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

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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)
C.2. PREDICTED CORRELATIONS BETWEEN THE FLARE PARAMETERS 133C.2 Predicted Correlations between the Flare Parameters• τ r vs τ dUsing Eqs.(C.8) and (C.11), we obtain a relation( τ ds ) = A(τ rs )1/2 ,(C.12)wheren cA ∼ = 640(10 9 cm −3 )−1/2 ( M A0.01 )1/2 . (C.13)Hence the positive correlation is expected between τ r and τ d .• < kT > vs τ rFrom Eqs.(C.6), (C.8) and (8.4), we obtainτ r∼ = 10 2.20 n c(10 9 cm −3 )( M A B0.01 )−1 (100 G )−4 ( < kT >keV )7/2 [s], (C.14)τ r∼ = 10 4.82 n c(10 9 cm −3 )1/3 ( M A L0.01 )−1 (10 11 cm )4/3 ( < kT >keV )−7/6 [s]. (C.15)These equations indicate that the positive and negative log-linear correlations of < kT > vsτ r are expected if B and L are equal for all sources, respectively.C.3 The kT –EM Scaling LawUsing Eqs.(C.4), (C.10), and assuming V ∼ = L 3 (V : plasma volume), Shibata & Yokoyama (2002)derived the relation between the flare maximum temperature (T ) and emission measure (EM, Eqs.5–6 in Shibata & Yokoyama 2002). Similar to Eq.(C.5), the time behavior of EM is also exponential(van den Oord et al., 1988), hence we assume EM ∼ = 1.6 . We thus replace Eqs.(5) and(6) in Shibata & Yokoyama (2002) asn c ∼ = 3.7 × 10 48 B(10 9 cm −3 )3/2 (100 G )−5 ( < kT >keV )17/2 [cm −3 ], (C.16)n c ∼ = 7.0 × 10 51 L(10 9 cm −3 )2/3 (10 11 cm )5/3 ( < kT >keV )8/3 [cm −3 ]. (C.17)We also derive the following two equationsB ∼ n c= 52(10 9 cm −3 )3/10 ( 10 50 cm −3 )−1/5 ( < kT >keV )17/10 [G], (C.18)n cL ∼ = 7.8 × 10 9 (10 9 cm −3 )−2/5 ( 10 50 cm −3 )3/5 ( < kT >keV )−8/5 [cm]. (C.19)
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Contents1 Introduction 12 Review of
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List of Figures2.1 The H-R diagram
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LIST OF FIGURESix6.17 Light curves
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List of Tables3.1 Multiwavelength s
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Chapter 1IntroductionStar formation
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Chapter 2Review of Low-mass Young S
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2.1. EVOLUTION OF LOW-MASS STARS 5w
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2.2. MOLECULAR CLOUDS 72.2 Molecula
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2.3. X-RAY OBSERVATIONS OF LOW-MASS
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16 CHAPTER 3. REVIEW OF THE ρ OPHI
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22 CHAPTER 4. INSTRUMENTATIONrespec
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24 CHAPTER 4. INSTRUMENTATIONangle
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26 CHAPTER 4. INSTRUMENTATIONThe AC
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30 CHAPTER 4. INSTRUMENTATIONTable
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32 CHAPTER 5. CHANDRA OBSERVATIONS
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62 CHAPTER 6. INDIVIDUAL SOURCES6.2
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Page 121 and 122: Chapter 8Systematic Study of YSO Fl
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Page 129 and 130: 8.6. EVOLUTION OF YSOS AND THEIR FL
Page 131 and 132: Chapter 9ConclusionWe summarize the
Page 133 and 134: Appendix AFlare Light CurvesFig. A.
Page 135 and 136: Fig.A.2 (Continued)121
Page 137: Fig. A.4.— Same as Figure A.1, bu
Page 140 and 141: 126 APPENDIX B. PHYSICAL PARAMETERS
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Page 145: Appendix CModeling of the FlareIn t
Page 149 and 150: BibliographyAgeorges, N., Eckart, A
Page 151 and 152: BIBLIOGRAPHY 137Feigelson, E. D., &
Page 153 and 154: BIBLIOGRAPHY 139Johnstone, D., Wils
Page 155 and 156: BIBLIOGRAPHY 141Rutledge, R. E., Ba
Page 157: BIBLIOGRAPHY 143Yokoyama, T. & Shib
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