Abstract

1 Introduction

M33 is a low-luminosity spiral galaxy in the Local Group. Its large angular extent and well determined distance make it ideal for a detailed study of the radial distribution of visible and dark matter. The interplay between the dark and visible matter as well as the nature and the radial distribution of dark matter are still open crucial issues with strong implications for galaxy formation and evolution theories. M33 has three advantages over its brighter neighbour M31: (i) being of lower luminosity, it is a dark-matter-dominated spiral in which it is easier to disentangle the dark and luminous mass components (Persic, Salucci & Stel 1996;Corbelli & Salucci 2000, hereafter CS); (ii) it has no prominent black hole or bulge component in the centre (Gebhardt et al. 2001) and therefore it can be used to test if the innermost dark matter density profile is consistent with cosmological models which predict a central density cusp; (iii) being at lower declination, high-sensitivity 21-cm observations with the Arecibo radio telescope made possible a velocity field map out to large galactocentric radii. CS have used this velocity field map, together with aperture synthesis data ofNewton (1980) and with the tilted ring model results ofCorbelli & Schneider (1997), to derive the rotation curve of M33 out to 13 disc scalelengths. This unveiled large-scale properties of the mass distribution in M33, such as the lack of correlation between the dark and visible matter, the dark halo mass and the possible radial slope of the dark matter density. However some ambiguities are left, especially in the inner regions, where the low spatial resolution of 21-cm observations and the possible role of a stellar nucleus or of the molecular gas component have not allowed us to establish the steepness of the dark matter density profile unambiguously.

Several authors, e.g.Bolatto et al. (2002), have shown the advantages of using CO observations for determining the central densities of dark matter haloes. A full map of the CO emission in M33 would not only provide higher spatial resolution data on the velocity field but would also give a first estimate of the molecular mass distribution in this galaxy. The very blue colour of M33 and the presence of prominent Hii regions indicate vigorous star formation activity in the nuclear region and along the spiral arms: here the molecular gas component might represent a substantial fraction of the total disc mass. CO observations of the innermost 500 pc of M33 (Wilson & Scoville 1989) have confirmed in fact that here the molecular gas has a higher surface density than the atomic gas. Recently Heyer et al. (in preparation) have fully mapped the CO emission in M33 using the FCRAO 14-m telescope. A detailed analysis of the molecular gas distribution and emission features will be presented elsewhere. In this paper we derive only the average radial distribution of the molecular gas mass (Section 2) and analyse the kinematics of the CO gas (Section 3). The implications for the dark matter halo models are shown inSection 4. An accurate knowledge of the velocity field and of the mass surface densities over a large disc area are useful for understanding the processes which trigger the star formation in this galaxy, such as disc instabilities, spiral arms, cooling and heating mechanisms. A possible link between the radial extent of the observed star formation region and the gravitationally unstable region in the disc is discussed inSection 5.

2 The Molecular Data

The nearby spiral galaxy M33 has been fully mapped in the COJ= 1 − 0 transition with a velocity resolution of 0.81 km s⁻¹ and a spatial resolution of 45 arcsec using the FCRAO 14-m telescope (Heyer et al, in preparation). A 22.5-arcsec spatial sampling is obtained over an area of about 900 arcmin² centred on and. For a distance to M33 of 0.84 Mpc (Freedman et al. 2001), which will be used throughout this paper, this means that the total disc area observed is about 44 kpc² at a spatial resolution of 165 pc. Here we shall use the data at the original spatial and spectral resolution to derive the kinematical information on the molecular gas. We analyse in detail each spectrum by assigning a spectral window ΔV=V^f−Vⁱ to any detectable emission feature. There are a few cases where more than one spectral window is needed but secondary signals are always quite low. If no emission is clearly present in a given spectrum, a spectral window is set based on the diffuse emission detectable when averaging nearby positions. We compute the integrated brightness temperatureF in K km s⁻¹ by integrating over ΔV the telescope temperature observed at each position in the sky, and dividing this number by the main beam efficiency of the telescope (which is 0.45 at 115 GHz). In order to derive the average radial distribution of the molecular gas we need a concentric ring model. 45 concentric rings of width 150 pc cover all the M33 surface area observed. They are tilted so as to best fit the neutral hydrogen distribution (Corbelli & Schneider 1997). Inside this area the position angle of the best fitting ring model changes only slightly, and therefore we consider it constant and equal to 22°. The inclination of the rings increases from 50° to 58° moving radially outwards, as described in details insection 3 of CS. We use the concentric ring model for assigning a face on radial distance to each observed position, and for deriving the inclination-corrected average flux in each ring. Following Wilson's results (Wilson 1995), we use the standard conversion factor between the CO luminosity and the line-of-sight column density ofH₂ (= 2.8 × 10²⁰×F cm⁻²= 4.5 ×F M_⊙ pc⁻²).

