A WIDE-FIELD HIGH-RESOLUTION H i MOSAIC OF MESSIER 31. I. OPAQUE ATOMIC GAS AND STAR FORMATION RATE DENSITY

The H i self-absorption (or HISA) phenomenon has been well documented in the Galaxy since the pioneering work of Radhakrshnan (1960) and more recently by Gibson et al. (2005) in a large swath of the Galactic plane (145°>l > 75°). The physical conditions required to witness HISA are a "sandwich" geometry in which a cooler spin temperature foreground gas is seen toward a higher brightness temperature background gas at the same radial velocity. The brightness decrement of an HISA feature is proportional to this temperature difference in the absence of additional complicating factors. The simplest circumstance to imagine is a single warm, semi-opaque background feature which has a brightness temperature comparable to its own spin temperature together with a single cooler, opaque feature in the foreground. In this simple case the HISA temperature decrement would give some information about the spin temperature of the foreground component. More complicated geometries are easily conceivable. Adding a third warm, semi-opaque foreground component to make a warm–cool–warm "sandwich" would significantly fill in the detected temperature decrement. Clearly, the interpretation of the HISA temperature decrements in physical terms is poorly constrained.

The largest individual features seen in the Gibson et al. (2005) Galactic HISA sample extend over about 5° at 2 kpc distance, and so have lengths as large as 175 pc. A subset of the Galactic HISA features, such as those kinematically associated with the Perseus spiral arm, is organized over tens of degrees making the complexes at least 1 kpc long. The departure from the Galactic edge-on geometry to the ∼78° inclination of M31 (Braun 1991) means that the necessary alignment conditions for witnessing HISA are essentially eliminated on large scales in M31. Instead, the "sandwich" geometry that can yield large-scale HISA features in the Galaxy would be projected into spatially resolved, parallel "slices" of semi-opaque gas of different spin temperature. We identify the filamentary local minima in peak brightness seen in M31 with the colder opaque features that are responsible for large-scale HISA in the Galaxy. We use the term "self-opaque" to emphasis the importance of internal optical depth effects in determining the profile shapes of these features, as distinct from "self-absorption" which implies a substantial temperature contrast of features that overlap both along the line of sight and in radial velocity.

By comparison to the kpc extent of Galactic HISA complexes, the self-opaque filamentary minima in M31 (for example the one running from (α, δ)∼ (00:45:25, +41:36) to (00:45:55,+41:50) in Figure 6) are often in excess of 10 arcmin in length corresponding to more than 2 kpc. One such linear feature is seen to cross very near the line of sight to the background continuum source J004218+412926 (=B0039+412), as seen in Figure 7. H i absorption measurements for this source and several others have been published previously in Braun & Walterbos (1992). We will comment further on these sources below. More complex local minima of all sizes can be seen in the M31 data wherever sufficient local intensity is present in both position and velocity. In contrast, the filamentary network of systematically broadened profiles (traced by the VC) are even more ubiquitous, since profile broadening can be detected independent of the emission characteristics of the surroundings.

A subset of Galactic HISA features is closely associated with molecular clouds as traced by OH and CO emission (e.g., Goldsmith & Li 2005), while the majority have no apparent CO counterparts (Gibson et al. 2000). The conjecture has been made (e.g., Gibson et al. 2005) that those HISA features without associated CO detections may represent an earlier evolutionary phase in molecular cloud formation and that such features may be organized over kpc-scale regions by the passage of spiral density wave (or other forms of) shocks. Only after subsequent cooling and compression would complex molecule– and star–formation proceed. Some of the most dramatic self-opaque features discernible in the M31 peak brightness images appear to lie on the leading edges of spiral arm structures, for example the feature extending from (α, δ)∼ (00:45:25, +41:36) to (00:45:55, +41:50) (we can deduce the upstream edge of bright H i features from the counter-clockwise sense of rotation that follows from the approaching major axis P.A. = 232° east of north and the near-side minor axis P.A. = 322°). This appears to be quite a general phenomenon; discernible as a sharp local minimum in peak H i along the leading edge of essentially every arm segment with a suitably bright background. Comparison of the peak H i brightness with the integrated CO(1–0) observed with the IRAM 30 m at 23 arcsec resolution (Nieten et al. 2006) in Figures 11 and 12 suggests a rather good correlation of the peak brightness of H i with integrated CO at this resolution of about 100 pc at radii less than about 12 kpc; with a distinct decline of associated CO to larger radii. In contrast, the filamentary self-opaque minima, particularly those associated with the leading edges of spiral features, have only very patchy correspondence with CO(1–0) features, even near 12 kpc.

