Issue 
A&A
Volume 617, September 2018



Article Number  A142  
Number of page(s)  8  
Section  Galactic structure, stellar clusters and populations  
DOI  https://doi.org/10.1051/00046361/201833510  
Published online  02 October 2018 
The constraining effect of gas and the dark matter halo on the vertical stellar distribution of the Milky Way
Department of Physics, Indian Institute of Science, Bangalore, 560012, India
email: suchira@iisc.ac.in ; cjjog@iisc.ac.in
Received:
28
May
2018
Accepted:
26
June
2018
We study the vertical stellar distribution of the Milky Way thin disk in detail with particular focus on the outer disk. We treat the galactic disk as a gravitationally coupled, threecomponent system consisting of stars, atomic hydrogen gas, and molecular hydrogen gas in the gravitational field of the dark matter halo. The selfconsistent vertical distribution for stars and gas in such a realistic system is obtained for radii between 4–22 kpc. The inclusion of an additional gravitating component constrains the vertical stellar distribution toward the midplane, so that the midplane density is higher, the disk thickness is reduced, and the vertical density profile is steeper than in the onecomponent, isothermal, starsalone case. We show that the stellar distribution is constrained mainly by the gravitational field of gas and dark matter halo in the inner and the outer Galaxy, respectively. We find that the thickness of the stellar disk (measured as the halfwidth at halfmaximum of the vertical density distribution) increases with radius, flaring steeply beyond R = 17 kpc. The disk thickness is reduced by a factor of 3–4 in the outer Galaxy as a result of the gravitational field of the halo, which may help the disk resist distortion at large radii. The disk would flare even more if the effect of dark matter halo were not taken into account. Thus it is crucially important to include the effect of the dark matter halo when determining the vertical structure and dynamics of a galactic disk in the outer region.
Key words: Galaxy: disk / Galaxy: halo / Galaxy: structure / galaxies: ISM / galaxies: kinematics and dynamics / galaxies: structure
© ESO 2018
1. Introduction
The stars in a spiral galaxy are distributed in a thin disk with a finite vertical extent. An important property that characterizes a gravitating disk is its vertical density profile, ρ(z), where ρ is the mass density and z is along the direction normal to the disk plane. This is because the mass distribution and dynamics in a gravitating system are interrelated (e.g., Binney & Tremaine 1987). For a onecomponent disk, the selfconsistent vertical density distribution for an isothermal disk is given as a sech^{2} (z) distribution (Spitzer 1942).
A real galaxy disk, however, consists of stars and interstellar gas. The gas contains 10–15% of the disk mass (Young & Scoville 1991; Binney & Merrifield 1998). However, because of its lower dispersion, the gas forms a thin layer and hence can affect the vertical distribution of stars significantly despite its low mass faction. The selfconsistent vertical structure of such a multicomponent disk (consisting of stars, atomic hydrogen gas, HI, and molecular hydrogen gas, H_{2}) where the components are gravitationally coupled was studied by Narayan & Jog (2002b) for the Galaxy for radii <12 kpc, where it was shown that the vertical distribution of each component is affected by the others. This approach has subsequently been used by many papers in the literature, including those that derive the disk properties from observations (e.g., Comeron et al. 2011) as it gives a more accurate result for the disk thickness.
In this paper, we adopt the method proposed by Narayan & Jog (2002b) and extend their study to the outer disk of the Milky Way between radii of 12–22 kpc. The aim is to study the dynamical effect of gas as well as of the dark matter halo on the vertical stellar profile of the thin disk. The density of the disk components as well as the dark matter halo density decrease with radius, but in the outer parts of a galaxy, the halo is likely to dominate the dynamics. Further motivation for our study comes from recent observations that show that the stellar disk flares in the outer Galaxy (e.g., Momany et al. 2006; LopezCorredoira & Molgo 2014). We show that the gravitational effect of any coupled component causes the stellar distribution to be constrained toward the midplane, thus reducing the disk thickness and making the density profile steeper than the singlecomponent starsalone case. We find that the constraining effect of dark matter halo starts to dominate beyond 14 kpc. The thickness of the stellar distribution increases with radius, flaring steeply beyond 17 kpc. If the effect of dark matter halo were not included, the stellar disk thickness would flare even more. Thus the dark matter halo plays a pivotal role in confining the stellar distribution closer to the midplane in the outer Galactic disk.
Section 2 contains the formulation and numerical solution of equations, and the input parameters used. Section 3 contains results for the constraining effect of gas and dark matter halo. Section 4 contains a brief discussion to show that the gas distribution is also vertically confined by the effect of stars and the halo. The main conclusions are given in Sect. 5.
2. Formulation of the equations and numerical solution
