Magnetic fields and radiative feedback in the star formation process
aa r X i v : . [ a s t r o - ph . GA ] F e b Magnetic fields and radiative feedback in the starformation process
Daniel J. Price ∗ ,† and Matthew R. Bate † ∗ Centre for Stellar and Planetary Astrophysics, School of Mathematical Sciences, MonashUniversity, Vic 3800, Australia † School of Physics, University of Exeter, Stocker Rd, Exeter EX4 4QL
Star formation is a complex process involving the interplay of many physical effects, includinggravity, turbulent gas dynamics, magnetic fields and radiation. Our understanding of the process hasimproved substantially in recent years, primarily as a result of our increased ability to incorporatethe relevant physics in numerical calculations of the star formation process. In this contributionwe present an overview of our recent studies of star cluster formation in turbulent, magnetisedclouds using self-gravitating radiation-magnetohydrodynamics calculations[1, 2]. Our incorporationof magnetic fields and radiative transfer into the Smoothed Particle Hydrodynamics method arediscussed. We highlight how magnetic fields and radiative heating of the gas around newbornstars can solve several of the key puzzles in star formation, including an explanation for whystar formation is such a slow and inefficient process. However, the presence of magnetic fieldsat observed strengths in collapsing protostellar cores also leads to problems on smaller scales,including a difficulty in forming protostellar discs and binary stars [3, 4], which suggests that ourunderstanding of the role of magnetic fields in star formation is not yet complete.
INTRODUCTION
Star formation is a ubiquitous phenomenon in spiral galaxies such as our very ownMilky Way. However the detailed process by which gas is converted into stars in suchgalaxies is still relatively poorly understood. One of the key open questions is why starformation is so remarkably inefficient, with only a few percent of the mass of gas in amolecular cloud ending up in stars. Recent observational results for nearby molecularclouds in the Spitzer cores-to-discs survey [5] find the mass of stars in a star-formingclouds typically around 3-6% of the mass of the parental molecular cloud, the latterestimated by multiplying the column density inferred from interstellar dust extinctionmaps by the area of the cloud (defined by an extinction threshold). Equivalently theobservational result can be restated as saying that only a few percent of molecular cloudgas is converted into stars per gravitational free-fall time.Thus star formation appears to be both inefficient — in the sense that not much gas hasbeen converted into stars — and slow , in the sense that, over the timescales necessaryfor gravity to act on the global cloud, not many stars have formed. It is also importantto note that the inefficiencies found by Evans et al. [5] refer to nearby molecular cloudsthat are actively forming stars. It is a further challenge to explain molecular clouds whererelatively little star formation occurs at all, such as the Pipe Nebula (efficiency ∼ . r -Ophiuchus cloud.he fact that star formation appears to occur on a slower timescale than the gravita-tional one indicates that the answer must lie in physics beyond gravity, or at least beyondthe self-gravity of the cloud. To achieve inefficiency of star formation over time scalesmuch greater than the dynamical time must further involve involve unbinding the cloudin some way – for example by internal driving of turbulence by jets and outflows [7] orinvoking tidal forces from the Galactic potential [8].Magnetic fields are, observationally, a good candidate for explaining why clouds inotherwise similar environments can have vastly different star formation efficiencies. Forexample, recent optical polarisation maps of the Pipe Nebula [9] reveal a remarkabledegree of uniformity in the magnetic field (as inferred from the polarisation angles), incontrast to the wide dispersion in polarisation angles (on large scales) seen in activestar forming regions like Orion [10]. The nearby Taurus molecular cloud, recentlysurveyed by Goldsmith et al. [11], forms an intermediate case, with relatively inefficientstar formation (but more efficient than the Pipe Nebula) and also a well-ordered largescale magnetic field (better ordered than Orion, though less well ordered than the Pipe),together with compelling evidence for magnetic fields strong enough to control the flowof gas in (relatively) low density outer regions [12].The effect of a magnetic field on the self-gravitating collapse of gas to form starscan be quantified in terms of the ratio of mass within a volume to the magnetic fluxthreading the surface of that volume. At a critical value magnetic fields are able toprevent collapse entirely, unless some decoupling of the magnetic field from the gasoccurs (i.e., ambipolar diffusion). Indeed, this theoretical understanding led