Flaring stellar disk in low surface brightness galaxy UGC 7321

Flaring stellar disk in low surface brightness galaxy UGC 7321

S. Sarkar Department of Physics, Indian Institute of Science, Bangalore 560012, India

11email: cjjog@iisc.ac.in suchira@iisc.ac.in
   C.J. Jog Department of Physics, Indian Institute of Science, Bangalore 560012, India

11email: cjjog@iisc.ac.in suchira@iisc.ac.in
Key Words.:
galaxies:halos - galaxies: ISM -galaxies:individual:UGC 7321- galaxies: kinematics and dynamics - galaxies:spiral - galaxies: structure
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We study the vertical structure of an edge-on low surface brightness galaxy UGC 7321 theoretically, which is one of the few well-observed LSBs. We model it as a gravitationally coupled disk system of stars and atomic hydrogen gas in the potential of the dark matter halo and treat the realistic case where the rotation velocity varies with radius while solving for the vertical disk structure. We calculate the thickness of stellar and HI disks in terms of half-width at half-maximum of vertical density distribution in a region of R=0 to 12 kpc using input parameters as constrained by observations. We obtain a mildly increasing disk thickness up to R=6 kpc, in a fairly good agreement with observations, and predict a strong flaring beyond that. To obtain this trend in thickness of the stellar disk, the velocity dispersion of stars should fall exponentially at a rate of , whereas the typically assumed value of gives decreasing thickness with radius. We also show that a compact and dense halo as implied by the observed rotation curve is needed to explain this trend. Interestingly both the stellar and HI disks show flaring at outer disk region despite being dynamically dominated by the dark matter halo from the very inner radii. The resulting vertical stellar density distribution cannot be fitted by a single function in agreement with observations. Hence invoking a double disk model to explain the vertical structure of LSBs as done in the literature may not be necessary.

1 Introduction

Low Surface Brightness (LSB) galaxies form a special class of galaxies that lie at the faint end of the galaxy luminosity function. These have a central surface brightness fainter than in the optical B-band, around times fainter than high surface brightness spirals. Their absolute magnitude lies between -17 to -19 which indicates the presence of a disk with low surface density (de Blok & McGaugh 1996; de Blok et al. 2001). Structurally and hence dynamically they are different from the high surface brightness (HSB) galaxies in a number of ways. Despite being rich in HI gas content they have a low star formation rate, have low metallicities and thus are under-evolved (Bothun et al. 1997). Most importantly, LSBs are dark matter dominated at all radii(Bothun et al. 1997; de Blok & McGaugh 1997; de Blok et al. 2001) compared to the HSB galaxies where dark matter halo becomes dominant only in the outer disk. Within the optical disk, almost 90 percent of the total galaxy mass is in dark matter, whereas in high surface brightness (HSB) galaxies the baryons and dark matter have a comparable mass at these radii (de Blok et al. 2001; Jog 2012). Dark matter dominance has also been shown to be the reason for lack of star formation and absence of strong spiral features in LSBs (Jog 2014;Ghosh & Jog 2014). All these properties make the dynamics and evolution of the low surface brightness galaxies distinct from that of the others.

Observationally it is challenging to study LSBs as even the brightness at their center is fainter than the night-sky background. Nevertheless in the last two decades, with the advent of wide-field surveys, observations are able to detect an increasing number of low surface brightness galaxies which also indicate that they may constitute a large fraction of galaxies in the local universe. Therefore, a number of studies have been done on surface photometry and determination of kinematical properties of LSBs (de Blok et al. 1995; Matthews et al. 1999; Du et al. 2015; Pahwa & Saha 2018; Honey et al.2018) and also on mass modeling using rotation curve decomposition methods (de Blok & McGaugh 1997; Kurapati et al. 2018). But very few studies have focused on vertical structure of such galaxies, till date (Matthews 2000; Bizyaev & Kajsin 2004; O’ Brien et al. 2010d). However, the vertical disk structure including the density profiles and the vertical thickness measurements can help us to study their stability and star formation history, and hence distinguish their evolution history from that of HSB galaxies.

In this paper we aim to study the vertical structure of a late-type (Sd), edge-on LSB galaxy, UGC 7321, through theoretical modeling. In a previous work (Sarkar & Jog 2018) we have studied the detailed vertical density distribution in the Milky Way for a multi-component disk, where the dark matter halo was found to have a significant constraining effect on the vertical stellar disk distribution in the outer disk, and despite this the disk was found to flare by a factor of from to 22 kpc. The LSBs are known to be dark-matter-dominated from the innermost radial region, so it would be interesting to study the vertical disk distribution in such a setting. This was the motivation for the present work. We show that the actual values of disk thickness depend critically on the variation with radius of the disk surface density, velocity dispersion; and the values of the dark matter halo parameters.

