Mass reconstruction in disc like galaxies using strong lensing and rotation curves: The Gallenspy package
MMass reconstruction in disc like galaxies using strong lensing and rotationcurves: The Gallenspy package
Itamar A. L´opez Trilleras a, ∗ , Leonardo Casta˜neda a a Observatorio Astron´omico Nacional, Universidad Nacional de Colombia, Carrera 30 Calle 45-03, P.A. 111321 Bogot´a,Colombia
Abstract
Two methods for mass profiles reconstruction in disc-like galaxies are presented in this work, the first is donewith the fit of the rotation curve based on the data of circular velocity which are obtained observationally ina stars system, while the other method is focused in the Gravitational Lensed Effect (GLE). For these massreconstructions, two routines developed in the language of programming python were used: one of themis
Galrotpy
Granados et al. [1], which was built by members of the Galaxies, Gravitation and Cosmologygroup from the Observatorio Astron´omico Nacional of the Universidad Nacional de Colombia and whosefuntionality is applied in the rotation curves, the other routine is
Gallenspy which was created in thedevelopment of this work [2] and it is focused in the GLE. It should be noted that both routines perform aparametric estimation from the Bayesian statistics, which allows obtaining the uncertainties of the estimatedvalues. Finally is shown the great power of combining galactic dynamics and GLE, for this purpose themass profiles of the galaxies SDSSJ2141-001 and SDSSJ1331+3628 were reconstructed with
Galrotpy and
Gallenspy where these results obtained are compared with those reported by other authors regarding thesesystems.
Keywords:
Mass reconstructions, GLE, rotational curves, mass profiles,
Gallenspy , Galrotpy .
1. Introduction
The study of mass distribution in galaxies allows obtaining valuable information about the universestructure on a large scale and the process of stellar evolution. For this reason, the rotation curves and theGLE present in galaxies are very important, because they let us the analysis of the distribution of barionicand dark matter in these systems and in this way we can obtain significant restrictions of values such ascosmological densities, the Hubble constant and the cosmological constant among others.The analysis of the rotation curves is done with base in the Newtonian gravitation theory, in this caseis important to point out that the flatness in the contours of these curves is the cause why the dark matteris included by other authors in different mass reconstructions [1, 3, 4, 5]. From this perspective, the darkmatter reconcile the keplerian decrease with the observations done along the astronomy history, whichforces to include the superposition of different mass components in the fitting of the rotation curves withobservational data.In addition to the rotation present in the galaxies, the GLE has been evidenced in many of them, whichhas to do with the deflection that the light coming from a background source presents due to gravitationalpotential of these stellar systems.Thanks to the achievements of the General Relativity Theory, is possibleto estimate the mass distribution of these galaxies based on the deflector angle of the light beams [6, 7] andfor this reason this effect can be complemented with the dynamical analysis in the reconstructions of massprofiles in galaxies. (cid:63) https://github.com/ialopezt/GalLenspy ∗ Corresponding author
Email addresses: [email protected] (Itamar A. L´opez Trilleras), [email protected] (Leonardo Casta˜neda)
Preprint submitted to Elsevier February 26, 2021 a r X i v : . [ a s t r o - ph . GA ] F e b etween the kinds of GLE observations present in galaxies and clusters, it is very common the forma-tion of Einstein rings and giant arcs which are evidenced in this work for the mass reconstructions of thegalaxies SDSSJ2141-001 and SDSSJ1331+3628, regarding to the rotational velocity data of this galaxies itis important to clarify that these were obtained of Dutton et.al. [3, 5] hence the mass profiles reconstructionwas done with both methods.Finally it is important to point out that in this work, the advantages of combining GLE and galacticdynamics are presented, where the complement between these methods is a powerful tool for the massreconstructions.
2. Galactic Dynamics
In the last time, has been evidenced that the disc-like galaxies have different mass components, these canbe classified into four kinds: dark matter halo, stellar halo, disc and bulge which interact between them inconcordance with the Newtonian dynamics, where each mass distribution is essential in the understandingof the functional form of the gravitational potential for these stellar systems.
Figure 1: Scheme of the main mass components of disc-like galaxies.
Since the gravitational force ( (cid:126)F ) present in galaxies is conservative [8], the relation between this and thegravitational potential (Φ) is: (cid:126)F = −∇ Φ , (1)this leads to the mass volumetric density ( ρ ) and Φ are related by means of the Poisson equation [1]: ∇ Φ( (cid:126)r, t ) = 4 πGρ ( (cid:126)r, t ) , (2)where G is the universal gravitation constant.Due to the linearity of the Poisson equation [9, 1], in the case of a galaxy with N components andrespective volumetric mass densities ρ , ρ , ... ρ N , the total density of this system is: ρ = N (cid:88) i =1 ρ i , (3)and this means that the total gravitational potential is expressed in this way Φ = N (cid:88) i =1 Φ i .As the circular velocity (in the equatorial plane) associated to a gravitational potential Φ( R, z = 0) indisc-like galaxies is described by: V c ( R ) = R ∂ Φ ∂r | r = R , (4)2he total circular velocity of this kind of galaxies, is expressed as a superposition of the velocities belongingto each gravitational potential, which is evidenced in the equation 5: V c = N (cid:88) i =1 V c ( i ) , (5)and therefore is possible to make reconstructions of mass profiles in disc-like galaxies through the fittingof rotation curves with the observationaly values of circular velocity, as such is evidenced in the figure 2. Figure 2: Fitting of rotational velocities with the rotation curve belonging to the galaxy NGC6361 (the observational data arethe black dots and the fitting curve is the continuous line), in this case the Miyamoto-Nagai profile was used for the bulge(gray dotted line) and the stellar disc (red dotted line), while the Navarro-Frenk-White (green dotted line) belongs to the darkmatter halo.
