinction. This step is only done for theMaNGA data cubes (Law et al. 2016), as such a correc-tion has been applied to the CALIFA cubes (S´anchezet al. 2016a). For this task, we use the E ( B − V )value obtained from the header (keyword: EBVGAL ) ofthe MaNGA DRP FITS file and then apply the dust ex-tinction correction adopting the Fitzpatrick (1999) red-dening law with R V = 3 . patially resolved SED fitting with P I X E D F I T Figure 2.
Example of the maps of multiband fluxes produced by the piXedfit images module. The target galaxy in thisexample is the NGC 309. The left most panel in the first row shows gri composite image. The 12 panels in the second to fourthrow show maps of flux at 12 wavelength bands from the FUV to W
2. The 12-band images are brought to the spatial resolutionof the W W vary across wavelength in an SED of a pixel, there arealso variations of the flux offset spatially. To get a sim-plified pattern of the variation of the flux offsets, first,we reconstruct g , r , and i ( g and r ) images from thepost-processed IFS data from MaNGA (CALIFA) byconvolving them with broadband filters. We then com-pare the reconstructed images with the real images. ForMaNGA, the mean log( f obs /f recons ) in g , r , and i are − . ± . − . ± . − . ± . f obs /f recons ) in g and r are − . ± .
141 and − . ± . f obs and f recons are flux from real image and thereconstructed image, respectively. These values are de-rived using a sample of 20 galaxies that will be used inthe analysis of Section 6.The mismatch between spectrum and photometricSED can be caused by at least two factors: systemat-ics in the data processing (PSF matching, spatial re-sampling, reprojection, etc.) of the broad-band imaging data and the IFS data, and the uncertainty in the fluxcalibration of the photometric and the IFS data. For de-tailed descriptions on the flux calibration in the MaNGAand CALIFA surveys, please refer to Yan et al. (2016)and Garc´ıa-Benito et al. (2015), respectively.In order to overcome the photometry–spectroscopyoffset, we multiply the spectrum with a wavelength-dependent smooth factor obtained from a third-orderLegendre polynomial function fit such that the spec-trum normalization become consistent with the normal-ization of the photometric SED. The polynomial orderof 3 is low enough to prevent the introducing of spec-tral breaks or artificial features to the spectrum. Tofind the smooth multiplicative factor, we first obtaina model spectrum that best describes the photometricSED using a χ minimization technique applied to a setof pre-calculated model SEDs (to be described in Sec-tion 4.2.2), then fit a third-order Legendre polynomial tothe ratio between the model spectrum and the observed0 Abdurro’uf et al. (IFS) spectrum. This method adopts the typical tech-nique used in the spectrum fitting that uses multiplica-tive polynomial function of a certain order ( ∼ −
8) tomake a model spectrum template fit the overall spectralshape of the observed spectrum (see e.g., Kelson et al.2000; Koleva et al. 2009; Emsellem et al. 2004; Newmanet al. 2014; Cappellari 2017; Westfall et al. 2019; Belfioreet al. 2019). Figure 3 shows examples of spectrophoto-metric SED data cubes of the galaxy NGC 309, whichis observed by the CALIFA survey (first row), and an-other galaxy, PLATE-IFU:8934-12702, observed by theMaNGA survey (second row). Regions in the galaxiesthat are covered by the IFU fiber bundle are shown bythe transparent hexagonal regions overlaid on top of the gri composite images (left panel in each row). Outside ofthese regions, we still have spatially resolved broad-bandphotometry data. In each row, the right panel showsSEDs of 4 randomly chosen pixels — three spectropho-tometric SEDs and one photometric SED. The gri com-posite images are made using the make_lupton_rgb function in
Astropy (Lupton et al. 2004).
Figure 3.
Examples of the spectrophotometric SED datacubes obtained with the piXedfit spectrophotometric module. The two galaxies are: NGC 309 (first row) which isobserved by the CALIFA survey, and a galaxy with PLATE-IFU:8934-12702 (bottom row) observed by the MaNGA sur-vey. The region covered by the IFU fiber bundle is plottedtransparently on top of the gri composite image (left panel ineach row). In each row, the right panel shows SEDs of 4 ran-domly chosen pixels — three spectrophotometric SEDs andone photometric SED (shown by the purple colored points). piXedfit bin : Pixel Binning
In most cases, fluxes measured in individual pixelshave a low S / N ratio. It is also common to find pix-els with missing or negative fluxes. In order to get anaccurate inference of the parameters in the SED fitting, typically one needs an observed SED with sufficient S / Nratio. For this reason, we do not apply the SED fittinganalysis to pixel-wise SED. Instead, we bin the data lo-cally before conducting further analysis to the data.Previous studies have applied pixel binning in spa-tially resolved SED fitting analysis (e.g., Wuyts et al.2013; Belfiore et al. 2019; S´anchez et al. 2018). A pop-ular pixel binning scheme is the Voronoi binning byCappellari & Copin (2003), who showed that, with theVoronoi tessellation technique, the bins can be made as‘compact’ as possible, no overlapping with each other,and having similar S / N ratio (in a particular band).In Abdurro’uf & Akiyama (2017), we developed a newpixel binning scheme that takes into account of the sim-ilarity in the SED shape among pixels. This new cri-teria is important especially for the spatially resolvedSED fitting analyses, because it is expected to preserveany important information from the SED at the pixelscale. While pixel binning is done to achieve a certainminimum S/N, at the cost of degrading the spatial res-olution, we can still preserve important information inthe SED at the pixel scale with this binning scheme.In the conventional pixel binning schemes that do notconsider the similarity of the SED shape, it is possiblethat neighboring pixels which have different SED shapes(likely having different properties) are binned together.This could smooth out the spatial variation of the stellarpopulation properties. piXedfit_bin is a module designed for performingsuch a binning scheme, and is built upon what was de-veloped in Abdurro’uf & Akiyama (2017). There arefour requirements in the pixel binning scheme: (1) prox-imity, such that only neighboring connected pixels arebinned together, (2) similarity of SED shape, (3) S/Nthreshold in each band, and (4) smallest diameter ofa bin ( D min , bin in pixel). The last requirement is anew parameter introduced with the current version of piXedfit_bin . This parameter prevents the binningprocess from picking a single bright pixel as a bin. Insome cases, a single bright pixel (typically around thecentral region) can exceed the S/N threshold such thatfurther binning with other pixels is not needed. Thesmallest diameter of the bin can be thought of as theFWHM of the PSF although the user is free to definethe diameter.The pixel binning scheme adopted in piXedfit_bin is a simple empirical one. Briefly speaking, a spatial binis obtained by first selecting a brightest pixel in a refer-ence band which is defined by the user (a band aroundthe middle of the rest-frame optical regime is recom-mended, e.g., the r band). Then pixels enclosed withina diameter of D min , bin from the brightest pixel are joined patially resolved SED fitting with P I X E D F I T dr = 2 pixels and pixels within the newannulus are examined to see if they have a similar SEDshape as the brightest pixel. Pixels that have similarSED shape are added into the bin and the total S/Nat each band is checked. If the total S/N in each bandis above the S/N threshold, the expansion of the bin isterminated. Otherwise, the expansion is continued untilthe S/N threshold at each band is reached. To proceedto the next bin, the brightest pixel among the remain-ing pixels is selected as the starting pixel, and the sameprocedure is applied again.The above procedure is applied until no more bins canbe made with the remaining pixels. In most cases, pixelsaround the outskirt are left without being binned. Thislikely caused by the insufficient number of those out-skirt pixels (which typically have low S/N) left over bythe previous binning process that makes binning some ofthem that have similar SED (within a certain χ , to bedescribed later) cannot reach the required S/N thresh-old. In this case, all the remaining pixels are finallybinned into one bin.The similarity of SED shape of a pixel with index of m to that of the brightest pixel with index of b is evaluatedwith the following χ formula χ = (cid:88) i ( f m,i − s mb f b,i ) σ m,i + σ b,i . (1) i in the above equation represents photometric band,and f m,i and f b,i are i -th band flux of a pixel m and b ,respectively. σ m,i and σ b,i are i -th band flux uncertaintyof the pixel m and b . s mb is a scaling factor that bringthe two SEDs into a similar normalization, and it canbe calculated using s mb = (cid:80) i f m,i f b,i σ m,i + σ b,i (cid:80) i f b,i σ m,i + σ b,i . (2)If χ is smaller than a certain value ( χ , bin which isdefined by the user), the pixels m and b are consideredto have a similar SED shape.Figure 4 shows two pixel binning results for the NGC309 obtained with binning requirements that only dif-fer in S / N thresholds for the three 2MASS bands. Thepixel binning results in the top and bottom panels use2MASS S / N thresholds of 1 and 3, respectively. TheS / N threshold for the rest of the photometric bands isset to 10 (see Figure 2 for the set of the photometric bands). The other requirements are the same for thetwo binning: D min , bin of 4 pixels and reduced χ , bin limit of 3 . / N ratios in the FUV and J of the original pix-els and bins are shown on the right side of each panel.The blue lines show S / N thresholds. The pixel binningscheme is able to meet the minimum S / N requirement.A general trend is that the bin size increases with radiusfrom the galaxy’s center, which can be understood be-cause the S / N of pixels decreases with radius, and thusmore pixels are needed in a bin to reach the S / N thresh-old. In this example, the 2MASS bands determine theoverall result of the pixel binning because they are theshallowest (i.e., having lowest S / N) among the photo-metric bands used in this analysis. Due to the similaritySED shape requirement, the pixel binning map roughlyreconstruct the spiral arms structure (where young stel-lar populations are), especially in the first binning anal-ysis (top left panel).For binning a spectrophotometric data cube, we usethe pixel binning map obtained with multi-band images(described above) as a reference to bin the spectropho-tometric SEDs of pixels, so that the spectroscopy andphotometry of a bin are consistent. For a bin in whichsome of the member pixels do not have spectroscopicSED, we only assign spectrophotometric SED to a binin which at least 90% of the member pixels have spectro-scopic SED. The derived spatial binning map togetherwith the fluxes and flux uncertainties are then saved intoa multi-extension FITS file.3.4. piXedfit model : Generating Model SEDs piXedfit_model is a module designed for gener-ating a model SED of a Composite Stellar Population(CSP) from a given set of input parameters.3.4.1.
