Solar Hard X-ray Albedo

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{{Infobox Nugget
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|name = Nugget
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|title = Nugget Details
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|number = 119
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|first_author = Ed Schmahl
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|second_author = Gordon Hurford
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|publish_date = 18  January 2010
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|next_nugget = [[Two phases of X-ray emission in a solar eruptive flare]]
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|previous_nugget = [[Cycle 24 has begun]]
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}}
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== Solar Hard X-ray Albedo ==
== Solar Hard X-ray Albedo ==
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In the early days (1972) of solar hard X-ray flare observations,  
+
In the early days (1972) of solar hard X-ray [http://en.wikipedia.org/wiki/Solar_flare flare] observations,  
Fred Tomblin [http://adsabs.harvard.edu/abs/1972ApJ...171..377T] published theoretical arguments  
Fred Tomblin [http://adsabs.harvard.edu/abs/1972ApJ...171..377T] published theoretical arguments  
that the hard X-ray spectrum of solar flares in the 5-40 keV  
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  
+
range must have an albedo component  due to [http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/comptint.html Compton] back-scattering (from the photosphere) of those primary [http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/xrayc.html#c2 bremsstrahlung] photons that were emitted downward. Later, John C. Brown and his colleagues[http://adsabs.harvard.edu/abs/1975A%26A....41..395B] derived analytically the size of the albedo source as a function of height.
-
in the photosphere of those primary bremsstrahlung  
+
In a thorough numerical
-
photons that are emitted downward. In a more complete
+
analysis,  Taeil Bai & Reuven Ramaty [http://adsabs.harvard.edu/abs/1978ApJ...219..705B]
analysis,  Taeil Bai & Reuven Ramaty [http://adsabs.harvard.edu/abs/1978ApJ...219..705B]
-
showed that this albedo component would be polarized and its size must depend on the height of the primary source.
+
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.  
+
The "reflected" photons form what is called an ''albedo patch''.  
-
For sufficiently high primary source altitudes, the albedo would  
+
Nugget #42 [http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=42]
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be much larger in extent than the primary source, with a size  
+
showed how the albedo patch "mirrors" the primary source in a distorted way.
-
scale that increases with source height. (See Fig. 1 below.)  
+
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).
-
Furthermore, the albedo source would be displaced toward  
+
In other words, this reflection is diffuse, and far from [http://en.wikipedia.org/wiki/Specular_reflection ''specular''].
-
disk center by a distance h sin θ,  
+
Furthermore, the albedo source must be displaced toward disk center by a distance h sin θ,  
where θ is the heliocentric angle.  
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
 +
[http://sprg.ssl.berkeley.edu/~tohban/wiki/index.php/RHESSI_Visibilities] (RHESSI Nugget # 39) of a number of such flares and applied [http://hesperia.gsfc.nasa.gov/rhessidatacenter/imaging/forward_fitting.html 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). 
-
<h3> Why albedo has not been imaged before </h3>
+
[[Image:albedo_patch_fig1.png|center|thumb|600px|Fig. 1. Model of a primary source (red) and its albedo patch (orange-yellow).]]
-
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 called the <i>albedo</i>, and it is remarkably difficult to observe because it is very diffuse with an intensity that is one or two
+
-
orders of magnitude smaller than the primary flare sources themselves.
+
-
Its importance for flare physics is that it both distorts the spectral interpretation of X-ray emission and offers a potentially powerful diagnostic of electrons accelerated in solar flares. Our study uses the unique capabilities of the Ramaty High Energy Spectroscopic Imager (RHESSI) to isolate this albedo component, determine its properties such as size, shape and centroid location as a function of energy. We have focused on single-component flares in the 12-30 keV range that appear within 45&deg; of disk center. Using standard techniques, we have obtained the X-ray visibilities
+
-
[http://sprg.ssl.berkeley.edu/~tohban/wiki/index.php/RHESSI_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 [http://en.wikipedia.org/wiki/Gaussian_function 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 [http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=8 RHESSI's coarsest grids].
-
[[Image:albedo_patch_fig1.png|right|thumb|400px|Fig. 1]]
+
The primary source is taken to lie at 35&deg;  longitude and the resultant albedo patch is assumed to contain 40% of the total flux.
-
Fig. 1.
+
The primary intensity is shown by solid contours (black) from  
-
Model of a circular primary source at 35&deg;  longitude and the  
+
-
resultant albedo patch, which contains 40% of the total flux.
+
-
The primary source (red disk) is taken to be a 2-D Gaussian, FWHM=5 arcsec,
+
-
at a height of 18 Mm.  Its intensity is shown by solid contours (black) from  
+
100% down to 1.5% of maximum. The albedo patch (yellow-orange) has  
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
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,
boundaries; the orange and red region together contain 80% of the total flux,
-
and the yellow regions contain the remainder. 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
+
and the yellow regions contain the remainder.  
-
be modulated by RHESSI's coarsest grids.
+
-
<h3>Sensitivity to source size</h3>
+
=== Sensitivity to source size ===
-
 
