A&A 381, 1110-1130 (2002)
B. Schulz 1,2 - S. Huth 2,3 - R. J. Laureijs 1,2 - J. A. Acosta-Pulido 2,3,4 - M. Braun 2,3,7 - H. O. Castañeda 2,3,4 - M. Cohen 10,11 - L. Cornwall 2,5 - C. Gabriel 1,2 - P. Hammersley 4 - I. Heinrichsen 2,6,8 - U. Klaas 2,3 - D. Lemke 3 - T. Müller 1,2,3 - D. Osip 9,12 - P. Román-Fernández 1,2 - C. Telesco 9
1 - ISO Data Centre, Astrophysics Division of ESA, Villafranca, PO Box 50727, 28080 Madrid, Spain
2 - ISO Science Operations Centre, Astrophysics Division of ESA, Villafranca, PO Box 50727, 28080 Madrid, Spain
3 - Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
4 - Instituto de Astrofisica de Canarias, 38200 La Laguna, S/C Tenerife, Spain
5 - Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK
6 - Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
7 - Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
8 - Infrared Processing and Analysis Center, California Institute of Technology, MS 100/22, Pasadena, CA 91125, USA
9 - 211 Bryant Space Science Center, PO Box 112055, Dpt. of Astronomy, Univ. of Florida, Gainesville, FL 32611-2055, USA
10 - Radio Astronomy Laboratory, 601 Campbell Hall, University of California, Berkeley, CA 94720, USA
11 - Vanguard Research, Inc. Suite 204, 5321 Scotts Valley Drive, Scotts Valley, CA 95066, USA
12 - MIT, Dept. of Earth, Atmospheric and Planetary Sciences, Bldg. 54-420, 77 Massachusetts Ave., Cambridge MA 02139, USA
Received 1 August 2001 / Accepted 29 October 2001
All observations by the aperture photometer (PHT-P) and the far-infrared (FIR) camera section (PHT-C) of ISOPHOT included reference measurements against stable internal fine calibration sources (FCS) to correct for temporal drifts in detector responsivities. The FCSs were absolutely calibrated in-orbit against stars, asteroids and planets, covering wavelengths from 3.2 to 240 m. We present the calibration concept for point sources within a flux-range from 60 mJy up to 4500 Jy for staring and raster observations in standard configurations and discuss the requisite measurements and the uncertainties involved. In this process we correct for instrumental effects like nonlinearities, signal transients, time variable dark current, misalignments and diffraction effects. A set of formulae is developed that describes the calibration from signal level to flux densities. The scatter of 10 to 20% of the individual data points around the derived calibration relations is a measure of the consistency and typical accuracy of the calibration. The reproducibility over longer periods of time is better than 10%. The calibration tables and algorithms have been implemented in the final versions of the software for offline processing and interactive analysis.
Key words: instrumentation: photometers - methods: data analysis - techniques: photometric - infrared: stars - infrared: solar system
The task of calibration can be divided into three parts: First,
development of the instrument model and determination of the
instrument parameters that are assumed to be unchanging and
that can be measured in the laboratory, e.g.
filter transmissions, aperture diameters, etc. Second, the
determination of the open parameters of the ideal
instrument model that can be determined only in-situ, i.e. with
the instrument built into the satellite and in the real
space-environment. And third, the determination of deviations
from the ideal instrument model that cannot be removed by
adjusting parameters, and require a new functional
property. These modifications to the instrument model are found
empirically, and generally originate in simplifications.
Because open parameters and non-ideal
instrumental effects are usually intertwined,
an iterative process is required to separate and
quantify all the contributing effects.
|Figure 1: Instrument schematic from a calibrator's point of view. The detector compares IR radiation directly from the sky and from the internal reference source. Stability of the detector is required only for the time interval of the two measurements.|
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This paper presents the photometric calibration of staring mode- and simple raster mode observations of point sources with the P- and C-sections of ISOPHOT (Lemke et al. 1996), which is one of four scientific instruments on board ESA's Infrared (IR) Space Observatory ISO (Kessler et al. 1996). Unlike CCD devices, the detectors for the Mid-IR (MIR) and Far-IR (FIR) are far less stable and exhibit a continuously changing relation between signal and incident flux. Thus the two Fine Calibration Sources (FCS) built into ISOPHOT played a crucial role as stable references for the photometry and most of this paper will describe their empirical calibration.
The data collected for this task represent the largest part of all specific calibration observations during the mission. This resulted in a fairly homogeneous block of data. Its analysis and comparison to modeled spectral energy distributions (SEDs), of the observed celestial standard sources yielded most of the results presented here and drove several refinements to the ideal instrument model. An additional difficulty was the large number of possible instrument configurations, which was limited somewhat by considering only one standard aperture for each filter band of the aperture photometer (see Table 2). The set of instrument configurations and modes we treat herein defines a well-understood baseline within the large parameter space, where absolute calibration errors are expected to be minimal. The calibration of further configurations and modes, like chopped observations, extended source photometry, or non-standard apertures, is left to future publications.
We start with a brief review of the instrumental design with some emphasis on the internal reference sources (Sect. 2), followed by an outline of the calibration strategy (Sect. 3). Section 4 continues with a description of the corrections applicable to the detector signal. Section 5 presents the celestial calibrators and Sect. 6 describes those corrections imposed by photometric constraints. We derive the FCS calibration tables in Sect. 7 and present the final flux calibration, with its mathematical description and a discussion on accuracy and reproducibility, in Sect. 8. A summary constitutes Sect. 9.