For a onecomponent disk, the thickness is determined by the balance between its selfgravity and vertical pressure (Spitzer 1942). To study the gravitationally coupled disk components (stars, HI and H_{2} gas) in the field of dark matter halo, we follow the treatment as given in Narayan & Jog (2002b) and solve the following coupled equations to obtain the net density distribution:
where i = 1, 2, and 3. Here ν_{z} is the velocity and K_{z} is the force per unit mass, both along the zaxis. The galactocentric cylindrical coordinates (R, z,ϕ) are used, and z = 0 denotes the galactic midplane. We assume each component to be isothermal so that the vertical velocity dispersion is taken to be constant along the z axis. This equation was solved with a fourthorder RungeKutta method in an iterative way for each component with fifth decimal convergence, following the same approach as in (Narayan & Jog 2002b).
The formulation described above is general. We next apply it to the Milky Way since various parameters such as the surface density and the velocity dispersion are known for it observationally. For the dark matter halo we consider a pseudoisothermal profile with the parameters (R_{c} = 5 kpc, and V_{c} = 220 km s^{−1}) taken as in Narayan & Jog (2002b), and the derived potential being
where r is the radius in spherical coordinates. Here R_{c} is the core radius and V_{c} is the limiting rotation velocity. Next, using cylindrical coordinates, we write d(K_{z})_{DM}/dz = −∂^{2}ψ_{DM}/∂z^{2} as the halo contribution in Eq. (1) (as given in Narayan & Jog (2002b) with a negative sign multiplying the total expression, since the negative sign was missed in the expression for the potential in that paper).
The stellar surface density values were taken from the Galaxy mass model of Mera et al. (1998), (see Table 1). The radial stellar velocity dispersion values were taken from the observational values obtained for the Galaxy by Lewis & Freeman (1989) up to 16 kpc. This follows an exponential curve:
From these the corresponding vertical velocity dispersion values σ_{z} were calculated by assuming the verticaltoplanar ratio to be 0.45 (Dehnen & Binney 1998; Mignard 2000). Beyond 17 kpc, the σ_{z} value was kept constant equal to its last value of 7.5 km s^{−1} so as to be higher than the σ_{z} value of HI. The physical reason for this assumption is given next. We note that stars form from the collapse of gas clouds, therefore the stellar velocity dispersion σ_{z} cannot be lower than that of the dispersion within the gas. Once the stars form, they gain random kinetic energy via gravitational scattering off clouds or spiral structure in the disk, hence the velocity dispersion shows a systematic increase with stellar age in the standard scenario of stellar heating (e.g., Binney & Tremaine 1987).
For HI gas between R = 4–14 kpc, the observed surface density values from Scoville & Sanders (1987) were used (see Table 1). Beyond that radius, the values were taken from Levine et al. (2006), who studied the outer Galactic HI disk and measured the disk surface density, Σ_{HI}, which they found was best fit by the following exponential curve:
The velocity dispersion σ_{z} for HI gas is measured only in the inner Galactic disk, see Narayan et al. (2005) for a detailed discussion, the salient points from which are mentioned here. From R = 4–12 kpc, the value of σ_{z} for HI is taken as 8 km s^{−1} based on the values given by Spitzer (1978) for the Galaxy and by Lewis (1984) for nearly 200 faceon galaxies. The velocity is observed to decrease and then saturate to around 7 ± 1km s^{−1} in the outer parts (e.g., Kamphuis 1993; Dickey 1996). Beyond R = 12 kpc, we therefore consider the velocity to taper off with a slope of −0.2 km s^{−1} kpc^{−1} up to 17 kpc and then assume it to be constant at the last value of 7 km s^{−1} for outer radii beyond that. Since the observed values of σ_{z} for HI and stars in the outer disk are not very welldetermined, we do not go beyond R = 22 kpc in this analysis. For H_{2} gas, the observed surface density and velocity dispersion values are taken from Scoville & Sanders (1987).
Surface density of stars and gas vs. radius in the Milky Way.
3. Results
3.1. Vertical density distribution of stars
The numerical solution of the coupled equation as given by Eq. (1) gives the selfconsistent vertical density distribution at a given radius. To study its nature in detail from the inner to the outer Galaxy, we performed the calculation from R = 4–22 kpc at an interval of 2 kpc. Instead of 8 kpc, we considered 8.5 kpc as it denotes the region of the solar neighborhood. The inner Galaxy limit was chosen to be 4 kpc because below this radius the bulge and the bar contribution may dominate the vertical luminosity. The resulting stellar profiles are shown in Fig. 1 for four selected radii, R = 6, 8.5, 18, and 22 kpc. This choice will help bring out the dynamical result that different components (gas or the dark matter halo) dominate at different radii, as discussed below.
Fig. 1. Vertical density distribution of stars vs. z at four different radii, R = 6 and 8.5 kpc in the inner Galaxy (panels a, b) and at R = 18 and 22 kpc in the outer Galaxy (panels c, d). The three curves represent the density distribution of stars in the gravitational field of starsalone, stars plus dark matter halo, and stars plus dark matter halo plus gas cases. The addition of other gravitating components (gas and halo) increases the midplane stellar density value that causes the scale height to become smaller and the curve to become steeper. The constraining effect is mainly due to gas in the inner Galaxy (panels a, b); while in the outer Galaxy (panels c, d), the dark matter halo has the dominant constraining effect. 