+
The modulation of RHESSI's count rates depends exquisitely on the [http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=8 grid pitch] and the source size.
-
The modulation of RHESSI's count rates depends exquisitely on the grid pitch and the source size, a fundamental property of Fourier imaging.  
+
This behavior is a fundamental property of Fourier imaging.  
-
When the source FWHM is less than the subcollimator angular
+
For a given source size, the modulation amplitude increases as the subcollimator angular resolution broadens.
-
resolution, the modulation amplitude falls rapidly. This is illustrated below
+
This is illustrated below
for three subcollimators with angular resolutions of 23, 69, and 217 arc seconds.  
for three subcollimators with angular resolutions of 23, 69, and 217 arc seconds.  
-
Put Fig 2 here.
+
[[Image:SizeSensitivity.jpg|center|thumb|600px|Fig. 2: Modulation by three [http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=8 RHESSI subcollimators].
 +
Given a primary source size of 20 arcsec, as a function of subcollimator the amplitude of the modulation is less than &#126; 7% for subcollimator 4 (and lower for the finer ones), but for the coarser subcollimators (6 and 8 in the above), the relative amplitude remains above 75%. This effect can be used to extract broad sources like albedo from small flare sources.
 +
]]
-
Caption:
+
Comparing this figure with Figure 1, one sees that the 1% intensity contour (where albedo is the
-
Modulation by 3 RHESSI subcollimators. The amplitude of the modulation drops to 7% for subcollimator 4 when the source size exceeds 20 arcsec, but for the coarser subcollimators (6 and 8 in the above), the relative amplitude remains above 75%. This effect can be used to extract broad sources like albedo from small flare sources.
+
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 attempts to infer albedo properties ==
+
=== Previous Non-imaging Albedo Studies ===
-
 
+
* '''Statistical center-to-limb variations''' Jana Kasparova, Eduard Kontar & John Brown [http://arxiv.org/abs/astro-ph/0701871] 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 [http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=74] illustrates albedo's anisotropy effect on the spectrum.
-
<ul>
+
   
-
<li> '''Statistical center-to-limb variations'''
+
* '''Spectroscopy of individual flares''' by Eduard Kontar and John Brown [http://www.iop.org/EJ/article/1538-4357/653/2/L149/21151.html] 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<sub>c</sub> and ignoring albedo  requires  an unusually high value of E<sub>c</sub> &#60; &#126; 30 &#177; 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.
-
  Jana Kasparova, Eduard Kontar & John Brown  
+
-
[http://arxiv.org/abs/astro-ph/0701871]
+
-
  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
+
-
  [http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=74]
+
-
  illustrates albedo's anisotropy effect on the spectrum.
+
-
 
+
-
 
+
-
  <li> '''Spectroscopy of individual flares'''
+
-
  Eduard Kontar and John Brown  
+
-
  [http://www.iop.org/EJ/article/1538-4357/653/2/L149/21151.html]
+
-
  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<sub>c</sub> and ignoring albedo  requires  an unusually high value of E<sub>c</sub> &#60; &#126; 30 &#177; 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.
+
-
  A related nugget (#42)[http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=42]  shows how the albedo "mirrors" the primary flux.
+
-
 
+
-
  <li> '''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
+
-
  [http://adsabs.harvard.edu/abs/2002SoPh..210..273S]  made a first step towards this by assuming circular symmetry. It is now possible
+
-
  to go beyond this, at least for some flares.
+
    
    
-
</ul>
+
* '''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 [http://adsabs.harvard.edu/abs/2002SoPh..210..273S]  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 ==
== Full Exploitation of Fourier methods ==
-
We have found 9 flares with reliable enough amplitudes and phases to compare models of simple sources with albedo patches. The flares all lie within 45&deg; 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.  
+
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&deg; 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.  
-
<h3>Forward Fitting</h3>
+
=== Forward Fitting ===
-
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 amplitudes. (For a discussion of Forward Fitting of amplitudes and phases, see RHESSI Nugget #35
+
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
[http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=35] )
[http://sprg.ssl.berkeley.edu/~tohban/nuggets/?page=article&article_id=35] )
-
<h3>Amplitude model of a primary source both with and without albedo</h3>
+
=== 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. Here we show two examples of flares
+
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.  
-
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.
+
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.
-
Put fig 3 here.
+
[[Image:Fig3Amplitudes.jpg|center|thumb|700px|Fig. 3. Comparison of observed and model amplitudes for two flares. The colors of Fig. 2 are represented here for subcollimators 4, 6 & 8.]]
-
Caption:
+
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).  
-
 
