4.1. Main observed facts
Observations of a starburst-merger connection The main and simplest observation probing the link between starbursts and mergers is that almost all starbursting galaxies in the nearby Universe are mergers. In particular, (Ultra-)Luminous Infrared Galaxies (LIRGs and ULIRGs) at low redshift are almost exclusively strongly interacting and merging systems (e.g., Duc et al. 1997). Note that at high redshift (z ~ 1 and above), LIRGs are not necessarily mergers because "normal" isolated disk galaxies are gas-rich enough to reach LIRG-like activity (Daddi et al. 2010a, Tacconi et al. 2010), but it remains true that the strongest starbursts (ULIRGs and Hyper-LIRGs) are predominantly merging systems (e.g., Elbaz & Cesarsky 2003).
It has been long known that mergers can trigger bursts of star formation (Sanders et al. 1988). But the fact that the most actively star forming systems at any given redshift are mostly mergers (LIRGs at z ~ 0, ULIRGs at z ~ 1) does not necessarily imply that mergers are always starbursting, or that all mergers even undergo a significant starburst phase - we here of course consider "wet" mergers where at least one galaxy contains a substantial amount of gas, not "dry" mergers of gas-free galaxies. Understanding whether all mergers are starbursting requires to be able to select mergers (at least major ones) in large surveys including star formation tracers. The main concern is that any technique selecting mergers based on morphology (for instance those by Conselice et al. 2003, Lotz et al. 2008) remains ambiguous, in particular at high redshift where morphological disturbances and asymmetries can arise internally (Bournaud et al. 2008). Surveys including spatially-resolved gas kinematics are certainly a more robust tool to identify on-going major mergers and (relatively) isolated disks (Shapiro et al. 2008) but such samples remain too limited to study the merger-starburst connection. Nevertheless, the most recent and detailed studies (Bell et al. 2006, Jogee et al. 2009, Robaina et al. 2009) are broadly in agreement that mergers do trigger star formation activity, and specific star formation rate (i.e., SFR divided by the stellar mass already in place) being increased by factor of about 3-4, on average: strongly starbursting mergers do exist but most mergers do not increase the SFR by a factor above about 10 at the peak, likely because strong starbursts are rare, or they are short phases compared to the duration of an interaction/merger until final relaxation of the post-merger early-type galaxy.
The factor 3-4 of increase in the SF activity is the average observed value for randomly interactions/mergers at any instant of the merging process, not at the peak of a starburst activity, so it likely indicates that peak SFR values are higher than this. Also, it is the typical measure for "major" interaction/mergers, including mass ratios of, say, 1:1 to about 3:1. There is no evidence of a variation with redshift, suggesting that the typical SF activity of "isolated" disks and merging systems increase with redshift in roughly similar proportions.
Morphology of merger-induced star formation Merger-induced star formation consists, for a large part, of nuclear starbursts taking place in the central 100-1000 pc (diameter). Nuclear activity in merger-induced starbursts has long been emphasized in studies of the merger-SF connection (e.g., Keel et al. 1985, Sanders et al. 1988, Duc et al. 1997, Soifer et al. 1984, Lawrence et al. 1989). Nevertheless, the importance of nuclear starbursts among merger-induced star formation in general has long been overstated in the literature. This may be partly because theorists have long been able to explain only nuclear starbursts, or because of using infrared and/or H luminosities to trace star formation - these are dependent on metallicity, dust properties, and threshold effect making them more sensitive to compact central starbursts than to spatially extended star formation with lower surface density.