3. Gravitational Lensing Effect
In the GLE, the relation between the coordinates of the images (cid:126)θ and the source (cid:126)β is given by theequation: (cid:126)β = (cid:126)θ − ∇ (cid:126)θ ψ ( (cid:126)θ ) , (6)where ψ is the deflector potential, which has the information of the lens and the cosmological distances.For the case of mass profiles with spherical symmetry, it is possible assumed that [10] ψ ( (cid:126)θ ) = 2 (cid:90) | (cid:126)θ | θ (cid:48) κ ( θ (cid:48) ) ln (cid:18) | (cid:126)θ | θ (cid:48) (cid:19) dθ (cid:48) , (7)for this case κ is the convergence, which is defined as κ = ΣΣ crit (8)where Σ is the superficial mass density and Σ crit is given by the relationΣ crit = c πG D s D d D ds , (9)in this case, the term c is the light velocity, D d , D s and D ds are the angular diameter distances betweenobserver-lens, observer-source and lens-source respectively.3dditionally in the GLE, the mapping between the planes of the lens and the source is done through thejacobian transformation matrix A i,j = ∂β i ∂θ j , (10)for i, j = 1 , µ = 1 | detA | , (11)and therefore the critical curve is the set of points in the lens plane where | detA | = 0. Note:
The set of points in the source plane, which through the equation 6 belongs to the critical pointsis denominated caustic curve.
4. Combining GLE and Galactic Dynamics
Due to the superposition between the different mass distributions in galaxies, there are mass profilesreconstructions done with galactic dynamics and other with lensing where the obtention of parameters doesnot occur within acceptable reliability regions [3, 4].A solution strategy for this problem has been proposed by other authors [11, 3 ? ], this has to do with thecombination of these mass reconstructions methods, which means taking advantage of the geometry usedfor each one of them.For this purpose it must be taken into account, that while the mass projection in the galactic dynamicsis done in the equatorial plane of the galaxy, in the case of the GLE the operator Σ is projected in the plane( θ , θ ) where the deflected images are formed. Figure 3: Illustration of the geometries used in galactic dynamics and GLE for the mass projection in 2-D based on what wasstated by Dutton et al [4].
In the figure 3, are illustrated the geometries belonging to the methods of mass reconstruction used inthis work, wherein the GLE is done the projection of mass in a cylinder through the line of sight z andwithin a radius R eins restricted by the position of the deflected images, which from the formalism of stronglensing is related to the formation of the Einstein ring [3] and it is described by the equation. R eins ≈ (cid:32) M eins π Σ crit (cid:33) / , (12)with M eins the mass projected in the cylinder of figure III.In the case of mass projection from galactic dynamics, it is corresponding to enclosed spheres in differentsradius due to the circular velocities estimated in the galaxy. This geometrical approach is optimal, as longas the disc has an inclination, so that the observer can see it from its edges.4ased on the exposed, the combination of GLE and galactic dynamics for this work is proposed aroundthe restrictions in the parameter space that both methods can provide, in such a way that it is possible todistinguish more clearly the gravitational contribution of each mass components in disc-like galaxies.
5. GallenspyGallenspy is an open source code created in python, designed for the mass profiles reconstruction indisc-like galaxies using the GLE. It is important to note, that this algorithm allow to invert numerically thelens equation (equation 6) for gravitational potentials with spherical symmetry, in addition to the estimationin the position of the source ( β , β ), given the positions ( θ , θ ) of the images produced by the lens.The main libraries used in this routine are: numpy [12] for the data hadling, matplotlib [13] regardingthe generation of graphic interfaces, galpy [14] to obtain mass superficial densities, as to the parametricadjust with Markov-Montecarlo chains (MCMC) is taken into account emcee [15] and for the graphics ofreliability regions corner [16] is used.The deflector potential is obtained numerically in Gallenspy by mean of the equation 7, which is veryhelpful because some mass distribution models do not have analytical solutions for the equations 3, 6 and 7.Also it is important to note others tasks of
Gallenspy as compute of critical and caustic curves andobtention of the Einstein ring. For a description more detailed, it is recommended to see the repository pagewhich the source code is available together with its requirements and instructions for its use. A way by which
Gallenspy was tested, is the comparison with the analytical solutions given by Hurtado[10] of the isotherm singular sphere (SIS) under certain specific conditions. In the figure 4 some of thesecomparisons are shown for the obtention of deflection angle, the images formation, and deflector potentialin the case of a circular source of radius 1 kpc to a distance of 2 kpc respect to the observer, where this massprofile was modeled with a dispersion velocity of 100 km/s .As it is possible to observe in each comparative graphic, the results obtained numerically with
Gallenspy present high reliability where the percentage error is of 0 . To start
Gallenspy , it is important to give the values of cosmological distances in
Kpc and critical densityin M (cid:12) /Kpc , which are introduced by means of a file named Cosmological_distancestxt . On the otherhand, it is the file coordinates.txt where the user must introduced the coordinates of the observationalimages (in radians).(
Note: for the case of a circular source it is present the file alpha.txt , where the usermust introduced angles value belonging to each point of the observational images.)
Gallenspy present a interactive fitting of parameters through a routine developed in Jupyter Notebookwith the name of
Interactive_data , in this case the user has the possibility of made a free choose of theparametric range for each value, however it is suggested a parametric space illustrated in table 1 whichis based on the values used by other authors and with the ones they modeled galaxies dwarfs and Milkyway-like [1].In figure 5 is illustrated the interactive panel, for the fitting of a Exponential Disk lens model and acircular source, this observational data belong to the galaxy J2141 which is analyzed later. https://github.com/ialopezt/GalLenspy igure 4: Comparison between the results obtained by the analytical solution of Hurtado (left side) and Gallenspy (right side).These results of deflection angle, images formation and deflection angle were evaluated in a grid of 100 X
100 points.