+
-
Comparison of observed and model amplitudes. Primary source amplitudes (blue crosses) are fit 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). <br>
+
Each vertical colored band is for one of the subcollimators (SC) 3-9.
Each vertical colored band is for one of the subcollimators (SC) 3-9.
-
Variations in the red curve at the top of each band are caused by rotation of the grids as the spacecraft spins. <br>
+
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&deg;. (Thus if x is, say, 3.4, SC=3 and PA = 0.4*180&deg; = 72&deg;.) The coarser subcollimators are to the right (SC=6-9), and these measure the flux coming from larger spatial scales.<br>
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&deg;. (Thus if x is, say, 3.4, SC=3 and PA = 0.4*180&deg; = 72&deg;.) The coarser subcollimators are to the right (SC=6-9), and these measure the flux coming from larger spatial scales.<br>
-
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 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.
the noise for the finer grids 3 to 5.
-
<h3> Model Albedo Visibility Back-Projection</h3>
+
=== 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 2 example "bpmaps" here for the flares and bands of Fig. 3.
+
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 [http://hesperia.gsfc.nasa.gov/rhessidatacenter/imaging/back_projection.html back projection].  
 +
We have done this for a number of flares, with two example maps here for the flares and bands of Fig. 3.
-
Put fig 4 here.
+
[[Image:Fig4bpmaps.jpg|center|thumb|700px|Fig. 4. [http://hesperia.gsfc.nasa.gov/rhessidatacenter/imaging/back_projection.html Back-projection] maps (color) of the model albedo
-
 
+
for the two flares of Fig 3.]]
-
Caption:
+
The black contours show the primary source mapped from the model amplitudes
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.
+
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 lies vertically below the primary. In each case, the model albedo patch was back-projected from a Fourier plane with uniformly spaced roll bins.
+
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.
-
<br>
+
== Our Initial Conclusions ==
== Our Initial Conclusions ==
-
Using Fourier amplitudes and phases for nine simple (single primary,  
+
Using Fourier amplitudes and phases for nine simple (single primary source,  
-
slowly varying) flares we have found evidence for X-rays back-
+
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  
scattered from the photosphere (the albedo patch).  Note that these  
results are only for the so-called "thermal phase" when a single
results are only for the so-called "thermal phase" when a single
component dominates the emission.  
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  
+
We have visualized the albedo patch by back-projecting our forward-fit model of the best-fit albedo parameters.   
-
process is isotropic (as we assume in the Forward-fit model), the  
+
If the back-scattering process is isotropic (as we assume following J.C. Brown's model), the  
-
albedo patch is displaced toward sun center from the projected  
+
albedo patch will be displaced toward sun center from the projected location of the primary source, such that it is vertically below the  
-
location of the primary source, such that it is vertically below the  
+
primary source.  
primary source.  
-
<h3> We make several inferences from our results for 9 flares:</h3>
+
=== We make several inferences from our preliminary study ===
-
<ul>
+
* 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).  
-
<li>The model fits of the amplitudes and phases to the observations  
+
* We have inferred primary heights ranging from about 10 to 30 Mm, in agreement with the range seen in limb flare 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.)
+
* 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.
-
<li> The albedo fraction increases with energy in the range 12-30 keV,
+
* Relative detector-to-detector responses affect these results, and improved calibration would improve our albedo measurements significantly.
-
    in qualitative agreement with theory (Bai and Ramaty 1978).  
+
* Extension of these methods to some two-component ("footpoint") flares may be possible, and this would have great significance for spectral work.
-
<li> We have inferred primary heights ranging from about 10 to 30  
+
-
    Mm, in agreement with the range seen in limb flare observations.  
+
-
<li> 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.
+
-
<li> Relative detector-to-detector responses affect these results, and  
+
-
    improved calibration would improve our albedo measurements  
+
-
    significantly.
+
-
<li> Extension of these results to some 2-component ("footpoint")  
+
-
    flares may be possible, and this would have great significance for  
+
-
    spectral work.
+
-
</ul>
+
== Acknowledgements ==
== Acknowledgements ==
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.
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.
+
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.
 +
 
 +
[[Category: Nugget]]

Latest revision as of 01:15, 17 March 2010


Nugget
Number: 119
1st Author: Ed Schmahl
2nd Author: Gordon Hurford
Published: 18 January 2010
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List all



Contents

Solar Hard X-ray Albedo

In the early days (1972) of solar hard X-ray flare observations, Fred Tomblin [1] 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[2] derived analytically the size of the albedo source as a function of height. In a thorough numerical analysis, Taeil Bai & Reuven Ramaty [3] 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 [4] 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 [5] (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).

Fig. 1. Model of a primary source (red) and its albedo patch (orange-yellow).

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.

Fig. 2: Modulation by three RHESSI subcollimators. Given a primary source size of 20 arcsec, as a function of subcollimator the amplitude of the modulation is less than ~ 7% for subcollimator 4 (and lower for the finer ones), but for the coarser subcollimators (6 and 8 in the above), the relative amplitude remains above 75%. This effect can be used to extract broad sources like albedo from small flare sources.

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

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.

Forward Fitting

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 [10] )

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.

Fig. 3. Comparison of observed and model amplitudes for two flares. The colors of Fig. 2 are represented here for subcollimators 4, 6 & 8.

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.

Fig. 4. Back-projection maps (color) of the model albedo for the two flares 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

Acknowledgements

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.

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