Tracing the Dynamical Mass in Galaxy Disks Using H i Velocity Dispersion and Its Implications for the Dark Matter Distribution in Galaxies (2024)

We start with the Jeans and Poisson equations for an axisymmetric system (Binney & Tremaine 1987). Assuming that the system is time independent and the velocities along the r, ϕ, z directions are independent, we have , where ρ_i represents the tracer density, and σ_z is the velocity dispersion in the vertical direction. Using the Poisson equation and assuming H i to be the tracer of the potential, we obtain the following equations:

where ρ_d is the disk stellar mass density, ρ_{H i} is the H i gas density, and σ_{zH i} is the H i disk velocity dispersion in the vertical or z-direction. Here we have assumed that the H i gas is a static, smooth atmosphere in equilibrium with the disk potential. The force in the vertical direction is given by

If we assume that the stellar and gas disks have exponential mass distributions given by and (van der Kruit 1988), where z is the distance from the z=0 plane, then

We have used an exponential form for the vertical mass distribution in the stellar disk and gas disk because it is the simplest form. Integrating both sides of the equation, from z=0 to , we obtain

The first integration on the right yields the term , while the second gives (where and ). For the left-hand side, we can assume that the gas density vanishes at large z. Then, we obtain the following equation,

The total disk surface densityΣ_d including the stellar and gas components, is given by the relation

The z-component of the H i velocity dispersion σ_{zH i} is hence related to the total stellar and gas mass disk surface densities, Σ_s and Σ_{zH i}, in the following way:

This relation can also be written as

Thus, the disk dynamical mass traced by σ_{zH i} is composed of two parts. The first is the sum of the stellar and gas mass surface densities, whereas the second is a coupling term that is due to the difference in the scale heights of the stellar and gas disks. For example, if z_e=z_g0, then the third term in Equation (9) vanishes, and we have

Until now, we have assumed that the vertical support is only due to the visible stellar disk and the H i gas mass. For the outer disks of galaxies where there is no visible stellar disk, Σ_s∼0. The left-hand side then represents the dynamical mass surface density of the disk and is given by

The mean H i velocity dispersion in the z-direction σ_{zH i} can be derived from the width of the H i velocity profiles of face-on galaxies (Petric & Rupen 2007; Ianjamasimanana et al. 2015). Then, using Equation (9), we can determine the dynamical mass in the disk (Σ_dyn). The value of Σ_dyn can be compared with the stellar and gas surface densities , which can be determined from the NIR images and H i intensity maps.

This comparison is especially important for the outer disks of galaxies where there is no visible stellar disk supporting the H i disk. In these regions, the equilibrium is given simply by taking Σ_s≈0, so that we get Σ_dyn=Σ_{H i}. It must be noted that in such cases when there is no visible stellar disk, the value of z_g0 will correspond to the scale height of the dynamical mass component that provides the binding gravity for the H i gas disk. This is important for understanding the nature of the dynamical mass component, whether it is part of an oblate halo or whether it is dark matter within the disk as discussed in Section 5.

However, for disk regions where there is no visible stellar disk and Σ_dyn>Σ_{H i}, it means that the disk dynamical mass is larger than the detected baryonic mass of the disk. This could indicate the presence of a very low-luminosity, undetected stellar disk; alternatively, there could be halo dark matter associated with the disk in the form of an oblate inner halo or a thick disk. The mass surface density of the dark matter associated with the disk, Σ_dm, can be determined from the difference between the dynamical and H i disk, i.e.,

In the following sections, we apply this method to the outer H i disks of four large galaxies that are close to face on and three dwarf galaxies that are also close to face on. Our main aim is to see if Σ_dyn is larger than Σ_{H i} in the outer regions of galaxy disks and if so, by how much.

This method can be applied only to face-on galaxies that have an inclination angle (i) close to 0. This is to minimize the rotational velocity component along the line of sight so that the velocity gradient is smaller within the finite beam of the telescope. It allows a more reliable estimate of σ_{zH i} from the width of the H i line profiles (van der Kruit & Shostak 1982). But such galaxies are hard to find, especially if high-resolution moment maps of the H i distribution are also essential. Hence, we have relaxed the criterion to include galaxies with i<40° (Table 1). The value of σ_{zH i} can then be approximately determined by averaging the values over concentric annuli in the disks. If we assume that (Dehnen & Binney 1998), and using cylindrical coordinates (r, ϕ, z) for the disk, then from Das et al. (2019),

where σ_{H i} is derived from the width of the H i profiles. The correction factor introduced in Equation (13) ranges in value from 1 (at 0°) to 0.45 (at 45°). It leads to a lower estimate of the disk dynamical mass, so not using the correction would lead to higher dynamical mass estimates.

