Nonthermal afterglow of GW170817: a more natural electron energy - - PowerPoint PPT Presentation

nonthermal afterglow of gw170817 a more natural electron
SMART_READER_LITE
LIVE PREVIEW

Nonthermal afterglow of GW170817: a more natural electron energy - - PowerPoint PPT Presentation

Nonthermal afterglow of GW170817: a more natural electron energy distribution leads to a new solution with radio flux in the low frequency synchrotron tail LIN, HAOXIANG (UTokyo) COLLABORATORS: TOTANI, TOMONORI (UTokyo), KIUCHI,


slide-1
SLIDE 1

Nonthermal afterglow of GW170817: a more natural electron energy distribution leads to a new solution with radio flux in the low frequency synchrotron tail

LIN, HAOXIANG 林淋 浩翔 (UTokyo) COLLABORATORS: TOTANI, TOMONORI (UTokyo), KIUCHI, KENTA (KyotoU)

LIN Haoxiang - High Energy Astrophysics 2018

slide-2
SLIDE 2

LIN Haoxiang - High Energy Astrophysics 2018

Nonthermal afterglow of GW170817

One-year afterglow of GW170817:

  • Nonthermal spectrum Fν ~ ν-0.6 : synchrotron

radiation from shock-accelerated electrons (electron spectral index p ≈ 2.2)

  • Slow rising (Fν ~ t0.8), peak ~ 160d, fast decay

(Fν ~ t-2). Simple pictures are challenged by slow rising pattern:

  • Uniform jet viewed off-axis (Fν ~ t3)
  • Single-velocity spherical outflow (Fν ~ t3)
  • Slow rising → Gradual input of energy →

more complicated outflow structure

Afterglow SED of GW170817 (Troja+18)

slide-3
SLIDE 3

LIN Haoxiang - High Energy Astrophysics 2018

Nonthermal afterglow of GW170817

Structured jet viewed off-axis (Jet, e.g. Lazzati+17, Margutti+18):

  • Ultra-relativistic core + slower wings
  • Initially beamed-away core emission gradually enters line of sight

to off-axis observer Radially-stratified (quasi-)spherical outflow (Sph, e.g. Mooley+18):

  • “Cocoon” created by a successful/chocked jet
  • Radially-stratified layers: fast layers traveling ahead gradually

caught up by slow layers and energized

Obs (Alexander+, Corsi+, Mooley+, Troja+18):

  • Fν ~ t-2 after ~ 160d
  • No cooling passage till ~ 360d
  • Not resolved by VLBI, but superluminal motion
  • ∏ < 12% (2.8 GHz) at 244d

“Cocoon” (Kasliwal+17)

Jet Sph Decay rate

Fν ~ t-2 Fν ~ t-1.2

Cooling passage ~ 104 d ~ 102 d Radio imaging Large axial ratio, flux centroid large offset Spherical, small

  • ffset

Linear polarization 10% – 60% < 10%

jet-dominated late-time emission

slide-4
SLIDE 4

LIN Haoxiang - High Energy Astrophysics 2018

Our research

Motivation Previous studies commonly apply a conventional GRB afterglow theory (e.g. Sari+, 1998, ApJ, 497, L17), which

  • assumed all electrons in the shock are accelerated (f* = 1), ignoring possible majority of thermal

electrons (e.g. normally observed in SNR)

  • formulated in ultra-relativistic limit, but BNS merger ejecta is mildly or non-relativistic
  • Degeneracy in f (Eichler & Waxman, 2005) and paucity of broadband GRB afterglow data

We re-examine afterglow modeling of GW170817 with a more natural electron spectrum:

  • f is allowed to vary freely
  • In conventional model, minimum electron energy γm is controlled by other model parameters. We

determine γm independently by 1 extra d.o.f. (electron-proton coupling level)

  • Applicable to trans-relativistic regime

We perform Markov-Chain Monte-Carlo (MCMC) analysis to estimate parameters of GW170817

*f injection efficiency, i.e. number fraction of accelerated electrons

slide-5
SLIDE 5

LIN Haoxiang - High Energy Astrophysics 2018

Modeling: electron spectrum

dNNT dγe ∝ γ−p

e ,

γe ≥ γm

<latexit sha1_base64="bGKFrE2tPTGjq+utLFMc9KgaPf4=">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</latexit><latexit sha1_base64="YSQfewFPksz/JGzAMkY1v9ylUw=">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</latexit><latexit sha1_base64="YSQfewFPksz/JGzAMkY1v9ylUw=">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</latexit><latexit sha1_base64="9jGmp5hxyxoR2bq1XdrizIghNew=">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</latexit>

(ii) energy as fraction ϵe of shock energy (iii) all electrons are accelerated

Conventional f = 1 model γm controlled by the spectrum energy ϵe and index p fmehei = ✏empΓs

<latexit sha1_base64="F4pOwqZkp3UKj4XsUVj2oprE7w=">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</latexit><latexit sha1_base64="F4pOwqZkp3UKj4XsUVj2oprE7w=">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</latexit><latexit sha1_base64="F4pOwqZkp3UKj4XsUVj2oprE7w=">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</latexit><latexit sha1_base64="F4pOwqZkp3UKj4XsUVj2oprE7w=">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</latexit>

m = ✏e p − 2 p − 1 mp me Γs

<latexit sha1_base64="j60ebAkEFdwnHC2niLAMv9x+mvE=">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</latexit><latexit sha1_base64="j60ebAkEFdwnHC2niLAMv9x+mvE=">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</latexit><latexit sha1_base64="j60ebAkEFdwnHC2niLAMv9x+mvE=">ACa3icbVDLSsNAFJ3Gd31V3amLwSK4sSQi6EYQXehSwT6gKWEyuamDM8kwMxFLyP/5C36ELtzo0mahbZeSDic+6dwklZ9q47lvNmZtfWFxaXqmvrq1vbDa2tjs6zRSFNk15qnoh0cBZAm3DIeVEBEyKEbPl2P9e4zKM3S5MGMJAwEGSYsZpQYSwWN0B8SIUg8AXO/fJeriAqfJCaceuAwo8VoflvURbHJ8U041VGEcjC/uzeTXlZB42m23LwbPAq0ATVXMXND79KWZgMRQTrTue640g5wowyiHou5nGiShT2QIfQsTIkAP8jJMgQ8tE+E4VfZLDC7Z3xs5EVqPRGidgphHPa2Nyf+0fmbi80HOEpkZSOjkoTj2KR4XCyOmAJq+MgCQhWzWTF9JLYRY+v/cyl6ZlJXqV8mseu2JG+6klnQOWl5bsu7P21eXlV1LaM9dICOkIfO0CW6RXeojSh6R/oC3X3p0dZ9fZn1idWrWzg/6Mc/gD3QPB4g=</latexit><latexit sha1_base64="j60ebAkEFdwnHC2niLAMv9x+mvE=">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</latexit>

(i) power-law energy spectrum with minimum injection energy

(ii) energy as fraction ϵe (rest mass excluded) (iii) degree of electron-proton coupling

Our free f model γm independently parametrized by ηe (as 1 new d.o.f) fme(hei1) = ✏emp(Γs1)

<latexit sha1_base64="ziguf3ehZeMYh64gA8lcGj/G4Cw=">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</latexit><latexit sha1_base64="ziguf3ehZeMYh64gA8lcGj/G4Cw=">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</latexit><latexit sha1_base64="ziguf3ehZeMYh64gA8lcGj/G4Cw=">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</latexit><latexit sha1_base64="ziguf3ehZeMYh64gA8lcGj/G4Cw=">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</latexit>

me(γm − 1) = ηemp(Γs − 1)

<latexit sha1_base64="8hnKcdgASGM1GKLmfoRSu2MHv9g=">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</latexit><latexit sha1_base64="8hnKcdgASGM1GKLmfoRSu2MHv9g=">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</latexit><latexit sha1_base64="8hnKcdgASGM1GKLmfoRSu2MHv9g=">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</latexit><latexit sha1_base64="8hnKcdgASGM1GKLmfoRSu2MHv9g=">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</latexit>

γm − 1 = ηe mp me (Γs − 1)

<latexit sha1_base64="QW07HYkIax1j2EKDqVRcDuRPBw=">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</latexit><latexit sha1_base64="QW07HYkIax1j2EKDqVRcDuRPBw=">ACR3icbVDLSsQwFE3H9/gadekmOAi6cGhF0I0iutClgqPCdCi3mdsxmLQlScWh9G/8Gd3qxq8QXIhLM50ufF1Icj3npvDCVPBtXHdV6c2Nj4xOTU9U5+dm19YbCwtX+okUwzbLBGJug5Bo+Axtg03Aq9ThSBDgVfh7fGwf3WHSvMkvjCDFLsS+jGPOANjqaBx4PdBSgjklkf3ae6XG/NQZFhQHw0EwzdSwHIZpIW9sNjwT0qJ3vI2g0bTbl0b/Aq0CTVHUWN79XsIyibFhArTueG5qujkow5nAou5nGlNgt9DHjoUxSNTdvHRV0HXL9GiUKHtiQ0v2uyIHqfVAhnZSgrnRv3tD8r9eJzPRXjfncZoZjNnoygT1CR0GBrtcYXMiIEFwBS3Xim7AZuKsdH+2NS746muXN+PbNdtSN7vSP6Cy+2W57a853m4VEV1zRZJWtkg3hklxySU3JG2oSRB/JEnsmL8+i8OR/O52i05lSaFfKjas4XnrOyoA=</latexit><latexit sha1_base64="QW07HYkIax1j2EKDqVRcDuRPBw=">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</latexit><latexit sha1_base64="QW07HYkIax1j2EKDqVRcDuRPBw=">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</latexit>

Given shock Lorentz factor Γs

average

f = 1

slide-6
SLIDE 6

LIN Haoxiang - High Energy Astrophysics 2018

Modeling: electron spectrum

Our free f model p ϵe γm(ηe) f = ϵe ηe p − 2 p − 1 (γm ≫ 1) γe dNNT dγe p ϵe

γm (ϵe p−2 p−1 )

f = 1 γe dNNT dγe Conventional f = 1 model How to parametrize γm? Spectral index p Injection efficiency f Total energy ϵe (as fractional shock energy) Degree of electron-proton coupling ηe (1 extra d.o.f.)

γm NOT independent parameter

slide-7
SLIDE 7

LIN Haoxiang - High Energy Astrophysics 2018

Modeling: electron spectrum

ηe ~ 1: equilibrium (heated Te ~ Tp) me/mp < ηe < 1: inefficiently-heated electrons

Degree of electron-proton coupling: ηe

ηe ≡ me(γm − 1) mp(Γs − 1)

<latexit sha1_base64="JR/z5DcvVq1j0kMr3mI0mBNyi0s=">ACPnicbZDLSgMxFIYz3q23qks3wSLUhWVGF2KLnRZwVahU4Yz6ZkaTGbGJFMsQ5/El9GtPoEv4E4EVy5N21l4OxD4+f5zkpM/TAXxnVfnInJqemZ2bn50sLi0vJKeXWtqZNMWywRCTqKgSNgsfYMNwIvEoVgwFXoY3J0P/sodK8yS+MP0U2xK6MY84A2NRUN730UCA1MfbjPeoHylguQyw6ndBSgjkjrc9sCt+qcjoIcgKFfcmjsq+ld4haiQoupB+cPvJCyTGBsmQOuW56amnYMynAkclPxMYwrsBrYsjIGibqdj743oFuWdGiUKHtiQ0f0+0QOUu+DG2nBHOtf3tD+J/Xykx02M5nGYGYzZ+KMoENQkdZkU7XCEzom8FMXtrpRdg03I2ER/3NTp8VQXW9+N1y7ZkLzfkfwVzd2a59a873K0XER1xzZIJukSjxyQI7IGamTBmHknjySJ/LsPDivzpvzPm6dcIqZdfKjnM8vdZCvjA=</latexit><latexit sha1_base64="JR/z5DcvVq1j0kMr3mI0mBNyi0s=">ACPnicbZDLSgMxFIYz3q23qks3wSLUhWVGF2KLnRZwVahU4Yz6ZkaTGbGJFMsQ5/El9GtPoEv4E4EVy5N21l4OxD4+f5zkpM/TAXxnVfnInJqemZ2bn50sLi0vJKeXWtqZNMWywRCTqKgSNgsfYMNwIvEoVgwFXoY3J0P/sodK8yS+MP0U2xK6MY84A2NRUN730UCA1MfbjPeoHylguQyw6ndBSgjkjrc9sCt+qcjoIcgKFfcmjsq+ld4haiQoupB+cPvJCyTGBsmQOuW56amnYMynAkclPxMYwrsBrYsjIGibqdj743oFuWdGiUKHtiQ0f0+0QOUu+DG2nBHOtf3tD+J/Xykx02M5nGYGYzZ+KMoENQkdZkU7XCEzom8FMXtrpRdg03I2ER/3NTp8VQXW9+N1y7ZkLzfkfwVzd2a59a873K0XER1xzZIJukSjxyQI7IGamTBmHknjySJ/LsPDivzpvzPm6dcIqZdfKjnM8vdZCvjA=</latexit><latexit sha1_base64="JR/z5DcvVq1j0kMr3mI0mBNyi0s=">ACPnicbZDLSgMxFIYz3q23qks3wSLUhWVGF2KLnRZwVahU4Yz6ZkaTGbGJFMsQ5/El9GtPoEv4E4EVy5N21l4OxD4+f5zkpM/TAXxnVfnInJqemZ2bn50sLi0vJKeXWtqZNMWywRCTqKgSNgsfYMNwIvEoVgwFXoY3J0P/sodK8yS+MP0U2xK6MY84A2NRUN730UCA1MfbjPeoHylguQyw6ndBSgjkjrc9sCt+qcjoIcgKFfcmjsq+ld4haiQoupB+cPvJCyTGBsmQOuW56amnYMynAkclPxMYwrsBrYsjIGibqdj743oFuWdGiUKHtiQ0f0+0QOUu+DG2nBHOtf3tD+J/Xykx02M5nGYGYzZ+KMoENQkdZkU7XCEzom8FMXtrpRdg03I2ER/3NTp8VQXW9+N1y7ZkLzfkfwVzd2a59a873K0XER1xzZIJukSjxyQI7IGamTBmHknjySJ/LsPDivzpvzPm6dcIqZdfKjnM8vdZCvjA=</latexit><latexit sha1_base64="JR/z5DcvVq1j0kMr3mI0mBNyi0s=">ACPnicbZDLSgMxFIYz3q23qks3wSLUhWVGF2KLnRZwVahU4Yz6ZkaTGbGJFMsQ5/El9GtPoEv4E4EVy5N21l4OxD4+f5zkpM/TAXxnVfnInJqemZ2bn50sLi0vJKeXWtqZNMWywRCTqKgSNgsfYMNwIvEoVgwFXoY3J0P/sodK8yS+MP0U2xK6MY84A2NRUN730UCA1MfbjPeoHylguQyw6ndBSgjkjrc9sCt+qcjoIcgKFfcmjsq+ld4haiQoupB+cPvJCyTGBsmQOuW56amnYMynAkclPxMYwrsBrYsjIGibqdj743oFuWdGiUKHtiQ0f0+0QOUu+DG2nBHOtf3tD+J/Xykx02M5nGYGYzZ+KMoENQkdZkU7XCEzom8FMXtrpRdg03I2ER/3NTp8VQXW9+N1y7ZkLzfkfwVzd2a59a873K0XER1xzZIJukSjxyQI7IGamTBmHknjySJ/LsPDivzpvzPm6dcIqZdfKjnM8vdZCvjA=</latexit>

