# NGC1300 dynamics - II. The response models

**ABSTRACT** We study the stellar response in a spectrum of potentials describing the barred spiral galaxy NGC1300. These potentials have been presented in a previous paper and correspond to three different assumptions as regards the geometry of the galaxy. For each potential we consider a wide range of Omegap pattern speed values. Our goal is to discover the geometries and the Omegap supporting specific morphological features of NGC1300. For this purpose we use the method of response models. In order to compare the images of NGC1300 with the density maps of our models, we define a new index which is a generalization of the Hausdorff distance. This index helps us to find out quantitatively which cases reproduce specific features of NGC1300 in an objective way. Furthermore, we construct alternative models following a Schwarzschild-type technique. By this method we vary the weights of the various energy levels, and thus the orbital contribution of each energy, in order to minimize the differences between the response density and that deduced from the surface density of the galaxy, under certain assumptions. We find that the models corresponding to Omegap ~ 16 and 22 kms-1kpc-1 are able to reproduce efficiently certain morphological features of NGC1300, with each one having its advantages and drawbacks. Based on observations collected at the European Southern Observatory, Chile: programme ESO 69.A-0021. E-mail: ckalapot@phys.uoa.gr (CK); patsis@academyofathens.gr (PAP); pgrosbol@eso.org (PG)

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**ABSTRACT:**Numerous solutions of the collisionless Boltzmann and Poisson equations, having the same flattened axisymmetric scale-free mass distribution, are surveyed using Schwarzschild's self-consistent field method. A great variety of the self-similar models of very flat elliptical galaxies or invisible spiral halos, which these solutions constitute, are found, because orbits remaining close to the plane (due to large angular momenta) may be replaced by orbits with little angular momenta which are still well confined by the third integral. All solutions have velocity ellipsoids approximately aligned with the cylindrical coordinate axes, since the local dynamics is dominated by orbits near peri.The Astrophysical Journal 05/1984; 281:100-111. · 6.73 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**ABSTRACTA dynamical mechanism is proposed that explains the spiral structure observed frequently as a continuation of the bars in barred spiral galaxies. It is argued that the part of the spirals attached to the bar is due to chaotic orbits. These are chaotic orbits that exhibit for long time intervals a 4:1-resonance orbital behaviour. They are of the same type of orbit as is responsible for the boxiness of the outer isophotes of the bar in cases like NGC 4314, as indicated by Patsis, Athanassoula & Quillen. The spirals formed this way are faint with respect to the bar, open as they wind out, and do not extend over an angle larger than π/2. A possible continuation of the spiral structure towards larger angles can be due to orbits trapped around stable periodic orbits at the corotation region. We present a family of stable, banana-like periodic orbits, precessing as EJ increases, that can play this role.Monthly Notices of the Royal Astronomical Society Letters 06/2006; 369(1):L56 - L60. · 5.52 Impact Factor - SourceAvailable from: arxiv.org
##### Article: Triaxial Galaxies with Cusps

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**ABSTRACT:**We have constructed fully self-consistent models of triaxial galaxies with central density cusps. The triaxial generalizations of Dehnen's spherical models are presented, which have densities that vary as 1/r^gamma near the center and 1/r^4 at large radii. We computed libraries of about 7000 orbits in each of two triaxial models with gamma=1 (weak cusp) and gamma=2 (strong cusp); these two models have density profiles similar to those of the core and power-law galaxies observed by HST. Both mass models have short-to-long axis ratios of 1:2 and are maximally triaxial. A large fraction of the orbits in both model potentials are stochastic, as evidenced by their non-zero Liapunov exponents. We show that most of the stochastic orbits in the strong- cusp potential diffuse relatively quickly through their allowed phase-space volumes, on time scales of 100 - 1000 dynamical times. Stochastic orbits in the weak-cusp potential diffuse more slowly, often retaining their box-like shapes for 1000 dynamical times or longer. Attempts to construct self- consistent solutions using just the regular orbits failed for both mass models. Quasi-equilibrium solutions that include the stochastic orbits exist for both models; however, real galaxies constructed in this way would evolve near the center due to the continued mixing of the stochastic orbits. We attempted to construct more nearly stationary models in which stochastic phase space was uniformly populated at low energies. These ``fully mixed'' solutions were found to exist only for the weak-cusp potential. No significant fraction of the mass could be placed on fully-mixed stochastic orbits in the strong-cusp model, demonstrating that strong triaxiality can be inconsistent with a high central density. Comment: 58 TEX pages, 14 PostScript figures, uses epsf.styThe Astrophysical Journal 11/1995; · 6.73 Impact Factor

Page 1

arXiv:1009.0383v1 [astro-ph.CO] 2 Sep 2010

Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 3 September 2010 (MN LATEX style file v2.2)

NGC 1300 Dynamics:

II. The response models

C. Kalapotharakos,1⋆P.A. Patsis,1,2,3and P. Grosbøl3⋆†

1Research Center for Astronomy, Academy of Athens, Soranou Efessiou 4, GR-115 27, Athens, Greece

2Observatoire Astronomique de Strasbourg, 11 rue de l’Universit´ e, 67000 Strasbourg, France

3European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany

Accepted ..........Received .............;inoriginal form ..........

