The Mechanism
The mechanism behind this scheme is that for an arbitrary source, the visibility
(via the Fourier transform) gives us:
- The amplitude and phase
- the location of a point source with the same phase,
- the flux of the equivalent point source with the same amplitude.
For all but the arcs, the Fourier transform is doable analytically, and
the variables produced go straight into Forward Fitting of the
count rate.
This method works most simply
for Gaussian sources and segments, for which
- the equivalent source location is the centroid of the source object itself;
- The flux of the equivalent source is independent of roll angle for:
- circular Gaussian,
- 360-degree arc,
- any circularly-symmetric source;
- The flux of the equivalent source is roll dependent for:
- orthogonal Gaussians
- elliptical Gaussians
- segments
- arcs
- almost anything else
For arcs and other such asymmetric shapes,
- the location of the equivalent point source is a function of roll angle;
- but this is not a complication, because so is the flux.
- The trick is to substitute a polygonal arc, and add up the contribution from segments;
- so the output of all the equivalent-source routines contains (at least) these vectors:
- equivalent_flux
- equivalent source_location
- roll angle
Convolved Sources
- Any Equivalent Point Source (EPS) may be convolved with a Gaussian, producing another
Equivalent Point Source .
- This is found very simply by a multiplication, angle bin by angle bin,
of the equivalent flux for the input source and the Fourier amplitude of the convolving
Gausssian.
- Any EPS can be convolved with a radially-symmetric source this way
(e.g. an arc with a segment)
- An EPS can also be convolved with a non-radially-symmetric source (such as an arc), but
- the phases must be included
- why bother?
Ed Schmahl
Last modified: Tue Mar 21 09:20:03 EST 2000