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In the inner part of the Galaxy, for instance, at R = 6 or 8.5 kpc, the surface density of stars is high and its selfgravity dominates the gravity of gas and the halo. Interestingly, even though the gas contains only ∼10–15% of mass in the disk, it has a significant constraining effect on the net stellar density profile as can be seen from the increased stellar midplane density and reduced stellar thickness, see the three distributions in Figs. 1a,b. In the outer Galaxy, for example, at R = 18 kpc, the stellar surface density is low, and the starsalone case therefore gives rise to an extended, diffuse vertical stellar distribution. When halo gravity is included, it strongly constrains the vertical stellar distribution, compared to the effect of gas (Figs. 1c,d). Even farther out at R = 22 kpc, the effect of dark matter halo is more prominent. Thus we conclude that with increasing radii, the dark matter halo is mainly responsible for shaping the stellar density distribution.
Thus our results show that an additional component that can exert an extra gravitational force increases the midplane density compared to the onecomponent case. For a constant surface density, it therefore also constrains the whole distribution to be closer toward the plane, which thus has a smaller thickness. The shape of the density profile is also altered and becomes steeper.
To give a physical explanation for the results shown in Fig. 1, we first calculated the vertical gravitational force per unit mass, K_{z}, exerted by the stars, HI, H_{2}, and the dark matter halo, treated as independent components, as a function of z. For a given disk component, K_{z} and the corresponding density distribution ρ(z) can be obtained analytically by solving the force equation and the Poisson equation together to be (Spitzer 1942)
Here ρ_{0} is the midplane density and the constant z_{0} is given by
For each disk component, using the observed surface density at a given radius as the boundary condition, the density ρ(z) Eq. (5) was integrated to obtain both ρ_{0} and z_{0} simultaneously. The force due to the dark matter halo in cylindrical coordinates was obtained from the expression for its potential Eq. (2) to be
In Fig. 2 we plot K_{z} for the various components at R = 6 kpc and R = 18 kpc. At R = 6 kpc, the gravitational force due to H_{2} gas is strong, ∼30% of that due to stars, for z < 150 pc. The force due to the dark matter halo is smaller, and we checked that it becomes more important than the force due to gas only beyond z ∼ 800 pc. On the other hand, in the outer Galaxy, at R = 18 kpc, the magnitude of the vertical force K_{z} due to the halo dominates that due to stars and HI gas from the lowest z values.
Fig. 2. Vertical gravitational force exerted by each component treated separately vs. z at R = 6 kpc (a) and 18 kpc (b). At R = 6 kpc, the stellar distribution is mainly affected by the gas gravity, while at R = 18 kpc, the force due to the dark matter halo dominates and hence strongly affects the stellar distribution. 

