Low Earth orbit (LEO) satellites are capable of gathering abundant Earth observation data (EOD) to enable different Internet of Things (IoT) applications. However, to accomplish an effective EOD processing mechanism, it is imperative to investigate: 1) the challenge of processing the observed data without transmitting those large-size data to the ground because the connection between the satellites and the ground stations is intermittent, and 2) the challenge of processing the non-independent and identically distributed (non-IID) satellite data. In this paper, to cope with those challenges, we propose an orbit-based spectral clustering-assisted clustered federated self-knowledge distillation (OSC-FSKD) approach for each orbit of an LEO satellite constellation, which retains the advantage of FL that the observed data does not need to be sent to the ground. Specifically, we introduce normalized Laplacian-based spectral clustering (NLSC) into federated learning (FL) to create clustered FL in each round to address the challenge resulting from non-IID data. Particularly, NLSC is adopted to dynamically group clients into several clusters based on cosine similarities calculated by model updates. In addition, self-knowledge distillation is utilized to construct each local client, where the most recent updated local model is used to guide current local model training. Experiments demonstrate that the observation accuracy obtained by the proposed method is separately 1.01x, 2.15x, 1.10x, and 1.03x higher than that of pFedSD, FedProx, FedAU, and FedALA approaches using the SAT4 dataset. The proposed method also shows superiority when using other datasets.