Kn L sc

RHESSI Hard XX-ray Imaging Spectroscopy of Extended Sources
and the Physical Properties of Electron Acceleration Regions
in Solar Flares
Yan Xu1, A. Gordon Emslie1, and Gordon J. Hurford2
Physics, Oklahoma State University
1.
2.
SSL, University of California at Berkeley
November 6 – 8, 2006
2
•
Ten Extended HXR sources were selected close to
the limb;
•
Visibilities + Forward Fit (Curved Elliptical
Gaussian)
Seven parameters + UNCERNTAINTIES!
3. Longitudinal one-sided emission centroid
∞
∞
0
0
v.s.
sc (ε ) = ∫ sI (ε , s )ds / ∫ I (ε , s )ds = 2 / π σ (ε )
November 6 – 8, 2006
E
3
November 6 – 8, 2006
4
August 23, 2005 Forward Fit Maps
CLEAN 10-15 keV
November 6 – 8, 2006
5
Other Nine Events (Images)
C =>
Clean Images
F => Forward Fit Maps
C
F
C
F
12-Apr-2002
15-Apr-2002
17-Apr-2002
17-Jun-2003
10-Jul-2003
02-Dec-2003
21-May-2004
31-Aug-2004
01-Jun-2005
November 6 – 8, 2006
6
Thermal
Non-Thermal
November 6 – 8, 2006
7
Theoretical Predictions for Sc v.s. E
Thermal
Source size should decrease with energy
T = T0 e − s
I=
2
/ 2σ 2
1 −ε / KT
e
εT 1 / 2
z = d ln s/d ln ε ~ - 0.5
November 6 – 8, 2006
8
Theoretical Predictions for Sc v.s. E
Non-Thermal (Point-Source-Injection)
-
dE
KE α −1
=−
ds
Eα
S c (ε ;α ) =
ε α +1
2(δ − 2)
×
2 KE- α −1 (δ − α − 2)(δ − α − 3)
z = d ln s / d ln ε = α + 1
α: single parameter
α=1: Coulomb collisions
z=2!
November 6 – 8, 2006
9
August 23, 2005
Mariska et al. 1989
November 6 – 8, 2006
10
Other Nine Events (Sc v.s. E)
November 6 – 8, 2006
11
Other Nine Events (Distribution of Slope)
November 6 – 8, 2006
12
Other Nine Events (Density profiles n(s) required by point-injection)
November 6 – 8, 2006
13
Non-Thermal
Thermal
November 6 – 8, 2006
14
Theoretical Predictions for Sc v.s. E
Non-Thermal (Extended Acceleration [Tenuous
Tenuous]/TEAR)
sc (ε ) = L +
ε2
(δ − 2)
Kn (δ − 3)(δ − 4)
×
ε << KnL ⇒ ζ = 0
ε >> KnL ⇒ ζ = 2
S c ∝ L + bn −1ε 2
November 6 – 8, 2006
15
Theoretical Predictions for Sc v.s. E
Non-Thermal (Extended Acceleration [Dense
Dense]/DEAR)
E0 = ( E 2 + 2 Kn s − x )1/ 2
I (ε , s ) = Γn
∫
L
−L
dx ∫
∞
ε
dE
( E + 2 Kn s − x ) (δ +1) / 2
2
∞
∞
dE
∫ sI (ε , s)ds = 0 − L ε ( E + 2Kn s − x )(δ +1) / 2
S c (ε ) = 0 ∞
∞
L
∞
dE
∫0 I (ε , s)ds ∫0 ds ∫− L dx ∫ε ( E 2 + 2 Kn s − x )(δ +1) / 2
∞
ß
1
2 Lε
∫
L
sds ∫ dx ∫
2
November 6 – 8, 2006
16
Theoretical Predictions for Sc v.s. E
Non-Thermal (DEAR continued)
November 6 – 8, 2006
17
Theoretical Predictions for Sc v.s. E
Non-Thermal (DEAR continued)
How to do the numerical fitting:
1.
Make a guess according to P16 for L and n
2.
Use the guess of L and n as the mean value to
generate an NxM array for several L and n
3.
For each pair of L and n, calculate the
Maximum Likelihood:
1
−( S
−S
)2 /σ 2
p=Π
e c _ act c _ exp ected i
2π σ i
4.
1
0.75
P
0.5
0.25
0
Fit p[N,M] with a 2D Gaussian and find the
peak L and n.
