Detailed Decomposition of Galaxy Images. II. Beyond Axisymmetric Models
aa r X i v : . [ a s t r o - ph . C O ] J a n Submitted to
The Astronomical Journal
Preprint typeset using L A TEX style emulateapj v. 7/8/03
DETAILED DECOMPOSITION OF GALAXY IMAGES. II. BEYOND AXISYMMETRIC MODELS
Chien Y. Peng , Luis C. Ho , Chris D. Impey , and Hans-Walter Rix Submitted to The Astronomical Journal
ABSTRACTWe present a two-dimensional (2-D) fitting algorithm (
Galfit , Version 3) with new capabilities tostudy the structural components of galaxies and other astronomical objects in digital images. Ourtechnique improves on previous 2-D fitting algorithms by allowing for irregular, curved, logarithmicand power-law spirals, ring and truncated shapes in otherwise traditional parametric functions likethe S´ersic, Moffat, King, Ferrer, etc., profiles. One can mix and match these new shape featuresfreely, with or without constraints, apply them to an arbitrary number of model components andof numerous profile types, so as to produce realistic-looking galaxy model images. Yet, despite thepotential for extreme complexity, the meaning of the key parameters like the S´ersic index, effectiveradius or luminosity remain intuitive and essentially unchanged. The new features have an interestingpotential for use to quantify the degree of asymmetry of galaxies, to quantify low surface brightnesstidal features beneath and beyond luminous galaxies, to allow more realistic decompositions of galaxysubcomponents in the presence of strong rings and spiral arms, and to enable ways to gauge theuncertainties when decomposing galaxy subcomponents. We illustrate these new features by way ofseveral case studies that display various levels of complexity.
Subject headings: galaxies: bulges — galaxies: fundamental parameters — galaxies: structure —techniques: image processing — techniques: photometric INTRODUCTION
Images of astronomical objects store a wealth of infor-mation that encodes the physical conditions and fossilrecords of their evolution. Over the past decade, theability of optical/near-infrared telescopes to resolve ob-jects improved by a factor of 10, and to detect faintsurface brightnesses by at least 2 orders of magnitude.These advances now enable the study of highly intricatedetails on subarcsecond scales (e.g., nuclear star clus-ter, spiral structure, bars, inner ring, profile cusps, etc.),and extremely faint outer regions of galaxies. Moreover,new integral-field imaging capabilities blur the tradi-tional boundary of obtaining, analyzing, and interpretingimaging and spectroscopic data. Faced with the conver-gence in volume, quality, and multiwavelength datasetslike never before, one of the main challenges toward mak-ing full use of the investments is developing sophisticatedways to extract information from the data to facilitatenew science.
Parametric and Non-Parametric Analysis
Analyzing astronomical images is challenging becauseof the diversity in object sizes and shapes, and nowhereis it more difficult than for galaxies. Since the early eraof photographic plates, one of the key methods for study-ing the light distribution of galaxies is to model it by us-ing analytic functions—a technique known as parametricfitting. This technique was first applied to galaxies by Herzberg Institute of Astrophysics, National Research Councilof Canada, 5071 West Saanich Road, Victoria, British Columbia,Canada V9E 2E7; [email protected] The Observatories of the Carnegie Institution forScience, 813 Santa Barbara St., Pasadena, CA 91101;[email protected] Steward Observatory, University of Arizona, 933 N. CherryAv., Tucson, AZ 85721; [email protected] Max-Planck-Institut f¨ur Astronomie, Koenigstuhl 17, Heidel-berg, D-69117, Germany; [email protected] de Vaucouleurs (1948) who showed that the light distri-bution of elliptical galaxies tended to follow a power-lawform of exp (cid:0) − r / (cid:1) . Subsequently, one of the break-throughs in our understanding of galaxy structure andevolution came when Freeman (1970) showed that dy-namically “hot” stars in galaxies make up spheroidalbulges having a de Vaucouleurs light profile, whereas“cold” stellar components make up the more flattened,rotationally supported, exponential disk region.From that simple beginning, parametric fitting hasbeen the mainstay for galaxy imaging studies, and ex-panding into many applications whenever the sciencecalls for detailed and rigorous analysis. Among some ofthe examples, past investigations delved into the struc-tural parameters of disk galaxies (e.g., de Jong 1996),the Tully-Fisher relation (e.g., Tully & Fisher 1977;Hinz et al. 2003; Bedregal et al. 2006), the evolution ofdisky galaxies (Simard et al. 2002; Ravindranath et al.2004; Barden et al. 2005), the cosmic evolution of galaxymorphology (e.g., Lilly et al. 1998; Marleau & Simard1998; Hathi et al. 2009) in both groundbased surveysand Hubble (Ultra-)Deep Fields (Williams et al. 1996;Beckwith et al. 2006), the morphological transformationof galaxies in cluster environments (e.g., Dressler 1980),the fundamental plane of spheroids (Djorgovski & Davis1987; Dressler et al. 1987; Bender et al. 1992), the redsequence of galaxies (Bell et al. 2004b,a; Faber et al.2007), morphological dissimilarities between spheroidalgalaxies and ellipticals (Kormendy 1985, 1987), the cen-tral structure of early-type galaxies (Kormendy 1985;Lauer et al. 1995; Faber et al. 1997; Ferrarese et al.2006b,a; Lauer et al. 2007) and implications for the for-mation of massive black holes (Ravindranath et al. 2002;Kormendy & Bender 2009), black hole vs. bulge re-lations (Kormendy & Richstone 1995) and their evolu-tion (Rix et al. 2001; Peng et al. 2006a,b), the “extralight” due to gas dissipation in galaxy centers (Kormendy1999; Kormendy et al. 2009; Hopkins et al. 2008b,a),quasar host galaxies (e.g., Hutchings et al. 1984;McLeod & Rieke 1994; McLure et al. 2000; Jahnke et al.2004; S´anchez et al. 2004; Kim et al. 2008b), gravita-tional lensing of quasar host galaxies (Rix et al. 2001;Peng et al. 2006b), and the clustering of dark matterthrough weak lensing (e.g., Heymans et al. 2006, 2008).Since the original development of galaxy fitting nearly70 years ago, where the analysis was performed onone-dimensional (1-D) surface brightness profiles (seealso Kormendy 1977; Burstein 1979; Boroson 1981; Kent1985; Andredakis & Sanders 1994; MacArthur et al.2003), newer techniques have emerged to directly analyzetwo-dimensional (2-D) images (e.g., Shaw & Gilmore1989; McLeod & Rieke 1994; Byun & Freeman1995; de Jong 1996; Moriondo et al. 1998; Simard1998; Ratnatunga et al. 1999; Wadadekar et al.1999; Simard et al. 2002; de Souza et al. 2004;Laurikainen et al. 2004; Gadotti 2008). The bene-fit of performing 2-D image analysis is to potentiallymake full use of all spatial information and to properlyaccount for image smearing by the point-spread function(PSF).Even though 2-D analysis can be quite sophisticated,there are legitimate questions about whether it is morebeneficial than 1-D for profile analysis. Proponents of the1-D technique are skeptical that perfect ellipsoid modelsare suitable to use for galaxies that show isophotal twists,or that are non-elliptical in shape. They note that notonly is 1-D analysis more appealing because it is morestraightforward to implement, the surface brightness pro-files serve as visual confirmation about the reality of fit-ting multiple component models.However, beneath the apparent simplicity there are anumber of important subtleties to weigh. For instance,the decision about how to extract 1-D profiles is oftennot unique, nor are there strong reasons to prefer majoror minor axis profiles, or a profile along some arc tracedby spiral arms or isophote twists that result from thesuperposition of multiple components oriented at differ-ent angles. When symmetry is broken it is also unclearthat there is an optimal or unique way to extract a 1-Dprofile, such as in irregular galaxies, overlapping galax-ies, and galaxies with double nuclei. Another factor toconsider is that the process of extracting 1-D profiles re-duces spatial information content: in many situations,a bulge, disk, and bar can all have different axis ratios,position angles (PAs), and profiles that help to breakmodel degeneracies, but this information is lost when thedata are collapsed into 1-D. Lastly, for compact galax-ies, 1-D profile fitting cannot properly correct for imagesmearing by the PSF because 1-D profile convolution isnot mathematically equivalent to convolution in a 2-Dimage. While some of the above concerns also affect 2-D analysis (i.e. irregular galaxies), most others benefitfrom treatment using 2-D techniques. When it comes tojudging which models are more plausible, there are fewdiagnostics more discerning than a moment’s glance at2-D models and residual images; a good fit in 2-D alwaysmeans that 1-D profiles are necessarily a good fit. Propo-nents of 2-D analysis therefore believe that the benefitsoutweigh the drawbacks. Moreover, many drawbacks canbe mitigated by breaking free from axisymmetry in 2-Danalysis, which is the purpose of this study to show. In the box of tools for morphology analysis, a compli-mentary approach is non-parametric analysis. While wedo not use non-parametric methods in this study, it isuseful to understand the conceptual differences betweenthe two approaches. We thus provide a brief overview.In contrast to function fitting, the non-parametric ap-proach does not involve deciding what functional form touse or how many. One method is to decompose an im-age into “shapelets” or ”wavelets” (e.g., Refregier 2003;Massey et al. 2004), which is analogous to taking a 2-D Fourier transform of an image using mathematicallyorthogonal basis functions. The main conceptual differ-ence with parametric fitting is that the shapelet basisfunctions do not represent physical subcomponents of agalaxy. Moreover, the power spectrum of the basis func-tions is quite useful for diagnosing the degree of galaxydistortions. There are also other non-parametric tech-niques (e.g., Abraham et al. 1994; Rudnick & Rix 1998;Conselice et al. 2000; Lotz et al. 2004). To measure con-centration non-parametrically, one way is to comparefluxes within apertures of different radii; whereas to mea-sure asymmetry one can rotate an image by 180 ◦ andsubtract it from the original image and measure the resid-uals (e.g., Abraham et al. 1994; Conselice et al. 2000).Toward the same goals, two studies, Abraham et al.(2003) and Lotz et al. (2004), introduce the Gini index tomeasure the concentration of a galaxy by comparing therelative distribution of pixel flux values within a certainarea. Lotz et al. (2004) also introduce a method for mea-suring asymmetry through the M parameter, which isthe second-order moment of the brightest 20% of the agalaxy’s flux.The application of non-parametric analysis has mostlybeen to quantify galaxy mergers (e.g., Conselice et al.2003; Lotz et al. 2008). These techniques are generallymuch simpler to implement than parametric fitting andhave a strong virtue that no assumptions are made aboutthe galaxy profiles and shapes. The tradeoff is that thetechniques often do not deal with image smearing by thePSF and different sensitivity thresholds between differentsurveys. Consequently, one has to take particular careto compare compact with extended objects, measured indifferent apertures, or measured from images of differentsurface brightness depths (Lisker 2008). One also shouldguard against contamination by intervening galaxies orstars because the techniques do not have a rigorous wayto separate overlapping objects. For separating objects,extracting structural components of a galaxy, and ex-trapolating galaxy wings well into the background noise,there are few, if any, alternatives to parametric analysisthat are more rigorous.When comparing the merits of non-parametric andparametric analysis, the idea of ellipsoid models in para-metric analysis is sometimes considered to be a weakness,because galaxies, after all, are not perfect ellipses in pro-jection. However, it is worth pointing out that the no-tion of there being a global average size inherently impliescomparison against some kind of approximate shape. In-deed, even in non-parametric techniques, to measure asize in an 2-D image, one assumes a basic shape eitherexplicitly (through using aperture photometry) or implic-itly (through calculating flux moments, which requires acenter to be defined). An ellipsoid is one of the sim-plest and most natural low-order shapes against whichall galaxies can be compared, especially for measuringan average size. This notion is useful: deviations fromthe basic ellipsoid shape can then be considered as higherorder modifications, even for highly irregular galaxies.Nevertheless, there are many situations where it is de-sirable to use models that deviate from ellipsoid shapes.Contrary to the common practice, there are numerousways to break from axisymmetry. However, the harderchallenge is to devise a scheme that is intuitive to graspand well motivated. Breaking free from axisymmetry al-lows for other interesting science applications, includinga promising new way to quantify asymmetry. The Next Generation of Parametric ImagingFitting
In this study we present, as a proof of concept, new ca-pabilities in 2-D image fitting that progress beyond thelimitations of traditional parametric fitting models. Onekey aspect of our approach is to first identify a mini-mum basis set of features that spans the range of galaxymorphologies and shapes. From experience, we deter-mine those four “basic elements” to be bending, Fourier,coordinate rotation, and truncation modes. Secondly,one of the main reasons why parametric fitting is use-ful is that the profile parameters are intuitive to grasp(e.g., concentration index, effective radius, total lumi-nosity, etc.). Therefore, another key requirement is thatthe traditional profile parameters must retain their origi-nal, intuitive, meaning even under detailed shape refine-ments, and even under such extreme cases as irregulargalaxies. This can be accomplished if the basic premisestarts with the traditional ellipsoid function, on top ofwhich one can add perturbations, rotations, irregulari-ties, and curvature. This is possible because of the factthat simple ellipsoid fits are a reasonable way to quan-tify global average properties, and other details can beconsidered to be higher order perturbations that may beof other practical interest.As we attempt to demonstrate, combining just the fourbasic mophology elements can quickly yield a dizzying ar-ray of possibilities for fitting galaxies. The end result canlook highly “realistic.” Indeed, it is now possible to fitmany spiral galaxies, asymmetric tidal features, irregu-lar galaxies, ring galaxies, dust lanes, truncated galaxies,arcs, among others (though, certainly, there are limita-tions). However, it is important to realize that “beingpossible” often does not mean “being necessary” or “be-ing practical.” Necessity ought to be judged in the sci-entific context of whether it is worth the extra effort toobtain diminishing returns. For instance, to measure to-tal galaxy luminosity, it is often unnecessary to fit highorder Fourier modes or spiral rotations. For many sci-ence studies interested in global parameterization, oftena single ellipsoid component would suffice. It is there-fore important to always let the science determine whatkind of analysis is required, rather than to use the newcapabilities in the absence of a clearly defined goal. Hav-ing provided some foregoing disclaimers, some of the keyscientific reasons motivating the new capabilities are to: • Quantify global asymmetry or substructure asym-metry. • Quantify bending modes for weak-lensing applica- tions, or fit arcs in the image plane for strong grav-itational lenses. • Obtain more accurate substructure decompositionin the presence of bars, spirals, rings, etc. • Obtain more accurate global photometry. • Quantify profile deviation in inner or outer regionsof a galaxy, such as disk truncations, deviationsfrom a S´ersic function, etc. • Extract parametric information to the limits im-posed by resolution, signal-to-noise, and othersmall scale fluctuations. • Quantify model-dependent errors in the decompo-sition.We thus begin by giving an overview of the
Galfit software ( § § § § §
6. We then apply thesenew features to real galaxies in §
7, followed by conclud-ing remarks ( § THE 2-D FITTING PROGRAM
Galfit
This study builds on an existing algorithm named
Galfit (Peng et al. 2002), which is a 2-D paramet-ric galaxy fitting algorithm, in the same spirit asother widely used 2-D image-fitting algorithms (e.g.,(GIM2D: Simard 1998; Simard et al. 2002); (BUDDA:de Souza et al. 2004)). Galfit is a stand-alone programwritten in the C language, and can be run on most mod-ern operating systems. To read and produce FITS im-ages,
Galfit calls upon the CFITSIO package (Pence1999).
Galfit is designed to allow for complex im-age decomposition tasks: by allowing for an arbitrarynumber and mix of parametric functions (S´ersic, Moffat,Gaussian, exponential, Nuker, etc.), it can simultane-ously fit any number of galaxies and their substructures.It is possible to use
Galfit for both interactive anal-ysis and galaxy surveys where complete automation isrequired. However, automation requires the use of anexternal “wrapping” algorithm written by the user thattakes care of both the pre-processing (object identifica-tion, initial parameter estimation) and post-processing(extracting and tabulating fitting parameters) of the fit-ting results. χ ν and Error Analysis Galfit is a non-linear least-squares fitting algorithmthat uses the Levenberg-Marquardt technique to find theoptimum solution to a fit. The Levenberg-Marquardtalgorithm is currently the most efficient one for searchinglarge parameter spaces, allowing for the possibility to fit http://users.obs.carnegiescience.edu/peng/work/galfit/galfit.html complex images with multiple components and a largenumber of parameters. Galfit determines the goodnessof fit by calculating χ and computing how to adjust theparameters for the next step. It continues to iterate untilthe χ no longer decreases appreciably. The indicator ofgoodness of fit is the normalized or reduced χ , χ ν : χ ν = 1 N dof nx X x =1 ny X y =1 ( f data ( x, y ) − f model ( x, y )) σ ( x, y ) , (1)where f model ( x, y ) = m X ν =1 f ν ( x, y ; α ...α n ) . (2) N dof is the number of degrees of freedom in the fit; nx and ny are the x and y image dimensions; and f data ( x, y )is the image flux at pixel ( x, y ). The f model ( x, y ) is thesum of m functions of f ν ( x, y ; α ...α n ), with n free pa-rameters ( α ...α n ) in the 2-D model. The uncertainty asa function of pixel position, σ ( x, y ), is the Poisson errorat each pixel, which can be provided as an input image.If no σ -image is given, one is generated based on thegain and read-noise parameters contained in the imageheader. Pixels in the image marked as being bad do notenter into the calculation of χ .In the Levenberg-Marquardt algorithm, the minimiza-tion process involves computing a Hessian matrix, whichis closely related to the covariance matrix of the param-eters (e.g., see Press et al. 1992). The covariance matrixis then directly related to the formal uncertainty in thefitting parameters that Galfit reports. However, theusefulness of the formal uncertainty is limited to idealsituations where the fluctuations in the residual imageare only due to Poisson noise after removing the model.This situation is mostly realized in idealized situations,such as image simulations. In real images, the residualsare due to structures like stars and galaxies that are notfitted, flat-fielding errors, and imperfect functional matchto the data. These factors cause formal uncertainties re-ported in numerical fits to be only lower estimates. Inimage fitting, more realistic uncertainties are necessarilyobtained by other processes, such as comparing fit resultsbased on different assumptions about the model ratherthan through a formal covariance matrix.In summary, the three images
Galfit takes as inputto calculate least squares are the data, a σ -image, and anoptional bad pixel mask. To account for image smearingby the PSF, Galfit will also require a PSF image.