Actually, there is a large fraction of systems in which merger-induced star formation is spatially extended, taking place outside the central kpc, and in these system the extended star forming component is not just a relics from extended SF in pre-merger spiral galaxies, but participates to the total starburst activity, even if there is also intense SF in the nucleus. The relatively large spatial extent of SF and the problem it poses for theoretical models (as we will review below) was probably first recognized by Barnes (2004) in the Mice (NGC 4676A + B, see also Chien & Barnes 2010 in NGC 7252). A well-know example if actually the Antennae system (NGC 4038 + 4039), where a large fraction of the starburst activity takes place at several kpc from the nuclei in big "super star cluster" (SSCs) formed along density waves throughout the interacting disks, and in the shocked overlap region between the two gas disks (Wang et al. 2004) - these star-forming components are more important than the dust-enshrouded nuclear star formation revealed in the infrared. There are many other examples of interacting/merging systems with spatially extended starbursts whose total SFR is far from being dominated by the nuclear SF activity, but rather by extended star-forming components: Arp 140 (Cullen et al. 2006); NGC2207 + IC2163 (Elmegreen et al. 2006) in which the starburst is almost exclusively non-nuclear with an SFR of 15 M yr-1 for global gas mass/size about similar to typical Milky Way-type spirals (i.e., a significant increase of SFR likely results from the interaction, but only in a spatially extended mode); the Antennae; the Cartwheel system - see also many tidally interacting pairs in Smith et al. (2008), Hancock et al. (2009), etc. There is to my knowledge no thorough census of the contributions of nuclear and extended star formation to merger-induced activity, probably because unveiling dust-enshrouded nuclear SF and detecting spatially extended SF at lower surface densities requires to employ different techniques at different wavelength. Nevertheless, eye examination of systems with UV imaging available in samples of interactions and mergers (e.g., the Arp Atlas) suggest that spatially extended (> 1 kpc) SF and nuclear activity (within the central kpc) each contribute to a rough 50% of star formation in mergers.
ISM properties in interacting starbursts Two fundamental properties of the interstellar gas in star-forming mergers majors will hold a fundamental role in our theoretical interpretation of the starburst activity:
gas velocity dispersions are higher in interacting/merging systems than in isolated galaxies. At redshift zero, the cold ISM of spiral galaxies has a typical velocity dispersion of, say, 5-10 km s-1, and only gas-rich disk at high redshift have higher dispersions. Major mergers can have gas velocity dispersions of several tens of km s-1, as first observed by Irwin (1994) and Elmegreen et al. (1995) (see also examples in Bournaud et 2004 although limited to star-forming clumps in H). These increased dispersions do not result from blending of several components in a large telescope beam (a.k.a. beam smearing), as they are also observed at high resolution, but trace a physical velocity dispersion on small scales. As this applies primarily to gas components observed through HI or CO lines, the thermal dispersion (sound speed) is lower than 10 km s-1 (for temperatures below 104 K), so most of the observed dispersion in isolated galaxies and merging systems consists in supersonic gas turbulence, which usually dominates the cold star forming phases of the ISM (Burkert 2006) (a warm, thermally-supported phase can fill a large volume fraction, but involves only a minor mass fraction of the ISM). Hence, the (supersonic) ISM turbulence is typically increased by a factor of a few in galaxy interactions and mergers. We will discuss later, using numerical and theoretical models, that this is probably a trigger of increased star formation (through compressive regions in the turbulent flow) rather than a consequence of the star forming activity itself (through stellar feedback impacting the surrounding gas).
Various density phases of the ISM are not filled and distributed in the same way as in isolated disk galaxies. There is often a spatial segregation between the moderate density, atomic phase, mostly found in the outer regions, and the denser molecular phases concentrated towards the few central kiloparsecs (e.g., Duc et al. 1997). More systematically, starbursting systems in the nearby Universe with ULIRG-like infrared luminosities (which are almost exclusively merger-induced starbursts, see above) exhibit enhanced HCN/CO luminosity ratios (Juneau et al. 2009). HCN is usually considered to be a tracer of the very dense gas (say, 105-6 cm-3) directly involved in actual star formation, while CO lines also (and mostly) trace lower-density regions (down to, say, 100 cm-3) that belong to star-forming clouds but are not directly star forming: hence observations point towards an excess of high-density gas in starbursting mergers compared to the amount of moderate-density molecular gas traced by CO.
As detailed in the next Section, if the density PDF of the ISM is driven by the supersonic turbulent support, then the increased velocity dispersion observed in mergers can directly result in a larger spread of the density PDF, which can result in an excess of high-density molecules compared to the total molecular gas mass. We will illustrate later, using hydrodynamic simulations, how these processes act in concert to trigger the star formation activity of interacting and merging galaxies.
