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THE LESSER ROLE OF SHEAR IN GALACTIC STAR FORMATION: INSIGHT FROM THE GALACTIC RING SURVEY

We analyze the role played by shear in regulating star formation in the Galaxy on the scale of individual molecular clouds. The clouds are selected from the 13CO J = 1–0 line of the Galactic Ring Survey. For each cloud, we estimate the shear parameter which describes the ability of density perturbations to grow within the cloud. We find that for almost all molecular clouds considered, there is no evidence that shear is playing a significant role in opposing the effects of self-gravity. We also find that the shear parameter of the clouds does not depend on their position in the Galaxy. Furthermore, we find no correlations between the shear parameter of the clouds with several indicators of their star formation activity. No significant correlation is found between the shear parameter and the star formation efficiency of the clouds which is measured using the ratio of the massive young stellar objects luminosities, measured in the Red MSX survey, to the cloud mass. There are also no significant correlations between the shear parameter and the fraction of their mass that is found in denser clumps which is a proxy for their clump formation efficiency, nor with their level of fragmentation expressed in the number of clumps per unit mass. Our results strongly suggest that shear is playing only a minor role in affecting the rates and efficiencies at which molecular clouds convert their gas into dense cores and thereafter into stars.


Introduction
The rate and efficiencies at which galaxies convert gas into stars determine their evolution and their observable properties. It is now well established that stars form in the densest regions of molecular clouds (Blitz 1993). It is therefore of prime importance to assess the relevance of the physical processes that affect the evolution of clouds and the rate at which they convert gas into stars. Supersonic turbulence produces local compressions, a fraction of which can be 'captured' by gravity and proceed to form stars (Klessen et al. 2000;Goodwin et al. 2004;Dib et al. 2007;Offner et al. 2008;Padoan & Nordlund 2011). Magnetic fields play an important role in determining the fraction of gravitationally bound gas in star forming clouds. Results from numerical simulations show that stronger magnetic fields lower the rate of dense core formation (Vázquez-Semadeni et al. 2005;Price & Bate 2008;Dib et al. 2008;Li et al. 2010). The regulation of the star formation rates on galactic scales have been explored through scenarios in which stars form as the result of gravitational instabilities in the disk (Madore 1977;Slyz et al. 2005;Li et al. 2006;Dobbs 2008). The gravitational instability can be mediated by thermal instabilities (Dib & Burkert 2005;Dib et al. 2006;Wada et al. 2000;Khesali & Bagherian 2007;Shadmehri et al. 2010;Kim et al. 2011) and influenced by turbulence (Shadmehri & Khajenabi 2012). Wong & Blitz (2002) and Blitz & Rosolowsky (2006) argued that the fraction of star forming gas in galaxies is related to the pressure of the interstellar medium. The role of stellar feedback in regulating the star formation rates (SFRs) and efficiencies (SFEs) in molecular clouds has been highlighted in a number of recent studies (Murray et al. 2010;Dib et al. 2009;Dib 2011). In particular Dib et al. (2011) and Dib (2011) showed that the SFEs and SFRs depend critically on the strength of the metallicity dependent, radiation driven winds. Weaker winds, associated with lower metallicities, allow for longer episodes of star formation in the clumps/clouds and lead to higher SFEs.
Another physical agent that has been suspected of participating in the regulation of the SFR is the level of shear in galaxies (Silk 1997), or shear induced cloud collisions (Tan 2000). the model of Tan (2000) predicts an enhanced/reduced SFR in region of high/low shear. On the observational side, Seigar (2005) used observations for 33 nearby galaxies and argued for the existence of a correlation between the shear rates of the galaxies and their ratio of far-infrared to K s -band luminosity which is a proxy for the specific star formation rates. However, he found a very weak correlation between the shear rate and the surface density of the SFR. Hunter et al. (1998) assessed the competition between self-gravity and shear in a number of Irregular galaxies. They found rather poor correlations between the shear strength and the SFRs. Elson et al. (2012) applied the Hunter et al. (1998) analysis to the blue compact dwarf galaxies NGC 2915 and NGC 1705. They found that the extent of the regions in which shear is important in these galaxies matches approximately the size of their stellar disks. However, they do not report on the quantitative relationship between the shear strength and the SFR. On the other hand, Weidner et al. (2010) presented numerical simulations of star cluster formation for clumps of masses 10 6 M ⊙ , sizes of 50 pc, and of varying rotational support. They found that higher initial shear levels expressed in the form of initially larger rotational energies lead to a reduction of the SFE and of the SFR. Hocuk & Spaans (2011) modelled star formation in molecular clouds within an AGN. They varied the level of shear by varying the mass of the black hole while keeping the cloud at the same distance. Hocuk & Spaans also report a reduction of the SFE and SFR as the shear induced by the black hole increases.
