Unified Relationship between Cold Plasma Ejections and Flare Energies Ranging from Solar Microflares to Giant Stellar Flares

<jats:title>Abstract</jats:title> <jats:p>We often find spectral signatures of chromospheric cold plasma ejections accompanied by flares in a wide range of spatial scales in the solar and stellar atmospheres. However, the relationship between physical quantities (such as mass, kinetic energy, and velocity) of cold ejecta and flare energy has not been investigated in a unified manner for the entire range of flare energies to date. This study analyzed the spectra of cold plasma ejections associated with small-scale flares and solar flares (energy 10<jats:sup>25</jats:sup>–10<jats:sup>29</jats:sup> erg) to supply smaller energy samples. We performed H<jats:italic>α</jats:italic> imaging spectroscopy observation by the Solar Dynamics Doppler Imager on the Solar Magnetic Activity Research Telescope. We determined the physical quantities of the ejecta by cloud model fitting to the H<jats:italic>α</jats:italic> spectrum. We determined the flare energy by differential emission measure analysis using the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory for small-scale flares and by estimating the bolometric energy for large-scale flares. As a result, we found that the ejection mass <jats:italic>M</jats:italic> and the total flare energy <jats:italic>E</jats:italic> <jats:sub>tot</jats:sub> follow a relation of <jats:inline-formula> <jats:tex-math> <?CDATA $M\propto {E}_{\mathrm{tot}}^{2/3}$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>M</mml:mi> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>tot</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjacac76ieqn1.gif" xlink:type="simple" /> </jats:inline-formula>. We show that the scaling law derived from a simple physical model explains the solar and stellar observations with a coronal magnetic field strength as a free parameter. We also found that the kinetic energy and velocity of the ejecta correlate with the flare energy. These results suggest a common mechanism driven by magnetic fields to cause cold plasma ejections with flares on the Sun and stars.</jats:p>