Observations of edge-on disk galaxies show that the cool H i disk can have an FWHM of 260pc in the central region of the disk but may flare out to an FWHM of 1.6kpc in the extreme outer parts (Matthews & Wood 2003; Zschaechner & Rand 2015). In the H i plots of Matthews & Wood (2003), we find that to a first approximation, the H i disk thickness appears to be fairly uniform and has a value of ≈±30'' or 0.75 kpc. The extended disks in these observations are similar to the H i-dominated outer disks of late-type galaxies and the diffuse disks of gas-rich dwarfs. Other observations of edge-on disk galaxies indicate that the H i disks have a z_1/2(H i) value of ≈0.5kpc as shown in Figure 25 of O'Brien et al. (2010), unless there is a warping or flaring of the outer disk.

In our calculations, the z_1/2(H i) value represents the disk height that contains most of the H i mass. Hence, we have assumed the disk thickness corresponds to the density 0.1ρ_g0, where . This height contains 90% of the H i surface density as shown below:

Hence, the exponential scale height of the vertical H i gas distribution z_g0 is related to the half z height z_1/2 by

An accurate estimate of Σ_dyn should include a radial variation of z_1/2 for both the stellar and H i disks as they both affect the values of Σ_dyn. But because this study is mainly aimed at comparing the Σ_dyn with the Σ_{H i} in the outer disks of galaxies, we will use constant dynamical disk width values of z_1/2=0.5kpc for the large disk galaxies (NGC 628, NGC 6946, NGC 3184, and NGC 4214). For the dwarf galaxies, because they have puffier stellar/H i disks (DDO 46, DDO 63, and DDO 187), we will use these two values of z_1/2=0.5 and 1kpc. The stellar disk thickness is relevant for only one galaxy (DDO 187) as discussed in Section 4. We will assume a similar relation as Equation (15) for the stellar exponential vertical scale length z_e.

We have chosen our galaxies based on the following criteria:

(i)The H i gas should extend well beyond their star-forming stellar disks. This is because the expression in Equation (9) holds for H i disks in hydrostatic equilibrium, and this might not apply to regions where there are AGN outflows or massive star-forming regions which can increase the turbulent gas pressure and destroy the disk equilibrium. However, on average, over sections of the disk, the mechanical energy input from star formation may still produce a H i gas layer in dynamical equilibrium with the gravitational potential. So, our assumption is that whatever causes the H i velocity dispersion, the gas is largely in equilibrium from a dynamical standpoint.

(ii)The galaxies should be nearby so as to obtain several beamwidths across the disks and hence several values of the azimuthally averaged H i velocity dispersion.

(iii)The galaxies should have low inclinations in order to trace the vertical motion (z-direction). We have finally selected seven galaxies (Table 1) that have relatively high H i gas masses, are at distances 2–11Mpc, and have inclinations less than 40°.

The first four galaxies, NGC 628, NGC 6946, NGC 3184, and NGC 4214, are large, late-type spirals that have H i to stellar disk mass ratios M(H i)/M(star)<1 (Table 1) and host star formation in their inner regions. But in all of them, the H i gas disk extends well beyond the R₂₅ radius of the galaxy and the old stellar disk. They are all part of the THINGS survey (Walter et al. 2008). The galaxies NGC 628, NGC 6946, and NGC 3184 are grand-design spirals with moderate star formation rates (SFRs) of values 1–5, but NGC 4214 has a much lower SFR of 0.05 (Walter et al. 2008). The size, gas-rich nature, and SFR of NGC 4214 is more akin to that of late-type dwarf galaxies (Das et al. 2019) than large spirals. The galaxies NGC 628 and NGC 6946 also have compact H ii regions in their outer disks (Ferguson et al. 1998; Lelièvre & Roy 2000), which is rare and often attributed to cold gas accretion from the intergalactic medium (IGM) which can trigger star formation (Thilker et al. 2007). Their outermost star-forming regions are associated with their inner spiral arms and can be traced into the outer disk using GALEX UV images (Gil de Paz et al. 2007).

Figure 1 shows an example of how the H i disk can extend well beyond the stellar disks of late-type galaxies such as NGC 628. The H i disk is about three times the size of the stellar disk. This is a well-known typeI extended UV (XUV) galaxy (Thilker et al. 2007); such galaxies are characterized by UV emission from star formation along spiral arms in their extended disks. Most of the UV emission coincides with the stellar disk but faint trails of UV emission extend into the outer H i disk, following the disk spiral structure.