While the results of Weidner et al. and Hocuk & Spaans show a clear trend in which the SFR and SFE decrease with increasing shear levels, their simulations do not include magnetic fields nor feedback, both of which act to reduce the SFR and SFE. Feedback from the first generation of stars formed in clouds with low levels of shear will disperse the remaining gas in the cloud and reduce the SFE. The observational characterisation of the role of shear in star formation (Hunter et al. 1998;Seigar 2005;Koda et al. 2009;Elson et al. 2012) has primarily investigated the ability of galaxies to convert diffuse gas into molecular clouds. While there is some observational evidence that high shear levels may prevent the formation of star forming molecular clouds (Elson et al. 2012), the role of shear in determining the SFR within these clouds is poorly understood. So far, there has been no observational tests to determine whether shear plays a significant role in star formation on the scale of individual clouds. In this work, we use data from the Galactic Ring Survey (GRS) in order to calculate the shear parameter, S g , which compares the level of shear to self-gravity on the scale of individual molecular clouds. We also investigate whether S g varies with Galactocentric radius and whether it correlates with the cloud masses and level of fragmentation. In §. 2 we briefly describe the selected sample of GRS clouds and in §. 3 we describe the method we employ to quantify the effects of shear. In §. 4 we present our results, and in §. 5, we conclude.

Data: The Galactic Ring Survey
The Boston University-Five College Radio Astronomy Observatory Galactic Ring Survey (GRS) is a 13 CO J = 1 − 0 emission line survey which covers the Galactic longitudes 18 • < l < 55.7 • and Galactic latitudes |b| ≤ 1 • with a spectral resolution of 0.21 km s −1 and a spatial resolution of 46 ′′ (Simon et al. 2001;Jackson et al. 2006). Rathborne et al. (2009) applied an adapted version of the CLUMPFIND algorithm (Williams et al. 1994) to the 13 CO data cubes and identified 829 clouds and 6124 clumps within them. Roman-Duval et al. (2009) measured the kinematical distances to 752 clouds from the Rathborne et al. (2009) sample assuming the Clemens (1985) rotation curve for the Galaxy with the parameters (R 0 , V 0 )=(8.5 kpc, 220 km s −1 ), where R 0 is the Galactocentric distance of the Sun, and V 0 is the rotation velocity of the gas at the position of the Sun. The masses of 749 of these clouds, their surfaces densities, and their velocity dispersions have been derived by Roman-Duval et al. (2010). The clouds in the GRS are affected by the Malmquist bias (Roman-Duval et al. 2010) and the masses of the nearby, low mass, molecular clouds are highly uncertain. We therefore select clouds whose mass is > 10 M ⊙ . We crossed matched the data of Rathborne et al. (2009), Roman-Duval et al. (2009), and Roman-Duval et al. (2010 and after eliminating a few clouds whose kinematical distances are degenerate, we obtained a sample which contains the masses of 727 clouds M, their surface densities Σ, velocity dispersions σ, sizes R, Number of clumps N cl , and kinematical distances D. Their masses fall in the range [14 − 9.92 × 10 5 ] M ⊙ , their surface densities between [4 − 601] M ⊙ pc −2 , their velocity dispersions between [0.38 − 6.70] km s −1 , and their sizes between [0.2 − 34.5] pc, with median values of 2.54 × 10 4 M ⊙ , 143.9 M ⊙ pc −2 , 2.27 km s −1 , and 7.5 pc, respectively. We calculate the clouds Galactocentric radius as being their Galactocentric distances projected onto the Galactic plane and which are given by:

Quantifying the Effect of Shear
Elmegreen (1993) and Hunter et al. (1998) argued that if condensations in the interstellar medium (ISM) accumulate mass because of streaming motions along magnetic field lines, then their growth rate is determined by the competition between their self-gravity and the local level of galactic shear. This competition will be more relevant than the one based on the competition between self-gravity, pressure, and the Coriolis force (which is represented by the Toomre Q parameter) because magnetic fields can transfer the angular momentum away from the cloud (Elmegreen 1987). The growth of the density perturbations against shear is given by (π G Σ)/σ, where Σ and σ are the local gas surface density and velocity dispersion, respectively. The growth of the perturbation is most effective between −1/A and 1/A, where A is the Oort constant given by: where Ω and V are the angular and rotation velocities of the gas, respectively. The amplitude of the growth from an initial perturbation of the surface density δΣ 0 will be given by: Hunter et al. (1998) argued that for a perturbation to be significant and not to be erased by shear, it must grow by a large factor, C, which they chose to be C = 100. A factor of ∼ 100 corresponds to the density contrast between the diffuse phase of the ISM with densities ∼ 0.1 − 1 cm −3 and the molecular phase with densities 100 cm −3 . This leads to a critical surface density Σ sh given by: where α A = ln(C)/2 (in Hunter et al. α A = ln(100)/2 ∼ 2.3). One can then define a shear parameter for gravitational instability, S g , which is given by: Shear will disrupt density perturbations when S g > 1 and would be ineffective in erasing them when S g < 1.