The telescope is a Ritchey-Chrètien design, diffraction-limited at 5 m, with an entrance aperture of 60 cm in diameter and an f-ratio of 15 (Kessler et al. 1996). Radiation reaches the chopper mirror within ISOPHOT via a pyramidal mirror, which is centred between the 4 ISO-instruments. The sky is projected onto the focal plane with a scale of 0.04363 mm/ .
The detectors, made from doped silicon or doped germanium, are extrinsic photoconductors operated at temperatures of 1.7 to 3.6 K, depending on material. Their response to IR radiation is characterised by the detector responsivity, R, which is the parameter most relevant to calibration. It is given by the relation between detector-signal and incident in-band power , where S is the detector signal expressed in V/s for an integrating amplifier. is the relaxed signal under dark conditions, is the integrating capacity (140 fF for C200, 90 fF for others) and P represents the in-band power in Watts.
The responsivity of these detectors is not constant. Depending on material, variations of a factor of 3 during one orbit were observed. Changes of R are induced by the ionizing radiation in space and by the previous sequence of IR fluxes to which the detector has been exposed. We consider R to be stable over typical time intervals of around 10 to 20 min. To bring R back to a nominal value, the detectors routinely underwent a curing-procedure after having crossed the Radiation Belts during perigee and before regular observations started ("science window") (see Laureijs et al. 1996, 2001). These procedures consisted of detector specific combinations of detector-heating, bias increase or illuminator flashes (Lemke at al. 1996) and brought the responsivity back to within 5% of its nominal value. The doped-Ge detectors, which showed the biggest responsivity changes, underwent a second curing procedure after 8 hours, close to apogee.
The radiation of both sources is combined in a diamond beam-splitter. The beam of TRS1 is reflected with practically no losses, while the beam of TRS2 is transmitted through the material and attenuated by a factor of 1000. Only one TRS is heated at a time. The attenuated TRS2 is operated with the doped-Ge detectors P3, C100 and C200, which show higher responsivities, whereas the doped-Si detectors P1 and P2 are used with TRS1 in the stronger reflected beam. Mirrors focus the emerging radiation in such a way that a beam with the same aspect ratio as the telescope is emitted, while the TRSs are imaged onto the chopper mirror, making the FCS appear to the detectors as a homogeneously illuminated source.
The in-band power provided by the FCS could be adjusted by changing its heating power. The fundamental calibration curves, in-band power versus FCS heating power, were initially determined by a grey body model (Schulz 1993), fitted to a few already known data points. The grey body is defined as , which is the Planck function multiplied by a wavelength-independent emissivity . The temperature, T, is linked to the FCS heating power, h, by an empirical function, , where is the temperature of the optical support structure (2.76 K), and , and are dimensionless constants. Typical values for these constants of FCS1/TRS2 are , and respectively. An attenuation factor which modifies the effective solid angle individually for each detector subsystem was introduced, describing flux losses that were not predicted by the grey body model.
The advantage of the grey body model was to allow predictions for other filter bands, when only a few data points per detector were available. To increase accuracy, once enough data points were available, the covered ranges were interpolated by smooth low-order polynomials and the model was used for extrapolations only.
The absolute calibration of these secondary standards was established in-orbit by comparing their output to known celestial standards, an activity that started during the performance verification phase (PV), but continued throughout the mission due to the visibility constraints of some of the sources. To minimize errors due to potential detector nonlinearities, the FCS heating power was adjusted so that the emitted FCS flux roughly matched the flux emitted by celestial source and background. The limited linearity of the system demanded celestial calibration sources at all flux levels in all 25 filter bands.
Integrating cold (3 K) readout electronics (CRE) (Dierickx et al. 1989) are used to amplify the currents of typically 10-16... A that flow through the ISOPHOT photoconductors (Lemke et al. 1996). The photocurrent is measured from the rate at which the voltage at the charge capacitor within the CRE increases with time. The voltage is sampled at regular time intervals for a given duration before it is reset (non-destructive readouts, NDR, and destructive readouts, DR). The readouts of all channels of a CRE unit are time-multiplexed and sent via a single line to the external electronics unit (EEU) outside the cryostat. There they are further amplified and digitized by a 12-bit analog-to-digital-converter (ADC). The sequence of samples between two resets is called an integration ramp and ideally fits a straight line. In the following we will refer to the slope of the fitted line as "signal'', measured in V/s. Multiplication by the integration capacity, for which the design value of 90 fF (for C200 140 fF) is adopted, leads to the photocurrent in Amps.
