Sophie Musset1, Eduard Kontar2, Nicole Vilmer1

15th RHESSI Workshop, 26-30 July 2016, Graz

1 LESIA, Observatoire de Paris 2 School of Physics and Astronomy, University of Glasgow

1. RHESSI Imaging Spectroscopy: a new tool to study electron

transport during solar flares

2. The 2004 May 21 solar flare

3. The diffusive transport model (Kontar et al, 2014)

4. Comparison between observations and model predictions

5. Conclusions

1

Battaglia & Benz (2006)

2

Battaglia & Benz (2006)

Photon spectral index

𝛾𝐿𝑇 − 𝛾𝐹𝑃 ≠ 2

Need additional

mechanism to

collisional transport

2

Battaglia & Benz (2006)

Simoes & Kontar (2013)

Photon spectral index

𝛾𝐿𝑇 − 𝛾𝐹𝑃 ≠ 2

Need additional

mechanism to

collisional transport

2

Battaglia & Benz (2006)

Simoes & Kontar (2013)

Photon spectral index

𝛾𝐿𝑇 − 𝛾𝐹𝑃 ≠ 2

Electron spectral index

𝛿𝐿𝑇 − 𝛿𝐹𝑃 < 0 Ratio of electron rate

𝑁 𝐿𝑇

𝑁 𝐹𝑃> 1

Need additional

mechanism to

collisional transport

2

Need additional

mechanism to

collisional transport

Radio observations : Kuznetsov & Kontar (2015)

3

Radio observations : Kuznetsov & Kontar (2015)

Spatial distribution of the density of

energetic electrons with E > 60 keV

𝛿 = 2.1

3

X-ray imaging spectroscopy

Musset et al, in prep

12-25 keV

25-50 keV

50-100 keV

4

Thick target

Thin target Thick target

X-ray imaging spectroscopy

Musset et al, in prep

5

X-ray imaging spectroscopy

Musset et al, in prep

𝛿 = 4.2 ± 0.2

𝑁 = 0.12 ± 0.03 × 1035 s−1

𝛿 = 4.4 ± 0.2

𝛿 = 5.2 ± 0.4

𝑁 = 0.06 ± 0.02 × 1035 s−1

12-25 keV

25-50 keV

50-100 keV

6

Thick target

Electron spectral index: 𝜹

Electron rate above 25 keV: 𝑁

Thin target

Electron spectral

index: 𝜹

Integrated electron

mean flux spectrum

above 𝐸0=25 keV

𝑛𝑉𝐹0 = 𝑛𝑉𝐹 𝐸 𝑑𝐸∞

𝐸0

𝑛𝑉𝐹0 = 0.46 ± 0.08 × 1055 𝑐𝑚−2𝑠−1

X-ray imaging spectroscopy

Musset et al, in prep

𝛿 = 4.2 ± 0.2

𝑁 = 0.12 ± 0.03 × 1035 s−1

𝛿 = 4.4 ± 0.2

𝛿 = 5.2 ± 0.4

𝑁 = 0.06 ± 0.02 × 1035 s−1

12-25 keV

25-50 keV

50-100 keV

6

Thick target

Electron spectral index: 𝜹

Electron rate above 25 keV: 𝑁

Thin target

Electron spectral

index: 𝜹

Integrated electron

mean flux spectrum

above 𝐸0=25 keV

𝑛𝑉𝐹0 = 𝑛𝑉𝐹 𝐸 𝑑𝐸∞

𝐸0

𝑛𝑉𝐹0 = 0.46 ± 0.08 × 1055 𝑐𝑚−2𝑠−1

Distance between the footpoints and the looptop source: ~17 and ~15 × 108 cm

Length of the loop 𝐿 ~ 3.2 × 109 cm

X-ray imaging spectroscopy

Musset et al, in prep

7

𝑁 𝐿𝑇 = 𝐴𝐿𝑇 𝐹 𝐸 𝑑𝐸 = 𝐴𝐿𝑇 𝑛𝑉𝐹(𝐸)

𝑛𝑉

∞

𝐸0

𝑑𝐸∞

𝐸0

𝑵 𝑳𝑻 = 𝟏

𝒏𝑳𝑳𝑻 𝒏𝑽𝑭 𝑬 𝒅𝑬∞

𝑬𝟎

X-ray imaging spectroscopy

Musset et al, in prep

7

𝑁 𝐿𝑇 = 𝐴𝐿𝑇 𝐹 𝐸 𝑑𝐸 = 𝐴𝐿𝑇 𝑛𝑉𝐹(𝐸)

