Empirical Brightening Model for Distant Long-Period Comets

Abstract

Recent studies have shown the traditional comet brightening model, $Hn$, with a constant activity index $n$ (i.e., single slope in $\log_{10}(r_{\rm h})$), fails to accurately describe heliocentric lightcurves at large distances, where water-ice sublimation is inefficient. We present a new empirical model, $Hab$, for distant comet brightening in which the effective brightening slope varies with heliocentric distance, $r_{\rm h}$. Using a multi-year pre-perihelion dataset from uniform instrumentation for 23 comets, we found that $Hab$ provides the best match to long inbound arcs spanning a wide range in $r_{\rm h}$. The best-fit coefficients $a$ and $b$ cluster tightly for our best-observed comets, suggesting a common brightening behavior; we therefore also tested one- and two-parameter $Hab$ variants in which $a$ and/or $b$ are fixed to anchor values derived from these comets ($A_1 = 0.40 \pm 0.09$, $B_1 = -1.78 \pm 0.68$), either individually, jointly, or via a linear relation. The three-parameter $Hab$ model is strongly favored by the Bayesian information criterion (BIC) for the full lightcurve, justifying its added complexity for characterization. We also tested predictive performance against $Hn$. For brightness forecasting from limited initial data ($Δr_{\rm h} \lesssim 2.5$ au, the heliocentric distance range spanned by observations), $Hab$ itself predicts poorly, whereas a one-parameter reduction, with both coefficients fixed, predicts more reliably, including outperforming $Hn$ with a fixed $n \sim 2-4$. Two-parameter reductions with only one coefficient fixed typically reach predictions within 1 mag of the observed brightness after only $Δr_{\rm h} \sim 1.0$ au of inbound coverage.