The resulting average surface density of molecular hydrogen, perpendicular to the plane of the galaxy (i.e. corrected for the disc inclination respect to our line of sight), is shown inFig. 1. The estimated total mass of molecular gas is 2 × 10⁸ M_⊙. This is a factor 6 higher than measured byWilson & Scoville (1989) in the smaller nuclear region. The molecular hydrogen mass is about 10 per cent the neutral hydrogen mass (2.2 × 10⁹ M_⊙ forD= 0.84 Mpc). The surface density distribution does not peak at the centre but shows a peak at a few hundred pc from the nucleus, as noted previously byWilson & Scoville (1989). The most prominent CO emission features are located in the innermost 1-kpc region and along the spiral arms. This can be seen already in the 1.3 × 7 kpc² preliminary emission map (Corbelli 2000) oriented along the M33 major axis. An exponential is fitted to the azimuthally averaged surface density data from 0 to 6 kpc. We leave out from the fit the last 6 points shown inFig. 1, which seem to indicate a plateau at the level of 0.6 M_⊙ pc⁻². In reality we do not have enough sensitivity and coverage in our survey to determine the slope of theH₂ distribution so far out. The best fit to is shown by the straight line inFig. 1 and is given by

(1)

Figure 1.

Average surface density of molecular hydrogen as a function ofR in the plane of the M33 disc. The straight line is the best fit to all data withR < 6 kpc (open dots). Larger radii were only partially covered by our survey.

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Therefore the CO radial scalelength in M33 is larger than theB- andK-band scalelengths (1.9 and 1.3 kpc respectively,Regan & Vogel 1994). InFig. 2 we show the Hi as well as the total (Hi+ H₂) gas distribution in this galaxy. The total gas surface density can be described by a Gaussian profile with half-width at half maximum of about 6 kpc. Note that the bump of the spiral arm feature around 2 kpc is visible in the azimuthal averages of the molecular and atomic gas shown inFig. 2.

Figure 2.

Average surface density of molecular hydrogen (filled triangles) and of atomic hydrogen (open circles) as a function ofR in the plane of the M33 disc. The solid line is the sum of the two surface densities.

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3 The CO Rotation Curve

AsWilson & Scoville (1989) have already pointed out from a comparison of interferometric and single-dish data, small clouds and diffuse emission in M33 contain a non-negligible fraction of the total molecular mass. Half of the molecular gas mass that we estimate in M33 resides in fact in regions with integrated flux less than 3F_σ≃ 1.33 K km s⁻¹.F_σ is the integrated noise defined in terms of the spectral window ΔV as

(2)

where σ_rms is the noise in the spectrum at the original resolution of 0.81 km s⁻¹. We fit single or double Gaussians to all signals with integrated fluxF >F_5σ with a success rate of 75 per cent. There is no significant shift between the Gaussian and the flux-weighted mean velocities (difference is ≲2 km s⁻¹ and is never above 5 km s⁻¹). In deriving kinematical information we shall consider only spectra withF > 5F_σ, well fitted by Gaussian profiles, from positions at angles α < 45° from the major axis. These conditions restrict the analysis to a smaller radial range than observed. Luckily the prominent CO features come from locations close to the major axis and therefore we do have many points to average for building up a rotation curve. Corrections for the inclination of the disc and for the angle α between the spectrum position and the major axis are applied according to the tilted ring model described in the previous Section. Radial bin averages of the flux weighted mean velocities are plotted inFig. 3, also for the northern and southern side of the galaxy separately. Error bars in each bin show the dispersion of the velocities about the mean.

Figure 3.

The northern and southern half data after averaging 10 adjacent points in each radial bin. Error bars show the dispersion in each bin. The rotation curve derived from averaging the northern and southern data is shown with open circles connected by a heavy solid line.

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In the innermost 1 kpc the south and north rotation curves coincide reasonably well. Beyond this radius there are some differences both in the dispersions and in the mean values ofV. Velocities in the southern half of M33 have higher dispersion than in the northern half. Also the southern side shows more clearly wiggles which can be interpreted as signs of density-wave perturbations associated with the spiral arms. The detection of such features in the northern side is more doubtful. Residual velocities, obtained by subtracting the average rotational velocityV(R) from the rotational velocity measured in each spectrum,V_obs(R), are shown inFig. 4. In the left-hand panel ofFig. 4 the map of residual velocities is superimposed to the used tilted ring model, after rotating the galaxy by 22° clockwise in the plane of the sky. Cross and square symbols are for positive and negative residuals respectively, and their size is proportional to the amplitude of the residual velocities. Since the approaching side of the galaxy is the northern side (y > 0), a positive residual there means that the local velocity is less than the average rotational velocity. For each quadrant of the galaxy, the right-hand panels ofFig. 4 show the residual velocities as function of the galactocentric radius. It hard to judge whether there are signs of non-circular motions and spiral density waves in the velocity residuals map. Part of the difficulty is due to the fact the bulk of the CO emission comes from localized regions and is not smoothly distributed throughout the disc. A better attempt can be made in the future with the use of a numerical simulation for any given model assumption.

