Protostar Formation in Magnetic Molecular Clouds beyond Ion Detachment: I. Formulation of the Problem and Method of Solution

We formulate the problem of the formation of magnetically supercritical cores in magnetically subcritical parent molecular clouds, and the subsequent collapse of the cores to high densities, past the detachment of ions from magnetic field lines and into the opaque regime. We employ the six-fluid MHD equations, accounting for the effects of grains (negative, positive and neutral) including their inelastic collisions with other species. We do not assume that the magnetic flux is frozen in any of the charged species. We derive a generalized Ohm's law that explicitly distinguishes between flux advection (and the associated process of ambipolar diffusion) and Ohmic dissipation, in order to assess the contribution of each mechanism to the increase of the mass-to-flux ratio of the central parts of a collapsing core and possibly to the resolution of the magnetic flux problem of star formation. We show how our formulation is related to and can be transformed into the traditional, directional formulation of the generalized Ohm's law, and we derive formulae for the perpendicular, parallel and Hall conductivities entering the latter, which include, for the first time, the effect of inelastic collisions between grains. In addition, we present a general (valid in any geometry) solution for the velocities of charged species as functions of the velocity of the neutrals and of the effective flux velocity (which can in turn be calculated from the dynamics of the system and Faraday's law). The last two sets of formulae can be adapted for use in any general non-ideal MHD code to study phenomena beyond star formation in magnetic clouds. The results, including a detailed parameter study, are presented in two accompanying papers.Comment: 17 pages, emulateapj; accepted for publication in the Astrophysical Journa

Mechanism of Magnetic Flux Loss in Molecular Clouds

We investigate the detailed processes working in the drift of magnetic fields in molecular clouds. To the frictional force, whereby the magnetic force is transmitted to neutral molecules, ions contribute more than half only at cloud densities $n_{\rm H} < 10^4 {\rm cm}^{-3}$, and charged grains contribute more than 90% at $n_{\rm H} > 10^6 {\rm cm}^{-3}$. Thus grains play a decisive role in the process of magnetic flux loss. Approximating the flux loss time $t_B$ by a power law $t_B \propto B^{-\gamma}$, where $B$ is the mean field strength in the cloud, we find $\gamma \approx 2$, characteristic to ambipolar diffusion, only at $n_{\rm H} < 10^7 {\rm cm}^{-3}$. At higher densities, $\gamma$ decreases steeply with $n_{\rm H}$, and finally at $n_{\rm H} \approx n_{\rm dec} \approx {\rm a few} \times 10^{11} {\rm cm}^{-3}$, where magnetic fields effectively decouple from the gas, $\gamma << 1$ is attained, reminiscent of Ohmic dissipation, though flux loss occurs about 10 times faster than by Ohmic dissipation. Ohmic dissipation is dominant only at $n_{\rm H} > 1 \times 10^{12} {\rm cm}^{-3}$. While ions and electrons drift in the direction of magnetic force at all densities, grains of opposite charges drift in opposite directions at high densities, where grains are major contributors to the frictional force. Although magnetic flux loss occurs significantly faster than by Ohmic dissipation even at very high densities as $n_{\rm H} \approx n_{\rm dec}$, the process going on at high densities is quite different from ambipolar diffusion in which particles of opposite charges are supposed to drift as one unit.Comment: 34 pages including 9 postscript figures, LaTex, accepted by Astrophysical Journal (vol.573, No.1, July 1, 2002

The Effect of the Hall Term on the Nonlinear Evolution of the Magnetorotational Instability: I. Local Axisymmetric Simulations

The effect of the Hall term on the evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks is investigated using local axisymmetric simulations. First, we show that the Hall term has important effects on the MRI when the temperature and density in the disk is below a few thousand K and between 10^13 and 10^18 cm^{-3} respectively. Such conditions can occur in the quiescent phase of dwarf nova disks, or in the inner part (inside 10 - 100 AU) of protoplanetary disks. When the Hall term is important, the properties of the MRI are dependent on the direction of the magnetic field with respect to the angular velocity vector \Omega. If the disk is threaded by a uniform vertical field oriented in the same sense as \Omega, the axisymmetric evolution of the MRI is an exponentially growing two-channel flow without saturation. When the field is oppositely directed to \Omega, however, small scale fluctuations prevent the nonlinear growth of the channel flow and the MRI evolves into MHD turbulence. These results are anticipated from the characteristics of the linear dispersion relation. In axisymmetry on a field with zero-net flux, the evolution of the MRI is independent of the size of the Hall term relative to the inductive term. The evolution in this case is determined mostly by the effect of ohmic dissipation.Comment: 31 pages, 3 tables, 12 figures, accepted for publication in ApJ, postscript version also available from http://www.astro.umd.edu/~sano/publications