We choose UGC 7321 for this detailed study because it is one of the few LSB galaxies whose vertical structure has been studied by observation. UGC 7321 was first studied in detail by Matthews et al.(1999) in optical and near infrared imaging. Matthews(2000) analyzed the H & R band data and the B-R color maps to probe its vertical structure, and thus measured the vertical scale height of the stellar disk at different galactocentric radii and noted a radially increasing scale height, but this fact was not explored further theoretically. Now with the advent of recent surveys there is increasing observational evidence that the thickness of both stellar and HI distributions in the disk increase with radius and thus both show flaring in the outer disk region of HSB spirals, e.g., Milky Way (Momany et al. 2006; Lopez-Corredoira & Molgo 2014; Wang et al. 2018). The flaring is shown to arise naturally in a theoretical study for the Milky Way (Sarkar & Jog 2018). Hence it is important and interesting to find out if flaring can be a generic result also for LSBs

With this aim, we solved for the vertical disk distribution for UGC 7321, using the observational data in the literature and also using the observed vertical scale height data up to 5.8 kpc as a constraint, to obtain the necessary input parameters for the model. We thus obtained the disk thickness as a function of galactocentric radius from R=0 to 12 kpc. For the sake of completeness, we also calculated the thickness of the HI disk in this radial range.

In an earlier study, Banerjee & Jog (2013) showed how the dark matter dominance can lead to the superthin nature of this galaxy. However, unlike that earlier paper, in this paper we have considered the more realistic case where the rotation curve varies with radius while solving for the vertical structure for any disk component. This is essential, as it can significantly affect the disk thickness for a galaxy in a region where the rotation curve is not flat, as we show here. Further we have used more physically justified input parameters, also constrained by the observed data.

This paper is organized as follows. The analytical model and its input parameters, as well as the numerical method used, are given in detail in Sect.2. The results for calculation of disk thickness and its dependence on various dynamical input parameters of the galaxy are explained in Sect.3. Finally, we present conclusions in Sect.4.

2 Formulation of the equations and numerical solution

We use the gravitationally coupled disk model consisting of stars and atomic hydrogen gas HI embedded in the field of the dark matter halo, as developed by Narayan & Jog(2002b). The molecular gas , being negligible in surface density for UGC 7321 (Banerjee et al. 2010), is not included in our model. We adopt the galactocentric cylindrical coordinate system () where denotes the mid-plane. The stellar and HI disks are taken to be coplanar.

The equation of hydrostatic equilibrium for any disk component in the direction is given by (Rohlfs 1977):

(1)

where represents mass density of stars or gas as a function of . Here is the joint potential of stars, HI and the dark matter halo. Here denotes the mean square velocity of the ith component along the direction, and is taken to be constant along , i.e. the disk is taken to be isothermal.

Now, the Poisson equation for this coupled disk plus halo model can be written as:

(2)

Combining the Poisson equation with the equation for vertical hydrostatic equilibrium, we obtain the following equation which describes the vertical distribution of stars and gas (where ):

(3)

Here, the radial variation of the joint potential is expressed quantitatively through the observed rotation velocity as : . To calculate the radial term in the Poisson equation, we need to know the rotation velocity at each radius. Therefore we fitted a curve through the observed data points (Uson & Matthews 2003) using the principle of polynomial fitting and obtained the best-fit rotation curve (Fig.1). Using this best-fit curve we determined the radial variation of rotation velocity and hence the radial term in the Poisson equation (the last term in Eq.(3)) at each radius. The rotation curve clearly shows that it is not flat over most of the radial range considered. We note that this approach was developed and used by Banerjee et al.(2011) to study the scale height in dwarf galaxies, but it was not included in the study of UGC 7321 by Banerjee & Jog (2013).

Figure 1: Rotation curve for UGC 7321 - the rotation velocity vs. radius. The circles represent the observed rotation velocity data obtained by Uson & Matthews(2003). The green curve plotted through the data points is the best-fit rotation curve. The curve increases quite steeply nearly up to 4 kpc, then has a slow increment and becomes almost flat around 10 kpc. At the edge, it shows a slight decline up to 12 kpc.

