1. Jaroslav Dudík 1,2 Elena Dzifčáková 3, Jonathan Cirtain 4 1 – DAMTP-CMS, University of Cambridge 2 – DAPEM, FMPhI, Comenius University, Bratislava, Slovakia 3 – Astronomical Institute of the Academy of Sciences, Ondřejov, Czech Republic 4 – NASA Marshall Space Flight Center, Huntsville, AL, USA 14 th European Solar Physics Meeting Dublin, Ireland, September 9 th, 2014 Area Expansion of Magnetic Flux-Tubes in Solar Active Regions

2. I.The Active Region Corona: Loops and what else? Observed non-expansion of coronal loops Do we understand the geometrical effects? Notes on active region modelling II.The Case of Quiescent AR 11482 Dudík et al. (2014) ApJ, submitted Hinode/SOT observations Potential extrapolation and approximation by magnetic charges Expansion with height Structure of area expansion factors: steep valleys and flat hills “Fundamental flux-tubes”: linguine rather than spaghetti III.Speculations: How to Create Coronal Loops The effect of expansion on density and total input heating Outline

3. Coronal Loops: Lack of Expansion Klimchuk et al. (1992), PASJ 44, L181 Klimchuk (2000), SoPh 193, 53 Watko & Klimchuk (2000), SoPh 193, 77 Aschwanden & Nightingale (2005), ApJ 633, 499 Brooks et al. (2007), PASJ 59, 691

4.  Closed magnetic flux – loops (widths, temperature profiles)  Spatial structure: – hot, X-Ray AR core, diffuse – warm EUV loops  – EUV moss – “bright points”  Could these observations be explained by ONE and UNIVERSAL heating function? Hot Core and Warm Periphery

5. Do We Understand the Geometry? DeForest (2007), ApJ 661, 532: Poorly resolved expanding structures may appear to be non-expanding

6. But some loops are resolved… Brooks et al. (2013), ApJ 772, L19 Peter et al. (2013) A&A 556, A104

7. Geometry… Part 2 Malanushenko & Schrijver (2013), ApJ 775, 120  No circular cross-sections  Introduces bias in loop selection

9. Expanding Loop: Thermal Struct. Peter & Bingert (2012), A&A 457, A1  MHD model of the solar corona  Magnetic flux-tube with expanding area (cross-section)  Interplay between temperature and density structure  Leads to apparently non-expanding AIA loop  Even if well- resolved

10. Dudík et al. (2011), A&A 531, A115

11. Area Expansion Factor: Structure  SOHO/MDI  2” spatial resolution  Expansion factor defined as: (flux cons.)

12. Hinode/SOT B Z : 0.3’’ resolution

13. Approximation by 134 charges

14. Area Expansion: General Properties  Flux-tubes in direct extrapolation expand more strongly  Rate of expansion increases with height of the starting point

15. Area Expansion Factor: Structure direct extrapolation magnetic charges significant structure little structure

16. “Steep Valleys and Flat Peaks” Direct extrapolation Approx. by charges

17. Steep Valleys: Coronal Loops? Periphery AR core

18. “Fundamental Flux-tubes”

19. Highly Squashed Cross-sections

20. Toy model: Density increase  Suppose heating depends on B, and B decreases exponentially:  The definition of the area expansion factor then gives  Electron density in the hydrostatic, steady-heating case can be obtained from the scaling laws: I.e., because of the definition of Γ. I.e., because of the definition of Γ.  Therefore, two field lines with different Γ 1, Γ 2 will produce density contrast:

21.  The flux-tube expansion is finely structured Even in potential fields – other fields likely even more complex.  Steep valleys with width of one or several 0.3’’ pixels  Prediction: Hi-C off limb may NOT see expanding loops  “Fundamental flux-tubes” have highly squashed cross-sections  Linguine rather than spaghetti  Combined with heating as a function of B, a structure of active region emission can emerge Dudík et al. (2011), Astron. Astrophys. 531, A115 Dudík et al. (2014), Astrophys. J., submitted Summary

22. Thank you for your attention

23. Hot Core and Warm Periphery EUV “warm” loops X-ray “hot” loops

24. Active Regions: The Scale Problem

25. The Heating Function  Unkown. Assumed to be exponentially decreasing & parametrized: C H0, ρ & τ – free parameters B 0 – footpoint magnetic field L 0 – loop half-length s H – heating scale-length  s H is determined from the rate of magnetic field decrease along a loop 

26. “DDKK” Generalized Scaling Laws  Non-uniform heating  Non-isothermal loops  Pressure stratification in non-isothermal loops  Parametrized form of radiative losses: R(T) = χT –σ n e 2 Dudík et al. (2009), Astron. Astrophys. 502, 957

27. Loop Temperature Profiles  Voxel position corresponding to a location s along the loop. Define  If the heating has L/s H < 3, then  Else (3 < L/s H < 25) Aschwanden & Schrijver (2002), ApJS 142, 269

28. AR 10963: Observations

30. The Temperature Structure

31. The Role of Heating Scale-Length

33. The Hinode/SOT Case

35. Area Expansion at 0.3’’ resolution Saturated to: Γ = 50 Γ = 150  XY plane, Z = 100*0.32’’ = 32’’ = 23.2 Mm

36. Area Expansion at 0.3’’ resolution Saturated to: Γ = 50 Γ = 150 YZ plane, X fixed

37. Area Expansion at 0.3’’ resolution Saturated to: Γ = 50 Γ = 150 XZ plane, Y fixed

38. Area Expansion at 0.3’’ resolution Volumetric rendering of the Expansion Factor

39. Area Expansion at 0.3’’ resolution Volumetric rendering of the (Expansion Factor) 1.5