Solar Hard X-ray Albedo
From RHESSI Wiki
|1st Author:||Ed Schmahl|
|2nd Author:||Gordon Hurford|
|Published:||18 January 2010|
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Solar Hard X-ray Albedo
In the early days (1972) of solar hard X-ray flare observations, Fred Tomblin  published theoretical arguments that the hard X-ray spectrum of solar flares in the 5-40 keV range must have an albedo component due to Compton back-scattering (from the photosphere) of those primary bremsstrahlung photons that were emitted downward. Later, John C. Brown and his colleagues derived analytically the size of the albedo source as a function of height. In a thorough numerical analysis, Taeil Bai & Reuven Ramaty  showed that this albedo component would be polarized and limb-darkened and would measurably change the flare spectrum.
The "reflected" photons form what is called an albedo patch. Nugget #42  showed how the albedo patch "mirrors" the primary source in a distorted way. For a point source at some altitude, the albedo would be extended on a size scale that increases with the source height (see Fig. 1 below). In other words, this reflection is diffuse, and far from specular. Furthermore, the albedo source must be displaced toward disk center by a distance h sin θ, where θ is the heliocentric angle.
Why albedo has not been imaged before
A significant fraction (possibly as high as 40%) of the X-ray flux from solar flares comes from X-rays that propagate back to the solar surface from coronal sources and "reflect" off the photosphere. This component of flares is remarkably difficult to observe simply because it is diffuse - this means that its intensity is correspondingly smaller than the primary flare source. The albedo distorts the X-ray spectrum, but it offers potentially important clues to the way electrons are accelerated in solar flares. Our study uses the unique capabilities of the Reuven Ramaty High Energy Spectroscopic Imager (RHESSI) to isolate this albedo component, and to determine its properties such as size, shape and centroid location as a function of energy. We have focused on simple flares in the 12-30 keV range that appear within 45° of disk center. Using standard techniques, we have obtained the X-ray visibilities  (RHESSI Nugget # 39) of a number of such flares and applied forward-fitting methods to determine the parameters of the primary component (position, flux, and size) and the albedo-related parameters (primary source height and albedo flux).
Figure 1 shows how broad and relatively faint an albedo patch is compared to a compact primary source. The primary source (red disk) is taken to be a 2-D Gaussian with width 5 arcsec, at a height of 18 Mm. The albedo patch is then computed under the assumption of isotropic scattering, a la Brown et al., in the photosphere. Note that, while the peak brightness of the albedo patch is less than 1% of the primary source, too faint for direct imaging methods, most of the albedo flux would still be modulated by RHESSI's coarsest grids.
The primary source is taken to lie at 35° longitude and the resultant albedo patch is assumed to contain 40% of the total flux. The primary intensity is shown by solid contours (black) from 100% down to 1.5% of maximum. The albedo patch (yellow-orange) has intensities below 1% of the maximum. The green contours show the flux boundaries; the orange and red region together contain 80% of the total flux, and the yellow regions contain the remainder.
Sensitivity to source size
The modulation of RHESSI's count rates depends exquisitely on the grid pitch and the source size. This behavior is a fundamental property of Fourier imaging. For a given source size, the modulation amplitude increases as the subcollimator angular resolution broadens. This is illustrated below for three subcollimators with angular resolutions of 23, 69, and 217 arc seconds.
Comparing this figure with Figure 1, one sees that the 1% intensity contour (where albedo is the dominant contributor) is about 20 arcsec wide, so subcollimator 4 (and other finer ones) would show insignificant modulation, while the coarser subcollimators (6 and above) would show strong modulation.
Previous Non-imaging Albedo Studies
- Statistical center-to-limb variations Jana Kasparova, Eduard Kontar & John Brown  demonstrated a center-to-limb variation of photon spectral indices in the 15-20 keV energy range and a weaker dependency in the 20-50 keV range, which is consistent with photospheric albedo as the cause. Nugget #74  illustrates albedo's anisotropy effect on the spectrum.
- Spectroscopy of individual flares by Eduard Kontar and John Brown  analyzed the 2002/08/20 and 2005/01/17 flares in terms of double-power-law fits. To fit the HXR spectrum with a low-energy cutoff Ec and ignoring albedo requires an unusually high value of Ec < ~ 30 ± 2 keV. This produces a clear gap in the range E = 15 to 30 keV, which is likely to be unphysical and suggests that albedo is important.
- Fourier methods The above statistical and spectral methods give no information about the spatial characteristics of albedo patches. Just as the exoplanet observers, forced to use indirect methods to infer the existence of planets, would like to have actual planetary images, we would like to get actual images of albedo patches. The only hope for getting such spatial information is by using the Fourier amplitudes and phases determined by RHESSI. In 2002, the authors  made a first step towards this by assuming circular symmetry. It is now possible to go beyond this, at least for some flares.
