Two-level (aka "Fluffy") Contamination Model

FAQs - last edited 11/19/03 - dd
What the heck is this all about?

The ACIS detector on the Chadra Observatory has a slowly growing layer of contamination on it. The contamination absorbs X-rays, especially lower energy ones. The absorption of X-rays at around 700 eV has been determined using an on-borad "External Calibration Source" or ECS. The absorption can also be seen and measured by looking at the size of absorption edges in the spectrum of bright continuum sources - a model with a calibration file has been constructed from these edge measurements. This "contamarf" edge-based model predicts an absorption at 700 eV that is not quite as much absorption as the ECS indicates - an additional factor of about 0.9 is needed in the contamarf model at 700 eV at time 2003.0 . How can this factor be included in a physical way without changing the size of the observed edges in the model?

Note the difference between these measurement methods: the ECS measures the absolute change in the QE of the system at 700 eV, whereas the edge depths are a relative measure or ratio of the QE above and below the edge, in particular the Carbon edge.

This work shows that if the contamination layer is non-uniform in thickness or density by about a factor of two or three then it is possible to create a physical model that has the same edge structure (depths) and yet an additional factor of 0.9 absorption at 700 eV. The model then predicts how this additional absorption factor will behave at other energies (the green curve in the figure below.)

Original contamarf model (black) and the modified "two-level" model (red).
Their ratio is shown in green. The two-level model adds more
absorption at ~700 eV (~18A) while retaining the same edge-jump
levels (shown by the smoothness of their ratio, the green curve.)

Is there an easy way to understand the basic idea?

Yes, maybe...

We'd like to add thickness to the edge-based model in order to match the transmision at 700 eV, but adding this thickness will make the Carbon edge (ratio) larger. If, however, we allow thin and thick regions in the contaminant then we have another two "knob"s to turn: an additional thickness and the filling-factor of the thick material. The depth of the carbon edge depends very much on the thickness of the "thin" component where as the absorption at 700 eV depends more on the total amount of material in thin and thick regions. In this way it is possible to match the transmission at 700 eV while retaining the same Carbon edge depth.

How could we physically have different thicknesses of material?

Well, the contaminant could vary in thickness with position as, for examples, drops on a surface or ripples on sand, giving actual thickness differences with position...

Another realization would be to have varying density of the contaminant, e.g., a fluffy contaminant perhaps like frost in a freezer or the web of warm-hot matter in the universe(!) The result is that X-rays traveling through it see differences in column density by ~factor of two or more from location to location.

Has a similar model ever been used in the past?

"There are no new ideas [including this one.]" - Keith Arnaud.

Well, the "ice" model on the Einstein Solid State Spectrometer, SSS, was modelled as "made up of two thicknesses of granules or snow flakes" in a partial covering model like this one! They even got more complex with a four parameter two-kinds-of-clumps model, see:
"The Einstein Solid State Spectrometer Calibration"
by Christian, Swank, Szymkowiak, and White for some details. Of course we have mostly Carbon building up so it may not be as flaky as H2O building up... or more so?

Does this two-level model reconcile the ECS curvature with the linear edge-depth growth?

[ The tau (thickness) of contaminant at 700 eV measured with the ECS is well fit to a step exponential-decay curve whereas the Carbon edge thickness has been fit with a straight line indicating constant thickness growth. ]

Answers: No. Perhaps. We'll see.

No: The two-level model itself does not introduce curvature in the 700 eV tau value with time if the two-level thicknesses themselves grow linearly with time, so curvature is not automatically added due to the model per se.

Comparing tau at 700 eV due to the contamarf model(green)
with the ECS best-fit curve (blue). The difference between
these two curves is what the two-level model is meant to fix (see next plot.)

Perhaps: Because the tau of the two measurements differed in value by a significant amount there was no impetous or ability to jointly fit them and see if a common thickness-vs-time curve would agree with both data sets. However, now that they can be made to agree in value it is worth trying to jointly fit them and see if a common thickness vs time is allowed by the data.

Comparing tau at 700 eV due to the Two-level model(green)
with the ECS best-fit curve (blue). Note that the two-level model,
a modification of the contamarf model, has not added any curvature
to the green curve so this difference remains. However the values
are very close and a common thickness-growth curve may be made to
agree with both data sets. A new Carbon-edge measurement at ~2004.0
may be a useful additional data point to settle this issue.