2.1 Input parameters, and numerical solution

To solve the coupled Eq.(3) we need to know the values of the input parameters such as the halo parameters, the vertical velocity dispersion and surface density of stars and HI as a function of radius. The surface density values of stars were taken as found from surface photometry by Matthews(2000), and deduced by Banerjee et al(2010) with the central value of 50.2 . We note that this is about 12 times smaller than the value for the Milky Way (e.g., Narayan & Jog 2002b). It is considered to fall off exponentially along radius with radial scale length (Matthews et al. 1999). The HI surface density values are from Uson & Matthews(2003), we have used the tabular form of this as provided by Matthews(personal communication) to us. These values are given in Table 1.

Radius Matthews(2000) Uson & Matthews(2003)
(kpc)
0.0 50.2 4.4
1.0 31.2 5.3
2.0 19.4 5.52
3.0 12.0 5.54
4.0 7.5 4.9
5.0 4.6 4.5
6.0 2.9 3.8
7.0 1.8 2.8
8.0 1.1 1.7
9.0 0.69 0.90
10.0 0.43 0.48
11.0 0.27 0.26
12.0 0.16 0.14
111
Table 1: Surface density of stars and HI as a function of radius

The dark matter halo is taken to be pseudo-isothermal with the density profile given as:

(4)

where and denote the central halo density and the core radius respectively. Using corresponding potential( see Ghosh & Jog 2014) we derived the halo contribution for the rotation curve( also see Ghosh et al. 2016) and the exponential disk contribution was taken from Binney & Tremaine (1987). We determined and by fitting net rotation curve to the observed data points taking the contribution of stars and the dark matter halo and adding these quadratically as shown in Fig.2. The gas contribution to the rotation curve was found to be negligible for this galaxy (Banerjee et al. 2010), and hence is not included here. The best-fit to the observed rotation curve gives the resulting halo parameters as  kpc.

Figure 2: This net rotation velocity curve is calculated by adding the contribution of the stellar disk and the dark matter halo quadratically as so as to get the best-fit to the observed rotation velocity data (Uson & Matthews 2003). This gives the best-fit parameter values for the dark matter halo (. This curve shows clearly that the dark matter halo dominates the dynamics of this galaxy almost at all radii.

To explain the scale height of the stellar disk as observed by Matthews(2000), we tried out a set of values for the input parameters. The detailed results of this parameter search are given in Sect.3. For some parameters such as the surface density the values are fairly well-known and the dark matter halo parameters are determined by fitting the rotation curve as above. For other parameters we tried a range of values as guided by observations, or as justified on physical grounds and then chose the ones that gave the best-fit to the scale height data. The details are given in Sects.3.1.1 and 3.1.2. All parameters have to be simultaneously varied to get the best-fit to the scale height data up to R= 5.8 kpc, hence we choose to give the sequence of best-fit values for all the parameters in this sect.

The central vertical velocity dispersion of stars is taken to be as discussed in Matthews(2000). It was measured indirectly by a simple analytical model for a single-component disk using the observed central stellar disk scale height. Here, for the best-fit case (found by trial and error) we have taken the velocity dispersion to vary with radius as:

(5)

Thus the dispersion is taken to fall off exponentially with radius with a scale-length instead of which is typically assumed in literature, for details see Sect.3.1.1. The stellar velocity is considered to be equal to HI velocity from 8 kpc onwards where it starts to go below , if taken to fall off as Eq.(5). This assumption is justified because stars are formed from gravitational collapse of gas clouds, hence they cannot have a lower velocity dispersion than the gas itself. With age, the stellar velocity dispersion is known to only increase, for a detailed discussion see Sarkar & Jog(2018). This transition radius depends on the central value of the stellar velocity dispersion and also on the rate at which it falls off with radius and the gas dispersion.

For the gas velocity dispersion we have used throughout the disk, as this value, corresponding to gas turbulence, was used to obtain the rotation velocity map of this galaxy by Uson & Matthews(2003).

We note that to find the density distribution of any component, Eq.(3) must be solved in a coupled state only, since the radial part of the potential has contribution from all the components; otherwise it may lead to non-convergence in the numerical solution. In particular we cannot obtain the disk thickness (HWHM) for say stars-alone even for the sake of illustration because the presence of dark matter halo gravity on the R.H.S. of Eq.(3) plays a significant role since the galaxy is dominated by the halo gravity at all radii in a typical LSB galaxy. This is different for the Galaxy case (which has a flat rotation curve and hence no contribution from radial term) where the disk thickness for stars-alone can be calculated and compared with that in a multi-component disk plus halo case (Sarkar & Jog 2018).

Using these input parameters we solved the coupled disk Eq.(3) numerically using 4th order Runge-Kutta method with 5th decimal convergence in solution using the same iterative procedure and the boundary conditions as described in Narayan & Jog (2002b).

