Magnetic Fields and Galactic Star Formation Rates
aa r X i v : . [ a s t r o - ph . GA ] J a n Draft version August 28, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
MAGNETIC FIELDS AND GALACTIC STAR FORMATION RATES
Sven Van Loo
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK andHarvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
Jonathan C. Tan
Departments of Astronomy and Physics, University of Florida, Gainesville, FL 32611, USA
Sam A. E. G. Falle
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Draft version August 28, 2018
ABSTRACTThe regulation of galactic-scale star formation rates (SFRs) is a basic problem for theories of galaxyformation and evolution: which processes are responsible for making observed star formation ratesso inefficient compared to maximal rates of gas content divided by dynamical timescale? Here westudy the effect of magnetic fields of different strengths on the evolution of giant molecular clouds(GMCs) within a kiloparsec patch of a disk galaxy and resolving scales down to ≃ . n H > cm − ) at an efficiencyof 2% per local free-fall time, we derive the amount of suppression of star formation by magnetic fieldscompared to the nonmagnetized case. We find GMC fragmentation, dense clump formation and SFRcan be significantly affected by the inclusion of magnetic fields, especially in our strongest investigated B -field case of 80 µ G. However, our chosen kpc-scale region, extracted from a global galaxy simulation,happens to contain a starbursting cloud complex that is only modestly affected by these magneticfields and likely requires internal star formation feedback to regulate its SFR.
Subject headings: galaxies: ISM - galaxies: star clusters: general - ISM: clouds - ISM: structure -methods: numerical - stars: formation INTRODUCTION
Understanding and thus predicting the star formationrate (SFR) that results from a galactic disk of given lo-cal properties, such as gas mass surface density, Σ g , andorbital timescale, is a necessary foundation on which tobuild theories of galaxy evolution. Global and kiloparsec-scale correlations between star formation activity, gascontent, and galactic dynamical properties have beenobserved (Kennicutt & Evans 2012). In molecular-richregions of normal disk galaxies, overall star formationrates are relatively slow and inefficient, i.e., a small frac-tion, ǫ orb ≃ .
04, of total gas is converted to stars ev-ery local galactic orbital time (Suwannajak et al. 2014).Most star formation is concentrated within ∼ −
10 pc-scale regions within GMCs, but even here star formationappears to be inefficient, with similarly small fractions, ǫ ff ≃ . − .
05, of total gas forming stars every localfree-fall time, t ff ≡ (3 π/ [32 Gρ ]) / , where ρ is gas density(Krumholz & Tan 2007; Da Rio et al. 2014). Low effi-ciencies of star formation may be the result of some com-bination of turbulence (Padoan et al. 2014), magneticfields (McKee 1989; Tassis & Mouschovias 2004) and starformation feedback (Krumholz et al. 2014). Here we con-sider the effects of magnetic fields on the evolution, col-lapse and SFR of a turbulent, shearing, kiloparsec-scaleregion of a galactic disk, extracted from the global galaxysimulation of Tasker & Tan (2009, hereafter TT09). [email protected]
Van Loo et al. (2013, hereafter Paper I) showed thatin the limit of purely hydrodynamic evolution with nostar formation feedback, GMCs forming, evolving andcollapsing in this same kiloparsec-scale environment pro-duce too much high density gas and, hence, the SFRis too high. Additional processes are needed to pro-vide extra support to or disruption of the GMCs. Onesuch process may be support by magnetic fields. Ob-servations show that clouds such as, e.g., Taurus and ρ Oph, are threaded by strong B -fields, (e.g., Heiles 2000;Goldsmith et al. 2008) and that the field is connectedon larger scales to galactic fields (Li & Henning 2011).The average solar-neighborhood Galactic magnetic fieldis 6 ± µ G (Beck 2001). Crutcher et al. (2010) arguefor a random distribution of field strengths from 0 to B max with B max = B = 10 µ G in low density gas( n H <
300 cm − ) and B max = B ( n H /
300 cm − ) . at higher denisties. Dynamo amplification to equiparti-tion values is a natural expectation, which has also beenseen in numerical simulations (e.g., Wang & Abel 2009;Pakmor & Springel 2013). The magnitude and directionof the B -field may play an important role in the for-mation of molecular clouds that are forming from com-pression of more diffuse atomic gas (Heitsch et al. 2009;van Loo et al. 2010). By potentially stabilizing and sup-porting GMCs, they may increase cloud lifetimes to thepoint that other processes such as GMC collisions (Tan2000) or spiral arm passage (Bonnell et al. 2013) becomeimportant.Here, we extend the model presented in Paper I toinclude B -fields, varying their strength to investigate theeffect on GMC evolution, especially the amount of high-density clump gas and SFR. § § § NUMERICAL MODEL
We start with the same initial conditions as Pa-per I, i.e., a kiloparsec-sized box at 4 .
