Jaroslav Dudík 1,2

Elena Dzifčáková 3, Jonathan Cirtain 4

1 – DAMTP-CMS, University of Cambridge2 – DAPEM, FMPhI, Comenius University, Bratislava, Slovakia

3 – Astronomical Institute of the Academy of Sciences, Ondřejov, Czech Republic4 – NASA Marshall Space Flight Center, Huntsville, AL, USA

14th European Solar Physics MeetingDublin, Ireland, September 9th, 2014

Area Expansion of MagneticFlux-Tubes in Solar Active Regions

I. The Active Region Corona: Loops and what else?Observed non-expansion of coronal loopsDo we understand the geometrical effects?Notes on active region modelling

II. The Case of Quiescent AR 11482Dudík et al. (2014) ApJ, submittedHinode/SOT observationsPotential extrapolation and approximation by magnetic chargesExpansion with heightStructure of area expansion factors: steep valleys and flat hills“Fundamental flux-tubes”: linguine rather than spaghetti

III. Speculations: How to Create Coronal LoopsThe effect of expansion on density and total input heating

Outline

Coronal Loops: Lack of Expansion

Klimchuk et al. (1992), PASJ 44, L181Klimchuk (2000), SoPh 193, 53Watko & Klimchuk (2000), SoPh 193, 77Aschwanden & Nightingale (2005), ApJ 633, 499Brooks et al. (2007), PASJ 59, 691

Closed magnetic flux – loops (widths, temperature profiles) Spatial structure: – hot, X-Ray AR core, diffuse

– warm EUV loops – EUV moss

– “bright points”

Could these observationsbe explained by

ONE and UNIVERSALheating function?

Hot Core and Warm Periphery

Do We Understand the Geometry?

DeForest (2007),ApJ 661, 532:

Poorly resolved expanding structures may appear to benon-expanding

But some loops are resolved…

Brooks et al. (2013), ApJ 772, L19 Peter et al. (2013) A&A 556, A104

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

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

Dudík et al. (2011), A&A 531, A115

Area Expansion Factor: Structure

0

1

B

B

SOHO/MDI 2” spatial

resolution

Expansion factor defined as: (flux cons.)

Hinode/SOT BZ: 0.3’’ resolution

Approximation by 134 charges

Area Expansion: General Properties, calculated for every voxel (volume pixel)

Flux-tubes in direct extrapolation expand more strongly Rate of expansion increases with height of the starting point

Area Expansion Factor: Structure direct extrapolation magnetic charges

significant structure little structure

“Steep Valleys and Flat Peaks”

Direct extrapolationApprox. by charges

Steep Valleys: Coronal Loops?

Periphery

AR core

“Fundamental Flux-tubes”

Highly Squashed Cross-sections

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.

Therefore, two field lines with different Γ1, Γ2 will produce density contrast:

The flux-tube expansion is finely structuredEven 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 structureof active region emission can emerge

Dudík et al. (2011), Astron. Astrophys. 531, A115Dudík et al. (2014), Astrophys. J., submitted

Summary

Thank you for your attention

Hot Core and Warm PeripheryEUV “warm” loops X-ray “hot” loops

Active Regions: The Scale Problem

The Heating Function Unkown. Assumed to be exponentially decreasing & parametrized:

CH0, ρ & τ – free parametersB0 – footpoint magnetic fieldL0 – loop half-lengthsH – heating scale-length

sH is determined from the rate of magnetic field decrease along a loop

0 0

H H

0H 0 H0 H0

0

( ) ,

s s s srefs s

ref

LBE s s E e C e

B L

0

0

HH

H 0

( )

( )

,1 1 /

BA s L

A B s L

ss

s L

0

HH,total H0 H 1

L s

sE E s e

“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 ne

2

Dudík et al. (2009), Astron. Astrophys. 502, 957

Loop Temperature Profiles Voxel position corresponding to a location s along the loop. Define

If the heating has L/sH < 3, then

Else (3 < L/sH < 25)

Aschwanden & Schrijver (2002), ApJS 142, 269

AR 10963: Observations

The Temperature Structure

2/55

1

1 2

( )

( )( ) ( )

ii

F T

CIFR TF T F T

The Role of Heating Scale-Length

The Role of Heating Scale-Length

The Hinode/SOT Case

The Hinode/SOT Case

Area Expansion at 0.3’’ resolution

Saturated to: Γ = 50 Γ = 150

XY plane, Z = 100*0.32’’ = 32’’ = 23.2 Mm

Area Expansion at 0.3’’ resolution

Saturated to: Γ = 50 Γ = 150

YZ plane, X fixed

Area Expansion at 0.3’’ resolution

Saturated to: Γ = 50 Γ = 150

XZ plane, Y fixed

Area Expansion at 0.3’’ resolutionVolumetric rendering of the Expansion Factor

Area Expansion at 0.3’’ resolutionVolumetric rendering of the (Expansion Factor)1.5