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 'N0001 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:
The IDL code used to create the plots above is given here:
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 'N0001 model (black) and modified "two-level" model (red).
Their ratio is shown in green.
- - -
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 'N0001 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.
I thank Amy Fredericks, Catherine Grant, Dave Huenemoerder, and Herman Marshall for useful inputs. 10/9/03 - Dan Dewey, firstname.lastname@example.org