ABSTRACT

We study the stellar response in a spectrum of potentials describing the barred spiral galaxy

NGC 1300. These potentials have been presented in a previous paper and correspond to three

different assumptions as regards the geometry of the galaxy. For each potential we consider

a wide range of Ωppattern speed values. Our goal is to discover the geometries and the Ωp

supportingspecific morphologicalfeatures of NGC 1300. For this purposewe use the method

of response models. In order to compare the images of NGC 1300 with the density maps of

our models, we define a new index which is a generalization of the Hausdorff distance. This

index helps us to find out quantitativelywhich cases reproducespecific features of NGC 1300

in an objective way. Furthermore, we construct alternative models following a Schwarzschild

type technique. By this method we vary the weights of the various energy levels, and thus the

orbital contribution of each energy,in order to minimize the differences between the response

density and that deduced from the surface density of the galaxy, under certain assumptions.

We find that the models corresponding to Ωp≈ 16km s−1kpc−1and Ωp≈ 22km s−1kpc−1

are able to reproduce efficiently certain morphological features of NGC 1300, with each one

having its advantages and drawbacks.

Key words: Galaxies: kinematics and dynamics – Galaxies: spiral – Galaxies: structure –

ISM:kinematics and dynamics

1 INTRODUCTION

In Kalapotharakos et al. (2010) (hereafter Paper I), we have pre-

sented three different general models representing the potential

of NGC 1300. The morphology we investigate can be observed

at a deprojected near-infrared image of the galaxy (Fig. 1). Each

of these model-potentials corresponds to different assumptions re-

garding the distribution of the luminous matter of the galaxy in the

third dimension, perpendicular to the equatorial plane. They repre-

sent limiting cases for the geometry of the system, and vary from

the pure 2D (Model A) to 3D cases. The two 3D cases correspond

either to a pure cylindrical geometry of a 3D disc with a constant

scale height (Model B) or to a combination of a spheroidal com-

ponent representing the central part of the bar with the cylindrical

geometry of a 3D disc for the rest of the luminous mass (Model C).

All of them have the option of the inclusion of two additional terms

in the potential representing the central mass concentration and the

⋆ckalapot@phys.uoa.gr (CK); patsis@academyofathens.gr (PAP); pgros-

bol@eso.org (PG)

† Based on observations collected at the European Southern Observatory,

Chile: program: ESO 69.A-0021.

dark halo component respectively. These terms are constrained by

the kinematical data derived by Lindblad et al. (1997).

We note that in our calculation we do not take into account

the outer spiral arms that exist beyond the edges of the frame of

Fig. 1. These spiral extensions are very weak in the near-infrared,

while they are conspicuous in the optical. This indicates that they

consist mainly of young objects, thus their contribution to the mass

distribution of the galaxy is small (Grosbøl et al. in preparation).

In the present paper we investigate the detailed dynamics in

the three cases using response models (hereafter RM). As we have

already mentioned in Paper I the global dynamical behavior in ro-

tating galaxies, is crucially determined by the assumed value of

the pattern speed Ωp. Thus, in this paper we use the method of

response models (Patsis 2006) in order to find which Ωpvalues, in

eachModel (A,B andC),areabletoreproduce thevarious morpho-

logical features of NGC 1300. We construct our models under the

assumption of a single pattern speed and we will assess the results

at the end. For this reason, we introduce an index that is a gener-

alization of the Hausdorff distance (see e.g. Deza & Deza 2009) so

that we can quantify the (dis)similarity between the density maps

of the response models and the K-band deprojected NGC 1300 im-

age. Comparisons that involve the surface density of the galactic

disc are done under the assumption of a constant M/L ratio (see

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C. Kalapotharakos et al.

Figure 1. The near-infrared surface brightness of NGC 1300, in gray scale

(color scale in online version). This image has been obtained after applying

all the necessary amendments (see Paper I) and after deprojecting the image

with the considered position and inclination angles (PA, IA)=(87◦,35◦).

The solid curves are six iso-density contours describing the basic contour

shapes of NGC 1300. The adopted distance is D=19.6 kpc.

Paper I). We remind also, that from the image we used (Fig. 1) we

have removed point-like sources, that are likely to be young stellar

clusters (Paper I).

This study allows also the comparison between various re-

sponse models and consequently indicates, which kind of geome-

try applies better to the NGC 1300 case. Nevertheless, our ultimate

goal is the detection of the dynamical mechanisms behind the ob-

served structures (bar and spiral arms). This needs a detailed anal-

ysis of the orbital stellar dynamics for each case and is presented in

Patsis et al. (2010) (hereafter Paper III).