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For the coupled system, the corresponding force components are not known, therefore we instead obtained the gradient of the force with z, that is, (dK_{z}/dz) in terms of the mass density ρ(z) for each disk component (Banerjee & Jog 2007). For the halo we plot the value of −(dK_{z}/dz) × (1/4πG). The resulting values are plotted for R = 6 kpc and R = 18 kpc in Fig. 3. Figure 3a confirms the strong effect of gas in constraining the vertical stellar distribution in the inner Galaxy. In an earlier study, a similar constraining effect on the disk around a molecular cloud complex (of ∼10^{7} M_{⊙} mass and with an extent of ∼200 pc) was shown by Jog & Narayan (2001). At R = 6 kpc, the dark matter halo density is lower than the gas density for z < 120 pc and lower than the stellar density until about 700 pc (not shown in the figure), and therefore it does not play an important role. On the other hand, in the outer Galaxy, at R = 18 kpc (Fig. 3b), the halo dominates the gas at all z values and stars at z > 200 pc. This shows that the halo has a very strong constraining effect that increases progressively at larger radii. This effect is also evident in the higher increase in the midplane density at larger radii that occurs when halo gravity is included (Figs. 1c,d).
Fig. 3. Density for stars and gas for the coupled system and the corresponding term due to the dark matter halo (dK_{z}/dz)/4πG (Eq. (1)) vs. z, at R = 6 kpc (panel a) and at R = 18 kpc (panel b). Because of its lower dispersion, the midplane H_{2} gas density is comparable to the stellar value, and hence despite its low surface density, it can significantly affect the stellar distribution at R = 6 kpc. In the outer Galaxy, the halo gravitational effect dominates, hence it is mainly responsible for the determination of the vertical stellar distribution. 

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3.2. Measuring the constraining effect: Disk thickness
3.2.1. Halfwidth at halfmaximum
To estimate quantitatively the constraining effect of different components, we next calculate the halfwidth at halfmaximum (HWHM) of the vertical stellar distribution, taken to be an indicator of disk thickness, see Table 2 and Fig. 4. This quantity is also sometimes referred to as the vertical scale height (e.g., Narayan & Jog 2002a,b). In the inner Galaxy, for instance, at R = 6 kpc, the HWHM is reduced by 6% when the halo is included compared to the starsalone case, while it is reduced by 26% in presence of both halo and gas. Thus the gravitational force due to gas is mainly responsible for decreasing the thickness. On the other hand, in the outer Galaxy, for example, at R = 18 kpc, including the halo has a very strong effect that leads to a decrease in the stellar HWHM by a factor of ∼2.3, while including gas decreases it by another ∼20% only. The stellar disk by itself would flare by a factor of 14 from R = 4–22 kpc. Thus the inclusion of halo mainly limits the disk to a net thickness of <1 kpc even at R = 22 kpc. This would help the disk resist distortion due to external perturbations.
Fig. 4. Results for the variation in HWHM of the vertical stellar density distribution in the joint potential of the disk and dark matter halo vs. radius in the Galactic disk. The stellar disk thickness increases gradually until about 17 kpc and then flares beyond that in the outer disk. 

