User's Guide to RHESSI Visibilities

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(Basic Properties of RHESSI Visibilities)
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The spatial frequency at which the visibility is measured is characterized either by its Cartesian coordinates (traditionally  
The spatial frequency at which the visibility is measured is characterized either by its Cartesian coordinates (traditionally  
-
labeled u, v) or by the polar equivalent as the magnitude of the spatial frequency and its orientation.  For RHESSI the  
+
labeled u, v) or by the polar equivalent as the magnitude of the spatial frequency and its orientation.  For RHESSI, the  
magnitude of the spatial frequency is determined by the subcollimator number (and harmonic) and is given by 1/(angular  
magnitude of the spatial frequency is determined by the subcollimator number (and harmonic) and is given by 1/(angular  
pitch) of the subcollimator.  For the first harmonic, this is equivalent to 0.5/(FWHM resolution)  (about 0.22 arcsec<sup>-1</sup> for  
pitch) of the subcollimator.  For the first harmonic, this is equivalent to 0.5/(FWHM resolution)  (about 0.22 arcsec<sup>-1</sup> for  
-
grid 1 of 0.0027 arcsec<sup>-1</sup> for grid 9).  Note that the courser grids have the smaller spatial frequency.   
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grid 1 to 0.0027 arcsec<sup>-1</sup> for grid 9).  Note that the courser grids have the smaller spatial frequency.   
{| border="1"
{| border="1"
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|9
|9
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|-
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|Resolution (FWHM)
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|Resolution (FWHM) [arcsecs]
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|2.26
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|3.92
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|6.79
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|11.76
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|20.36
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|35.27
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|61.08
 +
|105.8
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|183.2
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|-
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|Spatial frequency (first harmonic) [arcsec<sup>-1</sup>]
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|.22124
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|.12755
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|.07364
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|.04252
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|.02456
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|.01418
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|.00818
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|.00472
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|.00273
|}
|}

Revision as of 23:16, 17 September 2009

This write-up summarizes the properties of RHESSI visibilities and is based on a document written by Gordon Hurford.

Contents

Basic Properties of RHESSI Visibilities

A RHESSI visibility is a complex observable number that can be derived from RHESSI data and which represents a calibrated measurement of a single Fourier component of the source distribution measured at a specific spatial frequency and energy- and time-range.

As a complex number, it can be represented either by its amplitude and phase, or by the corresponding Cartesian representations, variously called the X and Y, cosine and sine, or real and imaginary components. In the context of visibilities, it traditionally labeled as u and v.

z = X + iY = r(\cos \varphi + i\sin \varphi ) = re^{i\varphi}

As a calibrated measurement, it incorporates all known spatial calibration information so that no further reference to aspect solution, grid response, etc is needed to interpret it. Note that the energy range of a freshly-measured visibility refers to the detected energy, not photon energy. As a ‘semicalibrated’ value, it therefore incorporates the diagonal elements of the detector response, but not the off-diagonal components. With that caveat, a measured visibility has units of photons cm2 second-1.

Except for contributing to statistical errors, background is inherently removed in the process of calculating visibilities. Background (in this context) refers to any photons that have not passed through both grids (i.e. non-imaging background).

The spatial frequency at which the visibility is measured is characterized either by its Cartesian coordinates (traditionally labeled u, v) or by the polar equivalent as the magnitude of the spatial frequency and its orientation. For RHESSI, the magnitude of the spatial frequency is determined by the subcollimator number (and harmonic) and is given by 1/(angular pitch) of the subcollimator. For the first harmonic, this is equivalent to 0.5/(FWHM resolution) (about 0.22 arcsec-1 for grid 1 to 0.0027 arcsec-1 for grid 9). Note that the courser grids have the smaller spatial frequency.

Grid 1 2 3 4 5 6 7 8 9
Resolution (FWHM) [arcsecs] 2.26 3.92 6.79 11.76 20.36 35.27 61.08 105.8 183.2
Spatial frequency (first harmonic) [arcsec-1] .22124 .12755 .07364 .04252 .02456 .01418 .00818 .00472 .00273


By convention, the orientation of a RHESSI visibility is defined by its position angle in the u,v plane, measured east from solar north. As RHESSI rotates, the magnitude of the spatial frequency measured by a given subcollimator remains fixed, while the position angle of the spatial frequency decreases as the measured visibility moves in a circle in the u,v plane.

Except for the units, RHESSI visibilities are equivalent to those measured by radio interferometers, a feature which opens the way to using full-featured image reconstruction software developed for radio applications.

Since the x-ray source is real (both practically and mathematically), visibilities measured at opposite points in the u,v plane are complex conjugates. This implies that the amplitudes of visibilities measured at opposite half rotations must be equal, and the phases of visibilities measured at opposite half rotations must sum to zero. To within statistical expectations, this condition should hold independent of source morphology. If this is not the case, then either the source time variability is relevant, or there is a calibration or calculation error. It also follows that such ‘conjugate visibilities’ can (and should) be combined without losing any imaging information.

As observables, visibilities are inherently linear both in terms of their dependence on the data and in terms of their dependence on the source. For example, the (complex) sum of two visibilities (at the same spatial frequency) measured over two separate time or energy intervals should give the same result as a measurement over the combined time or energy interval. Similarly, the visibilities of a multicomponent source should equal the sum of the corresponding visibilities of the individual components.

Some possible application of RHESSI visibilities

Software Overview

The calculation of visibilities is done by defining an image object, setting its parameters, calculating the calibrated event list, converting the calibrated event list to a stacked event list and then converting the stacked event list to a set of visibilities. These steps are transparent if the user makes the obvious inputs or edits to hsi_vis_usershell.

Visibility Cook book

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