Next: 5.3 Point Sources Versus Up: 5. Photometric Calibration Previous: 5.1 Overview

5.2 Calibration Strategy

5.2.1 Calibration involving the FCS

An observation with PHT-P or PHT-C always includes a photometric reference measurement using one or both internal fine calibration sources (FCSs, see also Section 2.7) which are assumed to be stable throughout the mission. The stability assumption is confirmed by monitoring measurements against celestial standards throughout the ISO mission. The emission from an FCS can be changed by adjusting the FCS heating power. The FCS reference measurement assesses the actual detector responsivity at the time of the measurement.

The crucial photometric calibration as contained in the Cal-G files (see Section 5.7.1) is the relationship between heating power applied to the FCS as commanded by the AOT logic, and the resulting in-band power received by the detector. This is the basis of the derivation of the detector responsivity which fixes the relation between signal and flux for the detector at a given time. The detector responsivity enables transfer of calibration between filters of the same detector and/or between apertures.

Transfer of calibration is in practice necessary in case of multi-filter photometry (PHT03, PHT17/18/19, PHT22, PHT37/38/39) or multi-aperture photometry (PHT04) where a single FCS measurement is used to calibrate combinations of several filters and apertures.

The FCS calibration itself is obtained empirically by comparing the signal of a designated celestial calibration target and signals of several FCS measurements. Based on the known spectral energy distribution (SED) of the calibration target, the known spectral instrument transmissions and the relative spectral response of the detector, the in-band power is calculated for each FCS measurement. These in-band powers are then related to the FCS heating powers via a comparison of the detector signals.

Three types of astronomical targets - all being point sources - have been used as prime calibrators for the FCSs:

Stars
Stellar calibrators are most suitable for the 2.5 to 35 $\mu$ m wavelength range. The brighter ones can still be extrapolated reliably out to 300 $\mu$ m. A dedicated ground-based preparatory programme for ISO calibration standards was carried out to provide the ISO mission with the most consistent and most accurate set of calibration stars, see Jourdain de Muizon & Habing 1992, [19], van der Bliek et al. 1992, [54], and Hammersley et al. 1998, [14]. Another set of calibrated stellar spectral energy distributions was provided by M. Cohen from an independent calibration programme (Cohen et al. 1995, [7]). The $1\,\sigma$ confidence in the absolute calibration of the calibration stars is about 3% in the 2.5 to 35 $\mu$ m wavelength range and better than 5-6% at longer wavelengths up to 300 $\mu$ m. The model spectral energy distribution for Vega ( ${\alpha}$ Lyr) has been used to define the zero point for both programmes.
Asteroids
Asteroids are used to cover the 45 to 200 $\mu$ m wavelength range at intermediate flux levels between bright stars and planets. An extensive ground and airborne based calibration programme was carried out to quantify the fluxes and to monitor the light curves of the prime asteroids 1 Ceres, 2 Pallas, 4 Vesta, and 10 Hygiea, and other asteroids that have been observed with ISO. For the current calibration the absolute fluxes of the asteroids were derived using a thermophysical model (Müller & Lagerros 1998, [40]). The accuracy of the predicted asteroid fluxes is in general $\pm$ 10% in the wavelength range longward of 24 $\mu$ m (Müller & Lagerros 2002, [43])
Planets
The outer planets Neptune and Uranus were used for calibration at the longer wavelengths in the high flux density range up to 1000 Jy. The model SED's of Uranus at a given time in the mission were derived from the model by Griffin & Orton 1993, [11] modified to match the Voyager IRIS data (see `The ISO Handbook: Vol. III, LWS - The Long Wavelength Spectrometer', [12]). Comparison with independent model predictions for Uranus showed a consistency of better than 10% in the wavelength range between 30 and 240 $\mu$ m.

The stability of both FCSs was monitored against the bright secondary standard NGC6543 as well as against a set of fainter stars.

The ISOPHOT calibration strategy is presented in detail by Klaas et al. 2001, [28]. A full description of the ISOPHOT calibration involving the FCS can be found in Schulz et al. 2002, [53].

5.2.2 Non-linear detector responsivity

Ignoring transient effects (see Section 4.2.3), the detector responsivity, which is the proportionality between in-band power and signal, can only be considered as constant under the following conditions:

the time between the responsivity measurement and the sky measurement is short. Over longer time periods the responsivity is a function of orbital phase and shows a steady increase for detectors P3, C100, and C200 as a function of time after curing.
the signal of the FCS measurement is close to the signal of the sky measurement. From calibration observations of standard stars it was found that the detector responsivity depends on the in-band power falling onto the detector $R(P) \propto S(P)/P$ , where the photo-current or signal has a non-linear dependence on power.

