NGC 1300 Dynamics:
I. The gravitational
potential as a tool for detailed stellar dynamics
In a series of papers we study the stellar dynamics of the grand design barred-spiral galaxy NGC 1300. In the first paper of this series we estimate the gravitational potential and we give it in a form suitable to be used in dynamical studies. The estimation is done directly from near-infrared observations. Since the 3D distribution of the luminous matter is unknown, we construct three different general models for the potential corresponding to three different assumptions for the geometry of the system, representing limiting cases. A pure 2D disc, a cylindrical geometry (thick disc) and a third case, where a spherical geometry is assumed to apply for the major part of the bar. For the potential of the disc component on the galactic plane a Fourier decomposition method is used, that allows us to express it as a sum of trigonometric terms. Both even and odd components are considered, so that the estimated potential accounts also for the observed asymmetries in the morphology. For the amplitudes of the trigonometric terms a smoothed cubic interpolation scheme is used. The total potential in each model may include two additional terms (Plummer spheres) representing a central mass concentration and a dark halo component, respectively. In all examined models, the relative force perturbation points to a strongly nonlinear gravitational field, which ranges from 0.45 to 0.8 of the axisymmetric background with the pure 2D being the most nonlinear one. The force perturbation in each model is found being robust to small changes of the required parameter values. We present the topological distributions of the stable and unstable Lagrangian points as a function of the pattern speed . The topological distribution found deviates in several cases from the classical paradigm with two stable Lagrangian points at the sides of the bar and two unstable ones close to the ends of the bar. In all three models there is a range of values, where we find multiple stationary points whose stability affects the overall dynamics of the system.
keywords:Galaxies: kinematics and dynamics – Galaxies: spiral – Galaxies: structure
Grand design galaxies are attractive objects for dynamical studies, because it is believed that behind their coherent global features are hidden the basic dynamical mechanisms that shape the morphology of galactic discs. Then, deviations from the grand design could be considered to reflect additional complications of this basic dynamical behavior. The result would be more complicated morphologies, like those encountered in multi-armed or flocculent spiral galaxies. The motivation of this study is to investigate the dynamics of NGC 1300, which is a well known barred-spiral grand design galaxy with a prominent bar. Its archetypical morphology promises a deeper understanding of the dynamics of a class of similar objects.
Barred-spiral systems offer many examples of grand design morphology, being characterized by a set of two spiral arms starting close to the ends of the bar. In some cases the spirals seem to emerge out of the bar as its continuation, while in others the beginning of the spirals is clearly displaced from the bar’s ends (e.g. compare the spirals of NGC 4535 and NGC 4548 in figure 5 in Grosbøl, 2008). Nevertheless the morphology of the arms in many of these barred-spiral systems is not as smooth as that of the arms of normal spiral galaxies (Grosbøl & Patsis, 1998), especially in near-infrared wavelengths. This can be characteristically seen by overplotting the component of the surface brightness on images of galaxies, as do e.g. Seigar et al. (2005). One can observe that their arms are in general asymmetric and have gaps. It is also clear that in many cases the surface brightness of the arms is larger close to the ends of the bar than at azimuths away from them. Characteristic cases are NGC 4314 and NGC 4665 (Gaddoti, 2008). In the case of NGC 4314, in near-infrared wavelengths, the arms hardly complete azimuthally a angle (Quillen et al., 1994).
In the last years the dynamics of barred-spiral systems have been revisited by several research groups. The reason is the investigation of the possibility that the stars on the spirals are in chaotic motion (see also articles by the same authors in Contopoulos & Patsis, 2008, and references therein). The motivation for this series of papers is exactly to test this hypothesis in one of the well known and well studied galaxies of this type, NGC 1300. The first step is to estimate its potential directly from observations (de Vaucouleurs et al., 1991). Since it is a well studied object, one can find observations from X-rays to radio wavelengths in order to understand the detailed morphology (e.g. Fabbiano et al., 1988; England, 1989). The composite color image STScI-PRC2005-01 (http://heritage.stsci.edu/2005/01/supplemental.html), highlights the characteristic asymmetry between the two sides of the galaxy.