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While some indications for self-opacity in the H i line are indicated by qualitative features, like flat-topped spectral profiles, obtaining quantitative estimates of the "hidden" mass in neutral hydrogen is considerably more challenging. The "classical" method of H i opacity correction (Schmidt 1957; Henderson et al. 1982; Braun & Walterbos 1992) has consisted of calculating the column density from

$$\begin{equation} N_{{\rm H}\,\hbox{\scriptsize {\sc i}}} = -1.823 \times 10^{18} T_{\rm S} \Delta V \ln \biggl (1 - {T_{\rm B} \over T_{\rm S}} \biggr) {\rm cm}^{-2}, \end{equation} \tag{ 1 }$$

where a single cool component spin temperature, T_S, is assumed to apply for all spectra. The relevant spin temperature has been chosen to be similar to the highest brightness temperature actually observed, for example T_S = 125 K for the Galaxy (Henderson et al. 1982), or has been estimated from a fit to the relation between observed absorption opacity and the emission brightness, T_S = 175 K for M31 (Braun & Walterbos 1992). In both cases, a single spin temperature is assumed to apply throughout the galaxy in question, which is clearly a poor assumption, and the highest observed brightnesses (those with T_B > T_S) have an undefined column density.

Rather than assuming that a single spin temperature might be appropriate for an entire galaxy, let us consider a more realistic scenario in which the major atomic clouds in a galactic disk are approximately isothermal on scales of 100 pc. From Braun (1997) we have an estimate of the face-on surface covering factor of such features within the star forming disk in a sample of nearby galaxies of about 15%. This implies that even at the moderately high inclination of M31, of 78°, we are unlikely to have more than a single major cloud along any line of sight in the absence of strong warping. (Even in the strongly warped case, there need not be widespread velocity overlap of these components.) In addition, the good correlation between absorption opacity and the emission brightness (Braun & Walterbos 1992) in both the Galaxy and M31 suggest that the contribution of warm (5000–10⁴ K), optically thin gas in the extended environment of major clouds to the emission brightness is only a few Kelvin, although distributed over the relevant line width of some tens of km s⁻¹. These considerations suggest that detailed modeling of the peak (and not the wings) of an emission profile as an isothermal cloud may provide a useful estimate of the underlying physical parameters, provided the physical and velocity resolution is sufficiently high (about 100 pc and 2 km s⁻¹) and the galaxy is viewed at an inclination of less than about 81°.

Such an approach has been considered previously by Rohlfs et al. (1972), who have documented the precision and expected biases that might pertain to parameter estimation as a function of the peak signal to noise for an unblended, isothermal cloud. We will return to the question of the expected uncertainties in such an approach below.

The line profile due to a spatially resolved isothermal H i feature in the presence of turbulent broadening can be written as

$$\begin{equation} T_{\rm B}(V) = T_{\rm S}\lbrace 1-\exp [-\tau (V)]\rbrace \end{equation} \tag{ 2 }$$

with

$$\begin{equation} \tau (V) = {5.49\times 10^{-19}N_{{\rm H}\,\hbox{\scriptsize {\sc i}}} \over T_{\rm S}\sqrt{2\pi \sigma ^2}} \exp \Biggl (-0.5{V^2 \over \sigma ^2}\Biggr), \end{equation} \tag{ 3 }$$

where the velocity dispersion, σ, has units of km s⁻¹ and is the quadratic sum of an assumed thermal and nonthermal contribution σ² = (σ²_T + σ²_NT) with the thermal contribution given by $\sigma _T=0.093 \sqrt{T}_k$, for a kinetic temperature, T_k. While such profiles have a Gaussian shape for low ratios of the H i column density (in units of cm⁻²) to spin temperature, $N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}/T_{\rm S}$ they become increasingly flat-topped when this ratio becomes high. For simplicity we assume that the H i spin and kinetic temperatures are equal, T_k = T_S, which should apply to regions of moderate volume density although not in general, see Liszt (2001). We do not consider the possibility of self-absorption (as discussed in Section 4.1) due to multiple temperature components along individual lines of sight, despite seeing occasional evidence for this phenomenon on small scales. Simple simulations to explore such multitemperature models demonstrate the great challenges of meaningfully constraining the absorber properties. Even negligible masses of cold H i can result in dramatic modulation of the composite spectrum. The additional free parameters associated with such a composite spectrum make for an intractable fitting problem.