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In a previous study, the dense, compact halo in low surface density galaxies such as UGC 7321 was shown to be mainly responsible for reducing the disk thickness (Banerjee & Jog 2013). Here we have shown that in general, when the halo term is larger than the stellar term in the r.h.s. of Eq. (1), the halo will have a dominant constraining effect as in the outer Milky Way disk.
Thus, we have shown that the detailed balance given by Eq. (1) in a multicomponent disk results in a moderate increase in disk thickness of 40% up to R = 12 kpc, and the disk flares steeply beyond R = 17 kpc. Since the dispersion does not fall rapidly at radii beyond R = 17 kpc (by assumption based on the physical argument, see Sect. 2), this results in a rapid flaring beyond R = 17 kpc. The stellar flaring obtained could be even higher if a higher gas dispersion value of 10 km s^{−1} as in Tamburro et al. (2009) is adopted. In our model, the coupled equations are such (see Eq. (1)) that the choice of the velocity dispersion, σ_{z}, mainly affects the density profile for that particular component, and has only a small indirect effect on the other components. However, because we have set the physical condition that the stellar velocity dispersion value should be greater than for gas (see Sect. 2), a higher gas dispersion will also imply a higher stellar σ_{z} as an input parameter, which would result in a higher value of stellar flaring in the outer disk. Thus, we note that the increase in stellar disk thickness with radius is a robust result from our work.
Our detailed predictions for stellar flaring in the outer disk would provide a motivation to compare these results with future data, from GAIA, for example. Interestingly, the various observational determination of the thickness with radius for the Milky Way outer disk so far do show flaring (Momany et al. 2006 using 2MASS data, LopezCorredoira & Molgo 2014 using SDSSSEGUE data, Wang et al. 2018 using LAMOST data), which agree reasonably well with our results for the net thickness (Col. 4, Table 2). However, the flaring values reported at larger radii by LopezCorredoira & Molgo (2014) are higher than our values. This difference could arise if the effect of warp has not been correctly removed from the data, or because the data is contaminated by the thick disk stars (Minchev et al. 2015). Furthermore, we point out that a tidal encounter with an external galaxy can cause heating of the stellar disk, preferentially in the outer parts (Walker et al. 1996). This could be one reason for the higher flaring seen in observed data. Finally, we note that Wang et al. (2018) use a sech^{2} profile to model the observed data to derive the scale height, while the real profile is different, as discussed above.
Thus our work has shown that the stellar disk thickness is not rigorously constant, as has long been assumed in the literature (van der Kruit & Searle 1981). Hence, the subsequent relation that σ_{z} falls off exponentially with radius with a scale length of 2R_{D} (where R_{D} is the exponential disk scale length) as proposed by them, which is routinely used in the literature, is not physically justified either.
We note that a moderate increase in stellar flaring in external galaxies has long been known from observations (de Grijs & Peletier 1997). This trend was also seen when the multicomponent model was applied and the results from it compared quantitatively with the intensity profiles for NGC 891 and NGC 4565 (Narayan & Jog 2002a). However, this has not received the attention it deserves. Our result for the flaring of the stellar disk in the outer Galaxy agrees with the trend noted and discussed by Kalberla et al. (2014), who have also questioned the constancy of the scale height with radius.
HWHM of vertical stellar density distribution vs. radius.
3.2.2. z_{1/2}, the halfmass scale height
We introduce a new parameter z_{1/2}, the halfmass scale height, measured from the midplane, as another indicator of the constraining effect of gas and the halo. Here 2z_{1/2} contains half the total stellar surface density. This will also implicitly depend on the shape of the density profile at larger z values, unlike the value of HWHM. Hence a smaller z_{1/2} indicates a distribution that is concentrated closer to the midplane. From an observational perspective, z_{1/2} can be thought of as an indicator of the halflight scaleheight of an edgeon galaxy with M/L considered to be constant.
The calculated values for z_{1/2} from R = 4 kpc to 22 kpc are given in Table 3 as determined for the three cases considered. This also shows the same trend, namely, gas and dark matter halo are more important at smaller and larger radii, respectively, and the values of z_{1/2} increase with radius. The actual values of z_{1/2} are lower than the corresponding HWHM values because their definitions are different.
z_{1/2} of vertical stellar density distribution vs. radius.
3.3. Measuring the constraining effect: increased midplane density
In a multicomponent disk in the field of the halo, the midplane density is higher than in the onecomponent case (see Sect. 3.1). The resulting midplane density values are given in Table 4. The increase in density is particularly important at larger radii where the dark matter halo force dominates the force due to the stellar component itself (see Fig. 2). While the midplane density is not likely to be directly observed, the values give an idea of how the midplane stellar density increases in a coupled system. The increase in the midplane density, which can be a factor of ∼4 in the outer disk, can have interesting dynamical consequences, for example, the vertical oscillation frequency given as (4πGρ_{0})^{(1/2)} (see Binney & Tremaine 1987) would be higher by a factor of ∼2 and could thus lead to a bettermixed vertical distribution in the outer disk.
Midplane density of stars vs. radius.
3.4. Measuring the constraining effect: stellar vertical profile
In real galaxies, the vertical stellar density profile close to the midplane does not follow the sech^{2} profile expected of a onecomponent, isothermal disk (Spitzer 1942), but instead is close to a sech distribution or lies between a sech and an exponential distribution, while at large z the profile tends to have an exponential form. van der Kruit (1988) suggested a family of curves that could fit these observational trends:
Here n = 1, 2 and n → ∞ correspond to a density profile with sech^{2}, sech and an exponential z distribution, respectively.
The parameters n, z_{e}, and ρ_{e} are obtained with a bestfit analysis of the data. We note that this profile is ad hoc and is not based on any physical principle, as was noted by van der Kruit (1988). Many observational studies have presented their results in terms of this function and have given the parameter n for the bestfit case (e.g., Bartledrees & Dettmar 1994; de Grijs et al. 1997).
We next try to fit the resulting stellar density profile from our work (Sect. 3.1) by the above function (Eq. (8)). We note that 2^{−2/n} ρ_{e} is the midplane stellar density, which is not known in observational studies, but we know it from our model calculations. For each radius, we obtain the bestfit value for n and z_{e}. Interestingly, we find that the bestfit value of the exponent 2/n is not robust, but instead depends on the zrange (Δz) chosen for the fitting, as shown in Fig. 5 for the case of R = 6 kpc. For example, the value of n thus obtained is ∼5 when Δz = 150 pc, while n ∼ 2 for Δz = 450 pc. These would physically imply very different density profiles of sech^{0.4} and sech respectively (see Eq. (8)). This clearly shows that the family of curves suggested by van der Kruit (1988) for the density profile is not valid physically for a realistic disk treated here. Hence for a working definition of disk thickness, we have used the HWHM value here, regardless of the actual shape of the density profile.
Fig. 5. Variation in bestfit value of the exponent 2/n as a function of fitting range Δz, when our results for stellar density distribution at R = 6 kpc are fit with a distribution of type sech^{2/n}. The value of the exponent 2/n varies with Δz, hence it cannot be regarded as a robust indicator of stellar density profile. 