Images in optical wavelengths, as well as BI color maps that cover an area of radius from the centre of the galaxy, reveal faint extensions of the main bisymmetric spiral structure (e.g. Elmegreen et al., 1996). These extensions, more clearly observed in bluer bands, consist of Population I objects.
In Fig. 1a we present the K band image of the galaxy, that we used in the present study. Images at near-infrared wavelengths depict the morphology of the old disc stellar population and are appropriate for dynamical studies (see Sect. 2 below). The axes labels are in kpc, assuming a distance to the galaxy Mpc. At this distance 1″kpc.
Our ultimate goal is to present dynamical mechanisms for the stellar component that lead to the development of the particular morphological features of the galaxy NGC 1300. The estimation of a suitable potential is the basis for all dynamical studies and this is done in the present paper, which is structured as follows: In Section 2, we present the observations we performed in order to obtain a reliable estimation of the potential of the galaxy. In Section 3 we describe the image processing techniques that allowed us to create images that will be used to compare our models with. The estimation of the potential is given in Section 4. The forcing in the models is presented in Section 5, while the properties of the corresponding effective potentials in Section 6. Finally we discuss our conclusions in Section 7.
The distribution of luminous matter in a galaxy is indicated by its surface brightness map. The translation from luminosity to mass, through a mass-to-light ratio , may vary significant depending on both the spectral band used and the underlying stellar population. The near-infrared K band at 2.2 is close to the emission peak of the old stellar disc and bulge populations which represent the major visual mass constituency of a spiral galaxy (Rix & Rieke, 1993). Although very young stars and red super giants also contribute to the K band flux, it remains a good indicator of the mass distribution of luminous matter outside star forming regions.
The K-map of NGC 1300 was observed on 2002-09-01 with SOFI at the 3.5 m NTT telescope, ESO La Silla. The total exposure time on target was 10 min in the K filter, at 2.162, with equal time for sky fields. A jitter pattern with shifts was used to allow removal of bad pixels through stacking, while sky frames were interleaved with offsets around from the galaxy. The SOFI instrument had a Rowckwell Hg:Cd:Te 10241024 detector with pixels on the sky. The field covered the main spiral structure of NGC 1300 but did not provide an accurate estimate of the sky background level. The final K band frame had a seeing of 0.8″and reached a surface brightness of K = 20.8 mag arcsec at a signal-to-noise (S/N) level of 3. Thus, the quality of our dedicated SOFI/NTT observations can be considered as an additional advantage for potential calculations, besides the known advantages of the K-band for studies related with the mass distribution in galaxies. For comparison, the 2MASS survey offers a seeing of 2-3″and the corresponding images reach 20 mag arcsec at a S/N level of 1 (see pages under http://www.ipac.caltech.edu/2mass/). The reduction of the SOFI data followed the procedure described by Grosbøl et al. (2004).
The sky projection parameters were adopted from Lindblad et al. (1997) who estimated the Position (PA) and Inclination Angle (IA) by fitting tilted ring models to their HI velocity data. The parameters values they found were (PA, IA) = . These values are different from those one finds by fitting an exponential disc to the regions outside the main bar in the K band image, as in Grosbøl et al. (2004), which are (PA, IA)=(106.6). This values are close to those found by Laurikainen & Salo (2002). However, since for barred spirals like NGC 1300, any determination of the sky projection based on surface photometry may be strongly biased due to the strong mode and the open spiral structure, we preferred to use for our study the values proposed by Lindblad et al. (1997). A disadvantage of the use of the near-infrared image is that one can hardly detect the axisymmetric disc in the outer regions. We have to note though, that for the response models and the orbital analysis, we investigated also the effect of the deprojection parameters in the potential, and the results are presented in the forthcoming papers of this series.
The deprojected with parameters (PA, IA) = image of NGC 1300, is shown in Fig. 1b and depicts the morphology we will try to model. The size of this image is 531531 since it has been re-binned to 0.5″pixel size. The standard background subtraction for the K-band frame could be slightly in error since it was based on separate sky exposures and not on the frame itself which was almost fully occupied by the galaxy.
3 Data reduction
We adopted as distance of the galaxy the value , according to the cosmology-corrected luminosity distance given in the NASA/IPAC Extragalactic Database (NED). This value may differ from the distance used in other papers on the dynamics of NGC 1300. In any case, the exact distance of a galaxy does not influence essentially the dynamical phenomena that shape its morphology (Patsis et al., 1991).