Investigation of the validity of the second condition has shown that the signal non-linearity can be corrected under the assumption that the shape of the signal non-linearity function is independent of time. Since the responsivity is proportional to signal, the responsivity can be linearised by introducing a linearisation correction such that:

$\displaystyle S'$	$\textstyle =$	$\displaystyle H(S(P))~~~~~~~{\rm [V/s]}$	(5.1)
$\displaystyle S'/P$	$\textstyle =$	$\displaystyle constant~~~~~{\rm for~all}~S$	(5.2)

where is the linearised signal. Time dependencies, such as the responsivity dependency on orbital phase, are treated as random scatter. The correction function has been determined empirically from observations of calibration standards. It is found that is not only a function of detector, but also of filter. The determination of is not straightforward due to the fact that all measurements of calibration standards include a background power which is not well known. In addition, time dependent variations of the responsivity introduces a substantial scatter in the data. A detailed description is provided by Schulz 1999, [51].

In the photometric calibration of PHT-P and PHT-C, the signal linearisation is applied to the mean signal per chopper plateau immediately after subtraction of the dark signal. In the remainder of this chapter we assume that the signals have been linearised unless explicitly stated.

5.2.3 Signal correction for chopped observations with PHT-P and -C

The photometric accuracy of chopped observations is completely determined by losses in the difference signal between the `source+background' (`on') and `background' (`off') signals. This is caused by detector transients with time constants greater or equal than the chopper modulation time (see Section 4.2.3).

Due to these transients the determination of the difference signal is not trivial. A method was developed which includes the determination of a generic pattern for chopped measurements (see Section 7.5). The signal difference is determined from the generic pattern depending on the transient behaviour of a given detector - for example, in case of detector P3 the signal difference is determined from the maximum of the on-signals and the minimum of the off-signals of the pattern, and in case of P1 and P2 the median signals of the on- and off-plateaux are used.

This analysis of chopped measurements applied to a sample of calibration standards yielded an empirical signal correction function $\zeta$ which corrects a given (linearised) difference signal $S_{src}$ :

$\begin{displaymath} S_{src}' = \zeta(S_{src},\,det,\,t_{plat})~~~~~~{\rm [V/s]}, \end{displaymath}$

(5.3)

where $S_{src}'$ is the signal corrected for losses due to the chopper modulation, is the detector used, and $t_{plat}$ is the chopper plateau time (in s). The dependency on and $t_{plat}$ is required to relate the chopper modulation frequency with the detector specific transient time scales.

The AOT PHT32 incorporated the chopper sawtooth mode to obtain oversampled maps with the PHT-C detector arrays (Sections 3.5 and 3.10.2). The movement of the chopper in PHT32 also caused signal transients. These cannot be corrected with $\zeta$ because there is no fixed on-source and background position as in the pointed chopped observations. Therefore no corrections are made in OLP for signal losses due to detector transients in PHT32.

The function $\zeta$ implicitly includes the correction for `chopper vignetting/offset' (Section 4.5.3) with PHT-C because the choice of available chopper throws with PHT-C was limited. For PHT-P the chopper offsets are small. In the case of PHT32 the signals per chopper plateau are explicitly corrected for chopper offsets:

$\begin{displaymath} S' = c_{chop}(f,i,\theta){\times}S~~~~~~~[{\rm V/s}], \end{displaymath}$

(5.4)

where is the filter, is the detector pixel and $\theta$ the chopper deflection with respect to the central field of view. The input signal is the total sky signal.

5.2.4 Absolute calibration of FCS

The FCS calibration against celestial standards has been determined for all 25 PHT-P and PHT-C filter bands over the full range of possible astronomical flux densities.

Fluxes were measured of prime calibrators in each available filter band spanning 3-5 decades of flux range per filter band. For a given calibration target, the following measurements were obtained in a given filter:

measurement of the FCS for which zero heating power is applied (also referred to as `cold FCS measurement' or `FCS straylight measurement'),
measurement of one or more background positions,
measurement of the calibration target, and
measurement of one to several FCS illumination levels.