The actual integration ramps are, however, not perfectly straight (see Fig. 2) (Schulz 1993). Two effects can be separated: i) The AC-coupled CRE circuit is not a perfect integrator. The output voltage can rise only to the level of the detector bias times the small amplification of the circuit (10). This saturation level is reached asymptotically via a typical RC-loading curve. High bias voltages above 10 V have sufficiently high saturation levels (100 V and more) that they show practically linear integration ramps within the CRE output range of 2 V. However, for the smaller biases of P3, C100 and C200, the saturation voltage drops such that the curvature of the loading curve appears within the dynamic range of the integration ramp (debiasing). ii) Secondly, all integration ramps show deviations from a straight line that always appear at the same CRE output voltage, regardless of the level from which the ramp started to integrate. Due to the direct link to the CRE output voltage we attribute this component to an intrinsic nonlinearity of the amplifier within the circuit.
|Figure 2: Ramp nonlinearity for pixel 8 of C100. Deviations of many integration ramps from a straight line are plotted against CRE output voltage. This relation is used for ramp-linearisation.|
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The correction algorithm is based on the observation that all integration ramps of one detector pixel can be matched by conserving the CRE output voltage of the individual readouts, while linearly stretching and shifting the time-axis of each ramp (Schulz 1993). In this way an average integration ramp was determined for each detector pixel from a number of measurements with ramps of large dynamic range. The difference between a straight-line fit to this ramp and the ramp itself (Fig. 2) is used to linearise individual ramps. The maximum absolute values range between 40 and 100 mV over the maximum CRE dynamic range of 2 V. The correction depends only on CRE output voltage, and has no further dependences on readout frequency, in-band power or time.
The reliability of the correction is deduced from the standard deviation of the individual ramps from the average integration ramp. Over most of the dynamic range it is below mV, with a tendency towards larger scatter for longer wavelength detectors. The highest standard deviation appears in the central pixel of C100, showing mV. The higher scatter towards long wavelength detectors with smaller biases is interpreted as resulting from a simplification in the correction algorithm. Relating the corrections only to the CRE output voltage neglects the fact that the curvature due to debiasing also depends on the reset level of the integration ramp, which shifts slightly w.r.t. the CRE output voltage. Hence an additional scatter appears, increasing with smaller bias. Considering the maximum standard deviation of mV and an average dynamic range of 1 V (half the maximum dynamic range), we estimate the typical residual systematic error to be about 2% after correction.
The output signal is disturbed by energetic particle hits (glitches), mostly protons and electrons. A typical distribution of fitted slopes after ramp linearisation is shown in Fig. 3. The asymmetry of the distribution is caused by glitches that instantaneously increase the charge on the capacitor during the integration, so that the average slope of the affected integration ramp is increased. Typically, the readouts following the glitch continue integrating as before; however, stronger hits can affect the detector, leading to long-lasting responsivity changes. A number of algorithms have been developed to remove signals affected by glitches (Gabriel et al. 1997).
|Figure 3: Histogram showing the signal distribution in a long measurement of the central pixel of C100. The tail towards higher signals is caused by high energy particle hits and can be removed by appropriate algorithms.|
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All deglitching algorithms still leave an asymmetry in the signal distribution. For stronger asymmetries the simple average is often quite far from the peak of the distribution. The median comes closer to the peak; however, it is a bad choice for very weak signals and short integration ramps, where the bins of the A/D-conversion become significant. The AC-converter covers the range from -10 to +9.995 V. Dividing by the gain factor of the analog electronics and 212 for 12 bits, the least significant bit is equivalent to 6.1 mV. For integration ramps of 1 second this would yield a detector current of A. As an example, the 100-m channel would produce discrete slope levels in about 20 mJy intervals. Since the result of the median is always an existing value of the sample, the accuracy is limited to the separation of the intervals. In this case, better results can be obtained by fitting a Gaussian to the signal distribution.
|Figure 4: Observations in "nodding mode'' with a linear 3-step raster used for faint sources. Several changes of the pointing between a strong background and a faint source on top of it make the source signal appear as a modulation on top of a strong transient. This is corrected by dividing by a baseline fitted to all background levels. The sequence starts and ends with FCS measurements.|
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(i) We calculate the average of the last 30% of the measurement and call it S30. The valid fit result must deviate by no more than 50% from S30.
(ii) A straight line is fitted to the last 40% of the measurement and extrapolated to a point at 2 times the measurement time. 80% of the difference between this point and S30 is the maximum difference allowed between the signal level predicted by the transient fit and S30.
(iii) For each fitted transient, the term (1-S1/S2)/(t1-t2) is evaluated twice, where S and t are the slopes and times of either the first two, or the last two, data points in the measurement. Absolute differences between these two results that are smaller than 0.001 s-1 are rejected. This and criterion (ii) buffer the curvature of the fit against small values since small curvatures result in very uncertain asymptotes.
In case these criteria fail, we take S30 as the best guess for but the result is flagged as being less reliable. This applied to about half the cases where transient fits were attempted. We caution that the criteria were tuned empirically to this large, but special, sample of FCS calibration measurements. We assume, therefore, that they work best for measurement times 64 s.
|Figure 5: The diagram shows the relations of signals of the same in-band power measured with different readout timing settings. The signals on the y-axis are measured using a 1/4 s reset interval and a data reduction factor of 1. The reset interval and data reduction factor, respectively, for the signals of the x-axis are given by the two numbers in each diagram. The data points follow a linear relation, with slope different from unity and non-zero x-intercept. This example was measured for detector C100.|
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The readout timing (ROT) of the CRE was selected individually for each measurement according to the observer's flux estimate, to give a maximum dynamic range without saturation and thus a minimum of readout and sampling-noise. It is controlled by 3 parameters: the time interval between two destructive readouts, ; the number of non-destructive readouts, ; and the data reduction factor. For the range of 128 s to 1/64 s for , depending on the detector, a total of 14 to 15 combinations was actually used during operations.
