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Hydrostatic model ling of active region EUV and X-ray emis sion

Hydrostatic model ling of active region EUV and X-ray emis sion. J. Dudík 1 , E . Dzif čáková 1,2 , A. Kulinov á 1,2 , M. Karlický 2 1 – Dept. of Astronomy, Physics of the Earth and Meteorology, FM Ph I , Comenius University , Bratislava

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Hydrostatic model ling of active region EUV and X-ray emis sion

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  1. Hydrostatic modelling of active regionEUV andX-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 I. 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)

  4. 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)

  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 heating II. 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)

  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): 1017 W, which isnine 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. III. Staticenergy equilibrium In following, we shall assume static energy equilibrium. In this case the losses due to radiation Erad and thermal conductionmust 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 k0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

  10. Radiative loss function Corona is optically thin environment, collisional excitation is compensated by spontaneous emission Total radiative lossesErad depend on square of electron density and through the statistical equilibrium equation also on temperature. They can be approximated : Erad = ne2Q(T) = cne2Ts 10–32ne2T–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)

  11. Heating function Not known. We parametrize it by the power-law function containing three parameters: whereCH, r, t = 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 sHcan be determined from the decrease of the field with height: , does not depend ont

  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-lengthL, heatingH, loop apex temperature T1and 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

  13. Scaling laws – derivation Generalized derivation containing radiative loss functionQ(T) = cTs 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:

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

  15. Scaling laws – derivation

  16. I(s, 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 temperatureT1 and base pressure tlak v ukotveníp0 as functions of L c= 10–32, s= –1/2, CH = 5.10–5  J.m–3.s–1, r= 1, B0 = 100, resp. 1000 G t= 1 t= 2 t= 0

  20. RTV approximation to Q(T)

  21. IV. Motivation Motivation: To study the dependency of coronal EUV and X-ray emission on heating function Task: „Assemble“ a model of EUV andX-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

  22. Loop geometry:Land H Informations about the magnetic field can be obtained by extrapolating the photospheric longitudinal magnetogram using force-free approximation (a = const.): using the method developed by Alissandrakis (1981) For a given grid point, we trace the field line passing through it and obtain the Land B0distributions The heating scale lenghtsH 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!

  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 Ta ne.

  24. Scheme longitudinal magnetogram Linear force-free extrapolation Field line tracing LaB0 Scaling laws T1 lpiterations T1andp0 DeterminesH T andne Comparison with observations, Constrains onH Filter response to emissivity Synthetic emisison

  25. AR 10963 – field andsH

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

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

  28. Filter ratios--> temperature 195/171, model c= 10–32, s= –1/2, CH = 5.105 J.m–3.s–1, r= 1, 195/171, observations t= 2 t= 1 t= 0

  29. Thank youfor your attention

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