to the so-called ‘standard model’ of star formation as a quasi-static diffusion process mediatedby magnetic fields [13]. However for all star formation to occur in this manner requiresthat all molecular cloud cores are sub-critical (magnetic fields able to prevent collapse),whereas Zeeman measurements of field strengths in cores indicate that they are generallymarginally supercritical (mass-to-flux ratios of few times critical, see Crutcher [14]). Afurther difficulty is the problem of how to maintain the observed turbulent motions inthe cloud, since supersonic turbulence decays rapidly with or without magnetic fields inthe absence of a continual driving mechanism [e.g., 15].This latter consideration in particular has led many to consider so-called ‘rapid’ or‘turbulent’ models of star formation, where the main controlling ingredient is turbulencerather than magnetic fields, with clouds that assemble, form stars and disperse withinroughly one crossing time [16]. Indeed, simulations based on simply the interaction ofturbulent gas dynamics and gravity alone [e.g. 17, 18, 19] do a remarkably good job ofreproducing many observed statistical properties of star formation, including the grosscharacteristics of the initial mass function, multiplicity as a function of mass and thefrequency of low mass binary stars [19]. However, the star formation efficiency in thesecalculations is much higher than observed, since the fraction of gas that is initially boundand will remain so in the absence of feedback processes, giving a SFE of order 50% (thetypical fraction of gas that is bound at the end of the calculations). Improved statistics inmore recent calculations of larger clouds [19] also suggest that there is also a problemin terms of an over-production of very low mass stars and brown dwarfs.Whilst observations indicate that magnetic fields are not strong enough to preventglobal collapse in typical clouds, they can nevertheless play a determining role in theinternal dynamics by acting as a source of pressure within the cloud (quantified by theatio of gas-to-magnetic pressure: The plasma b ) and by the magnetic braking of rotatingcores. Indeed, observationally typical values for b in molecular cloud cores are of order0 . NUMERICAL METHODS
Modelling star formation is made difficult by the tremendous range of length andtime scales involved. For example, to follow the collapse of a giant molecular cloudof size ∼ a few pc ( ∼ km) and containing up to 10 M ⊙ of material to a starthe size and mass of the Sun ( R ⊙ ∼ km) requires resolution over 6 orders ofmagnitude in length, around 14 orders of magnitude in density [10 M ⊙ / ( R ⊙ ) → M ⊙ / R ⊙ ] and roughly 11 orders of magnitude in timescale (from the dynamical time ofa GMC, ∼ N − body codes. By contrast, the solution of the equa-tions of magnetohydrodynamics (MHD) in SPH has proved more challenging, in partdue to the early discovery [24] of numerical instabilities associated with particular for-mulations of the MHD equations. Our goal over the last few years has been to developthe techniques for MHD in SPH sufficiently to be able to study the role of magneticelds in star formation problems. This has involved dealing carefully with many of thenumerical issues including the aforementioned instabilities [25], the treatment of MHDshocks [26, 27] and perhaps most importantly (and the main difficulty), exploring meth-ods for enforcing the (cid:209) · B = B = (cid:209) a × (cid:209) b . The corresponding induction equation for the magnetic fieldtakes the form d a dt = d b dt = , (1)corresponding to the advection of magnetic field lines by Lagrangian particles. TheEuler potentials are thus very naturally suited to a Lagrangian description, but thereare important limitations to their use. The main one is that fields with complicatedtopologies (such as a poloidal field wrapped by a toroidal one) cannot be represented byEuler potentials because they would become double-valued. A corollary to this is thatsuch fields also cannot be generated during the calculation and thus important dynamoprocesses are not captured. Another way to understand this is to appreciate that evolutionof a field using (1) is, in effect, a mapping of the field from the initial to final positions ofthe SPH particles, and requires a one-to-one mapping, after which the field winding willno longer be captured. A further issue is that it is difficult to formulate non-ideal MHDterms for the Euler potentials — although we add artificial dissipative terms to captureshocks it is clear that these do not and cannot be used to represent a correct physicaldissipation [see 29].Nevertheless, with the above caveats in mind, we have been able to study the effectof magnetic fields on the star formation process, mainly studying the influence of themagnetic field in supporting the cloud in the initial stages of collapse, and the effectof this on the subsequent star formation sequence. Rather