Before the K-band frame can be used to bootstrap the estimate of the potential, two possible corrections should be considered namely the population effect on the ratio and the exact background level. Although the major fraction of flux in the K-band originates from the old stellar, disc population (Rix & Rieke, 1993), young stars in the arm regions will bias the effective . The young stellar population may be present both as a diffuse, unresolved, component and as resolved young stellar complexes seen in the arms as bright knots (Patsis et al., 2001; Grosbøl & Dottori, 2008). In order to reduce the bias from young stars, we removed such bright knots by a digital filter. A correction that has to be done concerns the brightness of the outer parts of the galactic disc. This area is biased towards higher values due to the presence of young stellar clusters and stellar complexes, still present even in K band images (Patsis et al., 2001; Grosbøl & Dottori, 2008). For this purpose, in Fig. 1b we firstly removed the bright spots in the region with a radius . Bright spots, assumed to be young clusters, do not cover areas larger than pixels. We substituted the intensities of all extreme high (but also a few low) valued pixels with the median value of the corresponding pixels area centred at the very pixel under consideration. In Fig. 2a only the identified as high valued pixels are shown, while in Fig. 2b we see the rectified image of the galaxy after the removal of the bright spots. After these reductions the galaxy image seems quite smooth in the area of high valued pixels (Fig. 2b) but still presents significant noise in the area of low valued pixels. In Fig. 2c we see how noisy is the contour corresponding to the brightness value . In order to smooth out the image, we apply a low-pass smoothing filter which affects only the low valued pixels () getting a less noised image (Fig. 2d).
In Fig. 2d we indicate four radial directions A, A, B and B, centred at the axes origin, along which we plot the surface brightness after applying the smoothing filter. We have chosen the directions along the minor axis (A, A) which reveal better the exponential disc. The directions (B, B) are along the diagonal of the image nearest to the minor axis and provide data for longer distance. The surface brightness along these directions is given in Fig. 3a. It is obvious that no exponential profile is observed at the outer parts of the disc (beyond the region with the bar and spiral structures) as expected. This implies that the adopted zero value level for the background is not correct. Therefore, we decrease the zero value level by the smallest quantity so that the surface brightness profiles become exponential, i.e.
as shown in Fig. 3b (actually we increase the brightness values of all pixels by the same quantity). We consider the region with the spiral arms part of the exponential disc. We note that the applied correction is of the order of the background uncertainty. The corresponding scale length we find in that case is . By increasing the brightness by a smaller value than the one we finally applied, we do not get a well defined exponential profile. The value of 10 kpc we find is in agreement with the one given by Prieto et al. (2001) corresponding to observations of NGC 1300 in the I band. We note that Laurikainen & Salo (2002) and Laurikainen et al. (2004) find values close to 7 kpc in their 2MASS and H-band images respectively. The effect of such differences on the nonlinearity of models for the potential is investigated below (Sect. 5).
4 The estimation of the potential
After all the above amendments we result in an estimation for the surface density distribution of the luminous component in the reduced image (Fig. 2b), under the assumption of a constant mass-to-light ratio. In our approach we reconstruct the galactic potential assuming the maximum contribution of the luminous component to the observed rotation curve as given in Lindblad et al. (1997). For the cases that parts of the rotation curve will not be able to be entirely assigned to the luminous component we switch on two additional components in our modelling. They represent a dark halo and/or a central mass concentration. Since our goal is the study of the dynamics of the stars, treated as test particles moving in the estimated gravitational field, the total potential must be compatible with the rotation curve of the galaxy, in every model we study.
In order to proceed with the potential estimation of the luminous component we need to do an assumption about the distribution of the matter in the third dimension. Below, we study three different cases. The basic difference among these three general models is their geometry. We want to compare the marginal thin geometry of the 2D case (model A) with a thick disc case (model B) in order to see if the comparison between galaxy and models improves by varying our parameters towards the one or the other direction in the parameter space. Model C changes essentially the geometry of the bar from cylindrical to a combination of spherical with cylindrical and should be again considered as a limiting case. More specifically we consider:
- Model A.
