The sixth-generation (6G) wireless network is conceptualized to provide ubiquitous and reliable network access through effective inter-networking among space, air, and terrestrial networks, while posing considerable pressure on dynamic network orchestration. Digital twin (DT) provides an alternative approach to proactively make real-time resource allocation by mapping and learning the complex network topology. However, the dual challenges of limited energy capacity and insufficient computing power of unmanned aerial vehicles make it difficult to establish digital twin on aerial networks. In light of this, in this paper, we propose a lightweight DT empowered air-ground network architecture, where the DT modelling task is distributed to diverse ground devices based on federated learning. To improve the efficiency of DT modelling, we design a distributed incentive mechanism that incentivizes high-performance ground devices to take part in federated learning. Considering the computing burden and possible private disclosure caused by the incentive, we further solve the incentive scheme with a new distributed algorithm. The optimal solution, such as the local training rounds and required accuracy of terminal devices, could be calculated in parallel on ground devices. Simulation results show the effectiveness of the proposed lightweight DT scheme in terms of energy consumption and model accuracy.