Our paper is structured as follows: in Section 2 we present the

method of response models and the corresponding results. In Sec-

tion 3 we describe the generalized Hausdorff distance that helps us

quantify the similarity between the various response models and

the galaxy. In Section 4 we check the ability of the response mod-

els to describe better the morphological features of NGC 1300 by

varying the relative contribution of the energy levels. Finally, we

discuss our conclusions in Section 5.

2 RESPONSE MODELS

Thesemodels show theresponse of an initiallyaxisymmetric stellar

disc, with particles moving at the beginning of the simulation in

circular orbits determined in the axisymmetric part of the potential.

The steps we follow in our numerical experiments are:

(i) we choose a potential (among Models A, B, C) and the pat-

tern speed value Ωp,

(ii) we populate uniformly the disc up to Rmax = 15kpc (the

(xi,yi) of 106particle positions on the plane are taken at random

positions),

Figure 2. The RCR/Rbar ratio as a function of the pattern speed for our

models A (filled symbols) and B (empty symbols). Squares (blue colored in

the online version) and circles (red colored inthe online version) correspond

to the long and the short semi-major axis of the bar, respectively. Note that

there are ranges of Ωpvalues that give multiple Lagrangian points.

(iii) we set each particle in circular motion (in the rotating

(iv) we grow the non-axisymmetric terms of the potential lin-

early from 0 towards their full amplitude within two pattern rota-

tions,

(v) we integrate the particles’ orbits for 25 pattern rotations (we

consider that a particle has escaped and we stop the integration of

its orbit when it reaches a radius R > 22 kpc).

(vi) Finally we construct density maps, by converting our data

filesto images. For this we consider a grid and we take into account

the numerical density of the test particles on the disc.

frame) with velocity vcirc=

??????

dΦ0

dR

?????

1

R− Ωp

R, where Φ0is the

axisymmetric part of the potential,

In Paper I we have studied the distribution of the Lagrangian

points in all general models (A, B, C) and for an extended range

of Ωpvalues (see fig. 10 of Paper I). In Fig. 2 we plot the ratio

RCR/Rbarof the corotation radius RCRover the semi-major axis of

the bar Rbaras a function of the corresponding Ωpvalue. We have

omitted the points corresponding to Model C since they are very

close to those of Model B. We have considered the bar extending

up to the outer white isocontour of Fig. 1 (third white contour start-

ing counting from inside). This contour is not symmetric to the

origin, since the right semi-major axis of the bar is longer than the

left one. Thus, we set Rbar= 8.9 kpc for the right semi-major axis

of the bar and Rbar = 8.0 kpc for the left one. From the above it

is obvious that the ratio RCR/Rbardepends on the side of the bar.

The considered RCR value is the radius of the Lagrangian points

(L1,L2) lying in the corresponding side. In Fig. 2 the filled circles

and squares correspond toModel A, while theempty ones toModel

B. The squares (colored blue in the online version) and the circles

(colored red in the online version) represent the ratio RCR/Rbaron

the right and on the left side (L1and L2area), respectively. It is

argued that Rbar should not be longer than the corotation radius

Page 3

NGC1300 response models

3

Figure 3. The surface density in gray scale (color scale in the online version) of the RMs corresponding to the potential of the pure 2D case (Model A). At

the top of each panel is indicated the Ωpvalue. In each frame we plot over the density maps of the RMs six representative contours (solid lines) of the K-band

image of NGC 1300 (the same with those plotted in Fig. 1). The density scale is the same as in Fig. 1.

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C. Kalapotharakos et al.

Figure 4. As Fig. 3, but for the potential corresponding to the cylindrical geometry (Model B).

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NGC1300 response models

5

Figure 5. As Fig. 3, but for the potential corresponding to the (spheroidal + cylindrical) geometry (Model C).

Page 6

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C. Kalapotharakos et al.

RCR. This argument is sustained by dynamical studies according to

which the orbital content beyond corotation is not able to support

the bar shape (Contopoulos 1980). In our models, as we have men-

tioned already in Paper I, we see that there are ranges of Ωpvalues

with multiple Lagrangian points (especially in Model A). Particu-

larly, in some cases the ratios RCR/Rbarcorresponding to multiple

Lagrangian points are quite different when we consider as corota-

tion distance the distance of the one or the other Lagrangian point.

E.g. in Model A at Ω ≈ 25 km s−1kpc−1the ratio RCR/Rbarvaries

from ≈ 0.5 (unphysical value since it is significantly lower than 1)

up to ≈ 1.1 (reasonable value). These cases are quite complicated

and lie beyond the standard paradigm. Thus, there is no a priori

knowledge about the orbital behavior and the morphologies they

can support. In any case, the models with Ωp ? 28 km s−1kpc−1

have single Lagrangian points corresponding to significantly low

RCR/Rbarvalues.

In this study we present the results corresponding to the Ωp

values from 12 to 26 km s−1kpc−1. Empirically it has been also

realized that beyond the limits of this range the response mod-

els were particularly problematic in the sense that there was ob-

viously no good agreement with the image of the galaxy. The mod-

els with Ωp ? 17km s−1kpc−1correspond to “fast bars” as are

usually called these with RCR/Rbar < 1.4 (Rautiainen et al. 2008;

Debattista & Sellwood 2000).

