Point-Source Equivalents
To speed up and de-pixelize the Forward Fitting program (M. Aschwanden, 1999),
Gordon suggested we find the
"Point-Source Equivalents" for elementary extended sources.
Rationale
- Meshes well with existing simulation package, which accepts point sources as input
- Meshes well with visibility software
- Speeds up and generalizes existing Forward-Fitting code
- Permits elimination of pixelization in existing code
Exqmples of Elementary Extended Sources
- Gaussians
- Elliptical (flux, x_pos, y_pos, FWHM_1, FWHM_2, angle_orient)
- Orthogonal (flux, x_pos, y_pos, FWHM_1, FWHM_2)
- Circular (flux, x_pos, y_pos, FWHM)
- Segments
- Simple (flux, x_pos, y_pos, semi_len, angle_orient)
- Convolved (flux, x_pos, y_pos, semi_len, angle_orient, FWHM_1, FWHM_2)
- Arcs
- Simple (flux, x_pos, y_pos, arc_radius, arc_halfangle, angle_orient)
- Convolved (flux, x_pos, y_pos, arc_radius, arc_halfangle, angle_orient, FWHM_1, FWHM_2)
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 do-able analytically, and
the variables thereby 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
- Incomplete arcs
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 polyagonal 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