Shock frame

e−

<latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="Y8kWsa3LVxIPGd+N8eWsPoRkg3g=">AB+XicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hDmUwm7dDJMzclJbQvbj1P9yJ3+Jv+AVO2yxs64ELh3Pmce4JMykMet63U9nbPzg8qh67JzX39Oy8XuYNeMt1kqU90LqeFSKN5GgZL3Ms1pEkreDSePS7875dqIVL3gPONBQkdKxIJRtFJrWG94TW8Fskv8kjSgxLD+M4hSlidcIZPUmL7vZRgUVKNgki/cQW54RtmEjnjfUkUTboJiFXNBrq0SkTjVdhSlfr3RkETY+ZJaE8mFMdm21uK/3n9HOP7oBAqy5Ertv4oziXBlCx3JpHQnKGcW0KZFjYrYWOqKUPbzMZL0VRkpkw9W8d2bUf+diO7pHPb9L2m/+xBFS7hCm7Ahzt4gCdoQRsYRPAG786r8+F8rusOGWpF7AB5+sXb/6X2A=</latexit><latexit sha1_base64="TUfyXvNVqxM3lO4bE6dXs0uPZyw=">ACBHicbVDLTsJAFL31iYiKbt1MJCZuJK0bXZq4cYmJPBKoZDq9hQkzbTMzJZKGj9Ct/oc742/4G36BA3Qh4ElucnLOPM49QSq4Nq7Wxsbm3v7Jb2yvuVg8Oj6nGlpZNMWyRCSqE1CNgsfYNwI7KQKqQwEtoPR3cxvj1FpnsSPZpKiL+kg5hFn1Fip01Myx6fLab9ac+vuHGSdeAWpQYFGv/rTCxOWSYwNE1Truemxs+pMpwJnJZ7mcaUshEdYNfSmErUfj7POyXnVglJlCg7sSFz9e+NnEqtJzKwJyU1Q73qzcT/vG5mohs/53GaGYzZ4qMoE8QkZLY8CblCZsTEsoUt1kJG1JFmbEVLb0Ujnmqi9TPi9hlW5K3Wsk6aV3VPbfuPbhQglM4gwvw4Bpu4R4a0AQGAl7hDd6dF+fD+VzUueEUvZ7AEpyvX41nDM=</latexit><latexit sha1_base64="TUfyXvNVqxM3lO4bE6dXs0uPZyw=">ACBHicbVDLTsJAFL31iYiKbt1MJCZuJK0bXZq4cYmJPBKoZDq9hQkzbTMzJZKGj9Ct/oc742/4G36BA3Qh4ElucnLOPM49QSq4Nq7Wxsbm3v7Jb2yvuVg8Oj6nGlpZNMWyRCSqE1CNgsfYNwI7KQKqQwEtoPR3cxvj1FpnsSPZpKiL+kg5hFn1Fip01Myx6fLab9ac+vuHGSdeAWpQYFGv/rTCxOWSYwNE1Truemxs+pMpwJnJZ7mcaUshEdYNfSmErUfj7POyXnVglJlCg7sSFz9e+NnEqtJzKwJyU1Q73qzcT/vG5mohs/53GaGYzZ4qMoE8QkZLY8CblCZsTEsoUt1kJG1JFmbEVLb0Ujnmqi9TPi9hlW5K3Wsk6aV3VPbfuPbhQglM4gwvw4Bpu4R4a0AQGAl7hDd6dF+fD+VzUueEUvZ7AEpyvX41nDM=</latexit><latexit sha1_base64="pqsLzO5dThFfdYb4FR2/FjrhgYc=">ACD3icbVDLTgIxFO3gC/GFunTSEzcSGbc6JLoxiUm8khgJ3OHWhoO5O2QyQTPkK3+h/ujFs/wd/wCywCwFP0uTknHvb0xMknGnjut9OYW19Y3OruF3a2d3bPygfHjV1nCoKDRrzWLUDoEzCQ3DId2oCIgEMrGN5O/dYIlGaxfDjBHxB+pJFjBJjpXZXiQweLya9csWtujPgVeLlpIJy1Hvln24Y01SANJQTrTuemxg/I8owymFS6qYaEkKHpA8dSyURoP1slneCz6wS4ihW9kiDZ+rfjYwIrcisJOCmIFe9qbif14nNdG1nzGZpAYknT8UpRybGE8/j0OmgBo+toRQxWxWTAdEWpsRQs3hSOW6Dz10zx2yZbkLVeySpqXVc+tevdupXaT1VEJ+gUnSMPXaEaukN1EAUcfSCXtGb8+y8Ox/O53y04OQ7x2gBztcvVgOdqQ=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit>

p+

<latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit><latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit><latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit><latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit>

p+

<latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit><latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit><latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit><latexit sha1_base64="Qb40l26K+wOqs1V8I0Bpn6I08Nw=">ACD3icbVDLSgMxFL3js9ZX1aWbYBEocyIoMuiG5cV7APasWQymTY0mQlJpliGfoRu9T/ciVs/wd/wC0zbWdjWA4HDOfcmJyeQnGnjut/Oyura+sZmYau4vbO7t186OGzoJFWE1knCE9UKsKacxbRumOG0JRXFIuC0GQxuJ35zSJVmSfxgRpL6AvdiFjGCjZVaHSUy+Xg+7pbKbsWdAi0TLydlyFHrln46YUJSQWNDONa67bnS+BlWhFOx8VOqnEZIB7tG1pjAXVfjbNO0anVglRlCh7YoOm6t+NDAutRyKwkwKbvl70JuJ/Xjs10bWfsVimhsZk9lCUcmQSNPk8CpmixPCRJZgoZrMi0scKE2MrmrspHDKp89RPs9hFW5K3WMkyaVxUPLfi3V+Wqzd5XQU4hM4Aw+uoAp3UIM6EODwAq/w5jw786H8zkbXHynSOYg/P1C2YmnbY=</latexit>

e−

<latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="Y8kWsa3LVxIPGd+N8eWsPoRkg3g=">AB+XicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hDmUwm7dDJMzclJbQvbj1P9yJ3+Jv+AVO2yxs64ELh3Pmce4JMykMet63U9nbPzg8qh67JzX39Oy8XuYNeMt1kqU90LqeFSKN5GgZL3Ms1pEkreDSePS7875dqIVL3gPONBQkdKxIJRtFJrWG94TW8Fskv8kjSgxLD+M4hSlidcIZPUmL7vZRgUVKNgki/cQW54RtmEjnjfUkUTboJiFXNBrq0SkTjVdhSlfr3RkETY+ZJaE8mFMdm21uK/3n9HOP7oBAqy5Ertv4oziXBlCx3JpHQnKGcW0KZFjYrYWOqKUPbzMZL0VRkpkw9W8d2bUf+diO7pHPb9L2m/+xBFS7hCm7Ahzt4gCdoQRsYRPAG786r8+F8rusOGWpF7AB5+sXb/6X2A=</latexit><latexit sha1_base64="TUfyXvNVqxM3lO4bE6dXs0uPZyw=">ACBHicbVDLTsJAFL31iYiKbt1MJCZuJK0bXZq4cYmJPBKoZDq9hQkzbTMzJZKGj9Ct/oc742/4G36BA3Qh4ElucnLOPM49QSq4Nq7Wxsbm3v7Jb2yvuVg8Oj6nGlpZNMWyRCSqE1CNgsfYNwI7KQKqQwEtoPR3cxvj1FpnsSPZpKiL+kg5hFn1Fip01Myx6fLab9ac+vuHGSdeAWpQYFGv/rTCxOWSYwNE1Truemxs+pMpwJnJZ7mcaUshEdYNfSmErUfj7POyXnVglJlCg7sSFz9e+NnEqtJzKwJyU1Q73qzcT/vG5mohs/53GaGYzZ4qMoE8QkZLY8CblCZsTEsoUt1kJG1JFmbEVLb0Ujnmqi9TPi9hlW5K3Wsk6aV3VPbfuPbhQglM4gwvw4Bpu4R4a0AQGAl7hDd6dF+fD+VzUueEUvZ7AEpyvX41nDM=</latexit><latexit sha1_base64="TUfyXvNVqxM3lO4bE6dXs0uPZyw=">ACBHicbVDLTsJAFL31iYiKbt1MJCZuJK0bXZq4cYmJPBKoZDq9hQkzbTMzJZKGj9Ct/oc742/4G36BA3Qh4ElucnLOPM49QSq4Nq7Wxsbm3v7Jb2yvuVg8Oj6nGlpZNMWyRCSqE1CNgsfYNwI7KQKqQwEtoPR3cxvj1FpnsSPZpKiL+kg5hFn1Fip01Myx6fLab9ac+vuHGSdeAWpQYFGv/rTCxOWSYwNE1Truemxs+pMpwJnJZ7mcaUshEdYNfSmErUfj7POyXnVglJlCg7sSFz9e+NnEqtJzKwJyU1Q73qzcT/vG5mohs/53GaGYzZ4qMoE8QkZLY8CblCZsTEsoUt1kJG1JFmbEVLb0Ujnmqi9TPi9hlW5K3Wsk6aV3VPbfuPbhQglM4gwvw4Bpu4R4a0AQGAl7hDd6dF+fD+VzUueEUvZ7AEpyvX41nDM=</latexit><latexit sha1_base64="pqsLzO5dThFfdYb4FR2/FjrhgYc=">ACD3icbVDLTgIxFO3gC/GFunTSEzcSGbc6JLoxiUm8khgJ3OHWhoO5O2QyQTPkK3+h/ujFs/wd/wCywCwFP0uTknHvb0xMknGnjut9OYW19Y3OruF3a2d3bPygfHjV1nCoKDRrzWLUDoEzCQ3DId2oCIgEMrGN5O/dYIlGaxfDjBHxB+pJFjBJjpXZXiQweLya9csWtujPgVeLlpIJy1Hvln24Y01SANJQTrTuemxg/I8owymFS6qYaEkKHpA8dSyURoP1slneCz6wS4ihW9kiDZ+rfjYwIrcisJOCmIFe9qbif14nNdG1nzGZpAYknT8UpRybGE8/j0OmgBo+toRQxWxWTAdEWpsRQs3hSOW6Dz10zx2yZbkLVeySpqXVc+tevdupXaT1VEJ+gUnSMPXaEaukN1EAUcfSCXtGb8+y8Ox/O53y04OQ7x2gBztcvVgOdqQ=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit><latexit sha1_base64="q6xdIr3nUOgvmxLnF24HVq6NIfM=">ACD3icbVDLSgMxFM34rPVdekmWAQ3lhkRdFl047KCfUA7lkzmThuaZIYkUyxDP0K3+h/uxK2f4G/4BabtLGzrgcDhnHuTkxMknGnjut/Oyura+sZmYau4vbO7t186OGzoOFU6jTmsWoFRANnEuqGQ6tRAERAYdmMLid+M0hKM1i+WBGCfiC9CSLGCXGSq2OEhk8no+7pbJbcafAy8TLSRnlqHVLP50wpqkAaSgnWrc9NzF+RpRhlMO42Ek1JIQOSA/alkoiQPvZNO8Yn1olxFGs7JEGT9W/GxkRWo9EYCcFMX296E3E/7x2aqJrP2MySQ1IOnsoSjk2MZ58HodMATV8ZAmhitmsmPaJItTYiuZuCocs0Xnqp1nsoi3JW6xkmTQuKp5b8e4vy9WbvK4COkYn6Ax56ApV0R2qoTqiKMX9IrenGfn3flwPmejK06+c4Tm4Hz9AldDna0=</latexit>

Γs

Natural assumption: shock acceleration starts somewhere between Te and Tp

Cold ISM

Tp ∼ mp(Γs − 1) Te ∼ me(Γs − 1) ≪ Tp Te < me(γm − 1) < Tp me/mp < ηe < 1 Electrons further heated by protons up to a higher temperature up to equilibrium

ηe ~ me/mp: no energy transfer from protons to electrons

slide-8
SLIDE 8

LIN Haoxiang - High Energy Astrophysics 2018

Modeling: shock dynamics

Jet model Spherical model

Power-law radial profile in proper velocity u Continuous injection

Gaussian angular profile for energy Single injection (no radial profile)

