---
abstract: 'Star formation in galaxies is triggered by a combination of processes, including gravitational instabilities, spiral wave shocks, stellar compression, and turbulence compression. Some of these persist in the far outer regions where the column density is far below the threshold for instabilities, making the outer disk cutoff somewhat gradual. We show that in a galaxy with a single exponential gas profile the star formation rate can have a double exponential with a shallow one in the inner part and a steep one in the outer part. Such double exponentials have been observed recently in the broad-band intensity profiles of spiral and dwarf Irregular galaxies. The break radius in our model occurs slightly outside the threshold for instabilities provided the Mach number for compressive motions remains of order unity to large radii. The ratio of the break radius to the inner exponential scale length increases for higher surface brightness disks because the unstable part extends further out. This is also in agreement with observations. Galaxies with extended outer gas disks that fall more slowly than a single exponential, such as $1/R$, can have their star formation rate scale approximately as a single exponential with radius, even out to 10 disk scale lengths. H$\alpha$ profiles should drop much faster than the star formation rate as a result of the rapidly decreasing ambient density.'
author:
- 'Bruce G. Elmegreen'
- 'Deidre A. Hunter'
title: Radial Profiles of Star Formation in the Far Outer Regions of Galaxy Disks
---

Introduction
============

The outer disks of spiral galaxies have a low level of star formation (Ferguson et al. 1998; LeLièvre & Roy 2000; Cuillandre, et al. 2001; de Blok & Walter 2003; Thilker et al. 2005; Gil de Paz et a. 2005), even though the gas is gravitationally stable by the Kennicutt (1989) condition. Triggering by other mechanisms, such as turbulence compression (Mac Low & Klessen 2004), supernovae, and extragalactic cloud impacts (Tenorio-Tagle 1981), might be the reason. As a result, radial light profiles should not drop suddenly at the stability threshold, but should taper slowly as various star formation processes get more and more unlikely and the gas supply diminishes. The purpose of this paper is to investigate a simple model of star formation with generalized triggering in a smoothly varying gas disk. We seek to determine what the overall radial light profile might be.

The radial light profiles of spiral and irregular galaxies are typically exponential over 3 to 5 scale lengths (van der Kruit 2001) with rare examples, particularly among low-inclination spirals, going further (Courteau 1996; Barton & Thompson 1997; Weiner et al. 2001; Erwin, Pohlen, & Beckman 2005; Bland-Hawthorn et al. 2005). Some galaxies have another, steeper exponential in the inner disk bulge region (Courteau, de Jong & Broeils 1996), which does not concern us here as it may be the result of gas inflow or bar formation (Kormendy & Kennicutt 2004). Many galaxies also have a steep exponential in the far outer disk (Näslund & Jörsäter 1997; de Grijs, Kregel, & Wesson 2001; Pohlen et al. 2002). This outer exponential is the focus of our discussion. As the outer disk is significantly below the sky brightness and generally difficult to observe, its properties are not well known; it may not even be exponential. van der Kruit (1988) suggested that disk asymmetries can make what is really a sharp outer truncation appear much smoother when the light profiles are azimuthally averaged; he noted that very deep exposures of edge-on disks tend to show sharp edges instead of smooth outer exponentials. Florido et al. (2001) showed how a sharp function could be fitted to outer disk cutoffs. The outer disk profile also depends critically on the level and uniformity of the sky brightness subtracted from the image.

The transition from the main disk exponential to the outer disk profile has several observed characteristics. The outer disk scale length is about half that of the inner disk for both spiral and dwarf irregular galaxies (Hunter & Elmegreen 2006, hereafter Paper I). The ratio of the transition, or “break,” radius, $R_{br}$, to the main disk scale length, $R_D$, is 3 to 4 for spiral galaxies (van der Kruit & Searle 1981; Barteldrees & Dettmar 1994; Pohlen, Dettmar, & Lütticke 2000; Schwarzkopf & Dettmar 2000; Kregel, van der Kruit & de Grijs 2002) and $\sim2$ for dwarf and spiral Irregulars (Paper I). There is a slight increase in this ratio for decreasing $R_D$ among spirals (Pohlen, Dettmar, & Lütticke 2000; Kregel, van der Kruit & de Grijs 2002; Kregel & van der Kruit 2004), and another slight increase for increasing central surface brightness among spirals (Kregel & van der Kruit 2004). The first of these two correlations does not hold for dwarf Irregulars, which have both small disk scale lengths and small ratios $R_{br}/R_D$. The second correlation does hold for dwarf Irregulars. If there is a universal reason for outer disk transitions (as in the present model), then correlations which apply to both spirals and irregulars would seem to be most important. Thus the second correlation, in which $R_{br}/R_D$ increases with central surface brightness, should be viewed as fundamental, and the first simply a result of the second along with the independent correlation between scale length and central surface brightness found by de Jong (1996) and Beijersbergen, de Blok, & van der Hulst (1999). The apparent ratio $R_{br}/R_D$ should also depend slightly on galaxy inclination as a result of a tendency to overestimate $R_D$ for edge-on spirals where central extinction flattens the radial profile.

