The infrared lock on range equation: non-linear interplay of source, atmosphere and sensor

Abstract Infrared signature reduction is widely pursued under the assumption that lowering target radiant intensity directly reduces detection range. However, infrared detection is governed by a nonlinear interaction between target emission, atmospheric attenuation, background accumulation and sensor sensitivity. This study revisits the lock-on range problem by formulating the detection condition as a transcendental implicit equation derived from first-principles radiometric considerations. Analysis reveals the existence of a geometric contrast limit that imposes an absolute upper bound on detectability, independent of sensor gain. Sensitivity analysis further shows that detection-range responsiveness decreases progressively with increasing optical depth, leading to diminishing returns from both infrared signature reduction and sensor performance improvements in a strongly attenuating environment. Numerical evaluation using representative infrared search-and-track and missile-seeker systems confirms the transition between signal-limited and atmosphere-limited detection regimes. The results are further interpreted from both seeker designer and stealth designer perspectives, providing practical insight into how atmospheric conditions govern the effectiveness of sensor improvements and infrared signature suppression.