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Table of contents
Introduction
1 The interstellar medium
1.1 A variety of structures in our Galaxy
1.2 Where stars are formed
1.3 Density and chemical composition
1.4 Various phases
1.5 Magnetic field
2 Molecular clouds
2.1 Difficulties with observations
2.2 Transient structures
2.3 Characteristic scales and filamentary substructures
2.4 Complex thermodynamics
2.4.1 Heating
2.4.2 Cooling
2.4.3 Overall effects
2.5 Turbulent motions
2.5.1 Non thermal motions
2.5.2 Reynolds number and turbulence
2.5.3 Compressible, isothermal turbulence
3 Fundamentals of core and star formation
3.1 Newtonian Gravity
3.1.1 Attraction of point masses
3.1.2 Continuous media and Poisson equation
3.1.3 Gravitational potentials and fields of homogeneous distributions
3.2 Two descriptions of fluid motions
3.3 Homologous collapse: free-fall time scale
3.3.1 Lagrangian description and mass conservation
3.3.2 Solving the equation of motion
3.3.3 Dynamical regimes
3.3.4 « Standard » initially motionless free fall: the free fall time.
3.4 Virial equilibrium
3.4.1 Evolution of the moment of inertia
3.4.2 Virial parameter
3.5 The Jeans criterion
3.6 Overview of low mass star formation
4 Mass functions
5 Motivation for this work
Chapter 1 Tools
1 Elements of probability theory
1.1 A first approach to the notion of probabilities
1.2 Axiomatic of probability theory and set theory
1.2.1 Elements of set theory
1.2.2 Probability space
1.2.3 Simple properties
1.2.4 Conditional probability
1.2.5 Random variables
1.3 (Cumulative) Distribution function
1.3.1 Definition
1.3.2 Properties of the distribution function
1.4 Probability Density Function (PDF)
1.4.1 Defintion
1.4.2 Properties
1.5 Change of variables
1.6 Vector random variables
1.6.1 Distribution and density functions
1.6.2 Marginal distributions and densities
1.6.3 Independent variables
1.7 Characteristic numbers
1.7.1 Mathematical expectation
1.7.2 Moments
1.7.3 Variance and standard deviation
1.7.4 Uncorrelated variables
1.8 Gaussian random variables
1.9 Conditional statistics
1.9.1 Conditional Distribution and density
1.9.2 Conditional expectations
1.10 Practical application of the theory
1.10.1 Bienayme-Tchebychev’s inequality
1.10.2 Accuracy of estimators and prediction theory
2 Stochastic fields and analytical tools
2.1 One point statistics
2.1.1 One point distributions and densities
2.1.2 Moments
2.1.3 Commutation of the derivatives and the expectation
2.2 N-point statistics
2.2.1 N-point distributions and densities
2.2.2 Moments, correlation and covariance
2.3 Gaussian fields
2.4 Statistically stationary or homogeneous fields
2.4.1 Stationarity
2.4.2 Homogeneity
2.4.3 Isotropy
2.4.4 Structure functions
2.4.5 Properties of the ACF of homogeneous fields
2.4.6 Correlation coefficients
2.4.7 Power spectrum
3 Ergodic theory
3.1 Motivation
3.2 Frequency interpretation and repeated trials
3.3 Ergodic theorems. Autocovariance function. Correlation length
3.3.1 Ergodic theorems and the autocovariance function
3.3.2 Correlation length
3.3.3 Integral scale
3.3.4 Average size of most correlated structures
3.4 Estimates of the autocovariance function and correlation length
3.4.1 Reliability of the estimators of the auto-covariance and the power spectrum
3.4.2 Periodic estimators
3.5 Concluding remark
4 Observations
4.1 Elements of radiative transfer
4.1.1 Specific intensity
4.1.2 Equation for radiative transport
4.1.3 Formal solution
4.1.4 Planck’s law
4.1.5 Various definitions of temperature
4.2 Spectral lines
4.3 Extinction
4.3.1 Magnitude
4.3.2 The quantity extinction
4.4 Molecular lines
4.5 Dust emission
5 Numerical simulations
5.1 Numerical set up of Federrath & Klessen (2012)
5.2 Turbulent forcing
5.3 Handling high density regions: sink particles
5.4 List of models and general evolution of the simulations
5.5 Resolution and limits for resolving the PDFs
5.6 Concluding summary
A Ergodic estimate for a general control volume
Chapter 2 Column densities as tracers of the underlying density field: the link with observation
