Measuring stellar magnetic helicity density

Helicity is a fundamental property of a magnetic field but to date it has only been possible to observe its evolution in one star – the Sun. In this paper, we provide a simple technique for mapping the large-scale helicity density across the surface of any star using only observable quantities: the poloidal and toroidal magnetic field components (which can be determined from Zeeman–Doppler imaging) and the stellar radius. We use a sample of 51 stars across a mass range of 0.1–1.34 M⊙ to show how the helicity density relates to stellar mass, Rossby number, magnetic energy, and age. We find that the large-scale helicity density increases with decreasing Rossby number Ro, peaking at Ro ≃ 0.1, with a saturation or decrease below that. For both fully and partially convective stars, we find that the mean absolute helicity density scales with the mean squared toroidal magnetic flux density according to the power law: $|\langle {h\, }\rangle |$ ∝ $\langle {\rm {{\it B}_{tor}}^2_{}\, \rangle }^{0.86\, \pm \, 0.04}$. The scatter in this relation is consistent with the variation across a solar cycle, which we compute using simulations and observations across solar cycles 23 and 24, respectively. We find a significant decrease in helicity density with age.