Accounting for Telescope Optics and AtmosphericSeeing
The wavefront of light from distant sources is alwaysperturbed by the act of producing an image, distortionsdue to imperfect optics, and sometimes by the Earth’satmosphere, resulting in some blurring. To accuratelycompare the intrinsic shape of an object with a model,image blurring must be taken into account. In image fit-ting this is often done by convolving a model image withthe input PSF before comparing with the data. The pro-cess of performing convolution is mathematically rigor-ous, but the actual implementation has several subtleties. One consideration is the computation speed, as theprocess of convolving a model is frequently the most timeconsuming part of parametric fitting. The trade-off isthat the smaller the convolution region the faster thecomputation time, but also the less accurate. To achievea compromise,
Galfit allows the user to decide on thesize of the convolution region. This gives the flexibilityfor one to hone in on a solution quickly before trying toobtain higher accuracy in the final step.Another important issue to consider is whether to con-volve each component separately or all of them togetherjust once in the final image. This is an important con-sideration because even though the model functions areanalytic, they are resampled by discrete pixel grids, re-sulting in a “pixellated” profile instead of one that isinfinitely smooth. If an intrinsic model is sufficientlysharp, the curvature may not be critically subsampledby the pixels prior to convolution, regardless of whetherthe recorded data are Nyquist sampled. The resultingprofile after image convolution therefore can depend verysensitively on how the model is centered on a pixel. Ifsuch a model is created off-center, pixellation effectivelybroadens out the model ever so slightly more than normalonce convolution is applied, but the effect is noticeable inhigh-contrast imaging studies. Therefore, the better wayto deal with “pixellation broadening” is to convolve eachmodel component individually rather than the entire im-age at once. To do so,
Galfit creates every model on apixel center; the pixel fluxes near the center of the mod-els are integrated over the pixel area adaptively. Thento effect an off-centered model,
Galfit makes use ofthe convolution theorem by shifting the PSF by the re-quired amount before convolving it with the model. Thisprocess circumvents the problem of artificial pixellationbroadening because whereas the model core region maynot be sufficiently resolved, the PSF ought to be .Shifting the PSF, however, can be quite problematicwhen it is marginally Nyquist sampled, or if the diffrac-tion patterns are not critically sampled. Accurate shift-ing of the PSF is of basic importance in high contrastimaging studies. For instance, in the case of studyingactive galaxies with a strong central point source, issuesof contrast, resolution, and sampling all conspire to makethe PSF fitting crucial to deriving a reliable host model.In such situations, the standard interpolation techniques(e.g., linear or spline) tend to broaden out the PSF core,so they are only accurate in the extended outskirts wherethe gradient is shallow. One alternative is to interpolateusing the sinc kernel, which is theoretically the perfectinterpolation kernel for critically sampled images, andpreserves the intrinsic width of the data. However, signif-icant “ringing” appears around sharp features (i.e. PSFor galaxy core). This effect can be nearly as bad on afit as pixellation broadening. An improved solution isto taper the wing of the sinc kernel using a windowingfunction (e.g., Lanczos), but the ringing often may stillbe quite large beyond the PSF core, which must be fur-ther suppressed. Galfit seeks a compromise by using a hybrid schemewhere the interpolation in the PSF core is done by us-ing a sinc kernel with a Kaiser window function so as If the PSF is not resolved then the convolution process will notbe accurate regardless of the technique.
Fig. 1.—
Example of an input file. The object list is dynamic and can be extended as needed. Each model is modified by a mix of higherorder Fourier modes, bending modes, truncation, or spiral structure. These parameters produce the models shown in Figure 2.
Fig. 2.—
Shapes produced by parameters in the
Galfit inputfile of Figure 1.
Left : a S´ersic light profile modified by a singleFourier mode m = 5, creating the star shape. It is truncated inthe inner region by a truncation function, which is modified by abending mode m = 2, with a lopsided Fourier mode of m = 1. Right : a S´ersic light profile with Fourier modes m = 1 and m = 5is modified by a coordinate rotation function to create a lopsided,multi-armed, spiral structure. to faithfully preserve the width, but a bicubic splineinterpolation is used in the wings. The result of thisscheme is that for a Gaussian having a full width at half-maximum (FWHM) of 2 pixels, the interpolation is ac-curate to 0.1% in the center, and 0.03% at the distanceof the FWHM relative to the peak (or 1% relative tothe local flux). For oversampled PSFs, the interpola-tion is even more accurate. Compared to bicubic splineinterpolation, our scheme is about 20 times more accu-rate. From a more practical standpoint, the mismatchin the PSFs between data taken using the Hubble SpaceTelescope (HST) imaging cameras and synchronously ob-served PSFs is rarely better than 3% in the core. Forall practical purposes, our interpolation scheme there-fore will more than suffice for the most demanding highcontrast studies of quasar host galaxies at high redshift.When the data are undersampled, convolution of themodel can still be done correctly if the convolution PSFprovided to
Galfit is either critically sampled or over-sampled. In this situation,
Galfit will generate a modelon a finer grid, convolve it with the PSF, then bin theresult down to the resolution of the data for comparison.One way for users to obtain an oversampled PSF com-pared to the data is to dither the PSF observations byfractional pixels. Another way is to numerically recon-struct a more oversampled PSF star by extracting multi-ple stars from the data image itself (e.g., via DAOphot,Stetson 1987).However, lastly, we note that when the data and theconvolution PSF are both undersampled (i.e. with PSFFWHM < The Concept of a Model Component
Using the new features, each model can take on ashape that is completely unrecognizable from a tradi-tional ellipsoid shape. It is therefore necessary to clarifywhat constitutes a single model component. In
Gal-fit , each model component is referred to by the nameof the surface brightness profile , just as it is standard practice to call something a S´ersic, Gaussian or expo-nential component in traditional models. As implied bythis notion, no matter how complex the shape , the fluxdeclines monotonically (unless modified by a truncationfunction, Section 5) from a peak in every radial direc-tion in a non-rotating frame, or along an arc in a rotat-ing frame, strictly following the functional form specifiedby the user. The radial profile parameters are mathe-matically decoupled from the azimuthal shape becausethe radial profile functions are self-similar in the expres-sion of the radius parameter, i.e. with powers of ( r/r e ),whereas the complex azimuthal shapes are obtained bysimply stretching the coordinate metric into more exoticgrids than the standard Cartesian grid. This idea is infact implicit in all 2-D image-fitting algorithms, wherethe axis ratio parameter, q , turns a circular profile intoan ellipse by compressing the coordinate axis along onedirection, even though the functional form of the profileremains the same in every direction. In the same man-ner, the definition of a scale or effective radius in a com-ponent, no matter how complex the shape, correspondsclosely to that of the best-fitting ellipse in the directionof the semi-major axis.Figure 1 demonstrates how S´ersic profiles can be modi-fied by bending modes, Fourier modes, and a spiral rota-tion function in Galfit —the results of which are shownin Figure 2. In the example, there are only two S´ersicmodel components, despite the appearance of numerousparameters: both the “radial” and “power” functions aremodifications to the S´ersic profiles. Furthermore, theFourier and bending modes can modify the S´ersic pro-files, or modify the modifiers to the light profiles. Eachradial surface brightness profile has a single peak and theflux decline is monotonic radially (unless truncated by atruncation function called “radial” in Figure 1) or in arotating coordinate system (called “power” in Figure 1).Therefore, for each component, it is still meaningful totalk about, for example, an “average” light profile (e.g.,S´ersic), with an average S´ersic concentration index n —no matter what the galaxy may look like azimuthally. Inthis manner even irregular galaxies can be parameterizedin terms of their average light profile. When the averagepeak of an irregular galaxy is not located at the geomet-ric center, it has a high-amplitude m = 1 Fourier mode(i.e. lopsidedness).In such a way, no matter how complex the azimuthalshape, interpreting the surface brightness profile parame-ters is just as straightforward as the traditional ellipsoid. THE RADIAL PROFILE FUNCTIONS
The radial profile functions describe the intensity fall-off of a model away from the peak, such as the S´ersic,Nuker, or exponential models, among others. For exam-ple, early-type galaxies typically have steep radial pro-files whereas late-type galaxies have shallower intensityslope near the center. The rate of decline is governed bya scale-length parameter. The radial profile is often ofprimary interest in galaxy studies from the standpointof classification, and because the exact functional formmay have some bearing on the path of galaxy evolution.In
Galfit the radial profile can have the following func-tional forms, which are some of the most frequently seenin literature.
Fig. 3.—
The S´ersic profile, where r e and Σ e are held fixed.Notice that the larger the S´ersic index value n , the steeper thecentral core, and more extended the outer wing. A low n hasa flatter core and a more sharply truncated wing. Large S´ersicindex components are very sensitive to uncertainties in the skybackground level determination because of the extended wings. The S´ersic Profile
The S´ersic power law is one ofthe most frequently used to study galaxy morphology,and has the following functional form:Σ( r ) = Σ e exp " − κ (cid:18) rr e (cid:19) /n − ! . (3)Σ e is the pixel surface brightness at the effective radius r e . The parameter n is often referred to as the concen-tration parameter. When n is large, it has a steep innerprofile and a highly extended outer wing. Inversely, when n is small, it has a shallow inner profile and a steep trun-cation at large radius. The parameter r e is known as theeffective radius such that half of the total flux is within r e . To make this definition true, the dependent variable κ is coupled to n ; thus, it is not a free parameter. Theclassic de Vaucouleurs profile that describes a numberof galaxy bulges is a special case of the S´ersic profilewhen n = 4 (corresponding to κ = 7 . n = 1 and n = 0 .
5, respectively. As such the S´ersic profile is acommon favorite when fitting a single component.The flux integrated out to r = ∞ for a S´ersic profileis: F tot = 2 πr e Σ e e κ nκ − n Γ(2 n ) q/R ( C ; m ) . (4)The term R ( C ; m i ) is a geometric correction factor whenthe azimuthal shape deviates from a perfect ellipse. Asthe concept of azimuthal shapes will be discussed in Sec-tion 4, we will only comment here that R ( C ; m i ) is sim-ply the ratio of the area between a perfect ellipse with the area of the more general shape, having the sameaxis ratio q and unit radius. The shape can be modi-fied by Fourier modes ( m being the mode number) ordiskiness/boxiness. For instance, when the shape is mod-ified by diskiness/boxiness, R ( C ) has an analytic solu-tion given by: R ( C ) = π ( C + 2)4 β (1 / ( C + 2) , / ( C + 2)) , (5)where β is the Beta function. In general, when theFourier modes are used to modify an ellipsoid shape,there is no analytic solution for R ( m i ), and so the arearatio must be integrated numerically.In Galfit , the flux parameter that one can use for theS´ersic function is either the integrated magnitude m tot or some kind of surface brightness magnitude, for exam-ple at the center ( µ ), at the effective radius ( µ e ), or atthe break radius ( µ break ) for truncated profiles (see Sec-tion 5). The integrated magnitude follows the standarddefinition: m tot = − . (cid:18) F tot t exp (cid:19) + mag zpt , (6)where t exp is the exposure time from the image header.Each S´ersic function can thus potentially have 7 classicalfree parameters in the fit: x , y , m tot , r e , n , q , and θ PA . The non-classical parameters, C , Fourier modes,bending modes, and coordinate rotation may be added asneeded. There is no restriction on the number of Fouriermodes, and bending modes, but each S´ersic componentcan only have a single set of C and coordinate rotationparameters (see Section 4 for details). The Exponential Disk Profile
The exponentialprofile has some historical significance, so
Galfit is ex-plicit about calling this profile an exponential disk , eventhough an object that has an exponential profile need notbe a classical disk. Historically, an exponential disk hasa scale length r s , which is not to be confused with the ef-fective radius r e used in the S´ersic profile. For situationswhere one is not trying to fit a classical disk it would beless confusing nomenclature-wise to use the S´ersic func-tion with n = 1, and quote the effective radius r e . Butbecause the exponential disk profile is a special case ofthe S´ersic function for n = 1 (see Figure 3), there is arelationship between r e and r s , given by r e = 1 . r s (For n = 1 only). (7)The functional form of the exponential profile isΣ( r ) = Σ exp (cid:18) − rr s (cid:19) , (8)and the total flux is given by F tot = 2 πr s Σ q/R ( C ; m ) . (9)The 6 free parameters of the profile are: x , y , m tot , r s , θ PA , and q . The Gaussian Profile
The Gaussian profile is an-other special case of the S´ersic function with n = 0 . Fig. 4.—
The modified Ferrer profile. The black reference curvehas parameters r out = 100, α = 0 . β = 2, and Σ = 1000. Thered curves differ from the reference only in the α parameter, asindicated by the red numbers. Likewise, the green curves differfrom the reference only in the β parameter, as indicated by thegreen numbers. Figure 3), but here the size parameter is the FWHM in-stead of r e . The functional form isΣ( r ) = Σ exp (cid:18) − r σ (cid:19) , (10)and the total flux is given by F tot = 2 πσ Σ q/R ( C ; m ) , (11)where FWHM = 2.354 σ . The 6 free parameters of theprofile are: x , y , m tot , FWHM, q , and θ PA . The Modified Ferrer Profile
The Ferrer profile(Figure 4; Binney & Tremaine 1987) has a nearly flatcore and an outer truncation. The sharpness of the trun-cation is governed by the parameter α , whereas the cen-tral slope is controlled by the parameter β . Because ofthe flat core and sharp truncation behavior, historicallyit is often used to fit galaxy bars and “lenses.” The pro-file Σ( r ) = Σ (cid:16) − ( r/r out ) − β (cid:17) α (12)is only defined within r ≤ r out , beyond which the func-tion has a value of 0. The 8 free parameters of the Ferrerprofile are: x , y , central surface brightness, r out , α , β , q , and θ PA .It is worth mentioning that a S´ersic profile with lowindex n < . The Empirical (Modified) King Profile
The em-pirical King profile (Figure 5) is often used to fit the
Fig. 5.—
The empirical King profile. The black reference curvehas parameters r c = 50, r t = 100, α = 2, and Σ = 1000. Thered curves differ from the reference curve only in the α parameter,as indicated by the red numbers. Likewise, the green curves differfrom the reference only in the r c parameter, as indicated by thegreen numbers. Fig. 6.—
The Moffat profile. The black reference curve hasparameters n = 2, FWHM = 20, and Σ = 1000. The othercolored lines differ only in the concentration index n , as shown bythe numbers. The dashed line shows a Gaussian profile of the sameFWHM. light profile of globular clusters. It has the followingform (Elson 1999): Fig. 7.—
The Nuker profile. The black reference curve has parameters r b = 10, α = 2, β = 2, γ = 0, and I b = 100. For the other coloredlines, only one value differs from the reference, as shown in the legend. Σ( r ) = Σ (cid:20) − r t /r c ) ) /α (cid:21) − α × (cid:20) r/r c ) ) /α − r t /r c ) ) /α (cid:21) α . (13)The standard empirical King profile has a power law withindex α = 2. In Galfit , α can be a free parameter.In this model, the flux parameter to fit is the centralsurface brightness, µ , expressed in mag arcsec − (seeEquation 20). The other free parameters are the coreradius ( r c ) and the truncation radius ( r t ), in additionto the geometrical parameters. Outside the truncationradius, the function is set to 0. Thus, the total numberof classical free parameters is 8: x , y , µ , r c , r t , α , q ,and θ PA . The Moffat Profile
The profile of the
HST
WFPC2PSF is well described by the Moffat function (Figure 6).Other than that, the Moffat function (Moffat 1969) isless frequently used than the above functions for galaxyfitting. The functional profile isΣ( r ) = Σ [1 + ( r/r d ) ] n , (14)and the total flux is given by F tot = Σ πr d q ( n − R ( C ; m ) . (15)In Galfit the size parameter to fit is the FWHM, wherethe relation between r d and FWHM is r d = FWHM2 √ /n − . (16) The 7 free parameters are: x , y , m tot (i.e. total mag-nitude, instead of µ ) FWHM (instead of r d ), the con-centration index n , q , and θ PA . The Nuker Profile
The Nuker profile (Figure 7) wasintroduced by Lauer et al. (1995) to fit the central lightdistribution of nearby galaxies, and it has the followingform: I ( r ) = I b β − γα (cid:18) rr b (cid:19) − γ (cid:20) (cid:18) rr b (cid:19) α (cid:21) γ − βα . (17)The flux parameter to fit is µ b , the surface brightness ofthe profile at r b , which is defined as µ b = − . (cid:18) I b t exp ∆ x ∆ y (cid:19) + mag zpt , (18)where t exp is the exposure time from the image header,and ∆ x and ∆ y are the platescale in arcsec. The Nukerprofile is a double power law, where (in Equation 17) β is the outer power law slope, γ is the inner slope, and α controls the sharpness of the transition. The motivationfor using this profile is that the nuclei of many galaxiesappear to be fit well in 1-D (see Lauer et al. 1995) bya double power law. However, caution should be exer-cised when using this function because, for example, alow value of α ( α .
2) can be mimicked by a combina-tion of high γ and low β (compare Figure 7 c with theother two panels), which presents a serious potential fordegeneracy. In all there are there are 9 free parameters: x , y , µ b , r b , α , β , γ , q , and θ PA . The Edge-On Disk Profile
Both the S´ersic (Equa-tion 3) and exponential disk profile (Equation 8) aremerely empirical descriptors of a galaxy light profile.However, for edge-on disk galaxies, there is a more0physically motivated light profile: under the assump-tion that the disk component is locally isothermal andself-gravitating, the light profile distribution is given by(van der Kruit & Searle 1981):Σ( r, h ) = Σ (cid:18) rr s (cid:19) K (cid:18) rr s (cid:19) sech (cid:18) hh s (cid:19) , (19)where Σ is the pixel central surface brightness, r s isthe major-axis disk scale length, h s is the perpendiculardisk scale height, and K is a Bessel function. The fluxparameter being fitted in Galfit is the central surfacebrightness: µ = − . (cid:18) Σ t exp ∆ x ∆ y (cid:19) + mag zpt . (20)Note that if the disk is oriented horizontally the coor-dinate r is the x -distance (as opposed to the radius) of apixel from the origin. There are 6 free parameters in theprofile model: x , y , µ , r s , h s , and θ PA . The PSF Profile
For unresolved sources, one canfit pure stellar PSFs to an image (as opposed to func-tions with narrow FWHM convolved with the PSF). ThePSF function is simply the convolution PSF image thatthe user provides, so there is no prescribed analyticalfunctional form. This is also the only profile that is notconvolved in
Galfit . The PSF has only 3 free param-eters: x , y , and m tot . Because there is no analyticalform, the total magnitude is determined by integratingover the PSF image and assuming that it contains 100%of the light. If the PSF wing is vignetted, there will bea systematic offset between the flux Galfit reports andthe actual value.If one wants to fit this “function,” it is important tomake sure that the input PSF is close to, or super-,Nyquist sampled. The PSF interpolation used in shiftingis done by a sinc function with a Kaiser window, whichcan preserve the widths of the PSF even under subpixelshifting. This is in principle better then spline interpo-lation or other high-order interpolants. However, if thePSF is undersampled, aliasing will occur, and the PSFinterpolation will be poor. In this situation, it is bet-ter to provide an oversampled PSF to
Galfit (and tospecify the amount of oversampling), even if the dataare undersampled. With
HST data this can be done us-ing TinyTim (Krist & Hook 1997) or by combining stars.
Galfit will take care of rebinning during the fitting.Note that the alternative to fitting a PSF is to fit aGaussian with a small width (e.g., 0.4–0.5 pixels), which
Galfit will convolve with the PSF. This is generally notadvisable if a source is a pure point source because con-volving a narrow function with the PSF will broaden outthe overall profile, even if slightly. The convergence canalso be poor if the FWHM parameter starts becomingsmaller than 0.5 pixels. However, this technique can stillbe useful to see if a source is truly resolved.
