GalPaK3D: A BAYESIAN PARAMETRIC TOOL FOR EXTRACTING MORPHOKINEMATICS OF GALAXIESFROM 3D DATA

N. Bouché1,2, H. Carfantan1,2, I. Schroetter1,2, L. Michel-Dansac3, and T. Contini1,21 CNRS/IRAP, 14 Avenue E. Belin, F-31400 Toulouse, France

2 University Paul Sabatier of Toulouse/UPS-OMP/ IRAP, F-31400 Toulouse, France3 CRAL, Observatoire de Lyon, Université Lyon 1, 9 Avenue Ch. André, 69561 Saint Genis Laval Cedex, France

Received 2014 June 24; accepted 2015 June 16; published 2015 August 27

ABSTRACT

We present a method to constrain galaxy parameters directly from three-dimensional datacubes. The algorithmcompares directly the data with a parametric model mapped in x y, , l coordinates. It uses the spectral line-spreadfunction and the spatial point-spread function (PSF) to generate a three-dimensional kernel whose characteristicsare instrumentspecific or usergenerated. The algorithm returns the intrinsic modeled properties along with both an“intrinsic” model datacube and the modeled galaxy convolved with the 3Dkernel. The algorithm uses a MarkovChain Monte Carlo approach with a nontraditional proposal distribution in order to efficiently probe the parameterspace. We demonstrate the robustness of the algorithm using 1728 mock galaxies and galaxies generated fromhydrodynamical simulations in various seeing conditions from 0″. 6 to 1″. 2. We find that the algorithm can recoverthe morphological parameters (inclination, position angle) to within 10% and the kinematic parameters (maximumrotation velocity) to within 20%, irrespectively of the PSF in seeing (up to 1″. 2) provided that the maximum signal-to-noise ratio (S/N) is greater than ∼3 pixel−1 and that the ratio of galaxy half-light radius to seeing radius isgreater than about 1.5. One can use such an algorithm to constrain simultaneously the kinematics andmorphological parameters of (nonmerging) galaxies observed in nonoptimal seeing conditions. The algorithm canalso be used on adaptiveoptics data or on high-quality, high-S/N data to look for nonaxisymmetric structures inthe residuals.

Key words: methods: data analysis – methods: numerical – techniques: imaging spectroscopy

Supporting material: 3D figure

1. INTRODUCTION

Thanks to several studies using optical or near-infrared(NIR) integral field unit (IFU) spectroscopy of Hα emissionfrom local and high-redshift(z 1> ) galaxies (Förster Schreiberet al. 2006, 2009; Law et al. 2007, 2012; van Starkenburget al. 2008; Cresci et al. 2009; Lemoine-Busserolle et al. 2010;Contini et al. 2012; Epinat et al. 2012; Buitrago et al. 2014),our understanding of galaxy formation has changed signifi-cantly in the past decade. For instance, these surveys haveshown that a significant subset of high-redshift galaxies have adisklike morphology and showorganized rotation, with regularvelocity fields.

In contrast to low-redshift studies (e.g., Bacon et al. 2001;Cappellari et al. 2011), high-redshift ( z1 2 ) galaxies areobserved at a spatial resolution that is severaly limited by theseeing conditions owing to their small apparent angular sizes.In order to overcome the low spatial resolution, observationswith adaptive optics (AO) are often required (Law et al. 2007,2009; Wright et al. 2007, 2009; Genzel et al. 2008, 2011).However, observations with AO are expensive, with theadditional instrumental costs, and add strong observationalconstraints such as the additional exposure times required tocompensate for the loss in surface brightness (SB) sensitivity(Law et al. 2006). Indeed, the SB limit for AO observationstaken on smaller pixels is higher, leaving the current state-of-the-art observations to the objects with the highest SBs.

Given these challenges and the advancements in multi-plexing IFU observations with the Very Large Telescope(VLT) second-generation instruments like KMOS (Sharpleset al. 2006) and the Multi-unit Spectrograph Explorer

(MUSE;Bacon et al. 2006, 2015), it is important to havetools that can give robust estimates on the galaxy physicalproperties. In particular, KMOS will bring large statisticallysignificant samples of high-redshift galaxies as it can observe24 galaxies at a time, but this facility will always lack an AOunit. This could potentially be a serious limitation since therobustness of the derived kinematic parameters may depend onthe quality of the atmospheric conditions (seeing can rangefrom 0″. 4 to >1″. 0 in the NIR).In order to overcome these limitations, we present a new tool

named GalPaK3D (Galaxy Parameters and Kinematics) ,4

designed to be able to disentangle the galaxy kinematics fromresolution effects over a wide range of conditions. This is notthe first code to model galaxy kinematics from 3D data(e.g.,the TiRiFiC package,which performs tilted ring model fits tothree-dimensional radio data; Józsa et al. 2007), but this codeperforms disk model fits to three-dimensional data cubes, forthe first time,5whereas all other modelingof IFU data so farhasworked from the two-dimensional velocity field (e.g.,Cresci et al. 2009; Epinat et al. 2009; Davies et al. 2011;Andersen & Bershady 2013; Davis et al. 2013).This paper is organized as follows: we describe the

GalPaK3D algorithm in Section 2. We present some test caseexamples in Section 3. We present results from an extensiveanalysis of 1728 synthetic galaxies in Section 4, where wediscuss the impact of the accuracy in the point-spread function(PSF) characterization. In Section 5, we present an analysis ofdata cubes generated from hydrodynamical simulations of

The Astronomical Journal, 150:92 (15pp), 2015 September doi:10.1088/0004-6256/150/3/92© 2015. The American Astronomical Society. All rights reserved.

4 Available at http://galpak.irap.omp.eu/5 Law et al. (2012) made an attempt at 3D fitting, albeit not self-consistently.

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isolated disks from L. Michel-Dansac et al. (2015, inpreparation). We summarize this paper in Section 6. Through-out, we use the following cosmological parameters:H0 = 70km s−1, WL = 0.7, and MW = 0.3.

2. THE GALPAK3D ALGORITHM

In this sectionwe outline the algorithm principles, whicharedesigned to be able to determine galaxy morphokinematicparameters from the three-dimensional data cube directly. Wediscuss the merits of using the parametric forward fit and itslimitations.

2.1. A Parametric Galaxy Model in Three Dimensions

Traditionally, kinematic analyses use two-dimensional mapsgenerated by applying line-fitting codes to determine the linewavelength centroids and widths, which are only considered tobe reliable for spaxels with sufficiently high signal-to-noiseratios (S/Ns). This S/N condition is easily met at low redshifts,but it is harder to meet for small, high-redshift galaxies. Inprinciple, the choice to work in 2D or 3D space is equivalent,but we will show that our method can work in the regime (onthe spaxels) where the S/N per pixel (S/N pixel−1) is notsufficient for line-fitting codes, which require a minimum S/Non all spaxels.

When the PSF FWHM can be characterized to sufficientaccuracy6 (within 10% or 20%;see Section 4), one can take itscharacteristics, together with the instrumental line-spreadfunction (LSF), into considerationand recover the intrinsicmodeled galaxy parameters. The algorithm uses the spectralLSF and the spatial PSF to generate a three-dimensional kernelwhose characteristics are set for the given instrument (or a user-generated instrument module).

While a full deconvolution of hyperspectral cubes would bepreferred, it is usually a challenge mathematically (a newmethod has been proposed recently by Villeneuve &Carfantan 2014), and a forward convolution of a parametricmodel offers a very useful alternative. This forward convolu-tion gives us the opportunity to estimate intrinsic modeledkinematic parameters in a wider range of seeing conditions, asillustrated in recent papers (see Bouché et al. 2013; Pérouxet al. 2014; Schroetter et al. 2015, and Martin & Soto 2015 forfirst applications).

