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A Recalibration of the
EXOSAT
Diffraction Gratings
Frits B. Paerels, Debra N. Wolkovitch,
and Steven M. Kahn
Department of Physics
University of California, Berkeley
Abstract
A set of exceptionally high-quality spectra of Sco X-1 obtained with the
Transmission Grating Spectrometers on EXOSAT motivated the need for a detailed
model for the efficiency of the diffraction gratings on these instruments. We
have developed a physical model for the diffraction efficiency, based on
pre-flight calibration data. The model, without adjustments, quantitatively
accounts for the large count rates seen at large dispersion angles in the Sco
X-1 images. We describe this recalibration, and indicate to what extent
analysis of archival data may be affected by the new efficiencies.
1 Introduction
A first examination of the spectral images obtained on Sco X-1 with the two
Transmission Grating Spectrometers on EXOSAT revealed a systematic excess of
observed counts over predicted counts at large apparent wavelengths, for any
reasonable model for the incident spectrum (any reasonable continuum model with
non-zero interstellar absorption). The excess was highly significant, growing
to more than an order of magnitude at the longest wavelengths.
It appeared unlikely that these photons represent real first-order EUV
radiation, given the strong interstellar absorption at the longer wavelengths.
For example, after allowing for a reasonable fit to the short-wavelength
continuum, and assuming a reasonable interstellar column density of
NH ~ 1021 cm-2 the excess can be
described with a very soft
blackbody (TBB ~30 eV). However, the implied luminosity is an
unreasonably large LBB ~ few times 1037
erg sec-1! We therefore assumed that the excess was caused by
some systematic effect in
the instrument or reduction procedure. The reason this effect had not been seen
before in any other source had to be attributed to the extreme brightness of
Sco X-1, revealing low-intensity features that normally remain hidden in noise
or background.
The excess could not be attributed to imperfect background subtraction:
dispersed photons are clearly visible almost all the way out to the edge of the
detector, corresponding to a first order wavelength of ~200
Å (400 Å) for the 1000 (500) lines mm-1 grating (for
a general
description of the EXOSAT Low Energy Imaging Telescopes and Transmission
Grating Spectrometers, see de Korte et al., 1981; Brinkman et
al., 1980; Paerels et al., 1990).
Uncalibrated variations in the instrument efficiencies due to systematic
effects in the CMA quantum efficiency, the filter transmission, or the mirror
reflectivity could be excluded, as they would have been noticed before.
Similarly, an uncalibrated 'bump' in the grating diffraction efficiencies would
have stood out in the spectra of softer sources with a known spectrum,
(e.g., the isolated hot white dwarfs). The most likely explanation for
the apparent excess is that it is due to the cumulative effect of
short-wavelength (5-20 Å) photons, dispersed into high spectral orders
(m = 6 and beyond). These are presently not included in the efficiency
model for the grating; the calibration database has data up to spectral order
m = 5. This implied that a revision and extension of the grating
efficiencies was necessary. Of course, the very highest order dispersed photons
hardly yield significant additional constraints on spectral models for Sco X-1.
However, the shortest wavelength at which an excess was seen (depending on the
model for the incident spectrum) was close enough to the K edge of neutral
oxygen that we felt that a more robust model for the instrument efficiency was
needed in order to perform an unambiguous measurement of the depth of this edge
in the spectrum of Sco X-1.
2 Grating Efficiency Calibration Data
The grating efficiency, defined as the ratio of the flux into a given spectral
order at a given wavelength, to the flux incident on the detector without the
grating in the beam, was calibrated for both the 1000 and the 500 lines
mm-1 gratings in a series of X-ray exposures on the integrated
instrument package (mirror plus grating plus filter plus detector) in the long
beam test facility of the Max Planck Institut fur Extraterrestrische Physik at
Garching, Germany. Data were taken at a number of characteristic X-ray
wavelengths, the best sets being those at Si K, Al K, Cu L, F K, O K, C K, and
B K. The grating efficiencies were derived using an explicit model for the
mirror reflectivity, filter transmission, and detector quantum efficiency, as a
function of wavelength. Data up to and including spectral order m = 5
were compared with simple models for the grating efficiency as a function of
wavelength. For instance, at long wavelengths, the gold grating bars are
completely opaque to X-rays, and the grating efficiency is a unique function of
the slit-to-period ratio, a/d, and spectral order m. At
shorter wavelengths, there is interference of phase-shifted radiation passing
through the bars of the grating, with radiation passing through the slits,
causing the efficiency curve to exhibit a maximum in wavelength; the location
and shape of this maximum are a strong function of grating bar thickness,
cross-sectional shape, and spectral order.
