The low Earth orbit (LEO) satellite constellation in the terahertz (THz) band is one of the promising technologies to provide global coverage and high-rate transmission to the airplane, whose key performance evaluation is required before realistic deployment. In this paper, we consider a downlink THz LEO satellite-airplane network with directional antennas, where the satellites follow a binomial point process on a spherical surface. Incorporating the distinctive THz propagation properties and directional beams in the two scenarios with/without molecular absorption noise, we propose an analytical framework to evaluate the coverage probability and average achievable rate of an arbitrary airplane. The analytical results are expressed in terms of the Laplace transforms of the interference-plus-molecular absorption noise and interference. Specifically, considering a frequency reuse strategy, we first provide the probability distributions of the distances from an arbitrary satellite, the serving satellite and any non-serving satellite to the target airplane, and calculate the probabilities of whether an interfering satellite is blocked by the Earth. Numerical results verify the accuracy of the analytical expressions and show that fewer orthogonal channels and lower LEO satellite altitude lead to better coverage and rate performance. As a key parameter in network design, the number of satellites has two competing effects on coverage and rate, and an optimum satellite number exists. Overall, this study provides theoretical guidance for deploying and planning THz satellite constellations.