Numerical $\texttt{AXP4}$ Simulations of Pulse Profiles for Binary Accreting X-ray Pulsars $-$ II: A Case Study of Centaurus X-3

The pulse shapes simulated in the accompanying paper Part $-$ I are compared with observations of a model binary accreting X-ray pulsar, Centaurus X-3. With known Cen X-3 inclination angles provided as input to the $\texttt{AXP4}$ code, the generated pulse profile is suitably compared with the corresponding observed energy-resolved $\textit{AstroSat}$/LAXPC pulse profile. The pulsed fraction is proposed as a robust, quantitative measure for estimating the size of the emission region of Centaurus X-3 by extending the simulations to include spherical caps of varying fractional surface coverage of the neutron star $-$ over the full range of $0-100\%$, up to very large caps (with polar half angle $> 30^{\circ}$). The hotspot radius thus derived drops by an order of magnitude from $12.27$ km to $1.36^{+0.29}_{-0.26}$ km, within the ballpark of the standard model value of $\sim$$1$ km, after including the effect of gravitational light bending, lending further weight to its routine emphasis in the literature. The energy- and luminosity-dependence of the composite gravitationally bent and slab-integrated pulse profiles is further studied. As the pulse profile is sensitive to luminosity variations, the correlation of the size of a finite polar cap and its dependence on X-ray luminosity $-$ through the rate and subsequently, the geometry of accretion $-$ is discussed. Although a single, model pulsar was chosen for this work to exhibit the depth of the physical and astrophysical prospects of such a probe, this exercise can be extended to a wide range of existing X-ray pulse profiles of known binary accreting pulsars available in galactic catalogs, especially, with the possible inclusion of accretion columns (with cylindrical co-ordinate transformation) in the future.