The Background Sky
The background sky is a flatplane with flux gradient along x and y directions. Thusit has a total of 3 free parameters. The pivot point forthe sky is fixed to the geometric center ( x c , y c ) of the im-age, calculated by ( n pix + 1) /
2, where n pix is the number of pixels along one dimension. The tip and tilt are calcu-lated relative to that center. Because the galaxy centroidlocated at ( x, y ) is in general not at the geometric center( x c , y c ) of the image, the sky value directly beneath thegalaxy centroid is calculated by:sky( x, y ) = sky( x c , y c ) + ( x − x c ) d sky dx + ( y − y c ) d sky dy . (21) THE AZIMUTHAL SHAPE FUNCTIONS
Whereas the radial profile governs the decline of galaxyflux radially from a central peak, the azimuthal functionsgenerate the projected shape in the x − y plane of theimage. For instance, ellipsoidal, irregular, spiral, disky,and boxy shapes are all created by azimuthal functions.All traditional 2-D image-fitting techniques use an ellipseas the fundamental shape, which is obtained by stretch-ing the coordinate grid along one dimension compared tothe orthogonal direction. Indeed, all azimuthal functionsare coordinate transformations. Therefore, to change ashape from an ellipse into more exotic shapes, the coor-dinate system [ r ( x, y )] can be further stretched or shrunkradially from the peak, as a function of azimuth angle.This coordinate transformation preserves the functionalform of the surface brightness profile in every directionbecause the profiles are self-similar— that is, they arefunctions of ( r/r scale ). Thus defined, the radial profileparameters (e.g., r e , q , central concentration, etc.) re-tain their original meaning no matter the complexity ofthe azimuthal shape.We introduce four new ways to modify the azimuthalshape of a model, beginning with the traditional ellip-soidal model. On top of an ellipsoid, this section de-scribes how one can add Fourier modes, bending modes,and coordinate rotation functions (power law and loga-rithmic). Each component can be modified by any one orall of the azimuthal functions simultaneously, dependingon the complexity of the galaxy one is trying to analyze.The next section will cover truncation functions. Generalized Ellipses
The simplest azimuthal shapein
Galfit is the traditional generalized ellipse. This isthe starting point for all
Galfit analysis, no matter howcomplex is the final outcome. The radial coordinate ofthe generalized ellipse is defined by: r ( x, y ) = | x − x | C +2 + (cid:12)(cid:12)(cid:12)(cid:12) y − y q (cid:12)(cid:12)(cid:12)(cid:12) C +2 ! C . (22)Here, the ellipse axes are aligned with the coordinateaxes, and ( x , y ) is the centroid of the ellipse. Definedby Athanassoula et al. (1990), the ellipse is called “gen-eral” in the sense that C is a free parameter, whichcontrols the diskiness/boxiness of the isophote. When C = 0 the isophotes are pure ellipses. With decreasing C ( C < C in-creases ( C >
0) (see Figure 8). The major axis of theellipse can be oriented to any PA. Thus, there are a totalof 4 free parameters ( x , y , q, θ PA ) in the standard ellipseand an additional one, C , for the generalized ellipse.1 Fig. 8.—
Generalized ellipses with ( a ) axis ratio q = 1 and ( b ) axis ratio q = 0 .
5. Various values of the diskiness/boxiness parameter C are labeled. Fourier Modes
Few galaxies look like perfect el-lipsoids, and one can better refine the azimuthal shapeby adding perturbations in the form of Fourier modes.The Fourier perturbation on a perfect ellipsoid shape isdefined in the following way: r ( x, y ) = r ( x, y ) N X m =1 a m cos ( m ( θ + φ m )) ! . (23)In the absence of Fourier modes in the parenthesis, the r ( x, y ) term is the radial coordinate for a traditionalellipse, and θ = arctan (( y − y ) / (( x − x ) q )) defined inEquation 22. The Fourier amplitude for mode m is a m .Defined as such, a m is the fractional deviation in radiusfrom a generalized ellipse of Equation 22. The number ofmodes N is unrestricted, and any mode can be left out.The “phase angle,” φ m , is the relative alignment of mode m relative to the PA of the generalized ellipse; the phaseangle is 0 ◦ in the direction of the semi-major axis of thegeneralized ellipse (rather than up), increasing counter-clockwise. Figure 9 shows some examples of how Fouriermodes modify a circle and an ellipse into other shapes.Each Fourier mode has 2 free parameters, a m and φ m ,and the number of modes the user can add is unre-stricted. However, the most useful modes are low-orderones ( m = 1 , ... m = 2 mode ispartially degenerate with the classical axis ratio param-eter, q , for an ellipse. Therefore the use of m = 2 and q ,together, should be largely avoided except in some situ-ations (e.g., peanut-looking bulges).The phase angles of the Fourier modes are also usefulinformation to keep in mind. Modes with the followingphase angles have the following symmetry properties: • Symmetry about a central point: a = 0, regardlessof other mode phase and amplitude. • For all modes m , there is reflection symmetry at: φ m = 0 ◦ , ± ◦ m . For m = even, this symmetry isabout both the major and minor axes, whereas for m =odd, the reflection symmetry is only about themajor axis. • For odd modes of m , there is additional reflectionsymmetry about the minor axis at: φ m = ± ◦ m .An irregular galaxy has angles that are “out of phase”whereas regular galaxies have angles that are more “inphase” (i.e. reflectionally symmetric around either mi-nor or major axis). Therefore, it is possible to quantifyvarious forms and degree of asymmetry by constructingindices based on the amplitude and phase angles of theFourier modes. The most intuitively obvious asymmetryindex is the m = 1 mode, which captures the lopsided-ness ( A L ) of a galaxy, i.e. the positioning of the bright-est central region relative to the fainter outer region of agalaxy: A L = | a | . (24)Asymmetric galaxies are also characterized by overall de-viation from an ellipse; thus, another intuitively usefulquantity to measure is the sum of the Fourier amplitudes: A E = N X m | a m | . (25)Asymmetric galaxies by definition have high A E . How-ever, it is possible for galaxies with both high A E and A L to be reflectionally symmetric; the degree of reflectionalsymmetry may be an indicator for how well the galaxiesis relaxed. Reflection asymmetry is given by the index A R : A R = P m =even | a m | sin (cid:16) πm φ m ◦ (cid:17) + P m =odd | a m | sin (cid:16) πm φ m ◦ (cid:17) , (26)2where φ m is in degrees. In this formulation, the higherthe reflectional asymmetry, the higherthe index A R .Used together, these three descriptors provide highly use-ful ways to quantify the degree galaxies are irregular. Forinstance, high values of A R and A L most likely implyhigh global asymmetry in the intuitive sense. Whereasa high value of A E with low A R implies high regularity,but large deviation from an ellipse, such as edge-on diskygalaxies or a disky/boxy ellipticals. Bending Modes
Bending modes allow for power-law-shape curvatures in the model, as opposed to spi-ral windings. The coordinate transformation ( x, y ) = ⇒ ( x ′ , y ′ ) is obtained by only perturbing the y -axis (in arotated frame) in the following way: y ′ = y + N X m =1 a m (cid:18) xr scale (cid:19) m , (27)where x ′ = x , r scale is the scale radius of the model (i.e. r eff for S´ersic, r s for exponential, etc.). Some examplesof this perturbation are shown in Figure 10. Note that m = 1 resembles quite closely to the axis ratio parameter, q . However, the m = 1 bending mode is actually a shearterm, the effect of which is most easily seen when it op-erates on a purely boxy profile with C ≈ a ),shearing it into a more disky shape (see Figure 10 d ).The bending modes can be modified by Fourier modesor diskiness/boxiness to change the higher order shapeof the overall model. This kind of coordinate transfor-mation again preserves the original meaning of the radialprofiles. Here, the object size parameter refers to theunstretched size, i.e. projected onto the original ( x, y )Cartesian frame, as opposed to a length along the curva-ture. Coordinate Rotation: The Concept
Sometimesthe isophotes of a galaxy can rotate as a function of ra-dius, as in the case of spiral galaxies. To model spiral pat-terns, it is now possible to allow for coordinate rotation in
Galfit . Coordinate rotation in
Galfit means that theflux within circular annuli overlayed on a model rotatesas a function of radius, i.e. θ = f ( r ). The functional form f ( r ) can be fairly arbitrary but the most familiar patternin nature is that of a logarithmic spiral, i.e. θ ∼ log( r ).However, many spiral galaxies deviate from logarithmicwinding either in the inner region, for instance due tothe presence of a bar, or in the outer region, as mightbe due to tidal or non-relaxed features. These structurespose a problem when fitting galaxy images because onecannot simply mask out regions of non-interest when thegoal might be to obtain the cleanest separation betweena spiral and other embedded components. Therefore apure logarithmic spiral, while useful to trace segments ofa spiral, is often not ideal for fitting the whole galaxy,but ought to be modified in some ways. For this reasonwe introduce the concept of a hyperbolic-tangent (tanh)modification to a logarithmic or a power-law spiral.A pure tanh function looks like Figure 11 a , showingthat f ( r ) asymptotes to constant values at r → ±∞ ,which is a highly desirable feature. As shown in Fig-ure 11 a , the function can be scaled, stretched, and shiftedso that θ ( r ) ≈ r < r in : it is useful to model a bar-like feature, which, by definition, has a constant PA as a function of radius. A tanh function is also useful inthe upper asymptotic limit because f ( r ) at r > r out ,when multiplied by another function f ( r ), takes on theform of f ( r ), and the crosstalk within r in is minimal,as shown in Figure 11 b . In short, a tanh function al-lows for a transition between two functions: a constantfunction at r < r in and another r > r out , for example apower-law or a logarithmic function. Moreover, the rateof that transition can be cleanly managed and is easy tointerpret. For this reason a hyperbolic tangent is also afunction of choice later on in Section 5 when we presentthe idea of a truncation function. Galfit allows for twotypes of coordinate rotation functions, the power-law spi-ral ( α -tanh), and the logarithmic spiral (log-tanh), bothof which are motivated empirically. We note that eventhough the logarithmic spiral is favored more in the lit-erature, we find that the α -tanh spiral is better able tocapture the range of spiral behaviors found in nature be-cause of the one extra degree of freedom in α , which cansimulate the behavior of the log-tanh spiral over regimesof interest. We therefore tend to prefer use of the α -tanhcoordinate rotation by default. We now give an overviewof the two types of coordinate rotation: Coordinate Rotation I: Power-law - HyperbolicTangent ( α -tanh) The term “power law” refersto the fact that the pure tanh function of Figure 11 a ismultiplied by a function of the form ∼ r α . The exactfunctional form of the rotation function is lengthy (seeAppendix A), but the schematic functional dependenceof the power-law spiral on the parameters is given by thefollowing: θ ( r ) = θ out tanh (cid:16) r in , r out , θ incl , θ skyPA ; r (cid:17) × (cid:20) (cid:18) rr out + 1 (cid:19)(cid:21) α . (28)As defined, the power-law rotation starts to take holdbeyond r = r out , and below which the tanh transitiondominates. Figure 11 shows a pure hyperbolic tangentrotation function for several different values of the pa-rameter r in ( left ), and a combination of “bar” ( r in ) pa-rameter and the asymptotic power-law slope α ( right ),where r is the radial coordinate system and θ out is therotation angle roughly at r out . The inner radius, r in ,is defined to be the radius where the rotation reachesroughly 20 ◦ . This angle corresponds fairly closely to ourintuitive notion of bar length based on examining images,but is not a rigorous, physical definition. The angle θ incl is the line-of-sight inclination of the disk, where θ incl = 0 ◦ is face-on and θ incl = 90 ◦ is perfectly edge-on.To motivate intuition for the free parameters used inthe coordinate rotation definition, Figures 12 and 13show a progressive series of images for the spiral rotationfunction with different combination of parameter values.For instance, Figure 12 shows a series of images of purehyperbolic tanh spiral with increasing maximum rotationangle ( θ out ), all else being held constant at the values in-dicated at the top. The spiral arm winding increaseswith increasing θ out , and the winding gets tighter, butthe body does not expand wider because r out is fixed.It is also important to note that a face-on model doesnot necessarily mean that the outer-most isophotes areround. Rather, the ellipticity of the outer-most isophotesis related to the asymptotic behavior of the rotation func-3 Fig. 9.—
Examples of Fourier modes. (
Top ) Low-amplitude ( a m = 0 .
05) Fourier modes modifying a circular profile ( q = 1 .
0) with phaseangle φ m = 0 ◦ . ( Bottom ) High-amplitude ( a m = 0 .
5) Fourier modes modifying an elliptical profile ( q = 0 .
5) with phase angle φ m = − ◦ . tion, which asymptotes to a constant PA beyond a radiusof r out for a pure hyperbolic tangent ( α = 0, Figure 11 a ).The isophotes only appear circular in the main body ofthe spiral structure when it has a large number of wind- ings. Figure 13 shows several other examples of barredand unbarred spirals, with progressively different α val-ues, sky inclination angle, and rotated to different skyposition angles ( θ skyPA ). The parameters for each grey-4 Fig. 10.—
Examples of bending modes modifying a circular profile ( q = 1 .
0) with C = 0 (unless indicated otherwise). ( Top row )Low-amplitude ( a m = 0 . r m scale ) bending modes. ( Bottom row ) High-amplitude ( a m = 0 . r m scale ) bending modes. Bending modes can becombined with Fourier modes to change the higher order shape. Fig. 11.—
Hyperbolic tangent-power-law spiral angular rotation functions with outer spiral radius of r out = 100. ( a ) Examples of purehyperbolic tangent spirals ( α = 0) with different bar radii ( r in ). ( b ) Examples with different bar radii and asymptotic power law α , asindicated. See Figures 12 and 13 for examples of how these parameters translate into 2-D images. scale figure are shown at the top and to the right of thecorresponding (column, row). When the power-law in-dex α is negative, the spiral pattern can reverse courseafter reaching a maximum value (see right-most columnof Figure 13).In summary, the hyperbolic tangent power-law func- tion has 6 free parameters: θ out , r in , r out , α, θ incl , and θ skyPA . The thickness of the spiral structure is controlledby the axis ratio q of the ellipsoid being modified by thehyperbolic tangent, or by the Fourier modes that modifythe ellipsoid. To create highly intricate and asymmetric5 Fig. 12.—
Examples of pure (i.e. with power law α = 0 or without logarithmic function) hyperbolic tangent coordinate rotation modifyingan elliptical profile with axis ratio q = 0 .
4. Note that all panels share the same parameters as shown up top. The spiral model has no bar.The numbers within each panel show the amount of total winding (units in degrees) at the spiral rotation radius of 50. Notice that outside r = 50, the rotation angle becomes constant, due to the rotation function being a hyperbolic tangent, thereby creating the appearance ofa flattened disk, even though there is not a separate disk component involved in the model. spiral structures, Fourier modes can be used in conjunc-tion with coordinate rotation.We note that the “bar” radius ( r in ) is a mathematicaltool. Even though the r in term in the coordinate rota-tion does look like a bar when it is sufficiently positive,it should be regarded only as a mathematical construct to grant the rotation function as much flexibility as pos-sible. This construct can reflect reality, but it does nothave to. For instance, mathematically, a negative r in radius (Figure 11 b ) is perfectly sensible because of theway Equations 28 and 29 (for logarithmic spirals, be-low) are defined: a negative r in value just means thatthe spiral rotation function has a finite rotation angleat r = 0 relative to the initial ellipsoid out of which itis constructed. When there is clearly no bar, the r in pa- rameter can become quite negative; in this case, the fit isoften indistinguishable from one where the bar radius is0. Furthermore, often times, one may not wish to createa bar and a spiral out of one smoothly continuous func-tion for various reasons, for instance because they mayhave different widths, the spiral may not extend into thecenter, or the spiral may start off in a ring. In these situ-ations, one can “detach” the bar from the spiral by usinga truncation function (see § α -tanh rotation6 Fig. 13.—
Examples of power law - hyperbolic tangent ( α -tanh) coordinate rotation modifying a face-on ( θ incl = 0 ◦ ) elliptical profilewith axis ratio q = 0 .
4. The parameters of the rotation functions are shown on the top and right-hand side of the diagram. The top panelsshow the spiral rotation angle as a function of radius for the panels in the same column. In the right-most column, the spiral arms reversedirection at r = 30 because the spiral rotation function (top-right panel) decreases in rotation angle. function works surprisingly well for many spiral galaxies,the function is smooth, so “kinks” in the spiral struc-ture cannot yet be modeled, even though it is possibleto do so by allowing for “kinks” in the rotation function.Lastly, the spiral structure cannot wind back onto itself,because that would require the rotation function to bemulti-valued. Coordinate Rotation II: Logarithmic - Hyper-bolic Tangent (log-tanh)
The winding rate ofspiral arms in late-type galaxies is often thought to be logarithmic with radius rather than power law. Thus,
Galfit also allows for a logarithmic-hyperbolic tangentcoordinate rotation function, which is defined as: θ ( r ) = θ out tanh (cid:16) r in , r out , θ incl , θ skyPA ; r (cid:17) × (cid:20) log (cid:18) rr ws + 1 (cid:19) / log (cid:18) r out r ws + 1 (cid:19)(cid:21) . (29)Like the α -tanh rotation function, the log-tanh func-tion has a hyperbolic tangent part that regulates the7 Fig. 14.—
Logarithmic - hyperbolic tangent spiral angular rotation functions. ( a ) Examples of different bar radii, where the outerhyperbolic spiral radius is r out = r in + 10. The lower horizontal dashed line shows the rotation angle at the “bar” radius ( r in ). ( b )Examples with different “bar” radii ( r in ) and winding-scale radii r ws , as indicated, illustrating the degree of flexibility of the spiral rotationrate. The rotation angle at r out is fixed to 150 ◦ , as shown by the upper horizontal dashed line. The left-most, black curve is close to beinga pure logarithmic function, recasted so that at r = 0, the rotation angle θ = 0 ◦ . bar length and the speed of rotation within r out . Be-yond r out the asymptotic rotation rate is that of thelogarithm function, which has a winding scale radiusof r ws ; the larger the winding scale radius, the tighter the winding. Thus, like the α -tanh spiral, the log-tanh spiral rotation function also has 6 free parameters: θ out , r in , r out , r ws , θ incl , and θ skyPA . Note that in terms ofcapabilities, the α -tanh function can often reproduce thelog-tanh function and more. Therefore, the α -tanh isprobably a more useful rotation function in practice.Note that Galfit does not allow for a pure logarith-mic spiral because such a function has a negative-infinityrotation angle at r = 0. Therefore, in Galfit , at r = 0the rotation function reaches θ = 0 (Figure 14). Lastly,it is also important to keep in mind that the meaningof the “bar radius,” just as described in the section for α -tanh rotation function, is a mathematical construct. THE TRUNCATION FUNCTION
Truncation functions allow for the possibility of cre-ating rings, outer profile cut-offs, dust lanes, or a com-posite profile in the sense that the inner region behavesas one function and the outer behaves as another. Thetruncation function can modify both the radial profileand azimuthal shape. A ring can be created by truncat-ing the inner region of a light profile. Likewise, when agalaxy has spiral arms that do not reach the center, itcan be viewed as being truncated in the inner region.