Shock dynamics

M(R) = 4 3πR3nmp

<latexit sha1_base64="vtpRMEjQI67W/gZHBS2juhvM818=">ACHicbVDLSsNAFJ3UV62vqEsXDhahbkpiC7oRim7cCLXYBzSxTKaTduhkEmYmQglZuvFX3LhQxK2f4M6/cdpmoa0HLhzOuZd7/EiRqWyrG8jt7S8srqWXy9sbG5t75i7ey0ZxgKTJg5ZKDoekoRTpqKkY6kSAo8Bhpe6Orid9+IELSkN+pcUTcA049SlGSks98/Cm1DiBF9DxBcJNU0qRNR2LivQA6DXtQzi1bZmgIuEjsjRZCh3jO/nH6I4BwhRmSsmtbkXITJBTFjKQFJ5YkQniEBqSrKUcBkW4yfSFx1rpQz8UuriCU/X3RICKceBpzsDpIZy3puI/3ndWPnbkJ5FCvC8WyRHzOoQjhJBfapIFixsSYIC6pvhXiIdCRKZ1fQIdjzLy+S1mnZtsr2bVYu8ziyIMDcARKwAZnoAauQR0AQaP4Bm8gjfjyXgx3o2PWvOyGb2wR8Ynz+C/pe/</latexit><latexit sha1_base64="vtpRMEjQI67W/gZHBS2juhvM818=">ACHicbVDLSsNAFJ3UV62vqEsXDhahbkpiC7oRim7cCLXYBzSxTKaTduhkEmYmQglZuvFX3LhQxK2f4M6/cdpmoa0HLhzOuZd7/EiRqWyrG8jt7S8srqWXy9sbG5t75i7ey0ZxgKTJg5ZKDoekoRTpqKkY6kSAo8Bhpe6Orid9+IELSkN+pcUTcA049SlGSks98/Cm1DiBF9DxBcJNU0qRNR2LivQA6DXtQzi1bZmgIuEjsjRZCh3jO/nH6I4BwhRmSsmtbkXITJBTFjKQFJ5YkQniEBqSrKUcBkW4yfSFx1rpQz8UuriCU/X3RICKceBpzsDpIZy3puI/3ndWPnbkJ5FCvC8WyRHzOoQjhJBfapIFixsSYIC6pvhXiIdCRKZ1fQIdjzLy+S1mnZtsr2bVYu8ziyIMDcARKwAZnoAauQR0AQaP4Bm8gjfjyXgx3o2PWvOyGb2wR8Ynz+C/pe/</latexit><latexit sha1_base64="vtpRMEjQI67W/gZHBS2juhvM818=">ACHicbVDLSsNAFJ3UV62vqEsXDhahbkpiC7oRim7cCLXYBzSxTKaTduhkEmYmQglZuvFX3LhQxK2f4M6/cdpmoa0HLhzOuZd7/EiRqWyrG8jt7S8srqWXy9sbG5t75i7ey0ZxgKTJg5ZKDoekoRTpqKkY6kSAo8Bhpe6Orid9+IELSkN+pcUTcA049SlGSks98/Cm1DiBF9DxBcJNU0qRNR2LivQA6DXtQzi1bZmgIuEjsjRZCh3jO/nH6I4BwhRmSsmtbkXITJBTFjKQFJ5YkQniEBqSrKUcBkW4yfSFx1rpQz8UuriCU/X3RICKceBpzsDpIZy3puI/3ndWPnbkJ5FCvC8WyRHzOoQjhJBfapIFixsSYIC6pvhXiIdCRKZ1fQIdjzLy+S1mnZtsr2bVYu8ziyIMDcARKwAZnoAauQR0AQaP4Bm8gjfjyXgx3o2PWvOyGb2wR8Ynz+C/pe/</latexit><latexit sha1_base64="vtpRMEjQI67W/gZHBS2juhvM818=">ACHicbVDLSsNAFJ3UV62vqEsXDhahbkpiC7oRim7cCLXYBzSxTKaTduhkEmYmQglZuvFX3LhQxK2f4M6/cdpmoa0HLhzOuZd7/EiRqWyrG8jt7S8srqWXy9sbG5t75i7ey0ZxgKTJg5ZKDoekoRTpqKkY6kSAo8Bhpe6Orid9+IELSkN+pcUTcA049SlGSks98/Cm1DiBF9DxBcJNU0qRNR2LivQA6DXtQzi1bZmgIuEjsjRZCh3jO/nH6I4BwhRmSsmtbkXITJBTFjKQFJ5YkQniEBqSrKUcBkW4yfSFx1rpQz8UuriCU/X3RICKceBpzsDpIZy3puI/3ndWPnbkJ5FCvC8WyRHzOoQjhJBfapIFixsSYIC6pvhXiIdCRKZ1fQIdjzLy+S1mnZtsr2bVYu8ziyIMDcARKwAZnoAauQR0AQaP4Bm8gjfjyXgx3o2PWvOyGb2wR8Ynz+C/pe/</latexit>

Einj = M(R) (cus)2

<latexit sha1_base64="bEk2qZ05a4ioM+5809M63XJqj4=">ACB3icbVDLSsNAFJ34rPUVdSnIYBFakJIUQTdCUQ3QhX7gCaGyXTSjp1MwsxEKE7N/6KGxeKuPUX3Pk3TtstPXAhcM593LvPX7MqFSW9W3MzS8sLi3nVvKra+sbm+bWdkNGicCkjiMWiZaPJGUk7qipFWLAgKfUafv985DcfiJA04rdqEBM3RF1OA4qR0pJn7l14qSNCSPn9EJ7Cq+JNCTqHsIgT5buKp5ZsMrWGHCW2BkpgAw1z/xyOhFOQsIVZkjKtm3Fyk2RUBQzMsw7iSQxwn3UJW1NOQqJdNPxH0N4oJUODCKhiys4Vn9PpCiUchD6ujNEqienvZH4n9dOVHDipTHiSIcTxYFCYMqgqNQYIcKghUbaIKwoPpWiHtIKx0dHkdgj398ixpVMq2VbavjwrVsyOHNgF+6AIbHAMquAS1EAdYPAInsEreDOejBfj3fiYtM4Z2cwO+APj8wf9+pbP</latexit><latexit sha1_base64="bEk2qZ05a4ioM+5809M63XJqj4=">ACB3icbVDLSsNAFJ34rPUVdSnIYBFakJIUQTdCUQ3QhX7gCaGyXTSjp1MwsxEKE7N/6KGxeKuPUX3Pk3TtstPXAhcM593LvPX7MqFSW9W3MzS8sLi3nVvKra+sbm+bWdkNGicCkjiMWiZaPJGUk7qipFWLAgKfUafv985DcfiJA04rdqEBM3RF1OA4qR0pJn7l14qSNCSPn9EJ7Cq+JNCTqHsIgT5buKp5ZsMrWGHCW2BkpgAw1z/xyOhFOQsIVZkjKtm3Fyk2RUBQzMsw7iSQxwn3UJW1NOQqJdNPxH0N4oJUODCKhiys4Vn9PpCiUchD6ujNEqienvZH4n9dOVHDipTHiSIcTxYFCYMqgqNQYIcKghUbaIKwoPpWiHtIKx0dHkdgj398ixpVMq2VbavjwrVsyOHNgF+6AIbHAMquAS1EAdYPAInsEreDOejBfj3fiYtM4Z2cwO+APj8wf9+pbP</latexit><latexit sha1_base64="bEk2qZ05a4ioM+5809M63XJqj4=">ACB3icbVDLSsNAFJ34rPUVdSnIYBFakJIUQTdCUQ3QhX7gCaGyXTSjp1MwsxEKE7N/6KGxeKuPUX3Pk3TtstPXAhcM593LvPX7MqFSW9W3MzS8sLi3nVvKra+sbm+bWdkNGicCkjiMWiZaPJGUk7qipFWLAgKfUafv985DcfiJA04rdqEBM3RF1OA4qR0pJn7l14qSNCSPn9EJ7Cq+JNCTqHsIgT5buKp5ZsMrWGHCW2BkpgAw1z/xyOhFOQsIVZkjKtm3Fyk2RUBQzMsw7iSQxwn3UJW1NOQqJdNPxH0N4oJUODCKhiys4Vn9PpCiUchD6ujNEqienvZH4n9dOVHDipTHiSIcTxYFCYMqgqNQYIcKghUbaIKwoPpWiHtIKx0dHkdgj398ixpVMq2VbavjwrVsyOHNgF+6AIbHAMquAS1EAdYPAInsEreDOejBfj3fiYtM4Z2cwO+APj8wf9+pbP</latexit><latexit sha1_base64="bEk2qZ05a4ioM+5809M63XJqj4=">ACB3icbVDLSsNAFJ34rPUVdSnIYBFakJIUQTdCUQ3QhX7gCaGyXTSjp1MwsxEKE7N/6KGxeKuPUX3Pk3TtstPXAhcM593LvPX7MqFSW9W3MzS8sLi3nVvKra+sbm+bWdkNGicCkjiMWiZaPJGUk7qipFWLAgKfUafv985DcfiJA04rdqEBM3RF1OA4qR0pJn7l14qSNCSPn9EJ7Cq+JNCTqHsIgT5buKp5ZsMrWGHCW2BkpgAw1z/xyOhFOQsIVZkjKtm3Fyk2RUBQzMsw7iSQxwn3UJW1NOQqJdNPxH0N4oJUODCKhiys4Vn9PpCiUchD6ujNEqienvZH4n9dOVHDipTHiSIcTxYFCYMqgqNQYIcKghUbaIKwoPpWiHtIKx0dHkdgj398ixpVMq2VbavjwrVsyOHNgF+6AIbHAMquAS1EAdYPAInsEreDOejBfj3fiYtM4Z2cwO+APj8wf9+pbP</latexit>

us = ˙ R c 2 41 − ˙ R c !23 5

− 1

2

<latexit sha1_base64="PmhSq/TdqC/3IM2cuqmIaGMY1E=">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</latexit><latexit sha1_base64="PmhSq/TdqC/3IM2cuqmIaGMY1E=">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</latexit><latexit sha1_base64="PmhSq/TdqC/3IM2cuqmIaGMY1E=">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</latexit><latexit sha1_base64="PmhSq/TdqC/3IM2cuqmIaGMY1E=">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</latexit>

Energy equation Swept-up ISM mass Shock velocity

E(> u) ∝ u−k (umin < u < umax)

<latexit sha1_base64="mR0oLy1Wk8ZfgqI1Opc3puGKPxM=">ACO3icbVDLSgMxFM34tr6qLt0Ei9AuLDNF0EWRoguFawKbS2ZNK2hSbkUVqG6Xf4M7rVX3DtTlwJ7k3bEXxdCJyc+/NyQklo9r4/rM3NT0zOze/sJhZWl5ZXcub1zqyCpMqjhikboOkSaMClI1DByLRVBPGTkKuwej/SrHlGaRuLCDCRpcNQRtE0xMo5qZksn+UNbgHWpImkiaG/i3W4yHA7zthnXFYeciqRsy1831E8KzWzOL/rjgn9BkIcSOusmX2vtyJsOREGM6R1LfClacRIGYoZSTJ1q4lEuIs6pOagQJzoRjz+WwJ3HNOC7Ui5Iwcs98nYsS1HvDQdXJkbvVvbUT+p9WsaR80YiqkNUTgyUNty6BLYRQUbFsGEDBxBW1HmF+BYphI2L8emVo9KnbruT2xnXEjB70j+gstSMfCLwflernKUxrUAtsA2yIMA7IMKOAVnoAowuAMP4BE8efei/fqvU1ap7x0ZhP8KO/jE89Urts=</latexit><latexit sha1_base64="mR0oLy1Wk8ZfgqI1Opc3puGKPxM=">ACO3icbVDLSgMxFM34tr6qLt0Ei9AuLDNF0EWRoguFawKbS2ZNK2hSbkUVqG6Xf4M7rVX3DtTlwJ7k3bEXxdCJyc+/NyQklo9r4/rM3NT0zOze/sJhZWl5ZXcub1zqyCpMqjhikboOkSaMClI1DByLRVBPGTkKuwej/SrHlGaRuLCDCRpcNQRtE0xMo5qZksn+UNbgHWpImkiaG/i3W4yHA7zthnXFYeciqRsy1831E8KzWzOL/rjgn9BkIcSOusmX2vtyJsOREGM6R1LfClacRIGYoZSTJ1q4lEuIs6pOagQJzoRjz+WwJ3HNOC7Ui5Iwcs98nYsS1HvDQdXJkbvVvbUT+p9WsaR80YiqkNUTgyUNty6BLYRQUbFsGEDBxBW1HmF+BYphI2L8emVo9KnbruT2xnXEjB70j+gstSMfCLwflernKUxrUAtsA2yIMA7IMKOAVnoAowuAMP4BE8efei/fqvU1ap7x0ZhP8KO/jE89Urts=</latexit><latexit sha1_base64="mR0oLy1Wk8ZfgqI1Opc3puGKPxM=">ACO3icbVDLSgMxFM34tr6qLt0Ei9AuLDNF0EWRoguFawKbS2ZNK2hSbkUVqG6Xf4M7rVX3DtTlwJ7k3bEXxdCJyc+/NyQklo9r4/rM3NT0zOze/sJhZWl5ZXcub1zqyCpMqjhikboOkSaMClI1DByLRVBPGTkKuwej/SrHlGaRuLCDCRpcNQRtE0xMo5qZksn+UNbgHWpImkiaG/i3W4yHA7zthnXFYeciqRsy1831E8KzWzOL/rjgn9BkIcSOusmX2vtyJsOREGM6R1LfClacRIGYoZSTJ1q4lEuIs6pOagQJzoRjz+WwJ3HNOC7Ui5Iwcs98nYsS1HvDQdXJkbvVvbUT+p9WsaR80YiqkNUTgyUNty6BLYRQUbFsGEDBxBW1HmF+BYphI2L8emVo9KnbruT2xnXEjB70j+gstSMfCLwflernKUxrUAtsA2yIMA7IMKOAVnoAowuAMP4BE8efei/fqvU1ap7x0ZhP8KO/jE89Urts=</latexit><latexit sha1_base64="mR0oLy1Wk8ZfgqI1Opc3puGKPxM=">ACO3icbVDLSgMxFM34tr6qLt0Ei9AuLDNF0EWRoguFawKbS2ZNK2hSbkUVqG6Xf4M7rVX3DtTlwJ7k3bEXxdCJyc+/NyQklo9r4/rM3NT0zOze/sJhZWl5ZXcub1zqyCpMqjhikboOkSaMClI1DByLRVBPGTkKuwej/SrHlGaRuLCDCRpcNQRtE0xMo5qZksn+UNbgHWpImkiaG/i3W4yHA7zthnXFYeciqRsy1831E8KzWzOL/rjgn9BkIcSOusmX2vtyJsOREGM6R1LfClacRIGYoZSTJ1q4lEuIs6pOagQJzoRjz+WwJ3HNOC7Ui5Iwcs98nYsS1HvDQdXJkbvVvbUT+p9WsaR80YiqkNUTgyUNty6BLYRQUbFsGEDBxBW1HmF+BYphI2L8emVo9KnbruT2xnXEjB70j+gstSMfCLwflernKUxrUAtsA2yIMA7IMKOAVnoAowuAMP4BE8efei/fqvU1ap7x0ZhP8KO/jE89Urts=</latexit>

Einj = E(> ucol)

