HydrostaticHydrostatic model modelling ling of active region of active region EUV EUV andand X-rayX-ray

emisemissionsion

J. Dudík J. Dudík 11, E, E.. Dzif Dzifčáková čáková 1,21,2, , A. KulinovA. Kulinová á 1,21,2, M. Karlický , M. Karlický 22

11 – – Dept. of Astronomy, Physics of the Earth and Dept. of Astronomy, Physics of the Earth and Meteorology,Meteorology, FM FMPhPhII, Comenius University, Comenius University, Bratislava, Bratislava

22 – Astronomic – Astronomical Institute of the Academy of Sciencees al Institute of the Academy of Sciencees of the Czech Republic, of the Czech Republic, Ondřejov Ondřejov

Layout…

I. Solar corona and coronal loops – an obligate introductionTemperature and density structure of solar coronaCoronal loops and the geometrical structure of magnetic field

II. Coronal heatingEmpirical factsModels: nanoeruptions & braiding

III. Scaling laws – simple, analytical, static modelEnergy equilibrium in static caseDerivation: homogeneous vs. inhomogeneous heating

IV. Model description and preliminary results

Solar corona Highest, extended “layer” of solar atmosphere

Highly structured: – in visible light: coronal streamers (mostly open, static? structures)– in EUV and X-ray: coronal loops, coronal holes, bright points, flares (open and closed structures, sometimes highly dynamical)

I.

Solar corona Solar corona consists of hot and tenuous plasma (Edlén, 1943):

Tcor 106 – 107 K ne,cor 1015 – 1016 m–3

high ionisation degree, plasma and field “frozen-in” optically thin (collisional excitation / spontaneous emission)

Corona and magnetic field Momentum equation dominated by the Lorentz force

force-free approximation:

Geometry given by the magnetic field

Anisotropy – multitemperature corona

(e.g. thermal conduction differs greatly along and across the field)

( , ) ( , ) ( , )B r t r t B r t

171 (1 MK)195 (1.5 MK)284 (2 MK)

Coronal heatingII. Solar corona is ~ 50 times hotter than chromosphere, and more than

order of magnitude more tenuous Without energy source the corona would be in energy equilibrium, with

low, chromospheric-like temperature decreasing outwards Turning off the heating would result in radiation cooling (in EUV

and X – ray spectral domains) of the solar corona during ~ 101 hours

An energy source (at least one) must exist – coronal heating problem

(chromospheric heating problem, solar wind „heating“ problem)

3 3 3 134 104(3 ) 5.10

3 3cor e pM R R n m R M

Coronal heating An estimate of the supplied energy can be obtained if we take into

account that heating must compensate at least for the radiative losses:

The coronal heating mechanism must meet following criteria:

– energy buildup in lower layers of the atmosphere

– energy transport across the chromosphere and

– energy dissipation in corona

Clear relationship exists between X–ray emission and photospheric magnetic flux systems

(Fisher et al., 1998; Benevolenskaya et al., 2002)

35 2 5 3

5 3 7 2 2 4

10 10 .

2

. 10 . .2,5.10 2,5.10 . 4.10

H rad e

pH

H H H

E E n W m

F E W m m W m L

Braiding & nanoeruptions Close et al. (2003): most of the photospheric magnetic flux “closes”

before reaching coronal heights Photospheric fluxtube footpoints are subject to random motions due

to convection (granullation) Fluxtubes braid along one another due to the random motions

(Parker, 1972). The angles between the fluxtube and the photosphere change over time

If the angle reaches critical limit, local reconnection sets in, relasing the stored magnetic energy and simplifying the local field geometry

Estimate of the released energy (Parker, 1988): 1017 W, which is nine orders of magnitude less than the energy released in largest flares

– 10–9 : nanoeruptions Sturrock & Uchida (1981) + Rosner, Tucker, Vaiana (1978): heating

parametrisation for heating due to braiding & nanoeruptions:

12 / 70H B L

Static energy equilibrium

In following, we shall assume static energy equilibrium.