The remaining three galaxies, DDO 46, DDO 63, and DDO 187, are gas-rich dwarfs and are all part of the LITTLE THINGS survey (Hunter et al. 2012), which is a survey of nearby, gas-rich irregular/dwarf galaxies. All three galaxies have high H i to stellar disk mass ratios M(H i)/M(star)>1 (Table 1) but very low SFRs with values ranging from 0.001 to 0.005 (Hunter et al. 2012). Although they appear to be irregular in shape, they clearly have disks as indicated by their rotational to vertical velocity ratios v_rot/σ_z>1 (Johnson et al. 2015). These classical low-luminosity dwarfs are dark matter dominated, and their rotation curves indicate that their dark matter halos have flat cores rather than cuspy profiles (Oh et al. 2015).

The H i maps of NGC 628, NGC 6946, NGC 3184, and NGC 4214 were obtained from THINGS and have a spectral resolution of either 2.6 kms⁻¹ or 1.3 kms⁻¹ and spatial resolution of ∼6'' in the robust weighting scheme. This is important for obtaining a good H i surface density profile. For example, in NGC 628 (Table 1), the H i images have a spatial resolution of ∼6'' or 212pc assuming a distance of 7.3Mpc. The H i velocity dispersions were derived by fitting a single Gaussian function to the H i emission profiles. For the dwarfs (DDO 46, DDO 63, DDO 187), the H i data were obtained from LITTLE THINGS (Hunter et al. 2012), which has a spectral resolution of either 2.6kms⁻¹ or 1.3kms⁻¹ and a spatial resolution of ∼6'', which matches the resolution of THINGS. To derive the stellar masses, we used the mid-infrared (MIR, 3.6 μm) Spitzer IRAC images of the galaxies from the Spitzer Infrared Nearby Galaxies Survey (Kennicutt et al. 2003).

We have not included the molecular hydrogen gas (H₂) masses for the galaxies in this study mainly because we are interested in the dynamical masses of the outer disks of galaxies, where there is very little H₂ gas. These regions lie beyond the R₂₅ radius, which is the radius at the optical 25 mag arcsec⁻² isophote. The molecular gas in the galaxies NGC 628, NGC 6946, NGC 3184, and NGC 4214 lie well within their R₂₅ radii (Leroy et al. 2009). Molecular gas is not detected in DDO 63 (Leroy et al. 2009), DDO 46, or DDO 187 (Taylor et al. 1998).

To determine the disk surface density for the H i disk, we used the MIRIAD task ellint to determine the mean and total fluxes within annuli of width equal to the mean beamwidth of the observations (Sault et al. 1995). The annuli were centered at the optical center of the galaxy, and we used a fixed inclination angle and position angle (Table 1), which are the same as those of THINGS (Walter et al. 2008). The H i mass was calculated using the relation M(H i)=, where S_v is the flux in units of Jykms⁻¹, and the multiplicative factor of 1.4 is the correction for including helium (Table 1).

To determine the stellar mass, we used a mass-to-light ratio of 0.5 (McGaugh & Schombert 2014; Table 1). Total luminosities were determined directly from archival IRAC images using the techniques outlined in McGaugh & Schombert (2014) from the 3.6 μm curve of growth. Note that the H i mass is an order of magnitude larger than the stellar mass for the dwarf galaxies but is comparable to the stellar mass in the large spiral galaxies (Table 1). The stellar 3.6 μm surface brightness at the outermost disk radii lies between 24 and 26 mag arcsec⁻² for the dwarf galaxies, but for the larger disk galaxies, it varies between 21 and 27 mag arcsec⁻². However, in all galaxies, the surface brightness profile has a decreasing trend in the outer disk (Figures 2 and 3) and the stellar disk mass surface density appears to be falling rapidly.

The figures in Section 4 show the surface mass density of H i and stars (Σ(H i), Σ(s)) for our sample. The stellar disk lies well within the R₂₅ radii for all galaxies except for DDO 187. The H i gas disk is the most extended component, reaching out to twice the R₂₅ radius for nearly all the galaxies except NGC 4214.