4.2. Standard theory of merger-induced starbursts: gas inflows and nuclear starbursts
In an interacting galaxy pair, the gas distribution becomes non-axisymmetric, which results in gravitational torquing of the gas as detailed previously. Gas initially inside the corotation radius (typically a radius of a few kpc) undergoes negative gravity torques and flows inwards in a more and more concentrated central component (inside the central kpc). Any model for star formation will then predict an increase of the star formation rate (global Schmidt-Kennicutt law, models based on cloud-cloud collisions, etc). The result is thus a centralized, nuclear starburst. As the driving process is gravitational torquing, early restricted three-body models could already describe the effect (Toomre & Toomre 1972). Later models have added extra physical ingredients leading to more accurate predictions on the star formation activity: self-gravity (Barnes & Hernquist 1991), hydrodynamics and feedback processes (Mihos & Hernquist 1996, Cox et al. 2008), etc.
A large library of SPH simulations of galaxy mergers, in which the driving process is mostly the one presented above (tidal torquing of gaseous disk) was performed and analyzed by Di Matteo et al. (2007), (2008). This study highlighted various statistical properties of merger-induced starbursts. In particular, it showed that some specific cases can lead to very strong starbursts with star formation rate (SFRs) are increased by factors of 10-100 or more, but that on average the enhancement of the SFR in a random galaxy collision is only a factor of a few (3-4 being the median factor) and only lasts 200-400 Myr. These results were confirmed with code comparisons, and found to be independent of the adopted sub-grid model for star formation (Di Matteo et al. 2008). Models including an external tidal field to simulate the effect of a large group or cluster found that the merger-induced starbursts could be somewhat more efficient in such context - but the SFR increase remains in general below a factor of 10 or less (Martig & Bournaud 2008).
4.3. Limitations of the standard theory
The intensity of merger-induced starbursts Numerical simulations reproducing the interaction-induced inflow of gas and resulting nuclear starbursts can sometimes trigger very strong starbursts, but in general the SFR enhancement peaks at 3-4 times the sum of the SFRs of the two pre-merger galaxies. This factor of 3-4 seems in good agreement with the most recent observational estimates (e.g., Jogee et al. 2009, Robaina et al. 2009). In fact, there is substantial disagreement: the factor 3-4 in simulation samples is the peak amplification of the SFR in equal-mass mergers. In observations, it is the average factor found at random (observed) instant of interactions, and in mergers that are "major" ones but not strictly equal-mass ones. Given that typical duration of a merger is at least twice longer than the starburst activity in the models, and that unequal-mass mergers make substantially weaker starbursts (Cox et al. 2008), one would need a peak SFR enhancement factor of about 10-15 (as measured in simulations) to match the average enhancement of 3-4 found in observations. There is thus a substantial mismatch between the starbursting activity predicted by existing samples of galaxy mergers, and that observed in the real Universe - although the observational estimates remain debated and may depend on redshift.
The spatial extent of merger-induced starbursts The usual explanation of merger-induced starbursts accounts only for nuclear starbursts. While these are relatively frequent among mergers, they do not necessarily dominate interaction-triggered star formation, and spatially extended are actually common (see above, and two examples of Figure 5). Quantitative comparisons of the extent of star formation in observations to that predicted by "standard" models have shown a significant disagreement (Barnes 2004, Chien & Barnes 2010). These authors highlighted the failure of standard models to account for the extended morphology of (at least some) merger-induced starbursts. They also suggested that a sub-grid model of shock-induced star formation may better account for the spatial extent of merger-induced activity (see also Saitoh et al. 2009).
Figure 5. Two examples of starbursting interactions/mergers (the Antennae and the NGC2207+IC2163 pair), where a large part of the starburst activity is not limited to the central kpc regions, but also takes places in super star clusters and giant HII regions throughout the disturbed disks, and in overlap regions between the interacting galaxies.
4.4. Star formation in mergers: effect of the turbulent, cloudy ISM
The usually invoked mechanism of gas inflows for merger-induced star formation, as reproduced in low-resolution simulations, certainly takes place in real mergers - signs of nuclear merger-driven starbursts are a plenty. But it seems impossible to explain the typical intensity of merger-induced starbursts and their often relatively larger spatial extend, based on these "standard" models.
Traditional SPH simulations model a relatively warm gas for the ISM, because the limited spatial resolution translates into a minimal temperature under which gas cooling should not be modeled (it would generate artificial instabilities, Truelove et al. 1997). Modeling gas cooling substantially below 104 K requires "hydrodynamic resolutions" 4 better than 100 pc. Cooling down to 100 K and below can be modeled only at resolutions of a few pc. The vast majority of existing merger simulations hence have a sound speed of at least 10 km/s and cannot explicitly treat the supersonic turbulence that characterizes most of the mass in the real ISM (Burkert 2006). Turbulent speeds in nearby disk galaxies are of 5-10 km/s, for sound speed of the order of 1-2 km/s in molecular clouds, i.e. turbulent Mach numbers up to a few. These are even higher in high-redshift disks (Förster Schreiber et al. 2009), and in mergers (references above), but not necessarily for the same reason.