Exponential light profiles in galaxies have been attributed to several things. Cosmological collapse during galaxy formation, starting with a nearly uniform spheroid, can produce profiles that resemble exponentials out to $\sim 2-6$ scale-lengths (Freeman 1970; Fall & Efstathiou 1980). Exponential disks also arise through radial flows in viscously evolving disks if the star formation rate is proportional to the viscosity (e.g., Lin & Pringle 1987; Yoshii & Sommer-Larsen 1989; Zhang & Wyse 2000; Ferguson & Clarke 2001).

Double exponential profiles have no previous explanation (see review in Pohlen et al. 2004). van der Kruit (1987) proposed that outer disk truncations arise during galaxy formation and the break radius is determined by the maximum angular momentum of the proto-galactic cloud. Kennicutt (1989) suggested that truncation arises where the gas disk drops below the threshold for gravitational instabilities. Elmegreen & Parravano (1994) and Schaye (2004) proposed it arises when the ISM converts to a mostly warm phase, as observed in the outer regions of spirals (Dickey, Hanson & Helou 1990; Braun 1997) and dwarfs (Young & Lo 1996, 1997). Dalcanton et al. (1997), Firmani & Avila-Reese (2000), Van den Bosch (2001), Abadi, et al. (2003), Governato et al. (2004) and Robertson et al. (2004, 2005) simulated galaxy formation with threshold star formation and obtained exponential profiles with an outer disk cutoff. None of these models actually obtained double exponentials, only sharp outer disk truncations.

The theory of disk truncation is highly uncertain, however. The angular momentum in the outer parts of a galaxy can change over time during interactions. The gravitational stability threshold may not be sharp if the ISM cools (Elmegreen 1991) or magnetic forces remove angular momentum (Kim, Ostriker & Stone 2002) during compression. The phase transition may not occur if the outer gas disk tapers slowly, like $1/R$ (Wolfire et al. 2003). All of these uncertainties suggest that refined models may eventually obtain more gradual outer disk truncations.

The presence of double exponentials in dwarf galaxies (Paper I) places immediate constraints on the models. Most dwarfs have nearly solid body rotation curves throughout a large fraction of their optical disks. This means there is little shear, so viscous evolution should not play a significant role in structuring disk profiles. There is also no correlation in our Paper I sample between the break radius and the radius where the rotation curve changes from near solid body in the inner regions to near flat in the outer regions. Thus even the outer exponential is not likely to result from radial migration and evolution related to shear.

Collapse models could in principle be arranged to give the desired radial profiles, but the collapse models in cosmological simulations so far have just given inner exponential disks with relatively sharp outer cutoffs. There have been no suggestions yet about how conditions during galaxy formation could be tuned to give outer double exponentials. One possibility is that galaxy collapse gives a single exponential disk and then subsequent accretion of gas makes the far-outer disk with a different profile (Bottema 1996). This may explain a sudden decrement in the rotation speed at the optical disk edge of NGC 4013 (Bottema 1995; see also van der Kruit 2001), but the decrement could also come from the prominent warp in that galaxy. If the outer disk is accreted, then there is no obvious reason why the ratio of outer to inner scale lengths should be about the same from galaxy to galaxy, including the dwarfs (Paper I).

Here we consider a star formation model that includes turbulence and other compressions as cloud formation mechanisms, in addition to spontaneous gravitational instabilities (see also Kravtsov 2003). Observations of dwarf galaxies have shown that star formation is not simply regulated by a threshold column density (see review in Paper I). Star formation clearly occurs in clouds that stand above the threshold even if the average column density is below the threshold, and it persists far out in the outer disks of dwarfs as it does in spirals. It has also become clear that the ISM in both dwarfs and spiral galaxies is highly structured into clouds of all sizes, presumably as a result of turbulence and other processes. For the dwarf galaxies, this conclusion follows from the log-normal shape of the probability density function of H$\alpha$ emission (Hunter & Elmegreen 2004), and from the power-law power spectra of H$\alpha$ and stellar emissions (Willett, Elmegreen & Hunter 2005). The same power laws for star formation are seen in spiral galaxies (Elmegreen, Elmegreen, & Leitner 2003; Elmegreen et al. 2003). Dwarfs also show power law or fractal structure in the HI gas, as in the M81 dwarfs (Westpfahl et al. 1999) and in the Small and Large Magellanic Clouds (Stanimirovic et al. 1999; Elmegreen, Kim & Staveley-Smith 2001). The same is observed in local HI (e.g., Dickey et al. 2001). All of these distributions resemble the characteristics of compressible turbulence as illustrated in simulations (see review in Elmegreen & Scalo 2004).

