Self‐similar Collapse of Rotating Magnetic Molecular Cloud Cores

Abstract

We present self-similar solutions that describe the gravitational collapse of rotating, isothermal, magnetic molecular-cloud cores. These solutions make it possible, for the first time, to study the formation of rotationally supported protostellar disks of the type detected around many young stellar objects in the context of a realistic scenario of star formation in magnetically supported, weakly ionized, molecular cloud cores. This work focuses on the evolution after a point mass first forms at the center and generalizes previous results by Contopoulos, Ciolek, & Königl that did not include rotation. Our semianalytic scheme incorporates ambipolar diffusion and magnetic braking and allows us to examine the full range of expected behaviors and their dependence on the physical parameters. We find that, for typical parameter values, the inflow first passes through an ambipolar-diffusion shock (at a radius r a ), where the magnetic flux decouples from the matter, and subsequently through a centrifugal shock (at r c ), inward of which a rotationally supported disk (of mass M d ) is established. By the time (∼ 10 5 yr) that the central mass M c grows to ∼ 1 M ⊙ , r a 10 3 AU, r c 10 2 AU, and M d /M c 0.1. The derived disk properties are consistent with data on T Tauri systems, and our results imply that protostellar disks may well be Keplerian also during earlier phases of their evolution. We demonstrate that the disk is likely to drive centrifugal outflows that transport angular momentum and mass, and we show how the radially self-similar wind solution of Blandford & Payne can be naturally incorporated into the disk model. We further verify that gravitational torques and magnetorotational instability-induced turbulence typically do not play an important role in the angular momentum transport. For completeness, we also present solutions for the limiting cases of fast rotation (where the collapse results in a massive disk with such a large outer radius that it traps the ambipolar-diffusion front) and strong braking (where no disk is formed and the collapse resembles that of a nonrotating core at small radii), as well as solutions describing the rotational collapse of ideal-MHD and of nonmagnetic model cores.

The Earliest Stages of Star and Planet Formation: Core Collapse, and the Formation of Disks and Outflows

“…Krasnopolsky & Königl (2002) showed semi-analytically, using the 1D thin-disk approximation, that disk formation may be suppressed in the strongly magnetized post-shock region if the magnetic braking is efficient enough. The braking efficiency, parametrized in Krasnopolsky & Königl (2002), was computed self-consistently in the 2D (axisymmetric) simulations of Mellon & Li (2009), which were performed under the usual assumption of ion density proportional to the square root of neutral density. 3D simulations of AD were performed by Duffin & Pudritz (2009) using a specially developed, single fluid AMR code (Duffin & Pudritz 2008) as well as by a two fluid SPH code (Hosking & Whitworth 2004).…”

Star Formation and the Hall Effect

“…It is the type of diffusion most commonly included in simulations of star formation that go beyond ideal MHD (e.g. Krasnopolsky and Königl, 2002;Adams and Shu, 2007;Mellon and Li, 2009, see also Section 1.3). In both the Ohmic and ambipolar diffusion limits, when the magnetic field is globally reversed the magnetic response of the disc is unchanged, as demonstrated in Figure 1.1.…”

“…This behaviour is an oversimplification, as Ciolek and Mouschovias (1998) showed that for typical cloud and grain parameters the proportionality of the ion density cannot be parameterised by a single power law exponent k, as k > 1/2 for densities n H 10 5 cm −3 and k 1/2 for densities n H 10 5 cm −3 , but it is still a reasonable and widely-adopted approximation to the ion density in collapsing cores on scales 10 3 AU (see e.g. Shu et al, 1987;Galli and Shu, 1993a;Contopoulos et al, 1998;Krasnopolsky and Königl, 2002). This shall be discussed further in Chapter 2.…”

“…This MHD shock (referred to as the magnetic diffusion shock in the results presented in this work) takes the form of a continuous transition in the calculations of Desch and Mouschovias (2001) and Krasnopolsky and Königl (2002) because the shock structure depends upon ambipolar diffusion, which is explicitly determined in their equations.…”

Hubble Space Telescope scale 3D simulations of MHD disc winds: a rotating two-component jet structure

“…While this mechanism was originally developed in the context of AGN jets, its applicability to ubiquitous protostellar jets has been substantially developed for over 3 decades through detailed theoretical models (e.g. Pelletier & Pudritz 1992;Ferreira 1997; Krasnopolsky & Königl 2002), advanced sim-⋆ E-mail: jan.staff@mq.edu.au ulations in 2 dimensions (e.g. Ustyugova et al 1995;Ouyed et al 1997;Kato et al 2002;Ramsey & Clarke 2011) and 3 dimensions (e.g.…”