1. 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

2. 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

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

4. SoHO/EIT

5. 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= MK,
Half length: L=4-324 Mm
Loop width: w= Mm
Pressure: p= dyne cm-2
Scale height ratio: q=
Base density: ne=4*108-2*109cm-3
Apex density: ne=2*107-6*108 cm-3

6. 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 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)

7. 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.

8. 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

9. 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

10. 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,…)

11. 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)

12. 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

14. 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

15. - 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.

16. 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.

17. 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

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

19. -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.

20. Helical &
Sigmoidal
Loops

21. 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

22. 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.

23. 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:

24. 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)

25. 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

26. Conductive Loss Rate F = d/ds[-T5/2 dT/ds] Conductive loss rate:
Spitzer conductivity:
 = 9.2 x 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)]

27. 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

28. 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

29. Aschwanden, Nightingale, Alexander 2000, ApJ 541, 1059

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

31. - 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

32. - 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

33. 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

34. What did we learn ?

35. 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

36. 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)

37. 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

38. Loop Dynamics

39. 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.

40. 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)].

41. 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 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 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