These considerations lead to a model for star formation in a turbulent, self-gravitating medium. This model is more general than the instability model alone as it allows for more processes, including pressurized triggering of star formation, turbulence triggering, spiral density wave triggering, and swing-amplified instabilities. It should be useful for predictions of outer disk star formation rates and for semi-analytical models of star formation in cosmological studies.

Multi-component Model of Star Formation
=======================================

Many of the general properties of galaxy disks and star formation can be combined into a relatively simple model that gives the star formation rate as a function of radius. These properties lead to the basic assumptions of the model, as listed here:

-   Galaxies form with a smoothly distributed gas disk having an outer cutoff, as usually seen in cosmology simulations. This cutoff will enter the present discussion as the outermost point of the disk, significantly beyond any break radius that may appear. We assume in some models that the smooth gas disk is a single exponential, although other forms will have the same basic properties. It will be significant that the star formation rate takes an approximately double exponential profile even in a gas disk that is a single exponential. Other models assume an exponential gas disk with an outer $1/R$ extension. In these cases, the star formation profile can be a continuous exponential or a double exponential with the outer part flatter then the inner part. In all cases, the star formation profile will drop faster than the gas profile, but it will rarely truncate suddenly.

-   The ISM is turbulent and partly stirred by pressures related to existing stars. This means the velocity dispersion tends to decrease slightly with radius as the stellar disk decreases exponentially. Such a velocity dispersion decrease is observed for some spiral galaxies (Boulanger & Viallefond 1992). Theoretical discussions of the radial profile of gaseous velocity dispersion are in Jog & Narayan (2005). Equating the energy densities of turbulence and stellar energy input, this gives approximately $M^2\propto e^{-R/R_D}$ for Mach number $M$, radius $R$, and stellar exponential disk scale-length $R_D$. The precise form of this relation is not important to the model; other cases considered below use a constant Mach number and get about the same result. The exponential form assumes the gas density for stirring by supernovae and other stellar pressures is about constant with radius, as appropriate for the HI medium in the main disks of galaxies. Then the Mach number alone responds to the stellar energy density. This is in rough agreement with observations showing greater HI velocity dispersions for cool HI clouds near stellar associations (Braun 2005). This equation also emphasizes that the important Mach number for our model is the one that regulates the first step of cloud formation, i.e., the conversion of ambient gas into dense cloud complexes where stars form. Such emphasis places turbulence on an equal footing with large-scale gravitational instabilities. There may be other processes governing the radial profile of the Mach number inside individual dense clouds and the Mach number for the mass-weighted ISM as a whole.

-   The Mach number reaches a minimum value near unity in the outer disk as a result of either a transition to a warm-dominant thermal HI phase or a sustained low level of turbulence (Sellwood & Balbus 1999). In either case, cool clouds are still possible in the compressed regions, but turbulence compression is weak. Combined with the previous point, this means $$M^2\approx1+Ae^{-R/R_d}\label{eq:mach}$$ where $A$ is the square of the effective Mach number in the inner disk. We assume in some models below that $A=100$; the results do not depend on this value as long as the main part of the inner disk is Toomre unstable. The most important point for the model is that some level of turbulence remains in the outer disk so that turbulence-induced compression makes clouds even where the average disk is Toomre-stable. Thus $A=0$ gives acceptable results too. For the models shown below, Equation \[eq:mach\] will be used with either $A=100$ or $A=0$; a more detailed treatment might have the coefficient $A$ depend on the SFR per unit gas mass, or on other processes related to interstellar turbulence.

-   Isothermal turbulence produces clouds with a log-normal distribution of column density, as observed in simulations by Padoan et al. (2000), Ostriker et al. (2001), and Vázquez-Semadeni & García (2001). Then the probability of a region having a local column density $\Sigma_g$ is $$P(\Sigma_g)d\ln\Sigma_g=P_0 \exp\left(-0.5\left[\ln
    \Sigma_g/\Sigma_p\right]^2 /\sigma^2\right)d\ln
    \Sigma_g.$$ The column density at the peak of this distribution, $\Sigma_p$, will be determined at each radius to give the appropriate average column density (see below). The dispersion of the log-normal may scale with the Mach number of the turbulence, $$\sigma=\left(\ln\left[1+0.5M^2\right]\right)^{1/2}
    \label{eq:disp}$$ as in simulations by Nordlund & Padoan (1999). The log-normal is consistent with the pixel-to-pixel distribution of H$\alpha$ intensity in Im galaxies (Hunter & Elmegreen 2004).