In this case the losses due to radiation Erad and thermal conduction

must be compensated by energy source H:

. 0.rad condH E F

In solar corona the thermal conduction across the magnetic field is negligible. The thermal conduction tensor then has non-zero components only in the direction along the field and can be approximated as

where 0 9,2.10–11 W.m–1.K–7/2 is the Spitzer thermal conduction coefficient. This approximation allows us to solve the energy equilibrium equation in 1D Assumption: the coronal loop has constant cross section

.condF T

5/ 2 0 00 ,T B B

III.

Radiative loss function

Corona is optically thin environment, collisional excitation is compensated by spontaneous emission

Total radiative losses Erad depend on square of electron density and through the statistical equilibrium equation also on temperature. They can be approximated :

Erad = ne2Q(T) = ne

2T 10–32 ne2 T–1/2,

where the last expression holds in the temperature range of ~ 106 – 107.5 K.

Radiative loss function Q(T): RTV analytical approximation(Rosner, Tucker, Vaiana, 1978) Q(T) 10–32 T–1/2

(Priest, 1982: Solar MHD)

Heating function Not known. We parametrize it by the power-law function containing three

parameters:

where CH, ,= const. – free parameters

B0 – field strenght at loop footpoint,

L – loop half-lenght,sH – heating scale lenght

Bref = 10–2 T, Lref = 108 m – scaling constants

sH can be determined from the decrease of the field with height:

does not depend on

0

0 ,H H

s srefs s

Href

LBH H e C e

B L

( 1)( ) HH

ss

Scaling laws The solution of the energy equilibrium equation can be expressed in the

form of scaling laws, 1D analytical relations between loop half-length L, heating H, loop apex temperature T1 and base pressure p0 = p(s0) at the loop footpoint s0:

Rosner, Tucker & Vaiana (1978) – pionieer model, p = const, H(s) = const.

Serio et al. (1981) – inhomogeneous heating , hydrostatic decrease of gas pressure, no changes in gravitational acceleration :

Aschwanden & Schrijver (2002): linear terms added, constants as functions

1/31

7 / 6 5/ 6

1400( )

98000

T pL

H p L

/0 Hs sH H e

20.04

1/31 0

0.57 / 6 5/ 6

0

1400( )

98000

H p

H p

L L

s s

L L

s s

T p L e

H p L e

Scaling laws – derivation Generalized derivation containing radiative loss function Q(T) = T Assumptions: – p = qnekBT ; q = 23/12 H:He = 10:1, total ionisation

– Fcond (s=s0) 0 & Fcond (s=L) = 0 & Fcond (s s0,L) 0– symmetrical loop

Derivation:

0

1

2 /2/5/ 2 20

02 2

2 /2 2/5/ 2 1/ 2 5/ 20

02 2

2 /2/1/ 2 5/ 20

02 2

.

2

0

p

H

p

H

p

H

cond rad

s ss s

B

s ss s

B

T s ss s

BT

F E H

p ed dTT T H e

ds ds q k

p ed dT dTT T H T e

ds ds dsq k

p eT H T e dT

q k

0

0

0 0

/2 3/ 2 3/ 2/7 / 2 7 / 20 0 1

0 0 12 2

220

0 1 0 12 2

20

3/ 2 7

7 , 3 / 2

3 2

p

H

p H

L s sL s s

B

L s L ss s

B

p e T TH T T e

q k

pH T e e T T

q k

Scaling laws – derivation

We now write the integral as a function of the upper boundary T:

1

0 0

2 /2 2/5 1/ 2 5/ 20

02 2

( ) ( )332 25 7 / 2 7 / 20 22

0 112 2

2

4 4

7(3 2 )

p

H

p p

T s ss s

BT

L s L s

s s

B

p edTT T H T e dT

ds q k

pdTT T T e H T T e

ds q k

0

0 0

0

( )2 25 3/ 2 3/ 2 2 7 / 2 3/ 20

1 1 12 2

( ) ( )

3 3( )1/ 2 27 2 2 22042 2

1 1 1

4

(3 2 )