|Figure 6: The evolution of the dark signal with relative orbital position. The data points were taken randomly during the mission and plotted versus orbital position, where the range 0 to 1 corresponds to the full 24 hour orbit, starting at perigee.|
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|Figure 7: As for Fig. 6 but for the detectors C100 and C200.|
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To achieve consistency, we correct all signals so they appear as if measured with a common reset interval of s and a data reduction factor of 1. The signal, S, is corrected according to , where the parameters A0 and A1 depend on the ROT set-up. They were determined from a least-squares fit to the calibration data. An example is shown in Fig. 5. It should be noted that A0 is generally different from zero and the slope A1 is significantly different from unity. All pixels of equal detector-subsystems were found to follow the same relation, suggesting that a pure CRE-effect is observed. Again the residual errors of the correction are systematic in nature. The scatter of the calibration data around the fits suggests typical systematic errors of no more than 5% and probably better than 2%.
|Figure 8: The coverage of the flux domain accessible to ISOPHOT by the three kinds of celestial standard.|
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In Fig. 8 we illustrate the coverage of the flux domain accessible to ISOPHOT by celestial standard sources. The largest background components are zodiacal thermal re-emission, peaking around 20 m, and Galactic cirrus, contributing most at the longest wavelengths. Only at the shortest wavelengths does the dark current dictate the lowest signals. The dotted line shows typical surface brightnesses for celestial backgrounds, converted to an equivalent flux density of a point source, observed in the standard aperture without background. The upper absolute flux limits are plotted as dashed lines. Note that the limiting sky background for the overlapping wavelengths of PHT-P and PHT-C is different due to their different aperture areas. Examples of each of the three types of calibration source (planets, asteroids, stars) are plotted as solid lines.
|Stellar Celestial Standards||Solar System Standards|
|HR337*||And||MC||M0IIIv||192||-2.05||7.97||-2.17||1 Ceres||248 ...433||-6.46...-5.85|
|HR617*||Ari||MC||K2III||55.7||-0.76||2.33||-0.83||2 Pallas||55.3 ...208||-5.67...-4.23|
|HR1457*||Tau||MC||K5III||19.1||-3.12||3 Juno||37.6 ...62.2||-4.36...-3.81|
|HR1654||Lep||MC||K5IIIv||1.61||-0.43||4 Vesta||153 ...291||-6.03...-5.34|
|HR3748||Hya||MC||K3II-III||93.4||-1.31||3.92||-1.39||10 Hygiea||29.0 ...43.9||-3.97...-3.52|
|HR5340*||Boo||MC||K1III||506||-3.15||21.0||-3.22||54 Alexandra||20.0 ...26.1||-3.42...-3.13|
|HR5886||-||PH||A2IV||0.251||5.09||65 Cybele||13.4 ...22.3||-3.24...-2.69|
|HR5986||-||PH||F8IV-V||2.24||2.71||106 Dione||3.80 ...3.93||-1.35...-1.32|
|HR6514||-||PH||A4V||0.095||6.14||313 Chaldaea||7.74 ...8.14||-2.15...-2.10|
|HR6688||-||PH||K2III||0.463||0.92||532 Herculina||16.3 ...44.5||-3.99...-2.91|
The ISO Ground Based Preparatory Programme (GBPP) (Jourdain de Muizon & Habing 1992; van der Bliek et al. 1992; Hammersley et al. 1998) provided SEDs between 1 and 160 m by interpolating in the Kurucz stellar model grids (Kurucz 1993), using temperature, surface gravity and metallicity. The temperatures were derived either from the IR Flux Method (Blackwell et al. 1991, 1998) or the V-K versus relationship (Di Benedetto 1993, 1998). Surface gravities and metallicities were from Cayrel de Strobel et al. (1992). The spectral shapes derived from the grids were then normalized using near-IR (NIR) photometry (Kn or K-band). The attributed errors range between 3% and 5% which is dominated by the error in conversion to an absolute flux density scale. The random scatter between the measured and predicted fluxes at 10 m is shown to be better than 1.5% (Hammersley et al. 1998).
Another calibration programme by Cohen et al. (1996) provided empirical SEDs in the range from 1.2 to 35 m by splicing together measured spectral fragments of cool K- and M-giants and calibrating them by absolutely calibrated Kurucz models of Sirius and Vega. The measured "composite" SEDs are used to derive so-called "template" SEDs for fainter stars (Cohen et al. 1999), assuming that the intrinsic spectral shape depends only on spectral type and luminosity class. The absolute flux level is set by well-characterised NIR and MIR photometry, including IRAS data. To extend those composite-SEDs not observed as far as 35 m by the Kuiper Airborne Observatory, Engelke functions (Engelke 1992; Cohen et al. 1995, 1996), with effective temperatures from Blackwell et al. (1991), were used. To create FIR extensions to support the longest wavelengths of ISOPHOT (240 m), continuum model atmospheric spectra were attached to the empirical composites and extended to 300 m (Cohen at al. 1996). The typical absolute accuracy (i.e., in or at any wavelength) of the templates is about 3%, while the FIR extensions ranging from 25 to 300 m have computed uncertainties of about 6%.
Note that both programmes deliberately use the same Vega model SED for their zero point definitions. Furthermore, the measured zero points of the various photometric systems were determined to ensure that there would be no systematic differences depending upon which data were used (Cohen et al. 1999). Specifically for ISO, both programmes supplied electronic versions of their SEDs with spectral resolutions of typically 50 to 300. The objects are listed in Table 1.