𝑛𝑉

∞

𝐸0

𝑑𝐸∞

𝐸0

𝑵 𝑳𝑻 = 𝟏

𝒏𝑳𝑳𝑻 𝒏𝑽𝑭 𝑬 𝒅𝑬∞

𝑬𝟎

𝐿𝐿𝑇 = 9.6 × 108 𝑐𝑚

𝑉𝑡ℎ = 1.5 × 1026 𝑐𝑚3

𝑛 = 𝐸𝑀 𝑉𝑡ℎ = 1.2 ± 0.2 × 1011 𝑐𝑚−3

𝑵 𝑳𝑻 = 0.4 ± 0.2 × 1035 s−1 𝑵 𝑳𝑻

𝑵 𝑭𝑷= 2.2

Spatial distribution of electrons

<nVF(E,z)> at 25 keV

Electron mean spectra <nVF(E,z)>

in the different parts of the loop

Looptop source

Footpoints

8

𝑛𝑏𝐸𝑚𝑖𝑛 =

𝐹(𝐸)

𝑣(𝐸)𝑑𝐸

∞

𝐸𝑚𝑖𝑛

Energetic electron density above Emin cm-3

Energetic electron density

above 25 keV

9

𝑛𝑏𝐸𝑚𝑖𝑛 =

𝐹(𝐸)

𝑣(𝐸)𝑑𝐸

∞

𝐸𝑚𝑖𝑛

Energetic electron density above Emin cm-3

Energetic electron density

above 25 keV

Energetic electron density

above 60 keV

From radio (Kuznetsov

& Kontar 2015)

From

X-rays

9

Energetic electron density above Emin cm-3 𝑛𝑏𝐸𝑚𝑖𝑛 =

𝐹(𝐸)

𝑣(𝐸)𝑑𝐸

∞

𝐸𝑚𝑖𝑛

Energetic electron density

above 25 keV

Energetic electron density

above 60 keV

From radio (Kuznetsov

& Kontar 2015)

From

X-rays

Distribution deduced from gyrosynchrotron emission is more peaked than

the distribution deduced from X-rays

𝑛𝑏,𝐿𝑇25

𝑛𝑏,𝐹𝑃25 ~1.6 𝑒𝑡 3.8

𝑛𝑏,𝐿𝑇60

𝑛𝑏,𝐹𝑃60 ~7.7 𝑒𝑡 9

9

1

𝑣

𝜕

𝜕𝑧𝐷𝑧𝑧(𝑇) 𝜕𝐹

𝜕𝑧=

𝜕

𝜕𝐸

𝑑𝐸

𝑑𝑥𝐹 + 𝐹0𝑆(𝑧)

Diffusion Collisions Source

Kontar et al (2014)

𝐷𝑧𝑧(𝑇)

=λ𝑣

3 λ : mean free path

Strong pitch angle scattering due to small scale magnetic fluctuations

diffusive transport of energetic electrons

10

1

𝑣

𝜕

𝜕𝑧𝐷𝑧𝑧(𝑇) 𝜕𝐹

𝜕𝑧=

𝜕

𝜕𝐸

𝑑𝐸

𝑑𝑥𝐹 + 𝐹0𝑆(𝑧)

Diffusion Collisions Source

Kontar et al (2014)

𝐹𝐷 𝐸, 𝑧 =𝐸

𝐾𝑛0 𝑑𝐸′

𝐹0 𝐸′

4𝜋𝑎 𝐸′2 − 𝐸2 + 2𝑑2exp −

𝑧2

4𝑎 𝐸′2 − 𝐸2 + 2𝑑2

∞

𝐸

𝐷𝑧𝑧(𝑇)

=λ𝑣

3 λ : mean free path

Suppose λ constant

10

𝑖𝑛 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑐𝑚2 𝑠 𝑘𝑒𝑉

1

𝑣

𝜕

𝜕𝑧𝐷𝑧𝑧(𝑇) 𝜕𝐹

𝜕𝑧=

𝜕

𝜕𝐸

𝑑𝐸

𝑑𝑥𝐹 + 𝐹0𝑆(𝑧)

Diffusion Collisions Source

Kontar et al (2014)

𝐹𝐷 𝐸, 𝑧 =𝐸

𝐾𝒏𝟎 𝑑𝐸′

𝑭𝟎 𝐸′

4𝜋𝒂 𝐸′2 − 𝐸2 + 2𝒅2exp −

𝑧2

4𝒂 𝐸′2 − 𝐸2 + 2𝒅2

∞

𝐸

𝐷𝑧𝑧(𝑇)