In Fig. 3 we plot, in gray scale (color scale in online version),

the density maps we obtain from the RMs corresponding to the

potential of Model A (2D case) for the 12 < Ωp< 26 km s−1kpc−1

values. Here, we plot the RMs corresponding to the 8 indicated Ωp

values. However, we have data for models with totally15 Ωpvalues

within the above range of Ωp. On each panel of this figure we have

also overplotted the same (as in Fig. 1) six iso-density contours

(solid lines) of the K-band image of NGC 1300, outlining its major

morphological features.

In this figure we observe that the low Ωpvalues (first row)

fail to reproduce both the bar and the spiral structure. The response

bar is conspicuously broader than the galactic bar, while the spi-

ral arms are absent or flocculent. For higher values of Ωpwe get

spiral arms in good agreement with the real ones (see panel for

Ωp = 16 km s−1kpc−1) although the bar is still broad. For even

higher Ωpvalues (third row) the formed spiral arms of the models

do not match with the spirals of the galaxy. Nevertheless the bar

clearly gets an ansae character in agreement with the NGC 1300

bar morphology (see panel for Ωp= 22 km s−1kpc−1as best ex-

ample). Finally, the highest Ωpvalues (last row) produce very open

spiral arms and/or rings and a disintegrated bar.

Figure 4 presents the RMs for Model B (thick disc, cylindrical

geometry) at the same Ωpvalues as for Model A (Fig. 3). In this

case the response bars have always high density regions at their

ends, underlying a clear ansae-type character. A common feature

is also a density enhancement along the minor axis. The bar mor-

phology has a good relation with the galactic bar for low Ωpval-

ues (two first rows). This indicates that a proper selection of the

bar-supporting orbits might lead to a best fitting of the NGC 1300

bar. As Ωpincreases (panels in the third and forth rows) the re-

sponse bar becomes shorter than that of the galaxy. The response

spiral arms are not always well described but there is a range of

Ωp = 16 − 22 km s−1kpc−1, where occasionally we find high

densities in the models at the regions corresponding to the galactic

spiral arms.

Figure5issimilartoFigs.3and 4but now for Model C(spher-

ical and cylindrical geometry). The morphology of the response bar

resembles more the morphology we get in Model A especially for

the lower Ωpvalues (first two rows). However, the inner structure

of the NGC 1300 bar (R ? 2kpc) is fitted clearly better in this case.

The general spiral response is closer to that of Model B while it is

in general fainter.

Up to this point, the similarity between the various RMs and

the morphological features of the galaxy have been made by eye.

Although the eye is usually a good selecting tool it lacks of objec-

tivity and capability of quantification. For this reason, in the next

section, we introduce an index in order to quantify the comparison

between the NGC 1300 morphology and the response models.

3QUANTITATIVE COMPARISON

In Fig. 1 we have plotted six iso-density contours of NGC 1300

reproducing its basic morphological features. Our goal is to “mea-

sure” in an objective way the similarity of an RM density map with

the galactic morphology. Thus, we want to establish a quantitative

criterion for this resemblance.

The methodology of this study is related with image compar-

ison and pattern recognition techniques. A simple and well known

index used in such studies is the “Hausdorff Distance” dH, that is

defined as follows: let A, B be two non-empty subsets of a metric

space (M,d). In this case the Hausdorff distance dHbetween A and

B reads

dH= max

?

sup

a∈Ainf

b∈Bd(a,b), sup

b∈Binf

a∈Ad(a,b)

?

.

(1)

Considering two sets of points corresponding to two contour lines

and d being the Euclidean distance, dHmeasures the maximum of

the distances between the points of each curve from the other one

(as a whole). It is obvious from the above definition that dH = 0

corresponds to identical lines. The major issue about dHis its sen-

sitivitytooutliers. Let us suppose that a contour lineconsists of two

parts, the major one being identical to a contour line of NGC 1300,

while the minor one being a tiny part far away. In such a case al-

though we have a good matching of the response model with the

galaxy morphology (at least for the specific contour), dHreads a

false high value. There have been proposed many modifications of

dHthat aim to reduce this drawback (Zhao et al. 2005, and refer-

ences therein). Here we introduce an index which is a generaliza-

tion of dHand it is appropriate for our study.

Wecall thisindex“Generalized Hausdorff Distance”,dGH,and

itisdefinedasfollows: letC1,C2betwocurvesontheplaneofnon-

zero length LC1and LC2, respectively. The dGHbetween C1and C2

is

inf

dl +

dGH=

1

LC1

?

b∈C1

a∈C2d(a,b)

?b?