In the mass reconstructions,
Gallenspy allows assigning a mass profile to each component of the lensgalaxy, where each free parameter has a range of values possible for the obtention of a set of initial valuesin the parametric exploration.Also, it is important to point out that for the mass reconstruction of each component of the galaxy, it ispossible to choose between different profiles for the parametric fitting with
Gallenspy : for example, in thecase of galactic disc it is possible choose between the options of Miyamoto-Nagai and Exponential Disc [1][8], for the dark matter halo between Navarro-Frenk-White (NFW) and Burket [1] [8], while in the bulge isused the Miyamoto-Nagai even though in this profile are given two possible ranges of data.When the positions ( θ , θ ) of the GLE are known, the work with Gallenspy is to find the model andparameters set which can reproduce these provided data, for this reason in this routine the bayesian statisticsis not only based on the exploration of all possible positions of the source.For this parametric exploration,
Gallenspy implements the Metropolis-Hasting algorithm through theMCMC [17], where is obtained a posterior probability distribution P ( p | D, M ) for each parameter set of thelens model selected of the table 1, which is given by the relation: P ( p | D, M ) = P ( D | p, M ) P ( p | M ) P ( D | M ) , (13)with: • P ( p | D, M ) the probability that this parameter set p is appropriate for the model M and the data D .6 ange of values with GallenspyComponent Range of parameters Units Bulge I a = 0 kpc < b < kpc < M < M (cid:12) Bulge II 0.01 < a < kpc < b < kpc < M < M (cid:12) Disc thin 1 < a < kpc < b < kpc < M < M (cid:12) Disc thick 1 < a < kpc < b < kpc < M < M (cid:12) ExponentialDisc 2 < h r < kpc < Σ <
15 10 M (cid:12) /pc Halo NFW 0.1 < a < kpc < M <
10 10 M (cid:12) Halo Burket 2 < a < kpc < ρ <
10 10 M (cid:12) /kpc Table 1: Parametric space used in
Gallenspy • P ( D | p, M ) the probability that the data D are obtained with the model M and the parameter set p ,is known as likelihood [17] and it is denoted with L . • P ( p | M ) the prior and this is the reliability that the parameter set is correct for the model. • P ( D | M ) is denoted as Z , this is the normalization factor and is the probability of obtaining the data D with the model M .It is important to note that in Gallenspy the normalization factor is not considered, hence the fundamentalwork is the compute of the likelihood (this is because the prior for this parameter set have the same value).From this perspective the method for obtaining the initial values of P ( D | p, M ), is through a visual fittingin the Interactive_data routine from which it is possible to make a first approximation between the dataset D and the model values.Later of this process, the user must to execute the code created in the respective file.py (for the estimationof the source source_lens.py and in the case of mass reconstruction parameters_estimation.py ) where Gallenspy request to introduce the initial values obtained in the visual fitting. Next
Gallenspy let to theuser choose the number of steps and walkers, which is enough for the MCMC of this computational routine.In
Gallenspy the minimization function χ , for a source with a number n i of images in the GLE is givenby: χ i = n i (cid:88) j =1 | θ jobs − θ j ( p ) | σ ij , (14)where in this equation θ jobs is the position of observed image j in the data set D , σ ij the error in position θ jobs due to the noise in the image and θ j ( p ) the image j predicted by the mass model used with the parameterset.The index i appears in the function χ , this because in the GLE it is possible the formation of imagesfor various sources, for this reason the likelihood for each explored parameter set is expressed through the7 igure 5: Interactive panel with Gallenspy , for the Exponential disk lens model and the fitting of a circular source.
Gaussian distribution L = P ( D | p, M ) = N (cid:89) i =1 (cid:81) j σ ij √ π exp (cid:18) − χ i (cid:19) , (15)where N is the number of sources.In figure 6, is shown the algorithm of Gallenspy where in the next section is illustrated an example ofthis with the SIS profile.Finally it is important to note that
Gallenspy generate a file.txt with the final parameters obtained: inthe case of source estimation parameters_lens_source.txt and parameters_MCMC.txt , where these filesare request by
Gallenspy for other tasks as compute of Einstein ring and mass estimations.
Although the SIS is not included in the profiles of
Gallenspy , it is possible to show an illustration of themode which this routine performs the parameters exploration with this mass distribution.Because of the analytical solutions, shown by Hurtado [10] to the lens equation in the SIS profile, theformation of images in the GLE are given by the relations: | (cid:126)θ p | = | (cid:126)β | + 4 πσ D ds c D s , (16) | (cid:126)θ n | = −| (cid:126)β | + 4 πσ D ds c D s , (17)with σ the dispersion velocity, θ p and θ n are the positive and negative solutions respectively where theimages are formed. 8 igure 6: Process flow diagram of the parametric exploration with Gallenspy . In the figure 7 is evidenced the images formation, when σ = 1 X km/s for a circular source of radius r = 1arcs whose center has coordinates (h = 0.8arcs, k = 0.8arc) and where the cosmological distances are D ds = 1 Kpc and D s = 2 Kpc . Figure 7: Images formed with the SIS profile for | (cid:126)θ n | (left-down image) and | (cid:126)θ p | (right-up image) in the case of a circular source(blue image). If for this specific case is applied the parametric exploration with
Gallenspy , then it is necessary to supposethat the values of σ , r , h y k are not known and that the objective is the obtention of the parameters familywhich through of GLE can reproduce the images illustrated in figure VI, where in this example only valuesof | (cid:126)θ p | are supposed to be known. This time, the ranges established for the parametric exploration were10 km/s < σ < X km/s , 0 . < r < < h < < k < χ based on the equation 14, and with an error of 0 . likelihood for the MCMC. In the figure 8a, isevidenced the comparison between the images of | (cid:126)θ p | with those produced by the parameters obtained of χ .From these values obtained with the multiparametric minimization, the MCMC was executed with Gallenspy where the best result was obtained for a number of 100 walkers and 1000 steps. In the figure VIIIis shown the chain convergence in the obtention of parameter σ , where it is possible to observe that fromthe obtained data in step 400 can be done the estimation of the parameters.9 a) Red
Image formed in the GLE for a SIS with disper-sion velocity 0 . X km/s and a circular velocity ofradius 2arc, the center of this source is in (2 , arcseg . Black
Image to reproduce by means of
Gallenspy . (b) MCMC in the exploration of the σ parameter. Figure 8: Illustration of the arc belonging to the initial guess and evolution of the MCMC for the σ parameter, in this case thevalues converge from the 300 step approximately. (a) Results obtained with Gallenspy , in the exploration of parame-ters for the SIS profile. (b) Graph comparative between the produced images by the SISmodel and the images of | (cid:126)θ p | . Figure 9: Final results obtained with
Gallenspy , for the fitting of the arc generated with a SIS model. | (cid:126)θ p | . Parameter 95%SIS σ (cid:0) km/s (cid:1) . +4 . X − − . Source
Radio (arcseg) 1 . +0 . − . h (arcseg) 0 . +0 . . X − k (arcseg) 0 . +0 . . X − Table 2: Parameters obtained with
Gallenspy for the SIS model.
