GalPak3D: A Bayesian parametric tool for extracting morpho-kinematics of galaxies from 3D data

2.2 MB·English

by  N. Bouché

Received 2014 June 24; accepted 2015 June 16
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
GalPaK3D: A BAYESIAN PARAMETRIC TOOL FOR EXTRACTING MORPHOKINEMATICS OF GALAXIES
FROM 3D DATA
N. Bouche´1,2, H. Carfantan1,2, I. Schroetter1,2, L. Michel-Dansac3, T. Contini1,2
Received 2014 June 24; accepted 2015 June 16
ABSTRACT
We present a method to constrain galaxy parameters directly from three-dimensional data cubes.
The algorithm compares directly the data with a parametric model mapped in x,y,λ coordinates. It
5 uses the spectral line-spread function (LSF) and the spatial point-spread function (PSF) to generate
1 a 3-dimensional kernel whose characteristics are instrument specific or user generated. The algorithm
0 returns the intrinsic modeled properties along with both an ‘intrinsic’ model data cube and the mod-
2
eledgalaxyconvolvedwiththe3D-kernel. ThealgorithmusesaMarkovChainMonteCarlo(MCMC)

g approachwithanontraditionalproposaldistributioninordertoefficientlyprobetheparameterspace.
u Wedemonstratetherobustnessofthealgorithmusing1728mockgalaxiesandgalaxiesgeneratedfrom
A hydrodynamical simulations in various seeing conditions from 0(cid:48).(cid:48)6 to 1(cid:48).(cid:48)2. We find that the algorithm
can recover the morphological parameters (inclination, position angle) to within 10% and the kine-

6 matic parameters (maximum rotation velocity) to within 20%, irrespectively of the PSF in seeing
1 (up to 1(cid:48).(cid:48)2) provided that the maximum signal-to-noise ratio (SNR) is greater than ∼ 3 pixel−1 and

that ratio of galaxy half-light radius to seeing radius is greater than about 1.5. One can use such an
]
M algorithm to constrain simultaneously the kinematics and morphological parameters of (nonmerging)
galaxiesobservedinnonoptimalseeingconditions. Thealgorithmcanalsobeusedonadaptive-optics
I (AO) data or on high-quality, high-S/N data to look for nonaxisymmetric structures in the residuals.
.
h Subject headings: methods: dataanalysis—methods: numerical—techniques: imagingspectroscopy
p
-
o
1. INTRODUCTION Given these challenges and the advancements in mul-
r
t Thankstoseveralstudiesusingopticalornear-infrared tiplexing IFU observations with the Very Large Tele-
s scope (VLT) second-generation instruments like KMOS
a (NIR) integral field unit (IFU) spectroscopy of Hα emis-
(Sharples et al. 2006) and the Multi-unit Spectrograph
[ sion from local and high-redshift (z > 1) galaxies
Explorer (MUSE; Bacon et al. 2006, 2015), it is impor-
(F¨orster Schreiber et al. 2006; Law et al. 2007; van
3 tant to have tools that can give robust estimates on
Starkenburg et al. 2008; Cresci et al. 2009; F¨orster
v the galaxy physical properties. In particular, KMOS
Schreiber et al. 2009; Lemoine-Busserolle et al. 2010;
6 will bring large statistically significant samples of high-
Law et al. 2012; Contini et al. 2012; Epinat et al. 2012;
8 redshift galaxies as it can observe 24 galaxies at a time,
Buitrago et al. 2014), our understanding of galaxy for-
5 but this facility will always lack an AO unit. This could
mationhaschangedsignificantlyinthepastdecade. For
6
potentially be a serious limitation since the robustness
0 instance,thesesurveyshaveshownthatasignificantsub-
of the derived kinematic parameters may depend on the
. set of high-redshift galaxies have a disklike morphology
1 quality of the atmospheric conditions (seeing can range
andshoworganizedrotation,withregularvelocityfields.
0 from 0(cid:48).(cid:48)4 to >1(cid:48).(cid:48)0 in the NIR).
In contrast to low-redshift studies (e.g. Bacon et al.
5 2001; Cappellari et al. 2011), high-redshift (1 (cid:46) z (cid:46) 2) In order to overcome these limitations, we present
1 a new tool named GalPaK3D (Galaxy Parameters and
galaxiesareobservedataspatialresolutionthatissever-
v: aly limited by the seeing conditions owing to their small Kinematics) 4 designed to be able to disentangle the
i apparentangularsizes. Inordertoovercomethelowspa- galaxy kinematics from resolution effects over a wide
X tial resolution, observations with adaptive-optics (AO) range of conditions. This is not the first code to model
r are often required (Law et al. 2007, 2009; Genzel et al. galaxy kinematics from 3D data (e.g. the TiRiFiC
a 2008,2011;Wrightetal.2007,2009). However, observa- package, which performs tilted ring model fits to three-
tions with AO are expensive, with the additional instru- dimensional radio data; J´ozsa et al. 2007), but this code
mental costs, and add strong observational constraints performs disk model fits to three-dimensional IFU data
such as the additional exposure times required to com- cubes, forthefirsttime,5, whereasallothermodelingof
pensateforthelossinsurfacebrightness(SB)sensitivity IFU-data so far have worked from the two-dimensional
(Law et al. 2006). Indeed, the SB limit for AO observa- velocity field (e.g. Cresci et al. 2009; Epinat et al. 2009;
tions taken on smaller pixels is higher, leaving the cur- Davies et al. 2011; Andersen & Bershady 2013; Davis
rent state-of-the art observations to the objects with the et al. 2013).
highest SBs. This paper is organized as follows: we describe the
GalPaK3D algorithm in Section 2. We present some test
1CNRS/IRAP,14AvenueE.Belin,F-31400Toulouse,France case examples in Section 3. We present results from an
2UniversityPaulSabatierofToulouse/UPS-OMP/IRAP,F-
31400Toulouse,France 4 Availableathttp://galpak.irap.omp.eu/.
3CRAL, Observatoire de Lyon, Universit´e Lyon 1, 9 Avenue 5 Law et al. (2012) made an attempt at 3D fitting, albeit not
Ch. Andr´e,69561SaintGenisLavalCedex,France self-consistently.