than starting with globalturbulent-cloud star cluster formation calculations, we first studied the effect of magneticfields on the formation of individual stars at small scales, from which we have proceededto study star cluster formation on larger scales (see following sections).Alongside the development of the MHD algorithms, we have also developed an algo-rithm for incorporating the effect of radiation using the flux-limited diffusion approxi-mation. This is an approximation in radiation is assumed to be transported by diffusionthrough both optically thick and thin regions, but with the diffusion speed limited to thespeed-of-light in optically thin regimes. The key challenge for adapting grid-based flux-limited diffusion techniques into an SPH context was to develop an implicit integrationmethod that enables the radiative transport to (which is much faster than the gas dynam-ics, particularly in optically thin regions) to be computed on a timescale similar to thehydrodynamics [for details see 30, 31]. IGURE 1.
Effect of magnetic fields on the formation of circumstellar discs around young stars:Results of a simulation following the collapse of a rotating 1 M ⊙ spherical cloud core without (top) andwith (bottom) a uniform magnetic field threading the initial cloud. Field strengths are given in terms ofthe mass-to-magnetic flux ratio divided by the critical value at which magnetic fields prevent collapsealtogether. Despite the relatively weak field with respect to gravity the magnetic field is able to almostcompletely prevent disc formation due to a combination of magnetic braking and magnetic pressure in thecollapsing core. Time is shown in units of the gravitational free-fall time ( t f f ). SINGLE AND BINARY STAR FORMATION
Our first application of our MHD-SPH algorithm was to the collapse of a 1 M ⊙ , R = × cm molecular cloud core to form single and binary stars. As the initial conditionwe assumed a dense, spherical, cold ( T ∼ K ) core in pressure-equilibrium with awarm, low density medium with an initially uniform magnetic field threading the coreand the medium. The sphere was given an initial solid body rotation, of W = . × − rad s − for the case of a single star, and W = − rad s − for a binary, in the latter casealso imposing an initial m = r c = − g/cm ) and becomes polytropicwith g = . t / t f f ), for a calculation with no magnetic fields (top) and with a mass-to-flux ratio of 4 times the critical value — that is magnetic fields that are too weakto prevent collapse by a factor of 4). Despite the relatively weak field the effect on theformation of the circumstellar disc is catastrophic. In the absence of a magnetic fielda disc is formed that is sufficiently massive so as to become unstable to gravitationalnstability in the form of large scale spiral arms, yet with a magnetic field only thefaintest trace of a disc is visible even at the end of the calculations.Since our initial calculations by several other groups have found similar results basedon numerical simulations [e.g. 4, 32] and also semi-analytic calculations by Galli et al.[33] (see Galli, this volume). In fact [4] somewhat alarmingly discuss a ‘fragmentationcrisis’ and speculate further that, given the paucity of observational evidence for discs inthe earliest (class 0) phase of star formation, perhaps they do not exist (instead forminglater). More likely the solution lies in the fact that we have assumed ideal MHD ina regime where it is clear that non-ideal MHD effects are known to be important.Indeed later analysis by Shu et al. [34] suggests that Ohmic resistivity can provide asolution, though nonetheless requiring a diffusion parameter considerably higher thanthe microscopic value. We intend to explore non-ideal MHD effects in the near future,though it requires a shift away from the Euler potentials formulation [for recent progresson this, see 28].A similarly dramatic effect of magnetic fields on binary formation was also observed,though for the case of binaries the effect depended more strongly on the magneticfield configuration, since in certain circumstances the field configuration could assistbinary formation by forming a “magnetic cushion” between two overdense, collapsingregions. It was also found that a sufficiently large perturbation would produce a binaryregardless of the magnetic field strength. Nevertheless it is clear that the presence ofeven a relatively weak magnetic field in a molecular cloud core can drastically changethe star formation picture. EFFECT OF MAGNETIC FIELDS ON CLUSTER FORMATION
We have also considered the effect of magnetic fields on larger scales, important to theformation of whole star clusters [1, 2]. Our initial study was to evaluate the influenceof magnetic fields in star cluster formation calculations similar to those performed byBate et al. [17]. The initial conditions consist of a cold ( T = K ), 50 M ⊙ , uniformdensity cloud of radius ∼ . M = . t / t f f = .