Figure 4.

The left-hand panel shows the map of the residual velocities,V_obs−V, whereV is the average rotational velocity atR. It is superimposed to the used tilted ring model, after rotating the galaxy by 22° clockwise in the plane of the sky. Cross and square symbols are for positive and negative residuals respectively. The size of the symbols is proportional to the amplitude of the residuals. In the right-hand panels ofFig. 4 the residual velocities are shown as a function of galactocentric radius in each quadrant of the galaxy.

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Comparison of the rotation curve inFig. 3 with the 21-cm data ofNewton (1980) reveals a discrepancy in the northern inner region, where atR≲ 0.5 kpc the Hi velocities are lower by about 10 km s⁻¹ than the CO velocities. This might be due to the lack of atomic gas in the innermost region which, together with the larger 21-cm beam, limits the accuracy of the atomic line measure. Finite angular resolution is not affecting significantly the velocities measured in our CO survey. To estimate any possible ‘beam smearing’ effect we assume that the true velocity, for a given mass model, is that resulting from rotation curve and perform a numerical simulation to derive the observed velocities (van den Bosch & Swaters 2001). The observed velocities at each position on the plane of the sky are the flux-weighted means of the rotational velocities along the line of sight, corrected for the assumed ring model and convolved with the shape of the telescope beam. We then compare the binned data with equally binned values of the velocities resulting from the simulation (we exclude points which form an angle α > 45° with the major axis). Differences between the observed and the true velocities, for mass models which give a good fit to the data, are always smaller than 0.6 km s⁻¹ except in the innermost 0.5 kpc where the beam-smearing effect can lower the observed velocity as much as 2 km s⁻¹.

4 The Mass Components and Constrains on Cosmological Halo Models

Rotational velocities, which we use to derive the visible and dark matter distributions in M33, are shown inFig. 5. The innermost point atR≃ 150 pc is from Hα measurements ofRubin & Ford (1985). Our CO data extending over the interval 0.2 ≤R < 3.4 kpc have been averaged into the subsequent 11 bins. For 3.4 ≤R < 5.5 kpc data are taken from 21-cm observations ofNewton (1980) at the Cambridge Half-Mile Telescope while for larger radii we use Arecibo 21-cm data (CS and references therein). The plotted error bars in each bin are twice the dispersion of the velocities about the mean.

Figure 5.

The M33 rotation curve (points). (a) The best-fitting model (solid line) using Burkert profile for the dark halo density distribution, and the nuclear model (1). Also shown are the dark halo contribution (dot–dashed line), the stellar disc + nucleus (short-dashed line) and the gas contribution (long-dashed line). (b) The same as in (a) but for NFW dark halo profile withC₁₀₀= 8 and the nuclear model (2). The left panels show an enlargement of the inner fit.

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We consider the dynamical contribution of two visible mass components: stars, distributed in a disc and in the nucleus, and gas, in molecular and in neutral atomic form. For the stellar disc we consider that the mass follows the light distribution in theK band, well fitted by an exponential with scalelengthR_d∼ 1.3 ± 0.2 kpc (Regan & Vogel 1994). After subtracting the exponential disc from the surface brightness, a central emission excess remains atR≲ 0.5 kpc (Bothun 1992;Regan & Vogel 1994) which has been attributed to spiral arms, to an extended semistellar nucleus or to a diffuse halo or bulge. We shall refer to this excess as the ‘nucleus’ of M33 despite the fact that in the literature the word ‘nucleus’ is often used for the light excess observed only atR≲ 10 pc (Kormendy & McClure 1993). Nuclear rotational velocities have been observed in Hα along the major axis between 20 and 200 pc from the centre (Rubin & Ford 1985), and seem roughly constant at about ∼27 km s⁻¹. This flatness of the rotation curve is quite puzzling and needs further investigation once uncertainties on the nuclear rotational velocities will be established together with a knowledge of the velocity field along different directions. In this paper we model the rotation curve atR≳ 150 pc only, considering two different possibilities for the nuclear componentV_n at these radii.