25 kpc fromthe galactic center extracted from the TT09 simula-tion. We include the same physical processes, i.e., heat-ing/cooling functions developed in Paper I, and adopt astatic background potential yielding flat rotation curveof 200 km s − . As the TT09 simulation is nonmag-netic, no self-consistent initial magnetic field configura-tion is available. Therefore, for simplicity, we threadthe domain with uniform B -field along the shearing di-rection, i.e., B = B ˆ1 y . For B we assume 0 (no B -field), 10 or 80 µ G. The 10 µ G case represents fieldstrengths close to mean Galactic values expected at aninner ( ∼ µ G case exceeds ob-served kpc-scale B -fields by ∼ B max = 80 µ G corresponds to a density of n H ≃ − . The mean density of the two most mas-sive GMCs identified in Paper I is ≃
470 and 330 cm − ,but they contain substructures extending to more thanten times higher densities. Thus our adopted high mag-netic field case may be more appropriate for predictingthe dynamical evolution of these dense clumps and fila-ments. The above field strengths can also be comparedto the critical values needed to support idealized self-gravitating clouds of a given mass surface density B crit =21 . / (0 .
01g cm − ) µ G = 45 . / (100 M ⊙ pc − ) µ G (e.g.,Mouschovias & Spitzer 1976; McKee 1999). The TT09simulation GMCs have typical Σ ∼ few × M ⊙ pc − ,so only in the strongest field case do we expect signif-icant influence on global GMC dynamics. Indeed, onlyin this 80 µ G case are initial mass-to-flux ratios of thethree least massive GMCs below the critical value. Oth-erwise, the GMCs are supercritical by at least an orderof magnitude. A larger number of intermediate B -fieldstrengths and different initial geometries will be exploredin a future paper.To solve the ideal magnetohydrodynamics (MHD)equations we use the Adaptive Mesh Refinement MHDcode MG (Van Loo et al. 2006; Falle et al. 2012), ratherthan Enzo , which was used in Paper I. The basic al-gorithm is a second-order Godunov scheme with LocalLax-Friedrichs solver and piecewise-linear reconstructionmethod. Self-gravity is computed using a full approxi-mation multigrid to solve the Poisson equation. We alsosolve the non-conservative internal energy equation toensure positive pressures. We use the gas temperaturevalue from internal energy when this is < /
10 of totalenergy and from total energy otherwise. To ensure thatthe solenoidal constraint is met, divergence cleaning isimplemented following Dedner et al. (2002).Refinement is on a cell-by-cell basis, controlled by theTruelove et al. (1997) criterion. Note that, with the mag-netic Jeans length λ B = λ J p v A /c s (with v A the Alfv´en speed and c s the sound speed; e.g., Strittmatter1966), artificial fragmentation is less likely when a mag-netic field is included. We adopt the same boundaryconditions (pseudo-shearing box) and effective grid res-olution of 2048 × ×
256 with 4 refinement levels) as Paper I. Again,we follow evolution for 10 Myr, less than one shear-flowcrossing time over the domain.Inclusion of a uniform magnetic field in the domainintroduces a numerical issue: the simulation time stepis determined by the low density external medium andis < /
100 of that in the disk midplane (region of pri-mary interest). To speed up the simulation, we artifi-cially reduce the magnetosonic sound speed by adoptingthe Boris method with a maximum magnetosonic soundspeed of 2500 km s − (Boris 1970; Gombosi et al. 2002).Although this procedure is, in principle, only valid forsimulations of steady state problems, it is still applicablewhen the flow velocities are much smaller than the soundspeed.Following methods of Paper I, to model star forma-tion we allow collisionless star cluster particles, i.e., pointmasses representing star clusters or sub-clusters of min-imum mass M ∗ = 100 M ⊙ , to form. These particles arecreated when the density within a cell exceeds a star for-mation threshold value of n H , sf = 10 cm − . We assumea local SFR that converts a fraction, ǫ ff = 0 .
