Solar Tomography Nancy Mezo Jocelyn Renner

The Problem: Ummm. Maybe we should think about non-invasive imaging….

The Corona

The Solar Corona The corona is the outer region of the sun’s atmosphere Extends from chromosphere to several million km Generates and accelerates the solar wind.

Neat facts about the corona: The amount of light emitted by the solar corona compares to that of the full moon. Nearly a vacuum (very low density) VERY hot --- temperatures greater than 1 million degrees Celsius.

Why do we care? Solar winds affect satellites, astronauts, and power grids. Natural laboratory of super heated plasmas in strong fields. We want to understand the universe.

What do we study? Topology The magnetic field Electron density and temperature

Topology The study of the features of a place or object. Active regions, active region interconnections, coronal holes, bright points, loop structures, and filament structures.

Magnetic Field (Magnetic fields are at the root of virtually all of the features we see on and above the Sun.)

Electron Density Derived from coronography of scattered white light above the limb.

Types of Data SXR -- Soft X-Ray Higher temperatures EUV -- Extreme Ultraviolet Cooler Plasma, transition regions Radio Free-free emissions Magnetic field strength

What has been attempted... Bracewell Altschuler & Perry Hurlburt Frazin Komm Davila Fourier series Least Squares to find Fourier coefficients Filtered Back Projection Robust Regularized Positive Estimation Wavelets Algebraic Reconstruction Technique

Bracewell used Fourier series to reconstruct sun spots

Altschuler and Perry Least Squares to find Fourier Coefficients First to use solar rotation tomography (SRT) Reconstruction of electron density distribution One satellite observation Assumed static structures on the sun Two dimensional model--simple elliptic

Fourier Solutions Used least squares approach to find best set of coefficients. Does not guarantee positive solutions. Form of the basis functions imposes unrealistic constraints which are unrelated to the problem.

N.E. Hurlburt Reconstruction Using Filtered Back Projection Reconstruction of Magnetic Field One satellite observation -- used sun’s rotation Assumed static structures on the sun Two dimensional model--simple elliptic Used SXT data from the Yohkoh Satellite

Filtered Back Projection Accounts for the sun’s opacity Row of pixels as a function of time gives sinusoidal pattern.

Results: Reconstructed images on solar pole Equatorial reconstruction is complicated by highly active regions and faster rotation. Problems Topological change or rapid fluctuations lead to streaking or blurring. Sudden disappearance of features when they pass behind the x-ray opaque sun.

Hurlburt’s Reconstruction

R.W. Komm Wavelets Analysis of Magnetic Field Used magnetograms from the Vacuum Telescope at the Kitt Peak Solar Observatory

Why wavelets? Fourier transforms provide information about frequency, but not location in space. Wavelets provide both. Helps eliminate white noise Shows the smaller magnetic structures are more intermittent, less space filling than larger ones. Noise fills space at all length scales.

J.M. Davila Multiple Spacecraft Model Simulation Many spacecraft orbiting at 1 AU Two dimensional model--simple elliptic Created graph of source distribution Synthetically observed source by summing the emission along the line of sight (LOS)

Source Distribution

Algebraic Reconstruction Technique (ART) Choose any projection and spread the value evenly along LOS Chose a second projection angle and spread the reconstructed array along LOS Spread difference between actual and reconstruction along LOS Repeat for each angle

Advantages: Easy computationally Allows for unevenly spaced observation Number of observations can be varied easily Unequal pixel size easily accounted for by rebinning process Converges to reasonable likeness after one iteration Disadvantages: Weighted more strongly toward larger objects Tends to smear along LOS No natural way to update pixels that don’t fall along any LOS Depends on initial guess, unduly sensitive to out-lying points in data

Results: Angular placement: The optimal configuration occurs where the craft are equally spaced. For four craft, this occurs at 135º. Fourier space coverage Sufficient samples give full coverage of Fourier Space While theoretically no polar orbiting craft are needed, limitations on instruments provide a practical reason to employ them.

Number of spacecraft The larger number of spacecraft, the better the reconstruction. An arrangement with four spacecraft is optimal. The point source resolution improves with Gaussian Resolution Detector Noise Source of error Reconstruction is degraded only slightly at attainable signal to noise ratios (O ~ 200 pixels) Results:

R.A. Frazin Robust, Regularized, Positive Estimation Reconstruction of emissivity structures RRPE method is less sensitive to outlying points in data Removes problems of dependence on initial guess, non-positivity, non-robustness, and inappropriateness of Fourier expansion.

Question, comments, cheap shots or one-liners?