1 Motivation
2 One and two point statistics.
2.1 Inhomogeneity and anisotropy due to integration along the line-of-sight
2.2 Column-density field in a numerical simulation box
2.2.1 Statistical homogeneity
2.2.2 Auto-covariance function and variance
2.3 Decay length of correlations
3 Column density PDFs as tracers of the underlying density PDFs
3.1 Current status of research
3.1.1 Variance in column density and density PDFs
3.1.2 Exponents in power-law PDFs
3.1.3 Shape of the PDFs
3.2 Aim of this section
3.3 Critical densities for the onset of power-law tails
3.4 Numerical test
3.5 Model with one or two power-law tails
4 Conclusion
Chapter 3 Relevance of a statistical approach
1 Introduction
1.1 Motivation
1.2 Fair-sample hypothesis
1.3 Informations available from observations
1.4 Objectives of this chapter
2 Mathematical framework for a statistical approach
2.1 Ergodic theory
2.2 Expected fluctuations in repeated trials
3 Application to astrophysical density fields
3.1 Exact results regarding the properties of the auto-covariance function (ACF) of the density field
3.2 Phenomenology of (compressible) turbulence
3.3 Autocovariance function and correlation length in astrophysical situations
3.3.1 ACF from data
3.3.2 Usual estimates
3.4 Practical assumptions regarding the ACF
3.5 Homogeneity and correlation scales in cosmology
4 Application to observations of the Polaris cloud
4.1 Filtering out large scale gradients
4.2 Estimated ACF and correlation length
4.2.1 Correlation length from the ACF
4.3 Ergodic estimate and real error bars for the observed PDF
4.3.1 Reduced integration effects at high density contrasts
4.3.2 Statistical Errorbars
4.3.3 Gaussian approximation; effective error bars on the observed PDF
5 Applications to the Orion B cloud
6 Consequences for the estimation of the total gravitational energy of a cloud
6.1 Motivation
6.2 Correlation length and gravitational binding energy
6.2.1 Isolated cloud
6.2.2 Simulations in a periodic box
6.3 Virial parameter
7 Conclusion
A Homogeneity scale
B Ergodic estimators of the CMF and PDF
B.1 Cumulative Distribution Function (CMF)
B.2 Probability Density Function (PDF)
B.3 Gaussian process
B.3.1 Integrability of the ACF and short scale analysis
B.3.2 Exponential ACF
B.4 Deterministic function of a Gaussian field
B.4.1 Log-normal fields
C Orion B cloud
C.1 ACF of the square and filament region
C.2 Correlation length from the variance of the column densities
D Computation of the total potential energy on a control volume
Chapter 4 Evolution of the statistics of the density field in star forming clouds
