THE ROLE OF MHD WAVES AND AMBIPOLAR DIFFUSION IN THE FORMATION OF INTERSTELLAR CLOUD CORES AND PROTOSTARS BY CHESTER ENG B.S., Columbia University, 19

Transcription

1 cfl Copyright by Chester Eng, 2002

2 THE ROLE OF MHD WAVES AND AMBIPOLAR DIFFUSION IN THE FORMATION OF INTERSTELLAR CLOUD CORES AND PROTOSTARS BY CHESTER ENG B.S., Columbia University, 1992 M.S., University of Illinois, 1994 THESIS Submitted in partial fulllment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2002 Urbana, Illinois

3 Abstract How stars form out of their parent molecular clouds remains an unsolved fundamental problem of theoretical astrophysics. Magnetic elds are the dominant means of support against self-gravity and, therefore, important in regulating the rate at which stars form. We formulate the problem of the self-initiated formation and contraction of cloud cores due to ambipolar diffusion in isothermal, magnetic molecular clouds in the presence of MHD waves. The model clouds are initially in exact equilibrium states, with magnetic, thermal-pressure, and wave-pressure forces balancing self-gravity. An energy equation for MHD waves in a partially ionized medium is derived and solved numerically together with the two-fluid MHD equations appropriate for oblate (disklike) clouds about the mean magnetic eld. The evolution of the model clouds is initiated by the onset of ambipolar diffusion (the relative drift between neutral and charged particles), which is an unavoidable process in a self-gravitating, partially ionized, magnetic cloud. Redistribution of mass in the central flux tubes of a cloud leads to the relatively slow formation of a magnetically (and thermally) supercritical core, which then begins to contract dynamically while the cloud's envelope remains magnetically supported. We follow theevolution numerically up to central densities of about cm 3. The MHD waves do not affect the evolution in a signicant way, but they are themselves affected by the evolution. We nd that the physical processes that affect MHD waves in model clouds are damping by ambipolar diffusion, advection, escape through the cloud surface, energy input from the external medium, and compressive work done by the cloud's contraction. (Shocks are not important for the MHD waves accounted for in this investigation.) One or more of these processes become important at different stages of the evolution. The effect of the wave spectrum on the evolution and vice versa are investigated, as are other free parameters that enter the problem because of the presence of the MHD waves. (The dependence of the solution on the free parameters that appear in the two-fluid, four-fluid, and ve-fluid MHD equations in the absence of waves has been previously studied by Mouschovias and coworkers.) iii

4 acknowledgment Iwould like to thank Professor Telemachos Mouschovias for his dedication, guidance, and support during my stayattheuniversity of Illinois. This thesis could not have been completed without his encouragement, constructive criticism and keen physical insight whichcontinue to inspire me. I also thank the many people that I have hade the priviledge of calling a friend: Steven Desch, Glenn Ciolek, Kostas Tassis, Fabiano Oyafuso, Alan Wong, Chi-Yang, Terry Lee, the gang at the Forum and especially my buddies How-How" and Chia-Ning Liu. And to my parents and the rest of my family, whom have beenwaiting a long time for this day, Igive them a heartfelt thank you for believing in me. I am also thankful for support from a NASA GSRP Fellwship and a GAANN Fellowship. iv

5 Table of Contents 1 Introduction Properties of Molecular Clouds Distribution of Molecular Gas Magnetic Field Degree of Ionization Turbulence Mass-to-Flux Ratio Role of Magnetic Fields and Waves or Turbulence Critical Mass to Flux Ratio and Cloud Stability Role of Ambipolar Diffusion Properties of Alfvén Waves in a Partially Ionized Gas Source of the Waves Wave Reflection at the Cloud Surface Formulation Physics of the Basic Two Fluid Equations Ampere's Law and Charge Neutrality Newton's Second Law for the Ion Fluid Gravitational Field Faraday's Law of Induction Newton's Second Law for the Neutral fluid Collisional Drag Force Summary Effects of MHD Waves Cloud Model The Long Wavelength Cutoff Focusing of Alfvén Waves by Nonuniform Magnetic Field Transmission Coefcient Wavelength Limits of the Source Spectrum Incompressibility Assumption of Alfvén Waves Thin-Disk Approximation Thin-Disk Equations Gravitational Field Boundary Conditions Initial Conditions Reference State Initial Equilibrium State Dimensionless Problem and Free Parameters Basic Equations v

6 3.8 Free Parameters Results and Discussion Understanding the Evolution in Terms of Timescales Dependence of the Central Magnetic Field and Wave Pressure on Central Density Values of the Free Parameters Observational and Theoretical Constraints, Typical Values, and Scaling Laws Evolution of Model 1 - No Waves Evolution of Central Values Spatial Proles Evolution of Model 2 - The Effects of Waves Evolution of Central Values Timescales of Wave Processes Spatial Proles Parameter Study Effect of the Source Strength Effect of μ d;c Effect of the Wave Spectrum Effect of the Upper Wavelength Limit Summary and Conclusions A Alfvén Waves In A Partially Ionized Medium A.1 Eigenvalue and Eigenvector A.2 Calculation of the Relative Phase Difference B Interaction of the Waves With the Zero-Order Fields C Equation for the Evolution of the Wave Energy Density C.1 Assumptions C.2 The MHD Equations C.3 Equation for the Average Kinetic Energy Density C.4 Equation for the Average Magnetic Energy Density C.5 The Total Wave Energy Equation C.6 The Wave Energy Equation in Wavenumber Space C.7 The Wave Energy Equation in the Thin-Disk Approximation D Numerical Method of the Solution References Vita vi