Magnetorotational Instability in Protoplanetary Disks. II. Ionization State and Unstable Regions

We investigate where in protoplanetary disks magnetorotational instability operates, which can cause angular momentum transport in the disks. We investigate the spatial distribution of various charged particles and the unstable regions for a variety of models for protoplanetary disks taking into account the recombination of ions and electrons at grain surfaces, which is an important process in most parts of the disks. We find that for all the models there is an inner region which is magnetorotationally stable due to ohmic dissipation. This must make the accretion onto the central star non-steady. For the model of the minimum-mass solar nebula, the critical radius, inside of which the disk is stable, is about 20 AU, and the mass accretion rate just outside the critical radius is 10^{-7} - 10^{-6} M_{\odot} yr^{-1}. The stable region is smaller in a disk of lower column density. Dust grains in protoplanetary disks may grow by mutual sticking and may sediment toward the midplane of the disks. We find that the stable region shrinks as the grain size increases or the sedimentation proceeds. Therefore in the late evolutionary stages, protoplanetary disks can be magnetorotationally unstable even in the inner regions.Comment: 23 pages + 16 figures + 3 tables, accepted for publication in Ap

Evolution of Molecular Abundance in Protoplanetary Disks

We investigate the evolution of molecular abundance in quiescent protoplanetary disks which are presumed to be around weak-line T Tauri stars. In the region of surface density less than $10^2$ g cm$^{-2}$ (distance from the star $\gtrsim 10$ AU in the minimum- mass solar nebula), cosmic rays are barely attenuated even in the midplane of the disk and produce chemically active ions such as He$^+$ and H$_{3}^+$. Through reactions with these ions CO and N$_2$ are finally transformed into CO$_2$, NH$_3$, and HCN. In the region where the temperature is low enough for these products to freeze onto grains, considerable amount of carbon and nitrogen is locked up in the ice mantle and is depleted from the gas phase in a time scale $\lesssim 3\times 10^6$ yr. Oxidized (CO$_2$) ice and reduced (NH$_3$ and hydrocarbon) ice naturally coexist in this part of the disk. The molecular abundance both in the gas phase and in ice mantle varies significantly with the distance from the central star.Comment: 7 pages latex file (using aas2pp4.sty), 3 figures (ps file), to appear in the Astrophysical Journal Letter

Dead Zones and the Origin of Planetary Masses

Protoplanets accrete material from their natal protostellar disks until they are sufficiently massive to open a gap in the face of the disk's viscosity that arises from the magneto-rotational instability (MRI). By computing the ionization structure within observationally well-constrained disk models, we demonstrate that poorly ionized, low viscosity "dead zones" stretch out to 12 AU within typical disks. We find that planets of terrestrial mass robustly form within the dead zones while massive Jovian planets form beyond. Dead zones will also halt the rapid migration of planets into their central stars. Finally, we argue that the gravitational scattering of low mass planets formed in the dead zone, to larger radii by a rapidly accreting Jupiter beyond, can explain the distribution of planetary masses in our solar system.Comment: 10 pages, 4 figures, accepted for publication in ApJ

TMC-1C: an accreting starless core

We have mapped the starless core TMC-1C in a variety of molecular lines with the IRAM 30m telescope. High density tracers show clear signs of self-absorption and sub-sonic infall asymmetries are present in N2H+ (1-0) and DCO+ (2-1) lines. The inward velocity profile in N2H+ (1-0) is extended over a region of about 7,000 AU in radius around the dust continuum peak, which is the most extended ``infalling'' region observed in a starless core with this tracer. The kinetic temperature (~12 K) measured from C17O and C18O suggests that their emission comes from a shell outside the colder interior traced by the mm continuum dust. The C18O (2-1) excitation temperature drops from 12 K to ~10 K away from the center. This is consistent with a volume density drop of the gas traced by the C18O lines, from ~4x10^4 cm^-3 towards the dust peak to ~6x10^3 cm^-3 at a projected distance from the dust peak of 80" (or 11,000 AU). The column density implied by the gas and dust show similar N2H+ and CO depletion factors (f_D < 6). This can be explained with a simple scenario in which: (i) the TMC-1C core is embedded in a relatively dense environment (H2 ~10^4 cm^-3), where CO is mostly in the gas phase and the N2H+ abundance had time to reach equilibrium values; (ii) the surrounding material (rich in CO and N2H+) is accreting onto the dense core nucleus; (iii) TMC-1C is older than 3x10^5 yr, to account for the observed abundance of N2H+ across the core (~10^-10 w.r.t. H2); and (iv) the core nucleus is either much younger (~10^4 yr) or ``undepleted'' material from the surrounding envelope has fallen towards it in the past 10,000 yr.Comment: 29 pages, including 5 tables and 15 figure