1

LC2

?

b∈C2

inf

a∈C1d(a,b)

?b?

dl,

(2)

wheretheintegrationsareover thecurvesC1andC2and?·?denotes

the Euclidean norm. The integration adds the contributions from all

points, while the division with the curve lengths and the norm ?b?

aims to the normalization of the final dGHvalue so that it can be

comparable for two totally different contour lines. In practice, for

the computation of dGHwe sample both curves with a considerable

number of points NC1, NC2and we calculate the quantity

dGH=

1

NC1

NC1

?

i=1

1

?b?min

ai∈C1

b∈C2

d(ai,b) +

1

NC2

NC2

?

i=1

1

?b?min

ai∈C2

b∈C1

d(ai,b). (3)

What we do is to find which contour line of each RM corresponds

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NGC1300 response models

7

Figure 6. In the left-hand column we plot for the RM, the minimum generalized Hausdorff distance dGHvalue relative to the NGC 1300 contour line shown in

the right-hand column (same row). In the middle column we plot the minimum dGHvalues corresponding to SMs (see Sect. 4). Note that filled circles, empty

circles and empty squares (red, blue and green colored lines in the online version) correspond to pure 2D geometry (Model A), to cylindrical geometry (Model

B) and to spheroidal+cylindrical geometry (Model C), respectively. The star symbol (light blue line in the online version) in the left-hand column corresponds

to a scenario with time-dependent Ωp(see Sect. 5).

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C. Kalapotharakos et al.

Figure 7. The six representative contour lines of NGC 1300 (solid lines) together with the six contour lines corresponding to the minimum dGHvalues (dashed

lines) for all the SMs of Model A. The comparable lines have the same color. There are cases that these models are better than RMs in reproducing some

morphological features of NGC 1300.

to the minimum dGHvalue relative to each one of the contour lines

of NGC 1300 we have plotted in Figs. 1,3-5.

In Fig.6(left-hand column) wehaveplotted theminimum dGH

values corresponding to the NGC 1300 contour lines shown in the

right-hand column of the same figure. Filled circles, empty circles

and empty squares (red, blue and green colored lines in the online

version) correspond to Models A, B and C, respectively. Note that

in this figure we have also included the models corresponding to

Ωpvalues not plotted in Figs. 3-5. Based on this criterion, the basic

trends that are concluded by observing the behavior of dGHfor all

the Models (A, B, C) and for the studied range of Ωpvalues are:

• the lower Ωpvalues in Model A are more efficient (than the

higher ones) in portraying the outer contours corresponding mainly

to the spiral structure of NGC 1300 (see the two first rows of Fig. 6)

• the higher Ωpvalues in Model A are more efficient (than the

lower ones) in portraying the inner contours corresponding to the

barred structure of NGC 1300 (see third to fifth rows of Fig. 6)

• Model B is better than Model A in portraying the spiral struc-

ture in high Ωpvalues, while for low Ωp’s, Model A is in general

better than B, especially at the outermost isocontour (first row of

Fig. 6).

• Model C is in general worst than Models A and B in portray-

ing the spiral and bar structure, while it is definitely superior in

portraying the very inner area especially for the middle and high

Ωpvalues (see last row of Fig. 6).

Nevertheless, we should have in mind, that the quantification of the

similarity of the contours of the model with those of the galactic

image by means of dGH, cannot in general account for small mor-

phological differences, essentially in the phases, of the two curves.

Despite this drawback dGHis a reliable index in most cases encoun-

tered in the present study.

All the above results are related with the response models and

depend toalargeextent ontheirinitialsetup. Thesemodelsprovide

useful information about the underlying dynamics and the morpho-

logical features they can support. Despite the fact that the setting

up of the models is done in an unbiased way, there is no physical

reason to exclude weighting some energies more than others in an

effort to improve the similarity between RMs and galaxy morphol-

ogy. In order to explore this we modify our models by introducing

weights attached to each energy level. The energies of the particles

aredetermined by giving them circular velocities intheaxisymmet-

ric potential. This determines where they will be end up in the well

Page 9

NGC1300 response models

9

Figure 8. Similar to Fig. 7 but for the potential corresponding to the cylindrical geometry (Model B).

of the full potential. However, one can consider a totally different

energy distribution of the particles of the model. The procedure and

the corresponding results are described in the following section.

4 WEIGHTING ENERGY LEVELS

Schwarzschild (1979) in a pioneer study, presented a method

for constructing time independent

of specific models. His method has been widely used since

then (Richstone 1980, 1982; Schwarzschild 1982; Richstone

1984; Richstone & Tremaine

Merritt & Fridman1996; Cretton et al.

2009) mostly for models representing elliptical galaxies. The

basic idea of this method is very simple: One considers a density

distribution function and then finds the associated potential via the

Poisson equation. The next step is to calculate a library of orbits

corresponding to this potential for the desired energy range. By

assigning a weight to each orbit one tries with the superposition

of the weighted orbits to reproduce the initially imposed density.

For models of normal spiral galaxies, a similar method has been

used in the past by Patsis et al. (1991) for the investigation of the

self-consistentsolutions

1984; Schwarzschild

1999;

1993;

Antonini et al.

self-consistency in 12 galaxies, while in barred-spiral systems has

been applied by Kaufmann & Contopoulos (1996).

