Source-Free Domain Adaptive Object Detection (SF-DAOD) transfers knowledge acquired from the labeled source domain to the unlabeled target domain while preserving data privacy by restricting access to source data during adaptation. Existing approaches predominantly leverage the Mean Teacher framework for self-training in the target domain. The Exponential Moving Average (EMA) mechanism in Mean Teacher stabilizes training by averaging the student weights over training steps. However, in domain adaptation, its inherent lag in responding to emerging knowledge can hinder the student's rapid adaptation to target-domain shifts. To address this challenge, we propose the Dual-rate Dynamic Teacher (DDT) with an Asynchronous EMA (AEMA), which implements group-wise parameter updates. Unlike conventional EMA, which synchronously updates all parameters, AEMA dynamically partitions teacher parameters into two functional groups based on the contribution to capture the target domain shift. By applying a distinct smoothing coefficient to these groups, AEMA enables simultaneous fast adaptation and historical knowledge retention. Comprehensive experiments conducted on three widely used traffic benchmarks have demonstrated that the proposed DDT achieves superior performance, outperforming the state-of-the-art methods by a clear margin. The codes are available at https://anonymous.4open.science/r/Dual-Rate-Dynamic-Teacher-for-Source-Free-Domain-Adaptive-Object-Detection-17BF.