The ‘2D’ or ‘degenerate’ case in which all matter is considered lying on the () galactic plane.
- Model B.
The ‘disc’ case in which we assume a cylindrical geometry and that all matter is distributed in a 3D disc with scale height . Then the vertical dependence of the density reads (van der Kruit, 1988)
where is the vertical scale height and is an index (see Section 4.2, for its meaning).
- Model C.
The ‘spheroidal’ case in which we consider that the total observed brightness is the combination of two major components: i) a spheroidal component that accounts for the major part of the bar and ii) a 3D exponential disc with scale height as in model B. The spheroidal component includes the bulge (kpc, Prieto et al., 2001; Laurikainen & Salo, 2002; Laurikainen et al., 2004) and the axisymmetric contribution of the central part of the bar.
In Cartesian coordinates for a given spatial density the gravitational potential is
where is the gravitational constant. This means that is the convolution of the functions and . Thus the potential is written as , where denotes the convolution operator.
4.1 Model A
In model A, where all mass is assumed on the plane, the density can be written as , where is the surface density and is the delta function. Substituting the above density expression into (3) and integrating it over we get
which is the convolution of the functions and . Knowing the surface density at each grid point we can calculate the potential at the same grid points by using the convolution theorem
where and , denote the Fourier transform and inverse Fourier transform respectively. We use an extended square area 16 times larger than the original area of the image of the galaxy in order to avoid the contribution from periodic reflections of the signal from the galaxy’s pixels. Such an extended area will allow us to consider, in dynamical studies, stars that travel to large distances away from the centre of the galaxy. The surface density in the extended area is considered to be zero. We use the 2D fast Fourier transform (FFT) technique for the corresponding computations. Note, that in this calculation the adopted value is equal to since the softening length is of the order of the bin size. A detailed description of the whole procedure can be found in Hohl & Hockney (1969).
Given the potential values at the grid points one can calculate the potential everywhere in between these points using a 2D interpolation scheme. However, such a scheme, besides its complexity, provides the potential taking into account non-smooth, local density disturbances. For this reason we proceed as in Quillen et al. (1994) and we decompose the potential into its azimuthal Fourier components at every radius . Under this consideration the potential can be written in polar coordinates as
For the form (6) returns exactly the same values with those we have from the 2D interpolation scheme mentioned above. We cut off the high harmonics and consider all terms (even and odd) up to . Like this we are able to reproduce all the important morphological features of the galaxy (its asymmetries included). Quillen et al. (1994) calculated the amplitudes , for many rings and fitted them by 8th order polynomials. Despite the advantages of simplicity and analyticity of the expressions for each function that the use of polynomials secures, it is not capable of following the actual variations well. For this reason in the present study we apply the interpolation scheme with cubic splines to the , data sets. This scheme guarantees the continuity of the potential, of the forces and of derivatives of the forces, which are used in the variational equations providing indicators that will measure the chaoticity of the orbits (e.g. Lyapunov exponents). This is particular important for stellar dynamical investigations, which is our motivation for the estimation of the NGC 1300 potential. Note that we set in order to secure the continuity required at . Table 2 in Appendix111Appendix is only available in electronic form presents all the amplitude values, at 101 successive radii , used for the derivation of the interpolating splines.
Figure 4 shows the HI rotation curve (gray solid line with error bars together with optical velocity measurements (gray dots) from various slits aligned close to the major axis of the galaxy, as taken from Lindblad et al. (1997). In this figure we have also plotted a rotation curve along the major axis of the bar, corresponding to the potential (6) of the luminous matter only (dotted line). In this case it has been considered being (see Table 1). This curve fails reproducing the flat part of the rotation curve at large distances. This implies that some additional mass component exists besides the luminous component. The most reasonable candidate to be invoked is a dark halo (DH) term. As already mentioned our general model foresees the switching on of two additional components for a central mass concentration (CMC) and a DH. We model both of them with a Plummer sphere since it is simple, effective and widely used in the literature. The importance of this terms are controlled by the parameters of the Plummer spheres. Thus the total potential can then be written
where is the luminous matter component given by (6) and
is the potential for the Plummer spheres, with being a constant parameter. The adopted parameters are given in Table 1 (Model A). In Fig. 5 we see that the DH term is quite important and helps the rotation curve of the model reaching the level of the HI curve of Lindblad et al. (1997). On the other hand the CMC term is less important increasing the central peak by km/sec which eventually reaches the value km/sec. Note that the central peak value is not well defined by the observational data and therefore the adopted parameter values for the CMC term provide only a reasonable area in the parameter space.