The irradiances of a number of primary (Sirius, Vega), secondary (bright K- and M-giants) and tertiary (template-SEDs) calibration stars have been absolutely validated by a dedicated radiometric calibration experiment carried out on the US Midcourse Space Experiment (MSX: Mill et al. 1994) by Cohen et al. (2001). The 1.2-35 m spectra of seven of the stars in Table 1 have been validated by this means. We note that because of its observed deviation from the model SED longward of about 17 m, we do not use observations of Vega to calibrate any long wavelength filter.
Brighter flux levels in the FIR (i.e. filters with reference wavelengths 50 m) had to be calibrated by other objects, since even the brightest standard star measured by ISOPHOT ( Boo) drops to less than 2 Jy at 200 m (see Fig. 8).
In an effort to homogenize the calibration of ISOPHOT and the Long Wavelength Spectrometer (LWS) in the long wavelength range, for the final version of the offline processing software (OLP 10) we changed the planet models used in earlier versions (Schulz et al. 1999). The ones supplied by Abbas (1997, priv. comm.) were replaced by models produced by Griffin & Orton for LWS. The models are calibrated with 0.35 to 3.333 mm JCMT data (Griffin & Orton 1993), which in turn were calibrated against Mars (Wright 1976; Wright & Odenwald 1980). At the short wavelength end, the model was constrained by Voyager IRIS data from 25 to 50 m by Hanel et al. (1986) for Uranus, and Conrath et al. (1989) for Neptune. The temperature structure and composition of H2, He and CH4 was taken from Voyager radio occultation experiments by Lindal et al. (1987) for Uranus, and by Lindal (1992) for Neptune.
We compared the models of Griffin & Orton (GO) with the ones of Abbas (AB). Above 45 m the GO model of Neptune is 10% brighter than AB. In contrast, for Uranus the GO model fluxes are about 3 to 10% lower than AB above 50 m and deviate no more than 15% above 20 m. At shorter wavelengths the discrepancies between AB and GO rise dramatically for both planets. However, these are no longer relevant to the PHT calibration, since the C_50 filter passband starts only around 40 m. We take these differences as indicative of how well the fluxes from Uranus and Neptune are known at FIR wavelengths and conclude that, for our purposes, they are still within a 10% margin.
Some specific observational complications arose, however, since they are moving objects with respect to the sky background and show periodic variations of intensity due to rotation and varying distance from both the Earth and Sun. Selection criteria were: well understood rotational behaviour, small lightcurve amplitude, some form of independent size determination (either direct imaging or via occultation measurements), good visibility during the ISO mission, and availability of sufficient observational data from visible to submillimetre wavelengths. The selection resulted from a combination of extensive ground-based observing campaigns at thermal wavelengths from the IRTF, UKIRT, and JCMT, additional visible wavelength lightcurve measurements, and a series of FIR observations from the now retired KAO. These observations were used to confirm the validity of the subsequent thermophysical modeling effort.
Initially, modified versions of the Standard Thermal Model (Lebofsky 1989) were used. For the final ISOPHOT photometric calibration, a thermophysical model (TPM) assuming a rotating ellipsoid and parameterising heat conduction, surface roughness, and scattering in the regolith was adopted for 10 asteroids (Müller & Lagerros 1998). The TPM is capable of producing thermal lightcurves and spectral energy distributions for any time, taking into account the real observing and illumination geometry. The overall comparison between model predictions and our large sample of MIR, FIR, and submillimetre observations (about 700 individual measurements between 2 and 2000 m) demonstrated an accuracy of approximately 10% across a wide wavelength range from 10-500 m. In a few cases, existing direct measurements of shape significantly improved the model accuracies.
In a recent study of the accuracy of the TPM, ISOPHOT observations of asteroids were independently calibrated in the FIR (Müller & Lagerros 2001, 2001a). In this work, parts of the FCS power curves were established using measurements only of planets or stars. These were used then to calibrate the asteroid observations and compare the results to the TPM predictions. It was shown that observations and predictions for 1 Ceres, 2 Pallas and 4 Vesta agree within 5%. For 3 Juno, 10 Hygiea, 54 Alexandra, 65 Cybele, and 532 Herculina the agreement is still within 10 to 15%. For the 2 objects without independent ISO data the comparison to ground based thermal observations gave rms values of 14% for 313 Chaldaea and 29% for 106 Dione. Note that the rms values include also observational errors, which are dominated by the structured FIR sky background and calibration uncertainties. Modeling limitations are mainly due to uncertainties about the exact asteroidal shapes. The final modeling results were also confirmed to an accuracy of better than 10% via comparison with rotationally resolved target observations from the IRAS database. Table 1 lists the solar system objects that were used for ISOPHOT calibration.
The filter transmissions (including the out-of-band rejections over a wide wavelength range) and the relative response functions of the detectors were measured in the laboratory under "cold" conditions (at their anticipated operating temperatures in-orbit) with a Fourier-transform spectrometer (Schubert 1993). To calculate , we assumed the reflectance of each mirror surface to be 98% and wavelength-independent. Thus we obtain , with n being the number of mirrors in the optical paths, which is 6, 7, 7, 6 and 5 for the detectors P1, P2, P3, C100 and C200, respectively. For P1 an additional multiplicative factor of 0.95 was included to account for losses due to the Fabry lens.
