3-D Perception of Coronal Loop Structures American Geophysical Union (AGU) Meeting, San Francisco, Dec 15-19, 2000 Special Session SH05 : “A 3-D View of the Sun and Heliosphere” 3-D Perception of Coronal Loop Structures INVITED TALK Markus J. Aschwanden Lockheed Martin ATC, Solar & Astrophysics Lab., Palo Alto, USA

3-D Modeling of Coronal Loops OUTLINE of TALK : 1. Observations - Yohkoh/SXT - SoHO/EIT - TRACE 2. Geometric Data Analysis Techniques - Geometric Forward-Fitting - Static Solar-Rotation Stereoscopy - Dynamic Solar-Rotation Stereoscopy - Simultaneous 2-Spacecraft Stereoscopy - Magnetic Loop Rendering 3. Physical Data Analysis and Theoretical Interpretation - Hydrostatic/Hydrodynamic Models - Density and Temperature Structure ne(s), T(s) - Heating function EH(s), H - Flow dynamics v(s) - Coronal heating 4. Conclusions and Outlook for SECCHI

Yohkoh/SXT A) helmet-shaped arch B) arcade loops C) eruptive loop D) quadrupolar loops E) cusped loops F) double arcade G) sigmoidal loops

SoHO/EIT

TRACE - 171 A, Fe IX/X, T=1.0 MK - 195 A, Fe XII, T=1.5 MK - 284 A, Fe XV, T=2.0 MK Loop study by Aschwanden et al. (2000): Number: N=41 loops, Temperature: T=0.8-1.6 MK, Half length: L=4-324 Mm Loop width: w=1.5-7.5 Mm Pressure: p=0.15-0.60 dyne cm-2 Scale height ratio: q=0.5-4.2 Base density: ne=4*108-2*109cm-3 Apex density: ne=2*107-6*108 cm-3

Active Region … Postflare Loops TRACE 171 A observations (T=1.0 MK) on 1999-Nov-6, 02:30 UT at East limb. A flare occurred about 8 hours earlier, and intense heating is still going on in this active region. TRACE 195 observations (T=1.5 MK) of an X5.7 GOES-class flare on 2000-Jul-14, 10:03 UT, in AR 9077. The “slinky-like” appearance corresponds to a classical two-ribbon flare, with an arcade of cooling EUV loops over the magnetic neutral line. (FOV 230 Mm x 170 Mm)

Loop Detection TRACE 195 A, 1998-Aug-25, 04:00 UT 4-Gradient direction algorithm Local maximum criterion - Computer-aided data analysis of loops requires an interactive or automated pattern recognition algorithms. Automated algorithm are more objective and efficient than human interaction, but are more susceptible to confusion of structures. - The detection of coronal loops can mostly be done by tracing linear features (1-dimensional), e.g. see algorithms developed by Louis Strous (LMSAL) or Eric DeJong (JPL) - Problems occur for intersecting structures, which can only be disentangled with help of calculated projections from 3D models.

Geometric Data Analysis Techniques 1) Geometric Forward-Fitting 2) Static Solar-Rotation Stereoscopy 3) Dynamic Solar-Rotation Stereoscopy 4) Simultaneous 2-Spacecraft Stereoscopy 5) Magnetic Loop Rendering

Geometric Fitting Deduction of 3D geometry of loop from single image, using geometric a-priori constraints (e.g. semi-circular, symmetry, coplanarity, …) Coordinate transformations for coplanar loops (Loughhead et al. 1983, ApJ 74, 883) Application is useful to determine inclination angle of flare loops, where 3D loop geometry may change dynamically in every image. Yohkoh, 1992-Aug-22, 08:23 UT flare Nitta et al. 1999, Sol.Phys. 189, 181

Geometric Parameterization - One-dimensional parameterization with loop length coordinate s(x,y,z) - Projections in different stereo images s(x,y,z) and s(x’,y’,z’) - Semi-circular loop (6 parameters = position l,b,h, radius r; azimuth a, inclination  - Elliptical loop (7 parameters = … + eccentricity) - Helical loop (10 parameters = … + torus radius, number of twists, phase of twist angle) - Dynamical loops (12-20 parameters = … + first derivative of time-dependence (velocity v=dx/dt, rotation, twisting, tilting,…)

Static Solar-Rotation Stereoscopy Stereoscopic 3D reconstruction of loop structures can be achieved using the solar rotation, if the structure remains quasi-static during the stereoscopic time interval. 3D model of loop arcade containing 200 semi-circular loops, each one parameterized with 7 free parameters Observations: TRACE 171 A image of 1998-Sep-30, 14:30 UT postflare arcade (FOV=180,000 km)

Dynamic Solar-Rotation Stereoscopy - assumes that magnetic field is slowly changing during stereoscopic time interval, so that local ensemble of loops stays near-parallel - relaxes the assumption of static stereoscopy that loops cannot change. Magnetic field lines can be filled with hot plasma, can cool off, and new adjacent field lines may lighten up during stereoscopic time interval. Aschwanden et al. 1999, ApJ 515, 842

with dynamic stereoscopy method. 3D-Reconstruction of 30 Active Region loops from SoHO/EIT 171 A in NOAA 7986, 1996-Aug-29/30/31 with dynamic stereoscopy method. - Inclination angle of loop planes - De-projected density scale heights Aschwanden et al. 1999, ApJ 515, 842

- Column depth of plasma along loop can only be modeled properly by accurate knowledge of line-of-sight angle to loop. - The density scale height can only be measured properly with accurate knowledge of the loop plane inclination angle.

