Table 3 Model parameters of the outflow and wind

From: A wind environment and Lorentz factors of tens explain gamma-ray bursts X-ray plateau

GRB

 

ϵB = 10−2

ϵB = 10−3

ϵB = 10−5

 

Class I

\(\frac{{\epsilon }_{e}}{1{0}^{-2}}\)

Γi

\(\frac{{{{{{{{{\rm{A}}}}}}}}}_{\star }}{1{0}^{-2}}\)

Γi

\(\frac{{{{{{{{{\rm{A}}}}}}}}}_{\star }}{1{0}^{-2}}\)

Γi

A

Ratio νFν(X)/νFν(U)

080607

1.2

387

0.01

218

0.1

122

10−2

314 ± 74

091029

1.4

52

0.5

44

1

14

1.0

4.5 ± 1.4

110213A

8.3

136

0.1

91

0.5

43

0.1

45.4 ± 2.2

130831A

2.7

56

0.5

32

5

15

1.0

5.6 ± 1.3

  

ϵB = 10−2

ϵB = 10−3

 

Class II

\(\frac{{\epsilon }_{e}}{1{0}^{-2}}\)

Γi

\(\frac{{{{{{{{{\rm{A}}}}}}}}}_{\star }}{1{0}^{-2}}\)

Γi

A

Ratio νFν(X)/νFν(U)

060605

2.5

<51

> 1

< 28

> 0.1

0.21 ± 0.05

060614

0.49

<6

> 10

< 4

> 1.0

6.6 ± 2.0

060729

9.2

<8

> 10

< 5

> 1.0

1.5 ± 0.5

080310

1

<37

> 5

< 32

> 0.1

2.4 ± 0.8

100418A

17

<6

> 5

< 5

> 0.1

13.7 ± 4.6

171205A

2.3

<2

> 5

< 1.7

> 0.1

0.7 ± 0.2

 

ϵB = 10−2

ϵB = 10−3

ϵB = 10−5

 

Class III

\(\frac{{\epsilon }_{e}}{1{0}^{-1}}\)

Γi

\(\frac{{{{{{{{{\rm{A}}}}}}}}}_{\star }}{1{0}^{-4}}\)

\(\frac{{\epsilon }_{e}}{1{0}^{-3}}\)

Γi

\(\frac{{{{{{{{{\rm{A}}}}}}}}}_{\star }}{1{0}^{-3}}\)

\(\frac{{\epsilon }_{e}}{1{0}^{-7}}\)

Γi

\(\frac{{{{{{{{{\rm{A}}}}}}}}}_{\star }}{1{0}^{-2}}\)

Ratio νFν(X)/νFν(U)

050319

>31

>97

<3

>16

>61

<2

>21

>27

<5

1.03 ± 0.27

060714

>0.8

>147

<5

>0.55

>94

<3

>0.33

>39

<10

43.6 ± 14.8

061121

>2.8

>106

<5

>1.9

>70

<3

>1.1

>28

<10

37.1 ± 11.6

  1. In Column 1, GRB names are ordered by classes (I, II, III listed in Supplementary Tables 13 respectively). ϵe is the fraction of energy in the electrons, Γi is the initial jet Lorentz factor, A is the wind density. Direct value of ϵe is computed by using the information in the X-ray data for the GRBs listed in class I and II respectively. The values obtained (using the end of plateau time and X-ray flux in Supplementary Equation (19)) are surprisingly close to the fiducial values, ϵe 10−1 often obtained by fitting late time afterglow data61. In addition, direct values of A and Γi are obtained by assuming that the fraction of energy in the magnetic field is ϵB = 10−2, 10−3, 10−5 for the GRBs listed in class I. Moreover, an (external) upper limit on ϵB, (e.g., ϵB≤10−2) is used to compute an upper limit on Γi and lower limit on A for the GRBs listed in class II. Vice-versa, an external knowledge on lower limit (e.g., ϵB≥10−5) can be used to compute a lower limit on the value of Γi and an upper limit on A as well as a lower limit on ϵe for the GRBs listed in class III (see, Supplementary Method 2, Theoretical model). The ratio [1/(νFν(U)/νFν(X))] is such that νFν(U) is calculated at Tref.,U and νFν(X) is calculated at Ta,X (see Table 2). The errors correspond to a significance of one sigma. The ratios are consistent with the theoretical predictions in all three different classes. For the low luminous GRBs with the lowest Lorentz factor (GRBs 060614, 060729, 100418A and 171205A) in class II, when considering ϵB = 0.1, the obtained values are Γi = 11, 15, 9, 4 and A = 10−2, 10−2, 10−2, 5 × 10−3, respectively.