    These last two points (with $A>1$) make the ISM more clumpy in the inner regions than in the outer regions. For the unstable inner part of the disk, this clumpiness does not matter much for the star formation rate because it is relatively easy for $\Sigma_g$ to exceed $\Sigma_c$ and also because the instabilities themselves drive turbulence and cloudy structure. In the stable outer parts, however, the turbulence-formed clumps and any outward propagating spirals from the inner disk are the primary regions where $\Sigma_g>\Sigma_c$ and star formation occurs only in them. This makes star formation very patchy in outer disks, and it proceeds at a low average rate. The rate is not zero even though the average gas column density, $<\Sigma_g>$, is significantly less than $\Sigma_c$ because star formation persists in the tail of the $P(\Sigma_g)$ function.

    The log-normal form for $P\left(\Sigma_g\right)$ is not critical for the double exponential radial profile. It is used here primarily for convenience and because of its presumed connection with turbulence. The important point is that $P\left(\Sigma_g\right)$ has a tail at high $\Sigma_g$ that gets wider with increasing Mach number, and that some low level of turbulence compression remains in the gravitationally stable outer disk.

-   The critical column density for gravity to overcome Coriolis and pressure forces is the Toomre value appropriate for gas. The general concept that galaxy edges result from below-threshold $\Sigma_g/\Sigma_c$ dates back to Fall & Efstathiou (1980), Quirk (1972), Zasov & Simakov (1988) and Kennicutt (1989). We write $\Sigma_c$ here in terms of the epicyclic frequency $\kappa$ and the Mach number $M$ instead of the velocity dispersion, $$\Sigma_c=C M\kappa/\left(\pi G\right).
    \label{eq:thres}$$ The constant of proportionality, C (units of velocity dispersion), absorbs the fixed rms speed and effective adiabatic index that is in the usual expression because we replaced the dispersion with the radial-varying Mach number. In our model, $C$ determines where the break radius might occur in the original exponential; whether it breaks or not depends also on the run of Mach number with radius.

    The instability condition does not indicate only the onset of swing-amplified or shear instabilities in thin disks, or the onset of ring instabilities, as originally devised by Safronov (1960) and Toomre (1964). It is also the condition for the stability of giant expanding shells (Elmegreen, Palous, & Ehlerova 2002) and most likely relevant to the collapse of turbulence-compressed regions too (Elmegreen 2002). This is because all of these processes involve gravity, rotation, and pressure. When $\Sigma_g>\Sigma_c$, gravity overcomes the Coriolis force during the contraction of the largest cloud that is initially in pressure-gravity equilibrium. The origin of the cloud does not matter. If $\Sigma_g<\Sigma_c$, then Coriolis forces disrupt collapsing spiral arms, expanding shells, turbulence-compressed clouds, and ISM structures before much star formation begins in them. Thus, the Toomre condition should be a general condition for star formation, independent of the detailed triggering processes, which may be quite varied (Elmegreen 2002). The situation is the same if the ambient medium cools during the compression, but then $\Sigma_c$ should be set equal to $C \gamma_{eff}^{1/2}
    M\kappa/\left(\pi G\right)$ for effective adiabatic index $\gamma_{eff}=c^{-2}dP/d\rho$ (Elmegreen 1991), considering pressure $P$, density $\rho$, and velocity dispersion $c$. Most likely this occurs in the outer disks where compression can convert warm HI into cool diffuse clouds. We do not consider this additional factor here.

-   The star formation rate is proportional to some power of the [*local*]{} gas column density $\Sigma_g$ when the threshold is exceeded. A 1.4 power was observed by Kennicutt (1998) for a wide range of conditions. A lower limit to the power is $\sim1$, which also fits the data in some models (Boissier et al. 2003; Gao & Solomon 2004). Note that these power law observations differ significantly from what one would get for the Toomre instability alone, where the maximum growth rate, $\kappa\left(\Sigma_g^2/\Sigma_c^2-1\right)^{1/2},$ increases from zero rapidly as $\Sigma_g$ begins to exceed $\Sigma_c$, and then asymptotically levels off to a dependence on $\Sigma_g^1$. This makes the star formation rate per unit area, which is $\Sigma_g$ times the growth rate, proportional to $\Sigma_g$ raised to a power greater than or equal to 2. If $\Sigma_g/\Sigma_c\sim1.5,$ for example, then the star formation rate should scale with $\Sigma_g^{2.8}$. The Toomre condition alone is not appropriate for star formation because all of the other dynamical processes that are involved (such as turbulence driven by young stars and thermal cooling inside compressed regions) change both $\Sigma_g$ and $\Sigma_c$ locally. The Toomre condition assumes an isothermal uniform gas. If this isothermal assumption is relaxed, then galaxy disks can become unstable for a wider range of conditions (e.g., Elmegreen 1991). The origin of the observed power law is not fully understood, but it is probably related to star formation processes that operate at the local dynamical rate in a medium that is structured by turbulence (Elmegreen 2002). It should follow naturally from a full hydrodynamical model that includes these effects (Kravtsov 2003; Li, Mac Low & Klessen 2005).

