# Summary

In this thesis we develop approaches to studying two of the most longstanding theoretical problems in astrophysics, namely the nature and origin of astrophysical jets and the role of magnetic fields in star formation. The results are, however, more widely applicable to a range of problems in which magnetic fields are important.

For the former problem we employ a very simplified physical model of the jet acceleration process. We use time-dependent, spherically symmetric wind models in Newtonian and relativistic gravitational fields to ask whether the energy input rates required to produce the jet velocities observed in Young Stellar Objects (of about 2 $\times$ the escape velocity from the central object) can also produce Active Galactic Nuclei jet velocities (Lorentz factors of $\gamma \sim$ 10). Such a scaling would be expected if there is a common production mechanism for such jets. We demonstrate that such a scaling does exist, provided that the energy input process takes place sufficiently deep in the gravitational potential well, enabling physical use to be made of the speed of light as a limiting velocity, and provided that the energy released in the accretion process is imparted to a small fraction of the available accreting material.

For the latter problem we focus on developing accurate numerical methods for solving the equations of magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. The implementation of a `Smoothed Particle Magnetohydrodynamics' algorithm has previously been accompanied by numerous technical difficulties all of which are addressed at some level in this thesis in order to develop a robust and accurate method which can be applied to a wide range of problems of current theoretical interest. In the process we have undertaken a thorough review of the SPH method itself, from which several new results are derived. Amongst the technical issues addressed in the development of the SPMHD algorithm are the treatment of terms proportional to the divergence of the magnetic field in the MHD equations, the self-consistent formulation of the discrete equations from a variational principle, numerical stability of the algorithm and the self-consistent treatment of terms relating to the use of a spatially varying smoothing length. Considerable attention is paid to the ability of the algorithm to capture shocks for which artificial dissipation terms are formulated. Several methods are also examined for maintaining the divergence-free constraint in an SPMHD context. Perhaps most importantly the algorithm is benchmarked against a wide range of standard problems used to test recent high resolution shock-capturing grid-based MHD codes.

The entire thesis is available as: price_PhD.ps.gz (2.5Mb) or price_PhD.pdf (21.4Mb) or by chapter as below (as gzipped postscript or pdf). Postscript figures may be obtained by email request. All files are formatted for double sided printing. See also price_PhD_errata.pdf (25kb).

# Contents

• Declaration
• Acknowledgements
• Summary

• ### 2 Jet acceleration in YSOs and AGN (pp5-27) [ pdf (1.4Mb) ]

this chapter also published as a paper

• 2.1 Introduction
• 2.2 Non-relativistic (YSO) jets
• 2.2.1 Fluid equations
• 2.2.2 Numerical solution
• 2.2.3 Initial conditions
• 2.2.4 Results
• 2.2.6 Terminal wind velocities as a function of heating rate
• 2.3 Relativistic jets
• 2.3.1 Fluid equations
• 2.3.2 Scaling
• 2.3.3 Numerical Solution
• 2.3.4 Initial Conditions
• 2.3.5 Results
• 2.3.7 Terminal wind velocities and Lorentz factors as a function of heating rate
• 2.4 Discussion and Conclusions
• ### 3 Smoothed Particle Hydrodynamics (pp29-72) [ pdf (2.4Mb) ] [ Errata ]

• 3.1 Introduction
• 3.2 Basic formalisms
• 3.2.1 Interpolant
• 3.2.2 Errors
• 3.2.3 First derivatives
• 3.2.4 Second derivatives
• 3.2.5 Smoothing kernels
• 3.2.6 A general class of kernels
• 3.2.7 Kernel stability properties
• 3.3 Fluid Equations
• 3.3.1 Continuity equation
• 3.3.2 Equations of motion
• 3.3.3 Energy equation
• 3.3.4 Variable smoothing length terms
• 3.4 Alternative formulations of SPH
• 3.4.1 Variational principle
• 3.4.2 General alternative formulation
• 3.4.3 Ott & Schnetter formulation
• 3.5 Shocks
• 3.5.1 Artificial viscosity and thermal conductivity
• 3.5.2 Artificial dissipation switches
• 3.6 Timestepping
• 3.6.1 Predictor-corrector scheme
• 3.6.2 Reversible integrators
• 3.6.3 Courant condition
• 3.7 Numerical tests
• 3.7.1 Implementation
• 3.7.2 Propagation and steepening of sound waves
• 3.7.3 Sod shock tube
• 3.7.4 Blast wave
• 3.7.5 Cartesian shear flows
• 3.7.6 Toy stars
• 3.8 Summary
• ### 4 Smoothed Particle Magnetohydrodynamics (pp73-122) [ pdf (5.8Mb) ]

NB: parts of this chapter were published as two papers (paper I and paper II), although the thesis chapter presents a more up-to-date version of this work.

• 4.1 Introduction
• 4.2 Magnetohydrodynamics
• 4.2.1 Continuum equations
• 4.2.2 Conserved quantities
• 4.3 Smoothed Particle Magnetohydrodynamics
• 4.3.1 Induction equation
• 4.3.2 Equations of motion
• 4.3.3 Energy equation
• 4.3.4 Alternative formulations
• 4.3.5 Vector formulations of the magnetic force
• 4.3.6 Variable smoothing length terms
• 4.4 Stability
• 4.4.1 Anti-clumping term
• 4.4.2 Morris approach
• 4.4.3 Borve approach
• 4.4.4 Removing the constant component of magnetic field
• 4.5 Shocks
• 4.5.1 Artificial dissipation
• 4.5.2 Artificial dissipation switches
• 4.6 Numerical tests in one dimension
• 4.6.1 Implementation
• 4.6.3 Shock tubes
• 4.6.4 MHD waves
• 4.6.5 Magnetic toy stars
• 4.7 Summary
• ### 5 Multidimensional Smoothed Particle Magnetohydrodynamics (pp123-152) [ pdf (11Mb) ]

NB: A more up-to-date version of this work can be found here.

• 5.1 Introduction
• 5.2 Divergence correction techniques
• 5.2.1 Source term approach
• 5.2.2 Projection methods
• 5.2.3 Hyperbolic divergence cleaning
• 5.3 Numerical tests
• 5.3.1 Implementation
• 5.3.3 Circularly polarized Alfven wave
• 5.3.4 2.5D shock tube
• 5.3.5 Two dimensional shock tube
• 5.3.6 Spherical blast waves
• 5.3.7 Orszag-Tang vortex
• 5.4 Summary
• ### 6 Conclusions (pp153-157) [ pdf (33kb) ]

• 6.1 Summary
• 6.2 Future work: Applications
• 6.2.1 Star formation
• 6.2.2 Neutron star mergers
• 6.2.3 Accretion discs
• 6.3 Future work: Algorithms