It was found that the simplest models for the efficiency, based on a grating
model for rectangular bar cross-section, and characterized by a unique bar
thickness, b, and slit-to-period ratio a/d, did not
satisfactorily fit the measured efficiencies. Since the measured efficiencies
were estimated to be accurate to better than 10%, a simple spline fit to the
measured efficiencies was taken to be a sufficient model for interpolation
between the calibration wavelengths. Of course, in order to extrapolate in
spectral order, we do need a physical model for the grating efficiency.
The shortest wavelength calibration images actually show significant flux up to
order m = 10 (at the longer wavelengths, the highest orders are off the
detector), and we reanalyzed these data in order to build a model for the
efficiency extending out to high order. Figure 1 shows an example of such a
calibration spectrum (zero order has been subtracted), taken at 8.34
Å (Al K). Radiation is detected up to order m = 10.
Figure 1 EXOSAT 1000 lines
mm-1 Transmission Grating
Spectrometer calibration spectrum at Al K (8.34 Å); zero order and
background have been subtracted. Diffraction up to 10th order can be
seen.
3 A New Model for the EXOSAT Grating Efficiencies
The calibration dataset we used consisted of the efficiencies measured
previously, up to order m = 5. We reanalyzed two calibration images, at
Al K (8.34 Å) and Cu L (13.34 Å), extracting efficiencies for
orders m = 6 - 10. We first concentrated on the 1000 lines
mm-1 grating, because data taken with this grating will be the
most valuable in the case of Sco X-1 on account of their higher spectral
resolution.
The zero-order and background subtracted one-dimensional spectrum was fitted
with a model consisting of a set of discrete emission lines of variable
amplitude at the positions predicted from the known dispersion relation for the
grating. A simple power law, of the form was added, to account
for residual
zero-order contamination and continuum bremsstrahlung from the electron impact
X-ray source. This model spectrum was convolved with the spectrometer response
function, which at these short wavelengths is simply equal to the telescope
point-source response. At the largest dispersion angles, some degradation of
the response due to optical aberrations should set in; however, in the present
case the degradation remains small and does not critically affect our results.
Left- and right-order amplitudes were averaged after they had been found to be
consistent to within the errors. We devided the amplitudes in order m
by the amplitude in first order to obtain the efficiency ratios
to divide out the
efficiencies of the other instrument components. The resulting data were
compared with an explicit model for the grating efficiency.
We calculated the diffraction efficiency by numerically summing the complex
amplitudes of elementary waves along one grating period. The attenuation and
phase shifts incurred by radiation passing through the gold grating bars was
calculated using the optical constants compiled by Henke et al. (Henke
et al., 1981). The grating bars were assumed to have ellipsoidal cross
section, as suggested by microscope pictures of the actual grating. This
grating model has two free parameters: the slit-to-period ratio,
a/d, and the semiminor axis, b,of the ellipsoidal
cross-sectional shape.
In principle, one can fix these parameters separately. In the long wavelength
limit, the grating bars become opaque and the diffraction efficiency is a
unique function of a/d and spectral order,
and a single set of efficiencies is sufficient to determine a/d.
For a/d = 1/2, the design goal, dispersion into the even
orders vanishes. Comparison with the calibration data shows that
a/d is indeed close to 1/2, but no single value of
a/d adequately describes all orders simultaneously. The even
orders are very sensitive to small deviations from a/d =
1/2. Consequently, we assumed a distribution of a/d-values to exist
over the grating, peaked near a/d = 1/2.
For convenience, we
assumed the shape of the distribution to be Gaussian. The efficiency of the
grating was calculated by integrating the single-a/d efficiencies
over the distribution. The mean, (a/d)0, and width,
(a/d), of the distribution
were determined
from a -fit to all calibration
data. A well-defined minimum in
exists at
(a/d)0 = 0.46 and
(a/d) = 0.04.
Similarly, we assumed a Gaussian distribution of grating bar thicknesses,
b. The mean, b0, and width, (b), were determined with (a/d)0 and (a/d) held fixed at
the above values. A well-defined minimum exists at b0
= 2.9 x 10-5 cm and (b) = 2.0 x 10-5 cm.
The efficiencies calculated from this model agree reasonably well with the
calibration data. Figure 2 shows a comparison of model curves for
together with calibration
data.
Some discrepancies are evident, most prominently in the high orders, and in
second order at the shortest calibration wavelength (Si K). These discrepancies
indicate that additional effects not included in the grating model affect the
efficiency, such as perhaps non-Gaussian distribution of the grating
parameters, deviations from ellipsoidal bar shapes, and uncertainties in the
optical constants of gold.
