# Critical Height of the Torus Instability in Two-Ribbon Solar Flares

## TL;DR

This study investigates the background magnetic field's role in two-ribbon solar flares, revealing that the critical height for torus instability differs between eruptive and confined flares, influencing their eruptive behavior.

## Contribution

It introduces a method to estimate the critical height for torus instability using potential field extrapolation and demonstrates its correlation with flare eruptiveness.

## Key findings

- Confined flares have higher critical heights than eruptive flares.
- The distribution of critical heights is bimodal, depending on flare type.
- Background magnetic field configuration significantly influences flare eruption potential.

## Abstract

We studied the background field for 60 two-ribbon flares of M-and-above classes during 2011--2015. These flares are categorized into two groups, i.e., \emph{eruptive} and \emph{confined} flares, based on whether a flare is associated with a coronal mass ejection or not. The background field of source active regions is approximated by a potential field extrapolated from the $B_z$ component of vector magnetograms provided by the Helioseismic and Magnetic Imager. We calculated the decay index $n$ of the background field above the flaring polarity inversion line, and defined a critical height $h_\mathrm{crit}$ corresponding to the theoretical threshold ($n_\mathrm{crit}=1.5$) of the torus instability. We found that $h_\mathrm{crit}$ is approximately half of the distance between the centroids of opposite polarities in active regions, and that the distribution of $h_\mathrm{crit}$ is bimodal: it is significantly higher for confined flares than for eruptive ones. The decay index increases monotonously with increasing height for 86\% (84\%) of the eruptive (confined) flares but displays a saddle-like profile for the rest 14\% (16\%), which are found exclusively in active regions of multipolar field configuration. Moreover, $n$ at the saddle bottom is significantly smaller in confined flares than that in eruptive ones. These results highlight the critical role of background field in regulating the eruptive behavior of two-ribbon flares.