General Principle In Galfit each truncation function can modify one ormore light profile models. Also, any number of light pro-files can share the same truncation function. The trunca-tion function in
Galfit is a hyperbolic tangent function (see Equation 7 in Appendix B). Schematically, a trun-cated component is created by multiplying a radial lightprofile function, f ,i ( x, y ; ... ), by one or more truncationfunctions, P m or 1 − P n (depending on whether the typeis an inner or an outer truncation), in the following way: f i ( x, y ; ... ) = f ,i ( x, y ; x c,i , y c,i ... q i , θ PA ,i ) × (30) m Y P m ( x, y ; x c,m , y c,m , r break ,m , ∆ r soft ,m , q m , θ PA ,m ) × n Y [1 − P n ( x, y ; x c,n , y c,n , r break ,n , ∆ r soft ,n , q n , θ PA ,n )] . The break radius, r break , is defined to be the locationwhere the profile is 99% of the original (i.e. untruncated)model flux at that radius. The parameter ∆ r soft is thesoftening length, so that r = r break ± ∆ r soft is where theflux drops to 1% of that of an untruncated model at thesame radius (the ± sign depends on whether the trun-cation is inner or outer). The inner truncation function( P m ) tapers a light profile in the inner regions of a lightprofile, whereas the outer truncation function (1 − P n )tapers a light profile in the wings.The behavior of the hyperbolic tangent function is idealfor truncation because it asymptotes to 1 at the breakradius r & r break and 0 at the softening radius r < r soft ,and vice versa for the complement function. Thus, whenmultiplied to a light profile f ( r ), the functional behaviorexterior to the break radius has intuitively obvious mean-ings. For example, as shown in Figure 16 a , if a S´ersicfunction with n = 4 is truncated in the wings (shownin red), the core has exactly an n = 4 profile interior to r break (marked with a vertical dashed line), which is afree parameter to fit. Likewise, an n = 4 profile trun-cated in the core (green) has exactly an n = 4 profileexterior to the outer break radius. Thus, when one sums8 Fig. 15.—
Logarithmic - hyperbolic tangent spiral (log-tanh) angular rotation examples, all face-on ( θ incl = 0 ◦ ) and θ skyPA = 0 ◦ . Thetop-left panel shows the meaning of the rotation parameter values at the corners of each box. As with the α -tanh spirals, the log-tanhspiral can be tilted and rotated to any sky projection angle, or combined with Fourier modes to produce lopsided or multi-armed spiralstructures (not shown), and with truncation function to produce an inner ring or an outer taper. The top-left panel figure, for all practicalpurposes, is a pure logarithmic spiral with a winding scale radius r ws = 5. two functions of different S´ersic indices n (Figure 16 b )the asymptotic profiles of the wing and core retain theiroriginal meaning, and there is very little crosstalk out-side of the truncation region (denoted by vertical dashedlines in Figure 16).Use of the truncation functions is highly flexible. Therecan be an unrestricted number of inner and outer trunca-tion functions for each light profile model. Furthermore,multiple light profile models can share in the same trun-cation functions. This is useful, for instance, when tryingto fit a dust lane (inner truncation) in a fairly edge-ongalaxy that may affect both the bulge and the disk com-ponents. Just as with light profile models, the trunca-tion functions can be modified by Fourier modes, bending modes, etc., independent of the higher order modes forthe light profile they are modifying. Different Variations of the Truncation Function
Truncation models appear in many physical contexts,such as dust lanes, rings, spirals that do not reach thecenter, joining a spiral with a bar, or cut-off of the outerdisk. To allow the truncation parameters to be moreintuitive to understand given situations at hand,
Gal-fit offers several variations. In addition to inner andouter truncations, truncation functions can share in thesame parameters as the parent light profile. There areradial and length/height truncations, softening radius vs.softening length (default vs. Type 2), inclined vs. non-inclined (default vs. Type b) truncations, and, lastly,9
Fig. 16.—
Examples of hyperbolic truncation functions on n = 4 and n = 1 S´ersic profiles. ( a ) A continuous n = 4 model representedas two truncated models of otherwise identical r e , n , and central surface brightness, with truncation radii at r = 15 and r = 20, as markedby the vertical dashed lines. The black curve is the sum of the inner and outer functions. This shows that, outside the truncation region,there is very little “crosstalk” between the inner and outer components. ( b ) A composite profile made up of an n = 4 nucleus truncated inthe wings and an n = 1 truncated in the core, with truncation radii r = 10 and r = 20. Note that the hump in the summed model wouldgive rise to a ring in a 2-D model. four different ways to normalize the flux—the most sen-sible choice depends on how a profile is truncated. Wenow discuss each of these variations in more detail. Parameter Sharing.
In the most general form, eachtruncation function has its own set of free parameters: x , y , r break , ∆ r soft , q , and θ PA . However, by default, theparameters x , y , q , and θ PA are tied to the light profilemodel, and are activated only when the user explicitlyspecifies a value for them. Radial (“radial”) vs. Length (“length”) orHeight (“height”) Truncations.
The most use-ful type of truncation is one that has radial symmetryto first order, i.e. where it has a center, an ellipticity,and an axis ratio. However, in the case of a perfectlyedge-on disk galaxy (“edgedisk” model), an additionaltype is allowed that truncates linearly in length or inheight. For instance, a dust lane running through thelength of the galaxy has an inner height truncation. Forthe “edgedisk” profile,
Galfit also allows for a radialtruncation, as with all other functions. The one draw-back to height and length truncations is that they cannotbe modified by Fourier and higher order modes like theradial truncations.
Softening Length (“radial”) vs. Softening Radius(“radial2”).
Sometimes, instead of softening length (∆ r soft ), it is more useful for the fit parameter to bea softening radius ( r soft ), especially when one desires tohold the parameter fixed. That is also allowed in Galfit as a Type 2 truncation function, designated, for exam-ple, as “radial2.” The default option does not have anumerical suffix.
Inclined (default, “radial”) vs. Non-inclined(“radial-b”) Truncations.
A spiral rotation func- tion is an infinitesimally thin, planar structure. Never-theless, it should be thought of as a 3-D structure inthe sense that the plane of the spiral can be rotatedthrough three Euler angles, not just in position angleon the sky. When a truncation function is modifying aspiral model, it is therefore sometimes useful to thinkabout the truncation in the plane of the spiral model.When Fourier modes and radial truncations are modi-fying a spiral structure, the default (“radial”) is for themodification to take place in the plane of the spiral struc-ture. However, there are some instances when that maynot be ideal (e.g., a face-on spiral may actually be ellip-soidal). In those situations, one can choose “radial-b”,which would allow a truncation function to modify thespiral structure in the plane of the sky, even though thespiral structure can tip and tilt as needed.Lastly, the truncation function can be Type 2b (i.e.“radial2-b”) as well.
Flux Normalization.
The most intuitive flux nor-malization for a truncated profile is the total luminosity.Unfortunately, both the total luminosity and the deriva-tive of the free parameters with respect to the total lu-minosity are especially time-consuming to work out com-putationally and slow down the iteration process. Thereare generally no closed form analytic solutions to theproblem. Therefore, the alternative is to allow for differ-ent ways to normalize a component flux. The user maychoose whichever one is more sensible, given the situa-tion and the science task at hand. The default dependson the truncation type: • Inner truncation: the flux is normalized at thebreak radius. This is most sensible for a ring modelbecause this radius roughly corresponds to the peakflux of the ring.0
500 500 50050 50 50500505005050050 Truncation Examples a b cd e fg h i ( ) ( ) ( )( )( )( )( ) ( ) ( )
Fig. 17.—
Examples of truncation functions acting on a single-component light profile of various shapes. ( a ) Inner truncation of around profile, creating a ring. ( b ) The truncation function can be modified by Fourier modes, just like the light profile. ( c ) The truncationfunction can be offset in position relative to the light profile. ( d ) The truncation function can act on a spiral model. ( e ) The truncationcan tilt in the same way as the spiral. ( f ) The truncation function can be modified by Fourier modes while acting on a spiral model. ( g )A round light profile is being truncated in the wing by a pentagonal (Fourier mode 5) truncation function. ( h ) A round light profile isbeing truncated in the inner region by a triangular function (Fourier mode 3), and in the wing by a pentagonal function. ( i ) A three-arm,lopsided, spiral light profile model is truncated in the wing by a pentagonal function, and in the inner region by a triangular function. • Outer truncation: flux normalized at the center. • Both inner and outer truncation: same as the casefor inner truncation.However, there are many situations when the defaultis not desirable. Instead, the user can choose the radiuswhere the flux is normalized. To be pedagogical, we ex-plicitly show here the normalization for just the S´ersicfunction: • function : default (e.g., “sersic,” “nuker,” “king,”etc.). See the details for individual functions. • function1 : flux normalized at the center r = 0(i.e. Σ ). A function that is given originally by f orig ( r ) is now defined as f mod ( r ) = Σ f orig ( r ) f orig (0) . Forthe S´ersic profile (i.e. called “sersic1”), the profilefunction is redefined in the following way, writtenexplicitly: f mod ( r ) = Σ exp (cid:20) − κ (cid:18)(cid:16) rr e (cid:17) /n − (cid:19)(cid:21) exp [ κ ] . (31)For the Ferrer and King profiles, this normalizationis the same as the default normalization.1 • function2 : flux parameter is the surface brightnessat a model’s native size parameter (parameter 4 ofthe light profile model). For a S´ersic profile, called“sersic2,” this means the effective radius r e . So, f mod ( r ) = Σ e f orig ( r ) f orig ( r e ) . For example, a S´ersic profilenow has the following explicit form: f mod ( r ) = Σ e exp " − κ (cid:18) rr e (cid:19) /n − ! . (32)For the Nuker profile this normalization is the sameas the default normalization. • function3 : flux parameter is the surface brightness(Σ break ) at the break radius ( r break ). This is themost useful situation when a truncation results ina large-scale galaxy ring, so that the surface bright-ness parameter corresponds closely to the peak ofthe light profile model. When the truncation isnot concentric with the light profile model, thiskind of normalization is not very intuitive. For“radial” truncation, r break is parameter 4, whereasfor “radial2,” r break is parameter 4 for outer trun-cation and parameter 5 for inner truncation. Whenthe “sersic3” option is chosen, the r break parametercomes automatically from the first truncation com-ponent with which a certain light profile model isassociated.In our example of the S´ersic profile, f mod ( r ) =Σ break f orig ( r ) f orig ( r break ) . For example, a S´ersic profile nowhas the following explicit form: f mod ( r ) = Σ break exp (cid:20) − κ (cid:18)(cid:16) rr e (cid:17) /n − (cid:19)(cid:21) exp (cid:20) − κ (cid:18)(cid:16) r break r e (cid:17) /n − (cid:19)(cid:21) . (33)Figure 17 demonstrates just some of the possibilitiesallowed when fitting truncations. In addition to the reg-ular ellipsoid shape, the higher order modes like diski-ness/boxiness parameters, bending modes, and Fouriermodes can also modify the shape of the truncation func-tions. One can also use the truncation function on a spi-ral model, on models with Fourier and bending modes,and diskiness/boxiness models, some of which are shownin Figures 17 d , 17 e , 16 f , and 17 i . When a truncationfunction acts on a spiral component, it can do so eitherin the plane of the disk (“Type a”) or in the plane of thesky (“Type b,”; e.g., “radial-b”). While the default is inthe plane of the disk, the parameters are more intuitivein Type b cases when the disk is tilted and rotated. Caveats about using the Truncation Function
The use of truncation functions should be carefully su-pervised because unexpected things can happen, suchas the size or the concentration index of a componentcan grow without bound. This behavior is due to thefact that there are degeneracies between the sharpness oftruncation and the steepness/size of the galaxy. There-fore, truncation functions should only be used on objectsthat clearly have truncations. When two functions are joined by using a truncationfunction, the crosstalk region is located in between thetwo truncation radii: it is worth bearing in mind the defi-nition that at the break and softening radii, the fluxes are99% and 1% that of the same model without truncation,respectively. In other words, the larger the truncationlength, the larger the crosstalk region. Therefore, whenone (or more) of the parameters r break , r break + ∆ r soft ,or r soft is either too small ( . few pixels) or larger thanthe image size, it probably indicates that profile trun-cation parameters are not meaningful. Rather, it morelikely reveals the fact that there is a mismatch betweenthe light profile model and the actual galaxy profile. INTERPRETATION, PARAMETER DEGENERACIES,UNIQUENESS, LOCAL MINIMA, AND ERROR ANALYSIS
Now that we have introduced several ways to modify anellipse into more exotic shapes, a natural question to askis how unique or robust are the modifications. A single-component ellipsoid fit can often be used to quantify theglobal average profile of galaxies. However, beyond that,decisions about what procedure to use get to be morecomplicated. On the one hand, the science goal mightcall for fitting detailed structures inside a galaxy (e.g., abulge, bar, nuclear star cluster, etc.). On the other hand,doing so raises concerns about parameter degeneracies,uniqueness, and local minima solutions when the analysisbecomes complex. It is therefore useful to consider insome depth what causes degeneracies and the differentcontexts in which they appear. Doing so allows for betterunderstanding for how to deal with them and how toproperly interpret results from complex analysis. For,not all complex analyses are more suspect, nor are allsimple analyses more robust.The term “degeneracy” has a specific mathematicalconnotation, namely the relation of a + b = c is degen-erate in a and b for a constant value of c . In the galaxyfitting literature, “degeneracy” is often more loosely usedto also refer to “non-unique” or “local minimum” so-lutions (e.g., a fit oriented at 90 ◦ from the best orien-tation), or strong “parameter correlation” (e.g., sky isanti-correlated with the S´ersic index n ). We will mostlynot make the subtle distinctions here and instead will usethe term “parameter degeneracy” generically to refer toall such situations.However, when fitting galaxies, it is more important todistinguish between the aforementioned real degeneraciesfrom “pseudo” ones. Real degeneracies refer to correlatedparameters, local minima, and mathematically degener-ate solutions. By contrast, “pseudo” degeneracies involveconvergence issues when an algorithm is used beyond itstechnical limits, or when users provide bad input modelpriors to fit the data. They may have nothing to dowith real degeneracies, yet the behavior of convergencemay seem to suggest otherwise. Whereas problems withreal degeneracies are often resolvable by using full spatialinformation of 2-D images, pseudo-degeneracy problemsare solved through experience and by using sound scien-tific or technical judgment, as we elaborate further.In this section, we discuss how most of the parame-ter degeneracy problems are avoidable with proper inputpriors and proper fitting supervision, even when largenumbers of free parameters are involved. We also dis-cuss why, contrary to popular notions, when it comes to2avoiding model degeneracy and local minima, it is notsufficient to only choose a model that is the simplest.Rather, it is a judicious combination of simplicity and realism that make for the most robust solutions. Lastly,these discussions are intimately connected to the issueof error analysis because error measurements are nearlyalways dominated by systematic issues rather than pho-ton noise in galaxy fitting. We therefore discuss why it ismore important to quantify model-dependent systematicerrors rather than to rely on statistical estimates.We note that the discussions below are mostly basedon experience, which we present using practical exam-ples rather than to show using rigorous proof. Carryingout a rigorous proof is not only beyond the scope of thisstudy, but it is nearly impossible to do in a general man-ner because different scientific applications have differentsensitivities to different types of degeneracies. We arealso aware that presenting a full discussion of degeneracyissues lends credence to the common notion that multi-component analysis is dangerously complex. However,the reality is not nearly so dire when one has a properunderstanding of the underlying issues and causes. True Numerical Degeneracies Caused by Correlatedor Non-Unique Parameters
There are well-known situations when different param-eters in one or more functions are capable of modelingthe same profile behavior. This scenario is the one mostcommonly referred to in generic discussions about modeldegeneracies. For instance, very large S´ersic index val-ues ( n &
4) have highly extended wings, the presence ofwhich is non-unique with the sky parameter. A high n ,caused by profile mismatch or poor model prior, can oftensuppress the sky estimate. It is therefore advisable to es-timate the sky independent of the fit, and to hold it fixedto the best estimate. As a second example, in the Nukerprofile (Equation 17), there are three parameters ( α , β ,and γ ) that control the inner/outer slopes and sharpnessof the bending (Figure 7). When the break radius r b ofa Nuker profile is sufficiently small and profile mismatchsufficiently large, model discrimination relies entirely onthe power law γ − βα . Because there are numerous ways toyield a specific value for γ − βα in the model, it leads to adegenerate situation involving three parameters.As another example, a low-amplitude second Fouriermode and the first bending mode (shear) can both bedegenerate with the axis ratio q parameter of an ellipse,therefore they should not be used together except inobvious situations where doing so is useful. Lastly, inthe spiral rotation function, the periodicity of the rota-tion function can sometimes be a source of “degeneracy.”Multiple windings can approximate a smooth continuousmodel, whether or not there is a spiral structure present.For instance, a classical S´ersic ellipsoid can be simulatedby a spiral model with a very large θ out . While the fitis not good and easy to diagnose by an end user, it isnevertheless a numerically allowed solution.The above situations are not meant to be a completelaundry list, but they are the most common situations.In complex analysis, one always needs to be circumspectabout the potential hazards of mixing and matching dif-ferent functions whose parameters produce similar profilebehaviors. Even though Galfit allows for a great deal of flexibility in the analysis, it is ultimately up to the userto decide on what to allow, based on the goals of thescience, and to understand when potentially degenerateparameters may be used effectively.The above discussion may also seem to imply degen-eracies or non-uniqueness are too numerous for complexanalysis to be practical or reliable. That notion is onlytrue when it is not possible to verify the results of a fitand to try out other solutions. Such a scenario is morecommon for large scale galaxy surveys, in which auto-mated, detailed, analysis is admittedly quite difficult toconduct sensibly. However, even in those scenarios, thereare many situations where mutually coupled parametersdo not affect the other main parameters of scientific inter-est: degeneracies in the Fourier modes often do not haveany bearing on the total luminosity or size of a compo-nent. Moreover, when an analysis is done manually, itis reassuring that the problems are almost always easyto recognize and remedy when they do happen, even bysimple inspection of the model and residual images.
Pseudo-Degeneracies Caused by TechnicalConditions (e.g., Model Profile Resolution,Parameter Boundaries)
Occasionally, what appears to be numerical degener-acy problems may be caused by someone using a codeoutside the algorithm’s physical capabilities. As such,it is a pseudo-degeneracy. Different algorithms have dif-ferent limitations that affect convergence, whether thecode is gradient descent, Metropolis, or otherwise. Thissituation may appear like parameter degeneracy becauserestarting the fit does indeed yield a different solution,but in fact the code may be hamstrung in its conver-gence. For example, gradient descent algorithms requirethe calculation of a gradient, and thus can run into prob-lems when the gradient cannot be calculated properly. Insimulated annealing algorithms (e.g., Press et al. 1992),parameter boundaries and annealing speed control the al-gorithmic behavior: anneal too quickly, the solution maysettle into a local minimum. To search larger parameterspaces requires longer annealing times.While all algorithms have conditions under which theyperform poorly, pseudo-degeneracies can always be rec-ognized and mitigated.
Galfit is based on a Levenberg-Marquardt subroutine that performs the least-squaresminimization. In part a gradient descent algorithm, theconvergence behavior is affected by the calculation of gra-dient images that determines the direction of steepest χ descent. When the gradient images are problematic, theyaffect the convergence to a proper solution. There arethree main problematic situations. The first, and mostcommon, occurs when a model becomes extremely com-pact (FWHM . . q . .