In table 2 the parameters obtained are shown with its uncertainties for the quantile of 95%, thesevalues are consistent with those established for the obtention of the images of | (cid:126)θ p | , for this reason with thisillustrative example was possible to observe the form which Gallenspy proceeds and its efficiency in theestimation of the parameters from the GLE.Finally in figure 9b it is the superposition of images, where it is clear the reliability of results obtainedwith
Gallenspy . Another process that
Gallenspy performs, is the obtention of criticals and caustics curves for distinctlens model. To understand this method, it is important to remember that detA depends on the convergence κ and the shear γ , which have relation with the deflector potential computed by Gallenspy in the fitting ofimages ( θ , θ ) [7, 10].In this way, the first step given by Gallenspy is the obtention of γ and γ with the equation 18 and 19 γ (cid:16) (cid:126)θ (cid:17) = 12 (cid:32) ∂ ψ (cid:16) (cid:126)θ (cid:17) ∂θ − ∂ ψ (cid:16) (cid:126)θ (cid:17) ∂θ (cid:33) , (18) γ (cid:16) (cid:126)θ (cid:17) = ∂ ψ (cid:16) (cid:126)θ (cid:17) ∂θ ∂θ , (19)and from this results, Gallenspy computes the points of lens plane for those who detA = 0 based on thegiven relation by Hurtado [10] A = (cid:18) − κ − γ − γ − γ − κ + γ (cid:19) (20)As the obtention of this critical points is not a trivial process, Gallenspy makes this process with theBartelmann method [18], in which the critical curve in the lens plane is a border from which changes thesign of the determinant ( detA ) as illustred the figure 10.As is shown in this image, there are points of the plane ( θ , θ ) that although these do not belong tothe critical curve, can be considered close to this: from this focus, if S = Sign ( detA ) then a point S isconsidered adjacent to the curve in a grid when S changes between S and any of these neighbors points.In figure 10, is presented the estimation of each point for a grid with dimension N XN based on the sign S , this means that for positive values S i,j = 1 and in the case of negative values S i,j = −
1, where i and j igure 10: Illustration of the Bartelmann method for the obtention of critical points in Gallenspy , in this case is taken intoaccount the change in the sign of detA determinant from the critical curve. are in a range of 0 to N −
1. From this perspective, the established restriction in
Gallenspy to know if apoint S i,j is adjacent to the critical curve in the grid, is based on the condition: S i,j ( S i − ,j + S i +1 ,j + S i,j − + S i,j +1 ) < , (21)This method is very effective in the compute of the critical curve as the grid is refined since it allows toreduce the range in which each critical point can be.In figure 11 is the process flow diagram for the estimation of critical curves with Gallenspy , it is importantto highlight that the error range is low due to the grid has a refinement of 100 X
100 points. From thisperspective, the set of critical points with a percentage error of 0 . S i,j in whichthe condition of equation 21 is met. Figure 11: Process flow diagram in the compute of critical curves with
Gallenspy . When the critical points are computed by
Gallenspy , the obtention of the caustic points is an easy processbecause with the application of the equation 6 is enough.Going back to the example of SIS profile of the previous section, the critical curve was obtained with
Gallenspy where the result is shown in figure 12. 12 igure 12: (Red dots) Critical curve obtained with
Gallenspy for the case of SIS profile developed along this work. (Blue)Plotted circle with the averaged radius of the points obtained with
Gallenspy .Figure 13: (Right) Caustic curve and circular source (Left) Critical curve and images formed in the SIS model.
This graph evidences that the critical curve belongs to a circle with 6 .
278 arcs of the radius. In this case,it is important to consider the analytical solution to the SIS profile for detA = 0 [10], where the criticalcurve is a circumference of radius r given by the relation r = 4 πσ D LS c D OS , (22)Base on the data provided above regarding to the SIS, the radius obtained of the equation 22 is of 6 . Gallenspy , where it is possible to check a percentageerror of 0 . β , β )[10], this is evidenced in the figure 13 where the sourceis near to this caustic point and for this reason the magnification of the images is significant.
6. GALROTPYGalrotpy [1] is an interactive tool whose work is focused on the visualization and exploration of parametersthrough MCMC, in such a way that it is possible to make mass reconstructions from the fitting of rotationalcurves in disc-like galaxies.The main python packages used in this routine are: matplotlib [13] for the generation of graphic environ-ment, numpy [12] in the data mangement, astropy [19] which is useful for the units assignation,
Galpy [14] forthe construction of rotational curves with each mass profile, emcee [15] used in the exploration and fitting ofdata with MCMC, and corner [16] for the reliability regions of the parametric fitting.13 a) Panel for the selection of gravitational potentials. (b) Rotation curve created from the superposition of distinct massdistributions.
Figure 14: Graphic environment in
Galrotpy , wherein the left side it is possible the selection of potentials for the curve fitting.
The parametric exploration space in this routine is the same as
Gallenspy due to the reasons given inthe previous section. On the other hand, the rotational velocity data from which are done the parametricfitting should be consigned in a file denominated rot curve.txt , in which the units belonging to the radialcoordinates must be expressed in
Kpc and the velocities in
Km/s . In the figure 14a is shown the panel for theselection of gravitational potentials in
Galrotpy , and below these are sliders with the ones, the parametersvariation is done. Thus the user can do a visual fitting between the rotational curve and the observationaldata of rotational velocity (this is evidenced in figure 14b).
In the case of
Galrotpy the MCMC has similar characteristics with
Gallenspy even in the considerationof the prior and in their approach to evaluation of the likelihood .However, the process of parametric exploration with
Galrotpy presents greater facilities due to the visualfitting that is possible to do with this routine and for this reason, the parametric minimization done with
Gallenspy is not necessary.In this way, with the visual fitting of parameters
Galrotpy proceed to run the MCMC where as in
Gallenspy the user can choose the number of walkers and steps. It is important to point out that in thisroutine, the likelihood es given by the relation L = exp (cid:18) − N (cid:88) i =1 (cid:34) v iobs − v imodelo v errori (cid:35) (cid:19) , (23)with: • N the number of obser vationally obtained data. • v obs is the velocity observed. • v modelo the rotational velocity of the mass model chosen for the fitting.14 v error the error in the observational velocity data.Once Galrotpy do the parametric exploration, the behavior of the MCMC is illustrated and the valueswith these uncertainties of 68% and 95% are shown. Finally
Galrotpy generates two types of graphics, onein which is the fitting rotation curve and other where are the reliability regions.
M33 is a spiral galaxy without bar structure [1], and its data of rotational velocity were obtained fromCorbelli et al. [20]. For the mass reconstruction with
Galrotpy , it is important to point out that theparametric exploration was done with a number of 100 walkers and 3000 steps.
