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Until now, how the magnetic fields in M/X-class flaring active regions (ARs) differ from C-class flaring ARs remains unclear. Here, we calculate the key magnetic field parameters within the area of high photospheric free energy density (HED region) for 323 ARs (217 C- and 106 M$/$X-flaring ARs), including total photospheric free magnetic energy density E$_{free}$, total unsigned magnetic flux $\Phi$$_{HED}$, mean unsigned current helicity h$_{c}$, length of the polarity inversion lines $L$$_{PIL}$ with a steep horizontal magnetic gradient, etc., and compare these with flare/coronal mass ejection (CME) properties. We first show the quantitative relations among the flare intensity, the eruptive character and $\Phi$$_{HED}$. We reveal that $\Phi$$_{HED}$ is a measure for the GOES flux upper limit of the flares in a given region. For a given $\Phi$$_{HED}$, there exists the lower limit of F$_\mathrm{SXR}$ for eruptive flares. This means that only the relatively strong flares with the large fraction of energy release compared to the total free energy are likely to generate a CME. We also find that the combinations of E$_{free}$-$L$$_{PIL}$ and E$_{free}$-h$_{c}$ present a good ability to distinguish between C-class and M$/$X-class flaring ARs. Using determined critical values of E$_{free}$ and $L$$_{PIL}$, one predicts correctly 93 out of 106 M/X-class flaring ARs and 159/217 C-class flaring ARs. The large $L$$_{PIL}$ or h$_{c}$ for M$/$X-class flaring ARs probably implies the presence of a compact current with twisted magnetic fields winding about it., Comment: ApJ accepted for publication. 16 pages, 4 figures

With the aim of investigating how the magnetic field in solar active regions (ARs) controls flare activity, i.e., whether a confined or eruptive flare occurs, we analyze 106 flares of Geostationary Operational Environmental Satellite (GOES) class $\geq$M1.0 during 2010$-$2019. We calculate mean characteristic twist parameters $\alpha$$_{FPIL}$ within the "flaring polarity inversion line" region and $\alpha$$_\mathrm{HFED}$ within the area of high photospheric magnetic free energy density, which both provide measures of the nonpotentiality of AR core region. Magnetic twist is thought to be related to the driving force of electric current-driven instabilities, such as the helical kink instability. We also calculate total unsigned magnetic flux ($\Phi$$_\mathrm{AR}$) of ARs producing the flare, which describes the strength of the background field confinement. By considering both the constraining effect of background magnetic fields and the magnetic non-potentiality of ARs, we propose a new parameter $\alpha$/$\Phi$$_\mathrm{AR}$ to measure the probability for a large flare to be associated with a coronal mass ejection (CME). We find that in about 90\% of eruptive flares, $\alpha$$_\mathrm{FPIL}$/$\Phi$$_\mathrm{AR}$ and $\alpha$$_\mathrm{HFED}$/$\Phi$$_\mathrm{AR}$ are beyond critical values (2.2$\times$$10^{-24}$ and 3.2$\times$$10^{-24}$ Mm$^{-1}$ Mx$^{-1}$), whereas they are less than critical values in $\sim$ 80\% of confined flares. This indicates that the new parameter $\alpha$/$\Phi$$_\mathrm{AR}$ is well able to distinguish eruptive flares from confined flares. Our investigation suggests that the relative measure of magnetic nonpotentiality within the AR core over the restriction of the background field largely controls the capability of ARs to produce eruptive flares., Comment: 7 pages, 4 figures, 1 table and 1 online database

With the aim of understanding how the magnetic properties of active regions (ARs) control the eruptive character of solar flares, we analyze 719 flares of Geostationary Operational Environmental Satellite (GOES) class $\geq$C5.0 during 2010$-$2019. We carry out the first statistical study that investigates the flare-coronal mass ejections (CMEs) association rate as function of the flare intensity and the AR characteristics that produces the flare, in terms of its total unsigned magnetic flux ($\Phi$$_{AR}$). Our results show that the slope of the flare-CME association rate with flare intensity reveals a steep monotonic decrease with $\Phi$$_{AR}$. This means that flares of the same GOES class but originating from an AR of larger $\Phi$$_{AR}$, are much more likely confined. Based on an AR flux as high as 1.0$\times$$10^{24}$ Mx for solar-type stars, we estimate that the CME association rate in X100-class ``superflares" is no more than 50\%. For a sample of 132 flares $\geq$M2.0 class, we measure three non-potential parameters including the length of steep gradient polarity inversion line (L$_{SGPIL}$), the total photospheric free magnetic energy (E$_{free}$) and the area with large shear angle (A$_{\Psi}$). We find that confined flares tend to have larger values of L$_{SGPIL}$, E$_{free}$ and A$_{\Psi}$ compared to eruptive flares. Each non-potential parameter shows a moderate positive correlation with $\Phi$$_{AR}$. Our results imply that $\Phi$$_{AR}$ is a decisive quantity describing the eruptive character of a flare, as it provides a global parameter relating to the strength of the background field confinement., Comment: Accepted for publication in ApJ Letters; 16 pages, 4 figures