02, of gasabove the threshold density into stars per local free-falltime (Krumholz & Tan 2007). No mass-to-flux ratio cri-terion is included, so magnetically-subcritical cells arealso able to form stars, with necessary flux redistribu-tion assumed to be occurring at sub-grid scales. Particlemotions are calculated by using the gravitational field in-terpolated from the grid to the particle positions in theequation of motion. The mass of the particles is includedwhen solving the Poisson equation. RESULTS
Cloud Structure and Fragmentation
Figure 1 shows mass surface densities and tempera-tures within the numerical domain for models with dif-ferent initial magnetic field strengths. The pure hydro-dynamical model has a very fragmented structure, whichbecomes smoother as magnetic field strength increases.Significant differences are also seen in the temperaturedistributions, which are the result of different gas densi-ties (and thus different heating/cooling rates and equilib-rium temperatures) and also different amounts of shockheating from large scale flows and accretion heating fromdense clump formation. Figure 2 shows zoom-ins of twoselected regions, to illustrate more clearly the effects ofmagnetic field strength on resulting gas structures. Dif-ferences in star formation activity are discussed below( § n H = 100 cm − to define gas that is part ofmolecular “cloud” structures. For simplicity, all con-nected cells are counted as a single cloud. Using thisroutine, we find 287, 134 and 28 clouds in the 0 , µ G models, respectively. The mean[median] cloudmasses are 4.0 × [27] M ⊙ for 0 µ G, 7.3 × [100] M ⊙ Fig. 1.—
Logarithmic mass surface density (top) and logarithmic mass-weighted temperature (bottom) integrated along z -axis for (fromleft to right): initial conditions ( t = 0); final conditions ( t = 10 Myr) with B = 0 µ G, 10 µ G and 80 µ G. Lines indicate projected direction ofmass-weighted magnetic field and black dots show positions of star cluster particles. Boxes in the second top panel show different “region”(yellow) and “cloud” (white) selections. for 10 µ G and 3.1 × [5 . × ] M ⊙ for 80 µ G. Mostclouds forming in the 0 and 10 µ G models are thus ofvery low mass. As expected, increasing the magneticfield strength has a major impact on the fragmentationof the initial GMCs. The magnetic critical mass of spher-ical clouds, M B = 16 . B/ µ G) ( n H / − ) − (Bertoldi & McKee 1992), shows we expect a factor ∼
500 change in typical cloud mass comparing 10 and 80 µ Gcases, which is seen in the median cloud masses.In Figures 1 and 2 we also see that the magnetic field ismore distorted by gas motions in Region 1 than Region2. In the 10 µ G run, filaments predominantly lie parallelor perpendicular to field orientation. On ∼
100 pc scales,the 80 µ G fields are harldly affected by the cloud motions,but do become more distorted and disordered in regionsof high star formation activity.
Column and Volume Density Distributions
We also quantify ISM structure by examining the prob-ability distribution functions (PDFs) of Σ g (as viewedfrom above the disk) and number density of H nuclei, n H .Figure 3 shows these quantities weighted by area/volume(top row) and by mass (bottom row). The Σ g PDF is adirectly observable feature of clouds and galaxies, whilethe volume density PDF needs to be reconstructed usingdifferent techniques (e.g., Kainulainen et al. 2014).The Σ g PDFs show the typical combination of a log-normal with power-law tail at high surface densities (e.g.,Kainulainen et al. 2009). The transition occurs around10 − g cm − , corresponding roughly to the minimum Σ g of regions containing “cloud” (GMC) gas, which is essen-tially all part of self-gravitating structures. There is vari-ation from region to region, with Region 1 having its log-normal peaking at higher Σ g . The shape of the power-lawdistribution differs only slightly between these individual regions, roughly scaling as Σ − α Σ , PDF g with α Σ , PDF ≃ g is several times higher in Region 1than Region 2, i.e., the former has a higher dense gasmass fraction. This also correlates with its more frag-mented morphology and larger SFR (below).There are only modest differences seen in the PDFsas a function of magnetic field strength (although notethe large dynamic range in the figures). At higher Σ g ,these are more clearly seen in Region 2, which also showsthe greatest effect from B -field strength on its degree offragmentation and SFR (below).Comparing to observations, somewhat steeper indices,i.e., α Σ , PDF ≈ . − .