• (i)

  V_n∝R^0.5/(R+a). This rotational velocity approximates the dynamical contribution of a spheroidal mass distribution whose surface density can be fitted by a de VaucouleursR^1/4 law (Hernquist 1990). This scaling law has been used byRegan & Vogel (1994) to fit theJ-band nuclear surface brightness. The compatibility of the fit with photometric data in other bands (e.g.Bothun 1992) is somewhat uncertain, as is the value of the scalelengtha. Suggested values ofa lie in the range 0.1 ≤a≤ 1 kpc, which we will consider in this paper.

• (ii)

  V_n∝R^−0.115, as if core collapse has happened or is happening in the compact stellar nucleus (Cohn 1980;Kormendy & McClure 1993). The corresponding power-law mass density distribution develops up to a limiting outer radiusR_c, which should also be given. ForR >R_c,V_n∝R^−0.5. We consider as possibleR_c values the radius of each data point of the rotation curve inside the optical region i.e.R_c < 8 kpc. We consider also the caseR_c≪ 150 pc, which is appropriate if most of the nuclear mass resides at very small radii and the nucleus can be treated as a point source throughout the extent of the present rotation curve.

Hereafter, we will refer to these models as nuclear model (1) and (2) respectively. It has been established by CS that the visible baryons in M33 are only a small fraction of the total galaxy mass, and that a dark matter halo is in place. We will use the M33 rotation curve presented here to test in details the consistency of the required halo density profile with theoretical models which predict a well-defined dark matter distribution around galaxies. Namely, in addition to the widely used non-singular isothermal dark halo model we shall consider a spherical halo with a dark matter density profile as originally derived byNavarro, Frenk & White (1996, 1997, hereafter NFW) for galaxies forming in a cold dark matter scenario (hereafter CDM). We consider also the Burkert and the Moore dark matter density profile (Burkert 1995;Moore et al. 1998). All models have two free parameters to be determined from the rotation curve fit. The density distribution for a non-singular isothermal dark halo can be written as

(3)

The density profile proposed byBurkert (1995) has also a flat density core:

(4)

and a strong correlation between the two free parameters ρ_B andR_B has been found by fitting the rotation curve of low-mass disc galaxies.

The NFW density profile is

(5)

Numerical simulations of galaxy formation in CDM models find a correlation between ρ_NFW andR_NFW which depends on the cosmological model (Avila-Reese et al. 2001;Eke, Navarro & Steinmetz 2001). Often this correlation is expressed using the concentration parameterC_Δ≡R_Δ/R_NFW andM_Δ orV_Δ.R_Δ is the radius of a sphere containing a mean density Δ times the critical density,V_Δ andM_Δ are the characteristic velocity and mass insideR_Δ. We use a Hubble constantH₀= 65 km s⁻¹ Mpc⁻¹ and show results for Δ= 100, as in CDM halo models ofEke et al. (2001), unless otherwise specified.

A similar profile but with a different central slope has been suggested instead by numerical simulations ofMoore et al. (1998) for galaxies forming in CDM scenarios:

(6)

For each dark halo model and each nuclear model we shall determine five free parameters applying the least-squares method to the observed rotation curve. These free parameters are as follows: the dark halo core radiusR_iso,NFW,M,B, the dark matter density ρ_opt computed atR=R_opt≡ 3.2R_d≃ 4.1 kpc, the ratio between the stellar disc massM_d and the blue luminosity of the galaxyL (L≃ 5.7 × 10⁹ L_⊙), the nuclear massM_n, and the nuclear scalelengtha or the cut-off radiusR_c. The 1σ, 2σ and 3σ ranges are determined using the reduced χ² and assuming Gaussian statistics. They are computed by determining in the free parameters space the projection ranges, along each axis, of the hypersurfaces corresponding to the 68.3 per cent, 95.4 per cent and 99.7 per cent confidence levels.

Both the non-singular isothermal dark halo model and the Burkert dark matter density profile give good fits to the rotation curve since the minimum reduced χ² is 0.7 for the nuclear model (1) and 1.2 for the nuclear model (2). The best-fitting values of the free parameters area= 0.4 kpc,M_n= 3 × 10⁸ M_⊙,M_d/L= 1 M_⊙/L_⊙,R_iso= 7 kpc orR_B= 12 kpc, ρ_opt= 5 × 10⁻²⁵ g cm⁻³ for the nuclear model (1), andR_c= 0.7 kpc,M_n= 10⁸ M_⊙,M_d/L= 1 M_⊙/L_⊙,R_iso= 8 kpc orR_B= 13 kpc, ρ_opt= 5 × 10⁻²⁵ g cm⁻³ for the nuclear model (2). The 3σ ranges for the free parameters of both the isothermal and the Burkert model are