For the forward convolution, we need a parametric model,and we focus here on a galaxy disk model for emission-linesurveys, but the algorithm is adaptable to other situations. Inorder to construct a modeled galaxy in the observationalcoordinate systems (x, y, λ), we start by generating a three-dimensional galaxy model in aEuclidian coordinate system (x,y, z), where the z-axis is normal to the galaxy plane (x, y). Weapply a radial flux profile I(r), from one of the traditionalGaussian, exponential, and de Vaucouleur choices as para-meterized by the Sérsic (1963) profile:

I r I b r Rexp 1 1nn

e e1( )( )( ) ( )= - -⎡⎣ ⎤⎦

with n = 0.5, 1.0, and 4.0, respectively, where Re is theeffictive radius, ftot the total flux, and bn such that Re isequivalent to the half-light radius R1 2, and Ie the SB at Re. Forn = 0.5, 1.0, and 4.0, the constant bn is 0.69, 1.68, and 7.67,

respectively, from b n1.9992 0.3271n - . The Sérsic index nis kept fixed given the large degeneracies it creates with otherparameters, such as the galaxy half-light radius. Thisdegenaracy is due to the fact that the SB profiles around Re

are close to one another forn = 0.5, 1.0, or 4.0, as noted inGraham & Driver (2005).To this two-dimensional disk model, we add a disk thickness

hz. We adopt a Gaussian luminosity distribution perpendicularto the plane, I z z hexp 2 z

2 2( ) ( )µ - , defining hz as thecharacteristic thickness of the disk. GalPaK3D also allows theuser to choose an exponential I z z hexp z( ) ( ∣ ∣ )µ - or a sech2

distribution I z z hsech z2( ) ( )µ . We set the disk thickness to

h R0.15z 1 2= where R1 2 is the disk half-light radius. Thischoice corresponds to h 1z ~ kpc, typical of high-redshift edge-on/chain galaxies (Elmegreen & Elmegreen 2006). At thisstage, we have a disk model in Euclidean coordinates thataccounts for the flux distribution only.For the gas kinematics, we create three kinematic cubes in

the same spatial coordinate reference frame for the velocitiesv V V V, ,x y z( )= assuming circular orbits. The rotationalvelocity v(r) with a maximum rotation velocity Vmax can haveseveral functional forms: it can be an arctan velocity profile(e.g., Puech et al. 2008), an inverted exponential (Feng &Gallo 2011), or a hyperbolic tanh profile (e.g., Andersen &Bershady 2013):

v r V r r2

arctan “arctan” 2max t( )( ) ( )p

=

v r V r r1 exp “exp” 3max t( )[ ]( ) ( )= -

v r V r rtanh “tanh” 4max t( )( ) ( )=

where r is the radius in the galaxy x y, plane, rt is the turnoverradius, and Vmaxis the maximum circular velocity. Thesechoices are more extensively discussed in Epinat et al. (2010),but it is worth noting that the “exp” and hyperbolic rotationcurves have a sharper transition around the turnover radius. Westress that our parameter Vmax is not the projected asymptoticvelocity, but is the true asymptotic velocity irrespective of theinclination.Another option, called “mass,” assumes a constant light-to-

mass ratio and sets v(r) from the enclosed light/mass I r( )<profile

v rI r

r“mass” 5( ) ( ) ( )µ

<

where r is the radius in the galaxy x y, plane and Vmax

normalizes the profile. This option has a rotation curve thatpeaks at some radius (set by the half-light radius), decreases atlarger radii, and is to be preferred for nuclear disks or whenthere is no significant dark matter component.We then rotate the disk model around two axes according to

an inclination (i) and position angle (PA, anticlockwise from y)and create a cube in x, y, and λ using the three intermediate 2Dmaps: the flux map, the velocity field, and the dispersion map( tots ). The flux map is obtained from the rotated flux cubesummed along the wavelength axis. The velocity field isobtained from the flux-weighted mean Vz velocity cube. Thetotal (line-of-sight) velocity dispersion tots is obtained from thesum of three terms (added in quadrature). It includes (i) thelocal isotropic velocity dispersion ds driven by the disk self-

6 The PSF shape matters more than the level of accuracy on the FWHM, asdiscussed in Section 4.4.

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gravity, which is r h V r rzd ( ) ( )s = for a compact “thick” orlarge “thin” disk (Genzel et al. 2008; Binney & Tremaine 2008;Davies et al. 2011);(ii) a mixing term, ms , arising from mixingthe velocities along the line of sight for a geometrically thickdisk, which is obtained from the flux-weighted variance of thecube Vz; and (iii) an intrinsic dispersion ( os )—which isassumed to be isotropic and constant spatially—to accountfor the fact that high-redshift disks are dynamically hotter thanthe self-gravity expectation. Indeed, this turbulence term os isoften observed to be 50 –80 km s 1- in z 1> disks (Lawet al. 2007, 2012; Genzel et al. 2008; Cresci et al. 2009; FörsterSchreiber et al. 2009; Wright et al. 2009; Epinatet al. 2010, 2012; Wisnioski et al. 2011)and thus dominatesthe other two terms since the mixing term ms is typically ∼15km s 1- and the self-gravity term ds is typically 10–30 km s 1- .

To summarize, the flux profile can be chosen to be“exponential” (n = 1.0), “Gaussian” (n = 0.5), and “deVaucouleur” (n = 4.0);the velocity profile v(r) can be arctan(“arctan”), inverted exponential (“exponential”), hyperbolic(“tanh”), or that of the mass profile (“mass”);and the localdispersion can be that of the thin or thick disk. There are in total10 free parameters7 to be determined from the data. The 10parameters are the xc, yc, zc positions, the disk half-light radiusR1 2, the total flux ftot, the inclination i, position angle PA, theturnover radius rt, the maximum circular velocity Vmax, and theone-dimensional intrinsic dispersion os . We will refer to the lasttwo (Vmax, os ) as kinematic parameters. Finally, the simulatedgalaxy is convolved (in 3D) with the PSF and the instrumentalLSF specific for each instrument.8 The 3D convolution isperformed using fast Fourier transform (FFT) libraries.

2.2. The Markov Chain Monte Carlo (MCMC) Algorithm

In order to determine the 10 free parameters on hyperspectralcubes, one needs an algorithm that is independent of initialguesses on the parameters, that can converge even in thepresence of local minima, and that can handle low-S/N data.This is particularly difficult for traditional minimizationmethods because the 2c hypersurface is very flat (outside theshallow well near the optimum parameters), and as a result theminimization algorithm tends to not converge and be verysusceptible to local minima.

Herewe use an algorithm to optimize the parameters usingBayesian statistics with flat priors on bound intervals for eachof the parameters. The algorithm constructs MCMCs with aMetropolis–Hasting (MH) sampler (Hastings 1970). At eachiteration we compute the new set of parameters xi 1ˆ + from thelast xiˆ set with a proposal distribution P from which to draw:

x x h P x x , 6i i i i1 1( )ˆ ˆ ˆ ˆ ˆ ( )= ++ +

where the new set of parameters is accepted or rejected as inany MH algorithm. The new proposal set of parameters xi 1+ isthen accepted or rejected according to the posterior distribution,which amounts to the likelihood exp 2 cµ - in theconsidered case of flat priors on the parameters. In otherwords, we assume that the pixels are independent and that noiseproperties are Gaussian, which is appropriate for optical/NIRdata taken in the background-limited regime, and the user can

provide the full variance cube. More appropriate likelihoodfunctions for low counts with Poisson noise can be found inMighell (1999).The scaling vector h in Equation (6) is derived from the

variance on the flat (uniform) prior distributions, whoseboundaries are adjustable (the default values are listed inTable 1). The user may need to rescale the vector h in order tohave acceptance rates between 20% and 50%. Convergence isusually achieved in a few hundredto a few thousanditerations,even though we typically let the algorithm run for 15,000iterations.In principle, one has the freedom to use any proposal

distribution P (e.g., MacKay 2003). A Markov chain is said toconverge to a single invariant distribution (the posteriorprobability) when the state of the chain persits once it isreached and is said to be ergodic when the probabilities xnconvergeto that invariant distribution as n ¥, irrespec-tively of the initial parameters (Neal 1993).9 In addition, if thesampler satisfies the following conditions P x x P x x( ∣ ) ( ∣ )¢ = ¢ ,as we have used, the algorithm reverts to the Metropolismethod, which satisfiesthe two conditions. In practice,however, one also needs a distribution that probes theparameter space efficiently in order to convergein a reasonablenumber of cpuhours, regardless of the initial parameters.A common proposal distribution is the uniform distribution

that gives equal probabilities to all possible values. TheGaussian proposal distribution P x x x, 1( ∣ ) ( )¢ = is probablythe most commonly used and is popular but has one majordrawback: the Gaussian distribution is rather narrow such thatthe algorithm becomes sensitive to the initial conditions,making the time to convergence to the optimum values verysensitive to the initial guess. If the width of the proposaldistribution is small, the convergence is too slow/large, andwhen it is large (for convergence purposes), it will lead to lowacceptance rate and poor efficiencies for convergence. Toremedy this problem, one could use a mixed distribution with aGaussian draw, say, 90% of the time and a uniform draw 10%of the time, allowing the chain to escape from a localminimum. Compared to the Gaussian proposal, the mixeddistribution has one additional parameter that needs to be fine-tuned to the problem, such as the mixing ratio.A third option, as advocated by Szu & Hartley (1987), is to

use a draw from a Cauchy distribution that has by definitionlonger wings (i.e., P is a Lorentzian profile whereP x x x x2 2 2[ ]( ∣ ) ( )g g¢ µ + ¢ - ). The Lorentzian wings are

Table 1Default Range on Each Parameter

Parameter Min Max

xc 1/3 Npixx 2/3 Npixxyc 1/3 Npixy 2/3 Npixyzc 1/3 Npixz 2/3 NpixzFlux 0 v3 i j k i j k, , , ,( )´ SR1 2 0.2 spaxel 4″

Incl. (deg) 0 90PA (deg) −180 180rt 0.01 spaxel 1″Vmax(km s 1- ) −350 350

os (km s 1- ) 0 180

7 There are only nine free parameters when the “mass” profile is used for v(r)since the turnover radius rt is irrelevant.8 The user can choose a Gaussian PSF, a Moffat PSF. The PSF can be circularor elliptical with a user-defined axis ratio b a. 9 Available at http://www.cs.toronto.edu/~radford/res-mcmc.html

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important, allowing the chain to make large jumps during theinitial “burn-in” phase and ensuring rapid convergence of thechain with no sensitivity to the initial parameters. Anotheradvantage of a Cauchy proposal distribution is that it has onlyone parameter, γ, compared to the mixed one.

