On the frequency distribution of heating events in Coronal

Loops, simulating observations with Hinode/XRT

Patrick Antolin1, Kazunari Shibata1, Takahiro Kudoh2 , Daikou Shiota2, David Brooks3

1: Kwasan Observatory, Kyoto University 2: National Astronomical

Observatory of Japan 3: Naval Research Laboratory

IAUS 247, Isla de Margarita, Venezuela, 17 - 22 September 2007

IAUS 247

Universidad de Los Andes

International Astronomical Union Symposium 247: Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-Seismology.

How to heat and maintain the

plasma to such a high temperature?

The coronal heating problem

The Solar Corona

Grotrian, Edlén (1942)

point out existence of a 106

K plasma, 200 times hotter

than the photosphere.

Coronal Loops (~105 km long) seen by TRACE

Photosphere

coronaTransition region

height

T [

K]

Chromosphere

•Mode conversion (Hollweg, Jackson & Galloway 1980; Moriyasu et al. 2004).

•Phase mixing (Heyvaerts & Priest 1983; Sakurai & Granik 1974; Cally 1991).

•Resonant absorption (Goedbloed 1983; Poedts, Kerner & Goossens 1989)

• Alfvén wave model: (Alfvén 1947; Uchida & Kaburaki 1974; Wenzel 1974).

- Alfvén waves can carry enough energy to heat and maintain a corona (Hollweg, Jackson & Galloway 1982; Kudoh & Shibata 1999).

- Waves created by sub-photospheric motions propagate into the corona and dissipate their energy through nonlinear mechanisms:

Plausible models

• Nanoflare-reconnection model: (Porter et al. 1987; Parker 1988)

magnetic flux: current sheets

footpoint shuffling reconnection events

- Ubiquitous, sporadic and impulsive releases of energy (1024-1027 erg) that may correspond to the observed intermittency and spiky intensity profiles of coronal lines (Parnell & Jupp 2000; Katsukawa & Tsuneta 2001).

- However Moriyasu et al. (2004) showed that such profiles can also result naturally from nonlinear conversion of Alfvén waves.

Plausible models

((Parker, 1989)

How to recognize between the two heating mechanisms when they operate in the corona?

Yohkoh/SXT

Observational facts

• Energy release processes in the Sun, from solar flares down to microflares are found to follow a power law distribution in frequency, (Lin et al. 1984; Dennis 1985).

• Main contribution to the heating may come from smaller energetic events (nanoflares) if these distribute with a power

law index > 2 (Hudson 1991).• Initial studies of small-scale brightenings have shown a

power law both steeper and shallower than -2 (Krucker & Benz 1998, Aschwanden & Parnell 2002). Hence, no definitive conclusion has been reached at present.

= 1.4 - 1.6

Shimizu et al. 1995

Purpose• Propose a way to discern observationally between Alfvén

wave heating and nanoflare-reconnection heating.

• Diagnostic tool for the location of the heating along coronal loops.

Different X-ray intensity profiles & different frequency distribution of heating events between the models.

Link between the power law index of the frequency distribution and the mechanism operating in the loop.

Idea: Different characteristics of

wave modes

Different distribution of shocks and strengths

convective motions

Reconnection events

• Initial conditions: – T0=104 K : constant – ρ0= 2.5×10-7 g cm-3, – p0= 2×105 dyn cm-2, – B0=2300 G with apex-to-base area

ratio of 1000. – hydrostatic pressure balance up to

800 km height. After: ρ (height)-4 (Shibata et al. 1989).

• 1.5-D MHD code

• CIP-MOCCT scheme (Yabe & Aoki 1991; Stone & Norman 1992; Kudoh, Matsumoto & Shibata 1999) with conduction + radiative losses (optically thin. Also optically thick approximation).

• Alfven wave model: Torsional Alfvén waves are created by a random photospheric driver.

ss φφ

Numerical model

ChromospherChromospheree

100000 km

PhotospherePhotosphere

Nanoflare heating function

• Heating events can be:– Uniformly distributed along the

loop.– Concentrated at the footpoints

• Energies of heating events can be:– Uniformly distributed. – Following a power law

distribution in frequency.

Artificial energy injectionPhotospherePhotosphere

(Taroyan et al. 2006, Takeuchi & Shibata 2001)

= 1.5

Alfvén wave heating

Alfvén wave heating

Alfvén wave heating

• Heating mechanism

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Alfvén waves Slow/fast modes

Non-linear effects

Shock heating

Develop into shocks

Alfvén wave heating

For <v2>1/2 ≿ 2 km/s a corona is created.