The resulting 3σ range for the nuclear scalelengtha in the nuclear model (1) covers the whole range we have considered i.e. 0.1 ≤a≤ 1 kpc. For the nuclear model (2) we find a lower limit toR_c instead. This means that the nucleus cannot be considered a point source for any possible dark halo models we are examining. This ‘dynamical’ conclusion agrees with photometric data (Bothun 1992;Minniti, Olszewski & Rieke 1993;Regan & Vogel 1994) showing an excess of light inward of 0.5 kpc after subtracting the exponential disc. Unfortunately estimates of the nuclear luminosityL_n are still quite uncertain: in theB band 7 × 10⁷≤L_n,B≤ 3 × 10⁸ L_⊙ (Bothun 1992;Regan & Vogel 1994), and we cannot limit the nuclear stellar mass range further based on possible unrealistic values of the mass to light ratio (Bell & de Jong 2001). An extended nucleus is not required instead if the density of the dark matter scales asR^−1.3±0.1 throughout the halo; these type of solutions has been investigated by CS and provides also good fits to the rotation data presented in this paper. Both the Burkert and the isothermal model require a very extended core region of nearly constant dark matter density.Fig. 5(a) shows the best fit using the Burkert profile with the nuclear model (1). The 1σ and 3σ probability contours fora= 0.4 kpc in the ρ_opt–R_B plane are displayed inFig. 6(a). The correlation found by Burkert between ρ_B andR_B implies a relation between ρ_opt andR_B shown with crosses inFig. 6(a). We can see that the M33 acceptable parameters lie close to, but outside, the Burkert correlation line. So either the found scaling relation between ρ_opt andR_B should be slightly modified to include M33, or it changes for values ofR_B larger than those considered by Burkert. A Burkert model which fits M33 is indistinguishable from an isothermal model since both require a large constant-density core region and, even at the outermost fitted radius, the dark matter density does not decline yet as anR⁻² orR⁻³ power law. The dark matter density profile suggested by Moore fails to fit the observed rotation curve in M33.

Figure 6.

(a) The light dots indicate the 3σ confidence area for the Burkert dark halo model in theR_B log(ρ_opt) plane; the heavy dots are for the 1σ confidence area. The cross line indicates the correlation found by Burkert. The 3σ (light dots) and 1σ (heavy dots) confidence areas for the NFW dark halo model are shown in (b) in the lg(M_Δ)-lg(C_Δ) plane for Δ= 100. The dashed and dash–dotted lines indicates the numerical simulation results for modelW₈ andS_0.9 respectively ofEke et al. (2001).

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For the NFW dark matter density profile for Δ= 100 or 200 the minimum χ² is 0.7 for the nuclear model (1) and 1.0 for the nuclear model (2). The best-fitting values of the free parameters for the nuclear model (1) area= 0.4 kpc,M_n= 2 × 10⁸ M_⊙,M_d/L= 0.8 M_⊙/L_⊙,R_NFW= 140 kpc, ρ_opt= 6 × 10⁻²⁵ g cm⁻³. For the nuclear model (2)R_c= 0.7 kpc,M_n= 8 × 10⁷ M_⊙,M_d/L= 0.8 M_⊙/L_⊙,R_NFW= 300 kpc, ρ_opt= 6 × 10⁻²⁵ g cm⁻³. The 3σ ranges for the free parameters are

The corresponding 3σ upper limit for the concentrations isC₁₀₀ < 12.5 for nuclear model (1) andC₁₀₀ < 11 for nuclear model (2), while the corresponding best fits to the rotation curve requireC₁₀₀≃ 4 andC₁₀₀≃ 2.5 respectively. For all the dark haloes we are examining the nuclear model (2) provides a flatter rotation curve than model (1) inward of the centremost data point fitted, in agreement with the preliminary measures ofRubin & Ford (1985). The central steepness of the NFW dark matter density profile lowers the nuclear stellar mass required to fit the rotational data.Fig. 6(b) shows the 1σ and 3σ areas in theM₁₀₀−C₁₀₀ plane. The dashed and dash–dotted lines in the same figure are the fits to numerical simulation results for structure formation in the cosmological modelsW₈ andS_0.9 respectively ofEke et al. (2001). Both model assume matter density parameters Ω₀= 0.3 and Λ₀= 0.7, a power spectrum in the form given byBardeen et al. (1986) with σ₈= 0.9 and shape parameter Γ= 0.2.S_0.9 corresponds to this ‘standard’ΛCDM spectrum, whileW₈ mimics a warm dark matter power spectrum because in addition its power is reduced on scales smaller than that of the characteristic free streaming massM_f= 8 × 10¹¹ M_⊙. It can be easily seen inFig. 6(b) that the possibleC_Δ andM_Δ values for M33 are consistent with a wark dark matter cosmology but lie below the relation found in simulated standard CDM haloes. TheseM_Δ−C_Δ relations have however an associated scatter. Even a small scatter, σ(lgC_Δ) = 0.1 as suggested for galaxies which did not undergo major mergers (Wechsler et al. 2002), makes the M33 dark halo still compatible with standard ΛCDM galaxy formation models. We display inFig. 5(b) the fit to the M33 rotation curve using a NFW dark halo profile withC₁₀₀= 8, compatible with a standard ΛCDM and a warm dark matter cosmology. The nuclear model (2) is used. The reduced χ² and the values of the free parameters found from the best fit are χ²= 1.5,R_c= 0.7 kpc,M_n= 7.5 × 10⁷ M_⊙,M_d/L= 0.7 M_⊙/L_⊙,R_NFW= 35 kpc, ρ_opt= 7 × 10⁻²⁵ g cm⁻³.

