Small-scale dynamo action during the formation of the first stars and galaxies

I. The ideal MHD limit

¹ ESO, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany
e-mail: dschleic@eso.org
² Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
³ Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
⁴ Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
⁵ Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo Park, CA 94025, USA
⁶ Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands

Received: 9 June 2010
Accepted: 23 August 2010

Abstract

We explore the amplification of magnetic seeds during the formation of the first stars and galaxies. During gravitational collapse, turbulence is created from accretion shocks, which may act to amplify weak magnetic fields in the protostellar cloud. Numerical simulations showed that such turbulence is sub-sonic in the first star-forming minihalos, and highly supersonic in the first galaxies with virial temperatures larger than 10⁴ K. We investigate the magnetic field amplification during the collapse both for Kolmogorov and Burgers-type turbulence with a semi-analytic model that incorporates the effects of gravitational compression and small-scale dynamo amplification. We find that the magnetic field may be substantially amplified before the formation of a disk. On scales of 1 / 10 of the Jeans length, saturation occurs after  ~ 10⁸ yr. Although the saturation behaviour of the small-scale dynamo is still somewhat uncertain, we expect a saturation field strength of the order  ~ 10^-7n^0.5 G in the first star-forming halos, with n the number density in cgs units. In the first galaxies with higher turbulent velocities, the magnetic field strength may be increased by an order of magnitude, and saturation may occur after 10⁶ − 10⁷ yr. In the Kolmogorov case, the magnetic field strength on the integral scale (i.e. the scale with most magnetic power) is higher due to the characteristic power-law indices, but the difference is less than a factor of 2 in the saturated phase. Our results thus indicate that the precise scaling of the turbulent velocity with length scale is of minor importance. They further imply that magnetic fields will be significantly enhanced before the formation of a protostellar disk, where they may change the fragmentation properties of the gas and the accretion rate.