We show examples of model isothermal (half-)spectra in Figure 13. Shown in the figure are a sequence of increasing σ_NT = 2–12 by 1 km s⁻¹ at a fixed log($N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}$) = 22 and log(T_S) = 1.7 (on the left) as well as a sequence of increasing log($N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}$) = 21.5–23.5 by 0.2 at a fixed log(T_S) = 1.7 and σ_NT = 7 km s⁻¹ (on the right). We found the best-fitting model spectra for the brightest spectral feature along each line of sight using the 30'' data with full velocity resolution when the observed peak brightness temperature exceeded 10 K (10 times the rms noise) and the truncated peak (where brightnesses greater than 50% of the peak were encountered) spanned at least five independent velocity channels. This truncation was done to both isolate single spectral components from possibly blended features as well as eliminating potential broad wings from the fit. The data were compared to a precalculated set of model spectra spanning the range log($N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}$) = 20.0–23.5 by 0.01, log(T_C) = 1.2–3.2 by 0.01 and log(σ_NT) = 0.3–1.5 by 0.04. A search in velocity offset was done over displacements of −4 to +4 by 1 km s⁻¹ with respect to the line centroid as estimated by the first moment of each truncated peak. We purposely limited the range of log(T_C) to that expected for the cool neutral medium (CNM) where opacity effects might be expected to occur, but included sufficiently high temperatures (at least ten times the observed peak brightness temperature) that negligible opacity solutions are always available for the fit. lines of sight which might be dominated by the warm neutral medium (WNM) with spin temperature of perhaps 8000 K, will be fit by a moderately high T_S (since such profiles would not display opacity effects), combined with an apparent "nonthermal" dispersion of about 10 km s⁻¹. There is no method to discriminate between such intrinsically warm gas and cooler gas with a comparable nonthermal broadening of the profile. All that can be discriminated from the profile shape in the first instance is the presence or absence of opacity effects. Only if significant opacity effects are detected (implying τ ⩾ 1) can the physical parameters be constrained. However, since our primary goal with this procedure is the correction of apparent column density, this is precisely the regime in which we can expect some utility of this fitting method. This conclusion is consistent with the experience of Rohlfs et al. (1972) who indicate that useful parameter fits are only achieved with a peak signal to noise greater than about 10 (such as we require in our fitting) paired with peak opacities in excess of unity. Much higher signal to noise is required for a meaningful fit at lower peak opacities. Systematic biases of the fit parameters are expected to be below about 10% in this regime.

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Plausible spectral fits were possible for most lines of sight. However, there are locations in the M31 disk where emission features which are likely at significantly different distances along the line of sight become strongly blended in velocity. Such circumstances are most apparent in position–velocity images made parallel to the major axis, in which continuous spiral arm segments can be tracked as linear features with approximately constant orientation (as documented by Braun 1991). The gradient of velocity with major axis distance,

$$\begin{equation} {dV \over dX} = {V_c(R) \sin (i(R)) \over R } \end{equation} \tag{ 4 }$$

is proportional to the local projected rotation velocity divided by the radial distance of the feature within the galaxy. The occurrence of apparently intersecting linear features in position–velocity plots is a strong indication for velocity blending of physically unrelated features which reside at different radii in the galaxy. The most prominent instances of such velocity overlap occur in four locations within M31 at major axis distances of both ±30 arcmin and minor axis distances of both ±10 arcmin.

The challenges of velocity blending are illustrated by the sequence of individual spectra with overlaid fits shown in Figure 14. The sequence of spectra from left to right and top to bottom are ordered by decreasing quality of fit, as measured by the normalized χ² = Σ_i(D_i − M_i)²/(Nσ²). Our fitting method could deal successfully only with instances of sufficiently separated spectral features, where the local minima drop below 50% of the peak. More blended features are straddled with a single broad profile fit. From a careful examination of individual cases of blended and unblended profiles, based on the presence or absence of intersecting linear position–velocity features as described above, we established a cutoff in fit quality of χ² = 25 that most effectively separated these two regimes. Only those fits with a χ² below this cutoff were retained, resulting in rejection of some 4% of the total. Concentrations of rejected fits are located in the expected regions (noted above).