In practice what one is looking for is a way to minimize the

difference between the imposed density and that corresponding to

the superposed orbits. This detailed task is very demanding and is

beyond the scope of this paper. Here we will construct our “best”

models acting in a more rough way. We follow the rationale of

Schwarzschild’s method but we assign weights to the various en-

ergy levels. The density distribution for each energy level is con-

sidered being the one we get from the corresponding RM. This as-

sumption is true only for phase space domains where Chaos dom-

inates and for time scales for which the corresponding initial con-

ditions have been evolved and are close to a dynamical equilibrium

due to chaotic mixing. For phase space domains (of specific en-

ergy levels) with significant fractions of regular orbits the above

assumption is just a simplification since the distribution of the par-

ticles on the existing tori can be supposed to be different than that

corresponding to the RMs.

The determination of the weights is made by the following

procedure. Let Nebe the number of Jacobi constant (energy) inter-

vals, which we give to the particles of a response model and Ncthe

number of grid cells we divide the plane (x,y). In this situation we

derive the weights wi,(i = 1,...,Ne, w(i) ? 0), that minimize the

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C. Kalapotharakos et al.

Figure 9. Similar to Fig. 7 but for the potential corresponding to the (spheroidal + cylindrical) geometry (Model C).

quantity

Nc

?

j=1

Ne

?

i=1

wi

meij

mj

− 1

2

(4)

where meijis the mass contribution of the i−th energy interval on

the j−th grid cell and mjthe mass contribution we consider corre-

sponding to the same region at the NGC 1300 image on the j−th

grid cell. We consider the ratio meij/mjin order to treat equally

high and low density regions. The final considered surface density

distribution is given by

dj=

?Ne

i=1wimeij

Sj

,

j = 1,..., Nc

(5)

where dj, Sjare the density and the surface area of the j−th grid

cell, respectively. In the present study, we used Ne= 30 and Nc=

24 × 24 = 576 for the central 576 kpc2area.

Thereupon, we repeat the procedure of the determination of

the contour lines of each of these “Schwarzschild type” models

(hereafter SM) corresponding to the minimum dGHvalue relative

to each contour line of NGC 1300 plotted in Figs. 1 and 3-5.

Going back to Fig. 6, in the middle column, we plot the min-

imum dGH values, similar to the panels in the left-hand column

(RMs), but for the density distribution corresponding to Eq. (5)

(SMs). The comparison between the RMs and SMs behavior of

dGH, shows the following:

• In 2D geometry (Model A, filled circles or red color line in the

online version) the SMs with the high Ωpvalues are more efficient

(than the corresponding RMs) in portraying the spiral structure of

NGC 1300 (panels in the two first rows in Fig. 6).

• In cylindrical geometry (Model B, empty circles or blue color

line in the online version) the SMs with the low Ωp values are

more efficient (than the corresponding RMs) in portraying the bar

of NGC 1300 (third to sixth rows of Fig. 6).

• In spheroidal+cylindrical geometry (Model C, empty squares

or green color line in the online version) the SMs with the lowest

Ωpvalues (Ωp < 16 km sec−1kpc−1) are more efficient (than the

corresponding RMs) in portraying the NGC 1300 bar (third to sixth

rows of Fig. 6).

In Figs. 7-9 we plot the isocontours (dashed lines) corre-

sponding to the minimum dGH with respect to the isocontours of

NGC 1300 (solid lines), for a sample of the SMs of the Models A,

B and C, respectively. These samples include characteristic cases

such asthose withΩp= 16 kms−1kpc−1and Ωp= 22 km s−1kpc−1

. Note that the corresponding isocontour curves (of model and

Page 11

NGC1300 response models

11

Figure 10. The “global” dGHindex for all the six contour lines presented in Fig. 6. The left panel corresponds to RMs, while the right one to SMs. These dGH

values represent globally a model. The plotting styles are as in Fig. 6.

galaxy) are plotted in the same color. From these figures, it be-

comes evident the nice behavior of the dGH index in quantifying

the comparison between the models and the image of the galaxy.

In almost all cases low (or high) dGHvalues imply good (or bad)

resemblance between the corresponding curves.

Since the dGHvalues for the various isocontours are by defini-

tion comparable, we can add all the values of each model in order

to get a unique value describing globally the model. Such an in-

dex puts of course equally weights to all isocontours and it could

be low valued in models with no matching curves. Nevertheless,

it is reliable in cases with many model curves showing moderate

resemblance with those of the galactic image. In Fig. 10 we plot,

for all RMs and SMs, this “global” dGHindex for the isocontours

of Fig. 6.

The general conclusions we draw by observing Fig. 10 can be

summarized as follows:

• For Ωp< 20 km s−1kpc−1values, the RM (left-hand panel of

Fig. 10) with the lowest global dGHindex are those corresponding

to Model A.