7 List of Tables 4.1 Dimensionless Free Parameters for the Model Clouds vii

8 List of Figures 4.1 Central Neutral Density vstimeformodel Central Mass-to-Flux Ratio vs Central Neutral Density for Model Central Magnetic Field vs Central Neutral Density for Model Central Timescales vs Central Neutral Density for Model Central Accelerations vs Central Neutral Density for Model Density vsradiusformodel The z-component of the Magnetic Field vs Radius for Model Radial Component of the Magnetic Field vs Radius for Model Radial Neutral Infall Speed vs Radius for Model Radial Drift Speed vs Radius for Model Alfvén Speed vs Radius for Model Thermal-Pressure Force vs Radius for Model Ratio of Thermal-Pressure and Magnetic Force vs Radius for Model Total Force vs Radius for Model Mass-to-Flux Ratio vs Radius for Model Mass Infall Rate vs Radius for Model Central Neutral Density vstimeformodel Central Mass-to-Flux Ratio vs Central Neutral Density for Model Central Magnetic Field vs Central Neutral Density for Model Central Timescales vs Central Neutral Density for Model Central Accelerations vs Central Neutral Density for Model Central Wave Pressure vs Time for Model Central Wave Pressure vs Central Neutral Density for Model Central Timescales of Wave Processes vs Central Neutral Density at = A + for Model Central Timescales of Wave Processes vs Central Neutral Density at = mid for Model Central Timescales of Wave Processes vs Central Neutral Density at = l ref for Model Central Wave Magnetic Energy between = A! A + vs Central Neutral Density for Model Central Wave Magnetic Energy between = mid! mid + vs Central Neutral Density for Model Central Wave Magnetic Energy between = l ref! l ref + vs Central Neutral Density for Model Density vsradiusformodel Wave Pressure vs Radius for Model The z-component of the Magnetic Field vs Radius for Model Radial Component of the Magnetic Field vs Radius for Model Ratio of Wave Pressure to Magnetic Pressure vs Radius for Model viii

9 4.35 Radial Neutral Infall Speed vs Radius for Model Radial Drift Speed vs Radius for Model Reflection Coefcient vs Radius for Model Alfvén Speed vs Radius for Model The RMS Velocity vs Radius for Model Thermal-Pressure Force vs Radius for Model Ratio of Thermal-Pressure and Magnetic Force vs Radius for Model Total Force vs Radius for Model Magnetic Force vs Radius for Model Wave-Pressure Force vs Radius for Model Mass-to-Flux Ratio vs Radius for Model Mass Infall Rate vs Radius for Model Central Timescales vs Central Density for Model Central Accelerations vs Central Density for Model Central Neutral Density vstimeformodel Central Wave Pressure vs Time for Model Central Wave Pressure vs Central Density for Model Central Magnetic Field vs Central Density for Model Central Magnetic Field vs Central Density for Models 2, 4, and Density vsradiusformodel Radial Neutral Infall Speed vs Radius for Model Radial Drift Speed vs Radius for Model Central Density vstimeformodel Central Wave Pressure vs Time for Model Central Mass-to-Flux Ratio vs Central Density for Model Central Accelerations vs Central Density for Model Central Timescales vs Central Density for Model Central Timescales of Wave Processes vs Central Density at = l ref for Model Central Timescales of Wave Processes vs Central Density at = mid for Model Central Timescales of Wave Processes vs Central Density at = A + for Model Density vsradiusformodel Wave Pressure vs Radius for Model Ratio of Wave Pressure and Magnetic Pressure vs Radius for Model Alfvén Speed vs Radius for Model RMS Velocity vs Radius for Model Central Wave Magnetic Energy Spectrum vs Wavenumber for Model Central Wave Magnetic Energy Density vswavenumber for Model Central Wave Pressure vs Time for Model Central Timescales of Wave Processes vs Central Density at = l ref for Model Central Timescales of Wave Processes vs Central Density at = mid for Model Central Timescales of Wave Processes vs Central Density at = A + for Model 8152 ix