Figure 15: Behavior of the MCMC in the exploration of parameter h r . In figure 15, is presented the way of the MCMC in the exploration of parameter h r , where the convergenceis given in a number of steps less than in Gallenspy , this thanks to the visual fitting of
Galrotpy . Figure 16: Rotation curves and reliability regions of the Galaxy M33 with
Galrotpy , where the fitting was done with NFWprofile for the dark matter halo and Exponential Disc in the case of baryonic matter.
The figure 16 showed the fitting done with
Galrotpy , where the NFW profile was used for the darkmatter halo while the contribution of baryonic matter was analyzed with the Exponential Disc profile. Inthe right side of this figure are the reliability regions, and these values are consigned in the table 3.These values obtained and its uncertainties are in concordance with the results reported by L´opez Funeet.al. [21], where M (cid:63) (cid:0) X M (cid:12) (cid:1) = 4 . ± . M h (cid:0) X M (cid:12) (cid:1) = 5 . ± . Galrotpy and
Gallenspy are combined for themass reconstructions of two disc-like galaxies. 15arameter 68% 95%Exponential Disc h r (Kpc) 1 . +0 . − . . +0 . − . Σ (cid:0) X M (cid:12) pc − (cid:1) . +0 . − . . +0 . − . M (cid:63) (cid:0) X M (cid:12) (cid:1) . +0 . − . . +1 . − . NFW a (X10kpc) 1 . +0 . − . . +1 . − . M (cid:0) X M (cid:12) (cid:1) . +0 . − . . +2 . − . ρ (cid:0) X M (cid:12) Kpc − (cid:1) . +2 . − . . +6 . − . M h (cid:0) X M (cid:12) (cid:1) . +1 . − . . +2 . − . Table 3:
Estimated parameters with GalRotpy for the galaxy M33.
7. Mass Reconstruction of galaxies J2141 and J1331
The galaxies SDSSJ2141-001(J2141) and SDSSJ1331+3628(J1331) are systems that present the stronglensing effect, and its rotation velocities data were given by Dutton et.al. [3, 5]; for these mass reconstructionswere taken into account the profiles of Miyamoto-Nagai, Exponential Disc, and NFW [8, 1]. Regarding tothe GLE, in the case of J2141 it was modeled an extended circular source due to the deflected image is anarc, while for J1331 it was considered a punctual source in which four images are produced in the lens plane.As to the mass distribution of these galaxies, other authors have reported a high contribution of baryonicmatter [5, 3] and this coincides with the obtained results in the use of
Galrotpy and
Gallenspy . J2141 is a spiral galaxy of type S0, where its dominant gravitational contribution is from the disc[5].This object was initially observed in 2006 by mean of the Hubble Spatial Telescope(HST)[5], with a cameraACS in a filter F814 and an exposure time of 420 seconds. In 2009 the Keck telescope had again images ofthese object with a camera NIR and K filter.From these images the GLE in this galaxy was evidenced, with the formation of an arc belonging to asource with a redshift different to J2141 ( z L = 0 . z s = 0 . Figure 17: Images obtained of J2141 by mean of HST and KeckII telescopes with distinct filters.[5]
The rotational velocity data of this galaxy were derived of spectral lines Mg b 5177, Fe II5270, Na D5896, O II 3727 and H α H α a) Adjusted image by Dutton et.al. of the arc generatedby GLE in the J2141 plane. [5] (b) Spectral emission lines of H α belonging to J2141. [5] Figure 18: Observational data obtained of the galaxy J2141 in relation with lensing and rotation velocities.
For the coordinates of the arc in the lens plane ( θ , θ ), the restriction of its contours was made in theway of lower the computational cost in Gallenspy .This image treatment was made in python , trough a pixel to pixel discrimination based on to its positionand luminosity, which can be possible to get the contour showing in figure 19.As it is possible to observe, this obtained contour is incomplete given the noise of image 18a, even so,the number of gotten coordinates was enough for an appropriate adjustment with a circular source.In this case the equivalence between arc seconds and pixels was made based on the observed scale offigure 18a, and in this way it was possible to get each coordinate of the image in radians as observed infigure 19.For the determination of cosmological distances, the ΛCDM model was taken into account due to thefact that this one has been used by other authors in this galaxy [5]; in this way the current matter densityis Λ m = 0 . H = 70 kms − M pc − .Radio(arcsec) Radio(kpc) Velocidad(km/s) Error(km/s)0.000 0.00 3.5 5.30.593 1.45 114.1 5.81.185 2.89 153.8 2.91.778 4.33 212.7 2.62.370 5.78 243.8 2.62.963 7.22 259.8 2.34.148 10.11 254.9 7.54.740 11.56 263.4 2.35.333 13.00 265.9 3.5 Table 4: Rotational velocity values obtained of Dutton et.al. [5] for J2141. igure 19: Contours of arc generated by GLE in J2141. The scale of this plane is in 1 X radians. With this considerations, the cosmological distances given by Dutton et al. [5] are D OL = 497 . M pc , D OS = 1510 . M pc and D LS = 1179 . M pc ; what leads to Σ crit = 4285 . M (cid:12) pc − .In relation to the source, its ranges of possible values for the position and the radius are consigned intable 5. Parameter Range UnitsRadius 0.05 < r < < h < < k < Table 5: Range of values to explore for the source.
In this mass reconstruction, the profiles selected of table 1 were Bulge I, Exponential Disc, and NFW.The first parametric exploration was done with
Galrotpy , where the more reliable result for the MCMC wasobtained for a number of 100 walkers and 1500 steps so that the parametric exploration roads are shown infigures 20, 21 and 22.
Figure 20: Exploration roads for the parameters of the Miyamoto-Nagai profile. igure 21: Exploration roads for the parameters of the Exponential Disc profile.Figure 22: Exploration roads for the parameters of the NFW profile. In this graphics, the initial values of the MCMC are in red lines, where for the dark matter halo and bulgeis not evidenced a convergence of the values; this is pointed out by Dutton as the problem of degeneracyamong the disc and different mass components of the galaxy.[5]These authors, [5] affirm that the mean reason of this degeneracy is the gravitational dominance of discin this system, and for this cause, it is possible to adjust the rotation curve only with the profile of this masscomponent and therefore it is very difficult to know with clarity the circular velocity contribution of othercomponents.Because of this situation, this mass reconstruction was done through the integration of galactic dynamicsand GLE, where the restrictions of each method occur in different geometries and therefore this combinationis a powerful tool for the breaking of this degeneracy.In this way and with the arc coordinates, the parameters of the source were estimated as shown in figure23a under a lens model of Exponential Disk, where these values are presented in table 6 with a radius of thesource of 0.03 arcs . Parameter 68% 95% h (cid:0) X arcs (cid:1) . +1 . − . . +2 . − . k (cid:0) X arcs (cid:1) . +2 . − . . +4 . − . Table 6: Source values estimated by
Gallenspy for the case of J2141. . . . . k . . . . . .
