Evolving the Loeb Scale

TL;DR

The static Loeb scale provides a single-epoch anomaly-to-level mapping for interstellar objects, but fails to anticipate how observations will evolve as an ISO nears Earth. We formulate a radially evolving score by promoting the composite anomaly $S$ to a function of heliocentric distance via the relaxation equation $\frac{d S_{\mathrm{eff}}}{d r} = \frac{S_{\mathrm{inst}}(r) - S_{\mathrm{eff}}(r)}{L}$ with $S_{\mathrm{inst}}(r) = \sum_i w_i m_i(r) + \sum_{i<j} w_{ij} m_i(r) m_j(r)$ and distance-dependent metrics $m_i(r; \theta_i)$; this structure embeds memory and hysteresis, enabling early, stable forecasts of the Loeb level. Forecasts at Earth, $S_{\oplus} = S_{\mathrm{eff}}(r_{\oplus})$, are obtained by Green’s-function solutions and forward Monte Carlo over parameter posteriors $P(\theta|D)$ to quantify uncertainty. The framework supports automated monitoring pipelines, adaptive priors as population statistics tighten, and reinterpretation of anomalies for Earth-orbit or human-made objects, making it a versatile tool for rapid technosignature assessment and planetary defense planning.

Abstract

We develop a differential formulation of the Loeb Scale that extends the original static framework into a radially evolving, real time classification scheme for interstellar objects. By promoting each anomaly metric to a function of heliocentric distance and introducing a relaxation equation for the effective score, our method incorporates memory, hysteresis and predictive capability. This allows us to have early, stable forecasts of an object's eventual Loeb level based on sparse data obtained at large distances, which is more helpful to quantify its true nature when near Earth.