The higher-order discrepancies are less significant, since the experimental
errors become larger, and are progressively more subject to systematics. The
Si K data at 7.04 Å are close to the M absorption edge complex of gold
(3.62-5.62 Å). Close to the edges, the optical constants are the most
uncertain, so that the discrepancies between the efficiency model and the
calibration data is not too surprising. We therefore decided to compose our
final grating model of a spline fit to the experimentally determined
efficiencies in orders 0-5 (as before), and to adopt the grating model to
extrapolate to orders 6 and beyond.
Figure 2 Ratios of diffraction efficiencies relative to
first order, for the 1000 lines mm-1 grating. Black squares are
the
measured ratios, solid lines are calculated from the final model for the
grating efficiency, with (a/d)0 = 0.46,
(a/d) = 0.04; b0 = 2.9 x
10-5 cm, (b) = 2.0 x 10-5 cm. The numbers
indicate the various diffraction orders.
4 Comparison with Sco X-1
With the model efficiencies calculated as described above, we can now verify
that dispersion of short wavelength photons into a large number of high
spectral orders actually dominates the count rates at large dispersion angles
in the Sco X-1 images. We fitted a simple, rough continuum model to a spectrum
obtained in 1 hr of exposure on 1983/220 with the 1000 lines
mm-1 grating and thick Lexan filter combination, and compared models with and
without the spectral orders m = 6-20 included. Figure 3 shows the result
of this comparison. Clearly, the excess seen at
(E < 0.28 keV) is simply due to the higher orders of
short-wavelength flux. No modifications to the efficiency model have to be made
in order to obtain quantitative agreement with the measured fluxes.
The new efficiencies are not likely to affect any of the analysis done on
grating images other than those of Sco X-1. Probably only this source is bright
enough to show dispersion into sixth and higher order above the background; but
low-level 'excesses' at long wavelengths
should be treated with caution in any case. Discrepancies may persist even with
the new calibration, in view of the limitations of our model outlined in the
previous section.
The only other datasets that are affected by the new grating efficiencies are
the data taken with the 1000 lines mm-1 grating on AM Her. In
the
80-100 Å wavelength range, gold has a local minimum in the absorption
coefficient, and the bars of the 1000 lines mm-1 grating become
partially transparent again, giving rise to a resonance-like feature in the
efficiency curve. The exact shape and location of this feature is strongly
dependent on the grating parameters. With the grating model developed above, we
can now self-consistently calculate the efficiency curve in this wavelength
band as well. None of the other strong soft X-ray sources observed with EXOSAT
(the hot white dwarfs) were observed with the 1000 lines mm-1
grating.
Figure 3 1000 lines mm-1 TGS
spectrum, thick Lexan
filter, obtained in a 1 hr exposure on 1983/220; zero order and background
subtracted (crosses). The solid histogram is a thermal bremsstrahlung model, at
kT = 13.5 keV, column density of neutral absorbing material
NH = 1.6 x 1021 cm-2 with the first 5
diffraction orders only
included in the model (upper panel), and with 20 diffraction orders included
(lower panel).
5 Future Work
In principle, the same treatment should be applied to the 500 lines
mm-1 grating, and we plan to at least extend the efficiencies
for
this grating using a model based on the current best estimate for the
parameters of the grating.
A version of the code that produces a grating spectrometer matrix for XSPEC,
for an arbitrary number of spectral orders and each filter, will be implemented
at the HEASARC in the near future.
A paper describing the grating efficiency recalibration in more detail is in
preparation (Wolkovitch, Paerels, and Kahn 1992).
References
Brinkman, A.C., Dijkstra, J.H., Geerlings, W.F.P.A.L., van Rooijen, F.A.,
Timmerman, C., and de Korte, P.A.J., 1980, Appl. Opt., 19,
1601.
de Korte, P.A.J., Bleeker, J.A.M., den Boggende, A.J.F., Branduardi-Raymont,
G., Brinkman, A.C., Culhane, J.L., Gronenschild, E.H.B.M., Mason, I., and
McKechnie, S.P., 1981, Space Sci. Rev., 30, 495.
Henke, B.L., Lee, P., Tanaka, T.J., Shimabukuro, R.L., and Fujikawa, B.K.,
1981, in: D.T. Attwood and B.L. Henke (Eds.), 'Low Energy X-ray Diagnostics',
AIP Conference Proceedings No. 75 (AIP, New York).
Paerels, F.B.S., Brinkman, A.C., den Boggende, A.J.F., de Korte, P.A.J., and
Dijkstra, J., 1990, Astron. Astrophys. Suppl., 85, 1021.
Wolkovitch, D.N., Paerels, F.B., and Kahn, S.M., 1992, Appl. Optics, in
preparation.
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