4.2 Model B
In case B we assume that the density can be written as
van der Kruit (1988) proposed a family of models regarding the vertical dependence of the density which reads
where the normalization constant is given as
so that . is the Gamma function defined by
Varying from we get varying from
(isothermal disc) to
(exponential disc). de Grijs (1998) and Kregel et al. (2002) provide estimations for the mean values of the parameter and the ratio in samples of galaxies, which are and , respectively (for a review see van der Kruit, 2002). These studies have been based on observations in I and K bands.
where the function is given by
Thus, the potential on the galactic plane is the convolution of the functions and . We can again calculate it applying the convolution theorem , since we know the surface density at every grid point .
We note that Laurikainen & Salo (2002) and Laurikainen et al. (2004) introduced an improved version of the Quillen et al (1994) method, where the image pixels are first smoothed by calculating the azimuthal Fourier decompositions of the surface density in different radial zones in polar coordinates. This approach has some advantages, like a possibility to use more noisy images and to apply radially non-constant vertical models. However, for high quality images and for strongly barred galaxies (like our data for NGC 1300) the differences in the resulting potential should be negligible regardless of the integration method.
Working in the same way as in model A we obtain expressions mathematically similar to (6) for the potential corresponding to the luminous component and to (7) for the total potential. Table 3 in Appendix contains all the values of the amplitudes for calculated for 101 rings at . The specific values for correspond to for the mass of the luminous matter.
Finally, Fig. 5 presents the rotation curve corresponding to model B together with the observational data (Lindblad et al., 1997) as we did in Fig. 4 for the 2D case. The dotted line gives the rotation curve along the major axis of the bar when we take into account only the luminous component. We observe that this rotation curve reaches lower values than the observational one also at the central parts. For this reason we add a CMC term which is times more massive than the one we added in model A, i.e. now it is . We add also a DH term in order to get a better description at the outer parts of the disc. The dashed line gives the rotation curve of the total potential (7) with the corresponding parameter values given in Table 1.
4.3 Model C
In this case we consider that the main part of the bar can be described with a spherical component. This part could be vaguely described as “the bar without its ansae” (Fig. 1b). For this purpose we use the modified Hubble profile (MHP) (see Binney & Tremaine, 2008, p. 68)
The parameter is the scale length and the parameter determines the mass inside a specific radius (e.g. ). Note that the mass inside a radius, in MHP model, diverges logarithmically at large radii and therefore it is meaningless to refer to total mass. The corresponding potential via Poisson equation is
By integrating (15) over we take the expression for the surface density for MHP, which is
In order to find the values of the best fitting parameters of the MHP model we subtract the exponential disc with =10 kpc (see Fig. 3b) from the observational surface density so that the remaining luminosity at the central region ( kpc) is mainly due to the assumed spherical component. In Fig. 6 we plot the azimuthally mean surface brightness of the central area of the galaxy (solid line) after the subtraction of the exponential disc. We also give the curve corresponding to (18) (dashed curve) with the parameter values shown in Table 1. We observe that the MHP gives a reasonable fitting of the inner 3.5-4 kpc.
The total potential corresponding to the luminous mass is the sum of the MHP component (16) and the component corresponding to the remaining mass. The surface density of the latter component is derived by subtracting the surface density from the surface density we derive from the observations. Thus, the surface density (of the remaining disc) at each pixel is given by
where is the mean surface density inside the area of the squared pixel centered at . We consider this component as a 3D exponential disc with vertical distribution given by (10). In order to obtain an expression similar to (6) for the disc in model C we work in the same way as in model B. Table 4 in Appendix gives the values of the amplitudes of of the potential corresponding to the disc component.