<latexit sha1_base64="96Krz8fReuLR4qsiGnXpjNokOY=">ACJnicbVDLSgMxFM3UV62vUZciBItQN2VGBN0oRSm4rGAf0A5DJk3b2CQzJliGbryZ3Sr/+FOxJ3f4BeYtrOwrQcCh3NfJyeIGFXacb6szNLyupadj23sbm1vWPv7tVUGEtMqjhkoWwESBFGBalqhlpRJIgHjBSD/o343p9QKSiobjXw4h4HUF7VCMtJF8+7DsJy3JIRUPI3gJy4WreCqY1aMT3847RWcCuEjclORBiopv/7TaIY45ERozpFTdSLtJUhqihkZ5VqxIhHCfdQlTUMF4kR5yeQbI3hslDbshNI8oeFE/TuRIK7UkAemkyPdU/O1sfhfrRnrzoWXUBHFmg8PdSJGdQhHGcC21QSrNnQEIQlNV4h7iGJsDbJzWxqD2ikUtePU9s5E5I7H8kiqZ0WXafo3p3lS9dpXFlwAI5AbjgHJTALaiAKsDgCbyAV/BmPVv1of1OW3NWOnMPpiB9f0LhlOlew=</latexit><latexit sha1_base64="96Krz8fReuLR4qsiGnXpjNokOY=">ACJnicbVDLSgMxFM3UV62vUZciBItQN2VGBN0oRSm4rGAf0A5DJk3b2CQzJliGbryZ3Sr/+FOxJ3f4BeYtrOwrQcCh3NfJyeIGFXacb6szNLyupadj23sbm1vWPv7tVUGEtMqjhkoWwESBFGBalqhlpRJIgHjBSD/o343p9QKSiobjXw4h4HUF7VCMtJF8+7DsJy3JIRUPI3gJy4WreCqY1aMT3847RWcCuEjclORBiopv/7TaIY45ERozpFTdSLtJUhqihkZ5VqxIhHCfdQlTUMF4kR5yeQbI3hslDbshNI8oeFE/TuRIK7UkAemkyPdU/O1sfhfrRnrzoWXUBHFmg8PdSJGdQhHGcC21QSrNnQEIQlNV4h7iGJsDbJzWxqD2ikUtePU9s5E5I7H8kiqZ0WXafo3p3lS9dpXFlwAI5AbjgHJTALaiAKsDgCbyAV/BmPVv1of1OW3NWOnMPpiB9f0LhlOlew=</latexit><latexit sha1_base64="96Krz8fReuLR4qsiGnXpjNokOY=">ACJnicbVDLSgMxFM3UV62vUZciBItQN2VGBN0oRSm4rGAf0A5DJk3b2CQzJliGbryZ3Sr/+FOxJ3f4BeYtrOwrQcCh3NfJyeIGFXacb6szNLyupadj23sbm1vWPv7tVUGEtMqjhkoWwESBFGBalqhlpRJIgHjBSD/o343p9QKSiobjXw4h4HUF7VCMtJF8+7DsJy3JIRUPI3gJy4WreCqY1aMT3847RWcCuEjclORBiopv/7TaIY45ERozpFTdSLtJUhqihkZ5VqxIhHCfdQlTUMF4kR5yeQbI3hslDbshNI8oeFE/TuRIK7UkAemkyPdU/O1sfhfrRnrzoWXUBHFmg8PdSJGdQhHGcC21QSrNnQEIQlNV4h7iGJsDbJzWxqD2ikUtePU9s5E5I7H8kiqZ0WXafo3p3lS9dpXFlwAI5AbjgHJTALaiAKsDgCbyAV/BmPVv1of1OW3NWOnMPpiB9f0LhlOlew=</latexit><latexit sha1_base64="96Krz8fReuLR4qsiGnXpjNokOY=">ACJnicbVDLSgMxFM3UV62vUZciBItQN2VGBN0oRSm4rGAf0A5DJk3b2CQzJliGbryZ3Sr/+FOxJ3f4BeYtrOwrQcCh3NfJyeIGFXacb6szNLyupadj23sbm1vWPv7tVUGEtMqjhkoWwESBFGBalqhlpRJIgHjBSD/o343p9QKSiobjXw4h4HUF7VCMtJF8+7DsJy3JIRUPI3gJy4WreCqY1aMT3847RWcCuEjclORBiopv/7TaIY45ERozpFTdSLtJUhqihkZ5VqxIhHCfdQlTUMF4kR5yeQbI3hslDbshNI8oeFE/TuRIK7UkAemkyPdU/O1sfhfrRnrzoWXUBHFmg8PdSJGdQhHGcC21QSrNnQEIQlNV4h7iGJsDbJzWxqD2ikUtePU9s5E5I7H8kiqZ0WXafo3p3lS9dpXFlwAI5AbjgHJTALaiAKsDgCbyAV/BmPVv1of1OW3NWOnMPpiB9f0LhlOlew=</latexit>

ucol = R √ c2t2 − R2

<latexit sha1_base64="wnFtkuhrk5MhZaSATdkQN2wV5Jc=">ACMnicbVDLTgIxFO34RHyhLt0EhNdSGaIiW5MiG5cIpFHwgDplA40dB62d4hkMt/gz+hWv0N3xq0rv8ACsxDwJE1Ozn2dHicUXIFpvhtLyura+uZjezm1vbObm5v6aCSFJWpYEIZMhignusypwEKwRSkY8R7C6M7gZ1+tDJhUP/HsYhazlkZ7PXU4JaKmTO406sS09rDcl+ArbriQ0riSxrR4kxLRdhHbxrNIuJknlzcL5gR4kVgpyaMU5U7ux+4GNPKYD1QpZqWGUIrJhI4FSzJ2pFiIaED0mNTX3iMdWKJ19K8LFWutgNpH4+4In6dyImnlIjz9GdHoG+mq+Nxf9qzQjcy1bM/TAC5tPpITcSGAI8zgd3uWQUxEgTQiXjHtE50K6BRnNnWHPFSp68ep7awOyZqPZJHUigXLFh35/nSdRpXBh2iI3SCLHSBSugWlVEVUfSEXtArejOejQ/j0/iati4Z6cwBmoHx/QurX6tF</latexit><latexit sha1_base64="wnFtkuhrk5MhZaSATdkQN2wV5Jc=">ACMnicbVDLTgIxFO34RHyhLt0EhNdSGaIiW5MiG5cIpFHwgDplA40dB62d4hkMt/gz+hWv0N3xq0rv8ACsxDwJE1Ozn2dHicUXIFpvhtLyura+uZjezm1vbObm5v6aCSFJWpYEIZMhignusypwEKwRSkY8R7C6M7gZ1+tDJhUP/HsYhazlkZ7PXU4JaKmTO406sS09rDcl+ArbriQ0riSxrR4kxLRdhHbxrNIuJknlzcL5gR4kVgpyaMU5U7ux+4GNPKYD1QpZqWGUIrJhI4FSzJ2pFiIaED0mNTX3iMdWKJ19K8LFWutgNpH4+4In6dyImnlIjz9GdHoG+mq+Nxf9qzQjcy1bM/TAC5tPpITcSGAI8zgd3uWQUxEgTQiXjHtE50K6BRnNnWHPFSp68ep7awOyZqPZJHUigXLFh35/nSdRpXBh2iI3SCLHSBSugWlVEVUfSEXtArejOejQ/j0/iati4Z6cwBmoHx/QurX6tF</latexit><latexit sha1_base64="wnFtkuhrk5MhZaSATdkQN2wV5Jc=">ACMnicbVDLTgIxFO34RHyhLt0EhNdSGaIiW5MiG5cIpFHwgDplA40dB62d4hkMt/gz+hWv0N3xq0rv8ACsxDwJE1Ozn2dHicUXIFpvhtLyura+uZjezm1vbObm5v6aCSFJWpYEIZMhignusypwEKwRSkY8R7C6M7gZ1+tDJhUP/HsYhazlkZ7PXU4JaKmTO406sS09rDcl+ArbriQ0riSxrR4kxLRdhHbxrNIuJknlzcL5gR4kVgpyaMU5U7ux+4GNPKYD1QpZqWGUIrJhI4FSzJ2pFiIaED0mNTX3iMdWKJ19K8LFWutgNpH4+4In6dyImnlIjz9GdHoG+mq+Nxf9qzQjcy1bM/TAC5tPpITcSGAI8zgd3uWQUxEgTQiXjHtE50K6BRnNnWHPFSp68ep7awOyZqPZJHUigXLFh35/nSdRpXBh2iI3SCLHSBSugWlVEVUfSEXtArejOejQ/j0/iati4Z6cwBmoHx/QurX6tF</latexit><latexit sha1_base64="wnFtkuhrk5MhZaSATdkQN2wV5Jc=">ACMnicbVDLTgIxFO34RHyhLt0EhNdSGaIiW5MiG5cIpFHwgDplA40dB62d4hkMt/gz+hWv0N3xq0rv8ACsxDwJE1Ozn2dHicUXIFpvhtLyura+uZjezm1vbObm5v6aCSFJWpYEIZMhignusypwEKwRSkY8R7C6M7gZ1+tDJhUP/HsYhazlkZ7PXU4JaKmTO406sS09rDcl+ArbriQ0riSxrR4kxLRdhHbxrNIuJknlzcL5gR4kVgpyaMU5U7ux+4GNPKYD1QpZqWGUIrJhI4FSzJ2pFiIaED0mNTX3iMdWKJ19K8LFWutgNpH4+4In6dyImnlIjz9GdHoG+mq+Nxf9qzQjcy1bM/TAC5tPpITcSGAI8zgd3uWQUxEgTQiXjHtE50K6BRnNnWHPFSp68ep7awOyZqPZJHUigXLFh35/nSdRpXBh2iI3SCLHSBSugWlVEVUfSEXtArejOejQ/j0/iati4Z6cwBmoHx/QurX6tF</latexit>

Ek, iso(θ) = Ec, iso exp (− θ2 2θ2

c )

Γ0(θ) − 1 = (Γc − 1)exp (− θ2 2θ2

c )

(βcol = R/ct)

slide-9
SLIDE 9

LIN Haoxiang - High Energy Astrophysics 2018

Modeling: synchrotron flux

Comoving magnetic field strength (parametrized by ϵB) Comoving synchrotron spectral power Received flux

  • Doppler effect
  • Photon arrival time delay
  • Emitting angle from observer

adiabatic index (mono-energetic gas approx.)

peak frequency absorption frequency cooling frequency

ˆ γ = 4 3 + 1 3Γs

<latexit sha1_base64="IWwDMXdF1zXwy7Xa2JUkyUOIazI=">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</latexit><latexit sha1_base64="IWwDMXdF1zXwy7Xa2JUkyUOIazI=">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</latexit><latexit sha1_base64="IWwDMXdF1zXwy7Xa2JUkyUOIazI=">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</latexit><latexit sha1_base64="IWwDMXdF1zXwy7Xa2JUkyUOIazI=">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</latexit>

P 0

m = 8π

9 e3B0 mec2 M(R) mp f

<latexit sha1_base64="80dQ9kzrEROXBA46UD1DWIbl6UY=">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</latexit><latexit sha1_base64="80dQ9kzrEROXBA46UD1DWIbl6UY=">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</latexit><latexit sha1_base64="80dQ9kzrEROXBA46UD1DWIbl6UY=">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</latexit><latexit sha1_base64="80dQ9kzrEROXBA46UD1DWIbl6UY=">ACbXicbVDBThsxEHWtBAS6BCPbRCFlEVOBDtAhJwoEL0kultGoAKQkrzMbrNhey/ZSLSv/Tb+mV7j0K/gFnGQPDXQkS2/em9Ebv0RxZmwY/q0FCy9evlpcel1fXnzdrWxtn5uslxT6NKMZ/oyIQY4k9C1zHK4VBqISDhcJOMvE/3iBrRhmfxpCwUDQUaSpYwS6m48bnTikvh8Anup5rQ8qivmCuP3ayDq/2zlitFDJhe7blq5tv2j50JqVwaN5phO5wWfg6iCjRVZ14rfa+P8xoLkBayokxvShUdlASbRnl4Or93IAidExG0PNQEgFmUE4/6vAnzwxmn/pMVT9t+NkghjCpH4SUHstXmqTcj/ab3cpkeDkmVW5B0ZpTmHNsMT1LDQ6aBWl54QKhm/lZMr4kPw/ps51xMngxBeZs51rLx7W4KhREziu/iCn8ya7uo4yeBvcnO+1o7AdfT9onp5VoS6hD2gLbaMIHaJT9BV1UBdR9Bv9QXfovYQbAQfg83ZaFCrdt6huQpaj+sZvek=</latexit>

ν0

m = 1

2π eB0 mecγ2

m

<latexit sha1_base64="UYKo4KuCHildOG9vdB+lWGBxmjs=">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</latexit><latexit sha1_base64="UYKo4KuCHildOG9vdB+lWGBxmjs=">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</latexit><latexit sha1_base64="UYKo4KuCHildOG9vdB+lWGBxmjs=">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</latexit><latexit sha1_base64="UYKo4KuCHildOG9vdB+lWGBxmjs=">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</latexit>

ν0

c = 1

2π eB0 mec ✓ 6πmec σT B02Γst ◆2

<latexit sha1_base64="/7i6kZR2vRNvB3SzKhz2A+Peo=">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</latexit><latexit sha1_base64="/7i6kZR2vRNvB3SzKhz2A+Peo=">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</latexit><latexit sha1_base64="/7i6kZR2vRNvB3SzKhz2A+Peo=">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</latexit><latexit sha1_base64="/7i6kZR2vRNvB3SzKhz2A+Peo=">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</latexit>

P 0

ν0/P 0 m =

8 > > < > > :

  • ν0

c/ν0 m

(p1)/2 ν0/ν0

c

p/2 (ν0

c < ν0)

  • ν0/ν0

m

(p1)/2 (ν0

m < ν0 < ν0 c)

  • ν0/ν0

m

1/3 (ν0 < ν0

m)

<latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit><latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit><latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit><latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit>

δD = [Γs(1 − βsµ)]−1

<latexit sha1_base64="iNGdlGiua7RIlz4ie5Surjrxfc=">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</latexit><latexit sha1_base64="iNGdlGiua7RIlz4ie5Surjrxfc=">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</latexit><latexit sha1_base64="iNGdlGiua7RIlz4ie5Surjrxfc=">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</latexit><latexit sha1_base64="iNGdlGiua7RIlz4ie5Surjrxfc=">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</latexit>

ν0(µ) = ν/δD(µ)