23 in Fig. 2. This isbecause the fields are not weak with respect to gas pressure, so the magnetic field isable to act as the dominant source of pressure within the cloud, producing large-scalemagnetically-supported voids (middle and bottom rows) that are completely absent frompurely hydrodynamical calculations (top row).The means by which magnetic fields are able to act as a source of pressure on large
IGURE 2.
Effect of magnetic fields and radiation on the large scale structure of star-forming 50 M ⊙ molecular cloud cores. Showing calculations with no magnetic fields (top row), a mass-to-flux ratio of 5(middle row) and 3 times the critical value (bottom row). In the regime where magnetic pressure exceedsgas pressure the magnetic fields there is a dramatic influence on the global cloud structure, with theappearance of large-scale, magnetically supported voids. The large scale evolution of the cloud withradiative transfer explicitly calculated (right panels) is identical to that using an approximate, barotropicequation of state (left panels), at least for low mass star formation. cales is relatively simple to understand, since in ideal MHD the gas motions are tied tothe magnetic field lines. For a relatively strong field, this means that gas is channelledalong field lines as it collapses (rather than the gas dragging the field lines around inthe weak field case). Since in ideal MHD the mass-to-flux ratio is conserved along anygiven flux tube, any gas collapsing to form dense structures inevitably leaves behinda region evacuated of gas pressure but with the magnetic field strength (and magneticpressure) unchanged. Thus the ratio of gas-to-magnetic pressure decreases substantiallyaway from the densest gas. New material is prevented from re-entering the evacuatedregion because of the inability to cross magnetic field lines. Thus the region, onceevacuated, remains as a magnetic-pressure supported void. At a recent meeting the abovemechanism was paraphrased by Carl Heiles as “magnetic fields abhor a vacuum”, sinceit is easy to remove gas from a region of space along the magnetic field lines, but themagnetic fields themselves will remain.The effect of the support provided to the large scale regions of the cloud is a dramaticslow-down in the star formation rate with increasing magnetic field strength (discussedbelow, see Fig. 5), most effective in the regime where magnetic pressure exceeds gaspressure ( b <
1) and independent of the fact that the field may be weak relative togravity and/or turbulence. An unexpected finding from [1] was the resultant change tothe initial mass function of stars formed in the calculations, in the form of a reductionin the number of sub-stellar objects (i.e., brown dwarfs) relative to higher mass objects(i.e., stars). This occurs not because of some complicated influence of the magneticfields on the fragmentation — we do not resolve the magnetic fields structure on thesmallest scales in these calculations — but simply because of the overall slowdownin the star formation rate and a consequent reduction in the importance of dynamicalinteractions and the associated ejection of low mass objects from multiple systems.Given the low number of objects formed overall in the strong magnetic field calculations,it is not possible to state whether or not this effect is sufficient to resolve the statisticaldisagreement in the number of low mass objects and the observed IMF found by Bate[19], but the trend is certainly in the right direction.In the strongest field calculation, we also found that the expanding outer regions ofthe cloud started to show a ‘striped’ appearance as the gas was channelled along themagnetic field lines. This is strongly reminiscent of the ‘magnetically aligned striations’observed in the outer regions of the Taurus molecular cloud in the recent CO and CO molecular line emission maps by Goldsmith et al. [11], co-located with measurablevelocity anisotropy aligned with the global magnetic field [12]. This is a good indicationthat Taurus lies in a regime where the magnetic field is able to exert considerableinfluence on the star formation process.