05) and theobject is compact; here, the gradient does not exist alongone spatial direction because of a lack of pixel resolution.Another rare example occurs when the inclination angleof a spiral rotation component is close to perfectly face-on3( θ incl → Galfit is used for automated analysis(H¨aussler et al. 2007) . Pseudo-Degeneracies Caused by Bad Input ModelPriors
One of the most common causes of degeneracy prob-lems in galaxy fitting analysis comes from using inputpriors that are not well suited to the data. The mostcommon “input priors” involve the choice of the type orthe number of components in a model . Input priors areideal when the number of components of a model usedin a fit matches the number of luminous components ina galaxy. However, often times one may choose to use ei-ther fewer or more components than needed by the data.A common example where the input prior is bad iswhen one uses fewer model components than called forby the data. Two of the main reasons for doing so are toreduce the number of components/free parameters, or toallow automated analysis, where it is not yet possible totailor fits to individual galaxies. This approach is oftenan intentional course of action taken by many studies,especially when it comes to automating two componentanalysis; the goal, ostensibly, is to decompose a galaxyinto bulge and disk components. Seemingly reasonableand justifiable on the notion of reducing the potential fordegeneracy, the approach is generally regarded by mostpeople to be a positive attribute, rather than a sourceof problem itself. Yet, that intuitive notion conflictswith the basic principle of how least squares algorithmswork, and leads to perhaps the most common causes of(pseudo-)degeneracy problems cautioned by literature. While these conditions can always be anticipated in advance,implementing a solution in the code is more tricky, because the actof doing so may also induce other convergence difficulties. Thisleads to a false sense of security about the robustness of a solution. An input prior does not refer to the accuracy of the initialparameter guesses.
To understand why using fewer components than nec-essary is bad, it is important to appreciate that galaxy fit-ting analysis is fundamentally flux weighted. Thus, whena luminous structure is not accounted for, other subcom-ponents try to compensate, however imperfectly, for itspresence. For instance, one may use a two-componentmodel fit to a galaxy that has a bulge, disk, and bar.Doing so may have several different outcomes. One solu-tion is where one component is a sum of (disk+bar) whilethe other is the bulge. Another can be (bulge+bar) anddisk, or perhaps a compromise (e.g., bulge + 0.7 bar; disk+ 0.3 bar). Which scenario occurs depends on the rela-tive contrast (i.e. flux weighting) of the bar to the bulgeand disk, and potentially on the initial parameters of thethree components; small perturbations may “bump” thesolution out from one minimum into another. It is quitepossible for there to be a “global minimum” solution tothis problem. However, when the most meaningful solu-tion, physically, is simply dis-allowed by the input prior,a globally minimum χ cannot lend much credence to thereality of the model components.An input model prior might also be bad if the modelinvolves using more subcomponents than inherentlypresent in a galaxy. In this situation, the results de-pend strongly on the degree of profile mismatch betweenthe model function and the data. If there is significantmismatch, all the components cooperate to reduce theresiduals. For instance, it is always possible to fit mul-tiple exponential models to a single-component de Vau-couleurs profile. If the goal is to obtain the total flux,the sum would do a better job than using a single expo-nential. However, individually, the structural parametersmay not have much physical meaning.Another example involving model prior is in the areaof high-contrast imaging, where the goal is to deblenda central, unresolved, point source from a diffuse under-lying object (e.g., quasar and host galaxy). To do soreliably requires an accurate PSF model for the unre-solved source, or else the residuals may overwhelm theextended object, causing unreliable fits. Here, the prioris the PSF model. Quantifying how the prior affects thefitting results involves trying out different PSFs, or to in-clude extra components to account for the PSF residuals,depending on the science goal.These examples illustrate some of the most commonsituations where the reliability of a fit depends less on thenumber of free parameters, and more on having a propermodel to describe the data. Beyond a single-componentanalysis, the need to make such a decision means thatit will be difficult to automate highly detailed decompo-sitions of galaxies. However, while multi-subcomponentfitting is difficult to automate, it is reassuring that mak-ing a wise decision, interactively, is often not particularlydifficult when a science goal is clearly defined. More-over, for galaxy surveys, where the goal is to fit single-component profiles to galaxies, multi- object decomposi-tion is quite feasible to automate (e.g., M. Barden et al.2009, in preparation; H¨aussler et al. 2007).In summary, pseudo-degeneracy conditions exist be-cause least-squares fitting fundamentally involves fluxweighting: when luminous flux distributions are presentin an image, the models are attracted toward them toreduce the residuals. Therefore, when all components4are not properly modeled, the result may be tricky tointerpret not because of potential for model degenera-cies, but that the solution may have no physical meaningeven if there is a global minimum. The solution is to in-crease the complexity of the analysis progressively untilall luminous components are properly accounted. Thisprocess does not imply, however, that it is necessary toaccount for every component inside a galaxy for the so-lution to have any meaning, only that components ofsimilar flux ratios ought to be simultaneously accountedin detailed analysis; with a few exceptions (e.g. locallydominant features like nuclear star cluster, nuclear ring),components with low fluxes generally do not significantlydisturb the parameters of the much more luminous sub-components. Parameter Degeneracies Can be Broken by SpatialInformation in 2-D
One of the most common notions regarding fitting de-generacy is that the more free parameters there are thegreater is the potential for degeneracy problems. How-ever, the sheer number of parameters is often not itself anindication of a potential for parameter crosstalk. Con-sider, for instance, that it is equally robust to fit thou-sands of well-separated stars as it is to fit an isolatedone. The same is true for galaxies, even though they areconsiderably more extended and may overlap: in large-scale image simulations, H¨aussler et al. (2007) studiedautomated batch analysis of galaxies using one S´ersicprofile per galaxy. They find that simultaneously fittingoverlapping or neighboring objects using multiple com-ponents (often 3–10 S´ersic models at a time) recovers theinput simulated parameters more accurately than fittinga galaxy singly while masking out the neighbors.Indeed, spatially well-localized sources, like a bar, ring,or off-nuclear star clusters, are virtually free from degen-eracies caused by crosstalk with other components. Agalaxy bar is well determined because it is more elon-gated, has a flatter radial profile, and is more sharply de-fined than the surrounding bulge and disk components,despite being embedded within. Compact objects thatare off-centered may also be well determined if the restof the galaxy can be modeled accurately. Contrary tonotions that more model components lead to greater de-generacy, it is important to consider the qualitative as-pects of those components: not accounting for strongfeatures explicitly can yield a less reliable and less physi-cally meaningful fit because the solution is a compromisebetween the different subcomponents.
Measurement Uncertainties, ParameterCorrelation, and Parameter Degeneracies
The issue of parameter degeneracies closely ties intothe topic of measurement uncertainties, especially whenthe result of the analysis may depend on the input modelin fitting galaxies. When the model fits the data perfectly(i.e. the residuals are only due to Poisson noise) it ispossible to infer parameter uncertainties from the covari-ance matrix of free parameters, which is produced duringleast-squares minimization by the Levenberg-Marquardtalgorithm. In galaxy fitting, ideal situations are oftennot realized because the differences between the data andthe model profile involve not only random (e.g., Poisson)sources, but also systematics from non-stochastic (e.g., profile function or shape mismatch, neighboring galax-ies, etc.), and stochastic factors (overall smoothness, forinstance due to star clusters). The one exception is underlow signal-to-noise (S/N) situations, when Poisson noiseexceeds model imperfection. In most other situations,non-random factors dominate the residuals, causing un-certainties inferred from covariance matrices to be un-derestimated. Therefore, it is frequently not very mean-ingful in galaxy fitting to cite measurement uncertaintiesfor the fitting parameters in the traditional sense.One way to quantify uncertainties, possible in largegalaxy surveys, is to allow the scatter of the data points inphysical relations (e.g., radius vs. luminosity, luminosityvs. metallicity, etc.) to articulate the overall uncertaintyof the measurements, even if individual errors could notbe easily obtained. Such a scatter inherently involves aconvolution of several error sources: the intrinsic scatterpresent in a physical relation, Poisson measurement er-ror, stochastic and non-stochastic systematic errors dueto model imperfections. Intrinsic scatter, being a factof nature, remains present in physical relationships evenshould the data have infinite S/N, and even if the modelsare perfect fits to the data. Intrinsic scatter is often a sci-entifically interesting quantity, but it is difficult to differ-entiate from scatter caused by systematic and stochasticerrors, which do not vanish given infinite S/N.In the absence of large galaxy surveys, it is then impor-tant to quantify stochastic and non-stochastic systematicerrors for individual objects. Some example situationsinclude the black hole mass vs. galaxy relation stud-ies (Kormendy & Richstone 1995; Gebhardt et al. 2000;Ferrarese & Merritt 2000) and the fundamental plane(Djorgovski & Davis 1987).In general it is very difficult to pin-point all the causesof non-stochastic systematic errors in an analysis, and toquantify their magnitude. However, one common causeis profile model mismatch: to the extent that one doesnot know the intrinsic model of a galaxy a priori, the un-certainty in measuring the parameters is wedded to one’sassumptions about the model. Therefore, the process ofquantifying systematic, model-dependent errors involvesexploring the degree of parameter coupling, by tryingout different models and seeing how the parameters ofkey scientific interest change. Another source of system-atic error is due to comparing results from different al-gorithms. In this scenario, the most sensible practice istherefore to only compare parameters that are derivedusing the same fitting technique (rather than 1-D vs. 2-D), and using the same pixel and flux weighting scheme (instead of Poisson vs. non-Poisson) during analysis.In contrast, stochastic errors arising from general non-smoothness of a galaxy profile are caused by, for example,star forming patches, dust lanes, etc.. Existing on smallscales and widely dispersed, non-smoothness cannot beeasily identified and modeled in a practical manner usingmultiple components. Even if it is possible to do so,whether they ought to be fitted explicitly, masked, or nottreated at all, falls under the purview of the science goal.Stochastic fluctuations often influence the analysis in amanner analogous to having large correlated noise in thedata. If the fluctuations can be quantified, one possiblesolution is to include them in the fit as a variance termof χ (Equation 1). To estimate the fluctuations requiresfirst obtaining a smooth underlying model, which is not5always easy to do if galaxies have steep and/or irregularprofiles.While it is generally difficult to disentangle stochasticfrom non-stochastic sources of systematic errors, therealso do not seem to be obvious benefits for doing so froma scientific standpoint. For most applications, one shouldonly be interested in the overall magnitude of the sys-tematic errors in a collective sense. One way forwardis therefore to understand which parameters are moststrongly coupled, then compare the results of differentsolutions judging by which ones are physically plausible.For instance, one common interest in bulge-to-disk de-composition is to quantify the uncertainty of the S´ersicindex n . We know that the S´ersic index n takes on alarge value when a profile has both a steep core and anextended wing (see Figure 3). Therefore, quantifying sys-tematic errors in measuring the S´ersic n might involvemasking or fitting nuclear sources/neighboring contami-nation, trying out different PSFs, or fitting the disk byallowing for different disk S´ersic index values. Properlyjudging the causes of systematic errors and accountingfor them often would lead to more natural fits and moresensible parameter values, without the need to hold cer-tain parameters fixed to preconceived answers.In exploring the parameter space as described, thereis often a concern that parameter degeneracies are toonumerous or problematic to understand, which bringsthe discussion back full circle. As discussed in previ-ous sections, when the cause of parameter degeneracy isproperly identified, and when the model priors are wellconceived, our experience has been that spatial informa-tion in 2-D can often effectively break many potentialdegeneracies between the components. Even when thesize, luminosity, and central concentration, of the dif-ferent components correlate they often interact in fairlysuperficial ways, and do not dramatically change whatthe model components represent physically. However,in situations where cross-talk is significant and there isno reason to prefer one solution over another (when theinput prior is befitting), then differences in the answerspeak to the degree of the parameter uncertainty that isof key interest to quantify, rather than to avoid, becauseultimately the models are empirically motivated.In summary, to the extent that the results may de-pend on model assumptions, parameter exploration isthe only viable way to quantify true measurement errorsin the fit parameters. Thus, when used properly, param-eter coupling/degeneracy, rather than complicating theinterpretation, offers a deeper insight into the reliabil-ity of the overall analysis. We illustrate the above ideasmore explicitly in the following examples. EXAMPLES OF DETAILED DECOMPOSITION
To demonstrate how to use the new features to ex-tract complex structures, we analyze five galaxies thatare well resolved: IC 4710, an edge-on disk galaxy, Arp147, M51, and NGC 289. These galaxies are chosen be-cause they represent examples where traditional analysisusing perfectly ellipsoid models tend to leave some ques-tion as to what is physically being measured and to therobustness of the photometry and decomposition. Theprimary purpose here is to illustrate the basic buildingblocks of galaxy morphology, not to address what are the most “scientifically interesting” or useful applications—the scope of which is far too broad to address. As such,each individual example is not intended to necessarily be“interesting” in its own right. For instance, while param-eterizing a ring galaxy like Arp 147 may not itself be tooworthwhile scientifically, the concept has other relevanceto deblending Einstein rings from lensing galaxies in theimage plane of strong gravitational lenses, or separat-ing a ring from a bulge, disk, and bar in spiral galaxies.Indeed, these are heuristic examples meant to generatenew ideas for potentially interesting applications, and toillustrate the dynamic range of capabilities in our newapproach.Another goal of this section is to illustrate two seem-ingly contradictory notions when it comes to galaxy mor-phology analysis: • Sometimes it is not necessary to perform “full-blown” analysis, including spiral structures,Fourier modes, rings, etc..
The detailed analysisbelow will show when it is not necessary to utilizethe full machinery in order to meet the sciencerequirements, such as when the interest is to onlyquantify global properties. However... • Sometimes it is necessary to perform full-blownanalysis.
In situations where detailed decomposi-tion matters (e.g., quantifying bulge-disk-bar frac-tions) the most reliable analysis is to make full useof the machinery available.Indeed, the availability of new tools does not in anyway invalidate or weaken the conclusions of hundreds ofstudies that came before this one—quite the contrary.Rather, the main message is that given the new capabil-ities, it is more important now than ever to weigh therelative benefits of sophistication against the drawbackof increased difficulty and time, whereas no such optionsexisted before.
IC 4710
IC 4710 is an SB(s)m galaxy, which has a bar-likefeature in the middle of a roundish outer structure, asshown in the R -band image of Figure 18, which comesfrom the CINGS (Carnegie-Irvine Nearby Galaxy Sur-vey) project . Prior to the analysis, we masked out thestars using the SExtractor software (Bertin & Arnouts1996). We analyze this galaxy using, for comparison,both one- and two-component regular and higher ordermodels with Fourier modes, shown in Figures 18 b-i . Thebest-fit parameters are given in Table 1, which illustratesthree different sets of analysis parameters: best fit usingtwo components (Figure 18 c ), a model using just the tra-ditional ellipsoid component (Figure 18 g ), and the samesingle-component model with Fourier modes added (Fig-ure 18 b ). Figure 18 i shows the radial surface brightnessprofile of the data and the individual subcomponents ofthe best model.There are several points to understand from comparingdetailed and simple analyses. The best-fitting ellipsoidmodel (Figure 18 g ) is oriented more parallel to the bar-like, higher surface brightness component than the lower http://users.obs.carnegiescience.edu/lho/projects/CINGS/CINGS.html Fig. 18.—
Detailed analysis of IC 4710. ( a ) Original data. ( b ) Best single-component S´ersic profile fit with Fourier modes m = 1to m = 10. ( c ) Best two-component S´ersic profile fit each with Fourier modes, corresponding to the parameters shown in Table 1. ( d )Best-fit residuals. ( e ) Component 1 of 2 in the best-fit model of Panel ( c ). ( f ) Component 2 of 2 in the best-fit model. ( g ) A traditionalsingle-component ellipsoid fit. ( h ) Residuals from the model in Panel ( g ). ( i ) 1-D surface brightness profile. The individual components areshown as dashed lines, and the solid line coursing through the data is the sum of the two components. The lower panel shows the residualsof data − model. Table 1. IC 4710 Fitting Results x [ ′′ ] ∆ y [ ′′ ] mag r e [ ′′ ] n q θ PA [deg] Commentsfourier — mode: ampl. & phase [deg] mode: ampl. & phase [deg] mode: ampl. & phase [deg]Best fit 1 — sersic — 0 .
00 0 .
00 13 .
71 48 .
92 0 .
55 0 . − . .
16 0 .
08 0 .
00 0 .
17 0 .
00 0 .
00 0 . .
16 1: − .
40 3: 0 .
17 3: − .
95 4: 0 .
06 4: 17 . .
00 1: 1 .
18 3: 0 .
00 3: 0 .
21 4: 0 .
00 4: 0 . .
05 5: 18 .
37 6: − .
06 6: 13 .
22 7: 0 .
03 7: − .
20— 5: 0 .
00 5: 0 .
34 6: 0 .
00 6: 0 .
23 7: 0 .
00 7: 0 . .
05 8: 10 .
92 9: 0 .
01 9: 4 .
55 10: 0 .
03 10: − .
68— 8: 0 .
00 8: 0 .
17 9: 0 .
00 9: 1 .
63 10: 0 .
00 10: 0 . .
97 26 .
28 12 .
49 57 .
24 0 .
37 0 .
90 41 . .
24 0 .
19 0 .
00 0 .
09 0 .
00 0 .
00 0 . − .
31 1: − .
25 3: 0 .
03 3: 55 .
46 4: 0 .
03 4: − .
65— 1: 0 .
00 1: 0 .
86 3: 0 .
00 3: 1 .
31 4: 0 .
00 4: 0 . .
04 5: − .
73 6: 0 .
02 6: − .
47 7: 0 .
01 7: 16 .
56— 5: 0 .
00 5: 0 .
87 6: 0 .
00 6: 1 .
15 7: 0 .
00 7: 1 . − .
03 8: − .
49 9: 0 .
01 9: − .
63 10: 0 .
02 10: − .
64— 8: 0 .
00 8: 0 .
86 9: 0 .
00 9: 0 .
91 10: 0 .
00 10: 0 . χ = 167438.77 N dof = 127966 N free = 53 χ ν = 1.31Single 1 — sersic — 0 .
00 0 .
00 12 .
15 60 .
37 0 .
69 0 . − . .
06 0 .
05 0 .
00 0 .
12 0 .
00 0 .
00 0 . χ = 247304.81 N dof = 128009 N free = 10 χ ν = 1.93Single 1 — sersic — 0 .
00 0 .
00 12 .
15 59 .
00 0 .
69 0 . − . .
12 0 .
09 0 .
00 0 .
09 0 .
00 0 .
00 0 . − .
05 1: 72 .
02 3: − .
07 3: 27 .
55 4: − .
02 4: − . .
00 1: 2 .
39 3: 0 .
00 3: 0 .
31 4: 0 .
00 4: 0 . .
02 5: 5 .
54 6: − .
01 6: − .
72 7: − .
02 7: 15 .
40— 5: 0 .
00 5: 0 .
54 6: 0 .
00 6: 0 .
80 7: 0 .
00 7: 0 . .
01 8: 0 .
28 9: 0 .
01 9: 3 .
16 10: − .
02 10: 1 .
12— 8: 0 .
00 8: 0 .
50 9: 0 .
00 9: 0 .
63 10: 0 .