221

2 3

p

p p

p

L s

s

B

L s L s

s s

L s

s

B

pdTT e T T P T T T

ds q k

P e e

pdT T T TT e P

ds T T Tq k

0

0

1

1/ 2

2

( )1/ 2121

0 01 2 2

2 2

2 31 (1 )

7 2 30, 0, 2 0; 1

4 2

p

L s

s

BT

T

H p

x dxT p L s e

q kx P x P

P s s

Scaling laws – derivation

I(P)

1

0

7 2 3( , ) ; 0, 0, 2 0

4 21 (1 )

x dxI P

x P x P

Scaling laws Result:

0 0

0 00

( ) ( )2 22/ 74/ 7 7 74/ 7 2/ 7

1 0 0

4 2 11 24 82 2 8 472 77 70 00

2 22 1 (2 ) (2 )1 (2 )7 77

4( , )

7

3 2 4( , ) .

7 7

.

H H

p pH

L s L s

s s

B

L s L sL ss ss

T I P H L s e e

q kp I P H L s

e e e

Temperature and pressure Temperature – function of the position s

– analytical approximation (Aschwanden & Schrijver, 2002):

11

11

( )( )

10

( )( ) 50

1 100 0 0

0.3 ( ) 1

10.3 ( ) 1 1 log

2

b Ta T

H

b Ta T

H H

L L sT s T

s L s

s sL L s L L sT s T

s L s s L s L s

0 0

( ) 00 1

1 0

0

( ) ( ) ; ( )

( ) ( ) 1 1

p p

s s h h

s hp p

Bp e

H

L sp s p e p h e s h

h h

hqk hh T h

m g R R

pressure dependency – hydrostatic equilibrium:

confused by previous authors!

Scaling laws – Serio et al. (1981)

Apex temperature T1 and base pressure tlak v ukotvení p0 as functions of L

 = 10–32,  = –1/2, CH = 5.10–5  J.m–3.s–1,  = 1,

B0 = 100, resp. 1000 G

 = 0  = 1  = 2

RTV approximation to Q(T)

Motivation

Motivation: To study the dependency of coronal EUV and X-ray emission on heating function

Task: „Assemble“ a model of EUV and X-ray coronal emission of an active region in regular gridusing: magnetic field model (extrapolation)

radiative loss funcitonscaling lawspower-law heating function localised at loop footpoints

Results: (In)dependency of the heating scale length on the fieldEmission distributionConstrains to heating function

IV.

Loop geometry: L and H

For a given grid point, we trace the field line passing through it and obtain the L and B0 distributions

The heating scale lenght sH can be determined from the fall-off of the magnetic field induction with height

These values are then supplied into the scaling laws as the independent variables. We can in this way obtain the temperature and density distribution only from the knowledge of the photospheric longitudinal field!

Informations about the magnetic field can be obtained by extrapolating the photospheric longitudinal magnetogram using force-free approximation ( = const.):

using the method developed by Alissandrakis (1981)

B B

Filter response to emissivity Using the CHIANTI atomic database we compute the synthetic spectra (in a

given wavelenght range) for an entire range of temperature and density values. The spectra contain emission lines and continuum.

The synthetic spectra are then multiplied with the filter response functions and integrated. In this way we obtain the filter response to emissivity, which is a function of T a ne.

Scheme

Linear force-free

extrapolation

longitudinal magnetogram

Determine sH

Scaling lawsT1 p iterations

T1 and p0

Field line tracing L a B0

Synthetic emisison

T and ne

Filter response to emissivity

Comparison with observations,

Constrains on H

AR 10963 – field and sH

( 1)( ) HH

ss

AR 10963 – EIT observations

EIT 17,1 nm, linear scale EIT 19,5 nm, linear scale

AR 10963 – model & observationsEIT 171 EIT 195 EIT 284

model 171 model 195 model 284

Filter ratios --> temperature

195/171, observations195/171, model

 = 10–32,  = –1/2, CH = 5.105 J.m–3.s–1,  = 1,  = 2  = 1  = 0

Thank youThank youfor your attentionfor your attention