With central energy densities many times the density of atomic nuclei, and relatively low temperatures, neutron stars may be some of the best objects in nature from which we are able to study the properties of ultra-dense, cold nuclear matter. With advancements in observational astronomy, more high quality observed data is available than ever before. This study aims to establish a phenomenological model of the macroscopic properties of neutron stars which, when compared with observational data, can give some insight into the physics of the ultra-dense nuclear matter thought to exist in the cores of neutron stars, and help to provide a constraint on the poorly understood equation of state thereof. One example of a well documented neutron star is the Demorest pulsar PSR J1614-2230, which has a well measured mass of 1.97 ± 0.04 M_. Just from this mass measurement, we can set limits on the radius, central energy-density, and rotation for this pulsar. A parameter scan of neutron star properties is important because it allows us to establish limits on the possible exotic physical processes at work in the extreme environments of neutron star cores. Using the methods described in this thesis, it is possible to explore a wide range of possible physical limits to be set when studying the physics of neutron stars. The equation of state of ultra-dense matter is of key importance for determining the properties of neutron stars, however this equation of state is almost completely unknown. To produce the true equation of state for neutron stars constitutes a very complicated, many-body problem consisting of some 10__ particles. In addition, the nature of the matter existing at the core is a mystery. The basic building blocks of the ultra-dense matter are uncertain, and possible first order phase-transitions may exist. There are competing theoretical frameworks ranging from Schroedinger based, classical considerations, to relativistic quantum theory such as Dirac, to relativistic mean-field theory. These challenges have kept the equation of state of ultra-dense matter in neutron star cores from being fully realized thus far. For this reason, we employ a variational ansatz based on the minimal physical assumptions that the matter in the core of neutron stars exhibits microscopic stability, that causality is not violated, and Einstein's theory of general relativity is the correct theory of gravity. From this ansatz, we create a large number of physically allowed equations of state. Since we are not confined by the parameters of any one particular equation of state model, we have the freedom to run a complete parameter scan of allowed neutron star properties. The parameters established in this thesis give a wide range of physical constraints on the macroscopic properties of neutron stars.