Secure aggregation is a common technique in federated learning (FL) for protecting data privacy from both curious internal entities (clients or server) and external adversaries (eavesdroppers). However, in dynamic and resource-constrained environments such as low Earth orbit (LEO) satellite networks, traditional secure aggregation methods fall short in two aspects: (1) they assume continuous client availability while LEO satellite visibility is intermittent and irregular; (2) they consider privacy in each communication round but have overlooked the possible privacy leakage through multiple rounds. To address these limitations, we propose LTP-FLEO, an asynchronous FL framework that preserves long-term privacy (LTP) for LEO satellite networks. LTP-FLEO introduces (i) privacy-aware satellite partitioning, which groups satellites based on their predictable visibility to the server and enforces joint participation; (ii) model age balancing, which mitigates the adverse impact of stale model updates; and (iii) fair global aggregation, which treats satellites of different visibility durations in an equitable manner. Theoretical analysis and empirical validation demonstrate that LTP-FLEO effectively safeguards both model and data privacy across multi-round training, promotes fairness in line with satellite contributions, accelerates global convergence, and achieves competitive model accuracy.