We'll see: As the plot above shows, the straight-line and the curved ECS model agree at 2003.0 but should deviate significantly by 2004.0. So the next PKS data set should add a point to the Carbon edge growth which may show the Carbon edge thickness rolling off with a similar curvature as the ECS... or not!

Can we measure spatial variations in the density of the contaminant?


Since the ACIS has a pixel size of 24 um this sets a lower practical limit for spatial scales we can measure on the OBF. Of course it is worse than this because both the HRMA beams and ECS "beams" coming to a pixel actually sample a larger size on the OBF which is ~12 mm above the CCD surface. The "sampling foot-print" is described below for these two measurement cases.

- - -

The HRMA has an f-ratio of about f/8.3 for the outer shell 1. Thus the beam diameter intercepting the OBF is ~ 12.0/8.3 or 1.44 mm. Simulations for 700 eV X-rays and grating=NONE are presented here to show the specific distribution of events on the OBF without and with dither on. (Note that other energies and/or the use of the HETG/LETG may change the relative intensity of the different shell contributions.)

The figure below left shows a MARX simulation of photons being detected when the focal plane is moved 12 mm towards the HRMA, i.e., this is the "foot print" of rays going through the OBF from a point source instantaneously and/or with no dither. (Specifically, the plot here is of the Sky X and Y coordinates which have aspect motion removed.) The plot frame has a size of 2mm x 2mm here (+/-42 pixels on each axis) and the four HRMA shells are clearly visible.

The figure below right is the same simulation except that TDETX and TDETY coordinates are plotted (with one pixel randomization) and thus show the effects of dither motion in moving around the HRMA shell pattern on the OBF. A square of 2mm size is plotted as well for comparison with the frame of the figure at left. The 16 arc second peak-to-peak dither corresponds to a physical distance of almost 0.8 mm which causes a substantial spreading of the HRMA pattern on the OBF.

- - -

When looking at the ECS the amount of the OBF sampled depends on the sizes (1mm, 8mm) and distance (~500 mm) of the ECS sources, values provided by Mark Bautz. The result is that X-rays arriving at a specific point on the on the CCD array have sampled a circular region on the OBF of diameter 0.024 mm for the Mn/Fe source and 0.19 mm for the Al and Ti sources. For the Fe ~700 eV measurements 2x2 pixel cells would see largely independant OBF regions and so variations on scales of ~50 um and larger can be measured (subject to enough counts!)

Note that when looking at the ECS, if the ACIS is translated in SIM-Z by a distance of ~40 mm then the "shadow" of the OBF contamination pattern will be shifted on the devices by ~ 1 mm. By making ECS measurements with ACIS offset it may be possible to see the OBF variations shift with respect to the CCD coordinates and separate OBF variations (if visible) from CCD pixel-pixel/column-column variations...

Well, what next?

Another PKS 2155-304 observation with LETG-ACIS is set for Nov.22'03 and should add a useful point to the growth of the Carbon edge.

The specific models presented here are useful to demonstrate that this is one way to reconcile the two measurement methods, but a complete, self-consistant, final calibration has not been created - this is left to the meticulous CXC and ACIS cal folks to accomplish.

10/10/03 - ISIS version available below!

Two-level Contamination Model

Ratio of the "Two-level" model to the Nominal uniform-layer model of the contamination
(for 12/20/03.) The parameters of the two-level model have been adjusted to retain the
same Carbon-edge jump (ratio of 42A to 44A transmission) while also adding additional
absorption at 19 A (0.92) to agree with the ECS expectation.
Here 20% of the surface is 2.8 times the nominal thickness and the other 80% is 0.943 of the
nominal thickness. Other sets of parameters (e.g., 50% at x1.7 and 50% at x0.812)
give very similar curves.
Exact details at the C-edge are not shown here, but are shown on the ISIS-generated plot below.


Agreement between the LETG-ACIS edge-derived contamination model ("Herman's model") and the ECS measured data (Mn L/K ratio) can be achieved if the contaminant is assumed to be non-uniform in thickness, e.g., most simplisticly some fraction of the local surface (thkfill) is at a larger thickness (thkfac) and some fraction is near but below (thnfac) the nominal (edge-derived) thickness. The edge jump at Carbon K is very sensitive to the thickness of the thin component (thnfac) whereas the absolute absorption at the Mn L lines (or astrophysical O VIII) depends on the filling factor and thickness of the thick layer (thkfill, thkfac) as well. Hence it is possible to independantly retain the observed C-edge jump ratio and change the absolute absorption at ~19A to bring the two models into agreement.