Figure 7 shows the data of the observational rotational velocities (Lindblad et al., 1997) as in the two previous models together with the rotation curve of the luminous component (both MHP and disc) along the major axis of the bar represented by the dotted curve. Also in this case the activation of the two additional potential terms, especially the DH one, allows a better relation between the rotation curve of the model along its bars major axis and the inclination of the observed HI curve at large distances. The values of their parameters for model C are also shown in Table 1.
5 The Nonlinearity of the models
As a measure of the nonlinearity of the models we consider the quantity
which is the ratio of the modulus of the vectorial difference of the total force (axisymmetric and non-axisymmetric) from the axisymmetric force, over the modulus of the axisymmetric force. Thus, measures the non-axisymmetric force perturbation, normalized over the axisymmetric one. This parameter describes well the non-linear effect of the non-axisymmetric components but it is unable to determine by itself the global dynamics in a rotating system (see below the discussion about the importance of ).
In the past, the strength of bars and spirals in disc galaxies has been quantified by means of similar indices (e.g. Buta & Block, 2001; Vorobyov, 2006). The index introduced by Buta & Block (2001) has been widely used in large samples of galaxies (Laurikainen & Salo, 2002; Block et al., 2004; Laurikainen et al., 2004). This index is similar to the quantity defined in Eq. (20). The variation of in our models is presented in Fig. 8.
At the left-hand column the panels show in color diagrams the strength of the non-axisymmetric force, measured by , for the three models. Each model is indicated at the right edge of the figure. At the right-hand column we plot the maximum as a function of (black solid curves). We observe that the overall maximum perturbation strength reaches higher values () in model A (pure 2D), while in the other two models it is lower reaching and for models B and C respectively. The maximum values of models B and C are close to those presented by Laurikainen & Salo (2002), while the pure 2D case has a significant larger overall maximum value. In each panel of the right-hand column we have plotted additional maximum curves corresponding to different parameter values, which are indicated in the figure. The green curves give the maximum profiles without the DH component. We observe that the maximum values, with the DH component included, are approximately 0.8-0.85 of the corresponding values neglecting the DH component. Such values are within the range of values given by Buta et al. (2004), although towards the lower limit. We studied also the robustness of the maximum profile to the adopted orientation parameters. We considered two additional orientation parameter sets i.e. (PA, IA) = (dashed magenta curve, Fig. 8b) and (PA, IA) = (dashed blue curve, Fig. 8b). The first set of the orientation parameters is the one obtained from near-infrared data by Grosbøl et al. (2004). The second set has angles deviating from the values we finally used as much as the first one, but to the opposite direction. We observe, that the maximum value, in these cases, is affected by less than 10%. We also checked the dependence of the maximum value on the changes of the zero-value level, which yield to disc scale lengths of 7 kpc (dashed red curve, Fig. 8b) and 13 kpc (dashed-dotted orange curve, Fig. 8b), respectively. Note, that the exponential discs with these scale lengths do not fit the corresponding data as well as in the original =10 kpc case. We observe, that the maximum value, in these cases, is affected by less than 5%.
In Fig. 8d we see that the maximum value, in model B, varies approximately by 10% when the ratio varies from its mean value (see dashed-dotted orange and dotted red curves in Fig. 8d). We also certified, that the profiles almost coincide within variation of the exponent of Eq. (10) from its mean value.
Special attention has been given to the number and kind (even or odd) of the Fourier terms to be included in the potential. The morphological asymmetries observed in the galaxy underline the importance of the inclusion of the odd terms for dynamical studies. The number and the kind of the Fourier terms may affect the maximum value as well as the 2D distribution of the value on the galactic plane. In order to study these effects we constructed value maps on the galactic plane for different combinations of the Fourier terms (Fig. 9). We present the results for model A since it is the model with the highest non-axisymmetric amplitudes in the Fourier decomposition. The upper two panels (a,b) of Fig. 9 include terms up to in Eq. (6), while the two panels at the bottom (c,d) up to . The left panels (a,c) include all terms, while the right ones (b,d) only the even terms, as indicated at the top of each of them. By including all terms, the maximum value for the case we consider up to is 0.96 of the maximum value corresponding to . By including higher order terms , the maximum value does not get further refinement. On the other hand, by neglecting the odd terms, the map clearly fails to reproduce all the asymmetries and the maximum value becomes at least 10% lower than that with the odd terms even for the case. More specifically, the maximum values corresponding to panels (a), (b), (c) of Figure 9 are 0.96, 0.89, 0.84 of the maximum value we get in Fig. 9c. We can observe, that at the sides of the bar the prominent green ”tails”, which essentially describe the force field at the spirals’ region, are conspicuous only in panels (a) and (c). Also at radii around 8 and 9 kpc, at the ends of the bar, only panels (a) and (c) give a rather constant green shade, while at the same radii green color fades out in panel (b), but also in (d). Evident is that in terms of nonlinearity of the potential the odd terms are essential, while including terms does not improve the models at a level worth the additional computing time needed for the calculations. Thus, a reliable combination that can be used in dynamical studies is with all even and odd terms included (Fig. 9a).