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However, keeping in mind this caveat that n is not robust, we next try to use this function to fit our resulting stellar distribution vs. z (Sect. 3.1) within z < 150 pc so as to estimate the trend seen, and to be able to compare this with observational studies where data close to the midplane are studied. The resulting plot of 2/n vs. radius is shown in Fig. 6.
Fig. 6. Bestfit value of the exponent 2/n vs. radius when our results for the stellar density distribution for the Milky Way are fit by a distribution of type sech^{2/n} at each radius, for Δz < 150 pc. For radii > 14 kpc, a new regime of n < 1 is obtained. This corresponds to radii where the main constraining effect is due to the dark matter halo. 

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Interestingly, Fig. 6 shows that our results for the stellar density profiles gives n > 1 in the region R < 14 kpc when the constraining of the stellar profile is mainly due to gas (Sect. 3.2.2), whereas we find n < 1 in the outer Galactic disk when the constraining is due to the dark matter halo. The typical observed profiles for external galaxies are for the inner region of the disk where the constraining effect of gas dominates, and hence our calculations explain why the observed values measure n to be >1 with density profiles between sech and an exponential. When the additional gravitating component is more massive and extended along z than the stars (such as the halo), it can affect the stellar profile over a larger z range, and this results in n < 1. This regime of n < 1 has so far not been studied in the literature. An interesting point to note is that even if the stars are taken to be isothermal, the net stellar distribution is no longer given by a sech^{2} profile in a multicomponent disk and under the force of the dark matter halo.
In the literature the value n > 1 or correspondingly 2/n < 2 is often taken to be an indicator of the steepening of the stellar density profile (e.g., de Grijs et al. 1997; Banerjee & Jog 2007). We note that this labeling is misleading, since the case where n < 1 could also lead to a steeper profile. In reality, all the three parameters (the midplane density, width and n) decide how sharply the density profile falls with z. As we have shown in Sect. 3.1, for a given constant surface density, an additional component will always cause a steepening of the stellar profile. Similarly, the other disk components (HI and H_{2} gas) will also show a steeper profile than the corresponding onecomponent cases.
4. Vertical distribution of HI
Although the focus of this paper is on the resulting stellar density distribution, in this section we briefly consider the resulting HI gas density distribution in the outer disk for the sake of completeness. The HI scale heights (defined as the HWHM of the HI vertical density distribution) for the region inside of R = 12 were considered in detail by Narayan & Jog (2002b), hence we do not consider that range here.
The procedure for obtaining the stellar distribution (by solving the coupled Eq. (1)) also simultaneously gives the gas vertical distribution. We also consider the HI gas plus halo case. For comparison, we obtain the HIalone distribution analytically (Sect. 3.1). The corresponding plots for R = 18 kpc and R = 22 kpc are given in Fig. 7. The most striking result form this figure is that the gasalone case under its own gravity would be highly extended vertically with an HWHM of 1.8 kpc and 3.8 kpc at R = 18 kpc and 22 kpc, respectively. These are much higher than the observed values (see, e.g., Wouterloot et al. 1990). Including the gravitational field of the dark matter halo substantially reduces the HMHM values, by a factor of 2.7 and 4.3 at these two radii, respectively. The decrease is slightly stronger when the effect of stars is also included, so that the net HWHM values are 502 pc and 798 pc, respectively. Interestingly, these resulting values of the HWHM (in a multicomponent disk plus halo) up to R = 22 kpc agree fairly well with the observed values of gas flaring (Wouterloot et al. 1990; Kalberla & Dedes 2008), while Levine et al. (2006) give higher values. The HI gas thickness shows an azimuthal variation (Kalberla & Dedes 2008; Levine et al. 2006), which may be explained in terms of intergalactic accretion flows, as proposed by LopezCorredoira & BetancortRijo (2009).
Fig. 7. Vertical density distribution of HI gas under its own selfgravity, then also including the gravitational force of the dark matter halo, and then including the effect of both halo and stars, at R = 18 kpc (panel a) and at R = 22 kpc (panel b). The halo very strongly constrains the HI distribution in the outer Galaxy, and the effect is progressively higher at larger radii. 