These points incorporate the main processes that are believed to be involved with galactic-scale star formation: ISM turbulence, pressurized shell formation and other pressurized triggering, thermal equilibrium, and general gravitational instabilities. Turbulence and other compressions make the disk cloudy and this cloudy structure persists in the outer disk even where $<\Sigma_g>$ is less than $\Sigma_c$, allowing star formation to continue at large radii. The decrease in the Mach number means the cloudiness decreases with radius, so the ISM becomes less turbulent and more smooth in the outer regions, as observed for HI (Braun 1997). This combination of cloud-forming turbulence with a Mach number that converges asymptotically to near-unity, along with cloud-forming instabilities that become decreasingly important with radius, produces the transition from an inner near-exponential to an outer near-exponential in our model. The outer exponential is where the disk is Toomre-stable and the Mach number is of order unity. Both of these conditions are satisfied at about the same place when a clear double exponential appears (see below).

If there are spiral arms in the outer disk, even if they are generated in the inner disk and radiate dissipatively to the outer disk, then this model should not change much because these spirals provide only one more possible source of cloudy structure and triggered star formation. As long as the gas becomes gradually less compressive in the outer regions, the star formation rate tapers off smoothly until the physical edge of the disk (or ionized edge of the gas) is reached. Thus the Mach number in our model should be interpreted as the ratio of rms bulk motion to sound speed, regardless of whether the bulk motions occur in spiral shocks, turbulence, or pressurized shells.

Figure \[fig-sfr\] shows radial profiles of various quantities from models based on these principles. The models calculate the results in radial steps of $dR=0.1$ (arbitrary units) for a disk exponential scale length of $R_D=2.5$ and an outer disk cutoff of 20. At each radius, $R$, the average gas column density is determined from the initial exponential, $<\Sigma_g>=\Sigma_{g0}e^{-R/R_D}$, the Mach number is determined from Equation \[eq:mach\], and the dispersion of the probability distribution function for local column density is determined from the Mach number using Equation \[eq:disp\]. Then the peak column density in this distribution, $\Sigma_p$, is determined from the integral over $\Sigma_g P\left(\Sigma_g\right)$ by setting the average column density that results from this integral equal to $<\Sigma_g>$. The threshold column density, $\Sigma_c$, is also determined at this radius, from Equation \[eq:thres\].

After this setup for the average quantities, the model makes clouds and determines star formation rates. The local column density is determined by randomly sampling from the distribution function $P\left(\Sigma_g\right)$, and then the star formation rate is set equal to this local column density raised to a power of 1 or 1.5 in the two cases shown, provided the local column density exceeds $\Sigma_c$. If the local column density is less than $\Sigma_c$, then the star formation rate at this position is set to zero. To adequately sample the random assignments of column densities, we consider a number of azimuthal points at each radius equal to $R/dR$. That is, the size of a cloud is assumed to be constant with radius. When $R/dR$ random column densities and resulting star formation rates are determined at each $R$, we average together these column densities and star formation rates to give the plotted quantities.

Figure \[fig-sfr\] shows results for a rotation curve appropriate for most galaxies (the rotation curve affects only $\kappa$). This rotation curve is rising in the inner part and flat in the outer part: $V = V_0 (r/R_D)/ [ 1+ r/R_D]$. The star formation rate is shown on the left with dashed lines tracing two exponential profiles to guide the eye. Other quantities for the same models are shown on the right: critical column density, $\Sigma_c$ (magenta), Mach number (black dashed), average gas surface density, $<\Sigma_g>$ (red), and local gas column density, $\Sigma_g$ (green). Five cases are considered. In the top two and bottom two panels, the star formation rate scales with the local column density to the 1.5 power, while in the middle panel, the rate scales with $\Sigma_g$ to the first power. The difference is that when SFR$\propto\Sigma_g^{1.5}$, the inner exponential in star formation is steeper than the inner exponential in gas; otherwise the SFR and the gas have the same profiles. The break radius does not depend noticeably on whether the SFR scales with $\Sigma$ or $\Sigma^{1.5}$.