Generating Rest-frame Model Spectra
For generating model spectra, the Flexible StellarPopulation Synthesis (
FSPS ) package is used (Con-roy et al. 2009; Conroy & Gunn 2010). For interfaceto the Python environment, python-fsps packageis used (Foreman-Mackey et al. 2014). The FSPS pack-age provides a self-consistent modeling of galaxy’s SEDthrough a careful modeling of the physical componentsthat produce the total luminosity output of a galaxy.Those components consist of stellar emission, nebularemission, dust emission, and emission from the dustytorus heated by the AGN. We refer reader to Conroyet al. (2009), Conroy & Gunn (2010), and Leja et al. https://github.com/cconroy20/fsps http://dfm.io/python-fsps/current/ Abdurro’uf et al.
Figure 4.
Results of pixel binning for NGC 309 obtainedwith the piXedfit bin module. The top and bottom pan-els show results of pixel binning with requirements that onlydiffer in the S / N thresholds for the three 2MASS bands.The top (bottom) panel uses S / N thresholds of 1 (3) forthe 2MASS bands. The two pixel binning use the same S / Nthresholds of 10 for all other bands. The other requirementsare the same for the two binning: D min , bin of 4 pixels, re-duced χ , bin limit of 3 . (2017, 2018) for detailed description of the SED model-ing within the FSPS . For efficiency, we do not describein detail the ingredients of the SED modeling in thispaper but present the parameters in the SED modelingand fitting in Table 1 .In generating spectra of the Simple Stellar Population(SSP), piXedfit_model uses an option in the FSPS that allows interpolation of SSP spectra between the Z grids available in the isochrone and spectral libraries.The nebular emission modeling uses the CLOUDY code(Ferland et al. 1998, 2013) which was implemented inthe
FSPS by Byler et al. (2017). For the dust attenu-ation modeling, piXedfit_model allows two options:Calzetti et al. (2000) and the two-component dust modelof Charlot & Fall (2000). The dust emission modeling in
FSPS assumes the energy balance principle, where theamount of energy attenuated by the dust is equal to the A more detailed descriptions of the ingredients in the SED mod-eling and the parameters associated with it are available at https://pixedfit.readthedocs.io/en/latest/ingredients model.html amount of energy re-emitted in the infrared (da Cunhaet al. 2008).
FSPS uses the Draine & Li (2007) dustemission templates to describe the shape of the infraredSED. For the modeling of emission from the dusty torusheated by the AGN,
FSPS uses AGN templates from theNenkova et al. (2008a,b)
CLUMPY models.Due to the rare availability of the high spatial resolu-tion of imaging data in the infrared, the dust emissionand AGN dusty torus emission components are not ap-plicable in most of the spatially resolved SED fitting im-plementation. We still include dust emission and AGNdusty torus emission in the piXedfit_model becausethis module together with piXedfit_fitting can beused for fitting an integrated SED of a galaxy, not lim-ited to the spatially resolved SED. In case the sufficientlyhigh spatial resolution infrared imaging data is availableand the AGN component is necessary in the SED mod-eling, it is possible to include the AGN component to fitonly the SED of the central bin of a galaxy. Using the D min , bin parameter in the pixel binning (see Section 3.3),the minimum diameter of a bin can be set to be similarto the PSF FWHM of the images (which is implementedin the binning result that is shown in the top left panelof Figure 4). Thus, the central bin always larger thanthe PSF size, which supposed to enclose the AGN dustytorus component in the galaxy.Figure 5 shows an example of rest-frame modelspectrum (in black color) generated using the piXedfit_model module. The model spectrum isbroken down into its components: stellar emission (or-ange color), nebular emission (blue color), AGN dustytorus emission (green color), and dust emission (redcolor). Please refer to the caption for the values of theparameters used to generate the model spectrum.3.4.2. Choices for the Star Formation History (SFH)
In SED fitting, the assumed SFH is one of the fun-damental components yet difficult to constraint. As afundamental component, the assumed SFH and associ-ated priors are very influential to the inferred physicalproperties of galaxies, such that the robustness of theinferred parameters is dependent on whether or not theassumed SFH is flexible enough to reflect the true SFHof the galaxies (see e.g., Lee et al. 2009; Maraston et al.2010; Micha(cid:32)lowski et al. 2012, 2014; Conroy 2013; Iyer& Gawiser 2017; Carnall et al. 2019; Leja et al. 2019a;Lower et al. 2020).The recent developments in SED fitting enable theinference of SFH (i.e., SFH is not only an assumptionin the fitting). There have been many attempts thattry to infer SFH of galaxies using SED fitting (e.g., Dye2008; Smith & Hayward 2015; Pacifici et al. 2016; Iyer patially resolved SED fitting with
P I X E D F I T Figure 5.
Example of a rest-frame model spectrum (blackline) generated with piXedfit model . The decompositionof the spectrum to its components is shown with different col-ors: orange (stellar emission), cyan (nebular emission), red(dust emission), and green (AGN dusty torus emission). Themodel spectrum is generated assuming delayed tau SFH withlog( τ [Gyr]) = 0 . sys [Gyr]) = 0 . Z/Z (cid:12) ) = − . τ = 1 . M ∗ /M (cid:12) ) = 10 . f AGN ) = − . τ AGN ) = 0 . Q PAH ) = 0 . U min ) = 0 . γ e ) = − . & Gawiser 2017; Iyer et al. 2019; Carnall et al. 2018;Dressler et al. 2018; Leja et al. 2019a; Morishita et al.2019). In terms of the SFH modeling approach, theSED fitting techniques can be classified into two maincategories: parametric and non-parametric SFH. Theformer assumes a functional form for the SFH (e.g., Han& Han 2014; Carnall et al. 2018; Boquien et al. 2019;Zhou et al. 2020), while the latter do not, instead thelook-back time (i.e., stellar ages) is gridded and the SFRof each time grid is let free in the fitting (e.g., VESPA ,Tojeiro et al. 2007; Dressler et al. 2016; prospector ,Leja et al. 2017; Chauke et al. 2018; gsf , Morishitaet al. 2019;
Dense Basis , Iyer & Gawiser 2017, Iyeret al. 2019), or another way is using a set of SSPs withvarious ages and metallicities to fit the observed SED(typically a spectrum, e.g.,
STARLIGHT , Cid Fernandeset al. 2005;
STECMAP , Ocvirk et al. 2006;
FIREFLY ,Wilkinson et al. 2017).The parametric SFH approach has the advantage ofhaving fewer numbers of free parameters involved in thefitting and unlimited stellar age sampling (i.e., time res-olution in the SFH) compared to the non-parametricapproach. The non-parametric approach is expected tobe more flexible in reflecting the real SFH of galaxies(compared to the parametric one). However, this ap-proach has the disadvantage of the cruder sampling ofstellar ages and possibly complex degeneracies in thefitting due to large numbers of parameters involved.Recently, Carnall et al. (2018) have shown that us-ing the double power law SFH model can recover SFHs of simulated galaxies from the
MUFASA suite of cosmo-logical hydrodynamical simulations. The double powerlaw form has also been applied to fit the evolution ofthe cosmic SFR density (Behroozi et al. 2013). An-other study by Diemer et al. (2017) showed that thelog-normal SFH model can produce good fits to SFHsof simulated galaxies from the cosmological simulationIllustris. In piXedfit_model , we adopt the paramet-ric SFH approach, with 5 choices: exponentially declin-ing (i.e., tau model), delayed tau, log-normal, Gaussian,and double power law SFHs . The double power lawSFH has the following form, SF R ( t ) ∝ (cid:34)(cid:18) tτ (cid:19) α + (cid:18) tτ (cid:19) − β (cid:35) − , (3)where α and β are the falling slope, and the rising slope,respectively. The τ parameter controls the peak time.The t in the above equation represent the time since thestart of star formation (i.e., age of the system, age sys ).3.4.3. IGM Absorption, Cosmological Redshifting, andIntegrating through Photometric Filters
The rest-frame model spectra generated in the pre-vious step are then attenuated further to accountfor the absorption due to the intergalactic medium(IGM) between the galaxy and the observer. The piXedfit_model has two options for the IGM absorp-tion: Madau (1995) and Inoue et al. (2014). The effectof cosmological redshifting and dimming is then appliedto the model spectra. This will transform the spectra(that are still in unit of luminosity density, L λ ) into theobserver frame flux density ( f λ ). For this operation,redshift information of the galaxy is needed. However,if the redshift is unknown, it will be a free parameter inthe fitting. The calculation of the luminosity distanceuses the cosmology package in the Astropy . Thelast step in generating model photometric SEDs is toconvolve the model spectra with the set of filter trans-mission functions. The current vesion of piXedfit hasa library of transmission functions for 163 photometricfilters of ground-based and space-based telescopes. Theuser can also add a filter transmission function using aspecific function in piXedfit .Please refer to Table 1 for a compilation of the param-eters involved in the SED modeling and fitting. SED FITTING APPROACH IN
PIXEDFIT The functional forms of the SFH models are described in de-tailed at https://pixedfit.readthedocs.io/en/latest/ingredientsmodel.html Abdurro’uf et al.