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The corresponding midplane density for HI increases by a factor of 3.6 and 4.1, respectively, compared to the gasalone values, and the corresponding net midplane density values are 1.6 × 10^{−3} and 4.1 × 10^{−4} M_{⊙} pc^{−3}. Thus without the confining effect of halo, the vertical distribution of HI would be very extended, which would then be even more susceptible to disturbance by perturbations, including gas dynamical processes, than stars.
While the net gravitational force due to the halo decreases with radius, the gas velocity dispersion σ_{z} is nearly constant and hence the gas scale height shows huge flaring in the outer Galaxy (see Fig. 7). This is despite the strong constraining effect of the halo. This flaring is avoided in the inner Galaxy by the strong gravitational field of stars and gas, which results in a nearconstant HI scale height in the inner Galaxy, in agreement with observations, as was shown by Narayan & Jog (2002b).
5. Conclusion
In summary, we have studied the vertical stellar distribution of the Milky Way thin disk in the outer disk (R = 12–22 kpc) by applying the method of Narayan & Jog (2002b) for a multicomponent disk in the field of the dark matter halo. We have shown that the dark matter halo plays a dominant role in reducing the disk thickness in the outer Galaxy.
The main results are summarized below:
The thickness of the stellar disk (measured as the HWHM of the vertical density distribution) increases moderately by ∼40% up to R = 12 kpc, and flares at larger radii up to ∼1 kpc at R = 22 kpc. The starsalone disk would flare even more. Including the gravitational effect of the halo restricts the stellar distribution to be closer to the midplane at large radii, which may help the disk resist distortion due to external perturbations. The overall trend of flaring of the stellar disk in the outer Milky Way is a robust result from our work.
The midplane stellar density is higher in a multicomponent disk in the field of the halo than for the starsalone case. The increase is a factor of ∼4 in the outer disk.
Because of the gravitational effect of gas and dark matter halo, the stellar density distribution falls more steeply than for a starsalone disk. We showed that close to the midplane, the stellar density profile varies as sech^{(2/n)}, where n < 1 when constrained by the dark matter halo. The regime of n < 1 has so far not been studied in the literature.
The HI gas in the outer Galaxy also shows flaring, and without the dark matter halo, the flaring would be even larger. Such extended gas at higher z values would be liable to get disturbed more easily by perturbations.
The formulation in this paper is general. Hence we expect the trends in results, namely the flaring of the disk, and the dominant role played by the dark matter halo in constraining the outer disk, to be applicable for other galaxies as well.
Thus, the dark matter halo that is normally evoked to explain the nearly flat rotation curves in galaxies also strongly affects the stellar and gas distribution normal to the disk in the outer parts of the Milky Way, causing them to be confined closer to the disk plane. This dynamical effect of the halo on the disk distribution is progressively more important at larger radii. Thus it is crucially important to include the effect of the dark matter halo while studying the vertical structure and dynamics of a galactic disk.
Acknowledgments
We would like to thank the anonymous referee for constructive comments that have helped us improve the paper. 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/JCB31/2014).
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All Tables
All Figures
Fig. 1. Vertical density distribution of stars vs. z at four different radii, R = 6 and 8.5 kpc in the inner Galaxy (panels a, b) and at R = 18 and 22 kpc in the outer Galaxy (panels c, d). The three curves represent the density distribution of stars in the gravitational field of starsalone, stars plus dark matter halo, and stars plus dark matter halo plus gas cases. The addition of other gravitating components (gas and halo) increases the midplane stellar density value that causes the scale height to become smaller and the curve to become steeper. The constraining effect is mainly due to gas in the inner Galaxy (panels a, b); while in the outer Galaxy (panels c, d), the dark matter halo has the dominant constraining effect. 