In all examined cases the forcing of the models suggests that our systems are strongly nonlinear, since the perturbing term is at least higher than 40% of the axisymmetric background. This is a usual situation in models of strong barred galaxies (Kaufmann & Contopoulos, 1996).
We note that potential evaluations for NGC 1300 have been previously done by England (1989), Hunter (1990), Lindblad & Kristen (1996), Aguerri et al. (2001) and more recently by Laurikainen & Salo (2002) using a 2MASS K-band image and by Laurikainen et al. (2004) by means of OSUBGSG H-band data. These potentials have some features similar with our general model (for a discussion see Sect. 7). In our case, emphasis is given in constructing a potential function suitable for dynamical studies. We kept fixed the quantities that could be derived directly from our near-infrared K-band data, while the parameters that could be only confined within some range have been treated as free parameters. However, the most important parameter for the dynamics of a rotating disc galaxy is the pattern speed (for a review see Contopoulos, 2004), which is an unknown quantity. The most simple assumption is that the system we are studying has a unique pattern speed. We take it as a basic hypothesis, the validity of which will be evaluated from the feedback we will get from response models (to be presented in subsequent papers). At any rate, it is the effective potential that determines the force field in our calculations. The dynamics of two systems with the same potential but with considerably different pattern speeds are considerably different (see e.g. Patsis et al., 1991). An investigation of the basic categories of effective potentials is presented in Sect. 6 below.
6 Effective potentials
The locations of the stationary (stable and unstable) Lagrangian points strongly affect the dynamical properties of a rotating galaxy. The Lagrangian points and their stability structure the phase space at the corotation region. Given a mass distribution, the positions of the Lagrangian points are determined by the value of the pattern speed . In the rotating frame we have the conservation of the Jacobi integral
where is the effective potential
The stationary Lagrangian points at the equilibrium positions are defined by
This is the case of course, where the system rotates with a single pattern speed. In systems with multiple pattern speeds, multiple corotation radii will be defined on the disc. If bar and spirals rotate with different , each component will have its own corotation radius. In what follows we take the single pattern speed case as our initial assumption. Its validity and its limitation will be examined by means of models in subsequent papers.
Fig. 10 shows in a compact way how the number, the location and the stability of the Lagrangian points change on the disc for the various effective potentials. In Fig. 10a we give the positions of the Lagrangian points as a function of for model A. The calculation of the stability of the equilibrium points is done by following the standard procedure that can be found in relevant textbooks (see e.g. Binney & Tremaine, 2008, pp. 179-183). The red and blue points correspond to the unstable and stable Lagrangian points, respectively. At the basis of the cubes in all panels it is plotted the surface density of the galaxy together with the projections of the corresponding Lagrangian points. The projected orange and light blue points correspond to the unstable and stable Lagrangian points, respectively. Figure 10b is similar to Fig.10a and shows the positions of Lagrangian points in model A. We observe that there are ranges of values for which there are more than one points. Figures 10c,d and 10e,f are similar to Fig. 10a,b but for the model B and C, respectively. We see that the distribution of Lagrangian points have a very similar behavior in models B and C. Moreover, the variability of the positions of the Lagrangian points with seems smoother in models B and C. However, even in these cases we have ranges of values corresponding to multiple Lagrangian points (see Fig. 11c).