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The results of our spectral fitting were recorded as images of the physical parameters, $N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}^{\rm Fit}, T_{\rm S}$ and σ_NT together with the integral of brightness temperature over velocity that corresponds to the column density of the fit, ∫T^Fit_BdV. This allowed calculation of a total corrected column density image from,

$$\begin{equation} N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}^{\rm Tot} = N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}^{\rm Fit} + 1.823\times 10^{18}\biggl (\int T_{\rm B}^{\rm Tot}dV-\int T_{\rm B}^{\rm Fit}dV\biggr), \end{equation} \tag{ 5 }$$

where ∫T^Tot_BdV is just the usual image of integrated emission. In this way any spectral component not accounted for by the fit to the peak was retained, albeit with the assumption of negligible opacity for this residual. Those lines of sight with insufficient fit quality (χ² > 25) were assumed to have negligible opacity. The two column density images (assuming negligible optical depth and fitting for it) are contrasted in Figures 15 and 16. Substantially more fine-scale structure is apparent in the opacity-corrected column density, primarily in the form of compact features and narrow filaments. The narrow filaments, in particular, correspond to those seen in the images of VC, shown in Figures 6–8. Peak column densities are enhanced by about an order of magnitude over the uncorrected version.

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Given the uncertainties in the fit parameters, we do not attach undue significance to each pixel in the images of corrected column density, since they may easily be in error by a factor of 2 or more, but are encouraged by the spatial continuity of the solutions, since the fit to every line of sight is carried out independently of all others. Some indication for the relevance of the fit results can be obtained by comparison with high signal-to-noise observations of H i absorption toward background continuum sources. As noted previously, several bright background continuum sources that have been studied by Braun & Walterbos (1992) fall fortuitously near regions of significant self-opacity, particularly the sources B0039+412 and B0044+419 for which peak optical depths of τ_max = 0.46 ±  0.02 and τ_max = 1.06 ±  0.06 are observed. (Note that (1 − e^−τ) rather than τ is tabulated in the reference.) The model fits show very substantial structure in the vicinity of these background sources, making it difficult to estimate the likely line-of-sight parameters. We have determined the average peak model opacity over a 3 × 3 pixel² region that is offset azimuthally from the direction of each of these sources by one beamwidth. For these two lines of sight this yields τ_max = 0.4 ± 0.4 and 1.1 ± 0.8. Although the pixel-to-pixel variation is very large, and may well represent real structure rather than fitting errors, the order of magnitude of the opacity is estimated correctly in these cases.

The total detected H i mass of M31 is increased by the opacity corrections by about 30%, from 5.76 × 10⁹ to 7.33 × 10⁹ M_☉. In view of the necessity of discarding all blended profiles, this should be regarded as a lower limit to the actual mass correction (within the limitations of our approach).

The best-fit spin temperature and nonthermal velocity dispersion are shown in Figures 17 and 18. Regions of significant opacity (log($N_{{\rm H}\,\hbox{\scriptsize {\sc i}}}$) > 22) have spin temperatures of less than about 100 K and are often organized into patchy filamentary complexes. While there is some organized spatial structure in the distribution of nonthermal velocity dispersion, it is not obviously correlated with either the total column density or spin temperature. Not surprisingly, there is a significant correlation of the nonthermal velocity dispersion of the fits with the image of VC (shown in the right-hand panels of Figures 6–8).

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The basic radial trends of observed and derived quantities are illustrated in Figure 19. The peak observed brightness temperature (panel a) is seen to first increase dramatically with radius out to about 12 kpc and then subsequently to decline out to the largest detected radii. Even the median peak brightness exceeds 50 K near 12 kpc, while the distribution extends out beyond 100 K. This can be compared with the very general trend documented by Braun (1997) for a positive gradient in peak brightness to occur within the actively star forming disk of nearby galaxies. The unprecedented physical sensitivity obtained for M31 allows us to track the atomic gas properties out to more than three times larger radii. The apparent column density distribution in panel b follows a similar trend, illustrating the exponential character of both the increase to—and the subsequent decline from—the peak near 12 kpc. A 6 kpc scale length exponential function is overlaid on the plots in panels b, c, d, and h to illustrate how well this scale length describes the gas mass density. Intriguingly, the old stellar population of M31, as traced by 3.6 μm Spitzer imaging between 1 and 12 kpc radius is best fit with a 6.08 ± 0.09 kpc scale length exponential (Barmby et al. 2006). The corrected H i column density, as shown in panel c, has a median ridge line that closely tracks the apparent column, but also demonstrates that outliers to higher corrected column occur primarily at radii between about 7 and 30 kpc. Adding in the molecular contribution to mass surface density (as described in detail in Section 4.4) in panel d yields only minor changes to the radial trend of the atomic column density. The most significant is a slight addition of mass density near 12 kpc, where the molecular contribution permits the total to better track the 6 kpc exponential to slightly smaller radii. Inside of about 12 kpc is a sharp cutoff that can be described with a 3 kpc exponential scale length. Beyond about 28 kpc, where opaque H i is no longer found, there is a significant truncation of the disk exponential (neutral) mass density, corresponding to a 2 kpc scale length. The corresponding face-on column density (scaled by 1/cos(78°)) where this outer truncation sets in, is 5 × 10¹⁹ cm⁻².