Open with DEXTER  
In the text 
Fig. 2. Vertical gravitational force exerted by each component treated separately vs. z at R = 6 kpc (a) and 18 kpc (b). At R = 6 kpc, the stellar distribution is mainly affected by the gas gravity, while at R = 18 kpc, the force due to the dark matter halo dominates and hence strongly affects the stellar distribution. 

Open with DEXTER  
In the text 
Fig. 3. Density for stars and gas for the coupled system and the corresponding term due to the dark matter halo (dK_{z}/dz)/4πG (Eq. (1)) vs. z, at R = 6 kpc (panel a) and at R = 18 kpc (panel b). Because of its lower dispersion, the midplane H_{2} gas density is comparable to the stellar value, and hence despite its low surface density, it can significantly affect the stellar distribution at R = 6 kpc. In the outer Galaxy, the halo gravitational effect dominates, hence it is mainly responsible for the determination of the vertical stellar distribution. 

Open with DEXTER  
In the text 
Fig. 4. Results for the variation in HWHM of the vertical stellar density distribution in the joint potential of the disk and dark matter halo vs. radius in the Galactic disk. The stellar disk thickness increases gradually until about 17 kpc and then flares beyond that in the outer disk. 

Open with DEXTER  
In the text 
Fig. 5. Variation in bestfit value of the exponent 2/n as a function of fitting range Δz, when our results for stellar density distribution at R = 6 kpc are fit with a distribution of type sech^{2/n}. The value of the exponent 2/n varies with Δz, hence it cannot be regarded as a robust indicator of stellar density profile. 

Open with DEXTER  
In the text 
Fig. 6. Bestfit value of the exponent 2/n vs. radius when our results for the stellar density distribution for the Milky Way are fit by a distribution of type sech^{2/n} at each radius, for Δz < 150 pc. For radii > 14 kpc, a new regime of n < 1 is obtained. This corresponds to radii where the main constraining effect is due to the dark matter halo. 

Open with DEXTER  
In the text 
Fig. 7. Vertical density distribution of HI gas under its own selfgravity, then also including the gravitational force of the dark matter halo, and then including the effect of both halo and stars, at R = 18 kpc (panel a) and at R = 22 kpc (panel b). The halo very strongly constrains the HI distribution in the outer Galaxy, and the effect is progressively higher at larger radii. 

Open with DEXTER  
In the text 
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