The bottom two panels show the difference between models with high and low critical column densities. When $\Sigma_g/\Sigma_c$ is lower (bottom panel), less of the disk is unstable and the break radius is smaller. This is consistent with our observation that the relative break radius is smaller in dwarfs than in spirals (Paper I). It occurs because the average surface density is lower compared to the critical value in dwarfs than in spirals.

The top two panels in Figure \[fig-sfr\] show cases where the Mach number has constant values with radius: 1 (top) and 10 (second from top). When the Mach number is 10 throughout, cloud formation continues at a high rate in the outer part of the disk (i.e., $\sigma$ in the dispersion of $P(\Sigma_g)$ stays large), and there is no significant drop in SFR there. Consequently, there is no clear double exponential. When the Mach number is 1 throughout, the average SFR profile is almost exactly the same as for the exponential Mach number (compare to the second panel up from the bottom which has the same parameters except for the Mach number), but the rms scatter is much lower when $M=1$ than when $M\sim10$ in the inner disk. This illustrates how the Mach number is unimportant for the average star formation rate in the unstable inner part of the disk. That region is “saturated” with star formation from spiral shocks and instabilities and unable to produce more star formation even with more compression. However the Mach number in the inner disk is important for the detailed structure of star formation, i.e., for the variability of it and for the geometry of cloud structure.

These models illustrate how a combination of increasing disk stability and moderate Mach number can create an approximate double exponential in the star formation rate when the overall gas distribution is more uniform. The break radius occurs slightly beyond the point where $<\Sigma_g>\sim\Sigma_c$ if the Mach number is of order unity there. It can occur further out if the Mach number is still high.

The detailed profile of the star formation rate in regions where the average column density exceeds the threshold does not depend much on the rotation curve or Mach number. This is because once the threshold is exceeded, the threshold no longer enters into the star formation rate for the simple power law model.

Two other types of radial profiles are found in spiral and irregular galaxies: those which continue in an exponential fashion out to the largest measured radius and those which have a shallower exponential in the outer part (Erwin, Beckman, & Pohlen 2005; Paper I).

Galaxies of the first type, with a single exponential extending out to $\sim10$ scale lengths (Weiner et al. 2001; Bland-Hawthorn, et al. 2005), are difficult to understand with single-component star formation models because the outer disk should be far below the Kennicutt (1989) threshold. Our multi-component model can reproduce the observation, but the outer gas disk has to fall more slowly than an extrapolation of the inner exponential. The bottom panels in Figure \[fig-sfr2\] show an example. The average gas disk is exponential out to $6R_D$ and then it tapers beyond that as $1/R$ out to $10R_D$. This is consistent with the shallow outer HI profile in NGC 300 (Puche et al. 1990), which has its stellar disk extend continuously to $10R_D$ (Bland-Hawthorn et al. 2005). The profiles of Mach number (using $A=100$) and rotation speed are the same as in the previous examples, and the local star formation rate is $\propto\Sigma_g^{1.5}$. In the left-hand panel, the star formation rate becomes very patchy in the outer part, but the average rate (solid blue line) follows an overall exponential profile. In the right panel, the ratio of $\Sigma_c$ to $<\Sigma_g>$, which is the Toomre stability parameter, $Q$, is $5.7$ in the outer regions, indicating great stability on average. Still, there is a lot of cloud and star formation from local compressions.

The top part of Figure \[fig-sfr2\] shows the case if the $1/R$ part of the gas disk begins at a smaller radius, $5.2R_D$, with all else being the same. Then the profile in the outer disk can be shallower than in the inner disk. This is the second type of profile mentioned above. This explanation differs from that in Paper I, where we suggested that dwarf galaxies of this second type, with relatively flat outer parts, could have their steepening in the central regions because of enhanced star formation there. This was because the central regions tended to be bluer than the outer regions; most BCD galaxies were examples of this. The origin of the flat-outer exponential profiles in barred S0 galaxies (Erwin, et al. 2005) cannot be due to intense inner-disk star formation because their Hubble types are too early. In these S0 galaxies the isophotal contours tend to become more round with distance beyond the break radius. Radial profiles determined from deprojected circular averages at all radii could then introduce false inflections. Other flat-outer profiles in the Erwin et al. sample have break radii associated with outer rings or outer Lindblad resonances; these would not be connected with star formation changes either. More observations of the various types of radial profiles and their associated gas and star formation properties are necessary before the relative importance of these models can be understood.

