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8.4 Photometry

A large fraction of this section is dedicated to the processing of chopped measurements. Although the chopped observing mode is not scientifically validated for OLP Version 7, it is not expected that the algorithms for determining the background subtracted signal will change. A preliminary correction for signal losses in chopped mode with respect to staring mode has been implemented in this version of the software. This correction is not scientifically validated and is likely to change in a future version.

In this section the extraction of (point-)source emission from source+background emission in chopped and staring measurements (whenever possible) is described in detail. Subsequently the conversion from power on the detector in units of W to flux density in Jy or surface brightness in MJy/sr is presented. Finally, the writing of the resulting data to product files is briefly described.

8.4.1 Sorting out chopper plateaux

Operation: Sorts out the source and background records for each chopper cycle; possible data gaps from telemetry drops are handled.

In case the observation consists of one or more measurements, each valid chopper cycle should be determined in order to be able to perform the background subtraction.

The size of a chopper cycle depends on the chopper mode used:

1.: staring mode:
cycle = 1 record
2.: rectangular chopping mode:
cycle = 2 consecutive records
record 1 gives the background and record 2 gives [source+background] power.
3.: sawtooth chopping mode:
cycle = ${\rm 2{\times}NSTEP+1}$ consecutive records
NSTEP is the number of chopper steps; for NSTEP=1 record 1 and 3 give the respective first and second background powers, record 2 gives the [source+background] power.
4.: triangular chopping mode:
cycle = ${\rm 4{\times}NSTEP}$ consecutive records
NSTEP is the number of chopper steps; for NSTEP=1 record 1 and 3 give the respective first and second background powers, record 2 and 4 give the [source+background] power.

In case of a chopped measurement, the first chopper step is always on the background and in negative spacecraft Y-direction (cf. section 3.6). For all available AOTs except PHT32, NSTEP is set to 1 in case sawtooth or triangular chopping mode was selected. The PHT-S dark measurement is always treated as a staring measurement.

The presence of possible data gaps due to telemetry drop-outs is determined by checking whether the time difference between the current and the next record is according to the chopper dwell time. If this is not the case, the current and following records are skipped until the beginning of a new cycle is found. The start of a new cycle is inferred from the chopper step number. As each chopper cycle starts with the maximum negative offset, the step number should have the value:

1.: -1 for rectangular chopping mode
2.: -NSTEP for sawtooth or triangular chopping mode

As soon as a full valid cycle has been encountered, the data are passed to perform the background subtraction. In case of staring mode, this step is skipped and a flux calculation is performed.

8.4.2 Separation of source and background in-band power

Operation: Perform the background subtraction for a given chopper cycle and chopper mode.

Caveat: OLP Version 7 include weighting factors (cf. sections 8.4.2.3 and 8.4.2.2) which have not been scaled by the factor 2 in the average. This does not invalidate the weighting, but in the calculation of the $\sigma$ of the mean in section 8.4.3 the value of $\sigma$ will be too low by a factor 2. The observer should multiply the final uncertainty by 2 to obtain the correct values. In a future version of the software this omission will be corrected.

For a given chopper cycle the background in-band power is subtracted from the [source+background] in-band power to obtain the source in-band power. This operation is repeated until the end of a measurement is encountered. Weighting factors are derived from the uncertainties which are used for the averaging of all chopper cycles at the end of a measurement.

In the following sub-sections we describe the background subtraction method for the different chopper modes. Note that each cycle in triangular chopping mode consists of 4 plateaux referring to 2 [source+background] and 2 different background positions. In sawtooth mode there are 3 plateaux: 1 [source+background] and two different background positions. The following symbols are used:

${\rm P_{i}(s+b)_{1},~w_{i}(s+b)_{1}}$ : power and weight for first [source+background] plateau for chopper cycle i;
${\rm P_{i}(s+b)_{2},~w_{i}(s+b)_{2}}$ : power and weight for second [source+background] plateau (triangular mode only);
${\rm P_{i}(b1),~w_{i}(b1)}$ : power and weight for the background at the first reference positiont;
${\rm P_{i}(b2),~w_{i}(b2)}$ : power and weight for the background at the second reference position;
${\rm P_{i}(s+b),~w_{i}(s+b)}$ : average power and weight from source plus background plateaux;
${\rm P_{i}(s),~w_{i}(s)}$ : source power and weight;
${\rm P_{i}(b),~w_{i}(b)}$ : background power and weight.