Increased ISM turbulence in galaxy mergers is thus absent from the modeling used in most hydrodynamic simulations to date. Some particle-based models have nevertheless been successful in reproducing these increased gas dispersions (Elmegreen et al. 1993, Struck 1997, Bournaud et al. 2008), indicating that it is a consequence of the tidal interaction which triggers non-circular motions, rather than a consequence of starbursts and feedback. It should then arise spontaneously in hydrodynamic models, if these a capable of modeling gas below 103-4 K.
High-resolution simulations with multiphase ISM dynamics Adaptive Mesh Refinement (AMR) codes allow hydrodynamic calculations to be performed at very high resolution on adaptive-resolution grids. The resolution is not high everywhere, but the general philosophy is to keep the Jeans length permanently resolved until the smallest cell size is reached. That is, the critical process in the collapse of dense star-forming clouds, namely the Jeans instability (or Toomre instability in a rotating disk) is constantly resolved up to a typical scale given by the smallest cell size, or a small multiple of it. AMR simulations of whole galaxies have recently reached resolutions of a few pc for disk galaxies (e.g., Agertz et al. 2009) and even 0.8 pc lastly (Bournaud et al. 2010c). Such techniques have been first employed to model ISM dynamics and star formation in galaxy mergers by Kim et al. (2009), Teyssier et al. (2010). The main physical improvement is not necessarily directly the higher spatial resolution, but the possibility to model gas cooling down to low temperatures (below 300 K) while still resolving the typical Jeans length of gravitational instabilities. In theory, SPH techniques could also allow low temperature floor at high spatial resolution, but in practice this may be more expansive if not prohibitive, and has not been done for mergers and star formation (but see Hopkins et al. 2011 but high-resolution, low-temperature floor SPH disk models).
We here illustrate the results of such simulations using a sample of AMR simulations of 1:1 mergers of Milky Way-type spiral galaxies, performed with a resolution of 4.5 pc and a barotropic cooling model down to ~ 50 K, technically similar to the isolated disk simulation described in Bournaud et al. (2010c). Star formation takes place above a fixed density threshold and is modeled with a local Schmidt law, i.e. the local star formation rate density in each grid cell is SFR = ff / tff 1.5 where is the local gas density and tff the gravitational free-fall time, and the efficiency ff is a fixed parameter. Supernova feedback is included. Further details and results for whole sample will be presented in Powell et al. (in preparation). An individual model of this type (but at slightly lower resolution and without feedback), matching the morphology and star formation properties of the Antennae galaxies, was presented in Teyssier et al. (2010). More details on the simulations can also be found in Bournaud (2010b).
In the models, the pre-merger isolated spiral galaxies spontaneously develop ISM turbulence at a about 10 km s-1 under the effect of gravitational instabilities (and/or feedback), and most star formation takes place in dense complexes of dense gas along spiral arms. In some sense, the large-scale star formation process is not entirely sub-grid anymore in these simulations, as the first steps of star formation, namely are the development of ISM turbulence and the formation of dense molecular gas clouds in this turbulent ISM, are explicitly captured - the subsequent steps of star formation at smaller scales, inside the densest parts of these cold clouds, remain sub-grid.
A merger simulation is shown in Figure 8. The mass-weighted average of the gas velocity dispersion reaches 30-40 km/s (Fig. 6). This strong turbulence is consistent with the observations reviewed above. It induces numerous local shocks that increase the local gas density, which in turn triggers the collapse of gas into cold clouds. Also, gas clouds become more massive and denser than in the pre-merger spiral galaxies. The fraction of gas that is dense-enough and cold-enough to form star increases, and the timescale for star formation in these dense gas entities (the gravitational free-fall time) becomes shorter. As a result, the total SFR becomes several times higher than it was in the pre-interaction pair of galaxies. The standard process of merger-induced gas inflow towards the central kpc or so is also present, but the timescale is substantially longer, so this process dominates the triggering the star formation by enhancing the global gas density only in the late stages of the merger.
