Thermal Diagnostics of Elementary and Composite Coronal Loops with AIA

Markus J. Aschwanden & Richard W. Nightingale (LMSAL)

AIA/HMI Science Teams Meeting, Monterey, Feb 13-17, 2006Session C9: Coronal Heating and Irradiance (Warren/Martens)

A Forward-Fitting Technique to conduct Thermal Studies with AIAUsing the Composite and Elementary Loop Strands in a Thermally Inhomogeneous Corona (CELTIC)

Parameterize the distribution of physical parameters of coronal loops(i.e. elementary loop strands):

-Distribution of electron temperatures N(T)-Distribution of electron density N(n_e,T)-Distribution of loop widths N(w,T)

Assume general scaling laws:-Scaling law of density with temperature: n_e(T) ~ T^a-Scaling law of width with temperature: w(T) ~ T^b

Simulate cross-sectional loop profiles F_f(x) in different filtersby superimposing N_L loop strandsSelf-consistent simulation of coronal background and detected loops

Forward-fitting of CELTIC model to observed flux profiles F_i(x) in 3-6AIA filters F_i yields inversion of physical loop parameters T, n_e, was well as the composition of the background corona[N(T), N(n_e,T), N(w,T)] in a self-consistent way.

TRACEResponse functions171, 195, 284 AT=0.7-2.8 MK

Model:

Forward-Fitting to 3 filtersvarying T

171Aon

June 12 1998

12:05:20

Loop #3A T=1.39 MK w=2.84 Mm

Loop_19980612_A

Observational constraints:

Distribution of -loop width N(w), <w^loop> -loop temperature N(T), <T^loop> -loop density N(n_e), <n_e^loop> -goodness-of-fit, N(chi^2), <ch^2> -total flux 171 A, N(F1), <F1^cor> -total flux 195 A, N(F2), <F2^cor> -total flux 284 A, N(F3), <F3^cor> -ratio of good fits q_fit =N(chi^2<1.5)/N_det

Observables obtained fromFitting Gaussian cross-sectional profiles F_f(x) plus linear slope to observed flux profiles in TRACE triple-filter data (171 A, 195, A, 284 A)N_det=17,908 (positions)(Aschwanden & Nightingale 2005, ApJ 633, 499)

Forward-fittingof CELTIC Model:

Distribution of -loop width N(w), <w^loop> -loop temperature N(T), <T^loop> -loop density N(n_e), <n_e^loop> -goodness-of-fit, N(chi^2), <ch^2> -total flux 171 A, N(F1), <F1^cor> -total flux 195 A, N(F2), <F2^cor> -total flux 284 A, N(F3), <F3^cor> -ratio of good fits q_fit =N(chi^2<1.5)/N_det

With the CELTIC model wePerform a Monte-Carlo simulationof flux profiles F_i(x) in 3 Filters(with TRACE response functionand point-spread function) bysuperimposing N_L structureswith Gaussian cross-section andreproduce detection of loops toMeasure T, n and w of loop andTotal (background) fluxes F1,F2,F3

(Aschwanden, Nightingale, & Boerner 2006, in preparation)

Loop cross-section profileIn CELTIC model:

-Gaussian density distribution with width w_i n_e(x-x_i)

-EM profile with width w_i/sqrt(2) EM(x)=Int[n^e^2(x,z)dz] /cos(theta)

-loop inclination angle theta

-point-spread function w^obs=w^i * q_PSF EM^obs=EM_i / q_PSF

q_PSF=sqrt[ 1 + (w_PSF/w_i)^2]

Parameter distributions of CELTIC model: N(T), N(n,T), N(w,T)Scaling laws in CELTIC model: n(T)~T^a, w(T)~T^b

a=0b=0

a=1b=2

Concept of CELTIC model:

-Coronal flux profile F_i(x) measured in a filter i is constructed by superimposing the fluxes of N_L loops, each one characterized with 4 independent parameters: T_i,N_i,W_i,x_i drawn from random distributions N(T),N(n),N(w),N(x)

The emission measure profile EM_i(x) of each loop strand is convolved with point-spread function and temperature filter response function R(T)

Superposition of flux profiles f(x) of individual strands Total flux F_f(x)

The flux contrast of a detected (dominant) loop decreases with thenumber N_L of superimposed loop structures makes chi^2-fit sensitive to N_L

AIA Inversion of DEM

AIA covers temperature range of log(T)=5.4-7.0

Inversion of DEM with TRACE triple-filter data and CELTIC model constrained in range of log(T)=5.9-6.4 2 Gaussian DEM peaks and scaling law (a=1,b=2)

Inversion of DEM with AIA data and CELTIC model will extend DEM to larger temperature range3-4 Gaussian DEM peaks and scaling laws: n_e(T) ~ T^a w(T) ~ T^b

Constraints from CELTIC modelfor coronal heating theory

(1) The distribution of loop widths N(w), [corrected for point-spread function] in the CELTIC model is consistent with a semi-Gaussian distribution with a Gaussian width of

w_g=0.50 Mm which corresponds to an average FWHM <FWHM>=w_g * 2.35/sqrt(2)=830 km which points to heating process of fluxtubes separated by a granulation size.

(2) There is no physical scaling law known for the intrinsic loop width with temperature

The CELTIC model yieldsw(T) ~ T^2.0

which could be explained by cross-sectional expansion by overpressure in regions where thermal pressure is larger than magnetic pressure plasma-beta > 1, which points again to heating below transition region.

Scaling law of width with temperaturein elementary loop strands

Observational result from TRACETriple-filter data analysis of elementaryloop strands (with isothermal cross-sections):

97.1)( TTw

2

2/72/7

L

T

s

T

sFE cH

75.14/7)( TTTw

Loop widths cannot adjust to temperature incorona because plasma- << 1, and thuscross-section w is formed in TR at >1Thermal conduction across loop widthsIn TR predicts scaling law:

CONCLUSIONS

(1) The Composite and Elementary Loop Strands in a Thermally Inhomogeneous Corona (CELTIC) model provides a self-consistent statistical model to quantify the physical parameters (temperature, density, widths) of detected elementary loop strands and the background corona, observed with a multi-filter instrument.

(2) Inversion of the CELTIC model from triple-filter measurements of 18,000 loop structures with TRACE quantifies the temperature N(T), density N(n_e), and width distribution N(w) of all elementary loops that make up the corona and establish scaling laws for the density, n_e(T)~T^1.0, and loop widths w(T) ~ T^2. (e.g., hotter loops seen in 284 and Yohkoh are “fatter” than in 171)

(3) The CELTIC model attempts an instrument-independent description of the physical parameters of the solar corona and can predict the fluxes and parameters of detected loops with any other instrument in a limited temperature range (e.g., 0.7 < T < 2.7 MK for TRACE). This range can be extended to 0.3 < T < 30 MK with AIA/SDO.

(4) The CELTIC model constrains the cross-sectional area (~1 granulation size) and the plasma-beta (>1), both pointing to the transition region and upper chromosphere as the location of the heating process, rather than the corona!