We perform a linear analysis of the stability of isothermal, rotating, magnetic, self-gravitating sheets that are weakly ionized. The magnetic field and rotation axis are perpendicular to the sheet. We include a self-consistent treatment of thermal pressure, gravitational, rotational, and magnetic (pressure and tension) forces together with two nonideal magnetohydrodynamic (MHD) effects (ohmic dissipation and ambipolar diffusion) that are treated together for their influence on the properties of gravitational instability for a rotating sheetlike cloud or disk. Our results show that there is always a preferred length scale and associated minimum timescale for gravitational instability. We investigate their dependence on important dimensionless free parameters of the problem: the initial normalized mass-to-flux ratio μ 0, the rotational Toomre parameter Q, the dimensionless ohmic diffusivity η ˜ OD , 0 , and the dimensionless neutral–ion collision time τ ˜ ni , 0 , which is a measure of the ambipolar diffusivity. One consequence of η ˜ OD , 0 is that there is a maximum preferred length scale of instability that occurs in the transcritical (μ 0 ≳ 1) regime, qualitatively similar to the effect of τ ˜ ni , 0 , but with quantitative differences. The addition of rotation leads to a generalized Toomre criterion (that includes a magnetic dependence) and modified length scales and timescales for collapse. When nonideal MHD effects are also included, the Toomre criterion reverts back to the hydrodynamic value. We apply our results to protostellar disk properties in the early embedded phase and find that the preferred scale of instability can significantly exceed the thermal (Jeans) scale and the peak preferred fragmentation mass is likely to be ∼10–90 M Jup.
Ambipolar diffusion likely plays a pivotal role in the formation and evolution of dense cores in weakly ionized molecular clouds. Linear analyses show that the evolutionary times and fragmentation scales are significantly greater than the hydrodynamic (Jeans) values even for clouds with mildly supercritical mass-to-flux ratios. We use values of fragmentation scales and growth times that correspond to typical ionization fractions within a molecular cloud, and apply these in the context of the observed estimated lifetime of prestellar cores and the observed number of such embedded cores forming in a parent clump. By varying a single parameter – the mass-to-flux ratio – over the range of observationally measured densities, we fit the range of estimated prestellar core lifetimes (∼0.1 to a few Myr) identified with Herschel as well as the number of embedded cores formed in a parent clump measured in Perseus with the Submillimeter Array. Our model suggests that the prestellar cores are formed with a transcritical mass-to-flux ratio and higher densities correspond to somewhat higher mass-to-flux ratios, but the normalized mass-to-flux ratio μ remains in the range 1 ≲ μ ≲ 2. Our best-fit model exhibits B ∝ n0.43 for prestellar cores because of the partial flux-freezing caused by ambipolar diffusion.
We develop a semi-analytic formalism for the determination of the evolution of the stellar mass accretion rate for specified density and velocity profiles that emerge from the runaway collapse of a prestellar cloud core. In the early phase, when the infall of matter from the surrounding envelope is substantial, the star accumulates mass primarily because of envelope-induced gravitational instability in a protostellar disc. In this phase, we model the envelope mass accretion rate from the isothermal free-fall collapse of a molecular cloud core. The disc gains mass from the envelope, and transports matter to the star via a disc accretion mechanism that includes episodic gravitational instability and mass accretion bursts according to the Toomre Q-criterion. In a later phase, mass is accreted on to the star due to gravitational torques within the spiral structures in the disc, in a manner that analytic theory suggests has a mass accretion rate ∝t−6/5. Our model provides a self-consistent evolution of the mass accretion rate by joining the spherical envelope accretion (dominant at the earlier stage) with the disc accretion (important at the later stage), and accounts for the presence of episodic accretion bursts at appropriate times. We show using a simple example that the burst mode can provide a good match to the observed distribution of bolometric luminosities. Our framework reproduces key elements of detailed numerical simulations of disc accretion and can aid in developing intuition about the basic physics as well as to compare theory with observations.
Dissimilar flows can be compared by exploiting the fact that all flux densities divided by their conjugate volume densities form velocity fields, which have been described as generalized winds. These winds are an extension of the classical notion of wind in fluids which puts these distinct processes on a common footing, leading to thermodynamical implications. This paper extends this notion from fluids to radiative transfer in the context of a classical two-stream atmosphere, leading to such velocities for radiative energy and entropy. These are shown in this paper to exhibit properties for radiation previously only thought of in terms of fluids, such as the matching of velocity fields where entropy production stops.
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