INFLUENCE OF RADIATIVE FEEDBACK ON STAR CLUSTERFORMATION
A key limitation to all of the calculations discussed above was the approximate treatmentof the thermodynamics of the gas via the use of a barotropic equation of state wheregas pressure is a function of density alone, rather than being a function of density andtemperature. Naturally this assumption simplifies the calculations considerably, but it
IGURE 3.
Combined effect of magnetic fields and radiative feedback on star cluster formation. Theplots show a zoomed-in subsection of the clouds shown in Fig. 2 at 1.23 initial gravitational free-falltimes for the calculations of three different magnetic field strengths (top to bottom), without (left) andwith (right) a full modelling of radiative transport in the gas. The effect on the small scale fragmentationis dramatic: Once the gas becomes optically thick to radiation the heating effect provided to neighbouringmaterial completely inhibits any subsequent fragmentation within a radius of several AU. The result is atrend towards fewer but more massive stars and a further reduction in the overall star formation rate ontop of the large-scale effect provided by the magnetic field. isses important feedback processes, especially once the gas enters the optically thickregime. In particular, using the barotropic approximation the temperature is assumed torise strictly with density, but this neglects the fact that radiation in actual fact shoulddiffuse from the hot, dense, compressed gas into the less-dense surrounds, thus heatingit and preventing it from fragmenting further.In the initial phases of the cloud evolution and on the largest scales the radiation hasvery little influence, evident from Fig. 2 which compares the cloud structures using thebarotropic equation of state (left panels) with calculations incorporating the transportof radiation within the gas via the flux-limited diffusion approximation (right panels).This is partly the case because we form only low-mass stars in the calculations, but alsobecause we have neglected both the accretion luminosity within the sink radius and theluminosity of the protostars themselves. Thus the effect of radiation that we consider isvery much a lower limit to the true feedback effect. However, to capture as much of theradiative feedback as possible we have reduced the accretion radius on the sink particlesto a mere 0 . R r T dz / R r dz shown in Fig. 4 for thehydrodynamic calculations using the barotropic equation of state (left panel) and withradiative feedback included (right panel). With the barotropic equation of state (left)the temperature is high only at several discrete points corresponding to where the gasdensity exceeds the threshold for the polytropic index to change from g = g = . IGURE 4.
Effect of radiative feedback on star cluster formation: The plot shows the distribution ofaverage temperature R r T dz / R r dz from the hydrodynamic calculations shown in the top row of Figs. 2and 3 employing a barotropic (pressure-proportional to density) equation of state (left panel) compared tothe calculation (right) where the radiation is explicitly modelled and thus the transport of radiation fromhot to cold regions is captured. Whereas using the barotropic approximation only the material above thecritical density becomes hot, in the radiation hydrodynamics calculation an extended region surroundingeach protostar is heated and thus fragments no further (see Fig. 3). Offner et al. [35] using an adaptive mesh refinement code.The resultant effect on the initial mass function strengthens the trend already pro-duced by the magnetic field, namely towards producing fewer and more massive objects.As previously stated, this is in the right direction to resolve the discrepancy with the ob-served IMF found by Bate [19] in barotropic calculations, but given the very low numberstatistics — particularly with the overall reduction in star formation rate produced by thecombined influence of the magnetic fields and radiation — we are reluctant to draw firmconclusions in this regard.