We tested these various choices on simulated cubes andfound that the Cauchy proposal distribution converged fasterthan the other methods and was least sensitive to the initialparameters. In other words, with the Cauchy nontraditionalproposal distribution, a few hundredto a few thousandsteps ofthe MCMC are required to pass the burn-in phase depending onthe S/N of the data, and it is the user’s responsibility to confirmthat the MCMC chain has converged. Thus, we typically runthe chain through 10,000 or 15,000 steps to robustly sample theposterior probability distribution.

The “best-fit” values of the parameters are determined fromthe posterior distributions. We use the median and the standarddeviation of the last fraction (default 60%) of the MCMC chainto determine the “best-fit” parameters and their errors,respectively. One can also use a fraction (default 60%) of theMCMC chain around the minimum 2c . The full MCMC chainis saved such that the user can use his/her preferred technique.

The algorithm is implemented in Python and uses thestandard numpy and scipy libraires. In addition, it uses thebottleneck10 (Frigo & Johnson 2005) and FFTw11 libraries(Frigo & Johnson 2012) in order to speed up certain matrixoperations and the PSF+LSF convolution, respectively. Itrequires FITS files as inputs. The algorithm is modular so thatthe user can add specifications for other instruments. Theonline documentation describes the syntax, and it takes about 2,5, and 10 minutes on a laptop (at 2.1 GHz) to run 10,000iterations on a data cube with 303 pixels, 403 pixels, and 603

pixels, respectively. In other words, the computation timescales as t N Nlogpix pix( )µ where Npix is the number of pixels,showing that the FFT calculation dominates.

3. HIGHLIGHT APPLICATIONS

3.1. Example on 2D Data

Before applying the tool on 3D data, it is important tovalidate the method on simpler data sets, such as two-dimensional imaging data. We thus wrote a two-dimensionalversion of the algorithm, GalFit2D, one that does not include thekinematic, which is in essence similar to other parametricalgorithms (e.g., Simard et al. 1998; Peng et al. 2002), apartfrom the Bayesian approach.

Figure 1 shows a comparison between the derivedmorphological parameters from two data sets of very differentresolution. Panel (a) shows a Canada–France–Hawaii Tele-scope (CFHT) I-band image of the z 0.2~ galaxy SDSSJ165931.92+023021.92 (Kacprzak et al. 2014). Panel (e)shows anr-band image of the galaxy from the Sloan DigitalSky Survey (SDSS) at a spatial resolution of 1″. 1. For each dataset, we show the fitted (convolved) model, the residual map,and the one-dimentional SB profile. One sees that the intrinsicmodeled morphological parameters found from the SDSS data(PSF FWHM = 1″. 1) are in good agreement with the higher-resolution data (PSF FWHM = 0″. 7). Moreover, the residualsin both data sets show the spiral arms and a minor merger (or a

large clump) in the southern part of the galaxy, showing that asmooth axis-symmetric model can be used to unveil asym-metric features.

3.2. Example on a Mock Cube

Figure 2 shows an example of a mock disk model with a lowS/N (S/N pixel−1 of 4 in the central pixel) drawn from the setpresented in Section 4 and generated at 1″. 0 resolution. Thetop, middle, and bottom rows show the flux map, the velocitymap, and the apparent velocity profile Vz(r) across the majoraxis, respectively. From left to right, the panel columns showthe data, the convolved model, the modeled disk (free from thePSF), and the high-S/N high-resolution reference data(PSF = 0″. 15 and S/N = 100). In the bottom panels, thesolid red curves correspond to the reference rotation curve(obtained from the reference data set), and the trianglesrepresent the apparent rotation curve. These rotation curvesshow that the recovered kinematics from the modeled disk(intrinsic or unconvolved model) shown in the third column isin good agreement with the reference data (last column) in spiteof the low spatial resolution (1″. 0) and the low S/N in the mockdata set.This synthetic data cube was generated with a flux profile

with Sérsic index n = 1 and half-light radius R 0. 51 2 = ,corresponding to 2.5 MUSE/KMOS pixels), an “arctan”velocity profile with V 200max = km s 1- , a thick disk with avelocity dispersion 80os = km s 1- , an inclination i = 60°, aPA 130= , and with instrumental specifications for the newVLT MUSE instrument (0″. 2 pixel−1, 1.25 Å pixel−1,LSF = 2.14 pixels). The integrated total flux is 10−16

erg s−1 cm−2, and the synthetic noise per pixel is5 10 20s = ´ - erg s−1 cm−2 Å−1.

The synthetic data cube is also displayed in Figure 3, whichshows three one-dimensional spectra (a) taken at the threelocations labeled in the image shown in panel (b). Panel (c)shows a 3D representation of the data (blue) with the modeloverlaid (red) made with the “visit” software12, where thelight/dark areas correspondto two cuts at fluxes of 6 and8 × 10−20 erg s−1 cm−2 Å−1, i.e., anS/N pixel−1 of 1.2 and1.8, respectively.We ran the algorithm with 15,000 iterations, and Figure 4

shows the MCMC chains for the 10 free parameters along withthe 2c evolution in the bottom panel. The values of the fittedparameters (and their errors) shown by the black lines (graylines) are computed from the median (standard deviation) of thelast 60% iterations of the posterior distributions. The recoveredparameters are listed in Table 2and showgood agreementbetween the input and recovered values.Figure 5 shows the joint distributions for the radius, PA,

inclination, maximum velocity, and dispersion parameters. Theestimated parameters and their respective 1s error are shown asa solid line and dashed line, respectively. This figure shows aclear covariance between the turnover radius and theasymptotic velocity Vmax, and a small covariance between theinclination and Vmax. The users of GalPaK3D are stronglyadvised to confirm the convergence of the parameters usingdiagnostics like Figure 4 and to investigate possible covariancein the parameters, as these tend to be data specific, usingdiagnostics like Figure 5.

10 Available at https://pypi.python.org/pypi/Bottleneck.11 Available at https://pypi.python.org/pypi/pyFFTW. 12 Available at http://visit.llnl.gov/.

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4. TESTS WITH MOCK DATA CUBES

In order to characterize the performances and limitations ofthe GalPaK3D algorithm statistically, we generated a set of1728 cubes again with a MUSE configuration over a grid ofparameters listed in Table 3. The synthetic cubes weregenerated with noise typical to a 1 hr exposure with MUSEcorresponding to a pixel noise of 5 1020s = ´

erg s−1 cm−2 Å−1.We use a range of inclinations i from 20°to 80°. We use a range of disk sizes, with half-light radii R1 2 =0″. 3, 0″. 6, and 1″. 0 corresponding to a R1 2 of 2.5, 5, and7.5 kpc, covering the range of observed sizes at z 1~ (e.g.,Trujillo et al. 2006; Williams et al. 2010; Dutton et al. 2011).For each of the galaxy sizes, we use the Vmax–R1 2 scaling

relation (Equation (8) of Dutton et al. 2011) and its redshift

Figure 1. Application of the two-dimensional version of the MCMC algorithm (“GalFit2D”) on the z 0.2~ SDSS J165931.92+023021.9 with m 18.40r = mag fromKacprzak et al. (2014). Similarly to GalPaK3D, Galfit2D performs a parametric fit with an MCMC algorithm using set surface brightness profiles convolved with theseeing. The top row shows the result from archival CFHT Iband taken at a resolution of 0″. 7. The bottom row shows the result from the SDSS r-band image that has aresolution of 1″. 1. Panels (a) and (e) show the data. Using an exponential profile, panels (b) and (f) show the seeing-convolved model;(c) and (g) the residuals, i.e.,data-model normalized to the pixel noise σ; and (d) and (h) the one-dimensional SB profile. The recovered intrinsic disk scalelength Rd is about 1″ in both cases, inspite of the different spatial resolution.