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The radial distribution of VC (Figure 19(e)) shows a similar trend, rising rapidly out to about 10 kpc, then declining out to about 30 kpc and then plummeting. Both the derived nonthermal velocity dispersion (panel f) and spin temperature (panel g) show some broad systematic decline with radius beyond 10 kpc. For the spin temperature, it is only the lower envelope of the distribution which represents "opaque" fits. These "opaque" cool component temperatures are as low as 20 K near 7 and 25 kpc, but increase to about 60 K near 10 kpc. This is quite distinct from the most highly populated ridge line, which represents predominantly "transparent" lines of sight. The ridge line shows a similar trend, but is offset to higher temperatures by about an order of magnitude. Current massive SFR density (as defined in Section 4.4) is shown in panel h. The apparent nuclear peak in SFR density is due to other processes (as discussed below), while current coverage only extends to about 20 kpc. Current SFR density is highly peaked near 12 kpc, where gas mass densities are highest and does not track the 6 kpc exponential of the median gas mass density.

A particular form of correlation of observable parameters deserves special consideration, namely that between the surface density of gas mass and the SFR. Many authors have addressed the relationship between these properties on both a global and local level, beginning with the pioneering study of Schmidt (1959) and most recently by Kennicutt et al. (2007) and Thilker et al. (2007), who probe the resolved relationship of these parameters down to scales of 500 pc within NGC 5194 (M51a) and 400 pc within NGC 7331, respectively. Given the proximity of M31 we can extend this work by more than an order of magnitude to smaller surface area at spatial scales of 100 pc, albeit with a substantial inclination correction that introduces some potential linear smearing (by a factor of 4.8) in one dimension. Mass and star formation rates have been calculated after background subtraction and spatial smoothing of all relevant images to the same 30 arcsec (113 pc) Gaussian resolution as applies to the H i properties.

The mass surface density was calculated from the integrated CO(1–0) emission observed with the IRAM telescope by Nieten et al. (2006) assuming a mean atomic mass of 1.36m_H and an $N_{\rm H_2}$/I_CO conversion factor of 2.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. This differs from the procedure adopted by (Kennicutt et al. 2007, hereafter K07), who keep the comparison in terms of hydrogen nuclei (rather than including a Helium mass correction). The helium mass correction amounts to a shift by +0.13 dex in mass density. Note that the CO data only extend over the central 2° × 05 (or 28 × 32 deprojected kpc in diameter). Radii beyond about 15–20 kpc therefore only have an H i contribution to the calculated mass density. Although the CO brightness at the edges of the sampled region is generally faint and has a low surface covering factor, isolated pockets of CO may well be present at larger radii.

The star formation density was calculated following the methodology of Thilker et al. (2007) from the Spitzer Infrared Array Camera (IRAC) 8 μm and MIPS 24 μm surface brightness of Gordon et al. (2006) together with the GALEX FUV image of Thilker et al. (2005) and Gil de Paz et al. (2007). The "bolometric" luminosity was assumed to be L(bol) = ν_FUVL_ν,obs(FUV) + (1 − η)L(IR) after correcting the FUV image for foreground extinction by the Galaxy, and assuming η = 0.32. The term "bolometric" in this context is a misnomer, since the quantity is not assumed to represent a true integral of the entire SED, but merely to provide sensitivity to both directly emitted UV and reradiated infrared (IR) light. The value of η, which represents the fraction of the IR luminosity due to an older stellar population, is expected to vary substantially with environment within a galaxy. Ideally, this would also be modeled in a position dependent way, but that is beyond the scope of the current discussion.