10 1 Introduction How stars form remains a fundamental unsolved problem in theoretical astrophysics. Molecular cloud interiors are the birthplaces of most stars in our galaxy. These clouds typically have masses several orders of magnitude greater than the thermal Jeans mass. Observations also indicate smoothly varying magnetic elds (e.g. Hildebrand et al. 1999; Schleuning et al. 2000) strong enough to support the entire cloud against its self-gravity. Observations of molecular radiation reveal Doppler broadened linewidths (velocity dispersion, ff v ) which suggest that the internal motions are supersonic (C = 0:2 km/s isothermal sound speed) and slightly less than the Alfvén speed (ο 1 km/s). Large amplitude hydrodynamic motions (e.g., sound waves steepening into shocks) have the unattractive quality of high energy requirements to sustain the motions. Therefore, it is generally believed that nite amplitude, nonlinear, long-wavelength, Alfvén waves 1 are responsible for these internal motions because of their slow dissipation and thus low energy requirements (Arons and Max 1975). These waves can potentially affect the structure and dynamics of self-gravitating, magnetically supported clouds. The wave-pressure force can aid thermal-pressure and magnetic forces in supporting a molecular cloud against its self-gravity. This is revealed by comparing the wave, thermal, magnetic, and gravitational energy densities. For a cloud with a number density n = cm 3, temperature T = 10 K, magnetic eld strength B =10μG, and cloud radius ο1 pc, the thermal energy density P=(fl 1) = (3=2)nk B T= erg/cm 3, gravitational energy density ß GM 2 =RV = GMρ=R ß dyn/cm 2, and magnetic energy density B 2 =8ß ß erg/cm 3. Observations show that the energy due to nonlinear MHD waves is comparable to the energy density of the mean magnetic eld. For example, the speed of a particle due to a nonlinear (i.e. b ß B) Alfvén wave is v = V A b=b (Cowling 1976), where V A = B= p 4ßρ is the Alfvén speed, and b is the magnetic disturbance transverse to the mean eld B. Combining the equations for v and V A gives 0:5ρ( v) 2 ß B 2 =8ß. The fact that, for typical parameters, all of these energy 1 Alfvén waves are one of the magnetohydrodynamic (MHD) wave propagation modes in an ionized medium threaded by a magnetic eld. The Alfvén wave propagates at the speed V A = B= p 4ßρ where B is the magnetic eld vector and ρ is the gas density. 1

11 densities are within a factor of ten of each other suggests that any theory of star formation should account for all of them. Larson (1981) found that the observed velocity dispersion, ff v, and the size, R, of 54 gravitationally bound clouds, clumps and cores obey the relation: ff v / R 0:35, which was thought to be the signature of Kolmogorov turbulence. Leung, Kutner and Mead (1982) and Myers (1983) conducted a similar analysis of the observational data but subtracted the thermal contribution to the linewidth and found the turbulent" part to be given by ν turb ß 1:3 R 1=2 km s 1 : (1.1) 1pc Relation (1.1) was explained as a consequence of hydromagnetic waves in magnetically supported, self-gravitating clouds (Mouschovias 1987a; Mouschovias and Psaltis 1995). In such a cloud, the magnetic and gravitational energy densities are comparable, which leads to the relation 2GM 1=2 V A ß ß (2ßGffm R) 1=2 (1.2) R Thus, for hydromagnetic waves with ν turb ß V A, the relation ν turb / R 1=2 is seen as a consequence of the magnetic support of clouds having comparable column densities, ff m. Also, the second relation found by Larson (1981), ρr ß constant, namely, that the cloud column densities vary by less than a factor of ten, can also be explained as a consequence of magnetically supported self-gravitating clouds. These types of clouds have mass-to-flux ratios near the critical value, (M=Φ) crit =1=(63G) 1=2 (see x2.2.1). Since (M=Φ) crit =(ff m =B) crit, then self-gravitating clouds are expected to have column densities comparable to ff m;crit = (1=63G) 1=2 B, which depends only on the mean eld, B, which in turn is not expected to vary much from place to place in the interstellar medium under conditions suitable for the formation of self-gravitating clouds. To summarize, ff m (= 4ρR=3) ß constant, but only to the extent that the magnetic eld strength is constant for different self-gravitating clouds. Eliminating ff m in equation (1.2) using the critical mass-to-flux relation gives ν turb ß 1:4 B 1=2 R 1=2 km s 1 : (1.3) 30 μg 1pc 2

12 Mouschovias and Psaltis (1995) found excellent agreement between relation (1.3) and then current data on 14 objects (Myers and Goodman 1988) for which ν turb, R, andb are known. Relation (1.3) is expected to break down at high densities during the contraction of a protostellar fragment because the low degree of ionization allows ambipolar diffusion to damp the waves and therefore the linewidths reflect only Doppler broadening due to thermal motions. Evidence of thermalized linewidths in dense cores has been found by Baudry et al. (1981), Myers and Benson (1983) and Myers, Linke and Benson (1983). For an innite homogeneous turbulent medium, Chandrasekhar (1951) used a linear stability analysis to calculate the critical wavelength of a perturbation that would allow gravity toovercome the thermal and turbulent pressure. He found, 2 > (2ß) 2 C u2 4ßGρ : (1.4) This is similar to the Jeans length except for the presence of the mean-square turbulent velocity, u 2. From equation (1.4), one may conclude that turbulence does provide additional support against gravity. However, Bonazzola et al. (1987) and Vazquez-Semadini and Gazol (1995) performed an analysis similar to Chandrasekhar but with the turbulent pressure dependent on the size of the object relative to the lengthscales of the turbulent motion. They argue that turbulent motions with long lengthscales do not provide support to a cloud of a smaller radius (In the context of waves see Mouschovias 1987a). For R > the turbulent pressure is u 2 ( )rρ, where the mean square of the turbulent velocity, u 2 ( ), in terms of the energy spectrum of the turbulence, E( ), is u 2 ( ) Z 0 E( 0 )d 0 : (1.5) We caution that although their expression for the wavelength dependent turbulent pressure is intuitively appealing it is much harder to prove (Bonazzola 1992; Vazquez-Semadini and Gazol 1995). Numerical simulations of compressible hydrodynamic turbulence (no magnetic elds) by Klessen et al. (2000) provide evidence that cloud collapse can occur over lengthscales shorter than that given by Chandrasekhar's equation (1.4). The virial theorem has been used (Mestel and Spitzer 1965; Mouschovias and Spitzer 1976; 3

