> From belinda@head-cfa.harvard.edu Sat Aug 11 21:56:16 2001
> Date: Sat, 11 Aug 2001 21:54:24 -0400 (EDT)
> From: Roberts <roberts@hep.physics.mcgill.ca>
> X-Sender: roberts@jadzia.physics.mcgill.ca
> To: "Keith A. Arnaud" <kaa@genji.gsfc.nasa.gov>
> cc: chandra-users@head-cfa.harvard.edu
> Subject: Re: Effect of modest pile-up on spectra
> MIME-Version: 1.0
>
> Hi again,
> I downloaded the new xspec version, and have been trying
> the pileup model. I'm wondering if a fixed frame time of 3.2sec is
> correct, given that the spacecraft is dithering. Wouldn't the effective
> frame time be less (although probably somewhat complicated)? If so,
> any guesses as to what it might be?
> Thanks again,
> Mallory
>
Mallory, I found this to be an interesting problem that has, to
my knowledge, not been discussed within the CXC. Here is my 1st
level cut at examining how dithering affects the time used to estimate
the effects of pileup. I agree that the frame time is not the only
time scale that is relevant. For those who do not wish to review the
details, I estimate that dithering could reduce the effective frame
time by up to 30%.
--- Details ---
First off, it's clear that the dither rate *does* matter. Consider
an experiment where the frame time is very long. Then the dither
rate sets the length of time during which any region of the detector
will receive counts. Define a "dither time scale", td, by the time
it takes to dither across a telescope beam. For simplicity, I consider
the pixels to be "small" relative to the beam; obviously, the pixel
size matters as well because if there was just one huge pixel, then
the pileup timescale would just be the total length of the observation.
The dither crossing time depends on the angular dither rate, omega,
which changes throughout the dither pattern. This is an important
effect. At the corners of the lissajous pattern, the instantaneous
angular velocity is near zero. Near the center, however, the beam
is moving fastest across the detector. I simulated a simple example
because the pattern doesn't repeat precisely. For a 100 ks observation
with phases set so that the source starts in the center, the average
angular velocity varies by a factor of over 20 but stays within a
range between 40 and 100% of maximum for most of the pattern. Taking
these as the limits, I found that the average crossing time for a beam
size varies from 6.8 to 17 s with an average of 10 s. I assumed a
beam that has a Gaussian 1D profile with sigma = 0.32" and that the
average path through the beam is about 80% of the full beam width
(allowing for pixels that do not go through the middle of the beam).
Next, I assumed that there might be a simple form for the effective
pileup time, tp, which is related to the frame time, tf, and the
dither time, td. The asymptotic behavior is that when either of
the frame or dither times is much smaller than the other one, then
the pileup time should be the about equal to the smaller time. A
simple, symmetric formula that has this behavior is
1/tp = 1/td + 1/tf
Rearranging gives tp = tf /(1 + tf/td), so the term in parentheses
is a "correction" term that gives an approximate effect of dither on
the estimated pileup. I computed tp for a few typical frame times:
tf tp tp_range
3.2 2.43 0.26
2.5 2.00 0.18
1.7 1.46 0.093
1.1 0.99 0.043
0.003 0.003 3.6e-7
The last time is appropriate for continuous clocking, where the charge
is continuously shifted at 2.85 ms intervals. The quantity tp_range
gives the excursion of the computed pileup time. So, for a 2.5 s
frame time, I estimate that the appropriate pileup time is usually in
the range 1.82 - 2.18s.
I found some approximate forms that can be handy for other frame times:
tp = tf * exp(-0.088*tframe), tp_range = 0.036 * tf^(1.7).
In summary, it appears to me that dithering could change the effective
frame time by up to 30% and that it would broaden the distribution of
pileup times. A simulation approach would require dithering to create
an accurate representation of the effects of pileup.
Herman Marshall
CXC Calibration Scientist
PS. The effects of pileup on an ACIS spectrum are well documented in
the Chandra Proposer's guide. See figures 6.22-6.25.
This archive was generated by hypermail 2b29 : Wed May 15 2013 - 01:00:10 EDT