The cold FCS measurement provided the zero FCS signal level. All measurements of the celestial calibrators included an uncalibrated amount of background emission which had to be subtracted. The background level was determined by performing one or more separate background pointings in the same aperture. For the PHT-C arrays, small raster maps were often used to obtain the background level during the same measurement. In the case of multiple FCS measurements, the FCS illumination levels were tuned such that their in-band powers span a range of about one decade around the power expected from the calibration target plus background emission. It is assumed that the responsivity does not depend on signal in the measured power range (system linearity).

Using the total optical transmission of the instrument and detector chain (to determine and ) and the source SED, the in-band power from the calibration source on the detector was computed from Equation 5.10. By relating the in-band power of the calibration target to its corresponding signal, the FCS signals were converted to in-band powers. These are derived for a measurement in a given aperture in case of PHT-P or for a pixel in case of PHT-C, according to:

$\begin{displaymath} P_{fcs}^f(h) = P^f_{cal}{\cdot}\frac{(S_{fcs}^{f}-S^f_{str})}{S^f_{cal}} ~~~~~~{\rm [W]}, \end{displaymath}$

(5.5)

where

$P^f_{fcs}(h)$ : the in-band power in W for filter and FCS heating power in mW;
$P_{cal}^f$ : in-band power in W of the celestial calibrator;
$S_{cal}^f$ : measured signal in V/s of the prime calibrator after subtraction of the background signal;
$S_{fcs}^{f}$ : measured FCS signal in V/s;
$S_{str}^f$ : measured dark FCS signal (straylight) in V/s.

The FCS power calibration tables give the in-band power as a function of the commanded FCS heating power. It is assumed that the difference signal between calibration source and background scales linearly with the difference signal from the FCS measurements. Consequently, the accuracy of the absolute calibration of the background level depends on the linearity of the detector from zero to the background level and on the suppression of parasitic flux, such as straylight. Uncertainties become higher for flux levels close to the astronomical background because of the absence of reliable celestial calibrators that are faint.

For PHT-P the tables are normalised to the aperture area, and for PHT-C the tables refer to the average array. The power falling onto an individual pixel is derived by for inhomogeneous FCS illumination (see Section 5.2.5).

The FCS calibration makes the observation independent of long term responsivity drifts. However, on measurement time scales of order of 32-2048 s, detector transients (Section 4.2.3) due to flux changes occur and have to be corrected for. For the determination of the overall FCS calibration drift corrections have been applied to estimate the signal level of the asymptotic limit.

5.2.5 Photometry with PHT-P and PHT-C

The FCS measurement together with the FCS power calibration tables (see Section 14.12) are used to determine the detector responsivity. In the following derivations a separation will be made between single detector, multi-aperture, and multi-filter subsystems PHT-P and the multi-filter detector arrays PHT-C.

For a given PHT-P FCS measurement taken with detector , filter , and aperture , the responsivity $R_{det}$ is derived from:

$\begin{displaymath} R_{det} = \frac{S_{fcs}^{f'}(a')C^{det}_{int}} {P^{f'}_{fcs}(h)A(a'){\alpha}^{f'}(a'){\chi}^{f'}} ~~~{\rm [A/W]}, \end{displaymath}$

(5.6)

For a given PHT-C FCS measurement taken with detector and filter , and for pixel , the responsivity $R_{det}$ is derived from:

$\begin{displaymath} R_{det}(i) = \frac{S_{fcs}^{f'}(i)C^{det}_{int}} {P^{f'}_{fcs}(h){\Gamma}^{f'}(i){\chi}^{f'}(i)} ~~~{\rm [A/W]}, \end{displaymath}$

(5.7)

where the prime () indicates the FCS measurement configuration and

$S_{fcs}^{f'}$ : the signal in V/s from the FCS measurement in filter after subtraction of dark signal or FCS straylight (the latter one is only possible with PHT05 and PHT25);
$C^{det}_{int}$ : the integration capacity in A/(V/s) (=farad) associated with detector (see Figure 2.5 and Table 14.5);
$P^{f'}_{fcs}(h)$ : the in-band power in W for filter and FCS heating power in mW as obtained from the FCS power table, for PHT-P the power is per mm, for PHT-C per pixel;
: the area of aperture in mm;
${\alpha}^{f'}(a')$ : dimensionless correction for inhomogeneous illumination of PHT-P aperture by the FCS in filter band ;
${\Gamma}^{f'}(i)$ : dimensionless correction for PHT-C pixel due to inhomogeneous illumination of the array by the FCS;
${\chi}^{f'}(i)$ : dimensionless correction for non-uniform filter-to-filter responsivity of pixel of PHT-C or PHT-P where is dropped (single pixel).