Our discussion so far has concerned the radial profile of the star formation rate but not the radial profile of the resulting stars that form over a Hubble time. Our comparisons between the predicted star formation profiles and the observed surface brightness profiles are therefore premature. The next step in this analysis should be an integration of the star formation rate over time, but this requires some knowledge of the gas accretion rate, both as a function of radius and time. Two limiting cases may be discussed at this point. At the end of the star formation process, after all of the gas has been converted into stars, the stellar mass profile should reflect the total accreted gas profile, altered, if need be, by radial gas motions, stellar migration, minor mergers and tidal stripping. There is no reason to believe that this final stellar profile will resemble the star formation profile. The second limiting case is at the beginning of the star formation process, when the outer disk is still dominated by gas. Then the stellar component is only a small perturbation to the outer disk and the star formation profile should be about the same as the accumulated stellar mass profile, altered again by any accreted stars, radial migrations, and tidal effects. Fortunately the outer parts of late-type galaxy disks, beyond $R_{br}$, are usually in this second limit, i.e., gas-dominated. For dwarf galaxies, this was shown by Hunter, Elmegreen & Baker (1998). Thus we believe the star formation profiles derived here for the outer disk are a suitable explanation for the stellar surface density profiles.

To check this possibility, we ran two of the SFR models shown in Figure \[fig-sfr\] (the top panel and the second one up from the bottom) over a sequence of timesteps. At each timestep and at each radius, we deducted 1% of the instantaneous SFR from the average gas mass, and then added this mass to the integrated star mass. The initial conditions were the same as in Figure \[fig-sfr2\], with no accumulated stars at first. Figure \[fig-evol\] shows the resultant SFR, average gas surface density, and average stellar surface density at three times: the beginning of the run (blue curves), after 20 timesteps (green curves), and after 50 timesteps (red curves). We label these timesteps as 20% and 50% of the gas consumption time, respectively, where this consumption time is considered to be the inverse of the percentage (1%) deducted each step. There is no gas accretion from outside the galaxy. The panels on the left assume the same exponential Mach number profile as the second panel up from the bottom in Figure \[fig-sfr2\] (i.e., A=100 in Eq. \[eq:mach\]) and the panels on the right have the same constant Mach number profile as the top panel in Figure \[fig-sfr2\] (i.e., $A=0$ and $M=1$). Evidently, the gas gets depleted most quickly in the inner regions, as expected, and the stars build up an exponential profile over time. The double exponential is still present in the accumulated stars. The Mach number profile does not matter much, as discussed previously. At 50 timesteps, the star mass in the steep exponential part of the outer disk is still much less than the gas mass, in agreement with the observations cited in the previous paragraph.

We have not commented yet on the relation between star formation rate and H$\alpha$ surface brightness. The H$\alpha$ surface brightness should become an inadequate tracer of star formation in the very low-density regions of outer galaxy disks because the emission measure drops below detectability. The emission measure through the diameter of a classical Stromgren sphere with uniform density $n$ is $\left(6S_0n^4/\left[\pi\alpha\right]\right)^{1/3}$ for Lyman continuum photon luminosity $S_0$ and recombination rate $\alpha$. The scaling with $n^{4/3}$ implies that even if the $H\alpha$ flux does not escape the galaxy, the intensity of an HII region is at least $e^{16/3}=207$ times fainter at a position four scale lengths further out compared to the inner disk. This factor assumes the density is smaller by only the factor $e^{-4}$ without a corresponding increase in scale height. An increase in scale height with decreasing disk self-gravity, as $H=c^2/\left[\pi
G\Sigma_T\right]$ for velocity dispersion $c$ and total disk column density $\Sigma_T$, makes the midplane density roughly proportional to $\Sigma^2$, in which case the H$\alpha$ intensity can drop by a factor $e^{32/3}\sim4\times10^4$ in four scale lengths of an exponential disk. Thus there should be a significant drop in the radial profiles of H$\alpha$ even with outer disk star formation of the type discussed here. This type of $H\alpha$ drop is in agreement with observations (e.g., Thilker et al. 2005; Paper I).

The multi-component star formation proposed here also explains some of the irregularities with the Kennicutt (1989) prediction that were found in our previous papers on dwarf Irregulars. For example, star formation in dwarfs often occurs where the average column density is less than $\Sigma_c$, in contradiction to the Kennicutt model, as long as there are cool cloudy regions that locally have $\Sigma_g>\Sigma_c$. This peculiarity was noted before in many studies of dwarf galaxies (van der Hulst et al. 1993; Taylor et al. 1994; van Zee et al. 1997; Meurer et al. 1998; Hunter, Elmegreen, & van Woerden 2001). In the current star formation model, these cloudy regions form by turbulence and other processes (pressurized shells, external cloud impacts, end-of-bar flows, gaseous spirals, etc.) even in sub-threshold regions if the Mach number for the associated flows is still relatively high. The revised model also extends the Kennicutt result by allowing for a typical decrease in velocity dispersion with radius and by associating the threshold $\Sigma_c$ with any of a variety of cloud formation processes, and not just isothermal gravitational instabilities in initially smooth disks.