4
2
0 n
-5
-2.5
-2
0
L
2.5
-4
5
November 6 – 8, 2006
18
November 6 – 8, 2006
19
Comparing Two Fittings
In order to compare to kind of fits, we calculated
the reduced c2:
χ2 =
Σ ( S c _ act − S c _ exp ected ) 2 / σ i
2
N1 − N 2
s here is the measured error of Sc ;
N1 is the number of Sc ;
N2 is the number of independent variables.
In our case, N2 = 2
November 6 – 8, 2006
20
August 23, 2005
November 6 – 8, 2006
21
Other Nine Events (Best-fit of L and n under TEAR)
November 6 – 8, 2006
22
Other Nine Events (Best-fit of L and n under DEAR)
November 6 – 8, 2006
23
Distribution of Source Density n and Initial Length L for all Ten Events
November 6 – 8, 2006
23
Next Step
Time Dependent Variation
November 6 – 8, 2006
August 23, 2005
20 s Intervals from 14:27:00 UT
25
November 6 – 8, 2006
26
August 23, 2005
20 s Intervals from 14:27:00 UT
November 6 – 8, 2006
27
23-Aug-05
Time
Slope
L
n
d_slope
d_L
d_n
1
0.24
7.45
5.03
0.11
0.66
3.39
2
0.4
6.18
3.28
0.13
0.72
1.57
3
0.43
5.94
3.05
0.11
0.62
1.18
4
5*
6
0.38 0.06 0.4
5.86 7.15 5.74
2.91 33.63 2.59
0.04 0.14 0.16
0.06
0.66
0.69
0.1
0
0.93
7
0.52
6.29
2
0.08
0.53
0.4
8
0.2
7.41
7.24
0.14
0.72
0.67
9
0.33
6.65
3.6
0.08
0.46
1.2
10
0.3
6.9
4.39
0.12
0.68
2.64
11
0.64
5.28
1.97
0.17
0.82
0.65
12
0.42
5.65
3.34
0.14
0.68
1.46
13
0.43
6.04
2.9
0.08
0.53
0.9
14
0.25
6.62
4.49
0.08
0.41
1.67
15
0.41
5.97
3.35
0.12
0.64
1.47
STDEV
0.12
0.65
1.37
M EAN
0.36 ~ 33%
6.28 ~ 10%
3.58 ~ 38%
8
7
6
5
Slope
L
n
4
3
2
1
0
0
2
4
6
8
10
12
14
16
-1
November 6 – 8, 2006
April 15, 2002
20 s Intervals from 00:05:00 UT
November 6 – 8, 2006
28
29
April 15, 2002
20 s Intervals from 00:05:00 UT
November 6 – 8, 2006
30
15-Apr-02
Time
Slope
L
n
d_slope
d_L
d_n
1
0.81
2.89
2.22
0.13
0.44
0.44
2
0.57
3.33
3.43
0.08
0.32
0.77
3
0.59
3.42
3.014
0.07
0.27
0.54
4*
0.33
4.08
6.75
0.16
0.52
0
5
0.35
3.5
3.6
0.09
0.35
0.91
6
0.42
3.83
4.18
0.06
0.21
0.73
7
0.52
3.28
3.68
0.09
0.3
0.84
8
0.66
3.04
2.58
0.12
0.31
0.42
9
0.5
3.86
3.9
0.16
0.6
1.84
10
0.66
3.16
2.68
0.03
0.14
0.21
11
0.54
3.55
3.58
0.1
0.36
0.93
12
0.52
3.34
3.66
0.06
0.22
0.6
13
0.52
3.34
3.66
0.06
0.22
0.6
14
0.5
3.63
3.84
0.08
0.3
0.91
15
0.46
3.73
4.24
0.06
0.3
1.09
7
6
5
4
slope
L
n
3
2
1
0
0
2
4
6
8
10
12
November 6 – 8, 2006
14
16
STDEV M EAN
0. 11
0 .54 ~ 20%
0. 29
3 .42 ~ 8%
0. 60
3 .45 ~ 17%
33
Elementary Burst ?
2 s < dT < 20 s
November 6 – 8, 2006
34
Sui & Holman
2003
Sui L., Holman G. D., & Dennis B. R. 2004
November 6 – 8, 2006
35
April 15, 2002 (00:00:36 UT ~ 00:15:36 UT 1 min)
November 6 – 8, 2006
36
November 6 – 8, 2006
37
Sui & Holman
2003
Sui L., Holman G. D., & Dennis B. R. 2004
•Loop-top source moving upward
•Higher energy
Higher Altitude (But not thermal)
•Moving speed ~ 8 km/s
•UNCERNTAINTIES
•1 minute instead of 20 second
•Downward motion?
November 6 – 8, 2006
November 6 – 8, 2006