00 10: 0 . χ = 235140.44 N dof = 127991 N free = 28 χ ν = 1.84 Note . — Best-fitting parameters for IC 4710. The meaning of the object parameters is shown at the top for each model component. Thestatistical uncertainties for each model component, based on the covariance matrix of the fit, are shown in the row underneath the best-fittingmodel parameters. Systematic uncertainties due to imperfect model-data match are typically 1%–10% for the fluxes, 10%–20% for the sizes, and20%–30% for the S´ersic index. For the Fourier modes, the phase angle is relative to the major axis of the light profile component. Note that the skyparameters are not shown. The “
Best fit ” parameters (top section) correspond to Panel ( c ) in Figure 18, “ Single component ” parameters (middlesection) correspond to Panel ( g ), and “ Single component with Fourier modes ” parameters (bottom section) correspond to Panel ( b ). e ). This happensbecause a single-component fit is a compromise betweenthe various subcomponents of a galaxy, and, as such, itreflects neither one perfectly. Allowing the azimuthalshape to change by adding 9 Fourier modes results ina shape shown in Figure 18 b . Note that because theprofile is restricted to having a S´ersic functional form inevery direction radially from the peak, the shape doesnot have complete freedom to take on any shape, as op-posed to a shapelet or wavelet-type Fourier inversion: itis merely a higher order perturbation of the best-fittingellipse. Indeed, in comparing single-component fit pa-rameters in Table 1 for the two models, the main S´ersicstructural parameters hardly budged, despite the Fouriermodel having 18 more free parameters. Therefore, themarginal returns in using more free parameters is negli-gible in this situation when it comes to the main S´ersicstructural parameters. However, if the scientific interestis to quantify the global symmetry, then the higher ordermodes are of interest.Another point of interest is how higher order modelsaffect the accuracy of the global photometry. It is nat-ural to expect when a model is unrealistic for a galaxythat the photometry is also unreliable. In Figure 18 c ,it is evident that a two-component model is more ap-propriate than the single-component fits of Figure 18 b and 18 g . However, when the flux of the two-componentmodel is summed, one finds that the difference with thesingle-component fits is only 0.03 mag. This and subse-quent examples illustrate empirically that the process ofleast-squares minimization using even na¨ıve ellipsoids isoften capable of providing accurate photometry to within0.1 to 0.2 mag, even if the galaxy shape departs from el-lipsoid models quite drastically.Lastly, Figures 18 e and 18 f demonstrate that it is quitefeasible to unambiguously disentangle embedded com-ponents that have different shapes, using higher orderFourier modes. Despite there being a large number ofparameters, it is visually clear based on Figures 18 e and18 f that parameter degeneracy is not an issue, becausethe shapes of the components are quite different. In part,this is possible because of how Fourier modes are imple-mented in Galfit : the profile function is preserved inevery direction radially from the peak, even in situationswhere the shape is irregular, as in Figure 18 e . GEMS Edge-on Galaxy
This edge-on galaxy (Figure 19, Table 2) comes fromthe GEMS (Galaxy Evolution from Morphology andSED, (Rix et al. 2004)) project, which is an
HST imag-ing survey of the
Chandra
Deep Field-South. Belyinga benign morphological appearance is a dust lane (Fig-ure 19 e ) that courses through the center, complicatingthe traditional ellipsoid fitting technique.The analysis of even this simple object can be quiteinvolved. The best-fitting model involves three compo-nents: a fairly compact bulge, an edge-on disk compo-nent, and an puffy stellar halo enveloping both. Since thehalo component is more luminous than the bulge compo-nent, a two-component model fit would naturally ascribethe halo component to the bulge, despite there being adistinctly rounder component at the center. Like theprevious example, each of the three components (Fig- ure 19 f-h ) is modified by Fourier modes. Furthermore,the best fit includes an actual model for the dust lane(component 4, Table 2). The dust lane is modeled by aninner truncation function as discussed in Section 5.A truncation model is shown as a model “component”in the fit; it is unique because it is not a light profilemodel, and one cannot generate an image to see what itlooks like. Instead, its influence is to be seen on all thelight profile models (i.e. components 1–3; Figure 19 f-h ),where it reduces the light by the same fraction for allcomponents. In every other way, the truncation functionbehaves exactly like a light profile model: it can have itsown centroid (or not), and it can be modified by Fouriermodes, as shown in Table 2. The benefit of using a singletruncation model for all three light profile models is notonly to reduce the degrees of freedom, but it is also phys-ically motivated because foreground dust attenuates allbackground light sources by an equal fractional amount.Nevertheless, if desired, it is also possible to allow eachcomponent to be attenuated differently.This example also demonstrates how the result of theanalysis depends on the input prior of the model. In thefit using traditional ellipsoid parameters, a mask is usedto minimize the effect of the dust on the analysis. Yet,the effects cannot be completely removed. As shown inTable 2, the inclusion of the truncation model can sig-nificantly affect the structural parameters: the surfacebrightnesses can differ by 0.8 mag arcsec − , and the sizesby 10%–20%, even in this seemingly uncomplicated situa-tion. Moreover, the differences far surpass the statisticaluncertainties shown in Table 2. To the extent that it isnot possible to judge which model is more physically cor-rect, both measurements ought to be treated as equallyvalid. In that situation, the uncertainties, due entirelyto model assumptions, are roughly ∼ . ∼
10% in size.
Arp 147
The
HST /F814W image of the field Arp 147 containstwo ring galaxies (Figure 20, Table 3), one of which hasa bulge-like component with a tidally disturbed outer re-gion (Galaxy 1), and the other is a pure ring (Galaxy 2).The best-fitting model for Galaxy 2 is a single-componentring, modified by Fourier modes, as seen in Figure 20 b ,whereas Galaxy 1 requires two ring components, a bulge,and an inner fine-structure component (Figures 20 e-h ).The fine-structure component of Galaxy 1 can be easilyseen in the surface brightness profile as an upturn within r = 0 . ′′ i (left). In addition, the tidal com-ponent is slightly bent, which is modeled elegantly usingthe bending modes of Equation 27. As in the case ofa dust lane, the ring model comes about by truncatingthe inner region of a pure S´ersic profile (see Section 5.The only difference here is that the truncation radii are alarger fraction of the galaxy size. Whereas for the edge-on galaxy, it makes more sense to normalize the flux atthe effective radius (Equation 32), for ring galaxies, nor-malizing the flux at the break radius (Equation 33) ismore intuitive, because it is closer to the peak of theprofile model. In fact, the peak of the ring is about half-way between r break and r break + ∆ r soft , but the exactlocation depends on the profile type.It is again instructive to compare a traditional fit us-ing simple S´ersic ellipsoid models (Table 3, bottom) with9 Fig. 19.—
Detailed analysis of an edge-on disk galaxy from GEMS. ( a ) Original data. ( b ) Best two-component S´ersic profile fit each withFourier modes, corresponding to the parameters shown in Table 2. ( c ) Best-fit residuals. ( d ) The fit residuals using traditional (i.e. purelyellipsoid) models without masking the dust lane. ( e ) Residuals after subtracting the best traditional models, masking out the dust lane.( f ) The bulge component of the best-fit model. ( g ) The edge-on disk component of the best-fit model. ( h ) The extended halo componentof the best-fit model. ( i ) 1-D surface brightness profile. The individual components are shown as dashed lines, and the solid line coursingthrough the data is the sum of the different components. The lower panel shows the residuals of data − model. Table 2. GEMS Disk Galaxy Fitting Results x [ ′′ ] ∆ y [ ′′ ] mag/arcsec r e [ ′′ ] n q θ PA [deg] Commentsfourier — mode: ampl. & phase [deg] mode: ampl. & phase [deg] mode: ampl. & phase [deg] x [ ′′ ] ∆ y [ ′′ ] — r break [ ′′ ] ∆ r soft q θ PA [deg]Best 1 — sersic2 / — 0 .
00 0 .
00 19 .
80 0 .
40 1 .
60 0 .
72 44 .
00 Trunc. by comp.fit 0 .
00 0 .
00 0 .
01 0 .
00 0 .
01 0 .
00 0 .
16 inner: 4fourier — 1: − .
04 1: − .
16 3: − .
00 3: − .
36 4: 0 .
03 4: − .
61— 1: 0 .
00 1: 3 .
23 3: 0 .
00 3: 30 .
65 4: 0 .
00 4: 0 . − .
01 5: − .
74 6: 0 .
01 6: − .
31 — —— 5: 0 .
00 5: 2 .
35 6: 0 .
00 6: 1 .
74 — —2 — sersic2 / — { . } { . } .
19 2 .
29 0 .
85 0 .
31 41 .
30 Trunc. by comp. { . } { . } .
02 0 .
01 0 .
01 0 .
00 0 .
03 inner: 4fourier — 1: − .
04 1: − .
97 3: 0 .
02 3: 26 .
37 4: − .
02 4: − .
04— 1: 0 .
00 1: 1 .
82 3: 0 .
00 3: 1 .
03 4: 0 .
00 4: 0 . .
00 5: 6 .
19 6: − .
01 6: 6 .
62 — —— 5: 0 .
00 5: 5 .
63 6: 0 .
00 6: 1 .
35 — —3 — sersic2 / — { . } { . } .
92 4 .
45 1 .
08 0 .
49 41 .
06 Trunc. by comp. { . } { . } .
03 0 .
05 0 .
02 0 .
00 0 .
10 inner: 4fourier — 1: 0 .
04 1: 4 .
59 3: 0 .
01 3: − .
37 4: − .
01 4: 40 .
55— 1: 0 .
00 1: 1 .
16 3: 0 .
00 3: 1 .
99 4: 0 .
00 4: 3 . .
01 5: − .
19 6: − .
01 6: − .
25 — —— 5: 0 .
00 5: 1 .
40 6: 0 .
00 6: 0 .
73 — —4 — radial — 0 . − .
14 — 1 .
48 1 .
48 0 .
09 41 .
38 Truncates comp.0 .
00 0 .
00 — 0 .
01 0 .
02 0 .
00 0 .
05 inner: 1 2 3fourier — 1: − .
12 1: − .
83 3: 0 .
10 3: − .
62 4: 0 .
19 4: 9 .
20— 1: 0 .
00 1: 1 .
23 3: 0 .
00 3: 0 .
52 4: 0 .
00 4: 0 . .
10 5: 2 .
17 6: 0 .
14 6: 6 .
62 — —— 5: 0 .
00 5: 0 .
19 6: 0 .
00 6: 0 .
10 — —merit χ = 1474348.38 N dof = 1435846 N free = 64 χ ν = 1.03Tradit. 1 — sersic2 — 0 .
00 0 .
00 19 .
98 0 .
38 1 .
32 0 .
74 42 . .
00 0 .
00 0 .
00 0 .
00 0 .
01 0 .
00 0 . { . } { . } .
78 2 .
57 0 .
85 0 .
25 41 . { . } { . } .
03 0 .
02 0 .
01 0 .
00 0 . { . } { . } .
12 3 .
34 1 .
77 0 .
49 41 . { . } { . } .
04 0 .
04 0 .
03 0 .
00 0 . χ = 1478767.62 N dof = 1434511 N free = 18 χ ν = 1.03 Note . — Best-fitting parameters for an edge-on disk galaxy in GEMS. See Table 1 for details. The curly braces ( { ... } ) around parametersindicate that they are coupled relative to the first component. Note that the flux amplitude of sersic2 is normalized to the surface brigtness at r e ,as defined in Equation 32. The “ Best fit ” parameters (top section) correspond to Panel ( b ) in Figure 19, “ Traditional ellipsoid model ” parameters(bottom section) produce residuals shown in Panel ( c ), and the model is not shown. Fig. 20.—
Detailed analysis of Arp 147. ( a ) Original data. ( b ) Best S´ersic profile fits of the two galaxies, all with Fourier modes,corresponding to the parameters shown in Table 3. ( c ) Best-fit residuals. ( d ) The fit residuals using traditional, i.e. axisymmetricellipsoidal model components. ( e ) The bulge component of the right-hand galaxy in Panel ( b ). ( f ) The inner fine-structure component ofthe best-fit model. ( g ) The ring component of the best-fit model. ( h ) The extended tidal-feature-like component of the best-fit model. ( i )1-D surface brightness profile of the two galaxies. The individual components are shown as dashed lines, and the solid line coursing throughthe data is the sum of the different components. The lower panel shows the residuals of data − model.2
Detailed analysis of Arp 147. ( a ) Original data. ( b ) Best S´ersic profile fits of the two galaxies, all with Fourier modes,corresponding to the parameters shown in Table 3. ( c ) Best-fit residuals. ( d ) The fit residuals using traditional, i.e. axisymmetricellipsoidal model components. ( e ) The bulge component of the right-hand galaxy in Panel ( b ). ( f ) The inner fine-structure component ofthe best-fit model. ( g ) The ring component of the best-fit model. ( h ) The extended tidal-feature-like component of the best-fit model. ( i )1-D surface brightness profile of the two galaxies. The individual components are shown as dashed lines, and the solid line coursing throughthe data is the sum of the different components. The lower panel shows the residuals of data − model.2 Table 3. Arp 147 Fitting Results x [ ′′ ] ∆ y [ ′′ ] mag r e [ ′′ ] n q θ PA [deg] Comments x [ ′′ ] ∆ y [ ′′ ] mag/arcsec r e [ ′′ ] n q θ PA [deg]fourier — mode: ampl. & phase [deg] mode: ampl. & phase [deg] mode: ampl. & phase [deg]bending — mode: amplitude [ ′′ ] mode: amplitude [ ′′ ] mode: amplitude [ ′′ ] r break [ ′′ ] ∆ r soft q θ PA [deg]Galaxy 1 1 — sersic — 0 .
00 0 .
00 15 .
49 1 .
26 0 .
47 0 .
50 194 .
70 Bulge.0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 .
05 Inner fine2 — sersic — 0 .
07 0 .
13 17 .
39 0 .
35 0 .
43 0 .
60 150 .
75 structure.0 .
00 0 .
00 0 .
01 0 .
00 0 .
01 0 .
00 0 .
393 — sersic3 / — − .
08 0 .
33 23 .
84 1 .
21 0 .
96 0 .
18 187 .
42 Trunc. by comp.0 .
01 0 .
03 0 .
02 0 .
02 0 .
01 0 .
00 0 .
03 inner: 5 (ring).fourier — 1: 0 .
03 1: 49 .
11 3: 0 .
02 3: 15 .
69 4: 0 .
04 4: 4 .
24— 1: 0 .
00 1: 9 .
08 3: 0 .
00 3: 1 .
04 4: 0 .
00 4: 0 .
454 — sersic3 / — 0 . − .
34 22 .
04 3 .
81 1 .
94 0 .
42 184 .
08 Trunc. by comp.0 .
01 0 .
02 0 .
01 0 .
04 0 .
03 0 .
00 0 .
04 inner: 5 (Tidalfourier — 1: 0 .
16 1: 21 .
49 3: 0 .
09 3: 18 .
36 4: − .
01 4: 16 .
13 feature).— 1: 0 .
00 1: 0 .
82 3: 0 .
00 3: 0 .
17 4: 0 .
00 4: 1 . − .
14 — — — — —— 2: 0 .
00 — — — — —5 — radial — — — — 10 .
94 6 .
00 0 .
18 187 .
61 Truncates comp.— — — 0 .
02 0 .
06 0 .
00 0 .
03 inner: 3 4fourier — 1: 0 .
05 1: 39 .
00 3: 0 .
02 3: 19 .
88 4: 0 .
04 4: 3 .
88— 1: 0 .
00 1: 4 .
97 3: 0 .
00 3: 1 .
18 4: 0 .
00 4: 0 . − . − .
93 22 .
14 0 .
78 1 .
85 0 .
79 187 .
91 Trunc. by comp.0 .
01 0 .
01 0 .
00 0 .
01 0 .
01 0 .
00 0 .
12 inner: 7 (Ring)fourier — 1: 0 .
23 1: − .
97 3: 0 .
07 3: 15 .
27 4: − .
02 4: 23 .
33 mag tot = 14 .
90— 1: 0 .
00 1: 0 .
53 3: 0 .
00 3: 0 .
12 4: 0 .
00 4: 0 .
397 — radial — — — — 10 .
77 6 .
08 0 .
82 195 .
16 Truncates comp.— — — 0 .
01 0 .
01 0 .
00 0 .
19 inner: 6fourier — 1: 0 .
17 1: − .
22 3: 0 .
07 3: 4 .
98 4: 0 .
02 4: − .
52— 1: 0 .
00 1: 0 .
99 3: 0 .
00 3: 0 .
17 4: 0 .
00 4: 0 . χ = 714735.38 N dof = 357760 N free = 77 χ ν = 2.00Tradit. 1 — sersic — 0 .
00 0 .
00 15 .
33 1 .
07 0 .
90 0 .
62 193 .
49 Bulge.ellipsoid 0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 . − .
12 0 .
55 14 .
86 6 .
63 0 .
43 0 .
34 184 .
20 Disk.0 .
00 0 .
01 0 .
00 0 .
01 0 .
00 0 .
00 0 . − . − .
66 15 .
09 8 .
28 0 .
12 0 .
80 201 .
65 Ring0 .
01 0 .
01 0 .
00 0 .
01 0 .
00 0 .
00 0 .
29 galaxy.merit χ = 1435193.25 N dof = 357813 N free = 24 χ ν = 4.01 Note . — Best-fitting parameters for Arp 147. See Table 1 for details. Note that the flux amplitude of sersic3 is normalized to the surfacebrigtness at r break , as defined in Equation 33, whereas sersic magnitude means the total flux. The “ Best fit ” parameters (top section) correspondto Panel ( b ) in Figure 20, “ Traditional ellipsoid model ” parameters (bottom section) produce residuals shown in Panel ( d ), and the model is notshown. The free parameters for the sky are not listed. m = 14 .
18, compared to m = 14 . m = 14 .
15. For Galaxy 2, we knowfrom the outset that classical ellipsoid models are en-tirely inappropriate to use. Yet, despite every reasonto believe that the photometry would be inaccurate, wefind that the total flux of the traditional ellipsoid fitis only 0.2 magnitude different from the most realis-tic ring model. These two examples show once againthat a single-component S´ersic ellipsoid fit to compli-cated galaxies can produce quite accurate measurementof the total flux.It is sometimes desirable to conduct bulge-to-disk(B/D) decompositions, and Galaxy 1 is an ideal candi-date to conduct a comparison. In the traditional ellipsoidmodel (Table 3, bottom), the B/D ratio is 0.65. Themore sophisticated model (Table 3, top) requires sum-ming the ring+tidal feature components to obtain thedisk component, which yields 14.65 mag, thus a B/D ra-tio of 0.54. In this situation, most of the differences arisefrom measuring the disk component, which differs by 0.2mag, whereas the bulge component is quite robust, witha difference of only 0.01 mag.It is also of interest to understand how the structuralparameters are affected by different model choices, inparticular for the ring Galaxy 2. Whereas the effectiveradius for the ring model is only 0 . ′′
78, for the ellipsoidmodel it is 8 . ′′
28. This is understandable, bearing in mindthat the ring has a radius of nearly 8 ′′ . To a classicalS´ersic profile, the galaxy appears to have a very flat (infact, a deficit) core, which leads to a low S´ersic index of n = 0 .
