Quantification of Sub-Solar Star Ages from the Symmetry of Conjugate Histograms of Spin Period and Angular Velocity

Abstract

Empirical laws proposed for the decline in star spin with time have heretofore been tested using ambiguous fitting models. We develop an analytical inverse model that uses histogram data to unequivocally determine the physical law governing how dwarf star spin depends on time (t) and mass (M). We analyze shapes of paired histograms of axial rotation period (П) and angular velocity (ω = 2π/П) to utilize the fact that a variable and its reciprocal are governed by the same physics. Copious data on open clusters are used to test the formula ∂ω/∂t ∝ − ωn where n is unrestricted, and thus covers diverse possibilities. Histogram conjugates for each of 15 clusters with 120 to 812 measurements provide n = 1.13 ± 0.19. Results are independent of initial spin rate, bin size, cluster parameters, and star mass. Notably, 11 large clusters with mostly M-types yield fits with n = 1.07 ± 0.12. Associations behave similarly. Only exponential decay (n = 1) explains the similar shapes of the conjugate histograms for the spin period and angular velocity, despite the asymmetric (inverse) relationship of these variables. This rate law is consistent with viscous dissipation. Forward modeling confirms that n is near unity and further shows that coeval formation of all stars in a cluster does not occur. We therefore explore a constant rate of star production, which is reasonable for tiny stars. Inverse models show that episodic production increases with mass, but is unimportant below ~0.55 MSun. We infer star and cluster ages, and find that star production becomes less regular with time, as interstellar gas and dust are progressively depleted. Our new analytical approach of extracting a physical law from conjugate histograms is general and widely applicable.