2 Bouch´e et al.
extensiveanalysisof1728syntheticgalaxiesinSection4, with n = 0.5, 1.0, and 4.0, respectively, where R is the
e
wherewediscusstheimpactoftheaccuracyinthepoint- effictive radius, f the total flux, and b such that R
tot n e
spread-function (PSF) characterization. In Section 5, is equivalent to the half-light radius R , and I the SB
1/2 e
we present an analysis of data cubes generated from hy- at R . For n = 0.5, 1.0, and 4.0, the constant b is
e n
drodynamical simulations of isolated disks from Michel- 0.69, 1.68, and 7.67, respectively, from b (cid:39) 1.9992n−
n
Dansac et al. (in prep.). We summarize this paper in 0.3271. The Sersic index n is kept fixed given the large
Section6. Throughout,weusethefollowingcosmological degeneracies it creates with other parameters, such as
parameters: H =70 km s−1, Ω =0.7 and Ω =0.3. the galaxy half-light radius. This degenaracy is due to
0 Λ M
the fact that the SB profiles around R are close to one
e
2. THEGALPAK3D ALGORITHM another fo n=0.5, 1.0, or 4.0 as noted in Graham et al.
In this section we outline the algorithm principles, (2005).
which are designed to be able to determine galaxy To this two-dimensional disk model, we add a disk
morphokinematicparametersfromthethree-dimensional thickness h . We adopt a Gaussian luminosity distribu-
z
data cube directly. We discuss the merits of using the tion perpendicular to the plane, I(z) ∝ exp(−z2/2h2),
z
parametric forward fit and its limitations. defining h as the characteristic thickness of the disk.
z
GalPaK3D also allows the user to choose an exponen-
2.1. A Parametric Galaxy Model in Three Dimensions tial I(z) ∝ exp(−|z|/h ) or a sech2 distribution I(z) ∝
z
Traditionally, kinematic analyses use two-dimensional sech2(z/h ). Wesetthediskthicknesstoh =0.15R
z z 1/2
maps generated by applying line-fitting codes to deter- where R is the disk half-light radius. This choice cor-
1/2
minethelinewavelengthcentroidsandwidths,whichare responds to h ∼ 1 kpc, typical of high-redshift edge-
z
onlyconsideredtobereliableforspaxelswithsufficiently on/chain galaxies (Elmegreen & Elmegreen 2006). At
high signal-to-noise (S/N) ratios. This S/N condition is this stage, we have a disk model in Euclidean coordi-
easily met at low redshifts, but is harder to meet for nates that accounts for the flux distribution only.
small, high-redshift galaxies. In principle, the choice to Forthegaskinematics,wecreatethreekinematiccubes
work in 2D or 3D space is equivalent, but we will show in the same spatial coordinate reference frame for the
that our method can work in the regime (on the spax- velocities v = (V ,V ,V ) assuming circular orbits. The
x y z
els)wherethesignal-to-noiseratio(SNR)perpixel(SNR rotational velocity v(r) with a maximum rotation veloc-
pixel−1) is not sufficient for line-fitting codes, which re- ity Vmax can have several functional forms: it can be
quire a minimum SNR on all spaxels. an arctan velocity profile (e.g. Puech et al. 2008), an in-
When the PSF FWHM can be characterized to suffi- verted exponential (Feng & Gallo 2011), or a hyperbolic
cient accuracy 6 (within 10% or 20%; see Section 4), one tanh profile (e.g. Andersen & Bershady 2013) :
cantakeitscharacteristics,togetherwiththeinstrumen-
2
talline-spreadfunction(LSF),intoconsiderationandre- v(r)=V arctan(r/r ) ‘arctan’ (2)
cover the intrinsic modeled galaxy parameters. The al- maxπ t
gorithm uses the spectral LSF and the spatial PSF to v(r)=Vmax [1−exp(r/rt)] ‘exp’ (3)
generate a three-dimensional kernel whose characteris- v(r)=V tanh(r/r ) ‘tanh’ (4)
max t
ticsaresetfor the giveninstrument(orauser-generated
instrument module). where r is the radius in the galaxy x,y plane, rt is the
While a full deconvolution of hyperspectral cubes turnover radius, and Vmax is the maximum circular ve-
would be preferred, it is usually a challenge mathemati- locity. These choices are more extensively discussed in
cally (a new method has been proposed recently by Vil- Epinatetal.(2010),butitisworthnotingthatthe‘exp’
leneuve&Carfantan2014),andaforwardconvolutionof and hyperbolic rotation curves have a sharper transition
a parametric model offers a very useful alternative. This around the turnover radius. We stress that our parame-
forwardconvolutiongivesustheopportunitytoestimate ter Vmax is not the projected asymptotic velocity, but is
intrinsic modeled kinematic parameters in a wider range the true asymptotic velocity irrespective of the inclina-
of seeing conditions, as illustrated in recent papers (see tion.
Bouch´e et al. 2013; P´eroux et al. 2014; Schroetter et al. Another option, called “mass,” assumes a constant
2015; Martin & Soto 2015, for first applications). light-to-mass ratio and sets v(r) from the enclosed
For the forward convolution, we need a parametric light/mass I(<r) profile
model, and we focus here on a galaxy disk model for
(cid:114)
emission-line surveys, but the algorithm is adaptable to I(<r)
v(r)∝ ‘mass’ (5)
other situations. In order to construct a modeled galaxy r
intheobservationalcoordinatesystems(x,y,λ),westart
bygeneratingathree-dimensionalgalaxymodelinaEu- where r is the radius in the galaxy x,y plane and Vmax
clidian coordinate system (x, y, z), where the z-axis is normalizes the profile. This option has a rotation curve
normaltothegalaxyplane(x,y). Weapplyaradialflux that peaks at some radius (set by the half-light radius),
profile I(r), from one of the traditional Gaussian, expo- decreases at larger radii, and is to be preferred for nu-
nential, and de Vaucouleur choices as parameterized by clear disks or when there is no significant dark matter
the S´ersic (1963) profile: component.
We then rotate the disk model around two axes ac-
(cid:16) (cid:104) (cid:105)(cid:17)
I(r)=I exp −b (r/R )1/n−1 (1) cording to an inclination (i) and position angle (PA, an-
e n e
ticlockwisefromy)andcreateacubeinx,y,andλusing
6ThePSFshapemattersmorethanthelevelofaccuracyonthe the three intermediate 2D maps: the flux map, the ve-
FWHM,asdiscussedinSection4.4. locityfield,andthedispersionmap(σ ). Thefluxmap
tot
GalPaK3D: a 3D kinematic tool 3
converge even in the presence of local minima, and that
TABLE 1
Default Range on Each Parameter can handle low S/N data. This is particularly difficult
fortraditionalminimizationmethodsbecausetheχ2 hy-
Parameter Min Max
persurface is very flat (outside the shallow well near the
xc 1/3Npixx 2/3Npixx
yc 1/3Npixy 2/3Npixy optimum parameters), and as a result the minimization
zc 1/3Npixz 2/3Npixz algorithm tends to not converge and be very susceptible
Flux 0 3×Σ (v ) to local minima.