3 Results

3.1 Thickness of the stellar disk

Next we obtain the thickness of the stellar disk using our theoretical model, at an interval of 1 kpc in the radial range of R= 0-12 kpc, and compare these with the observed data. As mentioned earlier (Sect.2.1) these observations are used as a guideline to constrain the input parameters (velocity dispersion of stars and HI) for which the best-fit values from our model are obtained, along with the other input parameters which are justified on physical ground. The half-width half maximum (HWHM) of the vertical density distribution for the best-fit case is taken to denote the disk thickness, as was done in Narayan & Jog (2002b)- see Table 2. Instead of fitting the vertical density distribution to any function we choose to define the disk thickness in terms of HWHM because the fitting function is not robust and depends on the range over which the fitting is done, as shown later in Sect.3.4, also see Sarkar & Jog (2018) for our Galaxy.

To explain the observed light profile in the central region, Matthews(2000) suggested as a possibility the presence of a bulge or more likely a proto-bulge within the central 1 kpc region. But in our model we have considered UGC 7321 to be a bulgeless disk galaxy, as supported in a review by Kautsch(2009). If the effect of bulge was included then it would have changed the resulting disk thickness values within 1 kpc of the center.

Radius HWHM of HWHM of
(kpc) stellar disk(pc) HI disk(pc)
0.0 210.9 69.4
1.0 300.0 112.6
2.0 339.0 148.1
3.0 358.0 182.4
4.0 361.3 217.6
5.0 359.4 253.1
6.0 364.9 300.4
7.0 384.9 372.8
8.0 477.9 472.5
9.0 593.1 582.0
10.0 656.5 629.5
11.0 635.8 626.6
12.0 566.8 561.6
Table 2: HWHM of the stellar and HI disks from our calculation

The observed stellar disk scale height data of this galaxy is given by Matthews(2000) from to kpc, listed in Table 3 here. In that paper, the observed vertical surface brightness profile at each radius was fitted by either an exponential (at the center) or a function (at most other radii) to calculate the scale height values. Interestingly, an increase of scale height by over radius was noted and was even found to be statistically significant.

Radius Scale heightMatthews(2000). The central vertical density profile was fitted by an exponential function and rest are fitted by a distribution. Thickness (HWHM)Corresponding HWHM values.
(kpc) (pc) (pc)
0.0 140.0 97.1
0.73 119.0 157.1
1.45 143.0 188.8
2.91 168.0 221.8
4.36 175.0 231.0
5.82 204.0 269.3
222
Table 3: Scale height and thickness of the stellar disk from observed data

Since the observed data is not given at a regular interval of radius, a one-to-one comparison is not possible, but the overall trend of the radial variation in scale height can be studied.

To compare our results with the observed scale height data we used the conversion relation at the center and at all other radii shown in Table 3. The observed profile at 0.73 kpc was found to lie intermediate between exponential and by Matthews(2000) and hence the scale height corresponding to both of the fitting functions were given. Here we have given the value measured assuming a distribution for the sake of consistency in our calculations. When we compare the HWHM values for stars from Table 2 & Table 3 we note that our model is able to produce the mildly radially increasing HWHM values up to kpc as observed in the data, and our model shows flaring beyond that. Though the HWHM values differ in magnitude, the overall trend is the same as found in observations. But, interestingly, this observed trend can be reproduced only when the input parameter values are constrained in a certain way. We explain this in the following subsections.

3.1.1 Dependence on radial variation of stellar velocity dispersion

The central vertical velocity dispersion for stars was taken to be , as discussed in Sect.2.1. Typically, the vertical velocity dispersion is taken to fall off exponentially with radius with a scale length of , as was proposed by van der Kruit & Searle(1981a) to explain the constancy of disk scale height reported in their observations. They suggested that the velocity dispersion of stars should fall with radius in such a way so as to precisely compensate for the decline in the stellar disk surface density and hence its self-gravity. This has become an accepted paradigm though there is no physical justification of this assumption of an exponential fall-off with a scale length of 2. In fact the constancy of scale height itself has been questioned by many papers - for external galaxies (de Grijs & Peletier 1997; Narayan & Jog 2002a) and for the Milky Way (Kalberla et al. 2014; Sarkar & Jog 2018). When we applied the fall-off in velocity at this rate in our calculation we found that the thickness (HWHM) decreases steadily with radius, which is exactly opposite to the trend observed in data. We note that the same trend of radially decreasing thickness was also obtained by Banerjee & Jog (2013), who too used the same assumption of fall-off at .