Figure 2. Example of the algorithm application on a disk model simulated with a seeing of 1″. 0 (FWHM), a flux of 10−16 erg s−1 cm−2, and an S/N pixel−1 of ∼4 atthe brightest pixel. The top, middle,and bottom rows show the flux map, the velocity map, and the apparent velocity profile Vz(r) across the major axis, respectively.From left to right, the panel columns show the data, the convolved model, the modeled disk (free from the PSF), and the high-S/N, high-resolution reference data(PSF = 0″. 15 and S/N = 100). In the bottom panels, the solid red curves correspond to the reference case, the triangles represent the apparent rotation curve, and thedotted lines show the apparent V isinmax . One sees that the velocity profile from the modeled disk (third column) is in good agreement with the reference data.

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evolution (Equation (5) of Dutton et al. 2011) to set the rotationkinematics (Vmax). In particular, the sizes R1 2 = 2.5, 5, and7.5 kpc correspond to Vmax values ranging from ∼100 to 250km s 1- . We use “arctan” rotation curves to generate our mockdata cubes, and we have verified that our results remain thesame with “exponential” rotation curves.We use the scaling relation between the turnover radius rt

and the disk scalelength Rd that exists for disk galaxies (e.g.,Figure 1 of Amorisco & Bertin 2010) to set the turnover radiusrt. In particular, we set rt to R 1.8d where the 1.8 factor13 isdetermined empirically for the arctan rotation curve to satisfythe linear correlation between the galaxy disk scalelengthR R 1.68d 1 2= and RW, defined as the radius r whereV r V2 3 max( ) = (Amorisco & Bertin 2010).

For each of the galaxy sizes, the disk thickness ish R0.15z 1 2= , i.e., ranging from 0.4 to 1.3 kpc, bracketingthe average values of h 1z ~ kpc, found for high-redshift edge-on/chain galaxies (Elmegreen & Elmegreen 2006).We used fluxes for an [O II] (λ3727) emission line, expected

to lie in the MUSE spectral range at redshifts between 0.6 and1.2, with integrated fluxes from 3 × 10−17 erg s−1 cm−2 to3 × 10−16 erg s−1 cm−2 corresponding to the range of observedvalues (e.g., Bacon et al. 2015; Comparat et al. 2015, andreferences therein). We use a constant noise value per pixel of

5 10 20s = ´ - erg s−1 cm−2Å−1, in order to simulate the noiselevel of a 1 hr exposure, but we stress that the algorithm acceptsvariance/noise cubes to account for pixel-to-pixel noisevariations. In addition, we generated cubes with very high S/N (S/N = 100, flux 3 10 15= ´ - erg s−1 cm−2) and with aseeing typical of AO conditions, with a PSF FWHM of 0″. 15.These will serve as reference data sets.

4.1. Surface Brightness and Signal-to-noise Ratio

One could imagine that the S/N in the recovered parametersis a function of the average S/N pixel−1, or the apparent SBsince the observed central SB scales directly with the S/N inthe central pixel. But clearly the compactness of the object withrespect to the seeing plays a large role (as discussed in Driveret al. 2005; Epinat et al. 2010). Very compact objects(compared to the beam or the PSF) have high SB by definition(and high S/N pixel−1), but the morphology and/or kinematicinformation may be lost owing to the beam smearing. On theother hand, very extended objects have low SB (and low S/N pixel−1)but have many pixels in the outer regions (with lowS/N), where most of the information on the galaxy is locatedand not affected by the beam.Before illustrating this point, it is important to define

commonly used terms such as the SB of galaxies. From anylight profile I(r) such as given by Equation (1), there are manyways to define galaxy SB, such as Ie the SB at the effectiveradius Re, Io the intrinsic SB at the central pixel, Ao theobserved SB at the central pixel, and SB1 2 the average SBwithin the intrinsic half-light radius R1 2:

F

RSB

0.5. 71 2,conv

tot

1 2,conv2

( )p

º

Figure 3. For the synthetic data in Figure 2, we show three one-dimensionalprofiles (panels (a)) comparing the data (thin line) and the model (thick line)taken at the location labeled “1,” “2,” and “3” in panel (b). Panel (c) shows a3D representation of the data (blue) with the model overlaid (red) where weused two flux levels at 5 × 10−20 and1.5 10 19´ - erg s−1 cm−2 Å−1. The cubeorientation is shown, where the wavelength axis is the z-direction.

(An interactive version of this figure is available.)13 For “exponential” rotation curves, one should set rt to R 0.9d ´ in order tosatisfy the scaling relation; for “tanh” rotation curves, one should set rt toR 1.25d ´ .

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where Ftot is the galaxy total flux. A related quantity toEquation (7) is the observed SB, defined as

F

SSB

0.581 2,obs

tot

1 2,obs( )º

where Ftot is the galaxy total flux and S1 2,obs the galaxyapparent area given by a bpº where a and b are the observedmajor and minor semiaxes, respectively,of the galaxy. Therelations between these various definitions are described in theappendix.

To illustrate the point made at the beginning of thissection, we show in Figure 6(a) the relative errors p pd ºp p pfit in in( )- on some of the estimated parameters for ourmock data cubes generated in Section 4 as a function of centralSB, SB1 2,obs (defined in Equation (8)). Each row shows the

relative errors for the maximum circular velocity Vmax, the sizeR1 2, the PA, and inclination i from top to bottom, respectively.The crosses, squares, and circles represent the three subsampleswith sizes ∼2.5, 5, and 7.5 kpc, respectively. One sees that theerrors in the morphological parameters (size, PA, inclination)do increase toward low SBs, but the threshold point at whichthe relative errors reach ∼100% depends on the galaxy size,represented by the symbols. This illustrates the well-known factthat very extended objects have low SB (and low S/N pixel−1)but have many pixels in the outer regions thatcontain useful information.As argued at the beginning of this section and demonstrated

in Figure 6, SB alone might not be sufficient to determine theS/N in the fitted parameters, but the compactness of the galaxywith respect to the beam also plays an important role. InFigure 6(b), we show the relative error p pd with respect to theobserved SB1 2,obs times the size-to-PSF ratio R R1 2 PSF( )a.The symbols correspond to galaxy subsamples with varioussizes as in Figure 6(a). The index α was found to be empirically∼1 in order to have the relative errors for each of thesubsamples follow a similar trend and may differ sightly foreach of the parameters p. In fact, we find that α isapproximately 0.8, 1.2, and 1.4 for the size, PA, and inclinationparameter, respectively.These empirical results can be explained by the following

arguments. The apparent SB within the half-light radiusSB1 2,conv (Equation (7)) and the observed SB1 2,obs (Equa-tion (8)) are proportional to the SB (or S/N) of the centralpixel, Ao, as shown in the appendix (Equation (19)). In the caseof no PSF convolution, Refregier et al. (2012) showed that(their Equation (12)) the relative error a a( )s on morpholo-gical parameters (its major-axis a) scales inversely to the

Figure 4. Full MCMC chain for 15,000 iterations for the example shown in Figure 2. Each of the small panels corresponds to one parameter. One sees that the “burn-in” region is confined to the first 1000 iterations. The estimated parameters are shown with the red line and are calculated from the last 60% of the chain. The gray linesshow the 1s standard deviations, and the dotted lines show the 95% confidence interval. Note the fitted flux value is 1.06 0.03 10 16( ) ´ - erg s−1 cm−2, which isfound from the sum of the pixel values (here 8.5 10 17~ - erg s−1 cm−2 Å−1) times the 1.25 Å per spectral pixel. The bottom panel shows the 2c evolution relative tothe minimum, log 2

min2[ ]c c- . We use this nonstandard metric in order to show that the variations of the 2c around the minimum are 3–4 orders of magnitude

smaller, reflecting a very flat hypersurface. Hence, a plot of 2c or of the likelihood would show a straight line.