Figure 6. Radial profiles of the gas velocity dispersion (turbulent speed) in an isolated galaxy simulation, and three major mergers between two disks of the same initial type.
An example of star formation distribution is shown in Figure 8. Two consequences of modeling a cold turbulent ISM in merging galaxies are: (1) the peak intensity of the starburst can become stronger (as shown by Teyssier et al. 2010 for a model of the Antennae) although this is not a systematic effect, and (2) the spatial extent of star formation in the starbursting phase is larger. The radius containing 50% of the star formation rate (half-SFR radius) can more than double. This is because increased ISM turbulence is present throughout the systems (see Fig. 6): this triggers, through locally convergent flows and shocks, the formation/collapse of efficiently star-forming clouds even at relatively large distances from the nuclei. At least quantitatively, these results put the models in better agreement with observations. In these models, the increased ISM turbulence is also obtained without feedback, showing that it is not a consequence of the starbursting activity, but is rather driven by the tidal forces in the interaction.
Figure 7. Density PDFs of an isolated disk (left) and a starbursting major merger (right), in AMR simulations. The excess of very dense, star-forming gas results from the non log-normal shape of the PDF, induced by the rapidly increased turbulence and numerous local shocks compressing the gas, more than by a global shift of the initial PDF.
Figure 8. View of the very young stars (proxy for the instantaneous star formation rate) in a major merger model from Teyssier et al. (2010).
4.5. From triggered turbulence to dense gas excess and starbursts
Modern models are thus capable of resolving ISM turbulence and SF clouds in disk galaxies and starbursting mergers. ISM turbulence is self-regulated in isolated disk galaxies, and is significantly triggered in merging systems.
Our simulated of disk galaxies have quasi log-normal PDFs (see example on Fig. 7). The spatial resolution limit converts into a density limit at which the density PDF is truncated, and which corresponds to the smallest/densest entities that can be resolved. The maximal density resolution in the models presented here is around 106 cm-3, but parsec-scale resolution could capture even higher densities (as in Bournaud et al. 2010).
Representative density PDFs are shown for merger models on Figure 7 (see also Fig. 9) for a moderately starbursting merger (SFR increased by a factor ~ 3-4 compared to the two pre-merger galaxies taken separately), and a stronger merger-induced starburst (factor ~ 10). The PDFs of merging galaxies have a larger half-maximum width than those isolated disk galaxies, as expected from the higher turbulent speeds that result from the tidal interaction. Also, these PDFs are not log-normal, as would be the case for isolated disks with a self-regulated turbulence cascade in a steady state. Rapid perturbations of the velocity field in mergers result in a substantial excess of dense gas in the density PDF. Such density PDFs naturally imply high SFRs, independently from the local SFR prescription, since the fraction of efficiently star-forming gas is high.
Figure 9. Statistical distribution of the SFR/HI ratio (measured as the ratio of gas denser than 105 cm-3 as a proxy for SFR and gas less dense than 102 cm-3 as a proxy for HI (as illustrated on fig. 7), for isolated spirals and mergers at different stages. This quantified the excess of dense gas illustrated earlier by density PDFs.
High fractions of dense gas in mergers were already proposed by Juneau et al. (2009), based on detailed post-processing of merger simulations aimed at re-constructing dense molecular gas phases not resolved in these simulations (Narayanan et al. 2010). Here we obtain a qualitatively similar conclusion using simulations that explicitly resolve turbulent motions, local shocks and small-scale instabilities in cold ISM phases. This excess of dense gas should have signatures in molecular line ratios. If, in a rough approach, we assume that low-J CO lines are excited for densities of 100 cm-3 and above, and HCN lines for densities of 104 cm-3 and above, then the HCN/CO line ratios could be ~ 5-20 times higher in the starbursting phases of major mergers. Simulations with an somewhat higher resolution would actually be desirable to accurately quantify the emission of dense molecular tracers, and opacity effects may also affect the observed HCN/CO ratios.
4.6. Implications for star formation "laws"
The interpretation of merger-induced starbursts proposed from our high-resolution models is that it is not just a global gas inflow that increases the average gas density and increases the SFR, but also that there are strong non-circular motions, high turbulent velocity dispersions, causing many small-scale convergent flows and local shocks, that in turn initiate the collapse of dense star-forming clouds with high Jeans masses. The former "standard" process does take place, but the later can be equally important especially in the early phases of mergers.