COMBINED INFLUENCE OF MAGNETIC FIELD ANDRADIATIVE FEEDBACK ON THE STAR FORMATION RATEAND EFFICIENCY
Having assessed the effect of both magnetic fields and radiative feedback on the starcluster formation, we may return to our original question: namely, are these two piecesof missing physics the necessary and sufficient ingredients required to explain the kindof slow and inefficient star formation observed in real molecular clouds?The combined effect on the star formation rate is shown in Fig. 5, showing the totalmass accreted onto the sink particles for the full suite of calculations as a functionof time. The primary influence on the star formation rate is the strength of the initial
IGURE 5.
Combined influence of magnetic fields and radiative feedback on the star formation rate inour 50 M ⊙ model clouds. Line styles correspond to the four different magnetic field strengths employed:no magnetic fields (solid, black), and mass-to-flux ratios of 10, 5 and 3 in units of the critical value forpreventing collapse altogether (dotted red, dashed blue and dot-dashed magenta lines respectively), whilstthe line width shows whether (thick lines) or not (thin lines) radiative feedback was modelled (if not, abarotropic equation of state was employed). The magnetic field strength has the dominant influence onthe star formation rate, with a secondary effect due to radiative feedback occurring at later times. magnetic field, since it affects the large scale structure of the cloud and thus the amountof material that is able to later collapse and form stars. Radiative feedback enters as asecondary effect, reducing the star formation rate further, particularly at later times asthe radiation diffuses further from the protostars into the surrounding medium.It is notable that only the calculations employing the strongest magnetic fields (mass-to-flux ratio of 3 in units of the critical value) produce a star formation rate that iseven remotely close to the observed rate of 3-6% per gravitational free-fall time foundby Evans et al. [5]: The rate in the strongest field calculation with radiative feedbackis 0 . M ⊙ / . t f f / M ⊙ ≈
10% per free-fall time. This is not unreasonable sincemolecular cloud cores are indeed observed to have mass-to-flux ratios of a few times thecritical value, and radiative feedback is clearly an important effect. Relative differencesin the star formation rate across the Galaxy can also be explained as being due tovariations in the global flux threading individual star forming clouds. We can speculatethat the remaining discrepancy between our results and observational estimates of thestar formation rate is due to our neglect of additional feedback processes, namely theintrinsic and accretion luminosity from the protostars themselves as well as mechanicaleedback from jets and outflows.The question of the overall star formation efficiency in the presence of magneticfields and radiation transport is more difficult to answer given the limited time forwhich we have been able to evolve the calculations beyond one free-fall time. Ideallyone would continue the calculations over several global dynamical times until starformation activity has ceased, however this is currently prohibitively expensive in termsof CPU time. Observational estimates are limited in a similar manner because a starforming molecular cloud is defined as one in which star formation has initiated but notcompleted, and once completed one has little insight as to the initial mass of the parentalcloud. If we assume that star formation continues at the rate observed in Fig. 5 andthat the molecular cloud survives for 2-3 free-fall times beyond star formation, thenthe overall star formation efficiency in the strongest field case would be of order 20-30%. By contrast, for the calculations without magnetic fields the efficiency wouldbe close to 100% on a similar timescale. Since at supercritical mass-to-flux ratios thefield is relatively weak compared to gravity, the fraction of bound gas at the end of thecalculation remains relatively high even for the highest field strengths, of order 85%for the mass-to-flux ratio of 3 (times critical) calculation with radiative feedback, sothe main requirement for a low overall efficiency is that the cloud should be dispersedafter several dynamical times and that the star formation rate should not accelerateconsiderably with time.
ACKNOWLEDGEMENTS
DJP is supported by a Monash Fellowship, though much of this work was completedwhilst funded by a UK Royal Society University Research Fellowship at the Universityof Exeter. We thank the organisers for their hospitality in both Milan and Como, theopportunity to attend and present at the conference.