The total (3–1100 um) IR luminosity per pixel was estimated using the IRAC/MIPS data and the relation

$$\begin{equation} \log \bigg [{L({\rm IR}) \over \nu L_\nu (24\;\mu {\rm m})} \bigg ] = 1.06 + 0.475 \log \bigg [{F_\nu (8\;\mu {\rm m}) \over F_\nu (24\;\mu {\rm m})} \bigg ] \end{equation} \tag{ 6 }$$

and then converted to a SFR density using the projected pixel area and the SFR calibration:

$$\begin{equation} \log \, {\rm SFR}({\rm bol}) (M_\odot \;{\rm yr}^{-1}) = \log \ L({\rm bol}) (L_\odot) - 9.75. \end{equation} \tag{ 7 }$$

The 24 μm data only extend over the brighter disk region of ∼3° × 1°.

Only pixels with significant emission in the spatially smoothed images were used for the comparison, which resulted in thresholds at a brightness of 0.7 K km s⁻¹ for the CO emission, H i columns greater than 19 in the log and sky-subtracted IR brightness greater than 0.06 MJy sr⁻¹ at 24 μm and 0.1 MJy sr⁻¹ at 8 μm. As can be seen in Figure 19, the spatial coverage and sensitivity in the IR only extends out to a radius of about 20 kpc.

We contrast three different versions of the star formation law in Figure 20 utilizing the molecular only, the molecular plus apparent H i and the molecular plus opacity-corrected H i column density. The molecular-only plots are overplotted with both the relation of K07 with slope 1.37, a value which has also been found by Heyer et al. (2004) in M33, as well as a line with unit slope corresponding to a constant gas depletion timescale, as advocated by Wong & Blitz (2002; hereafter WB02). The slope of the molecular-only relationship is approximately matched by the simple linear dependency. The gas depletion timescale of the plotted line is 1 Gyr (for an $N_{\rm H_2}$/I_CO conversion factor of 2 × 10²⁰ cm⁻²/K km s⁻¹ as assumed by Wong & Blitz). This is similar to many of the galaxies analyzed in WB02, although occurring at mass and star formation densities that are at least an order of magnitude lower within M31. The M51a molecular-only relation of K07 is both steeper and shifted to higher mass densities by at least 0.3 dex.

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The total gas mass density plots are overplotted with the relation fit by K07 to the M51a data with slope 1.56. Note that we have plotted SFR density in units of M_☉ pc⁻² Gyr⁻¹ rather than the M_☉ kpc⁻² yr⁻¹ of K07 implying a shift of 3 dex. The different units of both axes (relative to K07) have been taken into account for the overplotted curves. Remarkably, the M51a relation provides a very good fit to the M31 data, despite the fact that it has been derived in the strongly molecular-dominated inner disk of M51 and is being applied throughout the atomic-dominated disk of M31. The highest contours of correlation probability are about twice as narrow for the total gas relation compared to the molecular-only one. In the relation using apparent H i, there is a very steep truncation to mass densities exceeding about 10 M_☉ pc⁻², together with a saturation that sets in near 16 M_☉ pc⁻². This truncation disappears in the opacity-corrected H i relation, together with a small shift of the peak correlation ridge line to higher mass densities. The saturation effect near 16 M_☉ pc⁻², is also alleviated although not entirely eliminated.

Data at the smallest radii (less than 5 kpc) yield the diffuse region of elevated apparent star formation in the total gas panels of the figure. The enhanced 24 μm emission from the nuclear regions of M31 is accompanied by both recombination- and forbidden emission line gas (as first documented by Jacoby et al. 1985) with line ratios suggestive of shock heating, and as such is not due to massive star formation. Essentially no CO is detected from the nuclear region, so this feature is not apparent in panels a and d of Figure 20. Beyond about 5 kpc, the same basic correlation is seen to apply at all radii, with only the range of the observables showing some systematic variation, but not the basic ridge line of their correlation. The sharp truncation of the molecular-only mass density near 1 M_☉ pc⁻² is a consequence of the limited sensitivity. The same is true for SFR densities below about 0.2 M_☉ pc⁻² Gyr⁻¹.

At SFR densities below about 0.4 M_☉ pc⁻² Gyr⁻¹ and total gas surface densities of about 5 M_☉ pc⁻² there appears to be a departure from the power-law relationship. The same feature can be seen in the NGC 7331 data of Thilker et al. (2007) at a similar location, although this is near the noise floor in both cases. This may indeed be a distinct threshold for the formation of molecular clouds from which stars will form. The most substantial contribution to this possible threshold comes from radii in excess of 15 kpc.

Perhaps the most surprising aspect of Figure 20 is the good statistical correlation of total gas mass and star formation density down to spatial scales of about 100 pc and gas masses of only 5 × 10⁴ M_☉, well inside the regime of individual, giant molecular clouds.