Nanoflare heating

20 Mm10 Mm~ 2

Conductive flux

Nanoflare heating

footpoint

uniform

energy input

Slow modes

Gas pressure

Fast dissipation

Shock heating

Top of TR Apex

Alfvén wave

Simulating observations with Hinode/XRT

1”x1” F.O.V.

Apex

Top of TR

Ubiquitous strong slow

and fast shocks

Nanoflare footpoint

Simulating observations with Hinode/XRT

1”x1” F.O.V.

Apex

Top of TR

Small peaks are levelled

out

Nanoflare uniform

Simulating observations with Hinode/XRT

1”x1” F.O.V.

Apex

Top of TR

Flattening by thermal conduction

Intensity histograms

I1

I2

Alfvén wave

Intensity histograms

1”x1” F.O.V.

Apex

Top of TR

= 2.53 = 2.44

Nanoflare footpoint

Intensity histograms

1”x1” F.O.V.

Apex

Top of TR

= 1.86 = 1.48

Nanoflare uniform

Intensity histograms

1”x1” F.O.V.

Apex

Top of TR

= 2.66 = 0.90

Power law index

~ close to the footpoint; decreases approaching the apex due to fast dissipation of slow modes & to

thermal conduction

= 2.1

Input:

Output:

- Footpoint- Power law spectrum in

energies

Measurement of power law index depends strongly on the location along the loop, hence on the formation temperature of the observed

emission line.

Conclusions• Alfvén wave heated coronas:

– Ubiquitous mode conversion -> ubiquitous fast and slow strong shocks.

– Intensity profiles are spiky and intermittent throughout the corona.– Power law distribution in energies. Steep index ( > 2), roughly

constant along the corona: heating from small dissipative events.

• Nanoflare heated coronas:– Uniform heating along the loop:

• weak shocks everywhere. • Flat, uniform intensity profile everywhere: Power law index ~ 1.

– Footpoint heating:• Strong slow shocks only near the transition region. Fast dissipation and

thermal conduction dampingonly weak shocks at apex.• Spiky intensity profiles near the transition region, flattening at apex:

power law index becomes shallower the farther we are from the transition region.

• If power law energy spectrum at input, the measured index matches original input power law index ( ~ ) only near the transition region.

– Measurement of power law index is strongly dependent on location along loops, hence on temperature.

Thank you for your attention!

• Mass conservation

• Momentum equation (s-component)

• Momentum equation (-component)

• Induction equation( -component)

• Energy equation

1.5-D MHD Equations

Alfvén wave

generator

• Radiative losses:- For T > 4 x 104 K:

Optically thin plasmas- For T < 4 x 104 K:

Optically thick plasmas

χTTQ =)(

(Landini & Monsignori-Fossi 1990, Anderson & Athay

1989).

Nanoflare heating function

Localization of heating

∑=

=

=n

ii

i

stHH

stH

1

),(

),(

0

,||

expsin0 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

h

i

i

i

ssstt

Eτ

π ti < t < ti + τi

otherwise

• Model parameters:

E0 ={ 0.01, 0.05, 0.5 } erg cm-3 s-1; sh={ 200, 500, 1000} km;

frequency:{ 1 / 50, 1 / 34, 1 / 7 } s; τi ={ 2η, 10η , 40η } s, with η a

random number in [0,1].• Heating events can be:

– Uniformly (randomly) distributed along the loop (above 2 Mm height).

– Concentrated at the footpoints: randomly distributed in [2,20] Mm, [2,12] or [1,10] Mm height.

• Heating events can have their energies:

– Uniformly distributed: <E0> = constant

– Following a power law distribution in frequency. <E0>

(Taroyan et al. 2006)

SXT/XRT Intensities

Filter Thin Be in XRT similar to filter Mg 3mm in SXT (used in Moriyasu et al. 2004).

2 = 2.08

1 = 1.84

Catastrophic cooling events

Intensity flux seen with Hinode/XRT

Loss of thermal equilibrium at apex due to footpoint heating (Mueller et al. 2005, Mendoza-Briceño 2005).

Power law index Alfvén wave

Nanoflare footpointNanoflare uniform

Power law index

Input:

Output:

= 2.2

= 2.1

= 2.0

= 1.9

= 1.8

= 1.7

= 1.6

= 1.5