i,j,k i,j,k
R 0.2spaxel 4(cid:48)(cid:48) Here we use an algorithm to optimize the parameters
1/2
Incl. (deg) 0 90 using Bayesian statistics with flat priors on bound inter-
PA(deg) −180 180 valsforeachoftheparameters. Thealgorithmconstructs
rt 0.01spaxel 1(cid:48)(cid:48) MCMCswithaMetropolis-Hasting(MH)sampler(Hast-
Vmax(kms−1) -350 350 ings 1970). At each iteration we compute the new set of
σo (kms−1) 0 180 parametersxˆ fromthelastxˆ setwithaproposaldis-
i+1 i
isobtainedfromtherotatedfluxcubesummedalongthe
tribution P from which to draw:
wavelength axis. The velocity field is obtained from the
flux-weighted mean V velocity cube. The total (line-of- xˆ =xˆ +hˆP(xˆ |xˆ ), (6)
z i+1 i i+1 i
sight) velocity dispersion σ is obtained from the sum
tot
where the new set of parameters is accepted or rejected
of three terms (added in quadrature). It includes (i) the
asinanyMHalgorithm. Thenewproposalsetofparam-
local isotropic velocity dispersion σ driven by the disk
d
eters x is then accepted or rejected according to the
self-gravity, which is σ (r)/h = V(r)/r for a compact i+1
d z
posterior distribution, which amounts to the likelihood
“thick” or large “thin” disk (Genzel et al. 2008; Binney
L ∝ exp−χ2 in the considered case of flat priors on the
&Tremaine2008;Daviesetal.2011);(ii)amixingterm,
parameters. In other words, we assume that the pixels
σ , arising from mixing the velocities along the line of
m
are independent and that noise properties are Gaussian,
sight for a geometrically thick disk, which is obtained
which is appropriate for optical/NIR data taken in the
from the flux-weighted variance of the cube V , and (iii)
z
background-limitedregime,andtheusercanprovidethe
an intrinsic dispersion (σ ) —which is assumed to be
o
fullvariancecube. Moreappropriatelikelihoodfunctions
isotropicandconstantspatially—toaccountforthefact
forlowcountswithPoissonnoisecanbefoundinMighell
that high-redshift disks are dynamically hotter than the
(1999).
self-gravityexpectation. Indeed,thisturbulencetermσ
o
is often observed to be (cid:39) 50–80kms−1 in z > 1 disks The scaling vector hˆ in Equation 6 is derived from the
(Law et al. 2007, 2012; Genzel et al. 2008; Cresci et al. variance on the flat (uniform) prior distributions, whose
2009; F¨orster Schreiber et al. 2009; Wright et al. 2009; boundaries are adjustable (the default values are listed
Epinat et al. 2010, 2012; Wisnioski et al. 2011) and thus in Table 1). The user may need to rescale the vector hˆ
dominatestheothertwotermssincethemixingtermσ inordertohaveacceptanceratesbetween20%and50%.
m
is typically ∼15 kms−1 and the self-gravity term σ is Convergence is usually achieved in a few hundred to a
d
typically 10–30kms−1. fewthousanditerations,eventhoughwetypicallyletthe
Tosummarize,thefluxprofilecanbechosentobe‘ex- algorithm run for 15,000 iterations.
ponential’ (n = 1.0),‘gaussian’ (n = 0.5), and ‘de Vau- In principle, one has the freedom to use any proposal
couleur’(n=4.0);thevelocityprofilev(r)canbearctan distribution P (e.g. MacKay 2003). A Markov chain is
(“arctan”), inverted exponential (“exponential”), hyper- said to converge to a single invariant distribution (the
bolic (“tanh”) or that of mass profile (“mass”); and the posterior probability) when the state of the chain per-
local dispersion can be that of the thin or thick disk. sits once it is reached and is said to be ergodic when the
There are in total 10 free parameters 7 to be determined probabilities xn converge to that invariant distribution
fromthedata. The10parametersarethex ,y ,z posi- as n→∞, irrespectively of the initial parameters (Neal
c c c
tions, the disk half-light radius R , the total flux f , 1993).9 In addition, if the sampler satisfies the following
1/2 tot
the inclination i, position angle PA, the turnover radius conditions P(x|x(cid:48))=P(x(cid:48)|x), as we have used, the algo-
r , the maximum circular velocity V , and the one- rithm reverts to the Metropolis method, which satisfies
t max
dimensional intrinsic dispersion σ . We will refer to the the two conditions. In practice, however, one also needs
o
lasttwo(V ,σ )askinematicparameters. Finally,the adistributionthatprobestheparameterspaceefficiently
max o
simulated galaxy is convolved (in 3D) with the PSF and inordertoconvergeinareasonablenumberofcpuhours,
the instrumental LSF specific for each instrument.8 The regardless of the initial parameters.
3DconvolutionisperformedusingfastFouriertransform A common proposal distribution is the uniform dis-
(FFT) libraries. tribution that gives equal probabilities to all possible
values. The Gaussian proposal distribution P(x(cid:48)|x) =
2.2. The Markov Chain Monte Carlo (MCMC) N(x,1)isprobablythemostcommonlyusedandispop-
Algorithm ular but has one major drawback: the Gaussian distri-
butionisrathernarrowsuchthatthealgorithmbecomes
In order to determine the 10 free parameters on hy-
sensitive to the initial conditions, making the time to
perspectral cubes, one needs an algorithm that is inde-
convergence to the optimum values very sensitive to the
pendent of initial guesses on the parameters, that can
initial guess. If the width of the proposal distribution is
small, the convergence is too slow/large, and when it is
7 There are only nine free parameters when the “mass” profile
isusedforv(r)sincetheturnoverradiusrt isirrelevant.
8 TheusercanchooseaGaussianPSF,aMoffatPSF.ThePSF 9 Available at http://www.cs.toronto.edu/~radford/
canbecircularorellipticalwithauser-definedaxisratiob/a. res-mcmc.html.