13 Mckee and Zweibel 1992; Nakano 1998) to gain insight into the energetics of molecular clouds. However, the virial theorem can only assess global properties because it involves integrals over the whole volume or surface of the cloud. For a non-rotating, self-gravitating, spherical cloud, immersed in a medium with an external pressure P ext, and with a frozen-in magnetic eld, one can write the virial theorem as (see Mouschovias and Spitzer 1976), 4ßR 3 P ext =3MC 2 + M( v) 2 1 R ψ! 3 5 GM 2 Φ2 B 4ß 2 (1.6) where R is the radius of the cloud, M is the mass of the cloud, C is the isothermal sound speed, v is the rms speed of the turbulent motions, and Φ B is the magnetic flux of the cloud. The rst and second terms represent the internal and turbulent energy. The third term is the gravitational potential energy. The fourth term is the magnetic energy of the cloud. Equation (1.6) is commonly used to analyze the stability of a cloud. For example, if the sum of the terms on the right hand side is negative then equation (1.6) requires that the external pressure, P ext have a negative value. Naturally, the pressure cannot be less than zero and thus the cloud will contract. One must be careful since the virial theorem is easily misused. The virial equation accounts for magnetic pressure support with the last term on the right-hand side of equation (1.6). However, the magnetic force is zero along magnetic eld lines and, thus, cannot provide any support against gravity along the direction of the eld. With the computing power at one's disposal now, one can more easily follow the formation of cores within molecular clouds using the appropriate MHD differential equations. Nakano (1998) uses the virial theorem to argue that molecular clouds, as a whole, are magnetically supercritical, i.e. the gravitational potential energy term in equation (1.6) is greater than the magnetic flux term. He states that a gravitationally bound cloud must have a mass nearly equal to the critical mass (for collapse). Since the magnetic eld and turbulent motions are observed to have nearly equal pressures in supporting the cloud against gravity, the magnetic eld strength must have avalue such that the cloud is magnetically supercritical (see also Crutcher 1999). However, the data used by Nakano are for dense cores in molecular clouds since it is easier to measure the magnetic eld there. These cores may already be on the verge of star formation and should therefore be magnetically supercritical. 4

14 Mckee and Zweibel (1995) and Mckee (1999) proposed to model the turbulent pressure with the polytropic relation, P turb / ρ fl. Polytropic models are primarily useful when little is known about the phenomenon in question. The value of fl may be attributable to a single physical phenomenon, e.g. fl = 1for isothermal or fl = 5=3 for adiabatic systems. For more complicated systems, one's ignorance of the physics could be lumped into an unphysical value for fl. In our work we derive an energy equation for the turbulence and, indeed, show thatseveral processes can signicantly affect the turbulent pressure (energy density) within a core. Each of these processes are signicant during different stages of the core formation and depend on the lengthscale of the turbulent motions. Computer simulations of MHD turbulence" within the context of molecular cloud formation or star formation have appeared within approximately the last decade (e.g. Carlberg and Pudritz 1990; Ballesteros-Paredes, Vazquez-Semadini, and Scalo 1999; Elmegreen 1999; Padoan and Nordlund 1999; Heitsch, Mac Low, and Klessen 2001; Ostriker, Stone, and Gammie 2001). In general, each of the simulations begin with a uniform magnetic gas in 2D or 3D. Moderate amplitude, v=v A ο 1, velocity perturbations are then introduced either uniformly throughout the cloud or at the grid boundary. Amajorgoalofeach study is to understand how clouds suitable for star formation are produced, out of the background medium, as a result of the perturbations. Many of the studies nd similar results such as the accumulation of gas into clumps due to shocks sweeping up matter along the magnetic eld lines. Shocks occur more easily along eld lines since the Lorentz force is zero along this direction and because the velocity of the driving perturbations is always several times greater than the sound speed, v >C. There are many publications of MHD simulations and the effects of random forcing. We will only select a few of these to describe in detail since some have been outdated by higher dimensional simulations or because they are not relevant to molecular clouds and star formation due to the parameters chosen by the authors or the exclusion of self-gravity. For example, Gammie and Ostriker (1996), and Ostriker, Gammie, and Stone (1999) have only one and two dimensional geometries, respectively, while Ostriker, Stone, and Gammie (2001) have three dimensions. Passot, Vazquez-Semadeni, and Pouquet (1995) and Ballesteros-Paredes, Vazquez-Semadeni, and Scalo (1999) have two dimensional geometries and use parameters for structures relevant to galactic scales but not to molecular clouds. 5