Table 1.
Description for the parameters involved in the SED modeling and fittingParameter Description M ∗ Stellar mass Z Stellar metallicity t Evolving age (age sys ) of the stellar population τ A parameter in the SFH that controls the duration of star formation T A parameter in the log-normal and Gaussian SFHs that controls the peak time α A parameter in the double power law SFH that controls the slope of the falling star formation episode β A parameter in the double power law SFH that controls the slope of the rising star formation episodeˆ τ Dust optical depth of the birth cloud in the Charlot & Fall (2000) dust attenuation lawˆ τ Dust optical depth of the diffuse ISM in the Calzetti et al. (2000) and Charlot & Fall (2000) dust attenuation laws n Power law index in the dust atttenuation curve for the diffuse ISM in the Charlot & Fall (2000) dust attenuation law U Ionization parameter in the nebular emission modeling U min Minimum starlight intensity that illuminate the dust γ e Fraction of total dust mass that is exposed to this minimum starlight intensity Q PAH
Fraction of total dust mass that is in the polycyclic aromatic hydrocarbons (PAHs) f AGN
AGN luminosity as a fraction of the galaxy bolometric luminosity τ AGN
Optical depth of the AGN dusty torus
The SED fitting in piXedfit is done by piXedfit_fitting module. This module can per-form SED fitting to a photometric SED as well as aspectrophotometric SED. The SED fitting approachadopted in piXedfit is described in the followingsections. 4.1.
Bayesian Inference Method
The piXedfit_fitting module uses the Bayesianinference technique for estimating the underlying pa-rameters of a galaxy’s SED. Two important compo-nents in the Bayesian inference process are the likelihood(i.e., P ( X | θ ), which is the probability of observing thedata X given the model θ ) and prior (i.e., P ( θ ), whichis the hypothesis on the probability of model θ beforefitting with the data). In SED fitting, the likelihood iscommonly given by the Gaussian function because of theassumption of a Gaussian form of noise. The Gaussianlikelihood form is used by the majority of Bayesian SEDfitting implementation, e.g., Kauffmann et al. (2003), MAGPHYS (da Cunha et al. 2008),
BayeSED (Han &Han 2014),
BAGPIPES (Carnall et al. 2018),
CIGALE (Burgarella et al. 2005; Noll et al. 2009; Boquien et al.2019).In Abdurro’uf & Akiyama (2017), we implemented adifferent likelihood function that make use of the Stu-dent’s t function. The new likelihood function has beenshown to be able to give a better recovery of the SFR inthe fitting tests using mock SEDs and better matchingto the SFR derived from the
Spitzer /MIPS 24 µ m flux (see Appendix A of Abdurro’uf & Akiyama 2017). Mo-tivated by this result, we implement two kinds of like-lihood functions in piXedfit : (1) Gaussian functionas mentioned above, and (2) Student’s t function whichhas the following form P ( X | θ ) = n (cid:89) i =1 Γ (cid:0) ν +12 (cid:1) √ νπ Γ (cid:0) ν (cid:1) (cid:18) χ i ν (cid:19) − ν +12 , (4)with χ i is given by χ i = f X,i − sf θ,i σ X,i . (5)The n represents number of bands (in case of photo-metric SED) or wavelength points (photometric bandsand wavelength grids of the spectrum, in case of spec-trophotometric SED), while f X,i and σ X,i represent theobserved flux and its associated uncertainty in a givenband or wavelength i , respectively. In case of fitting toa spectrum (or spectrophotometric SED), only the spec-tral continuum (or spectral continuum and photometricSED) is fitted. A certain window (default of ± FSPS ) is used to ex-clude emission lines in the fitting. The f θ,i and s are fluxof model SED in band or wavelength point i and a scal-ing factor that bring the model SED in overall similarnormalization as that of the observed SED, respectively.Since model SED generated with FSPS is normalized to1 M (cid:12) , so s corresponds to the stellar mass. patially resolved SED fitting with P I X E D F I T ν represents the degree of freedom which shouldbe specified by the user. A large value of ν will give alikelihood function similar to that of Gaussian, while asmall value of ν will give heavier tails in the likelihooddistribution (compared to the Gaussian one). In Ap-pendix B, we compare performances of various fittingapproaches and determine the best value for ν . We findthat ν ∼ − σ X,i ) is not just taken fromthe observational error, which is often an underesti-mation, but also consider the systematic uncertaintieswhich come from the observational procedure (e.g., as-sociated with image processing) and the SED modelingprocedures. We assume that the bulk of the systematicuncertainties is a multiplicative factor of the observedfluxes such that σ sys,i = err sys × σ X,i , following Han &Han (2019). We do not set the err sys as a free pareme-ter in the fitting considering that it can possibly add adegeneracy in the fitting process, instead we fixed it toa certain value that is obtained from a fitting test thatcan be done either to each individual galaxies or to onegalaxy representative of a whole sample. Practically, inthe fitting test we vary the err sys such that the reduced χ of the best-fit model SED is below ∼ .
0. Withoutadding such systematic uncertainties, it is quite oftento find cases where the reduced χ of the best-fit modelSED is large while the fluxes residuals are actually verysmall. From analysis of 20 local galaxies (to be pre-sented in Section 6), we find that err sys (cid:46) .
15 is enoughto reach the required reduced χ mentioned above.In the default setting and in the analysis throughoutthis paper, a flat prior over a certain range is assumed foreach parameter. For versatility, piXedfit_fitting can also adapt with the priors given by the user in arrayor a text file format.4.2. Posterior Sampling Method
The main task in Bayesian parameter inference is tosolve for the posterior probability distribution functionof each parameter. Commonly, a sampling method isused to reconstruct the posteriors. In the SED fittingapplication, there are at least three approaches adoptedfor the posterior sampling: the gridding method (e.g.,Boquien et al. 2019; Chen et al. 2020), MCMC (e.g.,Acquaviva et al. 2011; Leja et al. 2017; Morishita et al.2019), and nested sampling (e.g., Han & Han 2014; Car-nall et al. 2018; Leja et al. 2019b).In the gridding method, each parameter space is di-vided into a number of grids, then model SEDs are gen-erated for all the possible combinations of the parame-ters grids. One of the advantages of the gridding method is that it could fit a large number of SEDs quickly, es-pecially if the set of model SEDs (with many redshiftgrids) are generated before the fitting. The disadvan-tage of this method is that it typically requires a largenumber of parameters grids (and so the number of modelSEDs) in order for the sampling to be complete, es-pecially for high dimensional parameter space. In theMCMC fitting, the N dimensional parameters are ex-plored by random walks of sampler chains. Over time,the frequency of visited locations can in principle be arepresentative of the posterior probability function. Thedisadvantage of this method is that it is computationallyexpensive and typically slow.In piXedfit_fitting , we adopt two different pos-terior sampling methods: MCMC and random densely-sampling of parameter space (hereafter RDSPS). Eachof those methods is described in the following.4.2.1. Fitting with MCMC
For the MCMC sampling, we use emcee packageby Foreman-Mackey et al. (2013, 2018, 2019). Beforerunning the MCMC sampling, an initial fitting is doneusing the χ minimization technique to get an initialguess and set initial positions for the MCMC walkers.For this fitting, a set of pre-calculated model SEDs (tobe described in Section 4.2.2) is used. The initial po-sitions for the MCMC walkers are defined by a smallasymmetric Gaussian “ball” with a σ = 0 . × W aroundthe best-fit parameters obtained from the initial fitting.The W is the width (i.e., prior range) of a parameterspace.The next step is running the MCMC. The number ofMCMC walkers and steps should be defined by the user.When the MCMC is running, a model likelihood has tobe supplied for each ensemble of N parameter valuesthat are generated. In this case, we use the Gaussianlikelihood function for calculating the model likelihood.The MCMC sampling will finish when every walker hascompleted the specified number of steps. The resultsof MCMC sampling is the sampler chains which recordthe locations in the parameter space that are visited bythe walkers throughout the process. From these samplerchains, the posterior probability distribution of each pa-rameter can be constructed. The inferred value for eachparameter is then obtined from the median of the poste-rior, while the uncertainty is defined by the range givenby the 16th and 84th percentiles. In order to makethe calculation efficient, the parallelization module in emcee is implemented. https://github.com/dfm/emcee Abdurro’uf et al.