<latexit sha1_base64="+9WOsh8bmkv7qgmHpPrw1+ahHxU=">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</latexit><latexit sha1_base64="+9WOsh8bmkv7qgmHpPrw1+ahHxU=">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</latexit><latexit sha1_base64="+9WOsh8bmkv7qgmHpPrw1+ahHxU=">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</latexit><latexit sha1_base64="+9WOsh8bmkv7qgmHpPrw1+ahHxU=">ACTHicbVBNaxsxFNS6zZfTJE56KEURExJcoizWwrNpRDaHnpMIU4CXmPeat/GwpJ2kZ5atsan/Jpc2/Re/9HbqVQ2fGhTjIgGbeY54mq5R0FMe/o8aTp0vLK6trzfVnG5tbre2dc1d6K7ArSlXaywcKmwS5IUXlYWQWcKL7LRx6l/8RWtk6U5o7rCvoYrIwspgI0aL1Kjd/bT7U/4O954Edpjopg8GmDVrtuBPwB+SZE7abI7TwXb0Is1L4TUaEgqc6yVxRf0xWJC4aSZeocViBFcYS9QAxpdfz7x4S/DkrOi9KGZ4jP1P83xqCdq3UWJjXQ0N3puJjXs9TcdwfS1N5QiPugqvOJV8WgrPpUVBqg4EhJXhVi6GYEFQqG4hxfksxyrELKgkR98PC6yNBrPofANXh5MnzVBlcr+4h+T8TSeJO8mXt+2TD/NSV9lLtsv2WcLesRP2mZ2yLhPsmt2wH+xn9Cu6jf5Ef+9G9F85zlbQGP5Hy3Lsqk=</latexit>

T = t − R(t)µ c

<latexit sha1_base64="/S9tmXJ9hEJkcnrK2b2Cr2xgW4=">ACSXicbVDLSgMxFM3Ud321uhB0EyCLiwzIuhGEN24VLEqtKVk0jsamSG5I4yDgN+jVv9D7/Az3AnrkxrF1Y9EDicy/n5oSJFBZ9/80rjY1PTE5Nz5Rn5+YXFivVpUsbp4ZDg8cyNtchsyCFhgYKlHCdGAqlHAV9o7/tUdGCtifYFZAm3FbrSIBGfopE5l9YIeUKTbtBUZxvPzTdyiLZUWOS86lZpf9wegf0kwJDUyxGmn6q20ujFPFWjklnbDPwE2zkzKLiEotxKLSM9gNB3VTIFt54NPFHTDKV0axcY9jXSg/tzImbI2U6GbVAxv7W+vL/7nNVOM9tu50EmKoPl3UJRKijHtN0K7wgBHmTnCuBHuVspvmWsDXW8jKTYNu5C4mBEVRe9hO4JMK6ZHnXtmM3dyUXZVBr+L+0sud+qBXw/OdmuHR8NSp8kaWSebJCB75JCckFPSIJw8kifyTF68V+/d+/A+v0dL3nBnmYygNPYFwnaxhQ=</latexit><latexit sha1_base64="/S9tmXJ9hEJkcnrK2b2Cr2xgW4=">ACSXicbVDLSgMxFM3Ud321uhB0EyCLiwzIuhGEN24VLEqtKVk0jsamSG5I4yDgN+jVv9D7/Az3AnrkxrF1Y9EDicy/n5oSJFBZ9/80rjY1PTE5Nz5Rn5+YXFivVpUsbp4ZDg8cyNtchsyCFhgYKlHCdGAqlHAV9o7/tUdGCtifYFZAm3FbrSIBGfopE5l9YIeUKTbtBUZxvPzTdyiLZUWOS86lZpf9wegf0kwJDUyxGmn6q20ujFPFWjklnbDPwE2zkzKLiEotxKLSM9gNB3VTIFt54NPFHTDKV0axcY9jXSg/tzImbI2U6GbVAxv7W+vL/7nNVOM9tu50EmKoPl3UJRKijHtN0K7wgBHmTnCuBHuVspvmWsDXW8jKTYNu5C4mBEVRe9hO4JMK6ZHnXtmM3dyUXZVBr+L+0sud+qBXw/OdmuHR8NSp8kaWSebJCB75JCckFPSIJw8kifyTF68V+/d+/A+v0dL3nBnmYygNPYFwnaxhQ=</latexit><latexit sha1_base64="/S9tmXJ9hEJkcnrK2b2Cr2xgW4=">ACSXicbVDLSgMxFM3Ud321uhB0EyCLiwzIuhGEN24VLEqtKVk0jsamSG5I4yDgN+jVv9D7/Az3AnrkxrF1Y9EDicy/n5oSJFBZ9/80rjY1PTE5Nz5Rn5+YXFivVpUsbp4ZDg8cyNtchsyCFhgYKlHCdGAqlHAV9o7/tUdGCtifYFZAm3FbrSIBGfopE5l9YIeUKTbtBUZxvPzTdyiLZUWOS86lZpf9wegf0kwJDUyxGmn6q20ujFPFWjklnbDPwE2zkzKLiEotxKLSM9gNB3VTIFt54NPFHTDKV0axcY9jXSg/tzImbI2U6GbVAxv7W+vL/7nNVOM9tu50EmKoPl3UJRKijHtN0K7wgBHmTnCuBHuVspvmWsDXW8jKTYNu5C4mBEVRe9hO4JMK6ZHnXtmM3dyUXZVBr+L+0sud+qBXw/OdmuHR8NSp8kaWSebJCB75JCckFPSIJw8kifyTF68V+/d+/A+v0dL3nBnmYygNPYFwnaxhQ=</latexit><latexit sha1_base64="/S9tmXJ9hEJkcnrK2b2Cr2xgW4=">ACSXicbVDLSgMxFM3Ud321uhB0EyCLiwzIuhGEN24VLEqtKVk0jsamSG5I4yDgN+jVv9D7/Az3AnrkxrF1Y9EDicy/n5oSJFBZ9/80rjY1PTE5Nz5Rn5+YXFivVpUsbp4ZDg8cyNtchsyCFhgYKlHCdGAqlHAV9o7/tUdGCtifYFZAm3FbrSIBGfopE5l9YIeUKTbtBUZxvPzTdyiLZUWOS86lZpf9wegf0kwJDUyxGmn6q20ujFPFWjklnbDPwE2zkzKLiEotxKLSM9gNB3VTIFt54NPFHTDKV0axcY9jXSg/tzImbI2U6GbVAxv7W+vL/7nNVOM9tu50EmKoPl3UJRKijHtN0K7wgBHmTnCuBHuVspvmWsDXW8jKTYNu5C4mBEVRe9hO4JMK6ZHnXtmM3dyUXZVBr+L+0sud+qBXw/OdmuHR8NSp8kaWSebJCB75JCckFPSIJw8kifyTF68V+/d+/A+v0dL3nBnmYygNPYFwnaxhQ=</latexit>

ν0

a

<latexit sha1_base64="UKGXT2HWcO/lc0EvbB0Jo8iMgaM=">ACM3icbVDLSsNAFJ34rPXV6sKFm2AR3VgSEXRZdOygn1AW8rN5KYdO5mEmYkS/Brf6HyPuxK3/4PSxMK0XBg7n3Ms5c7yYM6Ud58NaWl5ZXVvPbeQ3t7Z3dgvFvbqKEkmxRiMeyaYHCjkTWNMc2zGEiH0ODa8wc1YbzyiVCwS9zqNsRNCT7CAUdCGqrdF0oWTbqHklJ3J2IvAnYESmU21W7QO2n5EkxCFphyUarlOrDtDkJpRjqN8O1EYAx1AD1sGCghRdYaTuCP72DC+HUTSPKHtCfv3YgihUmnomc0QdF/Na2PyP62V6OCqM2QiTjQKOjUKEm7ryB7/3faZRKp5agBQyUxWm/ZBAtWmoYyLSjwfY2OTYTUbPJ8FmIoQRFZ5ApWayKO8qdKdL24R1M/LrlN27y5KletZqTlySI7IKXHJamQW1IlNULJA3khr+TNerc+rS/re7q6ZM1u9klmrJ9fvQGrPw=</latexit><latexit sha1_base64="UKGXT2HWcO/lc0EvbB0Jo8iMgaM=">ACM3icbVDLSsNAFJ34rPXV6sKFm2AR3VgSEXRZdOygn1AW8rN5KYdO5mEmYkS/Brf6HyPuxK3/4PSxMK0XBg7n3Ms5c7yYM6Ud58NaWl5ZXVvPbeQ3t7Z3dgvFvbqKEkmxRiMeyaYHCjkTWNMc2zGEiH0ODa8wc1YbzyiVCwS9zqNsRNCT7CAUdCGqrdF0oWTbqHklJ3J2IvAnYESmU21W7QO2n5EkxCFphyUarlOrDtDkJpRjqN8O1EYAx1AD1sGCghRdYaTuCP72DC+HUTSPKHtCfv3YgihUmnomc0QdF/Na2PyP62V6OCqM2QiTjQKOjUKEm7ryB7/3faZRKp5agBQyUxWm/ZBAtWmoYyLSjwfY2OTYTUbPJ8FmIoQRFZ5ApWayKO8qdKdL24R1M/LrlN27y5KletZqTlySI7IKXHJamQW1IlNULJA3khr+TNerc+rS/re7q6ZM1u9klmrJ9fvQGrPw=</latexit><latexit sha1_base64="UKGXT2HWcO/lc0EvbB0Jo8iMgaM=">ACM3icbVDLSsNAFJ34rPXV6sKFm2AR3VgSEXRZdOygn1AW8rN5KYdO5mEmYkS/Brf6HyPuxK3/4PSxMK0XBg7n3Ms5c7yYM6Ud58NaWl5ZXVvPbeQ3t7Z3dgvFvbqKEkmxRiMeyaYHCjkTWNMc2zGEiH0ODa8wc1YbzyiVCwS9zqNsRNCT7CAUdCGqrdF0oWTbqHklJ3J2IvAnYESmU21W7QO2n5EkxCFphyUarlOrDtDkJpRjqN8O1EYAx1AD1sGCghRdYaTuCP72DC+HUTSPKHtCfv3YgihUmnomc0QdF/Na2PyP62V6OCqM2QiTjQKOjUKEm7ryB7/3faZRKp5agBQyUxWm/ZBAtWmoYyLSjwfY2OTYTUbPJ8FmIoQRFZ5ApWayKO8qdKdL24R1M/LrlN27y5KletZqTlySI7IKXHJamQW1IlNULJA3khr+TNerc+rS/re7q6ZM1u9klmrJ9fvQGrPw=</latexit><latexit sha1_base64="UKGXT2HWcO/lc0EvbB0Jo8iMgaM=">ACM3icbVDLSsNAFJ34rPXV6sKFm2AR3VgSEXRZdOygn1AW8rN5KYdO5mEmYkS/Brf6HyPuxK3/4PSxMK0XBg7n3Ms5c7yYM6Ud58NaWl5ZXVvPbeQ3t7Z3dgvFvbqKEkmxRiMeyaYHCjkTWNMc2zGEiH0ODa8wc1YbzyiVCwS9zqNsRNCT7CAUdCGqrdF0oWTbqHklJ3J2IvAnYESmU21W7QO2n5EkxCFphyUarlOrDtDkJpRjqN8O1EYAx1AD1sGCghRdYaTuCP72DC+HUTSPKHtCfv3YgihUmnomc0QdF/Na2PyP62V6OCqM2QiTjQKOjUKEm7ryB7/3faZRKp5agBQyUxWm/ZBAtWmoYyLSjwfY2OTYTUbPJ8FmIoQRFZ5ApWayKO8qdKdL24R1M/LrlN27y5KletZqTlySI7IKXHJamQW1IlNULJA3khr+TNerc+rS/re7q6ZM1u9klmrJ9fvQGrPw=</latexit>

B0 =  8⇡✏B ˆ Γs + 1 ˆ − 1 nmpc2(Γs − 1) 1/2

<latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">ACg3icbVFda9swFJW9teuyj3rbY18uC2MdpZkVCutLoWQP3WMHS1uIXCMrciIqyUKSyzLjH7q3/Yb9gimJC027C0KHc8/90FhpHA+TX9H8ZOnW9vPdp73Xrx89Xo3efP2wlW1ZXzMKlnZq4I6LoXmYy+85FfGcqoKyS+Lm6/L/OUt05U+odfGJ4pOtOiFIz6QOXJL7Lq0RSy5i2MPsIJEMlLP4FjYgThxgkZdCMgpaWsIXPqGzKjStGWnC2v3B3gdoM/xC1oULkBdj2E/TvZIf4ExIrZ3GfXDf48bPOknw7SVcBjgDvQR12c58lfMq1Yrbj2TFLnJjg1Pmuo9YJ3vZI7bih7IbO+CRATRV3WbN6XwsfAjOFsrLhaA8r9n5FQ5VzC1UEpaJ+7h7mluT/cpPal8dZI7SpPdsPaisJfgKlobDVFjOvFwEQJkVYVdgcxrM9OFbNjpNb4Vx3dY/12v3gkn4oSWPwcVwgNMB/n7UPx1du2gPfQe7SOMvqBT9A2dozFi6E+0He1GSbwVH8TD+GgtjaOu5h3aiPjkHyW0wq0=</latexit><latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit><latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit><latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit>

B0 = (32⇡✏Bnmp)1/2Γsc

<latexit sha1_base64="5H4dztMyiTi5ZVjTZUcYPil3TSA=">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</latexit><latexit sha1_base64="5H4dztMyiTi5ZVjTZUcYPil3TSA=">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</latexit><latexit sha1_base64="5H4dztMyiTi5ZVjTZUcYPil3TSA=">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</latexit><latexit sha1_base64="5H4dztMyiTi5ZVjTZUcYPil3TSA=">ACPnicbVDLSgMxFM34tr6qLt0Ei6ibOlMV3QhSF7pUsCp06pDJ3GowyYQkI5ahX+LP6Fa/wB9wJ4Irl6btLHwdCBzOufmcGLFmbG+/+INDY+Mjo1PTJampmdm58rzC2cmzTSFBk15qi9iYoAzCQ3LIcLpYGImMN5fHPQ89vQRuWylPbUdAS5EqyNqPEOikqb4f9G7mGpIvrq3trm7VQsRCUYdz5dSxFpNYv82Cj1g0PiRAkMphG5Ypf9fvAf0lQkAoqcByVP8IkpZkAaSknxjQDX9lWTrRlEO3FGYGFKE35AqajkoiwLTyfrQuXnFKgtupdk9a3Fe/b+REGNMRsZsUxF6b315P/M9rZra928qZVJkFSQcftTObYp7XeGEaCWdxwhVDOXFdNrogm1rtEfl5JbpkyR+m4Qu+RKCn5X8pec1aqBXw1Otir79aKuCbSEltEaCtAO2kdH6Bg1EX36BE9oWfvwXv13rz3weiQV+wsoh/wPr8Ap92vGQ=</latexit>

