X-ray Study of Low-mass Young Stellar Objects in the ρ Ophiuchi ...

Appendix CModeling of the FlareIn this appendix, we show details for the estimation of the flare parameters (τ r , τ d , and kT ), whichare used in chapter 8. These formulations are based on the standard magnetic reconnection model(Petschek, 1964), as well as the observed results and analysis of solar flares.C.1 Estimation of the Flare Parameters• Plasma temperature (< kT >)Fisher & Hawley (1990) showed that the flare maximum temperature, T max [K], is determinedby the balance between the heating rate (Q ergs s −1 cm −3 ) and conduction cooling(∝ T 7/2max/L 2 ), then obtained an equation,T max∼ = (2QL 2κ 0) 2/7 , (C.1)where L and κ 0 are the semi-length of the flare loop and Spitzer’s thermal conductivity (≡9.0×10 −7 ; Spitzer 1956). For the magnetic reconnection, Q is equal to the Poynting fluxdivided by L, i.e.,Q ∼ =c(B × v) × BE × B = = B2 v4πL 4πL 4πL ,(C.2)where E, B, and v are electric field strength, magnetic field strength, and fluid velocity,respectively. v is assumed to be the Alfvén velocity;v = v A ≡B√ 4πmnc,(C.3)where m and n c are the proton mass (≡ 1.67×10 −24 g) and pre-flare (coronal) density, respectively.Numerical simulations (Yokoyama & Shibata, 1998) further showed that the observed131