Imaging the X-ray Albedo with RHESSI

1.1 Solar Hard X-ray Albedo

The "reflected" photons form what is called an albedo patch. For sufficiently high primary source altitudes, the albedo would be much larger in extent than the primary source, with a size scale that increases with source height. (See Fig. 1 below.) Furthermore, the albedo source would be displaced toward disk center by a distance h sin θ, where θ is the heliocentric angle.

1.2 Why albedo has not been imaged before

Figure 1. Model of a circular primary source at 35° longitude and the resultant albedo patch. Solid contours (black) show the intensity of a primary source and its albedo where the albedo patch contains 40% of the total flux. The dashed contours (red) show the boundaries that contain 20, 30 and 36% of the total flux. The primary source is taken to be a 2-D Gaussian, FWHM=5 arcsec, at a height of 18 Mm. Note that the peak brightness of the albedo patch is less than 1% of the primary source, too faint for direct imaging methods, but nevertheless, most of the albedo flux would be modulated by RHESSI's coarsest grids.

2. Previous attempts to infer albedo properties

• 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.

• Spectroscopy of individual flares
  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 E_c and ignoring albedo requires an unusually high value of E_c< ~ 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. 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.

3. Full Exploitation of Fourier methods

3.1 Forward Fitting

3.2 Amplitude model of a primary source both with and without albedo

Figure 2. Comparison of observed and model amplitudes. Primary source amplitudes (blue crosses) are fit by a model of a non-backscattered elliptical source (dashed red curve), and a model including both the primary and its associated albedo patch (solid red curve). 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 < 6.

3.3 Model Albedo Visibility Back-Projection

Figure 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 using completely sampled roll bins. The arrow points from the primary to Sun center. The albedo patch centroid lies on this line because in three dimensions it lies vertically below the primary.

4. Three-Band Amplitude Fits For the Flare of 2002/04/10

e

Figure 4. Amplitudes vs subcollimator (SC) and grid position angle (PA) for flare models including back-scattered emission (solid) and without (dashed). The axes in these figures are the same as for Fig. 2. The black dashed curve is the difference between the models with and without albedo. Much of the errors in fitting are due to inter-detector calibration errors. Although these can be compensated for, they introduce uncertainties in the inferred albedo fraction and the source height.

Figure 5. Back-projection maps for three energy bands for the flare of 2002/04/10 (same time interval as Fig. 4.) As in Fig. 3, the arrows point from map center to Sun center. In each case, the model albedo patch was back-projected from a Fourier plane with uniformly spaced roll bins.

5. Conclusions

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 in the Forward-fit model), the albedo patch is 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 results for 9 flares:

• The model fits of the amplitudes and phases to the observations are significantly better when back-scattered emission is included than if it is not. (Compare the dashed and solid red curves in Fig 4.)
• The albedo fraction increases 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 results to some 2-component ("footpoint") flares may be possible, and this would have great significance for spectral work.

6. Acknowledgements

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, respectively.