µ = cos θ cos θv + sin θ cos ϕ sin θv

<latexit sha1_base64="u2uTaMR6YExIAtRUz07Gi6cb7iU=">ACU3icbVDNSgMxGEzXqrVarXr0EiyCIJRdEfQiFL2Ipwr2B7rLk3TNpjNLsm3xbLsO/kyCp70Gbx6MW30B8HAsPMfMmXCWLBNdj2Z8HaKG5ubZd2yrt7lf2D6uFRW0eJoqxFIxGpbkA0E1yFnAQrBsrRsJAsE7wcj/1O2OmNI/kM0xi5oVkKPmAUwJG8quPbpjgW+zSKcujBiQbIH74wxfYFdzuWyOiYpHPFswTNKv1uy6PQNeJ05OaihH06/+uP2IJiGTQAXRufYMXgpUcCpYFnZTSLCX0hQ9YzVJKQaS+d/TnDZ0bp40GkzJGAZ+riREpCrSdhYJIhgZFe9abif14vgcGNl3IZJ8AknT80SASGCE8LxH2uGAUxMYRQxc2umI6IhRMzUs39c81vnWr/O1y6YkZ7WSdK+rDt23Xm6qjXu8rpK6ASdonPkoGvUQA+oiVqIojf0gb7Qd+G98GtZVnEetQr5zDFaglX5A4o9t9c=</latexit><latexit sha1_base64="u2uTaMR6YExIAtRUz07Gi6cb7iU=">ACU3icbVDNSgMxGEzXqrVarXr0EiyCIJRdEfQiFL2Ipwr2B7rLk3TNpjNLsm3xbLsO/kyCp70Gbx6MW30B8HAsPMfMmXCWLBNdj2Z8HaKG5ubZd2yrt7lf2D6uFRW0eJoqxFIxGpbkA0E1yFnAQrBsrRsJAsE7wcj/1O2OmNI/kM0xi5oVkKPmAUwJG8quPbpjgW+zSKcujBiQbIH74wxfYFdzuWyOiYpHPFswTNKv1uy6PQNeJ05OaihH06/+uP2IJiGTQAXRufYMXgpUcCpYFnZTSLCX0hQ9YzVJKQaS+d/TnDZ0bp40GkzJGAZ+riREpCrSdhYJIhgZFe9abif14vgcGNl3IZJ8AknT80SASGCE8LxH2uGAUxMYRQxc2umI6IhRMzUs39c81vnWr/O1y6YkZ7WSdK+rDt23Xm6qjXu8rpK6ASdonPkoGvUQA+oiVqIojf0gb7Qd+G98GtZVnEetQr5zDFaglX5A4o9t9c=</latexit><latexit sha1_base64="u2uTaMR6YExIAtRUz07Gi6cb7iU=">ACU3icbVDNSgMxGEzXqrVarXr0EiyCIJRdEfQiFL2Ipwr2B7rLk3TNpjNLsm3xbLsO/kyCp70Gbx6MW30B8HAsPMfMmXCWLBNdj2Z8HaKG5ubZd2yrt7lf2D6uFRW0eJoqxFIxGpbkA0E1yFnAQrBsrRsJAsE7wcj/1O2OmNI/kM0xi5oVkKPmAUwJG8quPbpjgW+zSKcujBiQbIH74wxfYFdzuWyOiYpHPFswTNKv1uy6PQNeJ05OaihH06/+uP2IJiGTQAXRufYMXgpUcCpYFnZTSLCX0hQ9YzVJKQaS+d/TnDZ0bp40GkzJGAZ+riREpCrSdhYJIhgZFe9abif14vgcGNl3IZJ8AknT80SASGCE8LxH2uGAUxMYRQxc2umI6IhRMzUs39c81vnWr/O1y6YkZ7WSdK+rDt23Xm6qjXu8rpK6ASdonPkoGvUQA+oiVqIojf0gb7Qd+G98GtZVnEetQr5zDFaglX5A4o9t9c=</latexit><latexit sha1_base64="u2uTaMR6YExIAtRUz07Gi6cb7iU=">ACU3icbVDNSgMxGEzXqrVarXr0EiyCIJRdEfQiFL2Ipwr2B7rLk3TNpjNLsm3xbLsO/kyCp70Gbx6MW30B8HAsPMfMmXCWLBNdj2Z8HaKG5ubZd2yrt7lf2D6uFRW0eJoqxFIxGpbkA0E1yFnAQrBsrRsJAsE7wcj/1O2OmNI/kM0xi5oVkKPmAUwJG8quPbpjgW+zSKcujBiQbIH74wxfYFdzuWyOiYpHPFswTNKv1uy6PQNeJ05OaihH06/+uP2IJiGTQAXRufYMXgpUcCpYFnZTSLCX0hQ9YzVJKQaS+d/TnDZ0bp40GkzJGAZ+riREpCrSdhYJIhgZFe9abif14vgcGNl3IZJ8AknT80SASGCE8LxH2uGAUxMYRQxc2umI6IhRMzUs39c81vnWr/O1y6YkZ7WSdK+rDt23Xm6qjXu8rpK6ASdonPkoGvUQA+oiVqIojf0gb7Qd+G98GtZVnEetQr5zDFaglX5A4o9t9c=</latexit>

Fν(ν, T) = 1 4πd2 Z 2π dϕ 2π Z π sin θdθ 2 δ3

D(µ) P 0 ν0(ν0, t)

<latexit sha1_base64="7jgmzP1baAkH1lByzWY9lveY=">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</latexit><latexit sha1_base64="7jgmzP1baAkH1lByzWY9lveY=">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</latexit><latexit sha1_base64="7jgmzP1baAkH1lByzWY9lveY=">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</latexit><latexit sha1_base64="7jgmzP1baAkH1lByzWY9lveY=">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</latexit>

Trans-relativistic Relativistic limit (standard GRB theory)

slide-10
SLIDE 10

LIN Haoxiang - High Energy Astrophysics 2018

Best-fit model (Jet)

n=10-2.5 cm-3, ϵB=10-2.3, ϵe=10-1.5, p=2.13, Ec,iso=1051.70 erg, Γc=229, θc=0.11, θv=0.58 n=10-2.5 cm-3, ϵB=10-4.7, ϵe=10-1.2, p=2.18, Ec,iso=1052.59 erg, Γc=242, θc=0.09, θv=0.45, ηe=0.14 f = 1 with similar parameters of previous fits

Ndata = 54

<latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit><latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit><latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit><latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit>

χ2

min ∼ 137

<latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="hP+6LrUf2d3tZaldqaQvEKMXyw=">AB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i7+gsTLXTzQrMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GS7KDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RtRxzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D7odBu3wMYA6nMFXEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit><latexit sha1_base64="cK3jB5LP0pjDdN0rPK1f/ARgEbI=">AB93icbZBLSwMxFIXv1FetVUd34iZYBFdlpi7qUnDjsoJ9QGcMmDU0yQ5IRylDc+FfcuFDEf+LOf2P6WGjrgcDHOQk398QZ9p43rdT2tjc2t4p71b2qvsHh+5RtaPTXBHaJilPVS/GmnImadsw2kvUxSLmNuPL6Z5d1HqjRL5b2ZDQUeChZwg21orck4CMWFQESiDB5PShgQLNBPIvm5Fb8+reXGgd/CXUYKlW5H4Fg5TkgkpDONa673uZCQusDCOcTitBrmGyRgPad+ixILqsJivMEXn1hmgJFX2SIPm7u8XBRZaT0RsbwpsRno1m5n/Zf3cJFdhwWSWGyrJYlCSc2RSNOsDZixPCJBUwUs39FZIQVJsa2VrEl+Ksr0OnUfe9un/nQRlO4QwuwIcmXMtKANBJ7gBd7g3Xl2Xp2PRV0lZ9nbMfyR8/kDH/mUzg=</latexit><latexit sha1_base64="cK3jB5LP0pjDdN0rPK1f/ARgEbI=">AB93icbZBLSwMxFIXv1FetVUd34iZYBFdlpi7qUnDjsoJ9QGcMmDU0yQ5IRylDc+FfcuFDEf+LOf2P6WGjrgcDHOQk398QZ9p43rdT2tjc2t4p71b2qvsHh+5RtaPTXBHaJilPVS/GmnImadsw2kvUxSLmNuPL6Z5d1HqjRL5b2ZDQUeChZwg21orck4CMWFQESiDB5PShgQLNBPIvm5Fb8+reXGgd/CXUYKlW5H4Fg5TkgkpDONa673uZCQusDCOcTitBrmGyRgPad+ixILqsJivMEXn1hmgJFX2SIPm7u8XBRZaT0RsbwpsRno1m5n/Zf3cJFdhwWSWGyrJYlCSc2RSNOsDZixPCJBUwUs39FZIQVJsa2VrEl+Ksr0OnUfe9un/nQRlO4QwuwIcmXMtKANBJ7gBd7g3Xl2Xp2PRV0lZ9nbMfyR8/kDH/mUzg=</latexit><latexit sha1_base64="URbQW9lTcqliEjzDkQ6vCsr3xmY=">ACAnicbVA9SwNBEJ2LXzF+nVqJzWIQrMJdLGIZtLGMYBIhd4a9zSZsrt37O4J4Qg2/hUbC0Vs/RV2/hs3yRWa+GDg8d4M/OihDNtPO/bKaysrq1vFDdLW9s7u3vu/kFLx6kitEliHqu7CGvKmaRNwynd4miWESctqPR1dRvP1ClWSxvzTihocADyfqMYGOlrnsUkCHrZoESDA5ua+iQDOB/PNa1y17FW8GtEz8nJQhR6PrfgW9mKSCSkM41rje4kJM6wMI5xOSkGqaYLJCA9ox1KJBdVhNnthgk6t0kP9WNmSBs3U3xMZFlqPRWQ7BTZDvehNxf+8Tmr6F2HGZJIaKsl8UT/lyMRomgfqMUWJ4WNLMFHM3orIECtMjE2tZEPwF19eJq1qxfcq/o1Xrl/mcRThGE7gDHyoQR2uoQFNIPAIz/AKb86T8+K8Ox/z1oKTzxzCHzifP5Roljc=</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit><latexit sha1_base64="7ukQAwN5/soJrjdphHdIKUscX+c=">ACAnicbVDLSgNBEOyNrxhfU/iZTAInsJuFOIx6MVjBPOA7LrMTibJkJnZWZWCEvw4q948aCIV7/Cm3/j5HQxIKGoqb7q4o4Uwb1/12ciura+sb+c3C1vbO7l5x/6Cp41QR2iAxj1U7wpyJmnDMNpO1EUi4jTVjS8nvitB6o0i+WdGSU0ELgvWY8RbKwUFo98MmBh5iuBJPj+wryNRPIO6+GxZJbdqdAy8SbkxLMUQ+LX343Jqmg0hCOte54bmKCDCvDCKfjgp9qmAyxH3asVRiQXWQTV8Yo1OrdFEvVrakQVP190SGhdYjEdlOgc1AL3oT8T+vk5reZAxmaSGSjJb1Es5MjGa5IG6TFi+MgSTBSztyIywAoTY1Mr2BC8xZeXSbNS9tyd3tRql3N48jDMZzAGXhQhRrcQB0aQOARnuEV3pwn58V5dz5mrTlnPnMIf+B8/gCVqJY7</latexit>

χ2

min ∼ 108

<latexit sha1_base64="HP0O+f0Ra8/N/uZ2EukawKCxde4=">ACAnicbVA9SwNBEJ2LXzF+nVqJzWIQrMJdEwZtLGMYBIhd4a9zSZsrt37O4J4Qg2/hUbC0Vs/RV2/hs3yRWa+GDg8d4M/OihDNtPO/bKaysrq1vFDdLW9s7u3vu/kFLx6kitEliHqu7CGvKmaRNwynd4miWESctqPR1dRvP1ClWSxvzTihocADyfqMYGOlrnsUkCHrZoESDA5ua+iQDOBfK/WdctexZsBLRM/J2XI0ei6X0EvJqmg0hCOte74XmLCDCvDCKeTUpBqmAywgPasVRiQXWYzV6YoFOr9FA/VrakQTP190SGhdZjEdlOgc1QL3pT8T+vk5p+LcyYTFJDJZkv6qcmRhN80A9pigxfGwJorZWxEZYoWJsamVbAj+4svLpFWt+F7Fvzkv1y/zOIpwDCdwBj5cQB2uoQFNIPAIz/AKb86T8+K8Ox/z1oKTzxzCHzifP5Kdljk=</latexit><latexit sha1_base64="HP0O+f0Ra8/N/uZ2EukawKCxde4=">ACAnicbVA9SwNBEJ2LXzF+nVqJzWIQrMJdEwZtLGMYBIhd4a9zSZsrt37O4J4Qg2/hUbC0Vs/RV2/hs3yRWa+GDg8d4M/OihDNtPO/bKaysrq1vFDdLW9s7u3vu/kFLx6kitEliHqu7CGvKmaRNwynd4miWESctqPR1dRvP1ClWSxvzTihocADyfqMYGOlrnsUkCHrZoESDA5ua+iQDOBfK/WdctexZsBLRM/J2XI0ei6X0EvJqmg0hCOte74XmLCDCvDCKeTUpBqmAywgPasVRiQXWYzV6YoFOr9FA/VrakQTP190SGhdZjEdlOgc1QL3pT8T+vk5p+LcyYTFJDJZkv6qcmRhN80A9pigxfGwJorZWxEZYoWJsamVbAj+4svLpFWt+F7Fvzkv1y/zOIpwDCdwBj5cQB2uoQFNIPAIz/AKb86T8+K8Ox/z1oKTzxzCHzifP5Kdljk=</latexit><latexit sha1_base64="HP0O+f0Ra8/N/uZ2EukawKCxde4=">ACAnicbVA9SwNBEJ2LXzF+nVqJzWIQrMJdEwZtLGMYBIhd4a9zSZsrt37O4J4Qg2/hUbC0Vs/RV2/hs3yRWa+GDg8d4M/OihDNtPO/bKaysrq1vFDdLW9s7u3vu/kFLx6kitEliHqu7CGvKmaRNwynd4miWESctqPR1dRvP1ClWSxvzTihocADyfqMYGOlrnsUkCHrZoESDA5ua+iQDOBfK/WdctexZsBLRM/J2XI0ei6X0EvJqmg0hCOte74XmLCDCvDCKeTUpBqmAywgPasVRiQXWYzV6YoFOr9FA/VrakQTP190SGhdZjEdlOgc1QL3pT8T+vk5p+LcyYTFJDJZkv6qcmRhN80A9pigxfGwJorZWxEZYoWJsamVbAj+4svLpFWt+F7Fvzkv1y/zOIpwDCdwBj5cQB2uoQFNIPAIz/AKb86T8+K8Ox/z1oKTzxzCHzifP5Kdljk=</latexit><latexit sha1_base64="HP0O+f0Ra8/N/uZ2EukawKCxde4=">ACAnicbVA9SwNBEJ2LXzF+nVqJzWIQrMJdEwZtLGMYBIhd4a9zSZsrt37O4J4Qg2/hUbC0Vs/RV2/hs3yRWa+GDg8d4M/OihDNtPO/bKaysrq1vFDdLW9s7u3vu/kFLx6kitEliHqu7CGvKmaRNwynd4miWESctqPR1dRvP1ClWSxvzTihocADyfqMYGOlrnsUkCHrZoESDA5ua+iQDOBfK/WdctexZsBLRM/J2XI0ei6X0EvJqmg0hCOte74XmLCDCvDCKeTUpBqmAywgPasVRiQXWYzV6YoFOr9FA/VrakQTP190SGhdZjEdlOgc1QL3pT8T+vk5p+LcyYTFJDJZkv6qcmRhN80A9pigxfGwJorZWxEZYoWJsamVbAj+4svLpFWt+F7Fvzkv1y/zOIpwDCdwBj5cQB2uoQFNIPAIz/AKb86T8+K8Ox/z1oKTzxzCHzifP5Kdljk=</latexit>

New fit with radio in synchrotron tail (ν < νm) Best-fit free f model by MCMC

∆χ2

min ∼ 29 !