Random Densely-sampling of Parameter Space(RDSPS)
The second sampling method we adopt is the RDSPSmethod, which is a simple sampling method inspired bythe gridding method described previously. Unlike thegridding method which defines fixed grids of values foreach parameter, the RDSPS method draws random val-ues uniformly within the prior range in each parame-ter. For generating N mod number of model SEDs with N number of parameters, an N mod number of randomvalues are generated for each parameter. Then, those N arrays of parameters are randomly connected witheach other to construct the library of model SEDs. Thereason of using the RDSPS method over the griddingmethod is its efficiency. With a smaller number of gen-erated models (e.g., ∼ . s ) ofa model SED is calculated from the analytical solutionfor minimizing the χ (see e.g., Eq. 7 in Sawicki 2012).We do not set s as free in the fitting for the sake ofefficiency.After calculating the posterior probability of eachmodel, the inferred value of each parameter is obtainedfrom weighted averaging with model posterior serving asthe weight for the model. The uncertainty is estimatedfrom the weighted standard deviation. For fast fittingperformance, we have incorporated the parallel process-ing module, namely message passing interface (MPI) inthis SED fitting module. 4.3. piXedfit analysis : Visualization of FittingResult The output of the fitting process with the piXedfit_fitting module is a FITS file containingsampler chains (in the case of fitting with MCMC) orposterior probabilities of model SEDs (in the case of fit-ting with the RDSPS method). The FITS file can thenbe used for further analysis, such as deriving inferredvalues of parameters and visualization of the fitting,the latter task can be done with piXedfit_analysis module.For visualizing the fitting results with MCMC, 3 kindsof plots can be made using the piXedfit_analysis module: corner plot, SED plot, and SFH plot. Thecorner plot shows the posterior probability distributions(constructed from the sampler chains) of individual pa-rameters (as 1D histograms) as well as joint posteriorprobability distributions of every pair of two parameters(in 2D). In the corner plot, inferred values of parameters(from median of the posteriors), the uncertainty (16th–84th percentiles of the posteriors) are shown with blackvertical line and gray shaded area in the 1D histograms,respectively. For producing the SED plot, an ensem-ble of 200 sampler chains is randomly picked from thefull MCMC sampler chains, then their spectra are gen-erated. The median posterior model SED (spectrum aswell as photometric SED) and its uncertainty are thenobtained by taking median, 16th and 84th percentilesfrom the ensemble of spectra. The residual, which is( f X,i − f θ,i ) /f X,i , is also shown in the SED plot (seeSection 4.1 for the definitions of f X,i and f θ,i ). For pro-ducing the SFH plot, the inferred SFH is derived byfirst randomly picking 200 sampler chains from the fullMCMC sampler chains, then the SFHs associated withthe sampler chains are calculated. The median, 16th and84th percentiles are then calculated from the ensembleof SFHs at each time step. The median is then used asthe inferred SFH, while the area between the 16th and84th percentiles is used as the associated uncertainty.For fitting with the RDSPS method, currently, only theSED plot can be produced in which the best-fit modelSED is obtained from the model with lowest χ . Exam-ple of the corner plot, SED plot, and SFH plot can beseen in Figures 6 and 13. TESTING THE SED FITTING PERFORMANCEUSING MOCK SEDS OF ILLUSTRISTNGGALAXIESIn this section, we use FUV–NIR mock SEDs of the Il-lustrisTNG (hereafter TNG) galaxies to test the perfor-mance of the piXedfit_fitting module in terms ofits abilities in parameter inference and SFH reconstruc- patially resolved SED fitting with
P I X E D F I T
Generating Mock SEDs of TNG Galaxies
The IllustrisTNG simulations (Marinacci et al. 2018;Naiman et al. 2018; Nelson et al. 2018; Pillepich et al.2018; Springel et al. 2018; Nelson et al. 2019) are a suiteof cosmological hydrodynamical simulations that modela range of physical processes involved in the formation ofgalaxies. In order to test the performance of the SED fit-ting using piXedfit_fitting in inferring the galaxyproperties, we generate mock SED of TNG galaxies andthen fit them with the piXedfit_fitting module tosee whether the inferred parameters can recover the trueproperties of the TNG galaxies. Furthermore, havingrealistic SFH from the TNG galaxies, we can also testthe performance of the piXedfit_fitting module inreconstructing the SFH of a galaxy.For this fitting test, we use the fiducial TNG100 simu-lation, which has a volume of ∼ comoving Mpc anda baryon mass resolution of 1 . × M (cid:12) . We select 300galaxies from the TNG100 simulation. More specifically,we select 100, 80, 60, and 40 galaxies in every 0 . M ∗ between 10 and 10 M (cid:12) and other 20 galaxiesmore massive than 10 . M (cid:12) . The number is somewhatarbitrary, simply to reflect that there are more low-massgalaxies than high-mass ones. In each mass bin, we firstrank all TNG galaxies by their sSFR and choose targetnumber of galaxies equally spacing in terms of the per-centiles in sSFR. In this way, the selected galaxies coverthe entire sSFR range.The mock spectra of TNG galaxies are created byregarding a stellar particle as an SSP, then generat-ing the spectrum of each stellar particle using FSPS.In generating the SSP spectra, Padova isochrones (Gi-rardi et al. 2000; Marigo & Girardi 2007; Marigo et al.2008), MILES stellar spectral library (S´anchez-Bl´azquezet al. 2006; Falc´on-Barroso et al. 2011), and Chabrier(2003) IMF are assumed. The integrated spectrum of agalaxy is then obtained by summing up the spectra ofgravitationally-bound stellar particles in a subhalo asso-ciated with the galaxy. We assume a redshift of 0 . τ )and then apply the Calzetti et al. (2000) dust attenua-tion law to the galaxy’s spectrum. The random valuesof ˆ τ are uniformly distributed between 0 and 2 . photometric SEDs are obtained by convolving the syn-thetic spectra with 12 broad-band filters: GALEX(FUV, NUV), SDSS ( u , g , r , i , z ), 2MASS ( J , H , K s ), and WISE ( W W / N ratio of 10. We create themock FUV–NIR SEDs with the similar setting as thatprovided in the piXedfit_model because we only fo-cus on testing the performance of the fittting algorithmof piXedfit fitting module.5.2.
SED Fitting Analysis of TNG Galaxies
We fit the synthetic SEDs with the piXedfit_fitting module using the same assump-tions of the IMF, spectral library, isochrones, and dustattenuation law as those used for creating the syntheticSEDs. For the SFH, we use the double power law model.We choose double power law SFH form because of itsflexibility in the rising and falling phases. Since thewavelength of the mock SEDs ranges from FUV to NIR,we turn off the AGN dusty torus emission and the dustemission modeling in the fitting. This leaves us withseven free parameters: Z , τ , t (age sys ), ˆ τ , α , β , and M ∗ .Flat priors within a given range is assumed for all the pa-rameters. Logarithmic sampling is applied to all the pa-rameters, except for ˆ τ . The assumed parameters rangesfor the priors are as follows: log( Z/Z (cid:12) ) = [ − . , . τ ) = [ − . , . t ) = [0 . , . τ = [0 . , . α ) = [ − . , . β ) = [ − . , . M ∗ , we use a flat prior in logarithmic scale withina range of log( M ∗ ) = [log( s best ) − , log( s best ) + 2],with s best is the normalization obtained from the ini-tial fitting with the χ minimization technique (seeSection 4.2.1).In order to compare the performances of various fit-ting approaches provided within piXedfit_fitting ,we do the SED fitting with 8 different fitting ap-proaches. These cover the two posterior sampling meth-ods (MCMC and RDSPS), the two likelihood functions(Gaussian and Student’s t) in the RDSPS method, and6 values of degree of freedoms ( ν ) for the Student’s tlikelihood function: 0 .
3, 1 .
0, 2 .
0, 3 .
0, 5 .
0, and 10 .
0. Inthe MCMC fitting, the number of walkers and steps are100 and 1000, respectively.8
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Figure 6 shows fitting results to a mock photomet-ric SED (left) and spectrophotometric SED (right) ofa TNG galaxy. The fitting uses MCMC technique.In each side, three plots are shown: a corner plot,an SED plot, and an SFH plot. The three plots aremade using the piXedfit_analysis module (see Sec-tion 4.3). In the corner plot, the black vertical dashedlines and the shaded area represent median and 16th–84th percentiles. The red vertical lines show the truevalues. The SED plot shows the mock photometric SED(blue squares), the mock spectrum (red line, in caseof the right panel), the median posterior model spec-trum (black line), and the median posterior model pho-tometric SED (gray squares). The residual is given by( f X,i − f θ,i ) /f X,i (see Section 4.3). In the SFH plot, theblack line and gray shaded area represent the inferredSFH and its uncertainty, while the red line shows thetrue SFH. The figure shows that the SED fitting withthe piXedfit_fitting module can recover the trueproperties and the overall trend of the SFH of the TNGgalaxy well. The addition of the synthetic spectrumin the so-called spectrophotometric SED can add moreconstraining power in the fitting process and result inbetter contraints of the Z and SFH parameters (i.e., α , β , and τ ). The fitting with spectrophotometric SEDcan also reveal a bimodality in the posterior probabilitydistribution of metallicity.5.3. Recovering Physical Properties of the TNGGalaxies
For the first test, in this section we compare the in-ferred parameters obtained from fitting and the trueproperties of the TNG galaxies. The metallicity andage of a TNG galaxy are obtained by mass-weightedaveraging over the metallicities and ages of the stellarparticles, respectively. The SFR of a TNG galaxy is es-timated from the amount of stellar mass formed over thelast 50 Myr, based on the formation times of individualstellar particles.Figure 7 shows direct comparisons between the in-ferred parameters obtained from fitting to mock pho-tometric SEDs and the true values for two fitting ap-procahes: the RDSPS that uses Student’s t likelihoodwith ν = 2 . ν ∼ − µ ),scatter (i.e., standard deviation, σ ), and the Spearmanrank-order correlation coefficient ( ρ , which is calculatedusing the SciPy package, Virtanen et al. 2020). Thecoefficient ρ is a nonparametric measure of the mono-tonicity of the relationship between two datasets. Thehistogram for the logarithmic ratio and the associated µ , σ , and ρ values are shown along with the scattereddata.The figure shows that, overall, the inferred parame-ters by the fittting to the photometric SEDs with thetwo approaches can recover the true values quite well,except Z in which the true values are only broadly fol-lowed by the inferred values from fitting, though withsmall median offset ( ∼ . ∼ . M ∗ is the best recovered,corroborated by the small offset (absolute value of µ of (cid:46) .
01 dex), small scatter ( ∼ .