To resolve this issue we try to understand how an increasing disk thickness may arise for a single component stellar disk. We know that a flat disk can be observed if it satisfies the relation , where is the exponential radial scale length of the velocity dispersion of stars and is the scale length of surface density of the radially exponential disk. Now if the velocity dispersion falls off slower than the above required rate, i.e (), so that the pressure support is higher at a given radius then it can result in a flared disk. For a single component disk, increasing disk thickness can be observed at any value of . But in a gravitationally coupled multi-component system, no such simple relation exists between and that would give a disk with constant thickness or a value for which one would get a flared disk. The rate of increase of disk thickness will depend crucially on the value chosen for the ratio .

For example, Narayan & Jog(2002a) investigated the origin of such radially increasing scale height in two edge-on external galaxies, namely, NGC891 & NGC4565 using the same multi-component disk model. They found this ratio to be 2.5 and 3 respectively (or higher) which is able to explain the observed flaring in these two galaxies.

Hence in this paper, we applied this same idea and tried various values of the ratio . We found that an overall flaring in the stellar disk occurs only when or higher which corresponds to a slower fall-off in velocity dispersion (Fig.3). Though the values of disk thickness obtained are higher than the observed ones, we note that the right trend of overall increase in disk thickness with radius is obtained and thus gives a better overall fit to the data at all radii. Since any value of will give rise to even higher HWHM, hence we choose as the best-fit case and use it in all further calculations. Our findings once again show that there is no physical justification in taking , since, even in the case of a LSB it fails to produce the observed disk thickness. Further, we note that the disk thickness starts to increase radially significantly only when the velocity dispersion is kept constant with radius, i.e, beyond 7 kpc (when the stellar velocity dispersion becomes equal to the gas dispersion).

Figure 3: Resulting thickness of the stellar disk as a function of radius. When falls at a rate of , the HWHM values follow the trend of the given data, while results for an exponential fall-off at a rate of give a steady fall with radius, except at 5 & 6 kpc. There the velocity dispersion becomes equal to and hence kept constant thereafter.

3.1.2 Varying HI velocity dispersion

As mentioned earlier we solved for the stellar vertical distribution using HI velocity dispersion to be in the best-fit case. But since the typical observed range of is for outer disk region of galaxies (Kamphuis 1993; Dickey 1996), we also show the stellar disk thickness variation obtained using for completeness. Fig.4 shows clearly that the HWHM values corresponding to are higher than that obtained with at all points, the latter choice results in values closer to the observed ones and thus justifies our best-fit choice. Since the velocity dispersion of gas affects the stellar distribution only indirectly in the coupled disk system (Eq.(3)), hence the difference in resulting stellar disk thickness values are small.

Figure 4: Results for variation in stellar disk thickness on changing HI velocity dispersion from 7 to 8 . This causes the stellar disk to puff up a little and therefore the disk thickness is higher than that obtained with 7 . Since the values of thickness are closer to the data from , this is considered to obtain the best-fit model.

3.2 Effect of radial term in determining the disk thickness

In this paper, we have solved the full Poisson equation by including the contribution of radial term (Sect.2). As discussed earlier, this term is calculated by fitting the rotation curve to the available data. We note that, unlike our Galaxy, the rotation curve here is not flat over most of the radial range considered, with larger gradient towards the inner region. Hence the vertical distribution of stars and HI will be affected by the radial term mostly in the inner part.

Upto 10 kpc, the increasing rotation curve causes the radial term to add against the gravity of all the components in the coupled disk equation, thus causing the disk to puff up by decreasing the effective gravity. This effect becomes more prominent in the inner few kpc region of the disk.

The rotation curve declines near the outermost detected region of HI disk in UGC 7321 as reported by Uson & Matthews(2003). Though its origin is far from resolved, some possible interpretations were discussed. Truncation of mass distribution and hence probable decline in rotation curve is also discussed in a review by Sofue & Rubin (2001). Hence we considered this behavior in our calculation to study its possible effect on the disk thickness. We found that the radial term gets added to the Eq.(3) for R=11 & 12 kpc in such a way so as to add to the gravity of all other components. This causes the HWHM to fall at these two points.

Thus apart from the last two points, we found an overall increase in HWHM with radius for both stellar and HI disks.

Figure 5: Resulting variation of stellar disk thickness (HWHM of vertical density distribution) as a function of radius from 0 to 12 kpc for our best-fit model. The decline in the rotation velocity at the last two points causes the HWHM to decrease.

3.3 HI disk thickness as a function of radius

We also determined HI disk thickness (Fig.6) for this galaxy using the best-fit parameter values used to solve for stellar density distribution, which agrees fairly well with the observed values as discussed below.