Conclusions {#sec-disc}
===========

Many processes of star formation combine to give the radial profiles of galaxies. In the inner main-disk regions where the gas is usually gravitationally unstable in spite of the large Coriolis and pressure forces, star formation should saturate to its maximum possible rate. This is the gravitational collapse rate for the conversion of low density to high density gas, multiplied by the fraction of the high density gas that is suitable for star formation, i.e., the fraction in the form of stellar-mass globules with masses exceeding the local thermal Jeans mass (e.g., Elmegreen 2002; Kravtsov 2003). The actual dynamics involved with the first step, dense cloud formation, will be varied, involving swing-amplified spiral instabilities, spiral density wave shocks, compression or shell formation around existing star formation sites, and turbulence compression. In galaxies with strong stellar spirals, the spiral shocks may dominate dense cloud formation, making most clouds spiral-like, as in M51 (e.g., Block et al. 1997), while in galaxies without such strong spirals, another mechanism should dominate, making most clouds shell-like (as in the LMC), globular, or hierarchically fractal. In all of these cases, the same star formation rate per unit area arises, all from the saturation condition. Thus they all give the Kennicutt-Schmidt law or something like it in a regular fashion, regardless of the detailed processes involved.

In the outer parts of disks, some of these processes shut down completely. There should be no strong stellar spirals beyond the outer Lindblad resonance for the main (self-amplified) modal pattern speed, and there should be few swing-amplified stellar spirals if the Toomre Q parameter is high. Cold cloud formation should also be more difficult at low ambient pressure. However, a low level of star formation may sustain itself at large radii by driving shells and turbulence and by compressing the existing clouds. Also, gaseous spiral arms can propagate there from the inner disk, as they are able to penetrate the outer Lindblad resonance unlike the stellar spirals. Gaseous arms can also form by instabilities there if there is significant cooling during compression (because that lowers $\Sigma_c$ through the $\gamma_{eff}$ parameter). These processes maintain star formation at levels much lower than the saturation rate given above, and therefore lower than the Kennicutt-Schmidt law rate, primarily because an ever-decreasing fraction of the gas can make the transition from low density to high density in the first step.

In this paper, we modelled all of these processes in a general way using the few simple rules just mentioned. The transition from saturated star formation in the inner disk to unsaturated in the outer disk was followed, and radial profiles were obtained that look moderately close to real profiles. In the first case considered, the profile was exponential in the main disk and it tapered off beyond that with a form that also resembled an exponential, but steeper for several scale lengths. The ratio of the break radius to the inner scale length varied with the surface density (higher surface densities have higher ratios) because more unstable inner disks have their inner exponentials extend further out before the transition occurs. This correlation can explain the observation among both spirals and dwarfs that $R_{br}/R_D$ increases with main disk surface brightness. In two other cases with shallower outer gas profiles, the star formation profile varied between a nearly pure exponential out to $\sim10$ scale lengths and a shallow outer exponential, depending on where the transition between the inner and outer gas profiles occurred relative to the stability threshold radius. In all cases, the H$\alpha$ profile should drop much more suddenly with radius than the star formation profile as the emission measure of individual HII regions drops rapidly below the detectability limits.

The main ingredients of our star formation model are: a generally smooth decline of gas column density in the disk with a cutoff in the far outer part (usually beyond the observations), a turbulent Mach number that decreases with radius and then levels off to near unity, or remains near unity throughout, a distribution function for local column density with a high column-density tail and a dispersion that increases with Mach number, a column density threshold for self-gravity to overcome Coriolis and pressure forces, and a local star formation rate that increases with the local cloud density when the threshold is exceeded. For such a model, the inner exponential occurs where the average column density exceeds the threshold, almost regardless of Mach number or Mach number gradient. The outer profile occurs where the gas column density is sub-critical and the Mach number is relatively small but non-zero, e.g., near unity. The small but non-zero Mach number gives turbulence and other dynamical processes the ability to form clouds that locally exceed the stability threshold, but these processes are not likely to do this very often. As a result, star-forming clouds become very patchy in the outer disk, making the star formation gradient significantly steeper than the gas gradient. Only the peaks of the clouds stand above the stability threshold. Gas cooling during cloud formation is also an essential ingredient. Cooling to diffuse cloud temperatures and colder is assumed to follow any significant compression, as predicted elsewhere based on studies of interstellar thermal equilibrium.

We are grateful to the referee for useful comments. Funding for this work was provided by the National Science Foundation through grants AST-0204922 to DAH and AST-0205097 to BGE.

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