From a pragmatic point of view this model can agree with both the edge-derived data and the ECS data and so may be used to create a calibration product that is generally applicable without introducing new materials. If in addition the reality of thickness variations assumed by the model is granted then it suggests that there is a little more contaminant deposited: the "average thickness" of the contaminant is ~1.3 times the thickness currently given by the edge-derived model.

This brief exploration can be considered a 'proof of concept' that some assumption of a thickness distribution on the order of a factor of a few or so may be useful to include in the contamination modeling to reconcile the edge-data and ECS data. Detailed quantitative work especially around the Carbon edge is needed as a followup.


Specifically in IDL the two-level model is given as:

rat2 = (1.0-thkfill) * (rat)^thnfac + thkfill * (rat)^thkfac

where "rat" is the contamination correction array (vs wavelength or energy) that contamarf currently provides (i.e., Herman's arf appropriate for the date in question) and "rat2" is the modified correction based on the two-level model and parameters mentioned above.

Various combinations of the three parameters will match the Carbon-edge jump condition and the 19 A transmission value as listed here; note that the 50% case has the lowest "contrast ratio" of about a factor of 2 between the thick and thin regions. The unit of thickness here is the thickness in the nominal model.

   Thick filling     Thick        Thin       Thick/Thin   Average
   factor            thickness    thickness  ratio        Thickness
   (thkfill)         (thkfac)     (thnfac)

      10%            6.0          0.982       6.11           -
      20%            2.80         0.943       2.97          1.314
      30%            2.20         0.901       2.44           -
      40%            1.90         0.860       2.21           -

      50%            1.70         0.812       2.09          1.256

      60%            1.60         0.755       2.12           -
      70%            1.50         0.685       2.19           -
      80%            1.40         0.590       2.37          1.238
      90%            1.33         0.413       3.22           -

      95%            1.29         0.230       5.61
      97.5%          1.27         0.032      39.7
      99%            1.245        < 0.0       ---
Note that the total amount of contaminant, Average Thickness, varies only slightly with the filling factor - essentially determined by the 700 eV transmission. Plots for the 20%, 50% and 80% cases are given here: Note that the curves for the 20% to 80% of thick material are very similar with the biggest variation in the 30-40 A range - comparison with data in this range might favor a mostly thick or mostly thin model.

The IDL code used to create the plots above is given here:

ISIS version available

The two-level model (or modification) for the contamination effect can be implemented in ISIS by replacing "aciscontam(1)" in ones model with:

"(0.8*(aciscontam(1))^0.94 + 0.2*(aciscontam(1))^2.80)"
and slightly modifying the amounts of the O and F to retain their edge sizes by modifiying the model parameters from their 1.0 values:
isis> set_par("aciscontam(1).OK", 0.92);
isis> set_par("aciscontam(1).FK", 0.90);
This modifies aciscontam to be the sum of two different thickness aciscontam's: 80% at 0.94 of the thickness and 20% at 2.8 times the thickness. This modification, as shown in the figure below, makes a slight change to better agree with the External Cal Source measurements, introducing an additional factor of ~0.92 at 19 A but otherwise retaining the Carbon edge jump, etc.

The ISIS code used to create the models vs wavelength figure is given here:

Original contamarf model (black) and modified "two-level" model (red).
Their ratio is shown in green.

10/10/03 -dd
- - -

The ISIS code used to create the tau at 700 eV vs time figures is given here:

Comparing tau at 700 eV due to the Two-level model(green)
with the ECS best-fit curve (blue). Note that the two-level model,
a modification of the contamarf model, has not added any curvature
to the green curve so this difference remains. However the values
are very close and a common thickness-growth curve may be made to
agree with both data sets. A new Carbon-edge measurement at 2004.0
may be useful additional data to settle this issue.

11/14/03 -dd

I thank Amy Fredericks, Catherine Grant, Dave Huenemoerder, and Herman Marshall for useful inputs. 10/9/03 - Dan Dewey,