• The RMs corresponding to Model B and C improve their be-

haviour from low Ωpvalues towards high Ωpvalues (for Model C

only up to Ωp= 24 km s−1kpc−1.)

• The overall behavior of SMs corresponding to Model A are

improved (relative to those of the corresponding RMs) only for Ωp

values around Ωp= 22 km s−1kpc−1.

• The overall behavior of SMs corresponding to Model B are

improved (relative to those of the corresponding RMs) for Ωp ?

23 km s−1kpc−1.

• The overall behavior of SMs corresponding to Model C are

improved (relative to those of the corresponding RMs) only for

Ωp? 14 km s−1kpc−1.

5DISCUSSION AND CONCLUSIONS

We studied the stellar response, under the assumption of a single

pattern speed, for the potential models of the barred-spiral galaxy

NGC 1300 estimated in Paper I and for a wide range of Ωpvalues.

The three potential models of Paper I, correspond to three different

geometries as regards the mass distribution in the third dimension

(perpendicularly to the equatorial plane). Our goal was to find out,

which geometries and which values of Ωpwere able to reproduce

the fundamental morphological features of NGC 1300, i.e. the bar

and the spirals. We used the method of response models (RM), as

in Patsis (2006). So, we start with a uniformly populated disk of

stars moving initially in circular orbits, which are determined in

the axisymmetric part of the potential. We integrate the orbits of

the test particles for many pattern rotations under the full potential

and we get the density maps of the response models as described in

Sect. 2.

The “by eye” comparison between the RM morphologies and

the image of the galaxy gives useful qualitative information but

it lacks of objectivity and quantification. For this reason we in-

troduced a new index which is a generalized modification of the

Hausdorff distance. This index quantifies the resemblance of two

isocontour lines and enables the determination of the best matching

contour lines (of a model) to each one of the six preselected isocon-

tours of the deprojected K-band image of NGC 1300 (Fig. 1). This

new index testifies the general remarks made by eye and further-

more it describes the quality of the fitting for the individual com-

ponents of the galaxy. Moreover, we investigated possible improve-

ments of the response models as regards their resemblance with the

NGC 1300 morphology, by assigning weights to the contributions

of the various energy levels. This method follows the general con-

cept of Schwarzschild’s methods, but it is simpler in the sense that

it requires weights for the energy levels and not for individual or-

bits. Some of these modified Schwarzschild models (SM) improve

the similarity of specific features of NGC 1300 in comparison to

the original response models.

Below we enumerate our conclusions:

(1) The spiral structure of NGC 1300 is reproduced:

• Best by the RM corresponding to the potential of the pure

2D geometry and for Ωpvalues around 16 km s−1kpc−1. The big

advantage of thismodel is that the spirals of RM and galaxy have

similar pitch angles. The density enhancement of the the RM at

the spiral region is along the same direction as the corresponding

arms of NGC 1300.

• Acceptable by the SM corresponding to the potential of the

pure 2D geometry and for Ωpvalues around 22 km s−1kpc−1.

However, the drawback of this case is that the spirals, especially

the one appearing at the lower part of the figures, have different

Page 12

12

C. Kalapotharakos et al.

pitch angles than that of the galaxy. The spiral arms of the model

share common regions with the NGC 1300 arms but essentially

cross each other. They also add additional features beyond the

region within which the NGC 1300 barred-spiral structure ex-

tends.

• Moderately by the RM and SM corresponding to the po-

tential of the 3D cylindrical geometry and for Ωpvalues around

22 km s−1kpc−1. However, the remarks about the pitch angles of

the spirals we mention above for the 2D case, yield for the thick

disc 3D case as well. On top of that the upper arm is not nicely

reproduced by this model.

(2) The ansae-type bar of NGC 1300 is reproduced:

• Best by the RM and SM corresponding to the potential of

the pure 2D geometry, for Ωpvalues around 22 km s−1kpc−1

(very good is also the dGH index of the RM corresponding to

the potential of the 3D cylindrical geometry again for Ωparound

22 km s−1kpc−1. However, the ansae are reproduced in the cor-

rect regions only in the 2D case)

• Very well by the SM corresponding to the potential

of the 3D cylindrical geometry and for Ωp values around

17 km s−1kpc−1.

• Moderately by the SM corresponding to the potential

of the 3D cylindrical geometry and for Ωp values around

23 km s−1kpc−1.

(3) The inner oval-like shape structure of NGC 1300 is repro-

duced:

• Best by the RM and SM corresponding to the potential of

the 3D spheroidal+cylindrical geometry and for almost all Ωp

values

• Well by the SM corresponding to the potential of the 3D

cylindrical geometry and for Ωpvalues around 22 km s−1kpc−1.

• Moderately by the SM corresponding to the potential of the

pure 2D geometry and for Ωpvalues around 23 km s−1kpc−1.