Simultaneous Stereoscopy with 2 Spacecraft allows for simultaneous view of 2 projections and thus does not restrict the time evolution of structures (opposed to solar- rotation stereoscopy). - 3D reconstruction of loops yields unique geometric solution, provided there is no confusion problem with adjacent structures.

Magnetic Rendering - 3D Magnetic field lines are computed from extrapolation of photospheric field according to theoretical model. - Field lines are filled with plasma according to hydrostatic model - 3D magnetic field is “stretched” or transformed to match data Gary & Alexander 1999, SP 186, 123

Radial stretching of potential field lines provides a better fit to outlines of Yohkoh/SXT loops. Gary & Alexander 1999, Solar Phys. 186, 123

-The computed potential field B(x,y,z) based on SoHO/MDI magnetograms does not line out the EIT-traced loops observed in 171, 195, or 284 A. - A non-potential field model with alpha=0.045 matches the EIT data better.

Helical & Sigmoidal Loops

3-D Modeling of Coronal Loops OUTLINE of TALK : 1. Observations - Yohkoh/SXT - SoHO/EIT - TRACE 2. Geometric Data Analysis Techniques - Geometric Forward-Fitting - Static Solar-Rotation Stereoscopy - Dynamic Solar-Rotation Stereoscopy - Simultaneous 2-Spacecraft Stereoscopy - Magnetic Loop Rendering 3. Physical Data Analysis and Theoretical Interpretation\ - Hydrostatic/Hydrodynamic Models - Density and Temperature Structure ne(s), T(s) - Heating function EH(s), s_h - Flow dynamics v(s) - Coronal heating 4. Conclusions and Outlook for SECCHI

Are coronal loops in hydrostatic equilibrium ? - Dynamic coronal loops show strong deviations from hydrostatic equilibrium, in particular postflare loops. The example shown above exhibits up to 4 times larger density scale heights than expected in hydrostatic equilibrium.

THEORY: The mean exponential density scale height in the upper half of the loops is calculated for hydrostatic loops with various heating scale heights: OBSERVATIONS: The measured density scale height of EIT and TRACE loops:

Hydrostatic Equations Force (momentum) equation -dp/ds - mng(R/r)2 cos()= 0 Energy balance equation -FC + EH + ER = 0 Equation of state p = 2 n k T Solutions : L,EH0,sH,T1,n1 -> T(s), n(s), p(s)

Radiative Loss Function Radiative Loss: E R(s) = ne(s)2  [T(s)] erg cm-3 s-1 Radiative loss function [T(s)] depends on abundances and assumptions on ionization equilibrium e.g. chromospheric abundances (Meyer) vs. enhanced iron in coronal abundances (Feldman) Higher iron abundance increase radiative loss and thus yield lower densities

Conductive Loss Rate F = d/ds[-T5/2 dT/ds] Conductive loss rate: Spitzer conductivity:  = 9.2 x 10-7 erg s-1 cm-1 Boundary conditions: - F(s=L)=0 symmetric loops - F(s=0)=0 vanishing conductive flux at footpoint Observational constraints: - Lyman  in chromosphere proportional to conductive flux (Kankelborg et al. 1997) - Asymmetric loops constrain conductive flux near loop tops - Variation of loop cross-section A(s) affects conductive loss rate: 1/A(s)* d/ds [A(s) * F(s)]

Heating Function a) Uniform Heating EH(s)=const b) Footpoint heating EH(s)=EH0 exp(-h/sH) c) Lootpoint heating EH(s)=EH0 exp(+h/sH) Heating scale height: (exponential) : sH Rosner, Tucker & Vaiana (1978): Assumption of uniform heating Serio et al. (1981): Generalization of loop scaling laws for nonuniform heating Priest et al. (2000): Fitting of 5 different heating functions to data Aschwanden et al. (2000): Fitting of hydrostatic solutions w. variable sH to data

Hydrostatic Solutions - Temperature profiles T(s) are more iso-thermal for shorter heating scale heights - Densities n(s) and pressures p(s) are higher for shorter heating scale heights - Unstably stratified solutions (density inversion) occur for shortest heating scale heights - Unstably stratified solutions have (unobserved) steep temperature gradients

Aschwanden, Nightingale, Alexander 2000, ApJ 541, 1059

Density ne(s) and temperature T(s) analysis of 41 EUV loops observed with TRACE 171,195 A

- The temperature profiles T(s) of EUV loops are more iso-thermal than predicted by the (uniform-heating) RTV model - The density profiles ne(s) have higher densities than predicted by the (uniform-heating) RTV model

- The loop base pressure p(s=0) is up to a factor of 35 higher for the observed EUV loops than predicted by the (uniform-heating) RTV model - The loop base pressure p(s=0) is essentially independent of the loop length L

Key result: - All EUV loops are not consistent with the uniform-heating RTV model - Their base pressure is consistent with a heating scale height of sH=176 Mm

What did we learn ?