A few observations have no accompanying FCS measurement which can be used for the photometric calibration due to e.g. a failure during the observation or signal saturation. In those cases a default responsivity is applied which is a fixed value of $R_{det}(i)$ corrected for a predictable variation with orbital phase at the time of the observation.

Once the detector responsivity is known, measurements of a celestial source with filters of the same detector in a multi-filter photometry observation can be calibrated. For a given PHT-P measurement, the in-band power $P_{src}^{f}(a)$ for filter and aperture is:

$\begin{displaymath} P_{src}^{f}(a) = \frac{S_{src}(a)C^{det}_{int}} {R_{det}{\chi}^{f}}~~~~~{\rm [W]}, \end{displaymath}$

(5.8)

For a given PHT-C measurement, the in-band power $P_{src}^{f}(i)$ for filter and for pixel is:

$\begin{displaymath} P_{src}^{f}(i) = \frac{S_{src}(i)C^{det}_{int}} {R_{det}{\chi}^{f}(i)}~~~~~{\rm [W]}, \end{displaymath}$

(5.9)

where $S_{src}$ is the linearised source signal in V/s after dark signal subtraction. Note that the correction ${\chi}^f$ cancels out for a source measurement taken in the same filter band as for the FCS measurement, i.e. when . To convert from in band power to point source flux density on the sky, the following relation is used for PHT-P:

$\begin{displaymath} \frac{F_{\nu}^{f}}{k_f} = \frac{P_{src}^{f}(a)}{C1^f f_{PSF}^{f}(a)}~~~~~{\rm [Jy]}, \end{displaymath}$

(5.10)

and for PHT-C^5.1:

$\begin{displaymath} \frac{F_{\nu}^{f}}{k_f} = \frac{\sum_{i}P_{src}^{f}(i)}{C1^f f_{PSF}^{f}}~~~~~{\rm [Jy]}, \end{displaymath}$

(5.11)

where the summation is over all detector pixels and where:

$F_{\nu}^{f}$ : the source flux density in Jy for an assumed spectral energy distribution ${\nu}f_{\nu}$ = constant;
: the colour correction, for the spectrum of the observed celestial source;
: constant in ${\rm m^2Hz}$ describing^5.2 the transmission of the whole optical path for a given filter and assuming a ${\nu}F_{\nu}=$ constant reference source spectrum. The effective size of the primary mirror, reflection losses, filter transmission, spectral response of the detector, etc. are included;
$f_{PSF}^{f}$ : (dimensionless) fraction of the ISO point spread function entering the field aperture in case of PHT-P or falling onto the entire array in case of PHT-C for filter (details can be found in Salama et al. 2001, [47]).

$F_{\nu}^{f}$ is also referred to as the flux density at the reference wavelength of the filter band and is the value of the flux density computed by OLP. It is required that the constant is identical to the one used in the determination of the FCS power tables.

5.2.6 Spectro-photometry with PHT-S

At the shortest wavelengths below 10 $\mu$ m very high FCS heating powers are required to obtain a detectable in-band power. Due to the wavelength coverage of PHT-S no single FCS power can properly illuminate both SS and SL arrays without causing saturation in the SL array (a 300 K blackbody spans 5 orders of magnitude over the wavelength range 2.5-12 $\mu$ m). Therefore and due to the fact that the PHT-S detectors turned out to be very stable, the PHT-S measurements (PHT40) are not accompanied by any FCS measurement. Without any FCS measurement, the photometric calibration must be different from that of PHT-P or PHT-C.

Monitoring of the PHT-S responsivity at the beginning and end of each revolution showed that the daily responsivity variation is on average at most 10% depending on the space weather conditions, see Section 4.2.5. Shorter time scale responsivity variations are mainly due to transient behaviour of the detector responsivity following an illumination change. The PHT-S transient behaviour is corrected by applying the calibration method outlined in the remainder of this section.

The PHT-S measurement sequence as provided by the PHT40 AOT has always the same structure: an initial dark signal measurement of 32s followed by a sky measurement of s ( $5\,<~n\,<\,12$ ) either in staring or chopped mode. This measurement sequence causes the detector to be in a well defined (dark) state before the illumination by the sky starts.

Due to the long term responsivity stability, the photometric accuracy of the PHT-S observations is mainly determined by the uncertainties caused by the responsivity transients (Schulz 1999, [52]). The photometric calibration of PHT-S is therefore based on the following assumptions:

the transient response of an individual pixel depends on the illumination level.
the transient response for a given illumination level is reproducible as long as the flux history is similar.

