The rotation-magnetism relationship in solar-type stars. Constraining magnetic flux emergence rates

TL;DR

This study tackles how the emergence rate of magnetic flux in solar-type stars depends on rotation and how this drives the observed rotation–magnetism relation. Using the FEAT framework, which couples buoyant flux-tube rise, flux emergence, and surface flux transport, the authors decompose the surface magnetic field into a rotation-invariant small-scale dynamo component and a rotation-dependent large-scale, active-region component, then compare predictions with direct Zeeman measurements and spectropolarimetric data. They find that the emerging flux must scale steeply with rotation, with a power-law exponent of $p \approx 1.91\pm 0.02$, to match observed mean fields after correcting for metallicity and effective temperature; at rapid rotation, active-region fields can dominate the surface flux, while the SSD remains substantial at slower rotation. The results highlight metallicity and $T_{ m eff}$ as key systematic factors in the rotation–magnetism relationship, emphasizing the need for homogeneous samples or parameter corrections in stellar dynamo modelling and magnetic activity diagnostics.

Abstract

The rotation-activity relationship of G-type stars results from surface magnetic fields emerging from the interior. How the magnetic flux and its emergence rate scale with rotation rate are not well understood, both observationally and theoretically. We aim at constraining the emerging magnetic flux as a function of the rotation rate in solar-type stars by numerical simulations compared to empirical constraints set by direct measurements of stellar magnetic fields. We use our Flux Emergence And Transport (FEAT) model for stars with a range of power-law slopes for the dependence of emerging flux on rotation. Complementing this with a heuristic account of the main flux components, we model the resulting mean unsigned field strength as a function of the rotation rate. We compare the results with the Zeeman-intensification measurements and spectropolarimetric data of solar-type stars. Deviations of the model from observations of G stars correlate strongly with stellar metallicity ($r=0.83$) and effective temperature ($r=-0.76$), with a combined coefficient of 0.90, reflecting the dependence of magnetic activity on these two parameters. Correcting for these effects with multilinear regression, we find that magnetic flux emergence rates must scale steeply with rotation power-law exponent of about 1.9) to reproduce observed field strengths, significantly exceeding the estimates in the literature. We also provide correction factors for metallicity and temperature for measurements of early-G-type stellar magnetic fields. Stellar magnetic flux emergence rates scale steeply with rotation, requiring active-region fields to dominate the total surface flux on rapid rotators, whereas small-scale-dynamo fields dominate for slow rotators like the Sun. Metallicity significantly influences the rotation-magnetism relationship, necessitating sample-dependent corrections for accurate stellar dynamo modelling.

Figures (10)

• Figure 1: The rotation-rate dependence of the globally averaged unsigned field strength, modelled with FEAT simulations for solar-type stars (coloured diamonds with the flux injection exponent $p$ indicated in legend), scaling relations based on the solar simulation at $P_{\rm rot}=25$ d (dashed curves with the scaling exponent $q$ indicated in legend), in comparison to direct measurements using Zeeman intensification by Kochukhov20 for G stars (G8 and K2 types shown by red symbols) and Zeeman broadening for K5-M0 stars by Reiners22 -- see the legend in the figure. The dotted curve shows a least-squares fit to the G-star data (Sect. \ref{['ssec:scalings']}). Error bars denote the observed upper and lower margins for measurements and the $1\sigma$ levels for the simulations averaged over 100 d. The upper and lower horizontal lines denote the solar mean field level as observed by Kochukhov20 near the minimum and the mean SSD field strength adopted by Reiners22, respectively. The inset plot shows the mean SFT field for $p=1$ and $p=2$.
• Figure 2: Deviation of the observed mean field strength of the Kochukhov20 sample from the modelled values with $q=1.9$ in Eq. (\ref{['eq:Btotcurves']}), as a function of metallicity (left panel) and the effective temperature difference with respect to the Sun (right panel). The vertical error bars denote the error bounds given by the referred study, and the horizontal ones the standard deviation of the mean [Fe/H] and $T_{\rm eff}$ from the PASTEL catalogue PASTEL_Soubiran16. The colour scale shows the temperature (left panel) and metallicity (right panel). The regression lines are shown in red. The correlation coefficient and the $p$ values are given above each frame.
• Figure 3: Comparison between [Fe/H] estimates by Hahlin23 based on six NIR lines with magnetic-field inference and PASTEL averages. The Pearson coefficient $r$, rms and the mean differences are indicated. The lower panel shows the individual differences of the NIR estimates from PASTEL values.
• Figure 4: Correlation between effective temperature deviation and metallicity in our stellar sample ($r = 0.614$, $p = 0.034$). Points are coloured by the magnetic field residual $\Delta B = \langle B \rangle_{\rm obs} - \langle B \rangle_{\rm model}$. The moderate correlation between predictors creates multicollinearity in the multiple regression, inflating coefficient uncertainties. The color gradient shows that both stellar parameters influence $\Delta B$: metal-rich, cooler-than-solar stars (upper-right) show the largest positive residuals, while metal-poor, hotter stars show negative residuals.
• Figure 5: Corrected mean magnetic field strength as a function of the rotation period following temperature and metallicity corrections (Eq. \ref{['eq:Bcorr']}). See the caption of Fig. \ref{['fig:rot-b']} for the definitions.