To model existing or future low Earth orbit (LEO) satellite networks leveraging multiple constellations, we propose a simple analytical approach to represent the clustering of satellites on orbits. More precisely, we develop a variable-altitude Poisson orbit process that effectively captures the geometric fact that satellites are always positioned on orbits, and these orbits may vary in altitude. Conditionally on the orbit process, satellites situated on these orbits are modeled as linear Poisson point processes, thereby forming a Cox point process. For this model, we derive useful statistics, including the distribution of the distance from the typical user to its nearest visible satellite, the outage probability, the Laplace functional of the proposed Cox satellite point process, and the Laplace transform of the interference power from the Cox-distributed satellites under general fading. The derived statistics enable the evaluation of the performance of such LEO satellite communication systems as functions of network parameters.