We followed the procedure outlined in Ianjamasimanana et al. (2012, 2017) while deriving the radial variations of the H i velocity dispersion across the H i disk of the galaxies. In summary, we line up individual velocity profiles to the same reference velocity and coadd them to get azimuthally averaged high signal-to-noise ratio (S/N) profiles. Note that only individual profiles with an S/N better than three times the rms noise level in a single channel map were coadded. We then fit the high-S/N stacked profiles with a single Gaussian function. The width of the fitted Gaussians represents the velocity dispersion of the bulk of the H i gas. As studied in detail in Ianjamasimanana et al. (2017; see also Ianjamasimanana et al. 2012 and Mogotsi et al. 2016), the stacking procedure outlined above effectively reduces the effects of noise in the derivation of the velocity dispersion. In addition, a Gaussian fit to the profile gives a more robust estimate of the velocity dispersion as opposed to the more straightforward moment map analysis. This is because moment map values are more prone to the effects of low-level uncleaned fluxes than single Gaussian fitting values unless a very careful clipping is applied. For a full discussion of this, we refer the reader to Ianjamasimanana et al. (2017). Note that in their papers, Ianjamasimanana et al. (2012, 2015) separate the H i profiles in terms of broad and narrow Gaussian components. Here, our choice of a single Gaussian fit is motivated by the fact that we are mainly interested in the outer disks of galaxies where the contributions of the narrow components to the global shapes of the azimuthally averaged velocity profiles are not important (for details see Ianjamasimanana et al. 2015). Thus, a separation into narrow and broad Gaussian components is not relevant for the purpose of this work. Figure 2 shows the radial variation of the velocity dispersion across the galaxy disks. Note that the large galaxies, and especially NGC 6946, clearly show nondeclining H i velocity dispersion in the outer parts of their disks.

The nearly face-on orientation of the galaxies in our sample means that the H i velocity dispersion closely traces the velocity in the z-direction (σ_{H i}). We use Equation (13) with the respective inclination angles (Table 1) to obtain σ_{zH i}. As reported in the literature, the velocity dispersions have a much shallower profile than the SFR density profiles even in the outer disks where star formation is absent or unimportant. This indicates that mechanisms other than star formation are also important to drive the line width of the H i.

One of the main aims of this paper is to investigate what gravitationally binds the extended H i disks of galaxies in regions where there is no visible stellar disk. For example, Figures 3 and 4 show that the stellar disks traced by 3.6μm emission have smaller radii than the H i disks. Also, the stellar disks lie within the R₂₅ radii for all galaxies except for DDO 187. Within the R₂₅ radii of the large galaxies (Figure 1), the stellar mass surface density Σ_s is several magnitudes higher than Σ_{H i}. The gravity of the stellar disk maintains the vertical structure of the H i disk. The star formation within this region can give rise to turbulence in the gas, which can contribute to the gas pressure, which helps to balance the gravitational force of the disk.

But in the regions R>R₂₅, where the stellar disk surface density Σ_s is very low, there must be some other mass to gravitationally bind the H i disk because just the self-gravity of H i cannot maintain the vertical structure. We have applied our analytical results discussed in Section 2 to these regions, i.e., radii R>R₂₅, by assuming vertical equilibrium and applying Equation (11), i.e., . In Figure 3, we have compared Σ_dyn with Σ_{H i} for the normal galaxies using the vertical disk height z_1/2=0.5 kpc. Figure 4 shows the same for the dwarf galaxies but for two vertical disk scale heights, z_1/2=0.5 and 1kpc. Observations of edge-on galaxies suggest that z_1/2=0.5 kpc is a good approximation for the vertical height of the H i layer in large disk galaxies (Zschaechner & Rand 2015). But dwarf galaxies have puffier H i disks, hence z_1/2=0.5 kpc and 1kpc are suitable values for this pilot study (Kamphuis et al. 2011). The assumption here is that the dynamical mass has a scale height similar to the H i vertical scale height. The Σ_dyn decreases for larger vertical disk heights, which is to be expected as the smaller z_1/2 values mean larger disk gravity and hence larger dynamical mass (Figure 4).

From Figure 3, it is clear that Σ_dyn as traced by the H i velocity dispersion is at least 1.5 to 2 times the value of Σ_{H i} in the region R>R₂₅ for the galaxies NGC 628, NGC 3184, and NGC 4214. For NGC 6946, the difference between Σ_dyn and Σ_{H i} is not very significant but because the disk is very extended, the total dynamical mass beyond R₂₅ does become important. Table 2 shows the H i, stellar, and dynamical masses determined for the regions R>R₂₅. The upper radii for mass estimates are listed in column 6, and the R₂₅ radii (in the B band) are given in Table 1. The upper radius is limited by the signal to noise of the H i velocity dispersion, which traces the dynamical mass. In the case of the gas-rich dwarf galaxies, M(dyn) is much larger than M(H i) for nearly all radii (Figure 4) and especially for R>R₂₅. For radii R<R₂₅, the M(H i) is comparable to or larger than the stellar mass M(stars). This is not surprising as it is well known that the gas to stellar mass ratios are much higher for dwarfs, i.e., M(H i)/M(stars)≫1 (Table 1). But for the outer radii (R ≫ R₂₅), depending on the assumed vertical disk heights of the disks, the M(dyn) can vary from 2 to 14 times the value of M(H i). The most striking case is DDO 187, where M(dyn) is several times the value of M(H i) in the region R>R₂₅ for both z_1/2=0.5 and 1kpc (Table 2).

