1. Hydrostatic modelling of active region EUV and X-ray emission J. Dudík 1, E. Dzifčáková 1,2, A. Kulinová 1,2, M. Karlický 2 1 – Dept. of Astronomy, Physics of the Earth and Meteorology, FMPhI, Comenius University, Bratislava 2 – Astronomical Institute of the Academy of Sciencees of the Czech Republic, Ondřejov

2. Layout… I.Solar corona and coronal loops – an obligate introduction Temperature and density structure of solar corona Coronal loops and the geometrical structure of magnetic field II.Coronal heating Empirical facts Models: nanoeruptions & braiding III.Scaling laws – simple, analytical, static model Energy equilibrium in static case Derivation: homogeneous vs. inhomogeneous heating IV.Model description and preliminary results

3. 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.

4. Solar corona  Solar corona consists of hot and tenuous plasma (Edlén, 1943): T cor  10 6 – 10 7 Kn e,cor  10 15 – 10 16 m –3  high ionisation degree, plasma and field “frozen-in”  optically thin (collisional excitation / spontaneous emission)

5. 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) 171 (1 MK) 195 (1.5 MK) 284 (2 MK)

6. 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 ~ 10 1 hours  An energy source (at least one) must exist – coronal heating problem (chromospheric heating problem, solar wind „heating“ problem) 

7. 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)

8. 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): 10 17 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:

9. Static energy equilibrium In following, we shall assume static energy equilibrium. In this case the losses due to radiation E rad and thermal conduction must be compensated by energy source H :  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 III.

10. Radiative loss function  Corona is optically thin environment, collisional excitation is compensated by spontaneous emission  Total radiative losses E rad depend on square of electron density and through the statistical equilibrium equation also on temperature. They can be approximated : E rad = n e 2 Q(T) =  n e 2 T   10 – 32 n e 2 T – 1/2, where the last expression holds in the temperature range of ~ 10 6 – 10 7.5 K. Radiative loss function Q(T):  RTV analytical approximation (Rosner, Tucker, Vaiana, 1978)  Q(T)  10 – 32 T – 1/2 (Priest, 1982: Solar MHD)

11. Heating function  Not known. We parametrize it by the power-law function containing three parameters: where C H, ,  = const. – free parameters B 0 – field strenght at loop footpoint, L – loop half-lenght, s H – heating scale lenght B ref = 10 –2 T, L ref = 10 8 m – scaling constants  s H can be determined from the decrease of the field with height:   does not depend on 

12. 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 T 1 and base pressure p 0 = p(s 0 ) at the loop footpoint s 0 :  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

13. Scaling laws – derivation  Generalized derivation containing radiative loss function Q(T) =  T   Assumptions:– p = qn e k B T ; q = 23/12  H:He = 10:1, total ionisation – F cond (s=s 0 )  0 & F cond (s=L) = 0 & F cond (s  s 0,L)  0 – symmetrical loop Derivation:

14. Scaling laws – derivation  We now write the integral as a function of the upper boundary T:

15. Scaling laws – derivation

16. I(  P)

17. Scaling laws  Result:

18. Temperature and pressure  Temperature – function of the position s – analytical approximation (Aschwanden & Schrijver, 2002):  pressure dependency – hydrostatic equilibrium: confused by previous authors!

19. Scaling laws – Serio et al. (1981) Apex temperature T 1 and base pressure tlak v ukotvení p 0 as functions of L  = 10 –32,  = –1/2, C H = 5.10 –5 J.m –3.s –1,  = 1, B 0 = 100, resp. 1000 G  = 0  = 1  = 2

20. RTV approximation to Q(T)

21. 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 grid using:magnetic field model (extrapolation) radiative loss funciton scaling laws power-law heating function localised at loop footpoints Results:(In)dependency of the heating scale length on the field Emission distribution Constrains to heating function IV.

22. Loop geometry: L and H  For a given grid point, we trace the field line passing through it and obtain the L and B 0 distributions  The heating scale lenght s H 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)

23. 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 n e.

24. Scheme Linear force-free extrapolation longitudinal magnetogram Determine s H Scaling laws T 1  p iterations  T 1 and p 0 Field line tracing  L a B 0 Synthetic emisison T and n e Filter response to emissivity Comparison with observations, Constrains on H

25. AR 10963 – field and s H

26. AR 10963 – EIT observations EIT 17,1 nm, linear scaleEIT 19,5 nm, linear scale

27. AR 10963 – model & observations EIT 171EIT 195EIT 284 model 171model 195model 284

28. Filter ratios --> temperature 195/171, observations 195/171, model  = 10 –32,  = –1/2, C H = 5.10 5 J.m –3.s –1,  = 1,  = 2  = 1  = 0

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