Results
We calculate the rotational shear, A, affecting the sample of selected molecular clouds in the GRS by performing an interpolation of the rotational shear curve at their respective Galactocentric locations. Using their measured gas surface densities and velocity dispersions, we calculate S g using Eq. 5. In this work, we are concerned with the effects of shear on the scale of molecular clouds and the ability of shear to erase condensations before they are able to collapse into stars Therefore, we adopt a value of C = 10 3 (α A ∼ 3.45). This factor corresponds to the density contrast between the average molecular clouds densities (∼ 100 cm −3 ) and the densities of ∼ 10 5 cm −3 at which molecular gas becomes gravitationally bound (Dib et al. 2007;Parmentier 2011). Fig. 2 (full line) displays the distribution of S g values for the entire sample of selected clouds while the dashed line in the same figure displays the distribution of S g values for clouds that are considered to be gravitationally bound with their virial parameter 1 α vir < 1. Fig. 2 shows that almost all molecular clouds have S g < 1 and both distributions are peaked at a value of ∼ 0.065. This implies that Galactic shear is playing only a minor role in governing their evolution, mass growth, and the conversion of molecular gas into stars, and it also reflects the fact that almost all molecular clouds in the Galactocentric radius range covered by the GRS are prone to star formation. Fig. 3 (top) displays the S g values of the clouds versus their Galactocentric radius for gravitationally bound clouds (α vir < 1, diamonds) and unbound clouds (α vir > 1, triangles). This figure shows that there is no significant variation of S g with Galactocentric radius, particularly for the gravitationally bound clouds. In order to check for the correlation between the S g values and the Galactic SFR, we compare in Fig. 3 (bottom) the S g values normalised to the average value of S g for the sub-sample of gravitationally bound clouds to the Galactic radial profile of the SFR normalised to the SFR at the solar radius (the Galactic SFR data points are from Güsten & Mezger 1983, Lyne et al. 1985, and Guibert et al. 1978. The absence of a correlation 2 between the radial distribution of S g values on the scale of individual clouds and the radial profile of the Galactic SFR is an indication that other physical processes such as magnetic fields and stellar feedback (which are not taken into account in the simulations of Weidner et al. 2010) regulate the rate at which molecular clouds channel their gas reservoir into gravitationally bound cores and ultimately into stars.
Our results, which are obtained on the scale of individual molecular clouds (0.2 − 35) pc, suggest that the S g values derived by Elson et al. (2012) and Hunter et al. (1998) on larger scales ( 300 − 400 pc) are only able to describe in which regions of galactic disks molecular clouds are unable to form under the effect of shear without providing a direct link between the strength of shear and the SFRs within the clouds. The S g values measured on large scales are also likely to overestimate the true values of S g of individual clouds especially in the outer regions of these galaxies. The likely reasons for this are two fold. On the one hand, azimuthal averaging is likely to lower the surface density of molecular gas, especially in the outer regions where molecular clouds are more sparsely distributed. Furthermore, the S g values in Hunter et al. (1998) and Elson et al. (2012) use fixed values of the gas velocity dispersions σ of ∼ 5 km s −1 (e.g., Elson 2012) which are typical of the diffuse phase of the interstellar medium and which are substantially larger than the velocity dispersion of the gas on the scale of the clouds (see the values of σ for the GRS clouds reported above).