09 dex), and high value(close to unity) of ρ ( ∼ . τ and SFR are alsosuccessfully recovered by the fitting. While overall themass-weighted age is well recovered, there is a trend ofincreasing scatter toward the galaxies with young stel-lar populations. The color-coding indicates that thereis a notable relation between the relatively larger scat-ter around the low mass-weighted age region with theincrease of sSFR. The outshining effect by young stars(which is abundant in galaxies with high sSFR) may beplaying a role in this. In a high sSFR galaxy, which tendto have stellar population dominated by young stars, thelight from the young bright stars dominates the lightfrom the older ones, making it relatively easy to “hide”old stellar populations and consequently it is more diffi-cult to infer SFH of the galaxy (see e.g., Sawicki & Yee1998; Papovich et al. 2001; Maraston et al. 2010; Conroy2013). However, it is also revealed that the Z inferencetend to be better for the high sSFR galaxies than thatfor the low sSFR ones.The difficulty in inferring metallicities from SED fit-ting with photometric SED alone has also been reportedin the literature, e.g., Pacifici et al. (2012, their Fig. 11),who fit mock optical photometry with a set of modelSEDs of galaxies that are drawn from a semi analyt-ical model, which exhibit complex SFHs, Han & Han(2014, their Fig. 18), who fit mock FUV–NIR SEDs patially resolved SED fitting with P I X E D F I T Figure 6.
Example of fitting results to mock photometric SED (left) and spectrophotometric SED (right) of a TNG galaxy.The fitting uses MCMC method. In each fitting result, three kinds of plots are shown: corner plot, SED plot, and SFH plot.Please see the text for description of the symbols in each plot. with
BayeSED that uses the nested sampling method,and Smith & Hayward (2018, their Fig. 11), who em-ployed
MAGPHYS to perform pixel-by-pixel SED fittingto a set of FUV–FIR synthetic images constructed froma zoom-in simulation of an isolated disk galaxy. Despitethe wide wavelength coverage (which can be expected tobreak the well-known age–metallicity–dust attenuationdegeneracy) implemented in Smith & Hayward (2018),the inferred metallicity from SED fitting is systemati-cally underestimated compared to the true values.Micha(cid:32)lowski et al. (2014) evaluated the M ∗ inferenceof various SED fitting codes, which have various as-sumed SFH models, using the synthetic FUV–FIR pho-tometric SEDs of simulated galaxies. The median off-sets and scatters in the M ∗ comparisons have ranges of0 . − . . − . prospector to fit the synthetic FUV–FIR photomet-ric SEDs of simulated galaxies and obtained median off-set of 0 .
02 dex and scatter of 0 .
13 dex. Our M ∗ infer-ence has a smaller offset and scatter than that obtainedin the above studies. Though, the more comprehensive(i.e., realistic) simulation of dust component (throughthe radiative transfer technique) in the construction ofmock SEDs that is implemented in the above studiesmight add more complexity in the fitting test.In order to investigate the effect of the inclusion ofspectrum into the SED on the performance of the pa- rameters inference, we do the same fitting tests to themock spectrophotometric SEDs of the TNG galaxies. Infitting a spectrophotometric SED of a galaxy, only spec-tral continuum is fitted simultaneously with the photo-metric SED (see Section 4.1). Figure 8 shows the com-parison between the inferred parameters obtained fromthe fitting and the true values. The format of this figureis the same as that of Figure 7. Overall, we see im-provements on the inference of all the parameters (withthe two fitting approaches) over what is obtained withthe photometric data only, corroborated by the smallerscatter and higher ρ value, though slightly higher offset.The significant increase in ρ value of Z suggests that theinferred Z become better inline with the true Z . Thishappens at the same time with the decreasing scatterof ˆ τ , which suggests that the inclusion of spectrum canpotentially break the degeneracies in the fitting process.This result agrees with Pacifici et al. (2012) who foundthat SED fitting using mock optical spectroscopy signif-icantly improve the parameters inference over the onethat only use photometry.5.4. Recovering SFHs of the IllustrisTNG Galaxies
In this section, we test the performance of the piXedfit_fitting module in terms of its ability ofinferring the SFH of galaxies. The way we do this is bycomparing the inferred SFH with the true SFH of theTNG galaxies. Figure 9 shows examples of SFHs (black0
Abdurro’uf et al.
Figure 7.
Comparisons between the inferred parameters from fitting and the true properties of the TNG galaxies. Results fromfitting with two approaches (RDSPS that uses Student’s t likelihood with ν = 2 . µ , σ , ρ ) are shown. lines and gray shaded areas in the first row) inferred bythe MCMC fitting using the piXedfit_fitting mod-ule to spectrophotometric SEDs of three TNG galaxies.In each panel in the figure, the black line represents themedian, while the gray shaded area represents the un-certainty. The true SFHs of the TNG galaxies are shownby the red lines. The SFH of TNG galaxy is calculatedwith time steps of 100 Myr. In the second and thirdrows, the histories of the stellar mass growth ( M ∗ ( t ))and sSFR (sSFR( t )) are shown, respectively. They arederived from the inferred SFHs. Same as in the firstrow, the red and black lines here represent the true andinferred histories, respectively. The vertical red dashedlines in the M ∗ ( t ) plots are the true look-back timeswhen the galaxies were still having M ∗ of 30% ( lbt M ),50% ( lbt M ), 70% ( lbt M ), and 90% ( lbt M ) ofthe current M ∗ , while the vertical black lines are the val-ues inferred from the median M ∗ ( t ). The figure showsthat the inferred SFH, M ∗ ( t ), and sSFR( t ) can recover the overall shape (i.e., the rising and falling phases) ofthe true histories of these three TNG galaxies well.In order to quantitatively assess the performance ofthe piXedfit_fitting module in inferring the SFH,we compare the inferred and true values of the lbt M , lbt M , lbt M , and lbt M . Results from the fit-ting with the photometric SEDs are shown in Figure 10.This figure shows that overall, the true look-back timeepisodes in the M ∗ ( t ) can be recovered well using the piXedfit_fitting module with the two fitting ap-proaches. The earlier look-back time episodes seem tobe more difficult to recovered compared to the laterones, corroborated by the increasing ρ from lbt M to lbt M . The color-coding suggests that it is more dif-ficult to infer SFH of galaxies with high sSFR than thegalaxies with low sSFR. This may be in part causedby the outshining effect of young bright stars, which isabundant in the galaxies with high sSFR.By fitting synthetic FUV–FIR photometric SEDs ofsimulated galaxies using a modified version of MAGPHYS , patially resolved SED fitting with P I X E D F I T Figure 8.
Similar to Figure 7, but now the fitting is done to the mock spectrophotometric SEDs of the TNG galaxies. which assumes tau SFH model with random bursts su-perposed, Smith & Hayward (2015) tried to reconstructthe true SFH of the galaxies. They found that themedian-likelihood SFH (obtained by marginalizing overthe model libraries, which is similar to what is done inour work) can well recover the smoothly declining SFHof isolated disk galaxies, while it fails to recover thebursty episodes in the SFH of merging galaxies. Thisis likely caused by the assumed tau SFH model that isnot flexible enough to represent the general (i.e., real-istic) SFH of galaxies. Inline with our results, Carnallet al. (2018) showed that using the more flexible doublepower law SFH model can recover the overall shape ofthe true SFHs of simulated galaxies from
MUFASA , de-spite the narrower wavelength coverage (optical–NIR)of the mock SEDs used. While piXedfit_fitting module can recover the overall trend of rising and fallingepisodes (i.e., low frequency variation) in the true SFHsof TNG galaxies, however, it cannot recover the highfrequency variation in the true SFHs.It is interesting to see how the inclusion of the spec-trum into the SED can effect the SFH inference. Fig-ure 11 shows comparison between the inferred lbt M , lbt M , lbt M , and lbt M obtained from fitting tothe mock spectrophotometric SEDs. The format of thisfigure is the same as that of Figure 10. Overall, we seeimprovements made by the fitting with the spectropho-tometric SED, such that the scatters in the one-to-onecomparisons become smaller and the ρ values becomehigher compared to that obtained from the fitting withphotometric SED only. However, the offsets becomeslightly higher. We notice a flattening appears aroundthe highest and lowest ends of the correlation in case ofthe fitting with the RDSPS approach. This flatteningcan be caused by a multimodal posteriors distributions. TESTING THE PERFORMANCE OF
PIXEDFIT
USING SPATIALLY RESOLVEDSPECTROPHOTOMETRIC DATA OF LOCALGALAXIESIn this section, we analyze spatially resolved spec-trophotometric data of 10 galaxies observed by the CAL-IFA survey and 10 galaxies observed by the MaNGA sur-vey. The goals of this analysis are (1) to demonstratethe ability of the piXedfit in spatially matching theFUV– W Abdurro’uf et al.
Figure 9.
Comparison of the inferred SFH (first row), M ∗ ( t ) (second row), and sSFR( t ) (third row) obtained from fitting tothe spectrophotometric SEDs of 3 TNG galaxies using the piXedfit fitting module and the true histories. The vertical reddashed lines and black lines are the true and inferred look-back times when the galaxies were still having M ∗ of 30%, 50%, 70%,and 90% of the current M ∗ . (2) demonstrate the SED fitting analysis to the spatiallyresolved spectrophotometric dataset with piXedfit ,and (3) test the reliability of the SED fitting module bycomparing the inferred SFR from fitting with the SFRderived from H α emission. A more scientific-orienteddiscussion using larger samples is left for future works.6.1. Sample Selection and Data Reduction
First, we construct a catalog of galaxies that are ob-served by the medium imaging survey (MIS) of GALEX,SDSS, 2MASS, and WISE. We start from the cata-log of unique GALEX GR5 sources (i.e., eliminatingrepeated measurements) that has been matched withthe SDSS DR7 catalog by Bianchi et al. (2011) , thencross match it with the MPA-JHU (Max-Planck-Institutf¨ur Astrophysik-Johns Hopkins University) value addedgalaxy catalog (Kauffmann et al. 2003; Tremonti et al.2004; Brinchmann et al. 2004) to select only galax-ies and get their stellar masses. After that we crossmatch the catalog with the 2MASS extended sourcecatalog (Jarrett et al. 2000). Considering the all skycoverage of the WISE survey and its depth compared Available at http://dolomiti.pha.jhu.edu/uvsky Available at https://irsa.ipac.caltech.edu/Missions/2mass.html to 2MASS, we do not cross match the catalog furtherwith the WISE catalog. Once we get the catalog, wecross match it with the CALIFA DR3 and MaNGADRPALL (from SDSS DR15 ) catalogs separately. Asa result we get 41 galaxies matched with the CALIFAcatalog and 395 galaxies matched with the MaNGA cat-alog. Then we randomly select 10 galaxies that havelog( M ∗ [ M (cid:12) ]) > . z < .