1 Introduction
1.1 Context
1.2 Current status of research
1.2.1 Is gravity the only cause of these observed departures from lognormal like statistics ?
1.2.2 Gravitational infall as a mechanism producing PLTs
1.2.3 Statistical approach of turbulence based on first principles
1.3 Objectives of this chapter
2 Mathematical Framework
2.1 Description of a molecular cloud
2.2 Probability distribution functions (PDF)
2.3 Model for the statistics of a turbulent cloud
2.3.1 Homogeneous clouds
2.3.2 Accepted class of flows
2.4 Transport equations for the density PDF
2.4.1 Transport equations
2.4.2 Velocity divergence – density PDF relationship
2.5 Transport equations for the auto-covariance function of the density field
2.5.1 Transport equation
2.5.2 Correlation length and conserved quantity
3 Implications for astrophysical flows in star forming clouds
3.1 Velocity divergence – density PDF relationship
3.1.1 Physical class of flows
3.1.2 Separation of variables
3.1.3 Summary
3.2 Dynamical effects on the s-PDF shape
3.2.1 Stationary solutions
3.2.2 Density threshold for a gravity induced transitions of regime
3.2.3 Non-magnetized supersonic hydrodynamics
3.2.4 Magnetized supersonic hydrodynamics
3.2.5 Modified Jeans criteria
3.2.6 Overall effects of gravity
3.3 Time evolution of the density PDF in star forming clouds
3.3.1 Transient regime and short time evolution
3.3.2 Asymptotic case: evolution in regions of « free-fall » collapse (fully developed gravity induced dynamics)
3.4 Evolution of the correlation length of the density field in star forming clouds
4 Comparison with numerical simulations
4.1 Numerical set up
4.1.1 Methods
4.1.2 Resolution and limitations on the resolution of the PDFs
4.2 Short time evolution of the transport equations
4.3 Evolution of the s-PDF
4.3.1 Effects of gravity on the low-s-part of the s-PDF
4.3.2 Departure from lognormal statistics and formation of power law tails
4.3.3 Evolution of the -PDF
4.4 Evolution of the correlation length
5 Comparison with Observations
5.1 Clouds with active star formation
5.2 « Young » clouds with little or no sign of star formation activity
5.3 Polaris
5.3.1 Localisation of regions entailing the PLTs
5.3.2 Is gravity the origin of the PLT in Polaris ?
5.4 Draco
5.4.1 Other origin for departures from lognormal statistics in Draco
6 Discussion
6.1 The scale-dependent dynamics of molecular clouds and the suggested explanations
6.2 Physical picture: towards a consistent gravito-turbulent paradigm ?
6.2.1 Dynamical regimes
6.2.2 Size of the densest correlated structures
6.2.3 The averaged correlated mass and the averaged mass to form bound stellar cores
6.2.4 Concluding remarks
7 Conclusion
Chapter 5 Towards the stellar initial mass function (IMF)
1 Introduction
1.0.1 The observable luminosity function and the two regimes of the MF
1.0.2 Present day MF and the initial mass function (IMF)
1.1 Observational features
1.2 Motivation
2 Cosmological mass functions
2.1 Gaussian and homogeneous primordial fluctuations
2.1.1 Smoothed density field
2.1.2 Delta correlated Fourier modes
2.1.3 Particular choices of window functions
2.2 Press-Schechter formalism and mass functions
2.2.1 Initial volume fraction of region exceeding some density contrast at scale R
2.2.2 Mass attributed to unstable regions and the PS mass function
2.2.3 The cloud-in-cloud problem
2.2.4 Summary of the PS formalism
2.3 The excursion-set formalism and mass functions
2.3.1 Back to the cloud-in-cloud problem
2.3.2 Random walks
2.3.3 Brownian walks
2.3.4 The first crossing distribution
2.3.5 General moving barriers
2.3.6 The golden rule and the mass function
2.4 Summary and caveats
2.4.1 Relevance of the statistical procedures
2.4.2 Gaussian simplifications
2.4.3 Caveats regarding the various steps of the procedure
3 From the cosmological to the stellar mass functions
3.1 Step 1: Density threshold in a turbulent medium
3.2 Step 2 and 3: Statistical counting and the Press-Schechter approach of Hennebelle & Chabrier (2008)
3.2.1 Statistical counting
3.2.2 The cloud-in-cloud problem and the mass assignment procedure
3.2.3 Golden rule
3.2.4 Mass function
3.2.5 The variance at scale R
3.2.6 Dynamical regimes and properties of the CMF
3.2.7 From the CMF to the IMF
3.2.8 Summary of the model
3.3 Steps 2 and 3: Statistical counting and the excursion-set of Hopkins (2012a,b)
3.3.1 The cloud-in-cloud problem
3.3.2 The Hopkins (2012a,b) model and the variance problem
3.3.3 Proper lay out of the Hopkins (2012a,b) assumptions and overlooked issues
3.3.4 Success of the model and summary
3.4 Conclusion regarding these models
3.5 Additional potential flaws of these models
3.5.1 Conversion of the CMF into the IMF
3.5.2 Departures from lognormal statistics
4 Perspectives and suggested explanations
4.1 Corrected barrier
4.2 Meaning of the counting procedure for non Gaussian and non lognormal fields
4.2.1 Deterministic function of a Gaussian field
5 Conclusion
A A Volterra equation of the second kind for the first crossing distribution
A.1 Probability that a trajectory has never crossed the barrier at resolution S
A.1.1 Conditional probability of arriving at 2 [x; x + dx[ knowing that the trajectory started at x0
A.1.2 Probability (x; S) for the sharp k-space filter
A.2 The sharp k-space filter and the Volterra equation for the first crossing distribution
A.3 Key ingredients
Conclusion
Bibliography
