Variational problems in astrophysics

Presented in this thesis are three distinct astrophysical problems that are solved variationally. The first problem establishes the solar radial differential rotation in the outer third of the convection zone. Included is a means for determining the random component of the noise from the data itself; a new procedure for performing the inversion; and a method for using this knowledge in ascertaining the statistical significance of the result. Simulations are employed to illustrate the procedure. Applying this approach to existing data, we conclude that the sun's equatorial rotation rate decreases between 10 and 30 Mm below the temperature minimum. The implications of this finding to our understanding of solar structure are examined.

The second problem exploits line profile data to ascertain the rotational structure of astronomical objects. Analyzed in detail are radio observations of ammonia line inversions for studying protostar formation in dense molecular cloud cores. A new method for solving this problem is presented, along with simulations that demonstrate the power of this technique. Applications of this approach to the molecular cloud core G10.6-0.4 demonstrates a statistically significant infall velocity within the central 0.3 pc, and some knowledge of the rotational velocity is deduced. The majority of the discussion concerns future experiments and how they can be carried out so as to improve the statistical significance of the results.

The final problem applies variational techniques to finding the stable states of dynamical systems. Rather than explicitly integrating the equations of motion, which dictate the relaxational time scale, we opt to minimize a Lyapunov functional which characterizes the physical system. The induced dynamics are not necessarily physically realizable, but the stable states are guaranteed to be identical with those obtained by integrating the equations of motion. The advantage of this alternative approach to the problem is that the dynamical timescale has become decoupled from the solution; thus much faster convergence can be realized. As a secondary issue, we discuss how this type of problem can be efficiently implemented on a parallel processing computer. Finding the stable patterns in Rayleigh-Benard convection is used to illustrate the ideas. Applications to astrophysical problems are described.