08 0 .
06 0 .
04 0 .
02 0 . h h r .
000 0 .
004 0 .
008 0 . k .
05 4 .
20 4 .
35 4 .
50 4 .
25 50 75 100 125 h r (a) Estimation of circular source with Gallenspy for the galaxy J2141. Einstein RingLensed images (b) Deflected image and Einstein ring (scale in arcs).
Figure 23: Compute of
Gallenspy for source parameters (left side) and Einstein radius (right side).
With the position and size of the source, the mass reconstruction of J2141 based on the GLE was donewith the restrictions obtained of the rotation curves in relation with the parametric space. For this specificcase, the parametric exploration established was three times of the table 1.The combination of lensing and Galactic Dynamics was a process very efficient in the breaking of thedegeneracy between mass components of J2141, and this allowed a mass density minor value for the discas to is shown in figure 24. Based on the observed, it is important to point out, how the combination of
Galrotpy and
Gallenspy is a great alternative for galaxies where the gravitational contribution of each masscomponent is not easy to distinguish.In table 7 the parameters value and its uncertainties are consigned, where the parameters with majordispersion belong to NFW profile, which is related to the degeneracy of this system.Parameter 68% 95%NFW a (kpc) 47 . +8 . − . . +0 . − . m (cid:0) X M (cid:12) (cid:1) . +3 . − . . +6 . − . Exponential Disc h r (Kpc) 29 . +0 . − . . +0 . − . Σ (cid:0) X M (cid:12) pc − (cid:1) . +0 . − . . +0 . − . Miyamoto-Nagai b (Kpc) 1 . +0 . − . . +0 . − . M (cid:0) X M (cid:12) (cid:1) . +0 . − . . +0 . − . Table 7: Values of parameters obtained in the mass reconstruction for J2141. m o . . . . . . . . h R . . . . b
15 30 45 r s . . . . M mo .
32 4 .
36 4 .
40 4 . . . . . h R . . . . b . . . . M Figure 24: Contours obtained through the combination of restrictions between
Galrotpy and
Gallenspy for the galaxy J2141. θ Eins ).In figure 23b, this curve obtained by
Gallenspy is evidenced, where the Einstein radius θ Eins presented avalue of 0 . +0 . − . respectively.The compute of enclosed mass for J2141, was done taking into account the relation: M = Σ cr (cid:90) S ∇ ψ ( θ , θ ) d θ, (24)such that in table 8 are these estimated values.Parameter Log (cid:18) MM (cid:12) (cid:19) M Eins . +0 . − . Mbar
Eins . +0 . − . Table 8: Enclosed mass with Einstein radius, M is the total mass and Mbar the baryonic matter.
These results are in concordance with the values range given by Dutton et al. [5], where they reportedfor J2141 a value of
Log (cid:18) MbarM (cid:12) (cid:19) = 10 . +0 . − . within the Einstein radius.The results gotten in this work shown separately mass estimation of the bulge and disc, to differencewith results of Dutton et al., where they obtain the stellar mass without discriminating each contributionof these baryonic matter components. These mass values are shown in Table 9, and the fitting made to therotation curve and arc generated in the GLE with these results, it is illustrated in images 25a and 25b.Component of the galaxy log (cid:18) MM (cid:12) (cid:19) Bulge 8 . +0 . − . Disc 10 . +0 . − . Dark Matter Halo 7 . +0 . − . Table 9: Mass values for each component of J2141.
In the fitting of the rotational curve, it is possible to evidence how the dark matter halo is dominantgravitationaly in a radius minor to 1 . Gallenspy and
Galrotpy is a great option.
SDSSJ1331+3628 (J1331) is a spiral galaxy with a counter-rotating massive core [3], where just like J2141,the inclination of this galaxy allows to get its rotational velocity values in function of the galactocentricradius.J1331 is localet in RA = 202.9188 ◦ and DEC = 36.469990 ◦ , and in the observation of this system Treutet.al (2011) [22] observed 2 distinct redshifts within a radius of 1 arcs ( z L = 0 . , z s = 0 . SourceObservational dataModel values (a) Comparison of the observational data and lens model data fora circular source in galaxy J2141 with
Gallenspy , in this case thelens model choose let us obtained a set of images which overlapthe observational images. (b) Fitting of rotation curve with the model data.
Figure 25: Fitting obtained for lensing and rotation curves through the combination of restrictions between
Gallenspy and
Galrotpy . The produced images for the GLE are shown in figure 27a, which are pointed with letters A,B,C y D,also it is important to clarify that the other 3 unlabeled images do not belong to this group, since accordingto what indicates Trick et.al[3] these are part of a stellar formation region.Regarding to the dynamics aspect of J1331, Dutton et al.(2013) got its rotational velocity values withthe use of Keck I telescope by mean of a spectrograph LRIS (Low Resolution Imaging Spectrograph)[ ? ],where these data were obtained of the absortion lines M gb (5177 ˚A), F e II (5270 . F e II (5406 ˚A),while the gas velocity was estimed with emision lines H α (6563 ˚A) and N II (6583 ˚A).An important aspect of J1331 is its supermassive core, since due to the exposed by other authors [3]about half the brightness is enclosed in the effective radius illustrated in figure 27b. For this reason, thisgalaxy has been interest object in different works [ ?
22, 24], where it is speculated a possible merger eventin the past of this system which changes its structure and kinematics.
Unlike J2141, the mass profiles for J1331 within its core can not be used by
Galrotpy , due to the rotationalnegative velocity values present in this part of the galaxy.Because of this, the galaxy region enclosed in the effective radius was analyzed totally with lensing, suchthat the only restrictions applied in
Gallenspy are in the parametric ranges used with
Galrotpy for the fittingof the rotation curves in close radii to the galaxy periphery.In this way, other authors allow watching how the breaking of the degeneracy is not an easy task [3, 4],and even if different advances had been obtained this objective has not been achieved yet.The observational data of the images (A-D) are consigned in table 10, where it was necessary to expressthese coordinates in arcs for a scale 1 pixel = 0 .