4 Bouch´e et al.
large (for convergence purposes), it will lead to low ac- ilar to other parametric algorithms (e.g. Simard 1998;
ceptance rate and poor efficiencies for convergence. To Peng et al. 2002), apart from the Bayesian approach.
remedythisproblem, onecoulduseamixeddistribution Figure1showsacomparisonbetweenthederivedmor-
withaGaussiandraw,say,90%ofthetimeandauniform phological parameters from two data sets of very dif-
draw 10% of the time, allowing the chain to escape from ferent resolution. Panel (a) shows a Canada France
a local minimum. Compared to the Gaussian proposal, Hawaii Telescope (CFHT) I-band image of the z ∼
themixeddistributionhasoneadditionalparameterthat 0.2 galaxy SDSSJ165931.92+023021.92 (Kacprzak et al.
needstobefine-tunedtotheproblem,suchasthemixing 2014). Panel (e) shows an r-band image of the galaxy
ratio. from the Sloan Digital Sky Survey (SDSS) at a spatial
Athirdoption,asadvocatedbySzu&Hartley(1987), resolution of 1(cid:48).(cid:48)1. For each data set, we show the fit-
is to use a draw from a Cauchy distribution that has ted (convolved) model, the residual map, and the one-
by definition longer wings (i.e. P is a Lorentzian profile dimentional SB profile. One sees that the intrinsic mod-
where P(x(cid:48)|x) ∝ γ2/[γ2 +(x(cid:48) −x)2]). The Lorentzian eled morphological parameters found from the SDSS
wings are important, allowing the chain to make large data(PSFFWHM=1(cid:48).(cid:48)1)areingoodagreementwiththe
jumps during the initial “burn-in” phase and ensuring higher-resolution data (PSF FWHM=0(cid:48).(cid:48)7). Moreover,
rapid convergence of the chain with no sensitivity to the the residuals in both data sets show the spiral arms and
initial parameters. Another advantage of a Cauchy pro- aminormerger(oralargeclump)inthesouthernpartof
posal distribution is that it has only one parameter, γ, thegalaxy,showingthatasmoothaxis-symmetricmodel
compared to the mixed one. can be used to unveil asymmetric features.
Wetestedthesevariouschoicesonsimulatedcubesand
3.2. Example on a mock cube
found that the Cauchy proposal distribution converged
faster than the other methods and was least sensitive to Figure 2 shows an example of a mock disk model with
the initial parameters. In other words, with the Cauchy alowSNR(SNRpixel−1 of4inthecentralpixel)drawn
nontraditional proposal distribution, a few hundred to a from the set presented in § 4 and generated at 1(cid:48).(cid:48)0 reso-
few thousand steps of the MCMC are required to pass lution. The top, middle, and bottom rows show the flux
theburn-inphasedependingontheS/Nofthedata,and map, the velocity map, and the apparent velocity profile
it is the user’s responsibility to confirm that the MCMC V (r) across the major axis, respectively. From left to
z
chain has converged. Thus, we typically run the chain right, the panel columns show the data, the convolved
through 10,000 or 15,000 steps to robustly sample the model, the modeled disk (free from the PSF), and the
posterior probability distribution. high-S/N high-resolution reference data (PSF=0(cid:48).(cid:48)15 and
The“best-fit”valuesoftheparametersaredetermined S/N=100). In the bottom panels, the solid red curves
fromtheposteriordistributions. Weusethemedianand correspondtothereferencerotationcurve(obtainedfrom
the standard deviation of the last fraction (default 60%) the reference data set), and the triangles represent the
of the MCMC chain to determine the ‘best-fit’ param- apparentrotationcurve. Theserotationcurvesshowthat
eters and their errors, respectively. One can also use a the recovered kinematics from the modeled disk (intrin-
fraction (default 60%) of the MCMC chain around the sic or unconvolved model) shown in the third column is
minimum χ2. The full MCMC chain is saved such that in good agreement with the reference data (last column)
the user can use his/her preferred technique. in spite of the low spatial resolution (1(cid:48).(cid:48)0) and the low
The algorithm is implemented in Python and uses the SNR in the mock data set.
standard numpy and scipy libraires. In addition, it uses This synthetic data cube was generated with a flux
the bottleneck 10 (Frigo & Johnson 2005) and FFTw 11 profile with Sersic index n = 1 and half-light radius
libraries (Frigo & Johnson 2012) in order to speed up R1/2 =0(cid:48).(cid:48)5, corresponding to 2.5 MUSE/KMOS pixels),
certain matrix operations and the PSF+LSF convolu- an “arctan” velocity profile with V = 200 kms−1, a
max
tion, respectively. It requires FITS files as inputs. The thick disk with a velocity dispersion σ =80 kms−1, an
o
algorithm is modular so that the user can add specifica- inclination i = 60◦, a PA= 130◦, and with instrumental
tions for other instruments. The online documentation specifications for the new VLT MUSE instrument (0(cid:48).(cid:48)2
describesthesyntax,andittakesabout2,5,and10min- pixel−1,1.25˚Apixel−1,LSF=2.14pixels). Theintegrated
utesonalaptop(at2.1GHz)torun10,000iterationson
total flux is 10−16 ergs−1cm−2, and the synthetic noise
a data cube with 303 pixels, 403 pixels, and 603 pixels,
per pixel is σ =5×10−20 ergs−1cm−2 ˚A−1.
respectively. Inotherwords,thecomputationtimescales
The synthetic data cube is also displayed in Fig-
ast∝N log(N )whereN isthenumberofpixels,
pix pix pix ure 3, which shows three one-dimensional spectra (a)
showing that the FFT calculation dominates.
taken at the three locations labeled in the image
shown in panel (b). Panel (c) shows a 3D rep-
3. HIGHLIGHTAPPLICATIONS
resentation of the data (blue) with the model over-
3.1. Example on 2D data laid (red) made with the “visit” software;12 where the
Before applying the tool on 3D data, it is important light/dark areas corresponds to two cuts at fluxes of 6
to validate the method on simpler data sets, such as and 8×10−20 ergs−1cm−2 ˚A−1, i.e. an S/N pixel−1 of
two-dimensional imaging data. We thus wrote a two- 1.2 and 1.8, respectively.
dimensionalversionofthealgorithm,GalFit2D,onethat We ran the algorithm with 15,000 iterations, and Fig-
does not include the kinematic, which is in essence sim- ure4showstheMCMCchainsforthe10freeparameters
along with the χ2 evolution in the bottom panel. The
10 Availableathttps://pypi.python.org/pypi/Bottleneck.
11 Availableathttps://pypi.python.org/pypi/pyFFTW. 12 Availableathttp://visit.llnl.org/.
GalPaK3D: a 3D kinematic tool 5
Fig. 1.—Applicationofthetwo-dimensionalversionoftheMCMCalgorithm(‘GalFit2D’)onthez∼0.2SDSSJ165931.92+023021.9with
mr =18.40magfromKacprzaketal.(2014). SimilarlytoGalPaK3D,Galfit2DperformsaparametricfitwithanMCMCalgorithmusing
set surface brightness profiles convolved with the seeing. The top row shows the result from archival CFHT I-band taken at a resolution
of 0(cid:48).(cid:48)7. The bottom row shows the result from the SDSS r-band image that has a resolution of 1(cid:48).(cid:48)1. Panels (a) and (e) show the data.