15 Carlberg and Pudritz (1990) did one of the earliest simulations but did not solve the full MHD equations. For example, instead of solving the Faraday equation for the evolution of the magnetic eld they use a scaling law, B / ρ 2=3,whichisvalid only in the case of spherical cloud contraction with the magnetic flux perfectly coupled to the gas. Stone, Ostriker, and Gammie (1998) and Mac Low, Klessen, and Burkert (1998) have studied the decay of random perturbations in an MHD system without gravity and found that they decay on a timescale less than the initial free-fall time. Elmegreen (1997; 1999) did one and two dimensional MHD simulations (neglecting self-gravity) of Alfvén waves propagating along the mean magnetic eld direction, from the edge of the grid towards the center. The waves push the gas towards the center, where it accumulates. However, the wave amplitudes that he uses are too small compared to those commonly observed in clouds, so the density enhancement, compared to the initial uniform density, is only about a factor of two. Afactoroftwo increase in density does not justify the claim of cloud formation. In nearly every study, the full single-fluid MHD equations are used to follow the evolution of the perturbations and subsequent formations of clumps of gas. The velocity perturbations, v, driving the turbulence are applied either initially, in order to study the decay of the perturbations, or for some length of time, in order to study the affect of a source. We choose to describe the results of three groups Padoan and Nordlund (1999), Ostriker, Stone, and Gammie (2001), and Heitsch, Mac Low, and Klessen (2001). These calculations are not relevant to the star formation process itself, but they provide insight into the interactions of nonlinear MHD perturbations with the background. None of these calculations has followed the evolution of a self-gravitating core past a density enhancement of ten. (Note that the mean number density of a molecular cloud is about 10 3 cm 3 while the mean density of the Sun is about times greater.) In the two papers which include self-gravity, the nite grid spacing of their numerical codes cannot resolve the small dense cores and still capture the large scale dynamics of the random motions. Padoan and Nordlund (1999) performed three dimensional numerical experiments of driven and decaying MHD random perturbations that have either superalfvénic or Alfvénic fluid velocities. Their neglect of self-gravity implies that any density enhancements are temporary. Once the forces driving the MHD turbulence are turned off, the gas and magnetic elds return to their equilibrium initial state which has a uniform density and magnetic eld. The authors favor the superalfvénic 6

16 MHD turbulence model of clump formation citing that these simulations give some results that match a few observations better than the simulations with Alfvénic perturbations. For example, in their Ad2 model, their random driving forces produce large velocity perturbations that are almost thirty timesgreater than the Alfvén speed or sound speed. Initially, the fluid is unable to exert a signicant back-reaction to the distortions caused by the random driving perturbations. Therefore, gas and magnetic eld lines are initially swept up into sheets as shown in their gures. They claim that, initially, B / n, where n is the number density. This result expresses nothing more than mass and magnetic flux conservation when a slab of gas is compressed perpendicular to the eld lines. This implies that the magnetic eld begins to exert a back reaction on the shocks since the Alfvén speed (V A = B=(4ßρ) 1=2 ) increases, probably to a level where the gas velocity is comparable to the Alfvén speed. After some time, the gas and eld evolves to the point where the magnetic pressure and turbulent pressure of the driving forces are expected to equilibrate. Using their values for the maximum density and magnetic eld strength we can show that both pressures indeed equilibrate with one another. For example, B max = 100 μg, n = 1000 cm 3, and v = 7: cm s 1, leads to P B = P turb, where we have assumed that the mass of a particle is that of a hydrogen atom, and P B = B 2 =8ß is the magnetic pressure, and P turb = ρ( v) 2 =2 is the turbulent pressure. The fact that they nd B / n 1=2 for some parts of the gas can be explained as a consequence of the magnetic and turbulent pressures in equilibrium. The scatter in their B versus n plots is possibly the result of gas shocks along the magnetic eld lines which do not necessarily result in an increase of magnetic eld strength with density. In a different experiment (same paper), Padoan and Nordlund adjust the turbulence driving amplitude of the velocity perturbations to equal the initial Alfvén speed. They then turn off the driving force and allow the motions to decay via shocks. They nd that, after approximately one Alfvén crossing time, L=V A, where L is the length of their grid, the motions are mainly Alfvén waves (fluid motions transverse to the magnetic eld). They also nd that even though motions [transverse to] the eld lines are the most frequent, it is the less frequent motions along the eld lines that dominate the dissipation. The motions [transverse to] eld lines are subject to magnetic restoring [tension] forces and do not lead to substantial density enhancements." Waves in molecular clouds are likely to have sources from large distances away. Dissipation would damp the compressive 7

17 motions quickly, thus leaving Alfvén waves to affect the internal motions. This result was also found by Stone et al Because of their low damping rate, it may still be true that long-wavelength Alfvén waves are ideal candidates for explaining the Doppler-broadened linewidths (Arons and Max 1975; Zweibel and Jossafatson 1983). Stone, Ostriker, and Gammie (1998) solve the single-fluid MHD equations, without gravity, on a three dimensional grid. As in the case of Padoan and Nordlund (1999), they do not have a separate equation governing the evolution of the MHD turbulence" or waves. They apply random nonlinear perturbations to the system and let the basic equations handle both the mean and the random elds. This approach has advantages and disadvantages. The advantage is in its simplicity. One does not have to worry about how tohandle various interactions such as, shocks, wave steepening, or wave-wave interactions. The disadvantage is that one cannot account for the signicance of any individual process if more than one are acting at the same time. It is also harder to distinguish whether the motions are waves, shocks, oscillations, or the MHD analog of Kolmogorov turbulence since their driving force is random. They consider both decaying and forced turbulence. The dissipation of the energy is very high, with a timescale smaller than their flow crossing time, L= v, where v is the velocity dispersion caused by the source, and L is the size of the grid. However, the dissipation comes from articial viscosity. The MHD equations that they solve are ideal and, thus, contain no explicit resistivity, viscosity, or ambipolar diffusion terms. Therefore, it is not possible to determine the possible importance of any of these effects. Nevertheless, some general conclusions may be obtained from their work. In experiments with a source for the turbulence, they nd results similar to those of Padoan and Nordlund (1999) superalfvénic model when the source produced superalfvénic motions. The random motions amplify the magnetic energy tenfold and cause the magnetic eld to become highly tangled. When the source produces motions with subalfvénic but supersonic velocities, the magnetic eld lines appear well ordered, which is consistent with polarization measurements of the magnetic eld in molecular clouds (Schleuning et al. 2000; Hildebrand et al. 1999). Both the driving by the superalfvénic and subalfvénic sources produced equipartition between turbulent, magnetic and kinetic energy. SuperAlfvénic disturbances tend to fold the eld lines until the magnetic pressure can exert a backreaction force. SubAlfvénic perturbations directed perpendicular to the eld lines do the same 8