Figure 11 presents some characteristic cases of effective potentials with multiple Lagrangian points. Stable (unstable) Lagrangian points are plotted with yellow (green) big dots. Fig. 11a shows the isocontours of the effective (total) potential, in model A, for . In this figure we observe 5 Lagrangian points (3 unstable and 2 stable) at the region close to the end of the bar. Figure 11b corresponds to the same model A but for . We observe 3 Lagrangian points (2 stable, 1 unstable). Finally, Fig. 11c corresponds to model C for and shows 3 Lagrangian points (2 unstable, 1 stable).
7 Discussion and Conclusions
In this paper, we propose three different general models for the potential of the grand design barred-spiral galaxy NGC 1300, based on observations in the near-infrared. The ultimate goal of our work is the investigation of the stellar dynamics of this galaxy by means of response models and orbital theory. More precisely we want to reveal the dynamical mechanisms that shape the observed structure of the galaxy. This is done in the forthcoming papers of this series.
Observations in the near-infrared provide the most reliable potential models of disc galaxies, since they are excellent indicators of the distribution of mass associated to stellar matter. The needed additional assumptions refer mainly to the distribution of luminous matter in the third dimension (outside the galactic plane) and to the distribution of the non-luminous matter (dark halo and central mass concentration).
The three proposed models are: The pure 2D case (Model A), where all luminous matter is considered on the galactic plane; the thick disc case (Model B) where all luminous matter is considered in a 3D disc with a constant scale height; and the “spheroidal case” (Model C) in which the major part of the bar is a spherical modified Hubble profile. The remaining luminous matter is attributed to a 3D disc as in Model B. In all these cases the calculation of the potential is performed by means of an FFT transform technique, that has been introduced in various versions by (Hohl & Hockney, 1969; Quillen et al., 1994; Laurikainen & Salo, 2002).
These three general models can be considered as limiting cases in the parameter space of models whose responses and orbital content will be studied. Let us consider an unknown parameter, e.g. the disc thickness. If we find that we obtain better feedback in Model A than in Model B, we will conclude that we have more accurate dynamics closer to the thin disc limit. Only the study of the dynamics in a large number of effective potentials will show which configuration compares best with the observed morphology.
As we have seen in Section 5, all models are strongly nonlinear. The relative force perturbation (regarding the axisymmetric background) in all models is high reaching to 45%-50% in Models B and C and to even higher value 80% in Model A. We note that for normal (non-barred) spiral galaxies the models that reproduce successfully the morphology of this type of galaxies have maximum force perturbations typically of the order of 5-10% (Patsis et al., 1991; Laurikainen & Salo, 2002; Laurikainen et al., 2004; Vorobyov, 2006) and they are already characterized by nonlinear effects. Nonlinear effects are even stronger in larger perturbation characteristic of bars (see Kaufmann & Contopoulos, 1996; Laurikainen & Salo, 2002; Laurikainen et al., 2004; Block et al., 2004, and others). Thus, it is expected that all models will present a strongly nonlinear behavior with extended chaotic regions. Note that we certified the robustness of the force perturbation in each model to small changes of the various required parameter values (e.g. thickness of the disc).
The nonlinearity of a model, expressed as the relative force perturbation, is an important parameter for studying its dynamics, since it refers to the total distribution of matter (luminous and dark) in the galaxy. However, the estimation of the potential of a disc galaxy alone does not determine its dynamics. The main, and most important, parameter is the pattern speed and this cannot be unambiguously determined from observations. The Coriolis forces totally alter the landscape of the gravitational field, as we can see in the various effective potentials. A potential by itself can tell us whether or not nonlinear phenomena can play an important role in the dynamics of the system. However, for the same potential we can have totally different dynamics. By varying the pattern speed the equilibrium (Lagrangian) points are shifted and may alter their stability character. In our analysis we find cases with multiple Lagrangian points. This indicates that in real galaxies the existence of multiple Lagrangian points can be a common phenomenon. Our models give the opportunity to study the dynamics of such realistic systems, something that has not been done yet. The effect of the variation of the pattern speed on the dynamics of NGC 1300 will be presented in the forthcoming papers of this series.
We thank Prof. G. Contopoulos 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. We would also like to thank the anonymous referee for constructive comments that helped the clarity and the presentation of our paper.
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