Quadratic shift-and-stack for Ground-Based Optical Detection of Faint Cislunar Objects

Detecting faint objects in cislunar space using ground-based optical telescopes is difficult because of their low brightness, strong lunar background, and complex, nonlinear apparent motion. Traditional shift-and-stack techniques based on linear motion assumption suffer signal trailing loss due to significant nonlinear motion during long integrations, thus producing a degraded signal-to-noise ratio (SNR). In this paper, we first derive a theoretical criterion based on the point spread function to determine the maximum applicable integration time for linear-motion stacking. We then propose a quadratic shift-and-stack (QSS) method to correct for the first-order nonlinear motion, namely the angular acceleration of cislunar targets. Simulations of typical cislunar orbits verify this theoretical criterion and show that the QSS method significantly improves SNR from stacking and can enhance the detection limit by up to 1 stellar magnitude compared with the linear-motion stacking method. Furthermore, tests using observational data of the cislunar object Tiandu-1 confirm that while linear stacking degrades after a 29-minute integration due to trajectory curvature, the QSS method achieves continuous SNR improvement over a 46-minute integration, outperforming the peak SNR of the linear method by 31%.

💡 Research Summary

This paper addresses a significant challenge in space situational awareness: the ground-based optical detection of faint objects in cislunar space (the region between Earth and the Moon). These objects, such as satellites and debris, appear extremely dim and are often near the bright lunar background. Furthermore, their apparent motion across the sky is highly nonlinear due to the complex gravitational interplay of the Earth-Moon system. The conventional shift-and-stack technique—a standard method for detecting faint moving objects by aligning and co-adding short-exposure frames—assumes linear (constant-velocity) motion. When applied to cislunar objects, this assumption breaks down. Uncorrected angular acceleration causes the target’s signal to smear across multiple pixels during stacking, leading to “trailing loss” that ultimately degrades the signal-to-noise ratio (SNR) as integration time increases, imposing a hard limit on detectability.

The authors make two key contributions. First, they derive a theoretical criterion to predict when the linear-motion assumption fails. Based on the width of the telescope’s point-spread function (PSF), specifically its Full Width at Half Maximum (FWHM), they calculate the maximum integration time (T_linear) for which linear shift-and-stack remains effective. This time limit is inversely proportional to the square root of the target’s angular acceleration, providing a practical guideline for observers.

Second, and as the core innovation, they propose the Quadratic Shift-and-Stack (QSS) method. QSS models the object’s apparent motion as a quadratic function of time, explicitly incorporating angular acceleration parameters alongside velocity. During the stacking process, each frame is shifted not only by a velocity-dependent offset but also by an acceleration-dependent correction term (1/2 * a * t^2). This allows the signal from all frames to be accurately aligned onto a single, compact PSF, even over long integration periods, thereby eliminating trailing loss and restoring the ideal √N SNR growth with the number of frames.

The method is rigorously validated. Extensive numerical simulations of four representative cislunar orbits (Distant Retrograde Orbit, Earth-Moon resonant orbit, L2 halo orbit, and lunar orbit) demonstrate that QSS successfully overcomes the SNR plateau and decline seen in linear stacking. It is shown to potentially improve the detection limit by up to 1 stellar magnitude. The simulations also confirm the validity of the derived T_linear criterion.

Finally, the method is tested on real observational data of the Chinese cislunar test object Tiandu-1, acquired by the Multi-Application Survey Telescope Array (MASTA). The results are conclusive: while the linear stacking method reached its peak SNR (16.77) at 29 minutes of integration and then degraded, the QSS method achieved continuous SNR improvement up to 46 minutes, attaining a final SNR of 21.98—a 31% improvement over the linear method’s peak performance.

In conclusion, this research provides a crucial advancement for optical surveillance in the dynamically complex cislunar domain. By moving from a linear to a quadratic motion model, the QSS method effectively mitigates a fundamental source of signal degradation, enabling deeper detection of faint objects and enhancing future Cislunar Space Situational Awareness (CSSA) capabilities.