05 arcs.For the determination of the cosmological distances, the redshifts were taken into account in the numericalsolution to the Dyer-Roeder equation [10], which
Gallenspy made based on the Jimenez code [25]. For this23 igure 26: Images of J1331 obtained with the HST telescope in F450W and F814W filters.(Image take of Trick et.al [3]) (a) Quadruplet of images formed through the ELG for a pointsource, in this case G is the galactic center. (Image take of Tricket.al [3]). (b) Rotational velocity values of J1331, where the effective radiusis distinguished by Trick et.al with 2.6 arcs, within which is a su-permassive and counter-rotating core [3].
Figure 27: Observational data of lensing and rotational velocities for the galaxy J1331. case, it was used the cosmological model ΛCDM where the obtained results were D LS = 442 . X Kpc, D OL = 422 X Kpc and D OS = 817 . X Kpc.The next step in
Gallenspy was the estimation of the source position, and this obtained values are intable 11.In the parameters exploration for the mass reconstruction, the established ranges of the bulges I and IIbelonging to the table 1 were not enough for the fitting of the observational images, and this is shown infigure 28. For this reason a very massive bulge was considered, where the selected most appropriate profileis of the Miyamoto-Nagai with parametric exploration ranges of thick disc evidenced in table 1.24oordinates A B C D G θ θ Table 10: (In pixels) Positions of the images (A-D) and galactocentric center (G) given by Trick et al.[3]. The error in eachimage is of 0.05, while of the G is 0.07.
Parameter 68% 95% β (cid:0) X − arcseg (cid:1) . +0 . − . . +0 . . β (cid:0) X arcseg (cid:1) . +0 . − . . +0 . − . Table 11: Source position obtained with
Gallenspy for the case of the GLE in J1331.
This process was done in
Gallenspy with 100 walkers and 100 steps in the MCMC, in such a way that infigure 29 these contours are shown.Under this estimation, the parameters set more appropriate for the mass distribution of J1331 obtainedwith
Gallenspy is in table 12, and these values allow getting the fitting of figure 30b.Parameter 95% 68%NFW a (Kpc) 8 . +2 . − . . +8 . − . m (cid:0) X M (cid:12) (cid:1) . +2 . − . . +3 . − . Exponential Disc h r (Kpc) 9 . +5 . − . . +10 . − . Σ (cid:0) M (cid:12) Kpc − (cid:1) . +0 . − . . +1 . − . Miyamoto-Nagai b (kpc) 4 . +3 . − . . +8 . − . a (kpc) 2 . +1 . − . . +3 . − . M (cid:0) M (cid:12) (cid:1) . +4 . − . . +7 . − . Table 12: Parameters set obtained with
Gallenspy for J1331.
With this mass distribution, the critical and caustic curves and the Einstein ring are presented in figure31a,31b and 30a where the Einstein radius and critical radii have values of 0 . +0 . − . , 1 . +0 . − . and 0 . +0 . − . respectively. It is important to point out, that with this lens model was possible toestimate the mass within the effective radius under which Trick et al. [3] got restrictions for the massestimation through the luminosity of J1331.In table 13 the mass values restricted by the Einsten radius are consigned; when it is doing a reviewof the results reported by Trick et al. under the lens model that they assumed [3], it is found what theEinstein radius estimated is 0 . ± . . ± . M (cid:12) , and this beingin concordance with the results obtained in this work with Gallenspy .Regarding the restriction within the critical radius, the values obtained of baryonic and dark matterare 2 . +0 . − . X10 M (cid:12) and 2 . +0 . − . X10 M (cid:12) respectively, these results are also consistent with thereported by Trick et al. [5], where for the effective radius these values are 2 . ± . M (cid:12) for the totalmass and 1 . ± . M (cid:12) of baryonic matter, also it should be clarified that these authors obtain theirresults with alternative methods to the GLE of J1331[3].25 igure 28: Adjustment obtained by Gallenspy with the parametric space of the Bulbo II belonging to the table 1.
Mass components Mass value (cid:18) M (cid:12) (cid:19) Bulge 0 . +0 . − . Disc 7 . +1 . − . Dark matter halo 0 . +0 . − . Einstein Mass . + . − . Table 13: Mass Values within Einstein radius.
The results obtained with
Gallenspy , show that this is a very effective tool for the mass reconstructionswithin the critical curve and Einstein radius. However for J1331, the estimation is not enough with radiigreater than 2 . Galrotpy .In other’s works [3, 4] is evidenced how the mass reconstructions for J1331 from a dynamics analysishave many complications due to the complexity of its rotation curve. It is the reason which Trick et al. [3]restricted this mass reconstruction to the effective radius, while Dutton et al. in 2013 [4] dedicated efforts instudying the periphery of the galaxy. Based on the exposed, each routine in this work was applied separatelyin different galaxy regions.
The best result in the fitting of the rotation curve with
Galrotpy was for a number of 20 walkers and100 steps, wherein the image 32 the contours of each parameter obtained are presented and the figure 33the curve obtained for these observational data.In table 14 are presented these values and its uncertainties for each parameter. The estimated massdistribution with these indicated parameters was restricted to 7.56 arcs (in this radius are all data ofrotational velocity), and therefore the enclosed mass in this amplitude was estimated in log (cid:18) MM (cid:12) (cid:19) =11 . +0 . − . where the baryonic matter has a value of log (cid:18) MM (cid:12) (cid:19) = 10 . +0 . − . .The results given by Dutton et al. in 2013 [4] indicate that the baryonic matter in this radius is of log (cid:18) MM (cid:12) (cid:19) = 11 . ± .
07 which is in concordance with the result obtained through of
Galrotpy . Also, itis important to note, the great relation in the estimation of the bulge mass, where they report a value of26 m o h R b a r s . . . . M mo h R b a . . . . M Figure 29: Reliability regions of obtained parameters in the case of J1331. Einstein RingLensed images (a) Einstein ring obtained with the mass distribution of J1331, thelens plane is in an arcs scale. (b) Adjustment obtained with
Gallenspy for J1331,where it is important to highlight how the model im-ages are very closed with the observational images forthe lens model choose.
Figure 30: Comparison of observational images with Einstein ring and lens model images computed in
Gallenspy . (a) Critical curves obtained with the mass distributionof J1331, the lens plane is in an arcs scale.
10 5 0 5 10 (b) Caustic curves obtained with the mass distribu-tion of J1331, the source plane is in a scale of 1 X radians. Figure 31: Critical and caustic curves obtained with
Galle spy for the lens model selected. . . l o g ( b T k D ) . .
