# Estimating magnetic filling factors from Zeeman-Doppler magnetograms

## TL;DR

This study compares magnetic flux measurements from Zeeman-Doppler imaging and Zeeman broadening in low-mass stars, establishing a relationship to better estimate magnetic filling factors and their relation to stellar activity and rotation.

## Contribution

It introduces a power law relationship between ZDI and Zeeman broadening measurements, enabling improved estimation of magnetic filling factors from ZDI data alone.

## Key findings

- ZDI magnetic flux correlates with activity-rotation regimes.
- A power law links ZDI flux to Zeeman broadening flux.
- Estimated filling factors align with activity-rotation trends.

## Abstract

Low-mass stars are known to have magnetic fields that are believed to be of dynamo origin. Two complementary techniques are principally used to characterise them. Zeeman-Doppler imaging (ZDI) can determine the geometry of the large-scale magnetic field while Zeeman broadening can assess the total unsigned flux including that associated with small-scale structures such as spots. In this work, we study a sample of stars that have been previously mapped with ZDI. We show that the average unsigned magnetic flux follows an activity-rotation relation separating into saturated and unsaturated regimes. We also compare the average photospheric magnetic flux recovered by ZDI, $\langle B_V\rangle$, with that recovered by Zeeman broadening studies, $\langle B_I\rangle$. In line with previous studies, $\langle B_V\rangle$ ranges from a few % to $\sim$20% of $\langle B_I\rangle$. We show that a power law relationship between $\langle B_V\rangle$ and $\langle B_I\rangle$ exists and that ZDI recovers a larger fraction of the magnetic flux in more active stars. Using this relation, we improve on previous attempts to estimate filling factors, i.e. the fraction of the stellar surface covered with magnetic field, for stars mapped only with ZDI. Our estimated filling factors follow the well-known activity-rotation relation which is in agreement with filling factors obtained directly from Zeeman broadening studies. We discuss the possible implications of these results for flux tube expansion above the stellar surface and stellar wind models.