We note that the in-band power is only a fraction of the total incident IR power, since the formula includes the wavelength-dependent relative response function of the detector material. Normalizing this response function to 1.0 at its peak allows us to define the detector responsivity as characteristic of the detector pixel, independent of the filter used. Thus represents only the fraction of the in-band power that actually initiates a photocurrent in the detector material.
Since the wavelength dependence of the PSF within a filter band
is small, we simplified Eq. (1) to
For the initial calculations we used an analytical telescope model, taking into account the primary and secondary mirrors. Subsequently we replaced this by a more accurate model (Okumura 2000) based on numerical Fourier transformation, that also includes effects of the support structure of the secondary mirror. The PSF factors were calculated as the ratio of the in-band power resulting from Eq. (1), integrated over the detector aperture and the total in-band power in the PSF, i.e. Eq. (1) integrated over an infinite aperture. A spectrum was adopted as source SED.
Thus far, our intent has been to determine the
in-band fluxes of known standards to characterise the detectors
and to calibrate the FCS.
Once the FCS is calibrated, however, we can use
the instrument to determine the in-band powers of other celestial
sources. To convert these back to flux densities, we must invert
Eq. (2). For regular target observations
the source SED is generally unknown so we need a common reference
spectrum with a defined colour. We adopt a spectrum of the form
as Beichman et al. (1988) did
for IRAS. This enables us to replace
Eq. (2) by the much simpler form
|Filter||C1||Apert.||lo lim||hi lim|
To check the relative effective transmission of filter bands, measurement blocks of the same detector were further grouped together, so that responsivity drifts would be minimal during that time. We verified this by repeating the first measurement of the sequence at the end. The results lead to corrections to the filter transmissions, that are discussed in Sect. 6.2.
All detector pixels were calibrated individually. Therefore, measurements of the standard point sources were positioned at the centre of every pixel. To minimize the measurement time for the camera arrays, the spacecraft performed a raster mode pointing sequence with a step size equal to the distance between pixel centres (46 for C100, 92 for C200). Larger raster dimensions ( for C100 and for C200) were chosen to obtain the background zero level also. Where no raster measurement was required, background measurements were performed in staring mode on positions within a 5 radius of the source. For aperture-photometry of faint sources, a one-dimensional raster of 3 points with the source at the centre position was scanned back and forth (nodding-mode), to enable elimination of the baseline drift (see Fig. 4).
The use of a raster observation on a fixed sky position was also necessary because ISO could neither observe positions with a relative offset from a tracked Solar System Object (SSO), nor perform raster observations while tracking. Therefore, all raster observations on SSOs had to be scheduled at fixed times with the telescope pointing adjusted to encounter the moving object. We made sure that the proper motion of the SSO due to the relative motion w.r.t. ISO was always below /hr during observations. The contribution from ISO's motion relative to the centre of the Earth was only of the order of /hr, as science observations were made during the "slow'' part of the orbit. During the maximum duration of a raster measurement on an SSO with C100 (688 s), asteroids would travel only . Taking into account ISO's absolute pointing accuracy for fixed targets of about , the worst case pointing error amounts to , which is still much smaller than the large detector apertures or pixel sizes used at long wavelengths.
In its simplest form we calculate the responsivity from the
signals measured on-source
and the PSF-corrected in-band power,
|Figure 9: Naive plots of responsivities against total in-band power, assuming linearity (Eq. (4)) for representative filter bands of all detectors. The total in-band power includes the contributions from both source and background. Upper and lower diagrams show the responsivities before and after linearisation. The upper diagrams show a clear dependence on in-band power except for P3. The remaining scatter in the linearised data is due to the variation of the responsivity with time.|
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If the measured responsivity depends in some way on the
IR flux that falls onto the detector, R can also be
expressed as a function of the photocurrent or the detector
signal. We then rewrite Eq. (4) as
|Figure 10: Derivation of filter-to-filter corrections. We display the data for some representative detectors. Each diagram shows the normalized responsivities for the indicated detector, derived from various filter sequences. Each sequence was measured on a calibration standard within a time interval, sufficiently short to avoid long term drifts. Each sequence is connected by a line. The standard deviation of the scatter of the relative filter measurements is 2 to 9% for the P-detectors and 3 to 8% for the C-arrays.|
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All further data reduction steps assume a linear system, so we decided to linearise the detector signals, S, according to , where is a continuously rising transfer function. The condition for is such that the results of inserted into Eq. (4) should no longer correlate with the detector signal, but should be distributed around a constant responsivity with minimum dispersion. The absolute value of this constant is arbitrary because it cancels out in the flux calibration. We normalized it to the median of the nonlinearised responsivities calculated according to Eq. (4). It should be noted that application of the signal linearisation requires the prior subtraction of the dark signal.
Although the FCS calibration data comprise a large number of measurements, the number of independent responsivities distributed over the entire flux range per filter band ranges only between 14 and 28. We developed a manual fitting method to determine the linearisation tables, by displaying the results of Eq. (6) and interactively modifying the linearisation function, . This approach led to a better exploration of parameter space, a smoother transition at the boundaries, and a more tolerant treatment of outliers without the need to develop sophisticated fitting algorithms. We increased the objectivity of the method by keeping the shape of simple, generally close to a polynomial of second or third order. We also minimised differences between pixels of the array detectors by fitting all pixels of one detector in parallel on the same screen. In spite of the large scatter of responsivities, simple functions were found that significantly increase the consistency of data measured at different flux levels.
