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

132 APPENDIX C. MODELING OF THE FLAREmaximum temperature (T ) is a factor of ∼3 lower than T max . Finally, Shibata & Yokoyama(1999, 2002) obtained the following relation;T ∼ = 1 3 T max ∼ = 6.75 × 10 7 n cL B(10 9 cm −3 )−1/7 (10 11 cm )2/7 (100 G )6/7 [K], (C.4)As for the usual stellar flares, we can only determine the time-averaged temperature (< kT >)due to the limited statistics. However, van den Oord et al. (1988) showed that the behaviorof the flare temperature is exponential, hence we obtain the relationT =τ r + τ d∫ 0−τ re t/τ r dt +∫ τd0 e−t/τ d dt· < T > ∼ = 1.6 < T > (C.5)Using Eqs. (C.4) and (C.5), < kT > is determined as• Rise timescale (τ r )< kT > ∼ n cL B= 3.64(10 9 cm −3 )−1/7 (10 11 cm )2/7 (100 G )6/7 [keV]. (C.6)On the basis of a standard reconnection model, τ r is equal to the interval of magnetic reconnection.Petschek (1964) showed that the reconnection timescale is proportional to theAlfvén time, τ A ≡ L/v A . Hence τ r is,τ r = τ AM A=√ 4πmnc LM A B(C.7)The correction factor M A , sometimes referred as the reconnection rate, is estimated to be0.01–0.1 (Petschek, 1964), while the observations for the solar flares show that M A is in therange of 0.001–0.1 (Table 2 in Isobe et al. 2002), regardless of their flare size. τ r is therefore,• Decay timescale (τ d )τ r∼ = 1.45 × 10 4 ( M A n cL0.01 )−1 (10 9 cm −3 )1/2 (10 11 cm )( B100 G )−1 [s]. (C.8)We assume that τ d is nearly the same as the radiative cooling timescale (τ rad ), i.e,τ d∼ = τrad ≡3nkTn 2 Λ(T ) ,(C.9)where n and Λ(T ) are the maximum plasma density and radiative loss function given by10 −24.73 T 1/4 for T > 20 MK (Mewe et al., 1985). Furthermore, we assume that magneticpressure is comparable to plasma pressure at the flare peak;2nkT = B28π .Using Eqs.(C.4), (C.9), and (C.10), we obtain τ d as(C.10)τ d∼ = 7.75 × 10 4 n cL B(10 9 cm −3 )−1/4 (10 11 cm )1/2 (100 G )−1/2 [s]. (C.11)