Our main conclusion is that Galactic shear plays a minor role in opposing the clouds self-gravity in gravitationally bound clouds and thus does not influence substantially the SFR in the clouds. Nevertheless, it remains interesting to verify whether shear correlates with any of the clouds basic parameters. Weidner et al. (2010) and Escala (2011) argued that there is potentially a correlation between the mass of the most massive young cluster that can form in a galaxy, and therefore of its progenitor gaseous clump, and the level of shear. Fig. 4 displays the S g values of the gravitationally bound clouds (diamonds) and the unbound ones (triangles) versus their clouds masses (top) and sizes (bottom). The masses and sizes of the GRS clouds are strongly correlated with M = 228 ± 18R 2.36±0.04 . With the exception of two low mass clouds, Fig. 4 shows that for the bulk of the GRS molecular clouds, extending over 4 and 2.5 orders of magnitude in mass and physical size, there is no evidence of a correlation between their masses, sizes and their shear levels. In particular, the S g values of the gravitationally bound clouds which are the likely hosts of stellar clusters tend to increase with masses and sizes (S g ∝ M 1.30±0.19 , S g ∝ R 0.81±0.07 ). This implies, that physical processes other than shear such as magnetic fields, turbulence, gas accretion, and stellar feedback govern the mass growth of gravitationally bound clouds. However, for the population of unbound clouds, the upper envelope of the data may be suggestive of the role played by shear in setting the maximum mass of the cloud. The small number of clouds found at S g > 0.3 may also reflect an observational bias since the density of these unbound clouds which are likely to be dispersing would fall below the threshold density necessary to excite the 13 CO J = 1 − 0 line. Finally, we explore whether lower levels of shear are associated with higher levels of fragmentation of the clouds as would be expected if shear was playing a significant role in the dynamics of the clouds. Fig. 5 displays the shear parameter versus the number of clumps found in each cloud for the bound clouds (diamonds) and unbound clouds (triangles). Unsurprisingly, there is no correlation between the S g values of the clouds and the number of clumps they harbour. This suggests that shear plays only a minor role in the fragmentation of the clouds and in determining the SFR.

Conclusions
In this work, we use data from the Galactic Ring Survey to test the importance of Galactic shear in regulating star formation on the scale of individual molecular clouds. We calculate the shear parameter, S g , which is the ratio between a critical surface density for perturbations in molecular clouds to grow by a factor 10 3 (thus to reach densities of ∼ 10 5 cm −3 and become gravitationally bound) in the presence of shear to the actual surface density of the clouds. We find that the distribution of S g is peaked around a value that is much smaller than unity (∼ 0.065). This suggests that Galactic shear plays only a minor role in opposing self-gravity in Galactic molecular clouds. We find no dependence of the S g values calculated on the scale of clouds ([0.2 − 35] pc) on Galactocentric radius and no correlation with the radial profile of the Galactic SFR. We also find that for gravitationally bound clouds, higher shear values are not associated with lower masses (or sizes) which casts doubts on the idea that the maximum mass of a cluster is determined by the local level of shear. We also do not find a correlation between the clouds shear parameter values and their level of fragmentation. Other authors have calculated S g values on larger scales ( 300 − 400 pc; for the diffuse phase of the ISM). The calculation of the shear parameter by those authors assumed a growth factor for the density perturbations of 10 2 which corresponds to the typical density contrast between the density of the diffuse phase of the interstellar medium and that of molecular clouds. They found that regions of low shear correlate spatially with the size extent of the stellar disks in galaxies (Hunter et al. 1998;Elson et al. 2012). The picture that emerges from their and our study is the following: shear appears to play a significant role in regulating the formation of molecular clouds in galactic disks, and perhaps in establishing an upper limit on the masses of the initially unbound clouds as a function of the shear level. However, when clouds become gravitationally bound, our results suggest that the effects of shear are negligible in determining the rate at which clouds convert their gas into gravitationally bound cores and thereafter into stars. The rate and efficiency of star formation are likely to be be governed by other physical processes such as magnetic fields, turbulence, and stellar feedback as pointed out by several recent studies.   Empty diamonds designate the clouds with α vir < 1 and the empty triangles designate the clouds with α vir > 1. The lower panel displays the same S g values normalised to the mean value for the gravitationally bound sample < S g > (α vir < 1). The filled circles, triangles, and squares represent the radially determined star formation rates normalised to the the SFR at the position of the Sun (data taken from Lyne et al. 1985, Güsten & Mezger 1982, and Guibert et al. 1978.  -Correlation between the S g of the clouds in the GRS and the the number of clump they harbour. Empty diamonds designate the clouds with α vir < 1 and the empty triangle designate the clouds with α vir > 1.