<latexit sha1_base64="Rpu1DIJF1bO1A53jPa1NYOeikU=">ACLXicbVDNSgMxGMzWv1r/Vj16iVbBg5TdIqi3oh48VrA/0F2XbJq2oUl2SbLFsvQFfBm96nt4EMSrd5/AtN2DrQ4EhpnvSyYTxowq7TjvVm5hcWl5Jb9aWFvf2Nyt3fqKkokJjUcsUg2Q6QIo4LUNWMNGNJEA8ZaYT9q7HfGBCpaCTu9DAmPkdQTsUI2kwD70rgnTCHq4R4PUkxyKkb3ZegpymH5AnoncD+wi07JmQD+JW5GiBDNbC/vXaE06Exgwp1XKdWPspkpiRkYFL1EkRriPuqRlqECcKD+d/GYEj4zShp1ImiM0nKi/N1LElRry0ExypHtq3huL/3mtRHfO/ZSKONFE4OlDnYRBHcFxNbBNJcGaDQ1BWFKTFeIekghrU+DMTe0BjVW+mEau2BKcucr+Uvq5ZLrlNzb02LlMqsrD/bATgGLjgDFXADqAGMHgEz+AFvFpP1pv1YX1OR3NWtrMLZmB9/QDM+6eI</latexit><latexit sha1_base64="Rpu1DIJF1bO1A53jPa1NYOeikU=">ACLXicbVDNSgMxGMzWv1r/Vj16iVbBg5TdIqi3oh48VrA/0F2XbJq2oUl2SbLFsvQFfBm96nt4EMSrd5/AtN2DrQ4EhpnvSyYTxowq7TjvVm5hcWl5Jb9aWFvf2Nyt3fqKkokJjUcsUg2Q6QIo4LUNWMNGNJEA8ZaYT9q7HfGBCpaCTu9DAmPkdQTsUI2kwD70rgnTCHq4R4PUkxyKkb3ZegpymH5AnoncD+wi07JmQD+JW5GiBDNbC/vXaE06Exgwp1XKdWPspkpiRkYFL1EkRriPuqRlqECcKD+d/GYEj4zShp1ImiM0nKi/N1LElRry0ExypHtq3huL/3mtRHfO/ZSKONFE4OlDnYRBHcFxNbBNJcGaDQ1BWFKTFeIekghrU+DMTe0BjVW+mEau2BKcucr+Uvq5ZLrlNzb02LlMqsrD/bATgGLjgDFXADqAGMHgEz+AFvFpP1pv1YX1OR3NWtrMLZmB9/QDM+6eI</latexit><latexit sha1_base64="Rpu1DIJF1bO1A53jPa1NYOeikU=">ACLXicbVDNSgMxGMzWv1r/Vj16iVbBg5TdIqi3oh48VrA/0F2XbJq2oUl2SbLFsvQFfBm96nt4EMSrd5/AtN2DrQ4EhpnvSyYTxowq7TjvVm5hcWl5Jb9aWFvf2Nyt3fqKkokJjUcsUg2Q6QIo4LUNWMNGNJEA8ZaYT9q7HfGBCpaCTu9DAmPkdQTsUI2kwD70rgnTCHq4R4PUkxyKkb3ZegpymH5AnoncD+wi07JmQD+JW5GiBDNbC/vXaE06Exgwp1XKdWPspkpiRkYFL1EkRriPuqRlqECcKD+d/GYEj4zShp1ImiM0nKi/N1LElRry0ExypHtq3huL/3mtRHfO/ZSKONFE4OlDnYRBHcFxNbBNJcGaDQ1BWFKTFeIekghrU+DMTe0BjVW+mEau2BKcucr+Uvq5ZLrlNzb02LlMqsrD/bATgGLjgDFXADqAGMHgEz+AFvFpP1pv1YX1OR3NWtrMLZmB9/QDM+6eI</latexit><latexit sha1_base64="Rpu1DIJF1bO1A53jPa1NYOeikU=">ACLXicbVDNSgMxGMzWv1r/Vj16iVbBg5TdIqi3oh48VrA/0F2XbJq2oUl2SbLFsvQFfBm96nt4EMSrd5/AtN2DrQ4EhpnvSyYTxowq7TjvVm5hcWl5Jb9aWFvf2Nyt3fqKkokJjUcsUg2Q6QIo4LUNWMNGNJEA8ZaYT9q7HfGBCpaCTu9DAmPkdQTsUI2kwD70rgnTCHq4R4PUkxyKkb3ZegpymH5AnoncD+wi07JmQD+JW5GiBDNbC/vXaE06Exgwp1XKdWPspkpiRkYFL1EkRriPuqRlqECcKD+d/GYEj4zShp1ImiM0nKi/N1LElRry0ExypHtq3huL/3mtRHfO/ZSKONFE4OlDnYRBHcFxNbBNJcGaDQ1BWFKTFeIekghrU+DMTe0BjVW+mEau2BKcucr+Uvq5ZLrlNzb02LlMqsrD/bATgGLjgDFXADqAGMHgEz+AFvFpP1pv1YX1OR3NWtrMLZmB9/QDM+6eI</latexit>

with 1 more model parameter

Constant single power-law spectrum (ν > νm)

slide-11
SLIDE 11

LIN Haoxiang - High Energy Astrophysics 2018

Best-fit model (Sph)

n=10-3.2 cm-3, ϵB=10-1.1, ϵe=10-1.2, p=2.15, Ek,iso=1049.68 erg, umax=2.9, umin=1.7, k=5.7 n=10-1.8 cm-3, ϵB=10-2.2, ϵe=10-1.3, p=2.16, Ek,iso=1049.71 erg, umax=2.6, umin=1.2, k=6.2, ηe=0.3

χ2

min ∼ 127

<latexit sha1_base64="J0f+pmL/MP0xhHn4b0tckgLr1o=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdItRj0YvHCvYHunXJZrNtaJdkmyxLMWX0au+hzfxJPgUPoFpuwfbOhAYZr4vmUyQMKq043xZhbX1jc2t4nZpZ3dv/8A+PGqpOJWYNHMYtkJkCKMCtLUVDPSRBPGCkHQxvpn57RKSisbjX4T0OoLGlGMtJF8+8TDA/pQ9TNPcsipmEBPUQ7das23y07FmQGuEjcnZCj4ds/XhjlBOhMUNKdV0n0b0MSU0xI5OSlyqSIDxEfdI1VCBOVC+bfWECz40SwiW5gNZ+rfjQxpcY8MJMc6YFa9qbif1431dFVL6MiSTUReP5QlDKoYzjtA4ZUEqzZ2BCEJTVZIR4gibA2rS3cFI5ovLUj/PYJVOSu1zJKmlVK65Tce8uy/XrvK4iOAVn4AK4oAbq4BY0QBNg8ARewCt4s56td+vD+pyPFqx85xgswPr+BZ3vo/o=</latexit><latexit sha1_base64="J0f+pmL/MP0xhHn4b0tckgLr1o=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdItRj0YvHCvYHunXJZrNtaJdkmyxLMWX0au+hzfxJPgUPoFpuwfbOhAYZr4vmUyQMKq043xZhbX1jc2t4nZpZ3dv/8A+PGqpOJWYNHMYtkJkCKMCtLUVDPSRBPGCkHQxvpn57RKSisbjX4T0OoLGlGMtJF8+8TDA/pQ9TNPcsipmEBPUQ7das23y07FmQGuEjcnZCj4ds/XhjlBOhMUNKdV0n0b0MSU0xI5OSlyqSIDxEfdI1VCBOVC+bfWECz40SwiW5gNZ+rfjQxpcY8MJMc6YFa9qbif1431dFVL6MiSTUReP5QlDKoYzjtA4ZUEqzZ2BCEJTVZIR4gibA2rS3cFI5ovLUj/PYJVOSu1zJKmlVK65Tce8uy/XrvK4iOAVn4AK4oAbq4BY0QBNg8ARewCt4s56td+vD+pyPFqx85xgswPr+BZ3vo/o=</latexit><latexit sha1_base64="J0f+pmL/MP0xhHn4b0tckgLr1o=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdItRj0YvHCvYHunXJZrNtaJdkmyxLMWX0au+hzfxJPgUPoFpuwfbOhAYZr4vmUyQMKq043xZhbX1jc2t4nZpZ3dv/8A+PGqpOJWYNHMYtkJkCKMCtLUVDPSRBPGCkHQxvpn57RKSisbjX4T0OoLGlGMtJF8+8TDA/pQ9TNPcsipmEBPUQ7das23y07FmQGuEjcnZCj4ds/XhjlBOhMUNKdV0n0b0MSU0xI5OSlyqSIDxEfdI1VCBOVC+bfWECz40SwiW5gNZ+rfjQxpcY8MJMc6YFa9qbif1431dFVL6MiSTUReP5QlDKoYzjtA4ZUEqzZ2BCEJTVZIR4gibA2rS3cFI5ovLUj/PYJVOSu1zJKmlVK65Tce8uy/XrvK4iOAVn4AK4oAbq4BY0QBNg8ARewCt4s56td+vD+pyPFqx85xgswPr+BZ3vo/o=</latexit><latexit sha1_base64="J0f+pmL/MP0xhHn4b0tckgLr1o=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdItRj0YvHCvYHunXJZrNtaJdkmyxLMWX0au+hzfxJPgUPoFpuwfbOhAYZr4vmUyQMKq043xZhbX1jc2t4nZpZ3dv/8A+PGqpOJWYNHMYtkJkCKMCtLUVDPSRBPGCkHQxvpn57RKSisbjX4T0OoLGlGMtJF8+8TDA/pQ9TNPcsipmEBPUQ7das23y07FmQGuEjcnZCj4ds/XhjlBOhMUNKdV0n0b0MSU0xI5OSlyqSIDxEfdI1VCBOVC+bfWECz40SwiW5gNZ+rfjQxpcY8MJMc6YFa9qbif1431dFVL6MiSTUReP5QlDKoYzjtA4ZUEqzZ2BCEJTVZIR4gibA2rS3cFI5ovLUj/PYJVOSu1zJKmlVK65Tce8uy/XrvK4iOAVn4AK4oAbq4BY0QBNg8ARewCt4s56td+vD+pyPFqx85xgswPr+BZ3vo/o=</latexit>

χ2

min ∼ 115

<latexit sha1_base64="b3LeyzoYh7wcs6kszi8wYstj1Kc=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdouix6MVjBfsD3XJZtM2NMkuSbZYluL6FXfw5t4EnwKn8C03YNtHQgM9+XTCZMGFXacb6swsrq2vpGcbO0tb2zu2fvHzRVnEpMGjhmsWyHSBFGBWloqhlpJ5IgHjLSCgc3E781JFLRWNzrUJ8jnqCdilG2kiBfeThPn2oBpknOeRUjKGnKIeuexHYZafiTAGXiZuTMshRD+wfL4pxyonQmCGlOq6TaD9DUlPMyLjkpYokCA9Qj3QMFYgT5WfTL4zhqVEi2I2lOULDqfp3I0NcqREPzSRHuq8WvYn4n9dJdfKz6hIUk0Enj3UTRnUMZz0ASMqCdZsZAjCkpqsEPeRFib1uZuioY0UXnqx1nskinJXaxkmTSrFdepuHfn5dp1XlcRHIMTcAZcAlq4BbUQNg8ARewCt4s56td+vD+pyNFqx85xDMwfr+BZkCo/c=</latexit><latexit sha1_base64="b3LeyzoYh7wcs6kszi8wYstj1Kc=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdouix6MVjBfsD3XJZtM2NMkuSbZYluL6FXfw5t4EnwKn8C03YNtHQgM9+XTCZMGFXacb6swsrq2vpGcbO0tb2zu2fvHzRVnEpMGjhmsWyHSBFGBWloqhlpJ5IgHjLSCgc3E781JFLRWNzrUJ8jnqCdilG2kiBfeThPn2oBpknOeRUjKGnKIeuexHYZafiTAGXiZuTMshRD+wfL4pxyonQmCGlOq6TaD9DUlPMyLjkpYokCA9Qj3QMFYgT5WfTL4zhqVEi2I2lOULDqfp3I0NcqREPzSRHuq8WvYn4n9dJdfKz6hIUk0Enj3UTRnUMZz0ASMqCdZsZAjCkpqsEPeRFib1uZuioY0UXnqx1nskinJXaxkmTSrFdepuHfn5dp1XlcRHIMTcAZcAlq4BbUQNg8ARewCt4s56td+vD+pyNFqx85xDMwfr+BZkCo/c=</latexit><latexit sha1_base64="b3LeyzoYh7wcs6kszi8wYstj1Kc=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdouix6MVjBfsD3XJZtM2NMkuSbZYluL6FXfw5t4EnwKn8C03YNtHQgM9+XTCZMGFXacb6swsrq2vpGcbO0tb2zu2fvHzRVnEpMGjhmsWyHSBFGBWloqhlpJ5IgHjLSCgc3E781JFLRWNzrUJ8jnqCdilG2kiBfeThPn2oBpknOeRUjKGnKIeuexHYZafiTAGXiZuTMshRD+wfL4pxyonQmCGlOq6TaD9DUlPMyLjkpYokCA9Qj3QMFYgT5WfTL4zhqVEi2I2lOULDqfp3I0NcqREPzSRHuq8WvYn4n9dJdfKz6hIUk0Enj3UTRnUMZz0ASMqCdZsZAjCkpqsEPeRFib1uZuioY0UXnqx1nskinJXaxkmTSrFdepuHfn5dp1XlcRHIMTcAZcAlq4BbUQNg8ARewCt4s56td+vD+pyNFqx85xDMwfr+BZkCo/c=</latexit><latexit sha1_base64="b3LeyzoYh7wcs6kszi8wYstj1Kc=">ACInicbVDNSgMxGMzWv1r/Vj2Jl2ARPJXdouix6MVjBfsD3XJZtM2NMkuSbZYluL6FXfw5t4EnwKn8C03YNtHQgM9+XTCZMGFXacb6swsrq2vpGcbO0tb2zu2fvHzRVnEpMGjhmsWyHSBFGBWloqhlpJ5IgHjLSCgc3E781JFLRWNzrUJ8jnqCdilG2kiBfeThPn2oBpknOeRUjKGnKIeuexHYZafiTAGXiZuTMshRD+wfL4pxyonQmCGlOq6TaD9DUlPMyLjkpYokCA9Qj3QMFYgT5WfTL4zhqVEi2I2lOULDqfp3I0NcqREPzSRHuq8WvYn4n9dJdfKz6hIUk0Enj3UTRnUMZz0ASMqCdZsZAjCkpqsEPeRFib1uZuioY0UXnqx1nskinJXaxkmTSrFdepuHfn5dp1XlcRHIMTcAZcAlq4BbUQNg8ARewCt4s56td+vD+pyNFqx85xDMwfr+BZkCo/c=</latexit>