12. As most of the flux is at 8 ′′ , beyond which thering flux quickly fades, the ring radius is closely relatedto the effective radius for a classical S´ersic model. Forthe inner-truncated ring model (component 7), however,the physical size of the ring is captured by the break ra-dius r break parameter, whereas the r e term no longer hasthe classical meaning of the effective radius (i.e. half thelight is within r e ). Instead, r e for component 7 is essen-tially an exponentially declining scale length parameter,given by Equation 33. As the flux dies away quickly be-yond the peak, as shown in Figure 20 i , the scale length r e for the ring model must therefore be quite small. Thedifferences in the r e parameter between the traditionalmodel and the truncated model are therefore only due todefinitions, and not due to systematic or random mea-surement uncertainties. M51
The classical Whirlpool galaxy is a beautiful systemwhere a grand-design spiral, M51A, is interacting withanother spiral, M51B (Figure 21, Table 4). In additionto there being obvious spiral structures for both galax-ies, there are large tidal disturbances that emerge fromM51B, as seen in the SDSS r -band image provided byD. Finkbeiner. Because they are closely overlapping, adesirable goal is to deblend M51A and B, as well as tomodel the spiral and tidal structures, simultaneously.As with previous examples, we fit this galaxy usingboth the most sophisticated analysis (Table 4, top) inour toolbox, and comparing the results to the traditional axisymmetric ellipsoids (Table 4, bottom) analysis. Thetraditional analysis requires two components each, in or-der to decompose a galaxy ostensibly into a bulge and adisk. The reduction in χ ν between the two methods ismodest, because most of the residuals come from high-frequency starforming regions that are not removed bymodels which are fundamentally smooth, despite beingmodified by radial Fourier modes and spiral rotations.In the most detailed analysis of M51A, we use two spi-ral arm components and two components for the bulge.There is actually not a strong need to use two compo-nents for the bulge except to better capture the detailedprofile shape, which has an inflection at r ≈ . ′
4, as seenin Figure 21 i . On the other hand, the use of two spi-ral components is necessary because the spiral arm hasa “kink” in the rotation that cannot be created by us-ing a single smooth rotation function. The spiral struc-tures are modified by Fourier modes to create both aslight lopsidedness and other subtle features. Becausethere are more degrees of freedom in a two-arm spiral,the higher order Fourier modes also can “see” detailedstructures, like the reverse flaring of the spiral structurein Figure 21 f .For M51B, we employ three components in the fit, abulge (component 5 in Table 4, top), a tidal feature com-ponent (component 6), and a spiral function (component7), which model the three most visually striking compo-nents. The tidal feature is mostly obtained by using thesecond and third bending modes of Equation 27, as il-lustrated in Figure 10. However, bending modes 2 and3 are symmetric functions, so the high degree of asym-metry comes about because of combined action with theFourier modes, which is shown to have a high amplitudeof 0.23 for the m = 1 mode, as well as moderate valuesfor other modes. Incidentally, despite the complexity ofthe higher order structures, all the parameter values aredetermined automatically by Galfit without the needfor an user to provide initial guesses (i.e. initially all 0values) and without tweaking at any point in the analysis(which is hardly feasible anyhow).For even those who are experienced with detailed para-metric fitting, one of the alarming facts about this anal-ysis is that it employs 103 free parameters in the best-fit model. So there are natural concerns about param-eter degeneracies. However, as we have discussed inSection 6.5, parameter degeneracies do not arise purelybased on the number of free parameters, but rather onthe types of parameters involved. The availability ofspatial information in 2-D provides one of the most im-portant ways to break parameter degeneracies. We seethis explicitly in Figures 21 e-h , where there is little evi-dence that the subcomponents for M51A are strongly in-fluenced by M51B, and vice versa. Furthermore, withineach galaxy, the subcomponents are so different in shape,both qualitatively and quantitatively, that the amount ofcrosstalk between them is also not significant. Therefore,despite the extreme complexity of this system, and theuse of 103 free parameters, we find that degeneracies be-tween the parameters are not an issue. Or, if they exist,they do so at a low enough level that they do not sig-nificantly affect the main parameters of interest, like theluminosity of the subcomponents, or the profile shapesand sizes.There are , however, seemingly degenerate conditions4
Fig. 21.—
Detailed analysis of M51. ( a ) Original data. ( b ) Best S´ersic profile fits of the M51A and B, all with Fourier modes,corresponding to the parameters shown in Table 4. ( c ) Best-fit residuals. ( d ) The fit residuals using traditional, axisymmetric, ellipsoidalmodel components. ( e ) The extended grand-design spiral component of M51A model in Panel ( b ). ( f ) The inner fine-structure spiralcomponent of the best-fit model. ( g ) The spiral component of M51B. ( h ) The extended tidal feature-like component of M51B, usingsimultaneous bending and Fourier modes. A bulge component is present but not shown in the figures of M51A and B. i ) 1-D surfacebrightness profile of the two galaxies. The individual components are shown as dashed lines, and the solid line coursing through the datais the sum of the different components. The lower panel shows the residuals of data − model. Table 4. M51 Fitting Results x [ ′ ] ∆ y [ ′ ] mag r e [ ′ ] n q θ PA [deg] Commentspower — r in [ ′ ] r out [ ′ ] θ rot [deg] α θ incl [deg] θ sky [deg]fourier — mode: ampl. & phase [deg] mode: ampl. & phase [deg] mode: ampl. & phase [deg]bending — mode: amplitude [ ′ ] mode: amplitude [ ′ ] mode: amplitude [ ′ ]Best 1 — sersic — 0 .
00 0 .
00 13 .
09 0 .
04 1 .
18 0 . − .
25 Compoundfit 0 .
00 0 .
00 0 .
04 0 .
00 0 .
04 0 .
01 4 .
74 bulge.2 — sersic — { . } { . } .
49 0 .
26 0 .
67 0 . − .
31 CompoundM51A { . } { . } .
00 0 .
00 0 .
01 0 .
00 0 .
66 bulge.3 — sersic — { . } { . } .
50 2 .
78 0 .
35 0 . − .
22 Compound { . } { . } .
00 0 .
00 0 .
00 0 .
00 39 .
53 spiral.power — − .
29 4 . − .
11 0 .
29 40 . − .
20— 0 .
20 0 .
03 41 .
39 0 .
02 0 .
05 0 . − .
07 1: 109 .
10 3: 0 .
03 3: 4 .
07 4: 0 .
02 4: − .
57— 1: 0 .
00 1: 0 .
44 3: 0 .
00 3: 0 .
43 4: 0 .
00 4: 0 . .
02 5: 24 .
34 — — — —— 5: 0 .
00 5: 0 .
32 — — — —4 — sersic — { . } { . } .
06 1 .
88 0 .
14 0 .
39 5 .
45 Compound { . } { . } .
00 0 .
00 0 .
00 0 .
00 4190 .
51 spiral.power — 0 .
66 2 . − . − . − .
01 15 .
58— 0 .
02 0 .
01 3 .
51 0 .
01 1100 .
08 4190 . − .
15 1: 25 .
39 3: 0 .
02 3: − .
12 4: 0 .
15 4: 8 .
38— 1: 0 .
00 1: 0 .
62 3: 0 .
00 3: 1 .
50 4: 0 .
00 4: 0 . .
02 5: − .
02 — — — —— 5: 0 .
00 5: 0 .
82 — — — —M51B 5 — sersic — − .
19 4 .
44 12 .
06 0 .
05 0 .
89 0 . − .
22 Compound0 .
00 0 .
00 0 .
01 0 .
00 0 .
01 0 .
00 0 .
38 bulge.6 — sersic — − .
16 4 .
43 11 .
93 0 .
18 1 .
06 0 . − .
79 Compound0 .
00 0 .
00 0 .
02 0 .
00 0 .
03 0 .
01 1 .
45 bulge.7 — sersic — − .
45 5 .
19 9 .
93 2 .
51 [1 .
00] 0 . − .
19 Tidal0 .
01 0 .
01 0 .
00 0 .
01 — 0 .
00 0 .
36 structure.bending — 2: 0 .
03 3: − .
15 — — — —— 2: 0 .
02 3: 0 .
00 — — — —fourier — 1: 0 .
34 1: 17 .
20 3: − .
25 3: 32 .
55 4: 0 .
14 4: − .
73— 1: 0 .
00 1: 0 .
81 3: 0 .
00 3: 0 .
40 4: 0 .
00 4: 0 . .
03 5: 7 .
32 — — — —— 5: 0 .
00 5: 1 .
16 — — — —8 — sersic — − .
10 4 .
52 10 .
20 0 .
90 0 .
72 0 . − .
52 Bar and0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 .
56 spiral.power — 0 .
88 1 .
08 46 .
34 1 .
60 42 .
29 52 .
50— 0 .
00 0 .
00 0 .
66 0 .
01 0 .
16 0 . .
07 1: 103 .
72 3: 0 .
05 3: 28 .
79 4: 0 .
01 4: − .
11— 1: 0 .
00 1: 1 .
59 3: 0 .
00 3: 0 .
52 4: 0 .
00 4: 4 . .
01 5: 23 .
12 — — — —— 5: 0 .
00 5: 1 .
09 — — — —merit χ = 34279512.00 N dof = 632434 N free = 104 χ ν = 54.20Trad. 1 — sersic — 0 .
00 0 .
00 10 .
05 0 .
33 1 .
75 0 . − . .
00 0 .
00 0 .
00 0 .
00 0 .
01 0 .
00 0 . − . − .
13 8 .
47 2 .
21 0 .
33 0 .
75 26 . .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 .
00 0 . − .
18 4 .
40 8 .
93 2 .
54 8 .
02 0 . − . .
00 0 .
00 0 .
03 0 .
11 0 .
09 0 .
00 1 .
244 — sersic — − .
04 4 .
90 10 .
66 1 .
45 1 .
66 0 .
57 71 . .
00 0 .
00 0 .
02 0 .
02 0 .
02 0 .
00 0 . χ = 42126720.00 N dof = 632507 N free = 31 χ ν = 66.60 Note . — Best-fitting parameters for M51. See Table 3 for details. The “
Best fit ” parameters (top section) correspond to Panel ( b ) in Figure 21,“ Traditional ellipsoid model ” parameters (bottom section) produce residuals shown in Panel ( d ), and the model is not shown. The free parametersfor the sky are not listed. The parameter in square brackets, [...], is held constant in the fit. The curly braces ( { ... } ) around parameters indicatethat they are coupled relative to the first component. Galfit to “lock” on to them if the initialconditions happened to be sufficiently close. The con-sequences appear as degeneracies when, in fact, thereare many small local minima solutions. This graininessin the χ terrain introduces slight perturbations to themodels, and may even cause fairly large shape differ-ences in the final solutions. However, to a large extent,it rarely affects the main parameters of interest, suchas the luminosity of a particular component or its size,which are determined by much more global features thanthe nuisances of local fluctuations to which higher orderparameters are more sensitive.To gain some intuitive insight into the effects of com-plex analysis, it is instructive to compare simple andcomplex methods with regard to global and subcompo-nent properties. In terms of the total luminosity, here wefind excellent agreement between sophisticated and tradi-tional analysis, respectively, of m r = 8 .
24 vs. m r = 8 . m r = 8 .
80 vs. m r = 8 .
73 for M51B. Whilethis level of agreement may at first seem surprising, it isexpected given the basic premise of least-squares mini-mization. In fact, even a single-component fit to M51Ayields m r = 8 .
0, and for M51B m r = 9 .
0, which areboth quite close to the overall best-fit models, despitethe complications in the image. The main reason forthe discrepancy here is the uncertainty in the sky, dueto there being a large gradient. This fundamentally setsthe limit on the accuracy of the photometry to perhapsno better than 0.1 to 0.2 magnitude, independent of theanalysis method.The most sensitive benchmark for understanding dif-ferences in the analysis is in detailed decompositions.Here we compare the bulge-to-disk decomposition re-sults. In the traditional ellipsoid analysis, we find a B/Dratio of 0.23 for M51A and 4.9 for M51B. The large B/Dratio for M51B is clearly unphysical, and is driven bythe large S´ersic index ( n = 8 .
0) of the bulge component,which is increased to accommodate the flux in the out-skirts due to tidal features. In the most detailed analysis,the B/D ratio for M51A is 0.16, whereas for M51B it ismerely 0.17. Examining the bulge of M51A more closely,we find that the detailed analysis yields a total flux of10.38 mag, whereas the traditional analysis extracts abrighter bulge of 10.05 mag. The differences come fromthe fact that the light of the inner spiral is in part drivingup the S´ersic index of the bulge when it is not properlyaccounted. It is probably safe to conclude that a magni-tude of 10.05 is a firm upper limit to the bulge luminosity.Finally, it is worthwhile to compare how the disk pa-rameters differ between the analyses to gain an under-standing for how coordinate rotation affects the inter-pretation of the parameters for the spiral models. FromTable 4, we find that the S´ersic index of the simple andcomplex models are essentially identical for M51A, at n ≈ .
33. The interpretation for M51B is more compli-cated, because the “disk” in an ellipsoidal model is notqualitatively the same structure as the spiral analysis. Infact, it is necessary to hold the S´ersic index of the tidalcomponent 6 fixed in the analysis. Nevertheless there are clearly quantitative differences in that the simple anal-ysis is larger by 55% in n . With regard to the effectiveradius, the traditional analysis of M51A finds the disksize to be about 2 . ′
2, which compares favorably with thespiral model size of 2 . ′
8, or a 25% difference. Further-more, the disk magnitudes for M51A differ only by 0.03mag between simple and complex.These comparisons therefore demonstrate that despitethe complex analysis being much more realistic looking,fundamentally the meaning of the structural parameters(size, luminosity, concentration index) are unchangedfrom the original definition, even in the situation of spi-ral components. This is an useful fact because our priorintuitions, honed on fitting ellipsoidal models, continueto be applicable. We note that the generally good agree-ment between detailed and simplistic analysis witnessedhere and in previous examples is not entirely coinciden-tal. It so happens because all shapes are fundamentallyperturbations of the best-fitting ellipsoidal model, evenif the result bears no resemblance to the original ellipse.
NGC 289
NGC 289 is an SAB(rs)bc galaxy, with a weak barand a complex inner spiral system (Figure 22, Table 5)that resembles a ring. Upon closer examination, the ringappearance comes about because there exists a bifurca-tion in the spiral structure that connects up with theopposing spiral arm. Furthermore, the bar is also multi-component, with the inner component oriented at an an-gle nearly 45 ◦ from the strong outer bar.The best-fit analysis involves three spiral components,an inner and an outer bar component, and a bulge (Ta-ble 5, top). All except for the bulge component are modi-fied by five Fourier modes, and are shown in Figures 22 e-h . The requirement of components 3 and 4 (Figures 22 f and g ) is clear, because they are what form the moststriking and intricate patterns in the center, while therequirement of component 5 (Figure 22 h ) is only evidentin the residuals, and makes up some of the diffuse lightwithin the inner 60 ′′ region. Although it does not seemlike an essential component, the inner bar structure (Fig-ure 22 e ) qualitatively affects the detailed residual patternat the center, and is therefore included. When all the de-tailed inner structures are properly accounted for, it isstraightforward to infer the bulge component, and assessthe uncertainties by varying different parameters of thebulge. Doing so does not affect the inner fine-structuresbecause they are sharp and well localized.Conducting the same decomposition using traditionalellipsoid models (Table 5, bottom), we opted to fit threecomponents, ostensibly to model a bulge, disk, and a bar.The result produces residuals seen in Figure 22 d , reveal-ing the intricate details of the inner spiral system. Fromthe fit, even though the disk and the bar component aresensible, the bulge component is actually fitting a dif-fuse disk component, which, in retrospect, is that shownin Figure 22 h . Because that inner spiral component isquite luminous, and because there exists a bulge com-ponent superposed on top of it, this quasi-bulge modelis almost 0.7 mag brighter than that inferred throughthe detailed modeling above. Adding a fourth ellipsoidmodel is not possible, because the central spiral residualsare so great that they completely suppress the additionof another component, causing the flux to go to zero.7 Fig. 22.—
Detailed analysis of NGC 289 from CINGS. ( a ) Original data. ( b ) Best S´ersic profile fits with spiral rotation functionsand Fourier modes, corresponding to the parameters shown in Table 5. ( c ) Best-fit residuals. ( d ) The fit residuals using traditional,axisymmetric, ellipsoidal model components. ( e ) The fine details of the inner bar structure of Panel ( b ). ( f ) Spiral component 1 of 3 ofthe best-fit model. ( g ) The spiral component 2 of 3. ( h ) The spiral component 3 of 3. A bulge component is present but not shown in thefigures. ( i ) 1-D surface brightness profile of the galaxy. The individual components are shown as dashed lines, and the solid line coursingthrough the data is the sum of the different components. The lower panel shows the residuals of data − model. Table 5. NGC 289 Fitting Results x [ ′′ ] ∆ y [ ′′ ] mag r e [ ′′ ] n q θ PA [deg] Commentspower — r in [ ′′ ] r out [ ′′ ] θ rot [deg] α θ incl [deg] θ sky [deg]fourier — mode: ampl. & phase [deg] mode: ampl. & phase [deg] mode: ampl. & phase [deg]Best 1 — sersic — 0 .
00 0 .
00 11 .
69 64 .
01 1 .
72 0 .
78 61 .
27 Bulge.fit 0 .
09 0 .
10 0 .
01 0 .
75 0 .
03 0 .
00 0 .
432 — sersic — − . − .
92 13 .
27 6 .
05 1 .
02 0 .
51 77 .
83 InnerNGC 289 0 .
02 0 .
01 0 .
01 0 .
04 0 .
01 0 .
00 0 .
29 bar.fourier — 1: 0 .
10 1: 63 .
87 3: − .
05 3: − .
23 4: − .
05 4: − .
77— 1: 0 .
01 1: 4 .
87 3: 0 .
00 3: 0 .
92 4: 0 .
00 4: 0 . .
03 5: 4 .
62 6: 0 .
06 6: − .
09 — —— 5: 0 .
00 5: 1 .
32 6: 0 .
00 6: 0 .
61 — —3 — sersic — − . − .
77 12 .
30 32 .
86 0 .
54 0 . − .
94 Spiral0 .
02 0 .
02 0 .
01 0 .
15 0 .
00 0 .
00 0 .
53 comp. 1.power — 19 .
23 34 .
40 85 .
51 1 .
48 52 .
11 136 .
18— 0 .
14 0 .
14 0 .
94 0 .
02 0 .
08 0 . .
14 1: − .
28 3: − .
05 3: − .
55 4: 0 .
02 4: 3 .
41— 1: 0 .
00 1: 0 .
74 3: 0 .
00 3: 1 .
13 4: 0 .
00 4: 2 . .
02 5: − .
77 6: 0 .
01 6: 10 .
09 — —— 5: 0 .
00 5: 0 .
99 6: 0 .
00 6: 3 .
36 — —4 — sersic — − . − .
10 12 .
13 52 .
28 0 .
74 0 . − .
85 Spiral0 .
04 0 .
03 0 .
01 0 .
22 0 .
01 0 .
00 28 .
48 comp. 2.power — − .
32 71 .
31 450 .
01 0 .
77 53 .
30 140 .
24— 4 .