05 from each of the twocatalogs. We download the multiband images and theIFS data from the relevant survey websites, assisted bythe galaxies coordinates from the merged catalog.We require the galaxies to be covered by the GALEXMIS because the survey has relatively long exposuretime (typically 1500 s) so that we can have sufficientS/N ratio in the UV. Among the imaging datasets used,the 2MASS imaging data is the shallowest. However,it is still important to include the data because it com-plements the two WISE bands in putting strong con-straint in the NIR regime. In total, the photometrydata consists of 12 bands ranging from FUV to W piXedfit_images module to spatiallymatch (in resolution and sampling) the imaging data, Available at https://califaserv.caha.es/CALIFA WEB/publichtml/?q=content/califa-3rd-data-release patially resolved SED fitting with P I X E D F I T Figure 10.
Comparison of the inferred lbt M , lbt M , lbt M , and lbt M from fitting to the photometric SEDs of theTNG galaxies using the piXedfit fitting module and the true values. Two fitting approaches are used: the RDSPS thatuses Student’s t likelihood with ν = 2 . Figure 11.
Similar to Figure 10, but now the fitting is done to the spectrophotometric SEDs of the TNG galaxies. then use piXedfit_spectrophotometric moduleto spatially match the reduced imaging data cubes withthe IFS data. This processes produce spatially resolvedspectrophotometric data cubes that have spatial reso-lution similar to that of the W gri composite image and exampleof SEDs of four pixels are shown in the left panel and the4 Abdurro’uf et al. right panel, respectively. In the gri composite image,the transparent hexagonal area shows the area coveredby the IFU fiber bundle of the CALIFA and MaNGAsurveys. In the data cubes, only pixels covered withinthe region of the IFU fiber bundle have the spectra,while pixels outside of the region only have photometricSED.The spectrophotometric data cubes are then passed tothe pixel binning process. The pixel binning is done us-ing the piXedfit_bin module. See Section 3.3 for thedescription of the pixel binning scheme. In this analy-sis, the criteria for the binning are: S/N ratio thresholdof 10 . . D min , bin of4 pixels, and reduced χ limit ( χ , bin ) of 3 . SED Fitting Analysis
The reduced spectrophotometric data cubes (af-ter pixel binning) are then passed to the SED fit-ting process. The SED fitting is done using the piXedfit_fitting module (see Section 4) withMCMC approach. The SED fitting setup (IMF,isochrone, spectral library, SFH, and dust attenuationlaw) is the same as that for the fitting with the mockSEDs of the TNG galaxies (see Section 5.2), except forthe priors. Flat priors for all parameters are assumed,within the following ranges: log(
Z/Z (cid:12) ) = [ − . , . τ ) = [ − . , . − . , . τ =[0 . , . α ) = [ − . , . β ) = [ − . , . M ∗ prior is defined in the same way as that appliedin the fitting with the mock SEDs of TNG galaxies. Thereason of using different set of priors from those used forfitting the mock SEDs of the TNG galaxies is becausehere we analyze the spatially resolved SEDs, which comefrom stellar populations with wide range of ages (age sys ).In the MCMC fitting, we set the number of walkers andsteps as 100 and 1000, respectively.For spatial bins that have spectrophotometric SED(see Section 3.3 for the definition of spatial bins withspectrophotometric SEDs), two kinds of fitting aredone: a fitting to the photometric SED only anda fitting to the spectrophotometric SED. By default, piXedfit_fitting will fit both photometric SEDand spectrum (i.e., the spectral continuum) simultane-ously whenever it is fed with a spectrophotometric data.The aim of performing fitting to only the photometric SED is for conducting tests, which includes a reconstruc-tion of the observed spectral continuum, D n H α emission, and H β emission using model spectra obtainedfrom fitting to the photometry. These analyses will bediscussed in the next two sections.Figure 13 shows fitting results using MCMC for twospatial bins in the NGC 309, one located around thecenter and the other located in the spiral arms. Forcentrally-located bin, the fitting is done to both spec-trum and photometric SED simultaneously. The overallsymbols in the corner plot, SED plot, and SFH plot arethe same as that in the Figure 6. An obvious differencein the SED shape between the spatial bin around thegalaxy’s center (a red SED typical of old stellar popula-tion) and that in the spiral arms (a blue SED typical ofyoung stellar population) is shown in the SED plots. Inthe SED plot, the black spectrum and gray shaded areaaround it represent median posterior model spectrumand the associated uncertainty. The small residuals inthe SED plot indicate that the observed continuum andphotometric SED can be recovered well. The inferredSFH of the spatial bin located in the spiral arms indi-cates a steeply increasing SFR toward the observationaltime, while the inferred SFH of the spatial bin locatedaround the galaxy’s center indicates a gradual increaseof SFR from ∼
10 Gyr ago and reached peak around ∼ M ∗ (whichscaled with NIR bands) and SFR (which scaled with UVbands), the inferred value of the bin is divided into thepixels that belong to the bin by assuming that M ∗ isproportional to the W M ∗ and SFR maps. The other parametersare kept in the spatial bin space.Figures 14 and 15 show the maps of stellar popula-tion properties of the 10 galaxies from CALIFA and the10 galaxies from MaNGA, respectively. In the both fig-ures, the first 6 columns from the left show maps ofthe pixel binning, Z , mass-weighted age, A V dust at-tenuation, SFR surface density (Σ SFR ), and M ∗ surfacedensity (Σ ∗ ). The mass-weighted age is derived fromthe inferred SFH, while the A V can be calculated as A V = 1 . × ˆ τ . For comparison, in the rightmostcolumns of the both figures, we show Σ ∗ from the Py-CASSO data base (de Amorim et al. 2017) which is Available at http://pycasso.iaa.es/ patially resolved SED fitting with
P I X E D F I T Figure 12.
Compilation of the spatially resolved spectrophotometric data cubes of 18 galaxies from the sample analyzed inthis paper. The data cubes of the other 2 galaxies are shown in Figure 3. The left and right sides show 9 galaxies from theCALIFA and 9 galaxies from the MaNGA, respectively. For each galaxy, gri composite image (left panel) and SEDs of four pixels(right panel) are shown. In the gri composite images, the area covered by the IFU fiber bundle is shown with the transparenthexagonal area. Abdurro’uf et al.
Figure 13.
Examples of fitting results using MCMC for two spatial bins of the NGC 309, one located around the center andthe other located in the spiral arms. The fitting to the centrally-located bin is done to the spectrophotometric SED, while thefitting for the outter bin is done to the photometric SED. The overall symbols in the corner plot, SED plot, and SFH plot arethe same as that in the Figure 6. derived from the CALIFA data alone (in case of fig-ure 14) and Σ ∗ from the Pipe3D value added catalog (S´anchez et al. 2018) which is based on the MaNGAdata alone (in case of figure 15). The dimensions of theplotted maps in these rightmost columns correspond tothe same physical sizes as those of the dimensions of themaps in the other 6 columns. The fact that our datacubes have lower spatial sampling (i.e., larger pixel size;1 . (cid:48)(cid:48) pixel − ) than that of the CALIFA (1 . (cid:48)(cid:48) pixel − )and MaNGA (0 . (cid:48)(cid:48) pixel − ) data cubes makes our mapshave smaller total number of pixels.6.3. Reconstructing Observed Spectral Continuum withModel Spectra Obtained from Fitting toPhotometry
In this section and the next section, for spatial binsthat have spectrophotometric SEDs, we fit the photo-metric SEDs and then compare the median posteriormodel spectra with the observed spectra (see Section 5.2for the description on how the median posterior model spectra are obtained). We make the comparison by cal-culating the residual in spectral continuum (in this sec-tion) and directly comparing the D n H α and H β luminosities (in the next section). Thisanalysis can serves as an excellent test for piXedfit in terms of its SED modeling (based on FSPS) and thefitting performance. A similar exercise has been carriedout by Leja et al. (2017) with the prospector , but forgalaxies as a whole.We collected the spatial bins that have spectrophoto-metric SEDs in the CALIFA (560 bins) and MaNGA(145 bins) samples. To get the continuum from theobserved spectra and the median posterior spectra, weremove regions within ± patially resolved SED fitting with P I X E D F I T Figure 14.
Maps of the stellar population properties of the 10 galaxies from the CALIFA survey analyzed in this work. TheSED fitting uses the MCMC technique. The first 6 columns from the left show maps of the pixel binning, Z , mass-weightedage, A V dust attenuation, Σ SFR , and Σ ∗ . The righmost column show Σ ∗ from the PyCASSO data base (de Amorim et al. 2017)which is derived from the CALIFA data alone. The dimension of this map corresponds to the same physical size as those of thedimensions of the maps in the other 6 columns. Abdurro’uf et al.