5 Surface Densities of Visible Matter and the Global Stability of the Disc

In the previous section we have determined the dark matter and stellar contribution to the M33 rotation curve. Despite the fact that the radial scaling law of the dark matter density required for fitting the rotation curve is not unique, the total baryonic and dark mass up to the last measurable point (R= 17 kpc) are determined with small uncertainties. The total stellar mass in the M33 disc is estimated to be 3–6 × 10⁹ M_⊙. The mass of the dark halo at the outermost observable radius is about a factor 10 higher than the stellar disc mass. The neutral hydrogen plus the molecular gas mass is 2.4 × 10⁹ M_⊙. We don't have a firm estimate of the nuclear stellar massM_n, but all three possible dark matter models discussed in this paper require an extended distribution. Its mass lies in the range 5 × 10⁷≤M_n≤ 8 × 10⁸ M_⊙ if the dark halo has a large size core of constant density, while 3 × 10⁷≤M_n≤ 4 × 10⁸ M_⊙ if the dark halo has a central cusp as in the NFW model. We can estimate the total surface density of visible baryons in M33 neglecting at the moment the small contribution from the ionized gas.Fig. 7 shows the total baryonic surface density computed by adding the helium to the molecular and the neutral atomic gas (a 1.33 factor correction), and adding to this the nuclear and stellar disc surface density. We use the mass model displayed inFig. 5(b), for which we show also the dark matter surface density within 0.5 kpc from the plane of the galaxy. We extrapolate the fits of the stellar and molecular gas surface density out to 16 kpc but it is likely they experience a truncation at smaller radii. For the Hi surface density only we know that it declines sharply beyond 15–16 kpc owing to ionization effects by an extragalactic background (Corbelli & Salpeter 1993). The gas distribution is likely to extend further out but the dominant phase becomes ionized atomic hydrogen. Assuming a gaseous vertical extent of 0.5 kpc above and below the galactic plane, we can say that the surface density of visible baryons falls below the dark matter surface density contained in the M33 disc forR≳ 10 kpc. A similar conclusion applies also for the Burkert best-fitting model.

Figure 7.

The surface density of baryonic and dark matter in M33. The mass model used forFig. 5(b) has been used here to compute the stellar contribution from the nucleus and the disc (dot–dashed line) and the dark matter surface density within 0.5 kpc height from the galactic plane (continuous line). The short dashed line is the Hii= He surface density, the long-dashed line is the H₂+ He surface density, the solid heavy line is the sum of the two and of the stellar surface density.

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Using our accurate measurement of the rotational velocity gradient and of the surface density of the baryonic and non baryonic components, we can check if the onset of massive star formation in M33 is strongly related to a global instability of the disc. One of the most popular tools used to assess the gravitational stability of gaseous discs is the Toomre stability criterion:

(7)

where Σ_g is the gas surface mass density,ƒ is the disc thickness factor, κ is the epicyclic frequency (Toomre 1964), G is the gravitational constant,c_g is the gas velocity dispersion. We shall use bothƒ= 1 as for a thin disc andƒ= 0.7 to mimic instead a disc with a finite thickness (Romeo 1992). The criterion described byequation (7) is not at all related to the star formation efficiency but it correctly predicts the occurrence of a star formation threshold radius which, in many spiral galaxies, has been determined from the drop of the Hα surface brightness (e.g.Martin & Kennicutt 2001). In M33Kennicutt (1989) observed such a drop atR_*≃ 6.4 kpc. However, since the early work of Kennicutt it is well known that the simple Toomre stability criterion fails to account for the active star forming region in the disc of low-mass spirals such as M33 (Elmegreen 1993;van Zee et al. 1997;Wong & Blitz 2002). An alternative stability criterion includes the shear rate of the disc and it may be more appropriate for irregular galaxies (Elmegreen 1993;Hunter, Elmegreen & Baker 1998). It is based on the consideration that making new stars is possible only when there is enough time for the instability to grow, before shear stretches the perturbation apart. This corresponds to the following condition:

