Modeling the Galactic Compact Binary Neutron Star Population and Studying the Double Pulsar System

In this dissertation, we estimate the population of different classes of BNS systems that are visible to gravitational-wave observatories. Given that no ultra-compact BNS systems have been discovered in pulsar radio surveys, we place a 95\% confidence upper limit of $\sim$850 and $\sim$1100 ultra-compact neutron star--white dwarf and double neutron star (DNS) systems that are beaming towards the Earth, respectively. We show that among all of the current radio pulsar surveys, the ones at the Arecibo radio telescope have the best chance of detecting an ultra-compact BNS system. We also show that adopting a survey integration time of $t_{\rm int} \sim 1$~min will maximize the signal-to-noise ratio, and thus, the probability of detecting an ultra-compact BNS system. Similarly, we use the sample of nine observed DNS systems to derive a Galactic DNS merger rate of $\mathcal{R}_{\rm MW} = 37^{+24}_{-11}$~Myr$^{-1}$, where the errors represent 90\% confidence intervals. Extrapolating this rate to the observable volume for LIGO, we derive a merger detection rate of $\mathcal{R} = 1.9^{+1.2}_{-0.6} \times \left(D_{\rm r}/100 \ \rm Mpc \right)^3 \rm yr^{-1}$, where $D_{\rm r}$ is the range distance for LIGO. This rate is consistent with that derived using the DNS mergers observed by LIGO. Finally, we measure the sense of rotation of the older millisecond pulsar, pulsar A, in the DNS J0737--3039 system and find that it rotates prograde with respect to its orbit. This is the first direct measurement of the sense of rotation of a pulsar and a direct confirmation of the rotating lighthouse model for pulsars. This result confirms that the spin angular momentum vector is closely aligned with the orbital angular momentum, suggesting that kick of the supernova producing the second born pulsar J0737--3039B was small.