01 0 . l o g ( M T k D ) . . l o g ( h r ) . . . l o g ( ) . . . l o g ( a N F W ) . . log( a TkD ) .
51 1 . l o g ( M ) . . log( b TkD ) . . . log( M TkD ) . . log( h r ) . . . log( ) . . . log( a NFW ) . . log( M ) Figure 32: Credibility regions of obtained parameters with
Galrotpy for J1331. a (Kpc) 11 . +0 . − . . +0 . − . m (cid:0) X M (cid:12) (cid:1) . +0 . − . . +0 . − . Exponential Disc h r (Kpc) 1 . +0 . − . . +0 . − . Σ (cid:0) M (cid:12) pc − (cid:1) . +0 . − . . +0 . − . Miyamoto Nagai b (kpc) 7 . +0 . − . . +0 . − . a (kpc) 3 . +0 . − . . +0 . − . M (cid:0) M (cid:12) (cid:1) . +0 . − . . +0 . − . Table 14: Estimated parameters with
Galrotpy for J1331 galaxy. log (cid:18) MM (cid:12) (cid:19) = 10 . ± .
10 and in this work the obtained value is log (cid:18) MM (cid:12) (cid:19) = 10 . +0 . − . . R ( kpc ) V c ( k m / s ) Thick DiskExp. DiskNFW - HaloBest Fit
Figure 33: Rotation curve of J1331 obtained with
Galrotpy , in this case it is possible to observe how the bulge is dominantgravitationally which is in concordance with the analysis done with
Gallenspy from lensing.
In the mass reconstructions performed for this galaxy with
Galrotpy and
Gallenspy , it follows thatabout of the 78% of the mass of J1331 is enclosed in the effective radius, this confirmed the presence of asupermassive core, which to the presenting a negative direction in its rotation opens the possibility to thinkthat this galaxy is the result of a merger process between two stellar systems, with angular momentumoriented in distinct orientations.[3]Also it is important to mention the high effectiveness of
Galrotpy in this process, where the obtainedresults for radii close to the galaxy periphery were very successful in comparison with the results of Duttonet al. in 2013 [4], all this taking into account that these estimations were done with the fitting from just 3rotational velocity values. 29esides, it is important to remember that the degeneracy between the disc and halo is still a researchtopic [3, 4], and for this reason, the possibility of adjusting
Galrotpy for negative values of the rotationalvelocity is open, since this would allow combining lensing and galactic dynamics for similar galaxies to J1331.
8. HE 0435-1223 test case
An additional case of tested for
Gallenspy was the Quasar HE 0435-1223, in which is presented theGLE through a quadruply imaged belonging to a background source [26]. HE 0435-1223 was discovered byWisotzki et al. (2000) and from there has been research object in distinct works[26 ? ]. Figure 34: Quadruply imaged formed through the GLE on the case of quasar 0435-1223. (Image take of Courbin et.al.(2011)[26].
In figure 34 are shown these formed images by mean of the GLE, where the redshifts of the lens andsource are z s = 1 .
689 and z L = 1 . . Kpc for the Einstein radius.Based on these redshifts value, the cosmological distances estimated by
Gallenspy are D ds = 1070 . M pc , D d = 1163 . M pc and D s = 1700 . M pc . Also it is important to point out, that the positions of the imagesformed in the GLE were obtained of Courbin et al.(2011) [26], and of this manner it was possible to performthe mass reconstruction for this quasar with the Exponential Disc and NFW profiles, where the estimatedparameters are in table 15. Parameter 95% 68%NFW a (Kpc) 52 . +0 . − . . +0 . − . m (cid:0) X M (cid:12) (cid:1) . +0 . − . . +0 . − . Exponential Disc h r (Kpc) 11 . +0 . − . . +0 . − . Σ (cid:0) M (cid:12) pc − (cid:1) . +0 . − . . +0 . − . Table 15: Values of the obtained parameters with
Gallenspy
With these parameters, the obtained images are illustrated in figure 35, where the values of baryonic anddark matter are consigned in table 16. The results given by Courbin et al.(2011) for this system, reveal thatthe total mass of this quasar is of 3 . ± . X M (cid:12) which is very close to the obtained value in this work,other important aspect is the matter baryonic fraction which in this work is of 0 . ± .
15 while Courbin30t al.(2011) reported 0 . +0 . − . with the Sapelter IMF; in this way it is possible to confirm as Gallenspy isan efficient tool. Mass Value (cid:18) M (cid:12) (cid:19) Baryonic 2 . +0 . − . Dark 0 . +0 . − . Einstein Mass . + . − . Table 16: Mass Values within Einstein radius.Figure 35: Comparison between model and observational images formed through the GLE.
9. Conclusions
In this work was presented the
Gallenspy code, which is very useful in mass reconstructions based onGLE; in this way it is important to highlight, the manner which this routine allows to obtain the massdistribution of bulge and disc separately unlike to the methods of reconstruction applied by other authors[5, 3, 4].Also, the advantages of combining Lensing and Galactic Dynamics were illustrated with the use of
Galrotpy and
Gallenspy , in this case, with the restrictions given by each routine, it was possible to havesignificant progress in the breaking of the degeneracy in J1331 and J2141. Additionally, for J1331 was used
Gallenspy in the mass reconstruction within the critic radius, while with
Galrotpy the peripheral regionwas analyzed and although this degeneracy not could be break completely, the estimated parameters haveconcordance with the obtained results of other authors [3, 4], and this gives reliability to these routinesconstructed.On the other hand, it is important to highlight the use of mass models with spherical symmetry, the onesare used by distinct authors [27] [26], and which allows getting very good results as in mass reconstructionsas in estimations of Hubble parameter.Regarding future improvements for
Gallenspy , is considered the increase in the number of mass profilesused in this routine, besides there is the possibility of extending this code for reconstructions of superficial31rightness functions in lens galaxies, like the estimation of temporary cosmological delays for the study ofthe universe expansion.Finally it is important to mention the advantages of performing visuals fitting with
Galrotpy in therotation curves, since through this process it is possible get the initial values set for the MCMC in bothroutines. For this reason, a step to follow with
Gallenspy is related with its optimization, in such a way thatin this code would be possible to make interactives fitting, the ones allows that in similar galaxies to J1331can be given restrictions from the GLE for its dynamical analysis.
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