These assumptions are to a large extent fulfilled by the long term responsivity stability and the AOT PHT40 design. The only disturbing parameter is the operational temperature of the detector which affects the shape of the transient curves. It is, however, found that the temperature of the detector was quite stable (2.7-3.1K) throughout the routine phase, i.e. revolution 78 and later. During the Performance Verification Phase the PHT-S detector temperature was lower (2.4K) and the calibration methods outlined below are less accurate for observations obtained in that period.

5.2.6.1 PHT-S staring observations

By observing a large number of calibration stars covering a sufficient range in flux densities and integration time for each PHT-S pixel, one can construct an empirical transformation which directly relates the signal transient of a given pixel at time to a flux density $F^t_{\nu}$ :

$\displaystyle F^t_{\nu}(i)$	$\textstyle =$	$\displaystyle \varphi(i,\,S(i,t),\,t)~~~~~~~~~~{\rm [Jy]},$	(5.12)
$\displaystyle w(i,t)$	$\textstyle =$	$\displaystyle g(i,\,S(i,t),\,t),$	(5.13)

where $\varphi$ is the transformation and a function which gives the weight of the correction. We give the flux density a superscript to indicate that the flux itself is supposed to be constant and has no time dependence. The time resolution is determined by the reset interval length, i.e. the time between two consecutive signals. This calibration method is also called `dynamic calibration', because the spectral response function is adapted to the total brightness and transient behaviour of each pixel.

The weighting function has been introduced to account for the fact that the transient correction is usually large on short time scales and close to unity for very long integration times. It is assumed that the transient signal reaches the long term detector response asymptotically in time. The final flux $F_{\nu}(i)$ is the weighted average of the transformed signals $F^t_{\nu}(i)$ along a given measurement.

5.2.6.2 PHT-S chopped observations

PHT-S observations in chopped mode have the property that the background emission (off position) is usually the zodiacal light which is faint at the PHT-SS wavelengths, and less than 1 Jy at the longest SL wavelengths. In addition, the available range in chopper frequencies at low flux densities is small.

From the analysis of a large range of chopped PHT-S observations of calibration stars it is found that an accurate photometry can be accomplished by assuming two calibration components. One component is the average spectral response function which is the zero order approximation for the conversion from signal in V/s to flux density in Jy for each pixel . The second component involves the correction for flux dependent chopped signal losses due to signal transients and depends on the difference signal between the on-source signal and the background off signal:

$\begin{displaymath} C^{c,p}(i) = C^{c,p}_{ave}(i) (A_0(i)+ A_1(i){\rm log_{10}}(S_{on}(i)-S_{off}))~~~~~~ {\rm [V\,s^{-1}Jy^{-1}]}, \end{displaymath}$

(5.14)

where

$C^{c,p}(i)$ in (V/s)/Jy is the spectral response of pixel for a point source corrected for chopped signal loss;
$C^{c,p}_{ave}(i)$ in (V/s)/Jy is the PHT-S average spectral response function for a point source;
and are first order corrections on spectral response function for chopped measurements depending on the signal difference;
$S_{on}(i)$ and $S_{off}(i)$ (in V/s) are the on- and off-source signals.

The mean spectral response function was obtained by taking the average of the spectral response functions derived from a number of faint calibration stars.

5.2.6.3 PHT-S maps

PHT-S maps are obtained by observing a regular grid (raster) by moving the spacecraft from one grid point to the other. Although the measurement sequence in PHT40 mapping mode is the same as in staring or chopped mode (a dark measurement followed by the sky) the assumption of a flux history similar to that as for the calibration stars is not met for all raster points in a map except the first one. The photometric calibration is poorer than for staring or chopped mode observations because only a `static' response function not correcting for responsivity transients can be applied. The photometric calibration on each raster point is performed by direct conversion of the signal to a flux $F_{\nu}(i,k)$ :

$\begin{displaymath} F_{\nu}(i,k) = \frac{S(i,k)}{C^{s,p}_{ave}(i)}~~~~~~{\rm [V\,s^{-1}Jy^{-1}]}, \end{displaymath}$

(5.15)

where $C^{s,p}_{ave}(i)$ the average spectral response function for PHT-S staring observations derived from 40 observations of 4 different standard stars with different brightness.

Next: 5.3 Point Sources Versus Up: 5. Photometric Calibration Previous: 5.1 Overview

ISO Handbook Volume IV (PHT), Version 2.0.1, SAI/1999-069/Dc