Usinganexponentialprofile,panels(b)and(f)showtheseeing-convolvedmodel;(c)and(g)theresiduals,i.e. data-modelnormalizedto
thepixelnoiseσ,and(d)&(h)theone-dimensionalSBprofile. TherecoveredintrinsicdiskscalelengthR isabout1”inbothcases,in
d
spiteofthedifferentspatialresolution.
Fig. 2.—Exampleofthealgorithmapplicationonadiskmodelsimulatedwithaseeingof1(cid:48).(cid:48)0(FWHM)andafluxof10−16ergs−1cm−2,
andanS/Npixel−1of∼4atthebrightestpixel. Thetop,middle,andbottomrowsshowthefluxmap,thevelocitymap,andtheapparent
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(cid:48).(cid:48)15 and SNR=100). In the bottom panels,
the solid red curves correspond to the reference case, the triangles represent the apparent rotation curve, and the dotted lines show the
apparentVmax sini. Oneseesthatthevelocityprofilefromthemodeleddisk(thirdcolumn)isingoodagreementwiththereferencedata.
6 Bouch´e et al.
values of the fitted parameters (and their errors) shown
by the black lines (gray lines) are computed from the
1e 19
median (standard deviation) of the last 60% iterations
2 1
of the posterior distributions. The recovered parameters
1 are listed in Table 2 and show good agreement between
the input and recovered values.
0
250 250
1e 19 TABLE 2
2 2 Comparison between the model input values and the
recoveredvalueswith1σ errorsandconfidenceintervals
1 (CI) for the example shown in Figure 2.
0 Parameter Input Output [95%CI]
250 250
/A 1e 19 xc (pixel) 15 15.05±0.09[14.87;15.24]
2m2 3 yc (pixel) 15 15.06±0.09[14.89;15.23]
s/c zc (pixel) 15 15.05±0.07[14.92;15.19]
g/1 Flux(10−16) 1 1.06±0.03 [1.01;1.09]
er R (arcsec) 0.82 0.85±0.04 [0.78;0.95]
ux (0 In1c/l2. (deg) 60 62±3 [58;68]
fl 250 250 PA.(deg) 130 126±2 [123;130]
Vz (km/s) rt (pixel) 1.35 1.32±0.42 [0.8;2.47]
Vmax (kms−1) 200 202±22 [172;257]
(a) σo (kms−1) 80 82±5 [73;90]
Figure 5 shows the joint distributions for the radius,
PA, inclination, maximum velocity, and dispersion pa-
rameters. Theestimatedparametersandtheirrespective
1σerrorareshownasasolidlineanddashedline,respec-
1.5 3333333 tively. This figure shows a clear covariance between the
2222222
turnover radius and the asymptotic velocity V , and a
") 1111111 max
y ( 0.0 small covariance between the inclination and Vmax. The
δ users of GalPaK3D are strongly advised to confirm the
convergence of the parameters using diagnostics similar
1.5
to Figure 4 and to investigate possible covariance in the
parameters, as these tend to be data specific, using di-
agnostics similar to Figure 5.
1.5 0.0 1.5
δx (")
4. TESTSWITHMOCKDATACUBES
(b)
In order to characterize the performances and limita-
tions of the GalPaK3D algorithm statistically, we gen-
erated a set of 1728 cubes again with a MUSE configu-
ration over a grid of parameters listed in Table 3. The
synthetic cubes were generated with noise typical to a
1hrexposurewithMUSEcorrespondingtoapixelnoise
of σ = 5×1020 ergs−1cm−2 ˚A−1. We use a range of
TABLE 3
Range of parameters for the 1728 mock galaxies
Parameter GridValues
Flux(10−17ergs−1cm−2) 3,6,10,30
Seeing(”) 0.6,0.8,1.0,1.2
Redshift 0.6,0.9,1.2
R (kpc) 2.5,5,and7.5a
1/2
R (”) 0(cid:48).(cid:48)3,0(cid:48).(cid:48)6,and1(cid:48).(cid:48)0b
1/2
Incl. (deg) 20,40,60,80
PA(deg) 130
(c)
rt (”) 0.1–0.3c
Fig. 3.— For the synthetic data in Figure 2, we show three 1- Vmax (kms−1) 110,200,280
dimensional profiles (panels (a)) comparing the data (thin line) σo (kms−1) 20,50,80
andthemodel(thickline)takenatthelocationlabeled“1,”,“2,” aExact value to satisfy the size-velocity scaling relation (Dutton
and “3” in panel (b). Panel (c) shows a 3D representation of the etal.2011).
data (blue)with themodel overlaid(red) whereweused two flux bExactvaluewilldependontheredshift.
levels at 5×10−20 and 1.5×10−19 ergs−1cm−2 ˚A−1. The cube cExact value to satisfy the scaling relation between the galaxy
orientationisshown,wherethewavelengthaxisisthez-direction. size and the inner gradient (Amorisco & Bertin 2010) using rt =
(An interactive version of this figure is available in the published R /1.8.
d
onlineversion)
GalPaK3D: a 3D kinematic tool 7
x y z 1e 16 flux radius
15.4 15.4 15.3 0.95 5.0
15.3
15.2 15.2 15.2 0.90 4.5
15.1 15.1 0.85
1154..08 11115444....0897 111544...089 00..8705 43..05
14.6 0.70
0 7500 0 7500 0 7500 0 7500 0 7500
inclination pa turnover_radius maximum_velocity velocity_dispersion
135
70 3 300 100
65 130 2 250 90
80
60 125 1 200
70
55 120 0 150 60 Pˆ σ 95 % CI
1
0 7500 0 7500 0 7500 0 7500 0 7500
0 log [χ2−χm2in]
1
2
3
4
5
6
0 7500
Fig. 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
lineandarecalculatedfromthelast60%ofthechain. Thegraylinesshowthe1σstandarddeviations,andthedottedlinesshowthe95%
confidenceinterval. Notethefittedfluxvalueis(1.06±0.03)×10−16 ergs−1cm−2,whichisfoundfromthesumofthepixelvalues(here
∼8.510−17 ergs−1cm−2 ˚A−1) times the 1.25 ˚A per spectral pixel. The bottom panel shows the χ2 evolution relative to the minimum,
log[χ2−χ2 ]. Weusethisnon-standardmetricinordertoshowthatthevariationsoftheχ2aroundtheminimumwhichare3to4orders
min
ofmagnitudesmaller,reflectingaveryflathypersurface. Hence,aplotofχ2 orofthelikelihoodwouldshowastraightline.