18 thing except now the mean magnetic eld mainly determines the propagation properties of the disturbance, as it does for small amplitude MHD waves. Ostriker, Stone, and Gammie (2001) and Heitsch, Mac Low, and Klessen (2001) have done three dimensional single-fluid (no ambipolar diffusion) calculations. In addition to magnetic elds they also included self-gravity. Both found that magnetically subcritical clouds do not collapse in the presence and subsequent decay of turbulence since the mean magnetic eld can support the cloud against gravity. In magnetically supercritical clouds, Heitsch et al. (2001) point out that supersonic turbulence can prevent collapse on a global spatial scale but not locally. Ostriker et al. (2001) believe that the non-gravitationally bound clumps of gas that are commonly observed may be nothing more than a projection effect of physically disconnected regions along the line of sight. In this thesis, we assume that the turbulence is made up of a spectrum of Alfvén waves. We assume Alfvén waves instead of a random superposition of different kinds of waves (or turbulence) for several reasons. First, observations of dust alignment by magnetic elds show that eld lines are not tangled in molecular clouds (e.g. Hildebrand 1999, Schleuning et al. 2000). If the turbulent energy density is much greater than the mean magnetic eld energy density then the turbulence should tangle the eld lines. This was shown to happen in the superalfvénic models of Stone, Ostriker, and Gammie (1998) and Padoan and Nordlund (1999). If the turbulent energy density is in equipartion with the mean magnetic eld energy density, as observations suggest, then there is little turbulent amplication of the magnetic eld. This was demonstrated by the Padoan and Nordlund's (1999) equipartition model and to a lesser extent by Stone, Ostriker, and Gammie's (1998) strong eld (subalfvénic) model. Second, the numerical calculation of the decay rate of large-amplitude Alfvén waves has only been addressed by Gammie and Ostriker (1996). They nd a decay rate much smaller than subsequent calculations because the subsequent calculations use a driving force that excites random motions rather than Alfvén waves. Alfvén waves have very specic properties that are determined by the MHD equations. One property is that the magnetic and kinetic energies of an Alfvén wave are in equipartition at all times, ρ( v) 2 =2 = b 2 =8ß. Later publications of higher dimensional MHD turbulence use a random velocity perturbation which can excite many types of waves. Even though care is taken to select the excitation velocity v such that r v = 0, this does not 9

19 produce an Alfvén wave. This is demonstrated in Ostriker, Stone, and Gammie (2001) who nd an initial increase in the turbulent magnetic energy with time. Therefore, long-wavelength Alfvén waves propagating from sources over long distances may still be the source of waves that manifest themselves in the Doppler broadening of molecular linewidths. In this thesis, we extend the work by Fiedler and Mouschovias (1993) and Morton and Mouschovias (1991). They showed that ambipolar diffusion (the relative drift between neutral and charged particles) in the interiors of otherwise magnetically supported clouds is crucial to the formation and contraction of cores in molecular clouds. We include the averaged effects of waves in the formulation of the problem in a manner similar to that of Dewar (1970). The formation and contraction of a protostellar core in a non-rotating, magnetic, turbulent molecular cloud is then followed through both the quasistatic and dynamic phases of contraction. By following the time dependence of the average wave pressure, we can quantify the effectiveness of the wave pressure opposition to gravity at each stage of the core formation and contraction process. 10

20 2 Properties of Molecular Clouds In this section, we discuss the relevant observed properties of molecular clouds that form the theoretical basis of our model. 2.1 Distribution of Molecular Gas Molecular clouds consist of about 90% atomic hydrogen, by number, yet the gas is usually observed by using radio frequency spectral lines of trace molecules such asco, OH, NH 3, and H 2 CO. The symmetry of molecular hydrogen prevents it from having any permanent dipole moment. Its lowest rotational energy transition is the J = 2! 0. A gas temperature of 509 K is required to collisionally excite this transition. Molecular clouds have temperatures that are much lower, usually T ß 10 K. The tracer molecules listed above are more easily detected than molecular hydrogen. For example the CO molecule's lowest energy transition, J = 1! 0, corresponds to a temperature of 5.5 K. The CO spectral line intensityisthenconvertedtoah 2 column density, although there are uncertainties in the H 2 /CO ratio; a typical ratio is 10 6 (see Kutner 1984; Rohlfs and Wilson 1986). Molecular gas is found mostly in large molecular clouds which have masses ο M and diameters ο 1 pc to more than 10 pc. They are found in the galactic plane which is about 400 pc thick. High resolution observations of giant molecular clouds (GMC's) reveal smaller, denser clouds with sizes ranging from 1 to 5 pc and mean densities from cm 3. The cores of these clouds have densities of at least 10 4 cm 3. It is within these cores that young stars are usually found. Hence, the formation and evolution of these cores is the focus of our investigation Magnetic Field Polarization of starlight passing through a cloud can give information on the magnetic eld component, B?, perpendicular to the line of sight. Elongated dust grains spin around axes parallel to the magnetic eld (Davis and Greenstein 1951; Purcell 1979). Extinction of background starlight 11