New fit with radio in synchrotron tail (ν < νm) Best-fit free f model by MCMC

Ndata = 54

<latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit><latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit><latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit><latexit sha1_base64="I7aRM/dr0mS9n0SPejfubYtUG9E=">ACQHicbVDBatAEF2lbZo6TWKnhx56WIKvcRIwaW5BEJ6amkUCcB25jRapQs3l2J3VGNIvQpub/0b/oH+QWeu2pa8eHyu7AwO9Gd7jxbmSjsLwV7Dx5OmzedbL1rbL3d29qd/QuXFVbgQGQqs1cxOFTS4IAkKbzKLYKOFV7G049z/fIbWicz85XKHMcaro1MpQDy1KTd+TypRlbzBAhqfsLf9yftbtgLF8PXQbQEXbac80kneD1KMlFoNCQUODeMwpzGFViSQmHdGhUOcxBTuMahwY0unG1yF7zt5JeJpZv4b4gv3owLtXKljf6mBbtyqNif/pw0LSo/HlTR5QWjEo1FaKE4ZnxfBE2lRkCo9AGlz8rFDVgQ5OtquLgiTjD3Ng2W5PT2MXSaDBNZQau9JHrlq8yWi1uHVwc9aKwF3pd0/PlqVusTfsgL1jEfvATtknds4GTLAZu2Pf2Y/gZ3AfPAS/H083guXPK9aY4M9fAdqvOA=</latexit>

∆χ2

min ∼ 12

<latexit sha1_base64="B3U1VPCAdREFQWvahNn6V0udoI=">ACKHicbVC9TsMwGHTKXyl/AUYGLCokpiqpkGCsgIGxSPRHakLkOG5r1XYi26mo68DKzwHmyoK4/AE+C2GWjLSZOd9ny9MGFXacSZWYW19Y3OruF3a2d3bP7APj5oqTiUmDRyzWLZDpAijgjQ01Yy0E0kQDxlphYPbqd8aEqloLB71KCE+Rz1BuxQjbaTAPvXuCNMIerhPg8yTHIqxk9V6CnKoVsN7LJTcWaAq8TNSRnkqAf2jxfFOVEaMyQUh3XSbSfIakpZmRc8lJFEoQHqEc6hgrEifKz2UfG8NwoEezG0hyh4Uz9u5EhrtSIh2aSI91Xy95U/M/rpLp7WdUJKkmAs8f6qYM6hOW4ERlQRrNjIEYUlNVoj7SCKsTXcLN0VDmqg89fM8dsmU5C5Xskqa1YrVNyHy3LtJq+rCE7AGbgALrgCNXAP6qABMHgBb+AdfFiv1qf1ZU3mowUr3zkGC7C+fwFOgaZl</latexit><latexit sha1_base64="B3U1VPCAdREFQWvahNn6V0udoI=">ACKHicbVC9TsMwGHTKXyl/AUYGLCokpiqpkGCsgIGxSPRHakLkOG5r1XYi26mo68DKzwHmyoK4/AE+C2GWjLSZOd9ny9MGFXacSZWYW19Y3OruF3a2d3bP7APj5oqTiUmDRyzWLZDpAijgjQ01Yy0E0kQDxlphYPbqd8aEqloLB71KCE+Rz1BuxQjbaTAPvXuCNMIerhPg8yTHIqxk9V6CnKoVsN7LJTcWaAq8TNSRnkqAf2jxfFOVEaMyQUh3XSbSfIakpZmRc8lJFEoQHqEc6hgrEifKz2UfG8NwoEezG0hyh4Uz9u5EhrtSIh2aSI91Xy95U/M/rpLp7WdUJKkmAs8f6qYM6hOW4ERlQRrNjIEYUlNVoj7SCKsTXcLN0VDmqg89fM8dsmU5C5Xskqa1YrVNyHy3LtJq+rCE7AGbgALrgCNXAP6qABMHgBb+AdfFiv1qf1ZU3mowUr3zkGC7C+fwFOgaZl</latexit><latexit sha1_base64="B3U1VPCAdREFQWvahNn6V0udoI=">ACKHicbVC9TsMwGHTKXyl/AUYGLCokpiqpkGCsgIGxSPRHakLkOG5r1XYi26mo68DKzwHmyoK4/AE+C2GWjLSZOd9ny9MGFXacSZWYW19Y3OruF3a2d3bP7APj5oqTiUmDRyzWLZDpAijgjQ01Yy0E0kQDxlphYPbqd8aEqloLB71KCE+Rz1BuxQjbaTAPvXuCNMIerhPg8yTHIqxk9V6CnKoVsN7LJTcWaAq8TNSRnkqAf2jxfFOVEaMyQUh3XSbSfIakpZmRc8lJFEoQHqEc6hgrEifKz2UfG8NwoEezG0hyh4Uz9u5EhrtSIh2aSI91Xy95U/M/rpLp7WdUJKkmAs8f6qYM6hOW4ERlQRrNjIEYUlNVoj7SCKsTXcLN0VDmqg89fM8dsmU5C5Xskqa1YrVNyHy3LtJq+rCE7AGbgALrgCNXAP6qABMHgBb+AdfFiv1qf1ZU3mowUr3zkGC7C+fwFOgaZl</latexit><latexit sha1_base64="B3U1VPCAdREFQWvahNn6V0udoI=">ACKHicbVC9TsMwGHTKXyl/AUYGLCokpiqpkGCsgIGxSPRHakLkOG5r1XYi26mo68DKzwHmyoK4/AE+C2GWjLSZOd9ny9MGFXacSZWYW19Y3OruF3a2d3bP7APj5oqTiUmDRyzWLZDpAijgjQ01Yy0E0kQDxlphYPbqd8aEqloLB71KCE+Rz1BuxQjbaTAPvXuCNMIerhPg8yTHIqxk9V6CnKoVsN7LJTcWaAq8TNSRnkqAf2jxfFOVEaMyQUh3XSbSfIakpZmRc8lJFEoQHqEc6hgrEifKz2UfG8NwoEezG0hyh4Uz9u5EhrtSIh2aSI91Xy95U/M/rpLp7WdUJKkmAs8f6qYM6hOW4ERlQRrNjIEYUlNVoj7SCKsTXcLN0VDmqg89fM8dsmU5C5Xskqa1YrVNyHy3LtJq+rCE7AGbgALrgCNXAP6qABMHgBb+AdfFiv1qf1ZU3mowUr3zkGC7C+fwFOgaZl</latexit>

with 1 more model parameter

f = 1 with similar parameters of previous fits Constant single power-law spectrum (ν > νm)

slide-12
SLIDE 12

LIN Haoxiang - High Energy Astrophysics 2018

Synchrotron tail fit

New fit with early radio flux in the regime of synchrotron tail!

P 0

ν0/P 0 m =

8 > > < > > :

  • ν0

c/ν0 m

(p1)/2 ν0/ν0

c

p/2 (ν0

c < ν0)

  • ν0/ν0

m

(p1)/2 (ν0

m < ν0 < ν0 c)

  • ν0/ν0

m

1/3 (ν0 < ν0

m)

<latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit><latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit><latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">ADfnicdZJb9MwFMedhcsol23wAMvFhW0fWiTDCT2ANI0XnhCRaLbpKZUjnPSWrWdLZBJcrH4sMgXsf3wGmzqenGkSL/dS6/c3ziKONMad/7ey4d+7eu7/7oPXw0eMne/sHT09VanIKI5ryND+PiALOJIw0xzOsxyIiDicRYuPVfzsO+SKpfKrXmYwEWQmWcIo0dY1PXA+DzvTIpSmU3qVEiX+gMIZkwW1HJV2cLXFnJIdNcmT2nHqw7RCXM2m+vet6LfzfpBzsr5PWKXQjJbPhDVxFvLgwJMY1E7/HV0Ph+FtXf/fs8ERV5z1YblbwCJc7a2IuIHtcTfogfemAb7iCctrXiMEGdfbmu63/YG/MnxTBLVo9qG9g8D+OUGgFSU06UGgd+picFyTWjHMpWaBRkhC7IDMZWSiJATYrVDUr8ynpinKS5/aTGK+9mRUGEUksR2UxB9FxtxyrnbGx0cnRpGAyMxokXTdKDMc6xdUzwjHLgWq+tILQnNlZMZ2TnFBtH1ujizJRDJlt0/BqtvjZT2ApBZHNyA+ilnbksmVXGWwv7qY4PRwE/iD48rZ9fFIvdRe9QC9RFwXoHTpGn9AQjRB1fjl/nEvnr4vc127f9dapO05d8w1zD36By1UC4Q=</latexit><latexit sha1_base64="98wIWCm6daXat0S4LHndMqsQ8Mo=">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</latexit>

For reasonable range of parameters (e.g. ϵe < 1), synchrotron tail fit is uniquely found by free f model

Q: why new fit is found by allowing free f ?

Possible early νm passage in GHz!

νm is far below GHz with f = 1 fixed

ν′

m(free f ) ≃ 11.6 GHz η2 eϵ1/2 B,−2n1/2 −3 Γ3 s

(p = 2.17) ν′

m( f = 1) ≃ 2.2 MHz ϵ2 e,−1ϵ1/2 B,−2n1/2 −3 Γ3 s

(p = 2.17)

γm( f = 1) = ϵe p − 2 p − 1 mp me (Γs − 1) γm(free f ) = ηe mp me (Γs − 1)

ν′

m = 1

2π eB′ mec γ2

m

Emitted by low-energy electrons ~ γm

slide-13
SLIDE 13

LIN Haoxiang - High Energy Astrophysics 2018

MCMC

slide-14
SLIDE 14

LIN Haoxiang - High Energy Astrophysics 2018

MCMC: Table

Jet model Spherical model

aMedians with symmetric 68% uncertainties; bMaximum of the posterior probability density function

Jet: energy ~ other SGRBs (10^50–53 erg); Jet size (θc) & viewing angle (θv) strictly constrained θc ~ 0.09 (5°) and θv ~ 0.47 (27°). If + LIGO & VLBI, θc ~ 0.06 (3.4°) and θv ~ 0.3 (17°) Sph: energy ~ kilonova (10^51 erg), high umin ~ 1–2 (0.7–0.9c) disfavors dynamical ejecta (umin ~ 0.4, 0.4–0.5c) while cocoon remains as an possible option (umin >~ 1, 0.7c)

Common constraints by light curve turnover (Jet: Γc ~ 1/(θv - θc), Sph: us ~ umin)

slide-15
SLIDE 15

LIN Haoxiang - High Energy Astrophysics 2018

MCMC: Highlights

Higher density by 1—2 orders of magnitude Degree of electron-proton coupling ηe is strictly constrained, ηe ~ 0.1 (jet) and ~ 0.3 (sph) Close to simulation of non- or trans-relativistic shock acceleration (Park et al. 2015): ηe ~ 0.4 Jet: higher energy by nearly 2 order of magnitudes!

Other differences by Free f model Synchrotron tail solution

*Half-light radius re

Jet Sph

in tension with HI observation (< 0.04 cm-3) Make sense if hot gas is taken into account (0.01—0.1 cm-3 for offset ~ 0.64re*

  • f GW170817 in hot gas dominated giant

elliptical galaxies like NGC 4993) Lower and higher end of SGRB observations (1e50-53 erg)

slide-16
SLIDE 16

LIN Haoxiang - High Energy Astrophysics 2018

One more thing…

*best-fit parameters to GW170817 by sph model But in non-relativistic limit, νc almost stops evolving (~ t-0.2) after the shock becomes Newtonian ~ O(100d) Time of cooling break cannot be diagnostic btw jet/sph models Previous predictions: very early cooling passage in X-ray for Sph model Comoving cooling frequency νc' v.s. time

B0 = (32⇡✏Bnmp)1/2Γsc

<latexit sha1_base64="cmP6e0pTjWEbJReRaXadcMlG9l8=">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</latexit><latexit sha1_base64="cmP6e0pTjWEbJReRaXadcMlG9l8=">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</latexit><latexit sha1_base64="cmP6e0pTjWEbJReRaXadcMlG9l8=">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</latexit><latexit sha1_base64="cmP6e0pTjWEbJReRaXadcMlG9l8=">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</latexit>

B0 =  8⇡✏B ˆ Γs + 1 ˆ − 1 nmpc2(Γs − 1) 1/2

<latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit><latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit><latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit><latexit sha1_base64="enAOvF1ZPvfDMtdDlMHWV+JPJTA=">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</latexit>

Even if νc already reaches X-ray band, it stops to further decrease and cooling break becomes not observable

Trans-relativistic Relativistic limit (conventional GRB theory)

Non-passage for 1 yr disfavors Sph model

slide-17
SLIDE 17

LIN Haoxiang - High Energy Astrophysics 2018

Summary

A natural model for nonthermal electron spectrum is formulated:

  • Injection efficiency f is allowed to vary freely
  • Minimum energy γm is independently determined by electron-proton coupling level

A new fit to afterglow of GW170817 with early radio in synchrotron tail, in contrast to previous fits Parameter of GW170817 by free f model:

  • Higher ambient density by 1 – 2 orders of magnitude (but consistent with hot gas)
  • Jet energy increased by nearly 2 orders of magnitude

Time of cooling break at νc cannot be diagnostic btw jet/sph models Our prediction of “early radio in the synchrotron tail” may be tested in the future events by early multi-frequency radio observations