The intensity maps show presence of a warp and therefore an analytical form of a flared plus warped model was used in a Gaussian fit to find the scale heights as a function of radius, owing to the contribution of warp and flare separately (Matthews & Wood 2003). For the flaring part of the Gaussian the HWHM was found to increase from pc at to pc at the last measured point of 11.8 kpc which is in quite well agreement to our result.

Matthews & Wood(2003) also added a lagging halo to their model to account for the highly extended emission. But near mid-plane, the warped plus flared model is sufficient to account for the scale heights.

Figure 6: Resulting variation in HI disk thickness (HWHM of vertical density distribution) with radius. The best-fit model parameters taken for stars are used here. The HWHM of the HI disk increases till the drop in rotation velocity at 11 kpc.

3.4 Determination of vertical density distribution of stars

It is well-known that the vertical density distribution for a self-consistent, single-component isothermal disk obeys a distribution law (Spitzer 1942). But in a realistic coupled disk system (as described in Sect.2), the profile for any disk component would deviate from a simple behavior. Observationally it is found to lie intermediate between and exponential near the mid-plane and at larger , it extends to exponential distribution (van der Kruit 1988). Based on these observed results, van der Kruit(1988) proposed the following general form of density distribution function to fit along

(6)

Here and n correspond to density profiles with , and an exponential distribution, respectively.

We note that corresponds to the mid-plane density and is used as a measure of scale height. The parameters and are obtained with a best-fit analysis of the data.

Though this functional form was not derived on any physical ground, many authors have studied the observed vertical luminosity profile of galaxy disk using this form and have given the parameters for the best-fit case (e.g., Barteldrees & Dettmar 1994; de Grijs et al 1997). Here we first used this function to fit the vertical mass density distribution obtained from our model at different radii. We calculated the exponent 2/n both as a function of R and , the vertical interval taken for fitting.

Figure 7: Variation in best-fit value of the exponent as a function of radius is obtained when our results for stellar density distribution are fitted with a distribution of type for a range of pc. The value of is less than 2 (or ) for kpc and greater than 2 beyond that, i.e, in the outer disk region, which is fully dominated by dark matter halo.

In Fig.7, we have plotted the values of 2/n at all galactocentric radii obtained over a interval of 200 pc, as was done for the Galaxy in a previous study (Sarkar & Jog 2018). We note that, interestingly, the value of 2/n starts to be close to 2 or greater than 2, corresponding to from 5 kpc onwards, i.e., mainly in the outer disk region. This region is totally dominated by dark matter halo gravity. A similar trend in results was also noted for our Galaxy (Sarkar & Jog 2018). We note that this range of is new and was not considered by van der Kruit (1988).

Figure 8: Variation in best-fit value of the exponent as a function of fitting range at R=2 & 8 kpc, where our results for vertical stellar density distribution are fitted with a distribution of type . The value of increases with , irrespective of the disk region, in a similar way. Hence cannot be regarded as a robust indicator of density profile.

In Fig.8, we have shown the variation of the exponent with vertical range used for fitting at two different regime of the disk - at an inner radius of 2 kpc where the rotation curve rises steeply and at 8 kpc in the outer disk region, where the rotation curve is almost flat. In both cases 2/n value increases steadily with increasing . A similar result was also found for our Galaxy (Sarkar & Jog 2018). This shows that at any radius in the disk, a fitting function with fixed parameter values can not provide best fit to distribution over its full range of . Therefore we measure the disk thickness in terms of its HWHM instead of which changes with the value chosen for the range .

This could be explained as follows- when more than one isothermal disk components, e.g., stars and gas, each with a profile, with different surface density and velocity dispersion values, form a gravitationally coupled system in the field of dark matter halo which follows a vertical density distribution completely different to , the resultant disk distribution cannot be fitted by a function with constant best-fit parameter value along the full range of .

We note that Matthews(2000) tried to fit a function over the full vertical range of the observed surface brightness profile at any radius with a constant value of . The function fitted very well in the inner region, but failed at the profile wings at larger at almost all radii. In order to explain this, the presence of an additional disk, having a higher scale height, was invoked. Being thicker, this disk was supposed to contribute in the larger region more than the thinner disk, so as to explain the failure of a single fitting function as given by Eq.(6) at larger . The thick disk was also considered by Matthews(2000) to be a possible reason of the observed increase in scale height with radii.