(8)

where the Oort constantA=−0.5R dΩ/dR is the local shear rate. InFig. 8 we display Σ_T and Σ_A, usingc_g= 6 km s⁻¹. This value is at the lowest end of the cold gas velocity dispersion range estimated in M33 (6–9 km s⁻¹Warner, Wright & Baldwin 1973;Thilker 2000). The thin and thick continuous lines inFig. 8 show the mass surface densities forƒ= 1 and forƒ= 0.7 respectively. The arrow indicatesR_*. We can see inFig. 8(a) that atR≃R_* the azimuthally averaged surface gas density for a thin disc falls below the threshold for a gravitational instability to develop according to the shear rate criterion. We shall investigate now if considering additional sources of gravity in the disc, a modified version of the classical Toomre criterion can still give correct predictions for the star formation threshold radius.

Figure 8.

The stability of the M33 disc forc_g= 6 km s⁻¹. The dash–dotted line and the dashed line in each panel show Σ_T and Σ_A respectively, as a function of radius. An arrow has been placed at the observational value ofR_*. The thin and thick continuous lines are forƒ= 1 andƒ= 0.7 respectively. We plotƒΣ_g in (a) andƒ(Σ_g+Σ_*) in (b) forM_d/L= 0.7 and. In (c) we add the dark matter surface density within a 0.5 kpc layer above and below the galactic plane, while in (d) the dark matter is computed within a radially varying vertical height of the gas (see text for details).

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Since the masses of stars and gas in the M33 disc are comparable, we are encouraged to use a stability parameter for a two-component fluid. FollowingWang & Silk (1994) we define Σ_*≡Σ_s×c_g/c_s, where Σ_s andc_s are the surface density and velocity dispersion of the stars in the disc. Using the epicyclic frequency or the shear rate of the disc, the stability condition reads:

(9)

Fig. 8(b) showsƒ(Σ_g+Σ_*), usingM_d/L= 0.7 M_⊙/L_⊙ for deriving Σ_* (as for the mass model shown inFig. 5(b)), andc_g/c_s= 0.3. Forc_g= 6 km s⁻¹ this last ratio impliesc_s= 20 km s⁻¹, a value observed in the central regions of M33 (Kormendy & McClure 1993;Gebhardt et al. 2001). We can see that with a value ofM_d/L slightly higher than 0.7, the Toomre criterion for a thin disc predicts correctly a gravitationally unstable disc forR <R_*. If finite thickness effects are taken into account or ifc_g is higher than 6 km s⁻¹ atR≃R_* we need the shear rate criterion.

FollowingHunter et al. (1998) we consider also the possible influence of dark matter gravity. If Σ_dm is the dark matter surface density within the vertical extent of the gas, we can rewriteequation (9) as

(10)

InFig. 8(c) we showƒ(Σ_g+Σ_dm+Σ_*) assuming a gaseous extent of 0.5 kpc above and below the galactic plane and a NFW dark matter profile as used for the rotation curve fit displayed inFig. 5(b). In this case finite thickness effects, or equivalently a velocity dispersion as high as 9 km s⁻¹, make the disc gravitationally unstable, according to the Toomre criterion, exactly forR <R_*. This solution is not unique. If the gas is in hydrostatic equilibrium for example, the gas vertical extension and the velocity dispersion are not independent variables. An approximation to the scaleheight of the gas for a galactic disc of stars and gas in hydrostatic equilibrium in a dark matter halo is given by

(11)

Since Σ_dm is the dark matter surface density between +h_g and −h_g the above formula should be solved numerically for determiningh_g and Σ_dm. InFig. 8(d) we plotƒ(Σ_g+Σ_dm+Σ_*), for Σ_dm between ±h_g for the same dark matter density used inFig. 8(c). We find thath_g varies between 20 pc and 1 kpc through the radial extent of the disc analysed in this paper. This model reduces the radii of the unstable star forming regions respect to the constant scaleheight model displayed inFig. 8(c), making both Σ_A and Σ_T suitable indicators of the growth of instabilities in the disc. Which one it is better to use for defining a threshold radius in M33 is therefore still unclear since this depends on the radial variations of the involved parameters such as sound speed, thickness etc. which are not well determined yet. It is important to notice however that since the densities of visible baryons and dark matter are well constrained at intermediate radii, the intersections between Σ_A or Σ_T andƒ(Σ_g+Σ_dm+Σ_*) do not depend on which mass model we use, as soon as it is compatible with the rotational velocity data. The Burkert best-fitting mass model, for example, gives the same solutions displayed inFig. 8 for a NFW dark matter halo.