A Spherical Model for "Starless" Cores of Magnetic Molecular Clouds and Dynamical Effects of Dust Grains

In the standard picture of isolated star formation, dense ``starless'' cores are formed out of magnetic molecular clouds due to ambipolar diffusion. Under the simplest spherical geometry, I demonstrate that ``starless'' cores formed this way naturally exhibit a large scale inward motion, whose size and speed are comparable to those detected recently by Taffala et al. and Williams et al. in ``starless'' core L1544. My model clouds have a relatively low mass (of order 10 $M_\odot$) and low field strength (of order 10 $\mu$G) to begin with. They evolve into a density profile with a central plateau surrounded by a power-law envelope, as found previously. The density in the envelope decreases with radius more steeply than those found by Mouschovias and collaborators for the more strongly magnetized, disk-like clouds. At high enough densities, dust grains become dynamically important by greatly enhancing the coupling between magnetic field and the neutral cloud matter. The trapping of magnetic flux associated with the enhanced coupling leads, in the spherical geometry, to a rapid assemblage of mass by the central protostar, which exacerbates the so-called ``luminosity problem'' in star formation.Comment: 27 pages, 4 figures, accepted by Ap

Atomic Diagnostics of X-ray Irradiated Protoplanetary Disks

We study atomic line diagnostics of the inner regions of protoplanetary disks with our model of X-ray irradiated disk atmospheres which was previously used to predict observable levels of the NeII and NeIII fine-structure transitions at 12.81 and 15.55mum. We extend the X-ray ionization theory to sulfur and calculate the fraction of sulfur in S, S+, S2+ and sulfur molecules. For the D'Alessio generic T Tauri star disk, we find that the SI fine-structure line at 25.55mum is below the detection level of the Spitzer Infrared Spectrometer (IRS), in large part due to X-ray ionization of atomic S at the top of the atmosphere and to its incorporation into molecules close to the mid-plane. We predict that observable fluxes of the SII 6718/6732AA forbidden transitions are produced in the upper atmosphere at somewhat shallower depths and smaller radii than the neon fine-structure lines. This and other forbidden line transitions, such as the OI 6300/6363AA and the CI 9826/9852AA lines, serve as complementary diagnostics of X-ray irradiated disk atmospheres. We have also analyzed the potential role of the low-excitation fine-structure lines of CI, CII, and OI, which should be observable by SOFIA and Herschel.Comment: Accepted by Ap

The Origin of Jovian Planets in Protostellar Disks: The Role of Dead Zones

The final masses of Jovian planets are attained when the tidal torques that they exert on their surrounding protostellar disks are sufficient to open gaps in the face of disk viscosity, thereby shutting off any further accretion. In sufficiently well-ionized disks, the predominant form of disk viscosity originates from the Magneto-Rotational Instability (MRI) that drives hydromagnetic disk turbulence. In the region of sufficiently low ionization rate -- the so-called dead zone -- turbulence is damped and we show that lower mass planets will be formed. We considered three ionization sources (X-rays, cosmic rays, and radioactive elements) and determined the size of a dead zone for the total ionization rate by using a radiative, hydrostatic equilibrium disk model developed by Chiang et al. (2001). We studied a range of surface mass density (Sigma_{0}=10^3 - 10^5 g cm^{-2}) and X-ray energy (kT_{x}=1 - 10 keV). We also compared the ionization rate of such a disk by X-rays with cosmic rays and find that the latter dominate X-rays in ionizing protostellar disks unless the X-ray energy is very high (5 - 10 keV). Among our major conclusions are that for typical conditions, dead zones encompass a region extending out to several AU -- the region in which terrestrial planets are found in our solar system. Our results suggest that the division between low and high mass planets in exosolar planetary systems is a consequence of the presence of a dead zone in their natal protoplanetary disks. We also find that the extent of a dead zone is mainly dependent on the disk's surface mass density. Our results provide further support for the idea that Jovian planets in exosolar systems must have migrated substantially inwards from their points of origin.Comment: 28 pages, 10 figures, accepted by Ap