65, have been reported in dust-emission-derived Σ g PDFs by Schneider et al. (2014).We note that some extinction-based studies find PDFsthat can be fit by a single log-normal up to Σ ≃ . − (Butler et al. 2014a). Note, these observa-tions are derived from “in-plane” higher-resolution viewsof clouds, in contrast to the “top-down” views of thegalactic plane from the simulations. Direct observationalconstraints of top-down views are expected to be ableto achieved with high angular resolution observations ofnearby galactic disks, e.g., with ALMA .Volume density PDFs in the simulations show similartrends as the Σ g PDFs, with Region 1 having a largerfraction of gas at higher densities. For n H ≥
100 cm − ,i.e., the cloud gas, the distributions can be reason-ably well approximated with single power laws up to ∼ cm − with power law index of ≃ − Evolution of Dense Gas Mass Fractions & SFRs
Here we consider the time evolution of cloud ( n H >
100 cm − ) and clump ( n H > cm − ) mass fractions Fig. 2.—
Mass surface densities of Region 1 (top) and 2 (bottom) for 0 µ G (left), 10 µ G (middle) and 80 µ G (right). Black dots showstar cluster particles, each representing 100 M ⊙ of stars, while lines indicate direction of mass-weighted magnetic field. over the 10Myr timescale of the simulations (Fig. 4). Forthe whole kpc-sized region, cloud mass fractions startfrom relatively high values, ≃ .
6, inherited from theTT09 simulation. Over 10 Myr, they show a moderatedecline, partly because of the build up a mass fractionin stars, and partly in the magnetized cases from ex-pansion of cloud envelopes due to the introduced mag-netic pressure. Clump mass fractions grow to peak valuesof ∼ . − .
1, with smaller peak values reached moreslowly as B -field is increased. Since stars are only allowedto form from clump gas, SFR evolution closely tracksclump mass fraction evolution, although it exhibits morestochasticity. At late times the different models appearto converge in their clump mass fractions and SFRs, per-haps due to the exhaustion of most of the initially un-stable gas via star formation.Regions 1 and 2 follow similar trends. Indeed Re-gion 1’s area of 0.16 kpc contains large fractions of theclump mass and star formation of the whole kpc region.Region 2 starts with gas structures that are somewhateasier to stabilize with magnetic fields (at least in direc-tions perpendicular to the initial y -direction field orien- tation), so larger differences are seen in clump and SFRevolution as field strength is increased. Region 1 con-tains 3 × as much cloud gas as Region 2. It containstwo GMCs in the process of merging, and the resultingcloud has significant kinetic energy. The magnetic fieldtherefore plays a minor role in this case. The overallSFR surface densities are about 10 × larger in Region 1compared to Region 2.Some caveats are in order. First, the formation ofclumps is a response to simulation initial conditions,where dense, self-gravitating clouds are allowed to col-lapse to high densities as the resolution is suddenly in-creased. Second, once formed, star particles contributegravitationally, but stellar feedback, implementation ofwhich is much more uncertain and numerically challeng-ing, is not yet included in the simulation. Magnetic Fields and Average SFRs
The detailed time history of clump and star formationis sensitive to the choice of initial conditions. It is there-fore useful to compare the SFR averaged over the full10 Myr evolution to the observations since these are alsoaveraged over ∼
10 Myr. Here we consider the absolute
Fig. 3.—
Mass surface density PDFs weighted by area (top left) and by mass (bottom left) and volume density PDFs weighted by volume(top right) and mass (bottom right). In each panel, 0, 10, 80 µ G models are shown with black, blue and red lines, respectively. Solid linesshow the whole kpc-sized region, dashed lines Region 1, and dotted lines Region 2. Thin green lines in top left panel show log-normal andpower-law distributions fitted to these two PDFs. and relative average SFRs that are seen in the simula-tions on the kiloparsec, “region” and “cloud” (i.e., 100pc)scales.Figure 5 shows SFR surface density, Σ
SFR , versusΣ g . Observational results are those of ∼ kpc-scale re-gions of galactic disks from Bigiel et al. (2008) and of in-dividual Galactic GMCs from Heiderman et al. (2010).On the kpc region scale, the simulations all start withΣ g = 17 . M ⊙ pc − , decreasing to 13 . , . , . M ⊙ pc − after 10 Myr for 0 , , µ G cases, respectively, i.e.,Σ
SFR = 0 . , . , . M ⊙ yr − kpc − . On these scales,the magnetic field only has a modest impact on SFR,the values of which are much higher ( ∼ − × )than systems with equivalent Σ g observed by Bigiel et al.(2008) . However, as noted in § The average SFR derived in Paper I using