Figure 15.
Maps of the stellar population properties of the 10 galaxies from the MaNGA survey analyzed in this work. TheSED fitting uses MCMC technique. The first 6 columns from the left show maps of the pixel binning, Z , mass-weighted age, A V dust attenuation, Σ SFR , and Σ ∗ . The righmost column show Σ ∗ from the Pipe3D value added catalog (S´anchez et al.2018) which is based on MaNGA data only. The dimension of this map corresponds to the same physical size as those of thedimensions of the maps in the other 6 columns. patially resolved SED fitting with P I X E D F I T .
004 and 0 .
037 (for CALIFA) and − .
005 and 0 . u and g bands.6.4. Predicting H α , H β , and D n with ModelSpectra Obtained from Fitting to Photometry In this section, we try to predict H α luminosity, H β luminosity, and D n piXedfit_fitting with only photomet-ric SED. To measure luminosities of the H α and H β emission lines from the observed spectra, first, we sub-tract the observed spectra with the continuum of the me-dian posterior model spectra, generated from the modelposteriors. Then we fit the H α and H β emission lineswith Gaussian functions using fit_lines function inthe specutils (Earl et al. 2020) Python package.We visually inspect all the spatial bins to make sure thefitting work well. Uncertainties of the H α and H β lu-minosities are estimated based on the average S / N ratioof the observed spectral fluxes ± H α and H β luminosities of the medianposterior model spectra are derived from the posteri-ors distributions obtained from the MCMC fitting. Themedian, 16th, and 84th percentiles are calculated fromthe posteriors distributions. The median is then usedas the mean luminosity, while the 16th–84th percentilesare used as the uncertainty.The D n f ν in the narrow wavelength bandsof 3850–3950˚A and 4000–4100˚A. To estimate uncer-tainty for the D n n n H α luminosity (left), H β luminos-ity (middle), and D n https://specutils.readthedocs.io/en/stable/ piXedfit_fitting through fitting with photometricSEDs. In each panel, the histogram in the bottom rightcorner shows distribution of the logarithmic ratio be-tween the model predictions and the observed ones. Themean (or offset in dex, µ ), scatter ( σ ), and Spearmanrank-order coefficient ( ρ ) of the distribution are shown inthe top left corner. There is a good agreement betweenthe models and observations, especially for the H α and H β , corroborated by the small offsets (0 .
105 and 0 . H α and H β , respectively) and scatters (0 . .
342 dex for the H α and H β , respectively). The ρ values for H α and H β are relatively high (0 .
678 and0 . − .
024 dex) and scatter (0 .
036 dex), however,the ρ values for D n . n n n H α , H β ,and D n ρ values in all of the three comparisons.6.5. Comparison of SFR from piXedfit fitting with the SFR Derived from H α The Balmer emission lines, especially H α line (which isthe strongest) is a good indicator of instantaneous SFR.In addition to that, the Balmer decrement (i.e., ratio of H α /H β emission line fluxes) provides a good indicatorfor dust attenuation in the stars birth clouds. The H α -based SFR estimate has been widely used in the analy-sis of the IFS data in the CALIFA (e.g., S´anchez et al.2016b) and MaNGA (e.g., S´anchez et al. 2018; Belfioreet al. 2019) surveys.In this section, for spatial bins with spectrophoto-metric SEDs and the H α and H β S / N > . piXedfit_fitting module and the SFR derivedfrom the observed H α emission. Here, we use the H α Abdurro’uf et al.
Figure 16.
Merged residuals between the median posterior model spectra obtained from fitting to the photomeric SEDs andthe observed spectra for the CALIFA (top panel) and MaNGA (bottom panel) sample. The black lines and gray shaded areasare the median and 16th–84th percentiles of the residuals. The vertical cyan bands in the two panels show regions in the spectrathat are removed. and H β measurements from the analysis in the previ-ous section (Section 6.4). We do not use the publiclyavailable value added data cubes from the CALIFA andMaNGA surveys, because of the differences in spatialresolution and spatial sampling between their data cubesand our reduced data cubes. Spatially matching theSFR map or H α and H β maps from their data cubesto our data cubes will introduce some systematics thatcould dominate uncertainties in the comparison analy-sis. Moreover, the comparison can be made more self-consistent.In deriving SFR from the H α emission, first we cor-rect the H α luminosity for the dust attenuation associ-ated with the birth cloud. Balmer color excess is corre-lated with the ratio of the observed Balmer decrement( L H α /L H β ) obs and its intrinsic value ( L H α /L H β ) int through the following equation: E ( H β − H α ) = 2 . (cid:18) ( L H α /L H β ) obs ( L H α /L H β ) int (cid:19) . (6)The L H α and L H β are the luminosities of H α and H β ,respectively. The intrinsic Balmer decrement has a valueof 2 .
86 for the case B recombination (Osterbrock 1989).Once we have the Balmer color excess, the attenuationtoward H α can then be calculated as: A H α = E ( H β − H α ) k ( λ H β ) − k ( λ H α ) × k ( λ H α ) . (7) The k ( λ H α ) and k ( λ H β ) are the attenuation values atwavelengths of H α and H β , respectively. To get thesevalues, we assume the Calzetti et al. (2000) attenua-tion curve with R V = 3 .
1. It is important to notethat Calzetti et al. (2000) used two different attenuationcurves for the nebular and continuum. The two atten-uation curves have similar shapes but different normal-izations: R V = 3 . R V = 4 .
05 for the nebular andcontinuum, respectively. Once we have A H α , the dust-corrected H α luminosity can then be calculated via: L H α , corr = L H α , obs × . A Hα (8)For deriving the SFR from the dust-corrected H α lu-minosity, we use the Kennicutt (1998) prescription thathas been converted for Chabrier (2003) IMF as follows:SFR[ M (cid:12) yr − ] = 4 . × − L H α , corr [erg s − ] . (9)A division by 1 . H α is estimated using the bootstrap method.Figure 18, top panel, shows comparison between theSFR obtained from fitting to the photometric SEDs ofspatial bins and the SFR derived from the H α emission.There is a good agreement between the two SFR mea-surements, with a small offset of 0 .
127 dex and scatter patially resolved SED fitting with
P I X E D F I T Figure 17.
Comparison between the observed H α luminosities, H β luminosities, and D n µ ), scatter ( σ ),and Spearman rank-order correlation coefficient ( ρ ) are shown in the top left corner. of 0 .
389 dex. The Spearman ρ value is high (0 . − . . ρ value is reduced to 0 . ν = 2 . H α emission. Thereis a good agreement between the two SFR estimates, asindicated by the small offset (0 .
093 dex), small scatter(0 .
363 dex), and high Spearman ρ value (0 . ν = 2 . ∼ ∼
40 timefaster than MCMC, using the same number of cores)provides a great opportunity for an application to spa-tially resolved SED fitting analysis. In a future work,we will apply piXedfit to a large sample of galaxies. SUMMARYIn this paper, we present piXedfit , a Python pack-age that provides tools for analyzing the spatially re-solved properties (including stellar and dust compo-nents) of galaxies from broad-band imaging data ora combination of broad-band imaging and IFS data. piXedfit is designed to be modular, and consists of sixmain modules: (1) piXedfit_images is for the imageprocessing, (2) piXedfit_spectrophotometric isfor spatial matching between the imaging data and IFSdata, (3) piXedfit_bin is for pixel binning to maxi-2
Abdurro’uf et al.
Figure 18.
Comparison between the SFR derived from fit-ting with the MCMC method and the SFR derived from H α emission. In the top panel, the fitting is done to the pho-tometric SEDs of spatial bins, while in the bottom panel,the fitting is done to the spectrophotometric SEDs. In eachpanel, the histogram in the bottom right corner shows dis-tribution of the logarithmic ratio between the two SFR es-timates. The offset ( µ ), scatter ( σ ), and Spearman ρ valuesare shown in the top left corner. mize the S / N ratio of the spatially resolved SED, (4) piXedfit_model is for generating model SED, (5) piXedfit_fitting is for performing SED fitting, and(6) piXedfit_analysis is for visualization of fittingresults.We test the capabilities of piXedfit with two anal-yses in this paper: testing the SED fitting performanceusing mock FUV–NIR SEDs of IllustrisTNG galaxiesand testing piXedfit modules using spatially resolvedspectrophotometric data of local galaxies. Overall, thetesting results are summarized as follows:1. We test the performance of piXedfit_fitting module by fitting mock FUV–NIR SEDs (photo-
Figure 19.
Comparison between the SFR derived from H α emission and SFR from fitting using the RDSPS methodwith the likelihood function of Student’s t with ν = 2 .