us4.5 4.5 4.5 4.5 4.5
adi4.0 4.0 4.0 4.0 4.0
r
3.5 3.5 3.5 3.5 3.5
50.5 56.0 61.5 67.5 121.0126.0130.5 1.00.0 1.5 2.5 101.5156.0210.0264.0 66.0 79.0 92.0
inclination
n
natio6617..55 6617..55 6617..55 6617..55
ncli56.0 56.0 56.0 56.0
i50.5 50.5 50.5 50.5
121.0126.0130.5 1.00.0 1.5 2.5 101.5156.0210.0264.0 66.0 79.0 92.0
pa
130.5 130.5 130.5
pa126.0 126.0 126.0
121.0 121.0 121.0
116.5 116.5 116.5
1.00.0 1.5 2.5 101.5156.0210.0264.0 66.0 79.0 92.0
turnover_radius dius
er_ra 21..55 21..55
v
no 0.0 0.0
tur 11.001.5m1a5x6i.m02u1m0_.0ve2l6o4c.i0ty ocity 1.0 66.0 79.0 92.0
vel264.0
m_210.0
u
m156.0
maxi101.5 66.0 79.0 92.0
velocity_dispersion
Fig. 5.— Joint distributions for the radius, PA, inclination, maximum velocity, and dispersion parameters for the example shown in
Figure 2–4. The estimated parameters and their respective 1 σ error are shown as a solid line and dashed line, respectively. One sees
thatthetraditionaldegeneracybetweenVmax andtheinclinationiisbroken,thankstoour3Dmethod,buttheseeingleavesasignificant
correlation between Vmax and the turnover radius rt. The presence of this degenerancy is data specific and seeing specific, not a generic
featureofthealgorithm.
8 Bouch´e et al.
inclinations i from 20◦ to 80◦. We use a range of disk commonly used terms such as the SB of galaxies. From
sizes, with half-light radii R = 0(cid:48).(cid:48)3, 0(cid:48).(cid:48)6, and 1(cid:48).(cid:48)0 cor- any light profile I(r) such as given by Equation 1, there
1/2
respondingtoaR of 2.5,5,and7.5kpc,coveringthe are many ways to define galaxy SB, such as I the SB at
1/2 e
range of observed sizes at z ∼1 (e.g. Trujillo et al. 2006; the effective radius Re, Io the intrinsic SB at the central
Williams et al. 2010; Dutton et al. 2011). pixel,Ao theobservedSBatthecentralpixel,andSB1/2
Foreachofthegalaxysizes,weusetheVmax-R1/2 scal- theaverageSBwithintheintrinsichalf-lightradiusR1/2:
ing relation (Equation 8 of Dutton et al. 2011) and its
0.5F
redshift evolution (Equation 5 of Dutton et al. 2011) to SB ≡ tot . (7)
set the rotation kinematics (V ). In particular, the 1/2,conv πR2
max 1/2,conv
sizes R = 2.5, 5, and 7.5 kpc correspond to V val-
1/2 max
whereF isthegalaxytotalflux. Arelatedquantityto
ues ranging from ∼100 to 250 kms−1. We use “arctan” tot
Equation 7 is the observed SB, defined as :
rotation curves to generate our mock data cubes, and
we have verified that our results remain the same with 0.5F
SB ≡ tot (8)
“exponential” rotation curves. 1/2,obs S
We use the scaling relation between the turnover ra- 1/2,obs
dius rt and the disk scale length Rd that exists for disk whereFtot isthegalaxytotalfluxandS1/2,obs thegalaxy
galaxies(e.g. Figure1ofAmorisco&Bertin2010)toset apparent area given by ≡ πab where a and b are the
theturnoverradiusrt. Inparticular, wesetrt toRd/1.8 observed major and minor semiaxes, respectively, of the
where the 1.8 factor 13 is determined empirically for the galaxy. The relations between these various definitions
arctanrotationcurvetosatisfythelinearcorrelationbe- are described in the appendix.
tween the galaxy disk scale-length R = R /1.68 and To illustrate the point made at the beginning of this
d 1/2
R , defined as the radius r where V(r) = 2/3 V section,weshowinFigure6(a)therelativeerrorsδp/p≡
Ω max
(Amorisco & Bertin 2010). (pfit−pin)/pin on some of the estimated parameters for
For each of the galaxy sizes, the disk thickness is h = ourmockdatacubesgeneratedinSection4asafunction
z
0.15R , i.e. ranging from 0.4 to 1.3 kpc, bracketing of central SB, SB (defined in Equation 8). Each
1/2 1/2,obs
the average values of h ∼1 kpc, found for high-redshift row shows the relative errors for the maximum circular
z
edge-on/chain galaxies (Elmegreen & Elmegreen 2006). velocity Vmax, the size R1/2, the PA, and inclination i
We used fluxes for an [OII] (λ3727) emission line, from top to bottom, respectively. The crosses, squares
expected to lie in the MUSE spectral range at red- and circles represent the three subsamples with sizes ∼
shifts between 0.6 and 1.2, with integrated fluxes from 2.5,5,and7.5kpc,respectively. Oneseesthattheerrors
3 × 10−17 ergs−1cm−2 to 3 × 10−16 ergs−1cm−2 cor- in the morphological parameters (size, PA, inclination)
responding to the range of observed values (e.g. Ba- do increase toward low SBs, but the threshold point at
con et al. 2015; Comparat et al. 2015, and references which the relative errors reach ∼100% depends on the
therein). We use a constant noise value per pixel of σ = galaxy size, represented by the symbols. This illustrates
5×10−20ergs−1cm−2˚A−1,inordertosimulatethenoise the well-known fact that very extended objects have low
levelofa1hrexposure,butwestressthatthealgorithm surfacebrightness(andlowS/Npixel−1)buthavemany
acceptsvariance/noisecubestoaccountforpixel-to-pixel pixels in the outer regions that contain useful informa-
noise variations. In addition, we generated cubes with tion.
very high S/N (S/N=100, flux=3×10−15 ergs−1cm−2) As argued at the beginning of this section and demon-
and with a seeing typical of AO conditions, with a PSF strated in Figure 6, SB alone might not be sufficient to
FWHM of 0(cid:48).(cid:48)15. These will serve as reference data sets. determinetheS/Ninthefittedparameters,butthecom-
pactnessofthegalaxywithrespecttothebeamalsoplays
4.1. Surface Brightness and Signal-to-noise Ratio an important role. In Figure 6(b), we show the relative
error δp/p with respect to the observed SB times
One could imagine that the S/N in the recovered pa- 1/2,obs
the size-to-PSF ratio (R /R )α. The symbols cor-
rametersbeafunctionoftheaverageS/Npixel−1,orthe 1/2 PSF
respond to galaxy subsamples with various sizes as in
apparentSBsincetheobservedcentralSBscalesdirectly
Figure 6(a). The index α was found to be empirically
with the SNR in the central pixel. But clearly the com-
∼ 1 in order to have the relative errors for each of the
pactness of the object with respect to the seeing plays
subsamples follow a similar trend and may differ sightly
a large role (as discussed in Driver et al. 2005; Epinat
for each of the parameters p. In fact, we find that α
et al. 2010). Very compact objects (compared to the
is approximately 0.8, 1.2, and 1.4 for the size, PA, and
beam or the PSF) have high SB by definition (and high
inclination parameter, respectively.
