projct: project 3D ellipsoidal shells onto 2D elliptical annuli
This model performs a 3D to 2D projection of prolate ellipsoidal
shells onto elliptical annuli. The annuli can have varying
ellipticities and position angles but must have the same center. The
user should extract spectra in a series of annuli. Each spectrum needs
three additional keywords (as XFLTnnnn) in the spectrum
extension. These should be “major: x”, “minor: y”, “orient: z”
where x, y, z are the semimajor axis, semiminor axis, and position
angle (in degrees), respectively, for the outer boundary of the
annulus. (If an oblate model is required then switch the values of
major and minor and add ninety degrees to orient.) It is assumed that
the inner boundary is specified by the outer boundary of the previous
annulus. The lengths can be in any consistent units although for
numerical accuracy they should have reasonable values. Optional pairs
of extra XFLTnnnn keywords can be used to specify start and end angles
for a partial annulus. These have keys angleNl and angleNu so typical
keyword values might be “angle1l: 30.0” and “angle1u: 45.0”. These
angles should be given relative to the same zero as the position
angle.
The user reads in the spectra as separate datagroups and sets model parameters
for each datagroup. The model for datagroup J will be the model in the shell
whose outer boundary is a prolate ellipsoid of semimajor and semiminor axes
given by the semimajor and semiminor axes in the XFLT keywords for
dataset J. The projct model sums up the appropriate fractions of
each ellipsoid model to make the projected spectrum.
For example, suppose we extract spectrum from three elliptical regions defined
by (1,0.5,0), (2,1,0), (3,1.5,0). That is the first region is in an ellipse of
semimajor axis 1 and semiminor axis 0.5. The second region is an elliptical
annulus whose inner boundary has semimajor axis 1 and semiminor axes 0.5 and
whose outer boundary has semimajor axis 2 and semiminor axis 1. The third
region is defined similarly. The model fit has a temperature of 2 keV for the
first datagroup, 3 keV for the second, and 4 keV for the third. The actual
model fit to the first dataset has contributions from all three temperatures,
the second only from the 3 and 4 keV components, and the third only from the
4 keV component. The weighting is the fraction of the ellipsoidal volume
intersected by the elliptical annular crosssection. Thus the normalizations
correspond to the emission measure in each ellipsoidal shell.
If multiple observations are to be analyzed, data sets from different
observations corresponding to the same annulus should be part of the same
data group. For example, given the following 4 data files:
Data sets for obs 1: obs1_an1, obs1_an2
Data sets for obs 2: obs2_an1, obs2_an2
The proper data loading command is:
XSPEC12>data 1:1 obs1_an1 1:2 obs2_an1 2:3 obs1_an2 2:4 obs2_an2
The projct model has 3 (fixed) parameters, which can be used to define
the inner ellipse of the region being analyzed. For instance, in the example
above we could have only read in spectra for the outer two regions but then
set the projct parameters to (1.0,0.5,0.0). This would have allowed
us to determine the temperatures and emission measures of the outer two
annuli without having to worry about fitting a model to the central region.
par1 
semimajor axis of inner boundary ellipse 
par2 
semiminor axis of inner boundary ellipse 
par3 
position angle of inner boundary ellipse 
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Last modified: Friday, 23Aug2024 13:20:40 EDT