Aschwanden, Schrijver & Alexander (2001) ApJ Fitting of hydrostatic solutions to observed F171(s) and F195(s) fluxes varying the heating scale height H Best fits: H=12  5 Mm Diagnostic of hydrostatic loops: 30% hydrostatic loops Dynamic loops: 60% over-pressure loops 10% under-pressure loops Dynamic loops are also found to have super-hydrostatic density scale heights

Hydrodynamicequations Mass conservation dn/dt + (1/A) d/ds(nvA) = 0 Force (momentum) equation mn(dv/dt) + mnv(dv/ds) = -dp/ds - mng(R/r)2 cos() Energy balance equation (1/A)(d/ds)(nvA[eenth+ekin+egrav] +AFC) = EH + ER = 0 Enthalpy: eenth=(5/2) kBT Kinetic energy: ekin=(1/2)mv2 Grav.potential: egrav=mg(R/r)2 Solutions : L,EH0,sH,T1,n1,v1 -> T(s), n(s), v(s)

Loop Flow Models - Flows have been observationally detected by Doppler shifts and feature tracking - Lack of temperature transition zone at loop footpoints indicates flows - “Over-density” in coronal loops can only be supported by chromospheric upflows - Slow subsonic upflows have large enthalpy loss (cooling) during upflows - Fast subsonic upflows warrant near-isothermal temperature profile T(s) - Fast subsonic upflows become supersonic and form shocks near tops of large loops - Siphon flows produce asymmetric loops, with pressure difference betw. footpoints

Loop Dynamics

What Physics helps 3D Loop Modeling ? HYDROSTATICS : - Inclination angle of loop planes provides de-projection of density scale height - De-projected density scale height allows for tests of pressure equilibrium - Comparison of pressure scale height with temperature scale height provides diagnostic of static versus dynamic loops MAGNETIC FIELD: - Comparison of 3D loops traced out in EUV or soft X-rays with extrapolated magnetic field lines allows for test of theoretical coronal magnetic field models. - Diagnostic of potential vs. nonpotential field - Tracking evolution of twisting and shearing with time - Diagnostic on helicity, stable and (kink)-unstable magnetic configurations CORONAL HEATING: - 3D density and temperature profile constrains coronal heating function EH(h) - Localization of heating function allows to discriminate physical heating mechanisms.

Conclusions 3D loop modeling based on stereoscopic principles could only be done using the solar rotation so far. Those studies have proven to be very useful to explore the physics of the solar corona, but are restricted to quasi-static loop structures. STEREO/SECCHI (launch planned for 2004) will for the first time allow for true (simultaneous) stereoscopy, providing 3D information of dynamic loop structures without limitation on their time variability. Magnetic rendering techniques and testing of theoretical magnetic field models by EUV tracing is still in its infancy. Progress can be expected for automated, iterative fitting algorithms. More general modeling algorithms are anticipated that accomplish 4D-modeling of coronal loops, combining time dependence with 3D spatial coordinates, eg parameterized in terms of s[x(t),y(t),z(t)].

REFERENCES : Nitta,N., VanDriel-Gesztelyi,L., Harra-Murnion,L. 1999, Sol.Phys. 189, 181 Flare loop geometry Gary,A. and Alexander D. 1999, Sol.Phys. 186, 123 Constructing the coronal magnetic field by correlating parametrized field lines with observed coronal plasma structures Aschwanden,M.J., Newmark,J.S., Delaboudiniere,J.P., Neupert,W.M., Klimchuk,J.A., Gary,G.A., Portier-Fornazzi,F., Zucker,A., 1999, ApJ 515, 842 3D stereoscopic analysis of solar active region loops: I. SoHO/EIT observations at temperatures of 1.0-1.5 MK Aschwanden,M.J., Alexander,D., Hurlburt,N., Newmark,J.S., Neupert,W.M., Klimchuk,J.A., and Gary,G.A. 2000, ApJ 531, 1129 3D stereoscopic analysis of solar active region loops: II. SoHO/EIT observations at temperatures of 1.5-2.5 MK Aschwanden,M.J., Nightingale,R.W., Alexander,D. 2000, ApJ 541, 1059 Evidence for nonuniform heating of coronal loops inferred from multi-thread modeling of TRACE data Aschwanden,M.J., Schrijver,C.J., and Alexander,D. 2001, ApJ 550, (March 20 issue) Modeling of coronal EUV loops observed with TRACE: I. Hydrostatic solutions with nonuniform heating