S/N pixel−1), but the morphology and/or kinematic in-
Theseempiricalresultscanbeexplainedbythefollow-
formation may be lost owing to the beam smearing. On
ingarguments. TheapparentSBwithinthehalf-lightra-
the other hand, very extended objects have low surface
dius SB (Equation 7) and the observed SB
brightness (and low S/N pixel−1), but have many pixels 1/2,conv 1/2,obs
(Equation 8) are proportional to the SB (or S/N) of the
in the outer regions (with low S/N), where most of the
centralpixel,A ,asshownintheAppendix(Equation8).
information on the galaxy is located and not affected by o
InthecaseofnoPSFconvolution,Refregieretal.(2012)
the beam.
showedthat(theirEquation12)therelativeerrorσ(a)/a
Before illustrating this point, it is important to define
onmorphologicalparameters(itsmajor-axisa)scalesin-
13For‘exponential’rotationcurves,oneshouldsetrttoRd×0.9 versely to the central Io where Io is the intrinsic central
in order to satisfy the scaling relation; for ‘tanh’ rotation curves, SB(Equation1–2). InthepresenceofaPSFconvolution,
oneshouldsetrt toRd×1.25. Equation 16 of Refregier et al. (2012) —which applies
GalPaK3D: a 3D kinematic tool 9
here— shows that the relative errors on the major-axis (2012) formalism. Here we investigate whether the rela-
a scale as tiveerrorsdependonsomeoftheotherparameters,such
as inclination, seeing, and size.
σ(a)
∝A−1(1+R2 /R2 ). (9) Figure 7 shows the relative errors (p −p )/p for
a o PSF 1/2 fit in in
several key parameters p. The bottom (top) row shows
whereR istheradiusofthePSF(R ≡FWHM/2) the result for the size parameters R (inclination i),
PSF PSF 1/2
and R the intrinsic half-light radius. respectively,asafunctionofseeing,redshift,inclination,
1/2
In our cases, for high-redshift galaxies, the ratio and size-to-psf ratio R1/2/RPSF. The black curves with
R /R is (cid:39) 1.0 and after performing a Taylor ex- increasing thickness correspond to subsamples with dif-
PSF 1/2
pansion around R /R ∼ (1−x) with x ≡ (R − ferent SB levels (labeled) where the zero point (dotted
PSF 1/2 1/2
line)hasbeenoffsetforclaritypurposes. Thedatapoints
R )/R and |x|<<1, one finds that the factor (1+
PSF 1/2 represent the median, and the size of the error bars rep-
R2 /R2 )isapproximately∼2(1−x)∼2R /R .
PSF 1/2 PSF 1/2 resentthestandarddeviationforeachofthesubsamples,
Hence, Equation 9 on the errors in the major-axis a be- where we have typically ∼ 100 mock cubes per bin. We
comes in the regime where RPSF/R1/2 (cid:39)1.0: note that the median standard deviations on the param-
eters(fromtheposteriordistributions)tendtobewithin
σ(a) (cid:18)R1/2 (cid:19)−1 20% of these binned standard deviations.
∝ A
a R o Fromthisfigure,oneseesthattheGalPaK3Dalgorithm
PSF
recoverstheintrinsichalf-lightradiusR irrespectively
(cid:18)R (cid:19)−1 1/2
∝ 1/2 SB , (10) of seeing, redshift, and/or intrinsic size. Note that the
RPSF 1/2,obs relative errors with respect to size-to-seeing ratio at a
fixedSBfollowroughlytheexpectationfromEquation9,
which shows that the quality of the estimated morpho-
where the factor 1+(R /R )2 saturates to unity in
logical parameters will depend on both the pixel S/N PSF 1/2
(or SB) and the galaxy compactness with respect to the ourregimewithR1/2/RPSF ∼1to2.5. Theseresultsare
beam, R R , as shown in Figure 6(b) not affected by the choice of the SB profile (Sersic n).14
1/2 PSF
InbothFigure6(a)and6(b), thegraysolidlinesshow From the top row in Figure 7, one sees that the input
the expected behavior for the morphological parameters inclination is recovered except at the two smallest fluxes
(Equation 10) and one sees that they agree better with and for the more face-on cases. The reason that the
the mock data in the right panels for the morpholog- algorithm can recover the inclination well is that the al-
ical parameters. This shows that the Refregier et al. gorithm breaks the traditional degeneracy between Vmax
(2012)formalismdescribestherelativeerrorsonthemor- andiusingtheSBprofile(i.e. theaxisratiob/a)whereas
phological parameters (size, PA, and inclination) rela- traditional methods fitting the kinematics on velocity
tivelywell,asafirstapproximation. WenotethatEqua- fields have a strong degeneracy between Vmax and the
tion 10 is only an approximation to Equation 9 when inclination i.
R /R (cid:39)1andthattheremightbeotherdependen-
1/2 PSF 4.3. Reliability of kinematic parameters
cies for the other morphological parameters, namely, for
the PA and for the inclination. Here we refer the reader Figure8showstherelativeerrorsδp/p≡(pfit−pin)/pin
to Table 1 of Refregier et al. (2012) and their Appendix for the parameters Vmax (top row) and disk dispersion
forfurtherdetails; itisbeyondthescopeofthispaperto σo (bottom row) as a function of seeing, σo, inclina-
present a full 3D derivation of the Refregier et al. (2012) tion, and size-to-PSF ratio R1/2/RPSF. The curves as
formalism. afunctionofredshiftarenotshown,becausetherelative
Contrary to the morphological parameters, the errors errors do not depend on this parameter as in Figure 7.
in the kinematic parameter V show strong positive Theblackcurveswithincreasingthicknesscorrespondto
max
(negative) biases in the smallest (largest) mock galaxies, subsamples with different SB levels (labeled) where the
representedbythecrosses(circles)respectivelyinthetop zero point (dotted line) has been offset for clarity pur-
panel of Figure 6(b). The positive bias for for the most poses. The data points represent the median, and the
compact galaxies (crosses) with respect to the beam can size of the error bars represents the standard deviation
be understood because the V information is located for each of the subsamples, as in Figure 7.