Signal linearisation is performed by interpolation in a lookup table spanning the full range of possible signals. This table is calibrated only in the range covered by calibration standards but, in practice, signals occur outside this range that are higher, lower, or even negative in the case of noisy measurements close to the dark current. Reasonable entries for these cases had to be defined. For high signals beyond the point where data are available, we maintained a constant responsivity. Small signals are constrained so that an already dark-subtracted zero signal is not changed by signal linearisation. Therefore, we continue between the smallest point in the lookup table that is still determined by valid data, and zero, by a straight line. This assumes that the responsivity does not vary in that range. c) The most difficult cases are negative signals, because negative fluxes are simply undefined and appear only as a result of noise. The number of such cases actually encountered is low due to the reduced noise after signal averaging over periods of constant flux (SCP level). We decided to continue the transfer function such that , to cover all situations that occur.
The worst case is detector P1, where R varies by a factor of 4 between filters P_3.29 and P_16 (see Fig. 10). Currently, the wavelength-dependent flux losses of this detector remain unexplained. We speculate, however, that the sapphire substrate that mechanically supports the detector crystal within the kidney-shaped aluminium cavity might play a role. Sapphire is transparent in the NIR and starts to absorb beyond about 6 m onwards, in the middle of P1's wavelength range (Tropf & Thomas 1998). The absorption peak is at 17.5 m. Some fundamental lattice vibration modes occur in the reststrahlen region above 13 m. Top and bottom of both detector and substrate are gold-coated. The remaining effective area of the detector, where radiation from within the cavity can enter, is only twice that of the substrate. Given the high refractive index of Si, around 3.4, and the consequently higher reflectivity compared to Al2O3 (refractive index of only 1.7), we estimate that about the same number of photons enters both detector and substrate. The cavity was designed to counteract the high refractive index of the detector material by maximizing the number of reflections within it. Assuming that the sapphire substrate turns "dark" at longer wavelengths, it is conceivable that, with every reflection, a substantial fraction of the radiation enters the substrate and is attenuated. The effective response of the detector system would thus deteriorate. Another IR absorber present within the cavity is glue (Stycast). Unfortunately the question of P1's quantum efficiency remains largely academic, so that the effort of a much more detailed analysis is not justified.
The other detectors show less dramatic variations by factors up to 2. Moreover, the ratios found between different filters are not the same for different pixels of the same C-detector array. We attribute this to a projection of spatial nonuniformities of the filter surface onto the detector array, because the ISOPHOT filters are not located at the pupil of the optical path, but close to the detectors. The overall variation between filters of all detectors, except P1, is most likely due to optical misalignments and diffraction effects that are neglected in our ideal instrument model.
We applied a correction to the responsivity by introducing a matrix, . It works equally for P- and C-detectors. To obtain a wavelength independent detector responsivity, we split of Eq. (7) into filter-independent and filter-dependent parts, so that . The matrix, , was established from multi-filter measurements on calibration standards, executed so that the change of responsivity with time could be neglected for a filter sequence. After eliminating poor quality measurements, the responsivities measured for a detector pixel, p, in a filter sequence were renormalized. The factors per sequence were chosen to achieve the best match among all sequences for a given pixel. After calculating the average responsivities, , per filter, f, we normalized to the maximum responsivity found for that pixel using , thus assuming that the matrix describes flux losses w.r.t. the ideal model. The normalized data and the resulting filter-to-filter corrections, , for some detector pixels and filter bands, are shown in Fig. 10. The standard deviation of the scatter of the relative filter measurements is 3 to 8% for the C-arrays and 2 to 9% for the P-detectors.
|Figure 11: Constructing a power curve from calibration data. The names indicate the sources that have been used to calibrate the respective data points.|
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|Figure 12: Relative illumination of the array detectors C100 and C200 (illumination matrices) by FCS1 (valid after rev. 93). Pixel numbers and ISO coordinate axes are given.|
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FCS power curves for different pixels of the same C-array differ by constant factors. These factors result from both the nonuniform illumination of the detector arrays by the FCS and the putative spatial inhomogeneities in the filter. For each filter, these factors constitute the illumination matrix, , as shown in Fig. 12. Thus the FCS curve for each filter is split into a pixel-independent part, , and a pixel-dependent factor, , so that . The FCS curve is normalized such that , averaged over all pixels becomes 1, for a given filter. The resulting FCS power curves for FCS1 are shown in Figs. 13 and 14. Between revolution 93 and 94 the TRS2 of FCS1, which was routinely used with the long-wavelength detectors, abruptly increased its brightness by about a factor of 2. This change necessitated a recalibration of the corresponding filter bands and resulted in a second set of FCS calibration tables for P3, C100 and C200. Figure 14 shows the FCS power curves that apply to the period before revolution 94. The non-uniformity of the FCS illumination also affects the FCS signal measured in different apertures of the P-detectors. For a fixed heating power the FCS signal is not proportional to the area of the aperture. Strong deviations exist for apertures much larger or smaller than the standard apertures. The treatment of these cases is still poorly understood and is beyond the scope of this paper. For use with standard and very similar apertures, the in-band powers of the PHT-P power curves were divided by the area of the standard aperture of the measurement, and are expressed in units of W/mm2.