REFERENCES
1. D. J. Price, and M. R. Bate,
MNRAS , , 1820–1834 (2008).2. D. J. Price, and M. R. Bate, MNRAS , , 33–46 (2009).3. D. J. Price, and M. R. Bate, MNRAS , , 77–90 (2007).4. P. Hennebelle, and R. Teyssier, A&A , , 25–34 (2008).5. N. J. Evans, M. M. Dunham, J. K. Jørgensen, M. L. Enoch, B. Merin, E. F. van Dishoeck, J. M.Alcalá, P. C. Myers, K. R. Stapelfeldt, T. L. Huard, L. E. Allen, P. M. Harvey, T. van Kempen, G. A.Blake, D. W. Koerner, L. G. Mundy, D. L. Padgett, and A. I. Sargent, ApJS , , 321–350 (2009).6. J. Forbrich, C. J. Lada, A. A. Muench, J. Alves, and M. Lombardi, ApJ , , 292–305 (2009).7. F. Nakamura, and Z.-Y. Li, ApJ , , 395–412 (2007).8. J. Ballesteros-Paredes, G. C. Gómez, L. Loinard, R. M. Torres, and B. Pichardo, MNRAS , , L81–L84 (2009).9. F. O. Alves, G. A. P. Franco, and J. M. Girart, A&A , , L13–L16 (2008).10. H. Li, C. D. Dowell, A. Goodman, R. Hildebrand, and G. Novak, ApJ , , 891–897 (2009).11. P. F. Goldsmith, M. Heyer, G. Narayanan, R. Snell, D. Li, and C. Brunt, ApJ , , 428–445 (2008).12. M. Heyer, H. Gong, E. Ostriker, and C. Brunt, ApJ , , 420–427 (2008).13. F. H. Shu, F. C. Adams, and S. Lizano, Ann. Rev. Astron. Astroph. , , 23–81 (1987).14. R. M. Crutcher, ApJ , , 706–713 (1999).15. E. C. Ostriker, J. M. Stone, and C. F. Gammie, ApJ , , 980–1005 (2001).6. B. G. Elmegreen, ApJ , , 277–281 (2000).17. M. R. Bate, I. A. Bonnell, and V. Bromm, MNRAS , , 577–599 (2003).18. M. R. Bate, and I. A. Bonnell, MNRAS , , 1201–1221 (2005).19. M. R. Bate, MNRAS , , 590–616 (2009).20. T. L. Bourke, P. C. Myers, G. Robinson, and A. R. Hyland, ApJ , , 916–932 (2001).21. C. Heiles, and T. H. Troland, ApJS , , 271–297 (2004).22. D. J. Price, Magnetic fields in Astrophysics , Ph.D. thesis, University of Cambridge, Cambridge, UK.astro-ph/0507472 (2004).23. J. J. Monaghan,
Rep. Prog. Phys. , , 1703–1759 (2005).24. G. J. Phillips, and J. J. Monaghan, MNRAS , , 883–895 (1985).25. D. J. Price, and J. J. Monaghan, MNRAS , , 384–406 (2005).26. D. J. Price, and J. J. Monaghan, MNRAS , , 123–138 (2004).27. D. J. Price, and J. J. Monaghan, MNRAS , , 139–152 (2004).28. D. J. Price, MNRAS , , 1475–1499 (2010).29. A. Brandenburg, MNRAS , , 347–354 (2010).30. S. C. Whitehouse, and M. R. Bate, MNRAS , , 1078–1094 (2004).31. S. C. Whitehouse, M. R. Bate, and J. J. Monaghan, MNRAS , , 1367–1377 (2005).32. P. Hennebelle, and A. Ciardi, A&A , , L29–L32 (2009).33. D. Galli, S. Lizano, F. H. Shu, and A. Allen, ApJ , , 374–381 (2006).34. F. H. Shu, D. Galli, S. Lizano, and M. Cai, ApJ , , 382–389 (2006).35. S. S. R. Offner, R. I. Klein, C. F. McKee, and M. R. Krumholz, ApJ ,703