Table 2Comparison between the Model Input Values and the Recovered Values with1σ Errors and Confidence Intervals (CI) for the Example Shown in Figure 2

Parameter Input Output [95% CI]

xc (pixels) 15 15.05 ± 0.09 [14.87;15.24]yc (pixels) 15 15.06 ± 0.09 [14.89;15.23]zc (pixels) 15 15.05 ± 0.07 [14.92;15.19]Flux (10−16) 1 1.06 ± 0.03[1.01;1.09]R1 2 (arcsec) 0.82 0.85 ± 0.04[0.78;0.95]Incl. deg) 60 62 ± 3[58;68]PA deg) 130 126 ± 2[123;130]rt (pixels) 1.35 1.32 ± 0.42[0.8;2.47]Vmax (km s 1- ) 200 202 ± 22[172;257]

os (km s 1- ) 80 82 ± 5 [73;90]

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central Io, where Io is the intrinsic central SB (Equations (12)–13). In the presence of a PSF convolution, Equation (16) ofRefregier et al. (2012)—which applies here—shows that therelative errors on the major-axis a scaleas

a

aA R R1 . 9o

1PSF2

1 22( )( ) ( )s

µ +-

where RPSF is the radius of the PSF (RPSF º FWHM/2) andR1 2 the intrinsic half-light radius.In our cases, for high-redshift galaxies, the ratio R RPSF 1 2 is

1.0 , and after performing a Taylor expansion aroundR R x1PSF 1 2 ( )~ - with x R R R1 2 PSF 1 2( )º - andx 1∣ ∣ , one finds that the factor R R1 PSF

21 22( )+ is

approximately x R R2 1 2 PSF 1 2( )~ - ~ . Hence, Equation (9)

on the errors in the major-axis a becomes in the regime whereR RPSF 1 2 1.0 :

a

a

R

RA

R

RSB , 10

o1 2

PSF

1

1 2

PSF1 2,obs

1

( )

( )

sµ

µ

-

-

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

which shows that the quality of the estimated morphologicalparameters will depend on both the pixel S/N (SB) and thegalaxy compactness with respect to the beam, R R1 2 PSF, asshown in Figure 6(b).In both Figures 6(a) and 6(b), the gray solid lines show the

expected behavior for the morphological parameters (Equa-tion (10)), and one sees that they agree better with themockdata in the right panels for the morphological parameters. Thisshows that the Refregier et al. (2012)formalism describes therelative errors on the morphological parameters(size, PA, andinclination) relatively well, as a first approximation. We notethat Equation (10) is only an approximation to Equation (9)when R R 11 2 PSF and that there might be other dependen-cies for the other morphological parameters, namely, for the PAand for the inclination. Herewe refer the reader to Table 1 ofRefregier et al. (2012) and their Appendix for further details; itis beyond the scope of this paper to present a full 3D derivationof the Refregier et al. (2012)formalism.Contrary to the morphological parameters, the errors in the

kinematic parameter Vmax show strong positive (negative)biases in the smallest (largest) mock galaxies, represented bythe crosses (circles) in the top panel of Figure 6(b). Thepositive bias for the most compact galaxies (crosses) withrespect to the beam can be understood because the Vmaxinformation is located mostly in the outer parts of the galaxy,where the S/N is too low. The negative bias for the largest

Figure 5. Joint distributions for the radius, PA, inclination, maximum velocity, and dispersion parameters for the example shown in Figures 2–4. The estimatedparameters and their respective 1s error are shown as a solid line and dashed line, respectively. One sees that the traditional degeneracy between Vmax and theinclination i is broken, thanks to our 3D method, but the seeing leaves a significant correlation between Vmax and the turnover radius rt. The presence of thisdegenerancy is dataspecific and seeing specific, not a generic feature of the algorithm.

Table 3Range of Parameters for the 1728 Mock Galaxies

Parameter Grid Values

Flux (10−17erg s−1 cm−2) 3, 6, 10, 30Seeing (arcsec) 0.6, 0.8, 1.0, 1.2Redshift 0.6, 0.9, 1.2R1 2 (kpc) 2.5, 5 and 7.5a

R1 2 (arcsec) 0.3, 0.6 and 1.0b

i (deg) 20, 40, 60, 80PA (deg) 130rt (arcsec) 0.1–0.3c

Vmax (km s 1- ) 110, 200, 280

os (km s 1- ) 20, 50, 80

Notes.a Exact value to satisfy the size–velocity scaling relation (Dutton et al. 2011).b Exact value will depend on the redshift.c Exact value to satisfy the scaling relation between the galaxy size and theinner gradient (Amorisco & Bertin 2010) using r R 1.8t d= .

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galaxies (1″ in R1 2) at low SB is likely due to the spatial cut ofour mock cubes being too small.

We will return to the reliability of Vmax in Section 4.3and now turn to a more detailed discussion on the reliability ofthe parameters (size, inclination, disk velocity dispersion,and Vmax). Whilewe used an arctan rotation curve, we note thatthe following results were found to be identical when we usedan “exponential” rotation curve.

4.2. Reliability of Morphological Parameters

We have shown in the previous section with Figure 6 that therelative errors on the half-light radius follow appoximately theexpectation from the Refregier et al. (2012)formalism.Herewe investigate whether the relative errors depend onsome of the other parameters, such as inclination, seeing,and size.

Figure 7 shows the relative errors p p pfit in in( )- for severalkey parameters p. The bottom (top) row shows the result for thesize parameters R1 2 (inclination i), as a function of seeing,redshift, inclination, and size-to-PSF ratio R R1 2 PSF. The blackcurves with increasing thickness correspond to subsampleswith different SB levels (labeled), where the zeropoint (dottedline) has been offsetfor clarity purposes. The data pointsrepresent the median, and the size of the errorbars representsthe standard deviation for each of the subsamples, where wehave typically ∼100 mock cubes per bin. We note that themedian standard deviations on the parameters (from theposterior distributions) tend to be within 20% of these binnedstandard deviations.

From this figure, one sees that the GalPaK3D algorithmrecovers the intrinsic half-light radius R1 2 irrespectively ofseeing, redshift, and/or intrinsic size. Note that the relativeerrors with respect to size-to-seeing ratio at a fixed SB followroughly the expectation from Equation (9), where the factor

R R1 PSF 1 22( )+ saturates to unity in our regime with

R R 11 2 PSF ~ to 2.5. These results are not affected by thechoice of the SB profile (Sérsic n).14

From the top row in Figure 7, one sees that the inputinclination is recovered except at the two smallest fluxes andfor the more face-on cases. The reason that the algorithm canrecover the inclination well is that the algorithm breaks thetraditional degeneracy between Vmax and i using the SB profile(i.e., the axis ratio b/a), whereas traditional methods fitting thekinematics on velocity fields have a strong degeneracy betweenVmax and the inclination i.

4.3. Reliability of Kinematic Parameters

Figure 8 shows the relative errors p p p p pfit in in( )d º - forthe parameters Vmax (top row) and disk dispersion os (bottomrow) as a function of seeing, os , inclination, and size-to-PSFratio R R1 2 PSF. The curves as a function of redshift are notshown, because the relative errors do not depend on thisparameter as in Figure 7. The black curves with increasingthickness correspond to subsamples with different SB levels

Figure 6. Relative errors on the estimated parameters p pd , defined as p p pfit in in( )- . Each row shows p pd for the maximum circular velocityVmax, the size R1 2, thePA, and inclindation i from top to bottom, respectively. The crosses, squares, and circles represent the three subsamples with sizes ∼2.5, 5, and 7.5 kpc, respectively(Table 3). The relative errors for the morphological parameters (size, PA, inclination) are binned. (a)Relative errors as a function of central SBSB1 2,obs inerg s−1 cm−2 arcsec−2. (b)Relative error as a function of central SBSB1 2,obs times R R1 2 PSF( )a, where R1 2 is the galaxy intrinsic half-light radius and RPSF the PSFhalf-light radius. We found, empirically (see text), that α is approximately 0.8, 1.2, and 1.4 for the size, PA, and inclination parameter, respectively. These values areclose to the expectation of 1.0- of Equation (10) (gray lines) derived for morphological parameters in imaging data by Refregier et al. (2012). The relative errorinVmax doesnot follow the expected relationand is subject to strong systematics for the smallest and largest mock galaxies (crosses). This is due to Vmax beingconstrained in the outer parts of the galaxy, where the S/N is thus not sufficient for the compact galaxies or where the mock cube is too small for the largest galaxies.

14 A curve-of-growth analysis on the two-dimensional flux map can sometimesyield a constraint on the Sérsic index n and a more accurate determination ofthe intrinsic half-light radius (R1 2).

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Figure 7. Relative error p pd —defined as p p pfit in in( )- —for the parameters R1 2 (top) and inclination i (bottom),as a function of seeing, redshift, inclination, andsize-to-PSF ratio R R1 2 PSF (from left to right). The curves with increasing thickness correspond to subsamples with different SB levels from SB 2 10 17< ´ - ,2 10 17´ <- SB 5 10 17< ´ - , 5 10 17´ <- SB 1 10 16< ´ - , SB 1 10 16> ´ - erg s−1 cm−2 arcsec−2, respectively, where the zeropoint (dotted line) has beenoffsetfor clarity purposes. SB is the observed surface brightness within the observed half-light radius SB1 2,obs times the seeing-to-size ratio R R1 2 PSF, as in Figure 6.The data points represent the median, and the size of the errorbars represents the standard deviation for each of the subsamples. One sees that the GalPaK3D algorithmrecovers the morphological parameters irrespectively of seeing, redshift, and/or intrinsic size.