|Figure 13: FCS power curves of all filter bands. For P3, C100 and C200 we show the versions valid after the change of FCS1/TRS2 in rev. 94. The interpolated parts are indicated by solid or dashed lines with small squares at the boundaries, while the model extrapolations are indicated by dotted lines.|
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|Figure 14: FCS power curves of all filter bands of the long wavelength detectors before the change in FCS1/TRS2.|
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To obtain another quantitative representation, we considered the
deviations for individual filters and computed the maxima and
the rms. We also calculated the rms of the portions left
and right of the dotted lines in Figs. 15 and 16, to seek any differences between higher and
lower fluxes. Table 3 contains the values
applicable for most of the mission while Table 4
shows the values for the time before rev. 94. These figures and
tables give an overview of the accuracy expected for an
observation of a point source in the standard aperture,
including measurements of background, source, FCS and cold
FCS. Since the FCS straylight measurement can in most cases be
replaced by the dark signal, the validity of this
assessment extends to even a larger number of observations.
For multi-filter measurements, where an FCS measurement in only
a single filter is available to determine R (Eq. (9)),
this FCS calibration can be transferred to a sky measurement in
another filter of the same subsystem, because the ratios of
transmissions between filters have been determined
(see Sect. 6.2).
However, this transfer also accretes the uncertainty
of the factor
(10%) for any filter
different from the one that was actually used during the
|Figure 15: Calibration accuracy derived from calibration consistency. The diagrams show the ratio of the FCS power curves times the illumination matrices and the data points derived for the P-detectors, plotted against FCS in-band power. The residuals of all filters of a subsystem are combined within one plot. The scatter is a measure of the error budget for single observations. The vertical dashed line separates the low and high flux intervals.|
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These numbers serve as a guideline for the astronomer as to what accuracies to expect for a staring or raster observation that underwent a complete data reduction, involving all reduction steps and all necessary supporting measurements, i.e. background, FCS, FCS straylight. To assess the uncertainty of the FCS power curves alone, however, they only represent upper limits, since statistical uncertainties decrease with the number of data points used.
The uncertainties obtained are similar to or larger than the
accuracies quoted for the SEDs of the celestial standards.
This limits our ability to discriminate problematic
standards in comparison to others. A first analysis
is given by Schulz (2001).
|Figure 16: Calibration accuracy derived from calibration consistency. The diagrams show the ratio of the FCS power curves and the data points derived for the C-detectors, plotted against FCS in-band power. The residuals of all filters of a subsystem are combined within one plot. The scatter is a measure of the error budget for single observations. The vertical dashed line indicates the separation between low and high flux intervals.|
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The measurements on NGC 6543 were repeated every 5 weeks on average; the fainter sources were measured every 2 weeks. The planetary nebula was measured at the centre of each detector system, after performing a background measurement at a position about 5 away. The faint sources were observed either in nodding mode for the P-detectors or with a small raster of or for C100 and C200 respectively, to minimize the uncertainties due to background subtraction. We find the reproducibility of measurements on bright sources to be around 2-3%, except for P1 and P3 for which this is 6-7%. The fainter sources show larger scatter because uncertainties of source, background, FCS and FCS straylight contribute equally to the error, compared to the uncertainties only of source and FCS that contribute at higher fluxes. Photon noise and glitch noise also contribute relatively more at low fluxes. The reproducibility below 1 Jy ranges from 7 to 12%. A summary is given in Table 5.
We have described the photometric calibration of ISOPHOT using celestial standards. We used point sources either stars, planets or asteroids, to cover the full wavelength range. Standards at all flux levels were used to calibrate nonlinearities.
Because the detector responsivity varied with time, stable internal IR sources were calibrated against celestial standards and measurements of these references were included in each scientific observation. The reproducibility of results derived from the ratios of these signals was verified for high and low flux levels. We corrected for non-ideal effects like nonlinearities, imperfect optics and detector transients, and reduced their influence on the measurements to a minimum.
The photometric calibration of the internal sources initiated the definition of empirical FCS power curves and FCS illumination tables as well as the introduction of matrices of calibration factors, to account for pixel- and filter-dependent attenuations, which are probably caused by misalignments, diffraction and spatial gradients in filter transmissions. A major step in achieving a consistent picture was the correction for the flux dependence of the responsivity. A set of formulae describing the path from raw detector signals to flux densities was derived, which uses the detector responsivity to characterise the state of a detector pixel at the time of the observation.
To quantify the accuracies achievable by absolute photometry, a comparison was made of the derived calibration curves of the internal sources and the measured data. Depending mainly on flux level, the achievable uncertainties are around 10 to 20%, but can exceed 30% under exceptional conditions and at low flux levels. The filters of each subsystem were calibrated relative to each other with accuracies better than 10%.
The use of point sources and standard apertures for each filter for the aperture photometer defines a baseline within the parameter space, which serves as a reference for the calibration of further instrument modes like chopped measurements, multi-aperture measurements and mapping.
We would like to thank U. Grözinger for vital support. N. Lu and A. Wehrle are acknowledged for helpful comments on an early version of the manuscript. MC thanks the University of Florida for supporting his contributions to this work through subcontracts (under prime grants NAGW-4201 and NAG 5-3343) with VRI and, in later years, through subcontract UF99025 with Berkeley.