3-D Perception of Coronal Loop Structures

Markus J. AschwandenLockheed Martin ATC, Solar & Astrophysics Lab., Palo Alto, USA

American Geophysical Union (AGU) Meeting, San Francisco, Dec 15-19, 2000 Special Session SH05 : “A 3-D View of the Sun and Heliosphere”

INVITED TALK

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-D Modeling of Coronal Loops

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 archB) arcade loopsC) eruptive loopD) quadrupolar loopsE) cusped loopsF) double arcadeG) 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 MmLoop width: w=1.5-7.5 MmPressure: p=0.15-0.60 dyne cm-2

Scale height ratio: q=0.5-4.2Base density: ne=4*108-2*109cm-3

Apex density: nee=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 inthis 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 classicaltwo-ribbon flare, with an arcade of coolingEUV 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 geometrica-priori constraints (e.g. semi-circular,symmetry, coplanarity, …)

Coordinate transformations for coplanarloops (Loughhead et al. 1983, ApJ 74, 883)

Application is useful to determineinclination angle of flare loops,where 3D loop geometry may changedynamically in every image.

Yohkoh, 1992-Aug-22, 08:23 UT flareNitta 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

3D model of loop arcade containing200 semi-circular loops, each oneparameterized with 7 free parameters

Stereoscopic 3D reconstruction of loopstructures can be achieved using the solarrotation, if the structure remains quasi-staticduring the stereoscopic time interval.

Observations: TRACE 171 A imageof 1998-Sep-30, 14:30 UT postflarearcade (FOV=180,000 km)

Dynamic Solar-RotationStereoscopy

- 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

3D-Reconstruction of30 Active Region loopsfrom SoHO/EIT 171 Ain NOAA 7986, 1996-Aug-29/30/31with dynamic stereoscopymethod.

- 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

- 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.

MagneticRendering

- 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 potentialfield lines provides a betterfit to outlines of Yohkoh/SXTloops.

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

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-D Modeling of Coronal Loops

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 loopsis 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 FunctionRadiative 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 equilibriume.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 Conductive loss rate:

F = d/ds[-T5/2 dT/ds]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 Functiona) 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 heatingSerio et al. (1981): Generalization of loop scaling laws for nonuniform heatingPriest et al. (2000): Fitting of 5 different heating functions to dataAschwanden 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=176 Mm

What did we learn ?

Aschwanden, Schrijver & Alexander (2001) ApJ

Fitting of hydrostatic solutionsto observed F171(s) and F195(s) fluxesvarying the heating scale height H

Best fits: H=12 5 Mm

Diagnostic of hydrostatic loops: 30% hydrostatic loopsDynamic loops: 60% over-pressure loops 10% under-pressure loops

Dynamic loops are also found tohave super-hydrostatic densityscale heights

Hydrodynamicequations

Force (momentum) equationmn(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) kBTKinetic energy: ekin=(1/2)mv2

Solutions : L,EH0,sH,T1,n1,v1 -> T(s), n(s), v(s)

Mass conservation

dn/dt + (1/A) d/ds(nvA) = 0

Grav.potential: egrav=mg(R/r)2

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 ?

- 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

HYDROSTATICS :

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 doneusing the solar rotation so far. Those studies have proven to be very usefulto 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 allowfor true (simultaneous) stereoscopy, providing 3D information of dynamicloop 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 expectedfor automated, iterative fitting algorithms.

More general modeling algorithms are anticipated that accomplish4D-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, 181Flare loop geometry

Gary,A. and Alexander D. 1999, Sol.Phys. 186, 123Constructing the coronal magnetic field by correlating parametrized field lineswith 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, 8423D stereoscopic analysis of solar active region loops: I. SoHO/EIT observationsat 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, 11293D stereoscopic analysis of solar active region loops: II. SoHO/EIT observationsat temperatures of 1.5-2.5 MK

Aschwanden,M.J., Nightingale,R.W., Alexander,D. 2000, ApJ 541, 1059Evidence for nonuniform heating of coronal loops inferred from multi-threadmodeling 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 solutionswith nonuniform heating