Table 2.Approximate Baryonic and Dynamical Masses in the Region R>R_25B

Galaxy	M(H i)	M(stars)		M(dyn)	Relevant		Mdyn–[M(H i)+M(stars)]
Name	(M⊙)	(M⊙)	(kpc)	(M⊙)	Radii		(M⊙)
NGC 628	2.05×109	⋯	0.5	9.11×109	(1–1.9)R25B, (12.7–24.2) kpc	4.4	7.1×109
NGC 6946	2.05×109	⋯	0.5	9.62×109	(1–1.8)R25B, (14.2–25.6) kpc	4.7	7.6×109
NGC 3184	8.79×108	⋯	0.5	4.56×109	(1–1.6)R25B, (13.7–21.9) kpc	5.2	3.7×109
NGC 4214	9.16×107	⋯	0.5	4.33×108	(1–1.3)R25B, (4.8–6.2) kpc	4.7	3.4×108
DDO 46	1.23×108	⋯	0.5	5.71×108	(1–2.7)R25B, (2.0–5.3) kpc	4.6	4.5×108
		⋯	1.0	2.85×108		2.3	1.6×108
DDO 63	6.44×107	⋯	0.5	5.30×108	(1–1.9)R25B, (2.3–4.3) kpc	8.2	4.7×108
		⋯	1.0	2.65×108		4.1	2.0×108
DDO 187	5.20×106	0.96×106	0.5	8.81×107	(1–2.6)R25B, (0.58–1.5) kpc	14.3	8.2×107
			1.0	4.41×107		7.2	3.8×107

Note. The H i, stellar, and dynamical masses are derived for the regions R>R_25B. The lower and upper radii are listed in column 6, and the R_25B radii are given in Table 1.

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To compare the dynamical and baryonic masses (M(H i)+M(stars)) of our galaxies in the outer disk regions, we have summed over the surface densities Σ_dyn and + in the outer radii (Table 2). For all galaxies except DDO 187, we find that there is no stellar mass visible in the 3.6 μm NIR images at radii R>R₂₅. Hence, they have M(baryonic)=M(H i) for these regions. For the normal, large galaxies NGC 628, NGC 6946, NGC 3184, and NGC 4214, assuming a disk half-thickness or -height value of z_1/2=0.5kpc, the M(dyn)/M(baryonic) varies between 4.4 and 5.2. But for the dwarf galaxies, because their stellar disks are more diffuse and maybe thicker, we have calculated M(dyn)/M(baryonic) for z_1/2=0.5 and 1kpc (Table 2). It varies between 4.6 and 14.3 for z_1/2=0.5, but varies between 2.3 and 7.2 for z_1/2=1kpc. As the disk thickness is increased, the difference between the dynamical and baryonic masses becomes smaller, which follows from the form of the expression for Σ_dyn (Equation (11)). M(dyn)/M(baryonic) is largest for the gas-rich dwarf galaxy DDO 187. It is clear from Table 2 that for some of the dwarfs, the nonluminous mass associated with the disks is very significant and comparable to or more than the stellar disk mass of the galaxy (Table 1).

An estimate of the nonluminous mass associated with the disks for the region R>R₂₅ is shown in the last column of Table 2. The values of M_dyn–[M(H i)+M(stars)] are of the order of ∼10⁹M_⊙ for the larger galaxies (NGC 628, NGC 6946, NGC 3184) but around ∼10⁷–10⁸M_⊙ for the smaller/dwarf galaxies. In fact, for the dwarf galaxies, this mass is comparable to the stellar disk masses. The smaller galaxies are also the ones that are more gas rich and lower in luminosity. Hence, the outer disks of dwarf and low-luminosity galaxies maybe dark matter dominated as discussed in the next section. However, it must be noted that the mass in the disk region R>R₂₅ is in no way comparable to the total halo masses of the galaxies, which are typically 10–100 times the stellar masses in bright to low-luminosity galaxies. Our present sample consists of nearly face-on galaxies for which reliable rotation curves cannot be obtained and hence we cannot estimate the dark halo masses.