Figure 8. Relative error p pd —defined as p p pfit in in( )- —for the kinematic parameters Vmax (top) and os (bottom),as a function of seeing, disk dispersion os ,inclination, and size-to-PSFR R1 2 PSF (from left to right). The curves as a function of redshift are not shown because the relative errors do not depend on thisparameter, as in Figure 7. For the Vmax parameter, the black (red) curves show the results when R R1 2 PSF is less (greater) than 1.5. One sees that theGalPaK3D algorithm recovers the kinematic parameters irrespectively of seeing andredshift, provided that the galaxy is not too compact with R R1 2 PSF largerthan 1.5.

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(labeled), where the zeropoint (dotted line) has been offsetforclarity purposes. The data points represent the median, and thesize of the errorbars represents the standard deviation for eachof the subsamples, as in Figure 7.

Figure 8(top) shows that the GalPaK3D algorithm recoversthe maximum velocity Vmax irrespectively of seeing, diskdispersion, and redshift (now shown) provided that the galaxyis not too compact. For small galaxies with R R1 2 PSF less than1.5, the figure shows that it is increasingly difficult to estimatethe correct values for the most compact galaxies, with largeuncertainties and significant overestimations of this parameter.This result was already pointed out in Epinat et al. (2010, theirFigure 13) using 2D kinematic models. Epinat et al. (2010) alsonoted that using a simple flat rotation curve to model the disk,the maximum velocity can be recovered with an accuracy betterthan 25%, even when R R1 2 PSF is less than about ∼2.

Figure 8(bottom) shows that the GalPaK3D algorithmrecovers the disk dispersion irrespectively of seeingandredshift (not shown). Given the instrumental resolution ofMUSE used here (R 130 km s−1), small dispersions aremore difficult to recover. We note that the local dispersion israther sensitive to the instrument LSF FWHM, as one mightexpect. The user can specify more than one type of LSF(Gaussian or Moffat), and a user-provided vector can bespecified if the parametric LSF is not sufficient to describe theinstrument LSF.

4.4. A Note Regarding the PSF Accuracy

One could argue that our results are driven by the fact thatwe use the exact same PSF (in 3D) as the one used to generatethese modeled galaxies. To test the reliability of the algorithmin more realistic situations, when the PSF FWHM is not knownaccurately, we ran the algorithm on the same set of data cubeswith a random component added to the FWHM of the PSFgiven by a normal distribution with 0.1s = , corresponding touncertainties in the FWHM of ∼20%. We found that theaccuracy of the spatial kernel (PSF) has little impact on therecovered parameters. On the other hand, we find that the shapeof the PSF is more critical, especially for the morphologicalparameters such as the axis ratio b a (or the inclination). Wenote that sophisticated tools exist to determine the PSF fromfaint stars in data cubes, such as the algorithm of Villeneuveet al. (2011).

To conclude this section, our algorithm is able to recover themorphological and kinematic parameters from synthetic datacubes over a wide range of seeing conditions provided that thegalaxy is not too compact and has a sufficiently high SB. Thus,for galaxies to be observed with MUSE in the wide-fieldmodein 1 hr exposure and noAO, we find that the algorithm shouldperform well provided that the SB is greater than a few×10−17 erg s−1 cm−2 arcsec−2 and as long as the the size-to-seeing ratio R R1 2 PSF is larger than 1.5 (or R FWHM1 2

0.75> ).

5. APPLICATION ON HYDRODYNAMICALSIMULATIONS

In the previous section we validated the algorithm onsynthetic or mock data, which have by definition no defects,i.e., are perfectly regular and symmetric. In order to validate thealgorithm on more realistic data, we now analyze the

performance of the algorithm on data cubes created fromsimulated galaxies generated from a hydrodynamical simula-tion (L. Michel-Dansac et al. 2015, in preparation). This isintended to validate the algorithm in the presence of systematicdeviations from the disk model.

5.1. From Hydrodynamical Simulations to Data Cubes

The simulation used in this work comes from a set ofcosmological zoom simulations, each targeting the evolutionuntil redshift 1 of a single halo and its large-scale environment.The full sample of simulations is presented in detailinL. Michel-Dansac et al. (2015, in preparation). Here we focuson one output of one simulation to complement the test casesfrom Section 4 with a more realistic, intermediate-redshift, star-formingdisk galaxy.The simulations have been run with the adaptative mesh

refinement code RAMSES (Teyssier 2002) using the standardzoom-in resimulation technique to model a disk galaxy in acosmological context. Each simulation has periodic boundariesand nested levels of refinement in a zoom region around thetargeted halo, in both DM and gas. The refinement strategy isbased on the quasi-Lagrangian approach. The simulationzoomsin a dark matter halo inside a h20 1- Mpc comovingbox, achieving a maximum resolution of ∼200 pc. The virialmass of the dark matter halo is approximately M3 1011´ atz = 1, sampled with roughly 600,000 particles.The simulation implements standard prescriptions for

various physical processes crucial for galaxy formation: starformation, metal enrichment, and kinetic feedback due to TypeII supernovae (Dubois & Teyssier 2008);metal advection;-metallicity- and density-dependent cooling;and UV heatingdue to cosmological ionizing background (see Few et al. 2012for more details on similar simulations but focusing on z= 0Milky-Way-type galaxies).The simulated galaxy is a typical z = 1 star-forming galaxy

with M M3 1010 = ´ and a gas fraction of 0.33. The galaxy

exhibits a disk morphology with spiral arms, as seen inFigure 9 (top right panel).From the output of the hydro-simulation, we generated a data

cube with the Spectrograph for INtegral Field Observations inthe Near Infrared (SINFONI) instrumental resolution and pixelsize (0″. 125 pixel−1 and 2 Å pixel−1) using the star formationrate (SFR) and metallicity information in each cell. Toconstruct the mock data cube, the simulated galaxy isartificially placed at z = 1.3 ( 1.5 mcl m for Hα) and rotatedwith an inclination of 60°. Star-forming cells are selected bycomputing the mass of young stars inside each cell of thegalaxy. Then, we convert this star formation rate into Hα fluxusing the Kennicutt (1998) calibration. For each cell, we alsocompute the flux in the [N II] line from the values of theHα flux and the oxygen abundance following the calibrationgiven by Pérez-Montero et al. (2009). For each spatial elementor spaxel, we sum the contribution (to the spectrum) of eachcell along the line of sight. Each contribution has its own line-of-sight velocity, which blueshifts or redshifts the lines. Theline width in the spectrum is then due to the sum or integralover the cells, which is then convolved with the instrumentalprofile.We generated seeing-convolved cubes with seeing of 0″. 50,

0″. 65, 0″. 80, 1″. 0, and 1″. 2 (corresponding to typical values inthe NIR with SINFONI) and 0″. 15 (corresponding to AO-

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assisted observations) and added noise corresponding to agiven max S/N pixel−1. Cubes generated with a S/N equal to100 and a seeing of 0″. 15 are used as reference cubes. The finalcube size is 28 28 30´ ´ (in x, y, λ directions), but we alsoproduce another set of cubes of size 28 28 200´ ´ pixels toallow sufficient wavelength baseline for our custom line-fittingalgorithm that was used to produce the 2D velocity mapsshown in Figure 9.

5.2. Application of the Algorithm

Figure 9 shows the results of the GalPaK3D algorithm for aseeing of 0″. 8 and a minimum S/N pixel−1 of 3 in the brightestpixel. As in Figure 2, the top, middle, and bottom rows showthe flux map, the velocity map, and the apparent velocityprofile Vz(r) across the major axis, respectively. From left toright, the panel columns show the data, the convolved model,the modeled disk (free from the PSF), and the high-S/N, high-resolution reference data (PSF = 0″. 15 and S/N = 100). In thebottom panels, the solid red curves correspond to the referencerotation curve (obtained from the reference data set), and thetriangles represent the apparent rotation curve. By comparingthe two, one sees that the algorithm is able to recover thekinematics (third column) in a regime where traditional 2Dmethods (left most column) tendto be noisier. In other words,the recovered kinematics from the modeled disk (intrinsic orunconvolved model) shown in the third column is in goodagreement with the reference data (last column) in spite of thelower spatial resolution (0″. 8) and the lower S/N in thedata set.

