The Star Formation Rate of Turbulent Magnetized Clouds: Comparing Theory, Simulations, and Observations

Federrath, C. & Klessen, R. S., 2012

The Astrophysical Journal, 761, 156  [ ADS link ]  [ PDF ]


The role of turbulence and magnetic fields is studied for star formation in molecular clouds. We derive and compare six theoretical models for the star formation rate (SFR) — the Krumholz & McKee (KM), Padoan & Nordlund (PN), and Hennebelle & Chabrier (HC) models, and three multi-freefall versions of these, suggested by HC — all based on integrals over the log-normal distribution of turbulent gas. We extend all theories to include magnetic fields, and show that the SFR depends on four basic parameters: (1) viriaL parameter αvir; (2) sonic Mach number M; (3) turbulent forcing parameter b, which is a measure for the fraction of energy driven in compressive modes; and (4) plasma β=2MA2/M2 with the Alfvén Mach number MA. We compare all six theories with MHD simulations, covering cloud masses of 300 to 4x106Msol and Mach numbers M=3–50 and MA=1–infinity, with solenoidal (b=1/3), mixed (b=0.4) and compressive turbulent (b=1) forcings. We find that the SFR increases by a factor of four between M=5 and 50 for compressive turbulent forcing and αvir≈1. Comparing forcing parameters, we see that the SFR is more than 10x higher with compressive than solenoidal forcing for M=10 simulations. The SFR and fragmentation are both reduced by a factor of two in strongly magnetized, trans-Alfvénic turbulence compared to hydrodynamic turbulence. All simulations are fit simultaneously by the multi-freefall KM and multi-freefall PN theories within a factor of two over two orders of magnitude in SFR. The simulated SFRs cover the range and correlation of SFR column density with gas column density observed in Galactic clouds, and agree well for star formation efficiencies SFE=1–10% and local efficiencies ε=0.3–0.7 due to feedback. We conclude that the SFR is primarily controlled by interstellar turbulence, with a secondary effect coming from magnetic fields.

Solenoidal versus compressive turbulent forcing

These two movies show the gas column density of the simulations with sonic Mach number 10 and purely solenoidal versus purely compressive forcing of the turbulence. Star formation (modelled with sink particles and shown as small black-to-blue circles) is very slow in the solenoidal forcing run. In contrast, star formation is more than 10 times faster with purely compressive forcing, emphasizing the importance of the turbulent forcing mechanism for star formation.


Mach 10 sol, 56MB high-res mp4 ] [ Mach 10 comp, 13MB high-res mp4 ]

Effects of the magnetic field

These movies show the same as the ones above, but with mixed forcing. The first simulation has no magnetic field, while the second one includes a typical magnetic field of 3 micro Gauss. Star formation takes slightly longer in the magnetic-field run, and fragmentation is reduced (fewer sink particles form). The last movie shows the same as the second one, but with magnetic field vectors superimposed. The magnetic field is strongly amplified in regions of core and star cluster formation, and changes direction frequently due to turbulence.


Mach 10 mix B0, 30MB high-res mp4 ] [ Mach 10 mix B3, 34MB high-res mp4 ] [ Mach 10 mix B3vec, 64MB high-res mp4 ]

See also the related Paper II on the Star Formation Efficiency

See also the related Paper III on Physical Variations in the Star Formation Law


We thank Amanda Heiderman for sending us the observed SFR column densities measured in Galactic clouds shown in Figures 11 and 12, and we thank Patrick Hennebelle, Mark Krumholz, and Paolo Padoan for enlightening discussions and detailed comments on the manuscript. We also thank Chris McKee for a timely, detailed, and constructive referee report, which significantly improved this study. Stimulating discussions with Ben Ayliffe, Christian Baczynski, Robi Banerjee, Chris Brunt, Blakesley Burkhart, Michael Burton, Gilles Chabrier, Paul Clark, David Collins, Benoit Commercon, Timea Csengeri, Maria Cunningham, Bruce Elmegreen, Philipp Girichidis, Karl Glazebrook, Simon Glover, Nathan Goldbaum, Alex Hill, Alexandre Lazarian, Lukas Konstandin, Guillaume Laibe, Mordecai-Mark Mac Low, Faviola Molina, Joe Monaghan, Volker Ossenkopf, Daniel Price, Ralph Pudritz, Chalence Safranek-Shrader, Dominik Schleicher, Wolfram Schmidt, Nicola Schneider-Bontemps, Jennifer Schober, Martin Schrön, Rahul Shetty, Rowan Smith, Enrique Vazquez-Semadeni, and Mark Wardle during the preparation of this study are gratefully acknowledged. C.F. thanks for funding provided by the Australian Research Council under the Discovery Projects scheme (grant DP110102191). R.S.K. acknowledges subsidies from the Baden-Württemberg-Stiftung by contract research Internationale Spitzenforschung (grant P-LS-SPII/18). This work was supported by the Deutsche Forschungsgemeinschaft, priority program 1573 ("Physics of the Interstellar Medium") and collaborative research project SFB 881 ("The Milky Way system") in sub-projects B1, B2, and B5. Supercomputing time at the Leibniz Rechenzentrum (project pr32lo) and at the Forschungszentrum Jülich (project hhd20) are gratefully acknowledged. The software used in this work was in part developed by the DOE-supported ASC/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago.

© C. Federrath 2021