However, Fig.8 shows that a physically motivated multi-component disk in a dark matter halo as treated in our model shows that a single distribution does not provide a good fit to the vertical stellar density profile at all , as observed. Indeed our model gives a broader distribution at high as observed, without the necessity of invoking a second thicker disk as done by Matthews (2000). A similar second, thicker disk was also invoked to explain the intensity distribution in another LSB galaxy FGC1540 by Kurapati et al. (2018). Our model shows that idea of an additional thick disk as invoked in these papers is redundant.

We note that the formation of a thicker disk in a low surface brightness galaxy as proposed in these two papers would be problematic and less likely to occur as such galaxies are usually found in isolated environment (Rosenbaum et al. 2009) and therefore have a low probability of external interaction (Ghosh et al. 2016). Further they lack spiral arms and molecular clouds (e.g., Jog 2012; Ghosh & Jog 2014) so they do not have an internal source of stellar heating.

3.5 Resulting luminosity distribution

Now, similar to the mass density , the general form of luminosity density distribution proposed by van der Kruit(1988) is,

(7)

where is the luminosity density at the central point, i.e, and . The vertical surface brightness corresponding to this distribution for an edge-on galaxy can be calculated as,

(8)

where and is the modified Bessel function of the second kind. This equation can be rewritten in terms of the luminosity density and then by mass density as follows:

{align}

I(R,z) & = I(0,0)(R/R_D)K_1(R/R_D)[L(R,z)L0exp(-R/RD)]
& = 2RK_1(R/R_D)exp(R/R_D)[ρ(R,z)(M⊙/L⊙)]

Now in units of magnitude per the surface brightness profile can be written as (Binney & Merrifield 1998),

(9)

where denotes the absolute solar magnitude in the observed band. For H-band used by Matthews(2000) to observe the vertical surface brightness profiles of UGC 7321, the value of is (Binney & Merrifield 1998).

To find out the surface brightness profiles from our calculated data of using Eq.(9) and Eq.(10), we used a mass-to-light ratio of 1.9 (Bell & de Jong 2001) assumed to be constant at all radii and plotted the profiles all together in Fig.9. There it can be prominently seen that the vertical surface brightness profiles are getting flatter with increasing radii due to their increasing HWHM. In the inner disk region, namely, between 2 to 6 kpc, the increase is smaller than that observed at 9 & 10 kpc. At 11 and 12 kpc, the disk thickness falls as a consequence of the decline present in the rotation curve used in our model, as discussed earlier in Sec. 3.2. Thus, upto 10 kpc in the stellar disk, the surface brightness profiles appear to show flaring, which we have already seen in the measured HWHM in Table 2. Thus it becomes clear once again that invoking the idea of a second disk component to explain the observed increase in scale height as done by Matthews (2000) is not needed. A gravitationally coupled multi-component single disk is adequate to explain these observed properties through a physically justified model as done in the present paper.

Figure 9: Results for surface brightness distribution along for R = 2,6,9,10,11 & 12 kpc. The profiles are getting flatter with increasing radii from R=2 to 10 kpc, indicating increase in disk thickness and strong flaring at 9,10 kpc. The slight decrease in disk thickness seen at 11 & 12 kpc is due to the effect of observed decline in rotation curve.

Now, in some observations (e.g., van der Kruit & Searle 1981a) the vertical profiles of the surface brightness at different radii are presented together as a composite -profile by vertically shifting all the profiles to a certain value. From the slopes of those curves in that composite profile, it is inferred whether there is a significant increase in scale height or not, e.g., a constant scale height will be considered when all the curves can be identically superimposed on each other forming a single curve.
Here, in figures 10 & 11 we have shown the same vertical profiles as shown in Fig.9, but vertically shifted to a value of & respectively. We find that the vertical profiles deviate from each other in the figures. This shows that these vertical profiles have different slopes due to their different disk thickness which we have earlier found in our calculation. Therefore, in these figures, these profiles now show a range of spread in when stacked together. The deviation is more in the larger and lower in Fig.10 & 11 respectively since the points of stacking are different- one at a smaller , another at a larger value. Through these plots we make a note of caution that the spread in surface brightness can be controlled or even minimized by choosing the point of stacking accordingly. The observed vertical profiles can be stacked together in such a way which can mislead to the concept of a radially constant scale height, which is not true.

Figure 10: Results for surface brightness distribution along for R =2,6,9,10,11 & 12 kpc when the profiles are stacked together at to form a composite -profile. The profiles deviate from each other due to different slopes which correspond to different disk thickness. The deviation is larger at higher because the profiles are stacked at a lower value. If the profiles had the same disk thickness they would essentially have superimposed on each other forming a single curve.
Figure 11: Results for surface brightness distribution along for R = 2,6,9,10,11 & 12 kpc when the profiles are stacked together at to form a composite -profile. Similar to Fig.10, here also the profiles deviate from each other. The deviation is larger at smaller because the profiles are stacked at a higher value. If the profiles had the same disk thickness they would essentially have superimposed on each other forming a single curve.