45 0 .
59 25 .
64 0 .
06 0 .
05 0 . − .
12 1: 84 .
16 3: 0 .
06 3: 32 .
07 4: − .
06 4: − .
37— 1: 0 .
00 1: 0 .
64 3: 0 .
00 3: 0 .
45 4: 0 .
00 4: 0 . .
04 5: − .
47 6: 0 .
02 6: − .
35 — —— 5: 0 .
00 5: 0 .
32 6: 0 .
00 6: 0 .
81 — —5 — sersic — − . − .
90 11 .
82 50 .
67 0 .
46 0 . − .
45 Spiral0 .
04 0 .
04 0 .
00 0 .
30 0 .
00 0 .
00 35 .
81 comp. 3.power — − .
09 75 .
87 411 . − .
04 64 .
81 112 .
75— 4 .
42 0 .
69 36 .
56 0 .
01 0 .
13 0 . − .
11 1: − .
85 3: 0 .
01 3: 1 .
93 4: − .
00 4: 9 .
85— 1: 0 .
00 1: 1 .
08 3: 0 .
00 3: 3 .
10 4: 0 .
00 4: 6 . .
00 5: − .
26 6: 0 .
02 6: − .
02 — —— 5: 0 .
00 5: 10 .
08 6: 0 .
00 6: 0 .
58 — —Neighbor 6 — sersic — 67 . − .
06 14 .
69 20 .
03 1 .
93 0 . − . .
04 0 .
05 0 .
01 0 .
34 0 .
03 0 .
01 0 . χ = 158680.39 N dof = 150419 N free = 103 χ ν = 1.05Tradit. 1 — sersic — 0 .
00 0 .
00 11 .
03 41 .
90 1 .
62 0 .
74 54 .
96 “bulge”?ellipsoid 0 .
04 0 .
03 0 .
01 0 .
33 0 .
01 0 .
00 0 . . − .
52 11 .
69 36 .
98 0 .
29 0 .
55 23 .
85 disk0 .
04 0 .
05 0 .
01 0 .
06 0 .
00 0 .
00 0 .
123 — sersic — 0 . − .
10 12 .
70 10 .
83 1 .
24 0 .
43 67 .
37 bar0 .
02 0 .
01 0 .
03 0 .
08 0 .
01 0 .
00 0 . . − .
97 14 .
71 19 .
62 1 .
90 0 . − . .
04 0 .
05 0 .
01 0 .
34 0 .
03 0 .
01 1 . χ = 200361.19 N dof = 150491 N free = 31 χ ν = 1.33 Note . — Best-fitting parameters for NGC 289. See Table 1 for details. The “
Best fit ” parameters (top section) correspond to Panel ( b ) inFigure 22, “ Traditional ellipsoid model ” parameters (bottom section) produce residuals shown in Panel ( d ), and the model is not shown. The freeparameters for the sky are not listed. m = 10 .
37 mag vs. m = 10 . m = 10 .
46, despite themain structural details not being unaccounted. DISCUSSION AND CONCLUSIONS
This study is a proof of concept for how to conductmore realistic image-fitting analysis using purely para-metric functions, by breaking free from traditional as-sumptions about axisymmetry. We introduced severalnew ideas, including the use of Fourier and bendingmodes, spiral rotation functions, and truncation func-tions. These features can be used individually, or com-bined in arbitrary ways. While these features are individ-ually quite simplistic, used collectively they proliferate adizzying array of possibilities. Even so, the interpreta-tion of each component remains intuitive, down to themeaning of each fitting parameter. Indeed, the interpre-tation of the traditional ellipsoidal profile parameters,such as those for the S´ersic function, remains essentiallyunchanged under modification. We then applied thesetechniques to five case studies, illustrating that highlycomplex and intricate structures can be modeled usingfully parametric techniques.There are many practical applications for these tech-niques. For instance, the Fourier modes are useful forquantifying the average global symmetry of a galaxy,and can easily be automated for galaxy surveys. It isalso possible to disentangle bright from faint asymme-tries, and to conduct more robust bulge-to-disk decom-positions in some galaxies. It would be useful to quantifyhow much of the total flux is in a bulge versus the tidallydistorted component, which has implications for issuessuch as late- vs. early-stage mergers, or major vs. minormergers.More than just a presentation of new techniques, one ofthe main purposes of this study is to highlight a methodto more realistically quantify measurement uncertaintiesin high-S/N images. In galaxy fitting, the most desirablegoal will always be to obtain a fit with the lowest χ , us-ing the simplest model. In the past, this idea was closelytied to the practice of using one- or two-component el-lipsoid models, out of necessity. Simplicity is not nec-essarily congruent with propriety or reality. This studypromotes the notion that simple models can be realistic.It also opens up new possibilities for more detailed anal-ysis depending on the image complexity. However, thispossibility is both a blessing and a curse. For, the factthere is not one generic solution for any galaxy leads tothe following conundrum in image analysis, but one thatillustrates the merit of our approach: “What model should one adopt, how much detail isenough, and what about degeneracies?” We haveshown that detailed decomposition analysis can be arbi-trarily sophisticated. It is for that same reason there isoften not a single, unique answer. However, the essentialfact, as seen through our examples and other detailedanalysis outside of this work, is that the marginal returnof adding complexity quickly diminishes. Therefore, theabove conundrum is in practice easy to address by con-ducting analyses of varying sophistication without prej-udice, then judging the outcome by taking a clear view of what goal is to be achieved. If different solutions yieldthe same result for a desired science goal (e.g., bulgeluminosity, bulge-to-disk ratio, average size, total lumi-nosity, etc.), then it does not matter which model oneadopts. If they yield different outcomes, then the mostrealistic analysis ought to be the more true. However, ifit is not possible to decide on the correct model, the dif-ferent results by definition give an estimate of the model-dependent measurement uncertainty.
This last attribute,rather than being a perceived weakness, is fundamentallythat which makes the analysis quantitatively rigorous.Despite the flexibility allowed by the models, this pa-per is merely an initial demonstration of concept andleaves many issues unsolved. Currently, the formulationof the spiral rotation function is fairly rigid, and can-not produce arms that wind back onto itself (althoughthat can be approximated using the ring feature in
Gal-fit ). The amount of curvature in the bending modes canonly fit arcs and not fuller semi-circles (which can partlybe modeled using a lopsided ring). There remains sub-stantial room for future growth in profile types, shapedefinitions, and toward spatial-spectral decompositionsfor integral-field data.CYP gratefully acknowledges discussions with and sug-gestions from many people over the course of this work,including Lauren. A. MacArthur who greatly improvedthis manuscript, J. Greene, C. Brasseur, D. McIntosh, J.Hesser, T. Puzia, K. Jahnke, S. Zibetti, E. Bell, A. Barth,E. Laurikainen, M. Barden, B. H¨aussler, M. Gray, and
Galfit users over the years. We thank the anonymousreferee for an expeditious review that improved the dis-cussions. A number of individuals not only very kindlyprovided us high quality images, but also granted uspermission to analyze and use them for this study, in-cluding M. Seigar and A. Barth for IC 4710 and NGC289 from the Carnegie Irvine Nearby Galaxy Survey, D.Finkbeiner for the SDSS image of M51, J. Caldwell andthe GEMS collaboration for the edge-on galaxy fromtheir survey. Several others, including J. Lee, M. Kim,and H. Hernandez, contributed galaxy images which mo-tivated the development of certain
Galfit features. Thevarious contributions and input from all made the de-velopment of this software a greatly enjoyable endeavor,and to whom CYP is greatly indebted. This work wasmade possible by the generous financial support of theHerzberg Institute of Astrophysics, National ResearchCouncil of Canada, through the Plaskett Fellowship pro-gram, by STScI through the Institute/Giacconi Fellow-ship, and by visitor programs of the Max-Planck-InstitutF¨ur Astronomie, operated by the Max-Planck Society ofGermany.This research has made use of the NASA/IPAC Extra-galactic Database (NED) which is operated by the JetPropulsion Laboratory, California Institute of Technol-ogy, under contract with the National Aeronautics andSpace Administration.Based on observations made with the NASA/ESAHubble Space Telescope, and obtained from the Hub-ble Legacy Archive, which is a collaboration between theSpace Telescope Science Institute (STScI/NASA), theSpace Telescope European Coordinating Facility (ST-0 out θ rr in out Fig. 23.—
Appendix Figure A: Schematics of a hyperbolic tangent rotation function. r in is the inner radius where the rotation anglereaches 20 degrees relative to the PA of the best fitting ellipse of a component. Below r in the function flattens out to zero degrees. r out isthe outer radius, beyond which the function flattens out to a constant rotation angle, and θ out is the total amount of rotation out to r out . APPENDIX
A — HYPERBOLIC TANGENT ROTATION FUNCTIONThe hyperbolic tangent (tanh( r in , r out , θ incl , θ skyPA ; r )) portion of the α -tanh (Equation 28) and log-tanh (Equation 29)rotation functions is given by Equation 5 below. The constant CDEF is defined such that at the mathematical “barradius” r in , the rotation angle θ reaches 20 ◦ . This definition is entirely empirical. The above Figure shows a pure tanhrotation function, where the rotation angle reaches a maximum θ out near r = r out . Beyond r out , the rotation anglelevels off at θ out . This function is multiplied with either a logarithmic or a power-law function to produce the desiredasymptotic rotation behavior seen in more realistic galaxies (see Section 4).CDEF = 0 .
23 (constant for “bar” definition) (1) A = 2 × CDEF | θ out | + CDEF − . B = (cid:0) − tanh − ( A ) (cid:1) (cid:18) r out r out − r in (cid:19) (3) r = p ∆ x + ∆ y (circular − centric distance) (4)tanh( r in , r out , θ incl , θ skyPA ; r ) ≡ . × (cid:18) tanh (cid:20) B (cid:18) rr out − (cid:19) + 2 (cid:21) + 1 (cid:19) (5)1B — HYPERBOLIC TANGENT TRUNCATION FUNCTIONThe hyperbolic tangent truncation function (tanh( x , y ; r break , r soft , q, θ PA )) (see Section 5) is very similar to thecoordinate rotation formulation in Appendix A, except for different constants that define the flux ratio at the truncationradii: at r = r break the flux is 99% of the untruncated model profile, whereas at r = r soft the flux is 1%. With thisdefinition, Equation 7 is the truncation function: B = 2 . − . (cid:18) r break r break − r soft (cid:19) (6)tanh( x , y ; r break , r soft , q, θ PA ) ≡ . × (cid:18) tanh (cid:20) (2 − B ) rr break + B (cid:21) + 1 (cid:19) (7)Note that the radius r is a generalized radius (as opposed to a circular-centric distance), i.e. one that is perturbed by C , bending modes, or Fourier modes, of the truncation function. When the softening length (∆ r soft ) is used as a freeparameter, it is defined as ∆ r soft = r break − r soft . REFERENCESAbraham, R. G., Valdes, F., Yee, H. K. C., & van den Bergh, S.1994, ApJ, 432, 75Abraham, R. G., van den Bergh, S., & Nair, P. 2003, ApJ, 588, 218Andredakis, Y. C., & Sanders, R. H. 1994, MNRAS, 267, 283Athanassoula, E., Morin, S., Wozniak, H., Puy, D., Pierce, M. J.,Lombard, J., & Bosma, A. 1990, MNRAS, 245, 130Barden, M., Rix, H.-W., Somerville, R. S., Bell, E. F., H¨außler,B., Peng, C. Y., Borch, A., Beckwith, S. V. W., Caldwell,J. A. R., Heymans, C., Jahnke, K., Jogee, S., McIntosh, D. H.,Meisenheimer, K., S´anchez, S. F., Wisotzki, L., & Wolf, C. 2005,ApJ, 635, 959Beckwith, S. V. W., Stiavelli, M., Koekemoer, A. M., Caldwell,J. A. R., Ferguson, H. C., Hook, R., Lucas, R. A., Bergeron,L. E., Corbin, M., Jogee, S., Panagia, N., Robberto, M., Royle,P., Somerville, R. S., & Sosey, M. 2006, AJ, 132, 1729Bedregal, A. G., Arag´on-Salamanca, A., & Merrifield, M. R. 2006,MNRAS, 373, 1125Bell, E. F., McIntosh, D. H., Barden, M., Wolf, C., Caldwell,J. A. R., Rix, H.-W., Beckwith, S. V. W., Borch, A., H¨aussler,B., Jahnke, K., Jogee, S., Meisenheimer, K., Peng, C., Sanchez,S. F., Somerville, R. S., & Wisotzki, L. 2004a, ApJ, 600, L11Bell, E. F., Wolf, C., Meisenheimer, K., Rix, H.-W., Borch, A., Dye,S., Kleinheinrich, M., Wisotzki, L., & McIntosh, D. H. 2004b,ApJ, 608, 752Bender, R., Burstein, D., & Faber, S. M. 1992, ApJ, 399, 462Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393Binney, J., & Tremaine, S. 1987, Galactic dynamics (Princeton,NJ: Princeton University Press, 1987)Boroson, T. 1981, ApJS, 46, 177Burstein, D. 1979, ApJ, 234, 435Byun, Y. I., & Freeman, K. C. 1995, ApJ, 448, 563Conselice, C. J., Bershady, M. A., Dickinson, M., & Papovich, C.2003, AJ, 126, 1183Conselice, C. J., Bershady, M. A., & Jangren, A. 2000, ApJ, 529,886de Jong, R. S. 1996, A&AS, 118, 557de Souza, R. E., Gadotti, D. A., & dos Anjos, S. 2004, ApJS, 153,411de Vaucouleurs, G. 1948, Annales d’Astrophysique, 11, 247Djorgovski, S., & Davis, M. 1987, ApJ, 313, 59Dressler, A. 1980, ApJ, 236, 351Dressler, A., Lynden-Bell, D., Burstein, D., Davies, R. L., Faber,S. M., Terlevich, R., & Wegner, G. 1987, ApJ, 313, 42Elson, R. A. W. 1999, in Globular Clusters, ed. C. Mart´ınez Roger,I. Perez Fourn´on, & F. S´anchez , 209–248Faber, S. M., Tremaine, S., Ajhar, E. A., Byun, Y.-I., Dressler,A., Gebhardt, K., Grillmair, C., Kormendy, J., Lauer, T. R., &Richstone, D. 1997, AJ, 114, 1771Faber, S. M., Willmer, C. N. A., Wolf, C., Koo, D. C., Weiner,B. J., Newman, J. A., Im, M., Coil, A. L., Conroy, C., Cooper,M. C., Davis, M., Finkbeiner, D. P., Gerke, B. F., Gebhardt, K.,Groth, E. J., Guhathakurta, P., Harker, J., Kaiser, N., Kassin,S., Kleinheinrich, M., Konidaris, N. P., Kron, R. G., Lin, L.,Luppino, G., Madgwick, D. S., Meisenheimer, K., Noeske, K. G.,Phillips, A. C., Sarajedini, V. L., Schiavon, R. P., Simard, L.,Szalay, A. S., Vogt, N. P., & Yan, R. 2007, ApJ, 665, 265 Ferrarese, L., Cote, P., Blakeslee, J. P., Mei, S., Merritt, D., &West, M. J. 2006a, ArXiv Astrophysics e-printsFerrarese, L., Cˆot´e, P., Jord´an, A., Peng, E. W., Blakeslee, J. P.,Piatek, S., Mei, S., Merritt, D., Milosavljevi´c, M., Tonry, J. L.,& West, M. J. 2006b, ApJS, 164, 334Ferrarese, L., & Merritt, D. 2000, ApJ, 539, L9Freeman, K. C. 1970, ApJ, 160, 811Gadotti, D. A. 2008, MNRAS, 384, 420Gebhardt, K., Bender, R., Bower, G., Dressler, A., Faber, S. M.,Filippenko, A. V., Green, R., Grillmair, C., Ho, L. C., Kormendy,J., Lauer, T. R., Magorrian, J., Pinkney, J., Richstone, D., &Tremaine, S. 2000, ApJ, 539, L13Hathi, N. P., Ferreras, I., Pasquali, A., Malhotra, S., Rhoads, J. E.,Pirzkal, N., Windhorst, R. A., & Xu, C. 2009, ApJ, 690, 1866H¨aussler, B., McIntosh, D. H., Barden, M., Bell, E. F., Rix, H.-W.,Borch, A., Beckwith, S. V. W., Caldwell, J. A. R., Heymans,C., Jahnke, K., Jogee, S., Koposov, S. E., Meisenheimer, K.,S´anchez, S. F., Somerville, R. S., Wisotzki, L., & Wolf, C. 2007,ApJS, 172, 615Heymans, C., Bell, E. F., Rix, H.-W., Barden, M., Borch, A.,Caldwell, J. A. R., McIntosh, D. H., Meisenheimer, K., Peng,C. Y., Wolf, C., Beckwith, S. V. W., H¨außler, B., Jahnke, K.,Jogee, S., S´anchez, S. F., Somerville, R., & Wisotzki, L. 2006,MNRAS, 371, L60Heymans, C., Gray, M. E., Peng, C. Y., van Waerbeke, L., Bell,E. F., Wolf, C., Bacon, D., Balogh, M., Barazza, F. D., Barden,M., B¨ohm, A., Caldwell, J. A. R., H¨außler, B., Jahnke, K., Jogee,S., van Kampen, E., Lane, K., McIntosh, D. H., Meisenheimer,K., Mellier, Y., S´anchez, S. F., Taylor, A. N., Wisotzki, L., &Zheng, X. 2008, MNRAS, 385, 1431Hinz, J. L., Rieke, G. H., & Caldwell, N. 2003, AJ, 126, 2622Hopkins, P. F., Cox, T. J., Dutta, S. N., Hernquist, L., Kormendy,J., & Lauer, T. R. 2008a, ArXiv e-printsHopkins, P. F., Hernquist, L., Cox, T. J., Dutta, S. N., & Rothberg,B. 2008b, ApJ, 679, 156Hutchings, J. B., Crampton, D., & Campbell, B. 1984, ApJ, 280,41Jahnke, K., S´anchez, S. F., Wisotzki, L., Barden, M., Beckwith,S. V. W., Bell, E. F., Borch, A., Caldwell, J. A. R., H¨aussler,B., Heymans, C., Jogee, S., McIntosh, D. H., Meisenheimer, K.,Peng, C. Y., Rix, H.-W., Somerville, R. S., & Wolf, C. 2004,ApJ, 614, 568Kent, S. M. 1985, ApJS, 59, 115Kim, M., Ho, L. C., Peng, C. Y., Barth, A. J., & Im, M. 2008a,ApJS, 179, 283Kim, M., Ho, L. C., Peng, C. Y., Barth, A. J., Im, M., Martini, P.,& Nelson, C. H. 2008b, ApJ, 687, 767Kormendy, J. 1977, ApJ, 217, 406—. 1985, ApJ, 292, L9Kormendy, J. 1987, in Nearly Normal Galaxies. From the PlanckTime to the Present, ed. S. M. Faber, 163–174Kormendy, J. 1999, in Astronomical Society of the PacificConference Series, Vol. 182, Galaxy Dynamics - A RutgersSymposium, ed. D. R. Merritt, M. Valluri, & J. A. Sellwood,124–+Kormendy, J., & Bender, R. 2009, ApJ, 691, L1422