We ran the GalPaK3D algorithm on the data cubes, settingthe rotation curve v(r) to an arctan profile and setting the Sérsic

index n to 1.0.15 From the cube with an S/N of 100, theinclination found by the GalPaK3D algorithm is 58° ± 2°, andthe half-light radius R1 2 is 3.4 0.1~ kpc (or ∼0″. 4), and itsasymptotic maximum velocity Vmax is 215 10~ km s 1- ,placing it close to the z 1.5~ size–velocity relation of Duttonet al. (2011). The asymptotic maximum velocity is close to theoneextracted directly from the simulation, which is 235km s 1- .

We repeated the exercise on this simulated galaxy varyingthe luminosity (SFR in our case) where the noise level is set fora given exposure time corresponding to a 2 hr integration withthe SINFONI instrument. Figure 10 shows the maximum S/Nper pixel (solid lines) as a function of the seeing FHWM forfive fixed SFRs, 5, 10, 15, 30, and 60 M yr 1-

, respectively.The green region shows the parameter space where thealgorithm is able to recover the kinematics parameters within20%, from the value determined in the high-S/N cube. Theyellow region shows the parameter space where the algorithmis marginally able to recover the kinematics parameters, i.e.,within 20%–40% The red region shows the parameter spacewhere the algorithm is unable to recover the kinematicsparameter, where the relative error is larger than 40%. This plotshows that the kinematic parameters can be well estimatedirrespectively of seeing, provided that the S/N is above acritical value (3 in this case). Consequently, when the PSFFWHM is slightly below the original scientific goal, theoptimal observing strategy is to integrate longer.In the background-limited regime, the S/N per pixel scales

as texpµ , where texp is the exposure time. Given that the total

Figure 9. Application of the MCMC algorithm on a disk galaxy generated with the AMR code RAMSES (Teyssier 2002) and “observed” with a seeing of 0″. 8(FWHM). The maximum S/N is ∼3 in the brightest pixel. The top, middle, and bottom rows show the flux map, the velocity map, and the apparent velocity profileVz(r) across the major axis, respectively. From left to right, the panel columns show the data, the convolved model, the modeled disk (free from the PSF), and the high-S/N, high-resolution reference data (PSF = 0″. 15 and S/N = 100). In the bottom panels, the solid red curves correspond to the reference case, the triangles representthe apparent rotation curve, and the dotted lines show the apparent maximum line-of-sight velocity V isinmax . One sees that the velocity profile from the modeled disk(third column) is in good agreement with the reference data (solid line) at 0″. 15 resolution.

15 We also ran the algorithm with “Gaussian” profiles with n = 0.5, leading tovery similar results.

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flux of a circular extended source is F Aoobs2ps~ where

R R 1.1721 22

PSF2 2( )s = + , the S/N in the central pixel (i.e.,

the central SBc, or Ao) will scale as

AA t r

t r

t

R Rr

S NSB

11

oo exp pix

2

sky exp pix2

exp

PSF2

1 22 pix

( )

( )

µ

µ+⎡⎣ ⎤⎦

where RPSF the PSF radius, and R1 2 the object half-light andrpix the pixel size in arcseconds,such that a change of 0″. 2 inthe PSF FWHM (from 0″. 8 to 1″. 0)corresponds to a fractionchange of 15% in S/N for a galaxy of R 0. 61 2 = , andaccordingly 30%more exposure time would be required toreach the same S/N.

6. CONCLUSIONS

In this paperwe presented an algorithm to constrainkinematic parameters of high-redshift disks directly fromthree-dimensional data cubes. The algorithm uses a parametricmodel and the knowledge of the three-dimensional kernel toreturn a 3D modeled galaxy and a data cube convolved with the3D kernel. The parameters are estimated using an MCMCapproach with nontraditional sampling distributions in order toefficiently probe the parameter space.

In summary,

1. the 2D version of the algorithm is used on an SDSS r-band image of a z 0.2~ galaxy (Figure 1) taken at 1″. 1resolution. We find that the morphology is well recoveredcompared to a higher-resolution (0″. 7) CFHT image;

2. using a set of 1728 mock data cubes, Figure 6 shows thatthe accuracy on the recovered parameters depends on theproduct of the central SB, SB1 2,obs times the size-to-seeing ratio R R1 2 PSF

1( )~ , following approximately theanalytical expectation of Refregier et al. (2012);

3. from this set of mock data cubes, the morphologicalparameters do not depend on seeing, redshift, or the size-to-seeing ratio (Figure 7);

4. from this set of mock data cubes, the robustness of thealgorithm in recovering the kinematics parameters is alsoindependent of seeing and redshift,provided that the ratiobetween the galaxy half-light radius and the PSF radiusR R1 2 PSF( ) is larger than 1.5 (Figure 8);

5. we also find that the accuracy in the recovered parametersdoes not depend on the FWHM accuracy, but dependsmore critically on the shape of the PSF, except for thedisk dispersion os , which depends critically on theinstrument LSF;

6. using a simulated disk galaxy from the hydro-simulationof Michel-Dansec et al., which contains asymmetricdeviations, we found that the kinematic parameters can bewell estimated irrespectively of seeing, provided that theS/N is above a critical value (3 in this case; Figure 10).Consequently, when the PSF FWHM is slightly above theoriginal scientific goal (1″. 0 instead of 0″. 8), the optimalstrategy is to integrate 30% longer (Equation (11)) for agalaxy of size R 0. 61 2 = .

In conclusion, the GalPaK3D algorithm can provide reliableconstraints on galaxy size, inclination, and kinematics over awide range of seeing and of S/N. However, the algorithmshould not be used blindly, and we stress that users ofGalPaK3D are strongly advised (1) to look at the convergenceof the parameters (as in Figure 4);(2) to investigate possiblecovariance in the parameters (as in Figure 5), as these are ratherdata specific;and (3) to adjust the MCMC algorithm to ensurean acceptance rate between 30% and 50%, as discussed in theonline documentation.16

Recent applications of the GalPaK3D algorithm can be foundin Péroux et al. (2013), Bouché et al. (2013), Schroetter et al.(2015), and Bolato et al. (2015), which illustrate the potential inusing a global 3D fitting technique.

We are very grateful to the referee for his/her careful read ofthe manuscript and the detailed report that led to a significantlyimproved manuscript. L.M.D. acknowledges support from theLyon Institute of Origins under grant ANR-10-LABX-66.Numerical simulations used in this work were performed usingHPC resources from GENCI-CINES (Grant 2013-x2013046642). N.B. acknowledges support from a CarreerIntegration Grant (CIG) (PCIG11-GA-2012-321702) within the7th European Community Framework Program. We thankAntoine Goutenoir for his assistance in the current implemen-tation and web services.

APPENDIXSURFACE BRIGHTNESSES

For extended sources with total flux Ftot and exponentialprofiles, i.e., r I rSB( ) ( )º , one can define several measures ofSBs. We have the following:

1. The central SB Io, which is

IF

R212o

tot

d2

( )p

=

in the case of an exponential flux profile

Figure 10. S/N pixel−1 as a function of seeing (FWHM) for our SINFONI datacube simulated for a 2 hr exposure time. The lines correspond to a given SFR atz = 1.3 from SFR = 5 to 60 M yr 1-

. The simulated cubes (generated from thehydrodynamical simulation) can be reasonably well fitted by our algorithmprovided that the S/N pixel−1 (at the central region) is greater than 3,irrespectively of seeing. This diagram applies to galaxies with inclinationsaround 60~ .

16 http://galpak.irap.omp.eu/doc/overview.html

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I r I r Rexpo d( ) ( )= - since F I R2otot d2p= , where the

half-light radius R R1.681 2 d= . In the case of aGaussian flux profile I r I rexp 2o

2 2( ) ( )s= - , it is

IF

213o

tot2

( )ps

=

where the half-light radius R 1.171 2 s= .2. The average SB within the half-light SB1 2:

F

RISB

0.5, 14o1 2

tot

1 22

( )p

= µ

where R1 2 is the true or intrinsic half-light radius(R R1.681 2 d= ).

3. The central pixel SB, SBc:

ASB 15c o ( )=

where the observed SB profile F robs ( ) is the convolutionof I(r) with the PSF G(r), i.e.,F r A rexp 2oobs

2 2( ) ( )s- where now σ contains thecontributions from the intrinsic profile and from the PSFvia R R1.17 2

1 22

PSF2( )s = + ( R1 2,conv

2= ). RPSF is theradius of the PSF (RPSF º FWHM/2).

4. The apparent central SB within the half-light radius,SB1 2,conv:

F

R

F

R RSB

0.5 0.5161 2,conv

tot

1 2,conv2

tot

1 22

PSF2( ) ( )

p p=

+

where R1 2,conv is the convolved half-light radius.5. The observed central SB within the observed galaxy

surface area, SB1 2,obs:

F

S

F

abSB

0.5 0.5171 2,obs

tot

1 2,obs

tot ( )p

= =

where a and b are the observed major and minor axis,respectively.