max
mostly in the outer parts of the galaxy, where the S/N Figure 8(top) shows that the GalPaK3D algorithm re-
is too low. The negative bias for the largest galaxies (1” covers the maximum velocity Vmax irrespectively of see-
in R ) at low SB is likely due to the spatial cut of our ing, disk dispersion, and redshift (now shown) provided
1/2
mock cubes being too small. that the galaxy is not too compact. For small galaxies
We will return to the reliability of Vmax in section 4.3 with R1/2/RPSF less than 1.5, the figure shows that it
and now turn to a more detailed discussion on the reli- is increasingly difficult to estimate the correct values for
ability of the parameters (size, inclination, disk velocity the most compact galaxies, with large uncertainties and
dispersion,andV ). Whileweusedanarctanrotation significant overestimations of this parameter. This re-
max
curve,wenotethatthefollowingresultswerefoundtobe sultwasalreadypointedoutinEpinatetal.(2010, their
identical when we used an ‘exponential’ rotation curve. Figure 13) using 2D kinematic models. Epinat et al.
(2010) also noted that using a simple flat rotation curve
4.2. Reliability of morphological parameters to model the disk, the maximum velocity Vmax can be
We have shown in the previous section with Figure 6 14 A curve-of-growth analysis on the two-dimensional flux map
that the relative errors on the half-light radius follow cansometimesyieldaconstraintontheSersicindexnandamore
appoximately the expectation from the Refregier et al. accuratedeterminationoftheintrinsichalf-lightradius(R1/2).
10 Bouch´e et al.
x 2.5 kpc 5 kpc 7.5 kpc x 2.5 kpc 5 kpc 7.5 kpc
ma 1 ma 1 α=0.80
V V
/ 0 / 0
x x
ma 1 ma 1
V V
δ 10-17 10-16 δ 10-17 10-16
R 1/2 1 R 1/2 1 α=0.80
R/1/2 01 R/1/2 01
δ 10-17 10-16 δ 10-17 10-16
PA 1 PA 1 α=1.20
A/ 0 A/ 0
P P
δ 1 δ 1
10-17 10-16 10-17 10-16
1 1 α=1.40
δii / 0 δii / 0
1 1
10-17 10-16 10-17 10-16
SB1/2,obs [erg/s/cm2/arcsec2] SB1/2,obs.(R1/2/RPSF)α [erg/s/cm2/arcsec2]
(a) (b)
Fig. 6.— Relativeerrorsontheestimatedparametersδp/p,definedas(pfit−pin)/pin. Eachrowshowsδp/pforthemaximumcircular
velocity Vmax, the size R1/2, the PA, 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 in for the morphological parameters (size,
PA,inclination)arebinned. Left(a): RelativeerrorsasafunctionofcentralsurfacebrightnessSB inerg/s/cm2/arcsec2. Right(b):
1/2,obs
RelativeerrorasafunctionofcentralSB,SB1/2,obs,times(R1/2/RPSF)α,whereR1/2 isthegalaxy intrinsichalf-lightradiusandRPSF
the PSF half-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. Thesevaluesareclosetotheexpectationof−1.0ofEquation10(graylines)derivedformorphologicalparameters
in imaging data by Refregier et al. (2012). The relative error in Vmax does not follow the expected relation and is subject to strong
systematics for the smallest and largest mock galaxies (crosses). This is due to Vmax being constrained in the outer parts of the galaxy,
wheretheS/Nisthusnotsufficientforthecompactgalaxiesorwherethemockcubeistoosmallforthelargestgalaxies.
recovered with an accuracy better than 25%, even when To conclude this section, our algorithm is able to re-
R /R is less than about ∼2. cover the morphological and kinematic parameters from
1/2 PSF
Figure 8(bottom) shows the GalPaK3D algorithm re- synthetic data cubes over a wide range of seeing condi-
coversthediskdispersionirrespectivelyofseeingandred- tions provided that the galaxy is not too compact and
shift (not shown). Given the instrumental resolution of has a sufficiently high SB. Thus, for galaxies to be ob-
MUSE used here (R (cid:39) 130 km/s), small dispersions are served with MUSE in the wide-field mode in 1 hr ex-
more difficult to recover. We note that the local disper- posure and no AO, we find that the algorithm should
sionisrathersensitivetotheinstrumentLSFFWHM,as perform well provided that the SB is greater than a
one might expect. The user can specify more than one few ×10−17 ergs−1cm−2 arcsec−2 and as long as the
type of LSF (Gaussian or Moffat), and a user-provided the size-to-seeing ratio R1/2/RPSF is larger than 1.5 (or
vector can be specified if the parametric LSF is not suf- R /FWHM >0.75).
1/2
ficient to describe the instrument LSF.
5. APPLICATIONONHYDRODYNAMICALSIMULATIONS
4.4. A note regarding the PSF accuracy In the previous section we validated the algorithm on
synthetic or mock data, which have by definition no de-
Onecouldarguethatourresultsaredrivenbythefact
fects, i.e. are perfectly regular and symmetric. In order
that we use the exact same PSF (in 3D) as the one used
to validate the algorithm on more realistic data, we now
to generate these modeled galaxies. To test the relia-
analyze the performance of the algorithm on data cubes
bility of the algorithm in more realistic situations, when
created from simulated galaxies generatedfrom a hydro-
the PSF FWHM is not known accurately, we ran the al-
dynamical simulation (Michel-Dansac et al., in prep.).
gorithm on the same set of data cubes with a random
This is intended to validate the algorithm in the pres-
component added to the FWHM of the PSF given by a
ence of systematic deviations from the disk model.
normal distribution with σ = 0.1, corresponding to un-
certainties in the FWHM of ∼ 20%. We found that the
5.1. From Hydrodynamical Simulations to Data cCubes
accuracy of the spatial kernel (PSF) has little impact on
the recovered parameters. On the other hand, we find The simulation used in this work comes from a set of
that the shape of the PSF is more critical especially for cosmological zoom simulations, each targeting the evo-
the morphological parameter such as the axis ratio b/a lution until redshift 1 of a single halo and its large-scale
(ortheinclination). Wenotethatsophisticatedtoolsex- environment. Thefullsampleofsimulationsispresented
ist to determine the PSF from faint stars in data cubes indetailsinMichel-Dansacetal.,inprep. Herewefocus
such as the algorithm of Villeneuve et al. (2011). on one output of one simulation to complement the test