Full Exploitation of Fourier methods
We have found several flares with reliable enough amplitudes and phases to compare models of simple sources with albedo patches. The flares all lie within 45° of sun center, where albedo is expected to be strong. These flares form a small subset of RHESSI events for which MEM maps show only single, compact primary components in the range 12 to 30 keV.
Given the RHESSI amplitudes and phases for a flare, one may compare the observed values with those computed from a model. This process is called "Forward Fitting". At the present time, this method is reliable for computing albedo patch parameters only for single elliptical primary sources because the number of parameters (8 in this case) must be much smaller than the number of independent amplitudes. (For a discussion of Forward Fitting of amplitudes and phases, see RHESSI Nugget #35  )
Amplitude models of two primary sources, both with and without albedo
Modeling the albedo patch in addition to the primary source makes it possible to infer the height of the primary source, and the fraction of the total flux emitted by the reflected photons. Fig. 3 (beware! This is highly technical!) shows two examples of flares where the primary source amplitudes (blue crosses) are fit by a 6-parameter model (flux, position, ellipse FWHMs and orientation) with an albedo patch fit by 2 parameters (primary height and albedo fraction). In both cases, for comparison we show a fit made for a primary source without albedo.
The figure shows the fitting of primary source amplitudes (blue crosses) by a model of a non-back-scattered elliptical source (dashed red curve), and a model including both the primary and its associated albedo patch (solid red curve).
Each vertical colored band is for one of the subcollimators (SC) 3-9.
Wiggles in the red curve at the top of each color bar are caused by rotational modulation of
the elliptical primary source by the grids as the spacecraft spins.
The abscissa is a combination of subcollimator number (SC) and grid position angle (PA). The integer part of the x coordinate values is SC and the remainder is PA/180°. (Thus if x is, say, 3.4, SC=3 and PA = 0.4*180° = 72°.) The coarser subcollimators are to the right (SC=6-9), and these measure the flux coming from larger spatial scales.
The excess amplitudes of the solid red curve over the dashed red curve shown for subcollimators 6-9 represent the flux from the larger scales of the albedo patch. The black dashed curve shows the difference between the fitted amplitude profiles with and without albedo. Note how it drops into the noise for the finer grids 3 to 5.
Model Albedo Visibility Back-Projection
In those cases where the primary source and albedo patch are both well represented by a model, it is possible to display both sources using back projection. We have done this for a number of flares, with two example maps here for the flares and bands of Fig. 3.
The black contours show the primary source mapped from the model amplitudes and phases obtained by Forward Fitting, and the colors show a back-projection map made from the best-fit albedo model. The yellow arrow points from the primary to Sun center. The albedo patch centroid lies on this line because in three dimensions it sits vertically below the primary. In each case, the model albedo patch was back-projected from a Fourier plane with uniformly spaced roll bins.
Our Initial Conclusions
Using Fourier amplitudes and phases for nine simple (single primary source, slowly varying) flares in three energy bands we have found evidence for X-rays back- scattered from the photosphere (the albedo patch). Note that these results are only for the so-called "thermal phase" when a single component dominates the emission.
We have visualized the albedo patch by back-projecting our forward-fit model of the best-fit albedo parameters. If the back-scattering process is isotropic (as we assume following J.C. Brown's model), the albedo patch will be displaced toward sun center from the projected location of the primary source, such that it is vertically below the primary source.
We make several inferences from our preliminary study
- The model fits of the amplitudes and phases to the observations are significantly better when back-scattered (albedo) emission is included than if it is not. (Compare the dashed and solid red curves in Figure 3.)
- The albedo fraction tends to increase with energy in the range 12-30 keV, in qualitative agreement with theory (Bai and Ramaty 1978).
- We have inferred primary heights ranging from about 10 to 30 Mm, in agreement with the range seen in limb flare observations.
- In any given flare, the heights of the primary source determined by forward fitting do not significantly change with energy, consistent with a thermal interpretation, and also consistent with the single-component nature of our flares.
- Relative detector-to-detector responses affect these results, and improved calibration would improve our albedo measurements significantly.
- Extension of these methods to some two-component ("footpoint") flares may be possible, and this would have great significance for spectral work.
The RHESSI software team has given invaluable help in making visibility software available to the community. Without their continued support, this research would have been impossible.
Biographical notes: Ed Schmahl is a retired University of Maryland and GSFC scientist, currently employed at NWRA/CoRA, and Gordon Hurford is a senior RHESSI team member based at UC Berkeley.