=λ𝑣

3 λ : mean free path

𝒏𝟎 density of the medium

𝒅 size of the acceleration region

𝒂 α λ/𝑛0

𝑭𝟎 injected electron spectrum

Acceleration region

Suppose λ constant

10

1

𝑣

𝜕

𝜕𝑧𝐷𝑧𝑧(𝑇) 𝜕𝐹

𝜕𝑧=

𝜕

𝜕𝐸

𝑑𝐸

𝑑𝑥𝐹 + 𝐹0𝑆(𝑧)

Diffusion Collisions Source

Kontar et al (2014)

𝐹𝐷 𝐸, 𝑧 =𝐸

𝐾𝒏𝟎 𝑑𝐸′

𝑭𝟎 𝐸′

4𝜋𝒂 𝐸′2 − 𝐸2 + 2𝒅2exp −

𝑧2

4𝒂 𝐸′2 − 𝐸2 + 2𝒅2

∞

𝐸

𝐷𝑧𝑧(𝑇)

=λ𝑣

3 λ : mean free path

Spatial distribution of electrons at 20 keV

collisional

𝜆 = 109cm

𝜆 = 108cm

𝜆 = 107cm

𝑛𝑉𝐹𝐷

𝑛𝑉𝐹𝐷

𝑛𝑉𝐹𝐷

𝑛𝑉𝐹𝐶

10

1

𝑣

𝜕

𝜕𝑧𝐷𝑧𝑧(𝑇) 𝜕𝐹

𝜕𝑧=

𝜕

𝜕𝐸

𝑑𝐸

𝑑𝑥𝐹 + 𝐹0𝑆(𝑧)

Diffusion Collisions Source

Kontar et al (2014)

𝐹𝐷 𝐸, 𝑧 =𝐸

𝐾𝒏𝟎 𝑑𝐸′

𝑭𝟎 𝐸′

4𝜋𝒂 𝐸′2 − 𝐸2 + 2𝒅2exp −

𝑧2

4𝒂 𝐸′2 − 𝐸2 + 2𝒅2

∞

𝐸

𝐷𝑧𝑧(𝑇)

=λ𝑣

3 λ : mean free path

Spatial distribution of electrons at 20 keV Looptop and footpoint spectra

collisional

𝜆 = 109cm

𝜆 = 108cm

𝜆 = 107cm

𝜆 = 109cm

𝜆 = 108cm

𝜆 = 107cm

looptop

footpoint

𝑛𝑉𝐹𝐷

𝑛𝑉𝐹𝐷

𝑛𝑉𝐹𝐷

𝑛𝑉𝐹𝐶

λ Mean free path

𝑑 Size of acceleration region

𝐿 ~ 3.2 × 109 cm

Length of the loop

Spatial distribution at 25 keV Coronal and footpoint spectra

𝑛 = 1.2 ± 0.2 × 1011 𝑐𝑚−3; λ = 109 𝑐𝑚 ; 𝑑 = 3 × 108 𝑐𝑚

𝑛 = 3 × 1010 𝑐𝑚−3; λ = 3 × 108 𝑐𝑚 ; 𝑑 = 3 × 108 𝑐𝑚

11

Spatial distribution of energetic electron density

with energy > 25 keV

𝑛 = 1.2 ± 0.2 × 1011 𝑐𝑚−3 ; 𝑑 = 3 × 108 𝑐𝑚

𝑛 = 3 × 1010 𝑐𝑚−3 ; 𝑑 = 3 × 108 𝑐𝑚

λ = 109 𝑐𝑚 λ = 3 × 108 𝑐𝑚

12

Spatial distribution of energetic electron density

with energy > 25 keV

𝑛 = 1.2 ± 0.2 × 1011 𝑐𝑚−3 ; 𝑑 = 3 × 108 𝑐𝑚

𝑛 = 3 × 1010 𝑐𝑚−3 ; 𝑑 = 3 × 108 𝑐𝑚

λ = 109 𝑐𝑚 λ = 3 × 108 𝑐𝑚

12

The diffusive transport model

(Kontar et al. 2014) is

consistent with the X-ray

observations (spectral and

spatial distribution).

What about the radio

observations of Kuznetsov and

Kontar (2015) ?

