Low Earth Orbit (LEO) satellite networks connect millions of devices on Earth and offer various services, such as data communications, remote sensing, and data harvesting. As the number of services increases, LEO satellite networks will continue to grow, and many LEO satellite network operators will share the same spectrum resources. We aim to examine the coexistence of such a future heterogeneous LEO satellite network by proposing a tractable spatial model and analyzing the basic performance of downlink communications. We model a heterogeneous LEO satellite network as Cox point processes, ensuring satellites are located on various orbits. Then, we analyze two different access technologies for such a heterogeneous satellite network: closed access and open access. For both cases, we derive the coverage probability and prove that the coverage probability of the open access scenario outperforms that of the closed access, and this coverage enhancement applies to all users. By providing essential network performance statistics as key distributional parameters and by presenting the fact that the Cox point process approximates a forthcoming future constellation with slight variation, the developed framework and analysis will serve as a tractable instrument to design, evaluate, and optimize heterogeneous LEO satellite networks with numerous constellations.