On Approximation Methods in the Study of Boson Stars
  We analyze the accuracy of the variational method in computing physical
quantities relevant for gravitationally bound Bose-Einstein condensates. Using
a variety of variational ans\"atze found in existing literature, we determine
physical quantities and compare them to exact numerical solutions. We conclude
that a "linear+exponential" wavefunction proportional to $(1 + \xi)\exp(-\xi)$
(where $\xi$ is a dimensionless radial variable) is the best fit for attractive
self-interactions along the stable branch of solutions, while for small
particle number $N$ it is also the best fit for repulsive self-interactions.
For attractive self-interactions along the unstable branch, a single
exponential is the best fit for small $N$, while a sech wavefunction fits
better for large $N$. The Gaussian wavefunction ansatz, which is used often in
the literature, is exceedingly poor across most of the parameter space, with
the exception of repulsive interactions for large $N$. We investigate a "double
exponential" ansatz with a free constant parameter, which is computationally
efficient and can be optimized to fit the exact solutions in different limits.
We show that the double exponential can be tuned to fit the sech ansatz, which
is computationally slow. We also show how to generalize the addition of free
parameters in order to create more computationally efficient ans\"atze using
the double exponential. Determining the best ansatz, according to several
comparison parameters, will be important for analytic descriptions of dynamical
systems. Finally, we examine the underlying relativistic theory, and critically
analyze the Thomas-Fermi approximation often used in the literature.