Spatial distribution of energetic electron density

with energy > 25 keV with energy > 60 keV

𝑛 = 1.2 ± 0.2 × 1011 𝑐𝑚−3 ; 𝑑 = 3 × 108 𝑐𝑚

𝑛 = 3 × 1010 𝑐𝑚−3 ; 𝑑 = 3 × 108 𝑐𝑚

λ = 109 𝑐𝑚 λ = 3 × 108 𝑐𝑚

λ = 6 × 107 𝑐𝑚 λ = 2 × 107 𝑐𝑚

7

The distribution is more

peaked: need a smaller λ

Trapping can be caused by magnetic

field convergence.

The trapped fraction of particles is

And in the case of isotropic pitch-

angle distribution,

Where 𝛼0 is the loss cone angle

With 𝜎 = 𝐵𝐹𝑃 𝐵𝐿𝑇 the magnetic ratio.

(Simoes & Kontar 2013)

𝑇 = 1 −𝑁 𝐹𝑃

𝑁 𝐿𝑇

𝑇 = cos (𝛼0)

𝛼0 = sin−1 ( 1/𝜎)

13

Trapping can be caused by magnetic

field convergence.

The trapped fraction of particles is

And in the case of isotropic pitch-

angle distribution,

Where 𝛼0 is the loss cone angle

With 𝜎 = 𝐵𝐹𝑃 𝐵𝐿𝑇 the magnetic ratio.

(Simoes & Kontar 2013)

𝑇 = 1 −𝑁 𝐹𝑃

𝑁 𝐿𝑇

𝑇 = cos (𝛼0)

𝛼0 = sin−1 ( 1/𝜎)

𝑵 𝑭𝑷

𝑵 𝑳𝑻= 𝟐. 𝟐 𝝈 = 𝟏. 𝟒

13

Magnetic field strength along the

loop (Kuznetsov & Kontar 2015)

Trapping can be caused by magnetic

field convergence.

The trapped fraction of particles is

And in the case of isotropic pitch-

angle distribution,

Where 𝛼0 is the loss cone angle

With 𝜎 = 𝐵𝐹𝑃 𝐵𝐿𝑇 the magnetic ratio.

(Simoes & Kontar 2013)

𝑇 = 1 −𝑁 𝐹𝑃

𝑁 𝐿𝑇

𝑇 = cos (𝛼0)

𝛼0 = sin−1 ( 1/𝜎)

𝑵 𝑭𝑷

𝑵 𝑳𝑻= 𝟐. 𝟐 𝝈 = 𝟏. 𝟒

𝟏. 𝟒 < 𝝈 < 𝟒. 𝟕

13

Magnetic field strength along the

loop (Kuznetsov & Kontar 2015)

Trapping can be caused by magnetic

field convergence.

The trapped fraction of particles is

And in the case of isotropic pitch-

angle distribution,

Where 𝛼0 is the loss cone angle

With 𝜎 = 𝐵𝐹𝑃 𝐵𝐿𝑇 the magnetic ratio.

(Simoes & Kontar 2013)

𝑇 = 1 −𝑁 𝐹𝑃

𝑁 𝐿𝑇

𝑇 = cos (𝛼0)

𝛼0 = sin−1 ( 1/𝜎)

𝑵 𝑭𝑷

𝑵 𝑳𝑻= 𝟐. 𝟐 𝝈 = 𝟏. 𝟒

𝟏. 𝟒 < 𝝈 < 𝟒. 𝟕

But how to explain the spectral

hardening in the footpoints?

Imaging spectroscopy is used to study the spatial distribution of electrons and

the comparison of spectral distribution in different parts of the loop

Diffusive transport model (Kontar et al 2014) can explain the X-ray observations

Diffusive transport model can also explain the gyrosynchrotron observations,

but with a smaller mean free path

Mean free path is energy dependant

First comparison between radio and X-ray observations to probe energetic

electrons trapping in the corona

Allows to probe two energy domains

14

Imaging spectroscopy is used to study the spatial distribution of electrons and

the comparison of spectral distribution in different parts of the loop

Diffusive transport model (Kontar et al 2014) can explain the X-ray observations

Diffusive transport model can also explain the gyrosynchrotron observations,

but with a smaller mean free path

Mean free path is energy dependant

First comparison between radio and X-ray observations to probe energetic

electrons trapping in the corona

Allows to probe two energy domains

Need to further develop the diffusive transport model with energy-dependent

mean free path, and for relativistic electrons

With imaging spectroscopy, model predictions about the spatial evolution of the

electron distribution are useful to compare to observations

14