6 Summary

A full map of the nearby galaxy M33 in the COJ= 1 − 0 transition has given additional data for improving the accuracy of the rotation curve of this galaxy and the knowledge of the visible baryons distribution. The molecular gas peaks in the central regions with a surface density of about 7 M_⊙ pc⁻² and declines exponentially with a radial scalelength of about 2.5 kpc down to about 0.6 M_⊙ pc⁻² atR≃ 6 kpc. Our sensitivity was not sufficient to outline the molecular gas radial distribution further out and in particular to detect any possible sudden drop of the azimuthally averaged molecular gas surface density. The molecular gas is not smoothly distributed in the inner disc since there are remarkable mass concentrations along the spiral arms. Some wiggles of about 10 km s⁻¹ are visible in the rotation curves of the two separate halves of the galaxy, which are reminiscent of density wave perturbations. Details of atomic and ionized gas distribution in M33 can be found inHoopes & Walterbos (2000) and inThilker, Braun & Walterbos (2001). The rotation curve derived using Hα data for the innermost point, the CO data and the Hi 21-cm line data, has been used to constrain the visible baryonic mass and dark matter halo models. The total gas mass (Hi+ H₂+ He) is ∼3.2 × 10⁹ M_⊙. This is of the same order of the stellar disc mass, estimated to be between 3×10⁹ and 6×10⁹ M_⊙. The total visible baryonic mass of M33 can therefore be as high as 10¹⁰ M_⊙. The dark halo mass out to the last measured point (17 kpc from the centre) is ∼5 × 10¹⁰ M_⊙.

Both the non-singular isothermal halo and the Burkert halo model, give good fits to the M33 rotation curve when the flat density core is extended. The pureR⁻² orR⁻³ outer scaling law applies only beyond the outermost observed radius. Dark matter density profiles with an innerR^−1.5 cusp (Moore et al. 1998) are too steep to be consistent with the rotation of the central regions of M33 and are ruled out. Halo models with milder central cusps, such as the NFW radial profile, fit well all the observed rotational velocities. The mass of the virialized NFW dark halo for this galaxy is ≥5 × 10¹¹ M_⊙ and the halo size comparable to the distance between M33 and its bright companion M31 (∼180 kpc). The baryon fraction in M33 is then ≤0.02. The required concentrationsC_Δ are below the scaling laws found by numerical simulations of structures formation in a standard cold dark matter scenario (Ω₀= 0.3, Λ₀= 0.7, σ₈= 0.9) and favour cosmological models which predict less concentrated dark haloes (Eke et al. 2001;Zentner & Bullock 2002;van den Bosch, Mo & Yang 2003). We have shown however that if an associated scatter in the numerical simulation results is considered (Wechsler et al. 2002), the M33 dark halo is still compatible with galaxy formation models in a standard ΛCDM cosmology. The finding that dark matter haloes are consistent with constant density core models or with central cusps less steep thanR^−1.5, is known as the cusp–core degeneracy (van den Bosch & Swaters 2001). It is remarkable that this degeneracy holds also in M33 which is a galaxy with small uncertainties about its distance, its gaseous and stellar content, and with circular velocities sampled over a wide range of radii: from galactocentric distances of a few parsecs out to 13 disc scalelengths. For this galaxy it seems that the degeneracy holds mostly because of a nuclear stellar component, which regulates the dynamics of the innermost 0.5-kpc region. This extended ‘stellar nucleus’, with mass between 3 × 10⁷ and 8 × 10⁸ M_⊙, is not only dynamically required by the rotational data but it is also evident from photometric data (Bothun 1992;Minniti et al. 1993;Regan & Vogel 1994). We have considered a core collapse model and a de VaucouleursR^1/4 law for the nuclear stellar density but additional data on the kinematics and light distribution of the central few hundred pc region are needed to constrain better the mass distribution there.

Having a very accurate rotation curve and a good estimate of the gas, stellar and dark matter surface density in the M33 disc, we have addressed the question on the possible coincidence of the active star formation region in this galaxy with a gravitationally unstable region. Azimuthal averages of the observed gas surface densities are above the critical densities in the star-forming region of M33 only if one considers the minimum acceptable value of the gas velocity dispersion and a stability criterion based on the shear rate of the disc. However, the simple Toomre condition predicts a star formation threshold radius in agreement with the observed drop in the Hα surface brightness if the additional compression due to the stellar disc or to the dark matter surface density is considered.

I acknowledge Steve Schneider and Mark Heyer for having given me the opportunity to use the CO data prior to publication. I am also very thankful to the anonymous referee who made several excellent suggestions for improving the quality of the work presented in the original manuscript.

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