21 passing through a cloud is greater for light polarized perpendicular to the magnetic eld. Polarization maps of molecular clouds (Hildebrand et al. 1999; Rao et al. 1998; Schleuning et al. 2000; Lai 2001) reveal large-scale, ordered magnetic elds in the plane of the sky. Polarization maps, however, provide no reliable information on the magnetic eld strength (e.g. see review by Mouschovias 1981). The splitting of the 18-cm line of the OH molecule due to the Zeeman effect allows measurement of the magnetic eld strength component along the line of sight, B k. Observations detect magnetic eld strengths between 30 μg and 120 μg for densities between cm 3 and 10 5 cm 3 (e.g. Crutcher, Kazés, and Troland 1987). Observations of the B1 cloud (Crutcher et al. 1994) reveal a line-of-sight magnetic eld strength B k =16μG in the cloud envelope and 30 μg inthenh 3 core. The density in the B1 cloud envelope is about 10 3 cm 3 and increases to > cm 3 in the core Degree of Ionization Magnetic elds exert a Lorentz force on charged particles. The neutral particles are affected by magnetic forces indirectly, through collisions with charged particles. The degree of ionization plays a major role in determining the importance of the magnetic eld in molecular clouds. In the deep interiors of molecular clouds, cosmic ray ionization dominates other mechanisms, but UV ionization is important in cloud envelopes. Recombinations occur both in the gas phase and on the surfaces of grains. Observations reveal a degree of ionization > 10 4 in cloud envelopes but < 10 7 in dense cores (Caselli et al. 1998; Williams et al. 1998; Bergin et al. 1999) Turbulence Spectral linewidths indicate supersonic but usually subalfvénic gas motions in the envelopes of molecular clouds. Alfvén speeds are about 1 km s 1 for a cloud with a typical mean density cm 3 and large scale magnetic eld ο 30 μg. The cause of the Doppler broadening of these linewidths has remained elusive fora long time. On the other hand, linewidths observed in dense cores of the molecular clouds reveal thermalized linewidths (Myers, Linke and Benson 1983; Myers and Benson 1983; Tatematsu et al. 1993; Crutcher et al. 1994). 12

22 Some workers (e.g. Larson 1981; Leung et al. 1982; and Myers 1983) attribute the broad linewidths to supersonic turbulence (no magnetic eld). That deduction is based on the empirical fact that the correlation between the measured velocity dispersion and the square root of the cloud radius, i.e. v / R 0:5, has a form similar to the Kolmogorov spectrum, v / R 1=3, for subsonic turbulence. Observers have also measured a second correlation between the clouds' mean density and radius, i.e., ρr ß constant (to within a factor of ten) (Larson 1981; Myers 1983). They do not discuss how turbulence arguments might explain these observations. Large-amplitude longwavelength hydromagnetic waves are generally viewed as the cause of the broad linewidths (e.g., Arons and Max 1974; Mouschovias 1987; Crutcher 2000). The damping rate of long-wavelength Alfvén waves compared to the other MHD modes is smaller (Zweibel and Josaffatson 1983) Mass-to-Flux Ratio As we will show in the next section, the mass-to-flux ratio within a given region of a cloud is a measure of the ratio of the gravitational potential energy and the magnetic energy of the ordered eld. If the mass-to-flux ratio of an object exceeds a critical value, the object collapses against its magnetic forces. This condition is only a necessary condition for a core to collapse. Other factors, such as the external pressure and turbulent pressure, may aid or prevent the collapse, respectively. In nature, the mass-to-flux ratio of molecular clouds as a whole is not expected to stray very far from the critical value. If the ratio were much greater than the critical value, the entire molecular cloud would collapse allowing stars to form anywhere within the cloud, even in the cloud envelope. Velocities characteristic of this kind of collapse are not observed in any molecular cloud. If the mass-to-flux ratio of the cloud as a whole were much smaller than the critical value, then the cloud would probably never form any stars. Molecular clouds are typically observed to have mass-to-flux ratio close to the critical value. There is some debate regarding the observed mass-to-flux ratio of molecular clouds. Crutcher (1999) compiled measurements for a number of molecular clouds. He concluded that the typical cloud mass-to-flux ratio is a factor of two greater than the critical value. He also speculates that a wave or turbulent pressure comparable to the zero-order magnetic pressure may support the cloud against gravitational collapse. However, he also points out (e.g., Heiles 1987) that because only 13