We here note gas the average gas surface density of a galaxy. This is the quantity that observers would typically derive from the total gas mass and half-light radius, or similar quantities. The second mechanism above is a way to increase the SFR of a system, and its SFR surface density SFR, without necessarily increasing its average gas. Actually in our merger models gas does increase (as there are global merger-induced gas inflows), but SFR increases in larger proportions (as the starburst is not just from the global merger-induced inflow but also from the exacerbated fragmentation of high-dispersion gas). Going back to the density PDFs shown previously, one can note that the fraction of very dense gas (say, in the ~ 104-6 cm-3 range) can increase by a factor of 10-20 in mergers while the average surface density gas increases by a factor 3-5 (see also on Figure 4). As a consequence, the SFR activity of these systems is unexpectedly high compared to their average surface density gas.
Figure 4 shows the evolution of a system throughout a merger simulation in the (gas ; SFR) plane. While our pre-merger spiral galaxy models lie on the standard Kennicutt relation, starbursting mergers have high SFR / gas ratios. This is in agreement with observational suggestions that quiescent disks and starbursting mergers do not follow the same scaling relations for star formation, but could actually display two different star formation "laws" (Daddi et al. 2010b, Genzel et al. 2010). The offset between the disk and merger sequences proposed by Daddi et al. (2010b) is quantitatively recovered in our simulations (Figure 4). Post-starburst, post-merger systems lie back on the quiescent sequence, or even somewhat below it: these systems contain some dense gas which is somewhat stabilized by the stellar spheroid. This is another example of a "morphological quenching" effect in early-type galaxies (Martig et al. 2009), and the location of our post-merger early-type galaxies in the (gas ; SFR ) diagram may be consistent with observations of nearby ellipticals (Crocker et al. 2011, Saintonge et al. 2011), without having a substantial effect on the formation of the red sequence - at least at low redshift (Fabello et al. 2011).
The proposal that disks and mergers follow two different regimes of star formation by Daddi et al. (2010b) and Genzel et al. (2010) relies for a part (but not entirely) on the assumption that different CO luminosity-to-molecular gas mass conversion factors apply in quiescent disks and starbursting mergers. Interestingly, our simulations recover the two regimes of star formation without any assumption on such conversion factors since gas masses are directly known. But at the same time, excess of dense gas found in these merger models suggests that the excitation of CO lines would naturally be higher in mergers/starbursting phases (although this needs to be quantified in the models), which would mean that the assumption of different conversion factors by Daddi et al. and Genzel et al. could be physically justified. High molecular gas excitation in SubMillimeter Galaxies (SMGs, Tacconi et al. 1998) could then naturally result if these are starbursting major mergers with high gas surface densities and a clumpy turbulent ISM (e.g., Narayanan et al. 2010, Bournaud 2010b). However models do not predict a bimodality. Two regimes of star formation appear when mergers are only selected at the peak of their starburst activity, but merging systems spend over half of their time below the "starburst regime" on the diagram shown in Figure 10.
Figure 10. Kennicutt diagram comparing the surface density of gas and star formation in isolated disk galaxies (LTGs) and mergers near the peak of their starburst activity, as well as post-merger elliptical-like galaxies (ETGs). The "disks" and "starbursts" sequences are from Daddi et al. (2010). The low star formation efficiency in ellipticals is studied in Martig et al. (2009).
The triggering of star formation in merger largely results from tidal inflows driven by the interaction, which fuels a relatively concentrated or "nuclear" starburst. This is the main mechanism highlighted by resolution-limited simulations for two decades. Nevertheless it does not correctly account for observations of starbursting mergers: real mergers often have a nuclear starburst component, but also a relatively extended starburst activity, often taking place in massive star clusters. Recent simulations that resolve ISM turbulence and dense gas clouds show that the triggering of star formation in mergers is partly from gas inflows, but also from increased ISM turbulence, fragmentation into bigger/denser clouds, resulting in an excess of dense star forming gas compared to atomic gas or moderately-dense molecular gas. This can explain a "starburst regime" of star formation that does not follow the global scaling relations observed for isolated disk galaxies.
4 i.e., the average smoothing length in SPH codes, or the smallest cell size in AMR codes. Back.