0. Thehistogram in the bottom right corner shows distribution ofthe logarithmic ratio between the two SFR estimates. Theoffset ( µ ), scatter ( σ ), and Spearman ρ values are shown inthe top left corner. metric as well as spectrophotometric SEDs) of Il-lustrisTNG galaxies and then compare the inferredparameters from fitting with the true parame-ters. We implement various fitting approaches(the MCMC and RDSPS with likelihoods of Gaus-sian and Student’s t with various values of ν )provided within piXedfit_fitting to comparetheir performances. With photometric SED thatcovers FUV–NIR, piXedfit_fitting can wellrecover mass-weighted ages, dust optical depth, M ∗ , and SFR of the IllustrisTNG galaxies, for allof the fitting approaches (see Section 5 and Ap-pendix B). The fitting to mock spectrophotometricSED improve the parameters inference, especiallyfor the metallicity.2. Using the mock SEDs and SFHs of the Ilus-trisTNG galaxies, we test the performance of piXedfit_fitting in inferring the SFH of agalaxy. We quantitatively assess the performanceby comparing the true and inferred values of look-back times when the galaxies M ∗ were only 30%( lbt M ), 50% ( lbt M ), 70% ( lbt M ), and90% ( lbt M ) of the current values. With FUV–NIR photometric SEDs, piXedfit_fitting can well recover the lbt M , lbt M , lbt M ,and lbt M using all of the fitting approaches.The fitting to mock spectrophotometric SEDs im-proves the SFH inference. patially resolved SED fitting with P I X E D F I T piXedfit modules using spatially resolved spectrophotomet-ric data of 20 galaxies observed by the CALIFAand MaNGA surveys. The piXedfit_images and piXedfit_spectrophotometric are ca-pable of spatially matching (in resolution and sam-pling) of 12-bands imaging data from GALEX,SDSS, 2MASS, and WISE, and the IFS data fromCALIFA and MaNGA. piXedfit_bin is capa-ble of binning neighboring pixels with similar SEDshape and reach target S / N ratios in all bands.4. By fitting to photometric SED only, piXedfit can predict real spectral continuum, D n H α emission, and H β emission. The residuals betweenthe spectral continuum of the median posteriorand that of the observed spectra are flat over awide range of wavelength, in both CALIFA andMaNGA samples. The predicted H α , H β , andD n . . − .
024 dex, respec-tively.5. Using the H α and H β luminosities of the observedspectra, we derive the SFR. The dust attenuationcorrection based on the Balmer decrement is ap-plied. Then we compare that SFR with the SFRderived from the SED fitting. The SFR derivedfrom the SED fitting with piXedfit_fitting is consistent with the SFR derived from the H α emission.6. While most of the fitting approaches in the piXedfit_fitting give good inferences of stel-lar population properties and SFH, there are in-dications that the approach of RDSPS with likeli-hood of Student’s t with ν ∼
2, with proper priors,can give robust (and stable) parameters inference,as good as MCMC method. With its relativelyfast fitting performance ( ∼
40 time faster than theMCMC), this fitting approach can be a good op-tion for performing spatially resolved SED fittingfor a large sample of galaxies. piXedfit is a powerful tool for analyzing the spa-tially resolved properties of galaxies across wide rangeof resdshifts in the future era of big data in photome-try from the deep and high spatial resolution multibandimaging surveys. piXedfit will be made publicly avail-able on GitHub , archived in Zenodo (Abdurro’uf et al. piXedfit codebase: https://github.com/aabdurrouf/piXedfit , a community-developed core Python package for Astronomy (AstropyCollaboration et al. 2013, 2018). This research madeuse of Photutils , an
Astropy Abdurro’uf et al.
State University, Shanghai Astronomical Observatory,United Kingdom Participation Group, Universidad Na-cional Aut´onoma de M´exico, University of Arizona, Uni-versity of Colorado Boulder, University of Oxford, Uni-versity of Portsmouth, University of Utah, University ofVirginia, University of Washington, University of Wis-consin, Vanderbilt University, and Yale University. Thispublication makes use of data products from the TwoMicron All Sky Survey, which is a joint project of theUniversity of Massachusetts and the Infrared Process-ing and Analysis Center/California Institute of Tech-nology, funded by the National Aeronautics and SpaceAdministration and the National Science Foundation.This publication makes use of data products from theWide-field Infrared Survey Explorer, which is a jointproject of the University of California, Los Angeles, andthe Jet Propulsion Laboratory/California Institute ofTechnology, funded by the National Aeronautics andSpace Administration. This study uses data providedby the Calar Alto Legacy Integral Field Area (CALIFA) survey (http://califa.caha.es/). Based on observationscollected at the Centro Astron´omico Hispano Alem´an(CAHA) at Calar Alto, operated jointly by the Max-Planck-Institut f˝ur Astronomie and the Instituto de As-trof´ısica de Andaluc´ıa (CSIC).
Facilities:
GALEX, Sloan, CTIO:2MASS,FLWO:2MASS, WISE, Sloan (BOSS, MaNGA survey),CAO:3.5m (PMAS/PPAK, CALIFA survey)
Software:
Astropy (Astropy Collaborationet al. 2013),
Photutils (Bradley et al. 2019), reproject (Robitaille 2018),
SExtractor (Bertin& Arnouts 1996), sewpy , FSPS (Conroy et al. 2009), python-FSPS (Foreman-Mackey et al. 2014), emcee (Foreman-Mackey et al. 2013), matplotlib (Hunter2007),
SciPy (Virtanen et al. 2020),
NumPy (Harriset al. 2020), specutils (Earl et al. 2020)
APPENDIX A. COMPARISON OF THE EMPIRICAL PSF OF THE
SDSS
AND
WITH THE ANALYTICAL PSFFROM ANIANO ET AL. (2011)We construct empirical PSFs of the SDSS and 2MASS using the PSF modeling functions provided by
Photutils .The
Photutils package provides tools for building an effective PSF, which can represent the net PSF of a givencamera. The effective PSF is built based on the prescription in Anderson & King (2000). First, several images ofrandom fields are downloaded from the SDSS and 2MASS websites. Then background subtraction is done, especially for2MASS images (the SDSS image product is background free). After that, bright stars are collected using find_peaks function. The extract_stars function is used to extract cutouts of the stars. Then visual inspection is done toexclude “bad stars”, such as multiple stars in one cutout image and saturated stars. Finally, effective PSFs areconstructed using
EPSFBuilder function. In building the effective PSFs, number of stars selected for u , g , r , i , z , J , H , and K s are 103, 123, 143, 170, 268, 102, 118, and 94, respectively. The constructed effective PSFs of the SDSSand 2MASS are shown in Figure 20. We compare the empirical PSFs with analytical PSFs from Aniano et al. (2011).We found that the PSFs of u , g , and r are best represented by the double Gaussian with FWHM of 1 . (cid:48)(cid:48) ; PSFs of the i and z are best represented by the double Gaussian with FWHM of 1 . (cid:48)(cid:48) ; and PSFs of 2MASS are best representedby the Gaussian with FWHM of 3 . (cid:48)(cid:48) . This comparison is shown in the third and fouth rows in the figure. We alsoconstruct empirical PSFs of FUV and NUV bands of GALEX using the same procedure. The constructed empiricalPSFs of FUV and NUV are consistent with the PSFs from Aniano et al. (2011). The empirical PSFs from this analysiscan be found at https://github.com/aabdurrouf/empPSFs GALEXSDSS2MASS. B. COMPARISON OF THE PERFORMANCES OF VARIOUS FITTING APPROACHES PROVIDED IN
PIXEDFIT FITTING
MODULEIn Section 5, we fit the mock FUV–NIR photometric SEDs of the TNG galaxies using piXedfit_fitting modulewith 8 different fitting approaches, including the 2 posterior sampling methods (MCMC and RDSPS), the 2 likelihoodfunctions (Gaussian and Student’s t) in the RDSPS method, and 6 values of ν for the Student’s t likelihood function:0 .
3, 1 .
0, 2 .
0, 3 .
0, 5 .
0, and 10 .
0. The purpose of performing this fitting experiment is to compare the performances ofthe various fitting approaches provided within the piXedfit_fitting module.In this analysis, we compare the performances of those fitting approaches in inferring 5 key parameters: Z , dustoptical depth (ˆ τ ), mass-weighted age, M ∗ , and SFR. Similar to what we do in Section 5.3 and 5.4, for each parameter patially resolved SED fitting with P I X E D F I T Figure 20.
Comparison of the empirical PSFs of the SDSS and 2MASS with the analytical PSFs from Aniano et al. (2011).In the first and second rows, empirical PSFs of the SDSS and 2MASS are shown. Comparison of those empirical PSFs with theanalytical PSFs are shown in the third and fourth rows. obtained with each fitting approach, we calculate the offset ( µ ), scatter ( σ ), and Spearman ρ coefficient of the 1Ddistribution of the logarithmic ratios between the inferred values from fitting and the true values. Then here, for eachparameter, we compare the goodness of the recovery among the fitting approaches by directly comparing the µ , σ , and ρ values.Figure 21, left panel, shows a compilation of the values of µ (first row), σ (second row), and Spearman ρ (thirdrow). Different fitting parameters are shown with different symbols. The horizontal axis shows the various fittingapproaches. From this plots, overall, we can see that all the fitting approaches give good performances, indicated bythe low values of absolute µ ( (cid:46) .
13 dex) in all the parameters, low scatter ( (cid:46) . ν = 0 . ρ ( (cid:38) .
6) in all the parameters,except Z . The average absolute µ , σ , and ρ for the [ gauss , stdt dof03 , stdt dof1 , stdt dof2 , stdt dof3 , stdt dof5 , stdt dof10 , mcmc ] are [0 . . . . . . . . . . . . . . . . . . . . . . . . µ , σ , and ρ valuesassociated with each parameter are sorted and ranked from smallest (ranked as 0) to highest (ranked as 7). For the µ , the absolute value is considered. Different parameters are shown with circles of different colors and sizes. This plotindicates that the RDSPS method that uses Student’s t likelihood function with ν ∼ . − . χ ) in the Bayesian inference process compared to the Gaussian function.6 Abdurro’uf et al.
Figure 21.
Compilation of the µ , σ , and ρ for the 1D distributions of the logrithmic ratios between the inferred values fromfitting to the mock photometric SEDs of TNG galaxies and the true values. In the left side, real values of µ , σ , and ρ associatedwith the parameters and the fitting approaches are shown. Different symbols represent different parameters. In the right side,for each parameter, the µ , σ , and ρ values associated with the various fitting approaches are sorted and ranked from smallest(ranked as 0) to highest (ranked as 7). Different parameters are shown with circles of different colors and sizes. REFERENCES
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