The first three (Equations (12)–(14)) are not observablebutcan be derived from the total flux Ftot and from the galaxy’sintrinsic size Rd or R1 2. On the other hand, the other two(Equations (15)–(16)) are directly observable.

Naturally, the galaxy apparent area S1 2,obs is a bp ora b a2 ( )p ; thus, the face-on SB1 2,conv (Equation (16)) and

observed SB1 2,obs (Equation (17)) are related to one anothervia the axis ratio b a.17

From these definitions, we now derive relationshipsbetween these variants of SB and begin by noting that,typically for intermediate galaxies, the seeing radius RPSF andthe galaxy half-light radius R1 2 are of the same order, i.e.,R R 1PSF 1 2 ~ . Hence, one can write R R x1PSF 1 2 ( )~ - withx R R R1 2 PSF 1 2( )º - and x 1∣ ∣ .Since the total flux F A2 otot

2p s= is also R I2 od2p , we

have

R I AR R

I AR

R

2 22 ln 2

1.68

ln 2, 18

o o

o o

d2 1 2

2PSF2

2PSF

1 2

( )

( )( )

p p=+

which relates the observed S/N in the central pixel Ao to theintrinsic central SB Io.The average central SBSB1 2,conv within R1 2,conv is

F

R R R

F

Rx

R

RI

R

RA

SB0.5 1

1

0.5 1

21

0.25 SB

0.5ln 2

19

o

o

1 2,convtot

1 22

PSF 1 22

tot

1 22

1 21 2

PSF

1 2

PSF

( )( )

( )( )

p

p

=+

+

´ µ ´

which shows that the observed central SBSB1 2,obs directlymaps onto the S/N in the central pixel.

REFERENCES

Amorisco, N. C., & Bertin, G. 2010, A&A, 519, A47Andersen, D. R., & Bershady, M. A. 2013, ApJ, 768, 41Bacon, R., Copin, Y., Monnet, G., et al. 2001, MNRAS, 326, 23Bacon, R., Bauer, S., Böhm, P., et al. 2006, Msngr, 124, 5Bacon, R., Brinchmann, R., Richard, J., et al. 2015, A&A, 575, A75Binney, J., & Tremaine, S. 2008, Galactic Dynamics (2nd ed.; Princeton, NJ:

Princeton Univ. Press)Bolatto, A., Warren, S. R., Leroy, A. K., et al. 2015, ApJ, in press

(arXiv:1507.05652)Bouché, N., Murphy, M. T., Kacprzak, G. G., et al. 2013, Sci, 341, 50Buitrago, F., Conselice, C. J., Epinat, B., et al. 2014, MNRAS, 439, 1494Cappellari, M., Emsellem, E., Krajnovic, D., et al. 2011, MNRAS, 416,

1680Comparat, J., Richard, J., Kneib, J. -P., et al. 2015, A&A, 575, A40Contini, T., Garilli, B., Le Fèvre, O., et al. 2012, A&A, 539, A91Cresci, G., Hicks, E. K. S., Genzel, R., et al. 2009, ApJ, 697, 115Davies, R., Förster Schreiber, N. M., Cresci, G., et al. 2011, ApJ, 741, 69Davis, T. A., Alatalo, K., Bureau, M., et al. 2013, MNRAS, 429, 534Driver, S. P., Liske, J., Cross, N. J. G., De Propris, R., & Allen, P. D. 2005,

MNRAS, 360, 81Dubois, Y., & Teyssier, R. 2008, A&A, 477, 79Dutton, A. A., van den Bosch, F. C., Faber, S. M., et al. 2011, MNRAS,

410, 1660Elmegreen, B. G., & Elmegreen, D. M. 2006, ApJ, 650, 644Epinat, B., Amram, P., Balkowski, C., & Marcelin, M. 2010, MNRAS,

401, 2113Epinat, B., Contini, T., Le Fèvre, O., et al. 2009, A&A, 504, 789Epinat, B., Tasca, L., Amram, P., et al. 2012, A&A, 539, A92Feng, J. Q., & Gallo, C. F. 2011, RAA, 11, 1429Few, C. G., Gibson, B. K., & Courty, S. 2012, A&A, 547, A63Förster Schreiber, N. M., Genzel, R., Lehnert, M. D., et al. 2006, ApJ,

645, 1062Förster Schreiber, N. M., Genzel, R., Bouché, N., et al. 2009, ApJ, 706,

1364Frigo, M., & Johnson, S. G. 2005, in Proc. IEEE 93, Program Generation,

Optimization, and Platform Adaptation, 216Frigo, M., & Johnson, S. G. 2012, ascl soft: 1201.015Genzel, R., Burkert, A., Bouché, N., et al. 2008, ApJ, 687, 59Genzel, R., Newman, S., Jones, T., et al. 2011, ApJ, 733, 101Graham, A. W., & Driver, S. P. 2005, PASP, 22, 118Hastings, W. K. 1970, Biometrika, 57, 97Józsa, G. I. G., Kenn, F., Klein, U., & Oosterloo, T. A. 2007, A&A, 468, 731Kacprzak, G. G., Martin, C. L., Bouché, N., et al. 2014, ApJL, 792, L12Kennicutt, R. C. 1998, ARA&A, 36, 189Law, D. R., Shapley, A. E., Steidel, C. C., et al. 2012, Natur, 487, 338Law, D. R., Steidel, C. C., & Erb, D. K. 2006, AJ, 131, 70Law, D. R., Steidel, C. C., Erb, D. K., et al. 2007, ApJ, 669, 929Law, D. R., Steidel, C. C., Erb, D. K., et al. 2009, ApJ, 697, 2057Lemoine-Busserolle, M., Bunker, A., Lamareille, F., & Kissler-Patig, M. 2010,

MNRAS, 401, 1657MacKay, D. 2003, Information Theory, Inference, and Learning Algorithms

(Cambridge: Cambridge Univ. Press)Martin, C. L., & Soto, K. 2015, ApJ, submitted

17 Generally speaking, a R R R1 2,conv 1 22

PSF2 0.5( )º + and b

R i Rcos1 22 2

PSF2 0.5( ( ) )+ .

14

The Astronomical Journal, 150:92 (15pp), 2015 September Bouché et al.

Mighell, K. J. 1999, ApJ, 518, 380Neal, R. 1993, Probabilistic Inference Using Markov Chain Monte Carlo

Methods (Department of Computer Science, Univ. of Toronto)Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266Pérez-Montero, E., Contini, T., Lamareille, F., et al. 2009, A&A, 495, 73Péroux, C., Bouché, N., Kulkarni, V. P., & York, D. G. 2013, MNRAS,

436, 2650Péroux, C., Kulkarni, V. P., York, D. G., et al. 2014, MNRAS, 437, 3144Puech, M., Flores, H., Hammer, F., et al. 2008, A&A, 484, 173Refregier, A., Kacprzak, T., Amara, A., Bridle, S., & Rowe, B. 2012, MNRAS,

425, 1951Sérsic, J. L. 1963, BAAA, 6, 41Schroetter, I., Bouché, N., Péroux, C., Murphy, M., & Contini, T. 2015, ApJ,

804, 83Sharples, R., Bender, R., Bennett, R., et al. 2006, NewAR, 50, 370

Simard, L. 1998, in ASP Conf. Ser. 145, Astronomical Data Analysis Softwareand Systems VII, ed. R. Albrecht, R. N. Hook & H. A. Bushouse (SanFrancisco, CA: ASP), 108

Szu, H., & Hartley, R. 1987, PhLA, 122, 157Teyssier, R. 2002, A&A, 385, 337Trujillo, I., Förster Schreiber, N. M., Rudnick, G., et al. 2006, ApJ, 650, 18van Starkenburg, L., van der Werf, P. P., Franx, M., et al. 2008, A&A, 488, 99Villeneuve, E., & Carfantan, H. 2014, ITIP, 23, 4322Villeneuve, E., Carfantan, H., Serre, D., et al. 2011, in 3rd Workshop on

Hyperspectral Image and Signal Processing, Evolution in Remote Sensing(WHISPERS), (Lisbon, Portugal)

Williams, R. J., Quadri, R. F., Franx, M., et al. 2010, ApJ, 713, 738Wisnioski, E., Glazebrook, K., Blake, C., et al. 2011, MNRAS, 417, 2601Wright, S. A., Larkin, J. E., Barczys, M., et al. 2007, ApJ, 658, 78Wright, S. A., Larkin, J. E., Law, D. R., et al. 2009, ApJ, 699, 421

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