All the above indicate the tendencies of the models to improve

their similarity with the K-band image of the galaxy for a given ge-

ometry or for a certain Ωprange. However, the relation between

“good” models and the NGC 1300 morphology is not unambigu-

ous. It is obvious that some geometries support better certain mor-

phological features. This may suggest the appropriate geometry for

the description of specific regions of the galaxy (e.g. the central

area should be considered as a spheroidal). Another issue is that

a morphological feature may be reproduced equally well by mod-

els that differ in their pattern speeds. In addition we remind our

assumption of the time-independency of the potential and a con-

stant time-independent Ωp. The study of the stellar dynamics of

our system under these assumptions not only helps us see some ba-

sic trends in the dynamical behaviour that favours the formation of

the one or the other morphological feature, but after all it will in-

dicate the necessity of different assumptions. The evolution of the

models under a time-dependent potential and Ωpis very dubious in

the response model approach, since it requires specific assumptions

for the evolution laws for both potential and Ωp. The assumptions

are free and therefore the number of the possible combinations is

large. Such a study is beyond the scope of this paper and should be

attacked by means of N-body simulations. However, in order to get

a crude impression about how different can be the resulting mor-

phologies if a basic parameter varies with time, we test a scenario

with time dependent Ωpin a fixed potential. We have chosen for

our calculations a pure 2D geometry. The choice is justified by the

fact that the RM corresponding to the pure 2D geometry portrays

on the one hand the spiral structure for Ωp≈ 16 km s−1kpc−1and

on the other hand the ansae-type bar for Ωp ≈ 22 km s−1kpc−1.

We adopted a rather arbitrary evolution law for the increase of Ωp,

which reads

Ωp(t) = 19 + 3tanh

?1

5(t − 15)

?

km s−1kpc−1,(6)

where the time t is measured in periods corresponding to a pat-

tern rotating with Ωp = 23km s−1kpc−1. The integration time

of the orbits was up to t = 30. In this numerical experiment the

initial value of Ωp = 16 km s−1kpc−1, while the final one is

≈ 22 km s−1kpc−1. In the left-hand columns of Figs. 6 and 10

we have plotted (with star symbol or light blue lines in the on-

line version) the dGHvalues corresponding to each snapshot’s Ωp

value. In these figures we observe that for the time corresponding

to Ωp≈ 17 km s−1kpc−1and Ωp≈ 22 km s−1kpc−1we get a good

resemblance with many isocontours of NGC 1300. This could be

an indication that the present NGC 1300 structure is just a snapshot

in an evolutionary scenario, according to which the observed mor-

phology undergoes successive transient phases. However, whatever

good results we find, we find for increasing Ωpfrom one value to

the other. The opposite procedure, i.e. decreasing Ωpaccording to

the law in Eq. 6 does not lead to equally nice results. Decrease of

Ωpduring an N-body simulation is frequently observed due to dy-

namical friction (Chandrasekhar 1943; Donner & Sundelius 1993;

Debattista & Sellwood 1998), but there is no obvious physical rea-

son for the opposite. In our model the particles that participate in

the spiral structure stay for longer times at the outer regions of the

disc, i.e in regions that evolve dynamically slower than particles

at the central region that contribute to the bar structure. So, by in-

creasing Ωp, the particles that stay longer at the outer disc do not

have enough time to feel the potential. On the other hand particles

staying at the central region of the system respond fast during the

period our simulation reaches the fast Ωpdomain.

Concluding, we remark that taking into account all the models

we constructed as well as the techniques we used to assess them,

there are clearly two Ωpintervals, where we find morphological

features of the models resembling the structures of NGC 1300.

These are Ωpvalues close to 16 and 22 km s−1kpc−1.

Comparing this result with pattern speeds proposed by

other authors for NGC 1300 (England 1989; Lindblad & Kristen

1996), there is a relative agreement between the fast model

of Lindblad & Kristen (1996) and the group of models around

Ωp=22 km s−1kpc−1, which we find matching the NGC 1300

bar (Lindblad & Kristen (1996) propose Ωp=20 km s−1kpc−1for

a distance of the galaxy D=20 Mpc). The alternative solution pro-

posed by the same authors (Ωp=12 km s−1kpc−1) is in a range

of Ωpvalues, where the bars of our RMs are much thicker than

the one of NGC 1300 and the stellar spirals are absent. As re-

gards the work of England (1989), the proposed pattern speed gives

equilibrium points in a narrow zone compatible again with the

Ωp=22 km s−1kpc−1values, as it will become evident in Paper

III.

In any case, the understanding of the observed NGC 1300

morphology goes through the understanding of the underlying dy-

namical mechanisms and this can be done only by studying the

orbital behavior of the “successful” models. This will allow us also

the comparison among them, and is done in Paper III in this series.

Page 13

NGC1300 response models

13

ACKNOWLEDGMENTS

We thank Prof. G. Contopoulos and Dr. Ch. Efthymiopoulos for

fruitful discussions. P.A.P thanks ESO for a two-months stay in

Garching as visitor, where part of this work has been completed.

This work has been partially supported by the Research Committee

of the Academy of Athens through the project 200/739.

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