Enzo is 2 . × theequivalent hydrodynamic run with MG , which we attribute to ourimplementation of improved methods for staggering introductionof AMR at early times in the simulation. Focusing on smaller 100 pc-size “Cloud” scales(see Fig. 1), we find simulated SFRs overlap withobservational rates in Galactic GMCs derived byHeiderman et al. (2010). Stronger magnetic fields againalmost always lead to lower SFRs. One reason for thedecrease in SFR with increasing B -field is that clumpsform later in models with stronger fields ( § B = 0 µ G run.Our simulations thus reproduce SFRs similar to someobserved local Galactic GMCs. Note the Heiderman etal. GMCs do not contain especially vigorous regions ofmassive star formation, so may be relatively less affectedby internal star formation feedback. On the kpc scale,even our strongly magnetized simulation has values ofΣ
SFR that are too large. However, this is largely a con-sequence of the “starburst” of Region 1. Inclusion ofstar formation feedback is expected to have an impactin reducing this activity, to be investigated in a futurepaper. SUMMARY AND DISCUSSION
We have studied the effects of magnetic fields on molec-ular clouds extracted from a global galaxy simulation,especially for cloud structure and SFRs. Magnetic fields
Fig. 4.—
Time evolution of mass fraction (left) in clouds (solid), dense clumps (dashed) and star cluster particles (dotted) and SFR perunit disk area (right) for different values of magnetic field, i.e., 0 µ G (black), 10 µ G (blue) and 80 µ G (red). Top panel is for full domain,middle for Region 1 and bottom for Region 2. suppress fragmentation, as expected from considerationof the magnetic critical mass. Their effects on Σ g and n H PDFs are more modest.In the context of models of star formation from gasabove a threshold density ( n H ≥ cm − ), we find thaton the largest kpc scale, average SFRs are only modestlysuppressed by magnetic fields. However, this is due tothe presence of a starbursting region (Region 1) in thesimulation domain. Considering other regions, includingdown to “GMC”-scales of ∼
100 pc, we find a variety of Σ
SFR s, with values quite similar to those of some GalacticGMCs. Average suppression factors of ǫ B = 0 . , .
54 for B = 10 , µ G compared to the nonmagnetized case areseen in the clouds, but in some cases there can be muchmore dramatic effects, including complete suppression inone cloud.As a numerical experiment, this study highlights sev-eral important aspects, including issues of B -field initial-ization when overdense structures are already present ininitial conditions. Stochastic effects are important, i.e., Fig. 5.—
SFR surface density versus total gas mass surfacedensity showing kiloparsec-scale data (Bigiel et al. 2008) (opengrey circles) and GMC data (Heiderman et al. 2010) (open or-ange circles). Solid symbols show mean values for full kpc domain(squares), Region 1 (circle) and Region 2 (triangle), while selectedcloud regions are indicated with the letter assigned in Fig. 1. Col-ors indicate magnetic field strength: 0 µ G (black), 10 µ G (blue)and 80 µ G (red). variation in behavior of individual GMCs, and kpc-scaleSFRs can be influenced by a single GMC complex. Asub-grid model for star formation is needed: here via achosen threshold density and an empirically motivatedefficiency per local free-fall time, ǫ ff = 0 .
02. Such low efficiencies may require the effects of local star formationfeedback, like protostellar outflows, to maintain turbu-lence and prolong star cluster formation (Nakamura & Li2007). And/or they may result from the influence ofmagnetic fields themselves. Note, we have not includedmagnetic field effects on the star formation sub-gridmodel, rather keeping its empirically-based parametersfixed for simplicity. There is scope in future work for ex-ploring magnetic sub-grid star formation models, wherecell mass-to-flux ratio is also an input, which should leadto a more natural threshold criterion for star formationactivity.Other effects of local star formation feedback, such asstellar winds, ionization and supernovae, have not yetbeen included in these simulations, but are expected toact to further reduce SFRs. To disentangle the relativeimportance of these effects and of magnetic fields willrequire testing many properties of the simulated cloudsand young stellar populations against observed systems(e.g., Butler et al. 2014b).SvL acknowledges support from the SMA Postdoc-toral Fellowship (SAO). JCT acknowledges support fromNASA grant ATP09-0094. Resources supporting thiswork were provided by NASA High-End Computing Pro-gram, the
Smithsonian Institution High PerformanceCluster and the High Performance Computing facilitiesat the University of Leeds.and the High Performance Computing facilitiesat the University of Leeds.