23 the line-of-sight component of the magnetic eld strength is measured (Zeeman effect), the actual magnetic eld strength must on average be greater than the measured value by a factor of two. Also, the average column density measured is greater than the actual value by a factor of two becauseofageometrical effect. Applying these corrections, Shu etal. (1999) nd that the clouds are typically subcritical. In either case, the mass-to-flux ratio is approximately within a factor of two of the critical value. In astronomy, a factor two is usually not a very big deal. However, in regards to star formation, two different scenaria for star formation are possible. Which scenario is chosen depends on whether the mass-to-flux ratio is less than or greater than the critical value (Mouschovias 1987; Shu etal. 1987). If the initial mass-to-flux ratio of a cloud as a whole is subcritical, then stars can only form if the fluid particles can cross the magnetic eld lines and accumulate to form a core whose mass-to-flux ratio exceeds the critical value (see section 1.2.1). If the initial mass-to-flux ratio of a cloud as a whole is supercritical, then stars can form anywhere within the cloud as long as other means of support have sufciently abated. 2.2 Role of Magnetic Fields and Waves or Turbulence In this thesis, we take the view that molecular clouds are initially magnetically subcritical. Observations indicate that, although cloud masses exceed the thermal Jeans mass by several orders of magnitude, only a small fraction of the mass is found in stars (Straw and Hyland 1989; Loren et al. 1990; Bachiller et al 1990; Tapia et al. 1991; Eiroa and Casali 1992; Lada 1992). The inefciency of star formation has been attributed to the magnetic support of molecular clouds against gravitational collapse (Mouschovias 1976, 1978). Stars form only when self-gravity drives neutral matter past magnetic eld lines (and the plasma) to form fragments (or cores), whose mass-to-flux ratio eventually exceeds the critical value (Mouschovias 1977, 1978, 1979; see also review 1996). This process is referred to as ambipolar diffusion. This point of view is substantiated by numerical simulations (Paleologou and Mouschovias 1983; Mouschovias and Morton 1991, 1992a, 1992b; Fiedler and Mouschovias 1992, 1993; Ciolek and Mouschovias 1993, 1994, 1995; Basu and Mouschovias 1994, 1995a, 1995b; Desch and Mouschovias 2001). 14

24 2.2.1 Critical Mass to Flux Ratio and Cloud Stability The stability of an isothermal, nonrotating, nonmagnetic spherical cloud was rst examined by Bonnor (1956) and Ebert (1957). They calculated the critical mass, M BE, required for self-gravity to balance the thermal pressure forces. A cloud with a mass greater than M BE collapses. The Bonnor-Ebert mass is given by C 4 M BE = 1:18 (G 3 P ext ) 1=2 (2.1) ψ! T 3= cm 3 1=2 = 5:8 M ; (2.2) 10 K n n where C is the isothermal sound speed, P ext the external pressure, T the temperature, and n n the density of the neutral fluid. Mouschovias (1976a,b) calculated two-dimensional equilibria of initially uniform, cold, spherical clouds embedded in a constant-pressure external medium and threaded by a frozen in magnetic eld. They are oblate and perpendicular to the eld lines. Mouschovias and Spitzer (1976) used these equilibrium states to nd the critical mass-to-flux ratio M ΦB crit = 1 1=2 (2.3) 63G above which the magnetic eld cannot support the cloud against its self-gravity. This condition can be rewritten to give a critical mass, M crit, in terms of the magnetic eld strength B and the neutral-particle number density n n : M crit = B 3 30 μg n 2 n 10 3 cm 3 M : (2.4) Notice that equation (2.4) suggests that, under typical molecular cloud conditions, the mean magnetic eld can support cloud masses two orders of magnitude greater than that supported by thermal pressure alone. In the presence of waves or turbulence (but T = 0 and B = 0), a relation for the critical mass required for gravity to overcome an isotropic wave or turbulent pressure can be derived in 15

25 a manner similar to the derivation of the Bonnor-Ebert critical mass or the Jeans mass. This is possible since the expression for the isotropic turbulent pressure force, r(ρu 2 n), has a form similar to the expression for the thermal-pressure force, r(ρc 2 ), where u n is the velocity of the fluid due to turbulence, and ρ is the fluid density. From a linear stability analysis of the single-fluid MHD equations one can derive a Jeans-like lengthscale for the turblence (see eq.[1.4]), However, caution must be used when applying this expression. Turbulence can provide pressure support against gravity only when acting within structures larger than the smallest turbulent lengthscale. 1 Turbulence of long lengthscales tends to push smaller structures around as a whole (Bonazzola et al. 1987; Vazquez-Semadeni and Gazol 1995; Klessen et al. 2000) Role of Ambipolar Diffusion Ambipolar diffusion was rst proposed by Mestel and Spitzer (1956) as a means by which an interstellar cloud can reduce its magnetic flux. They found that the dominant forces on the ions were the Lorentz force and the collision force with neutrals. They then argued that these two forces will nearly balance each other and thus estimated the terminal drift velocity between the ions and neutrals, v D = (r B) B 4ßρ n n i ff in v n;t ; (2.5) where ff in and v n;t are the collision cross section between ions and neutrals and the thermal velocity of the neutral particles. Since the ions are attached to the magnetic eld lines, they viewed the eld lines as moving outward and leaving the cloud, thus allowing the neutral particles to collapse on the free-fall timescale ff = 3ß 32Gρ n 1=2 : (2.6) Spitzer (1968, 1978) realized that in a self-gravitating cloud where magnetic forces balance gravity the neutral particles would be in force balance as well. He derived a timescale for quasistatic ambipolar diffusion in a cylindrical cloud, AD r v D = hffwi ih 2 2ßGm n x i 1:4 ; (2.7) 1 In hydrodynamics, turbulence usually refers to the superposition of fluid eddys of many different sizes. Here, we will refer to turbulence as the superposition of MHD waves. 16