Low Earth orbit (LEO) Mega-constellations have emerged as a transformative approach to realize enhanced system capacity and improved coverage to satisfy the ever-increasing global demand for data services. Subsequently, the high density of satellites in a confined orbital region poses challenges, including potential interference among neighboring satellites. Further, it is vital to adequately address the impacts of safety distances in satellite communication systems on ensuring proper operation, collision avoidance, and interference management. Inspired by these observations, this work proposes a novel analysis tool, the Poisson arc hardcore point process (PAHPP), by extending the traditional Poisson line hardcore point process to characterize the unique orbiting properties of the satellites in LEO mega-constellations, accounting for factors such as the orbit, the satellite density, and spatial distribution. Specifically, this paper presents the PAHPP by enforcing a minimum separation between satellites operating in the same circular orbit to reflect the practical LEO mega-constellations. The imposed minimum inter-satellite separation in the proposed PAHPP model has also been applied to multi-orbit multi-satellite communication cases. Moreover, the discretization approximation technique is employed to analyze system performance, focusing on serving distance and outage probability. Numerical results provide valuable insights and conclusions for uncovering and recognizing LEO mega-constellations.