A similar kind of composite -profile was obtained by Narayan & Jog(2002a) who studied the vertical structure of two edge-on external galaxies, NGC891 & NGC4565. The spread in surface brightness of around in the vertically shifted and stacked -profiles, which is seen in the observed data, could be reproduced only when a varying scale height with radius was considered in the theoretical model, instead of a constant one which resulted into a single curve, and this required the velocity to fall exponentially with a scale length between 2.5 and . This shows that any kind of spread observed in the composite -profile of a galaxy could be naturally attributed to varying disk thickness and flaring.

4 Conclusions

We have studied the vertical disk structure of UGC 7321, a low surface brightness galaxy, by treating the galaxy as a coupled system of stellar and HI disks embedded in the dark matter halo. We have obtained the vertical stellar distribution and measured the disk thickness in terms of the HWHM of this distribution.

1. Our model can explain the slight flaring of the stellar disk as observed in the data given upto kpc, predicts even stronger flaring at larger radii upto 12 kpc (upto which the model can be applied as the rotation curve is known upto this radius) and the disk thickness increases by a factor of 3 upto 10 kpc.

For completeness we also studied the HI disk thickness and found a similar kind of flaring all throughout the disk.

The vertical distribution is strongly determined by the dark matter halo in this LSB galaxy as it dominates at all radii. Despite this, the stellar and HI disks show flaring. The actual values of disk thickness depend on the values of the input parameters used, however the trend of a flaring disk appears to be a generic result. Thus flaring appears to be a common phenomenon in the outer disks of galaxies, in both the HSBs (Narayan & Jog 2002a; Kalberla et al. 2014; Sarkar & Jog 2018) as well as the LSBs. The LSB disk has very low surface density and the velocity dispersion falls at a slower rate than (of to explain the scale height data as we have shown) and the stellar velocity dispersion is taken to be constant once it equals the gas dispersion value, all these features together give rise to a lower disk gravity compared to the pressure support and hence the disk shows flaring at large radii despite the dark matter halo becoming more important at large radii.

The following two interesting, general dynamical points emerge from this study:

2. We use the observed scale height data between to 5.8 kpc to constrain the radial variation of the stellar velocity dispersion and find that the best-fit to the trend in the scale height data is obtained when the velocity dispersion falls exponentially with radius with a scale length of 3.2 . This corresponds to a much slower fall-off in stellar velocity dispersion than the rate of typically assumed that gives a decreasing disk thickness in the coupled disk system. This reinforces the point that even for the LSBs, the assumption of fall-off at routinely used in the literature is not physically justified, and indeed it should be used with caution in dynamical modelling of a multi-component disk in a dark matter halo.

3. We have shown that taking account of the radial variation of the joint potential (which corresponds to the case when there is radial variation in rotation curve) can affect the disk thickness of both stars and gas to a great extent at those radii where the rotation curve of the galaxy rises steeply or even declines shortly. This shows that in general it is essential to consider the radial variation of potential in studying vertical structure of galaxies with non-flat rotation curves such as dwarf galaxies.

4. Apart from the vertical disk thickness, the vertical density distribution of stars in LSBs too has received very little attention to date. Both the density distribution as well as the composite luminosity profiles obtained from our model show that a single function does not fit the data at all , in agreement with observations which show wings at higher . To explain this behavior in data, a second, thicker disk has been invoked (Matthews 2000; Kurapati et al. 2018). We point out that such a disk is hard to explain in LSBs which are isolated and also lack internal sources of stellar heating like spiral arms and molecular clouds. Our model results show that it is not necessary to invoke such a second disk to explain the data. A gravitationally coupled multi-component single disk is adequate to explain these observed properties through a physically justified model as done in this paper.

We hope that our results, which can predict the nature of the density profiles within and beyond the observed radii of UGC 7321, may motivate the observers to study this galaxy and other LSBs as well, in detail and will help us to understand their dynamics better.

Acknowledgments:SS would like to thank CSIR for a fellowship. CJ would like to thank the DST, Government of India for support via a J.C. Bose fellowship (SB/S2/JCB-31/2014). We thank Lynn Matthews for providing through personal correspondence a tabular form for the data for the rotation curve and the surface density of HI for UGC 7321.


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