All in-band powers are given in Watts, the weights are dimensionless.

8.4.2.1 Rectangular mode

Each cycle contains only 1 [source+background] plateau and 1 reference background position. For chopper cycle i:

$\begin{eqnarray*} {\rm P_{i}(b)} = & {\rm P_{i}(b1)}\\ {\rm P_{i}(s)} = & {\rm P_{i}(s+b)_1 - P_{i}(b1)}\\ \end{eqnarray*}$

The weighting factor is determined from the in-band power uncertainties:

$\begin{displaymath} w_{i}(s+b) = \frac{1}{\sigma_{i}^{2}(s+b)}\end{displaymath}$

$\begin{displaymath} w_{i}(b) = \frac{1}{\sigma_{i}^{2}(b)}\end{displaymath}$

$\begin{displaymath} w_{i}(s) = \frac{1} {\lbrack \sigma_{i}^{2}(s+b) + \sigma_{i}^{2}(b) \rbrack}\end{displaymath}$

where $\sigma_{i}(s+b)$ is the uncertainty in power for the measurement on [source+background], etc.

8.4.2.2 Sawtooth mode

Each chopper cycle contains 1 [source+background] chopper plateau and 2 reference positions. For chopper cycle i:

$\begin{displaymath} P_{i}(b) = \frac {P_{i}(b1) + P_{i}(b2)}{2}\end{displaymath}$

P_i(s) = P_i(s+b) - P_i(b).

With weighting factors:

$\begin{displaymath} w_{i}(s+b) = \frac{1}{\sigma_{i}^{2}(s+b)}\end{displaymath}$

$\begin{displaymath} w_{i}(b1) = \frac{1}{\sigma_{i}^{2}(b1)}\end{displaymath}$

$\begin{displaymath} w_{i}(b2) = \frac{1}{\sigma_{i}^{2}(b2)}\end{displaymath}$

$\begin{displaymath} w_{i}(b) = \frac{\gamma} {\lbrack\sigma_{i}^{2}(b1) + \sigma_{i}^{2}(b2)\rbrack}\end{displaymath}$

$\begin{displaymath} w_{i}(s) = \frac{1} {\lbrack \sigma_{i}^{2}(s+b) + (\sigma_{i}^{2}(b1) + \sigma_{i}^{2}(b2))/\gamma \rbrack}\end{displaymath}$

where $\gamma$ = 1 in OLP7, but the correct value should be 4; $\sigma_{i}(s+b)$ is the uncertainty in power for the measurement on [source+background], etc.

8.4.2.3 Triangular mode

Each chopper cycle contains 2 [source+background] chopper plateaux and 2 reference positions. For chopper cycle i:

For chopper cycle i:

$\begin{displaymath} P_{i}(s+b) = \frac {P_{i}(s+b)_{1} + P_{i}(s+b)_{2}}{2}\end{displaymath}$

$\begin{displaymath} P_{i}(b) = \frac {P_{i}(b1) + P_{i}(b2)}{2}\end{displaymath}$

P_i(s) = P_i(s+b) - P_i(b)

A weighting factor is also determined from the power uncertainties:

$\begin{displaymath} w_{i}(s+b)_{1} = \frac{1}{\sigma_{i}^{2}(s+b)_{1}}\end{displaymath}$

$\begin{displaymath} w_{i}(s+b)_{2} = \frac{1}{\sigma_{i}^{2}(s+b)_{2}}\end{displaymath}$

$\begin{displaymath} w_{i}(b1) = \frac{1}{\sigma_{i}^{2}(b1)}\end{displaymath}$

$\begin{displaymath} w_{i}(b2) = \frac{1}{\sigma_{i}^{2}(b2)}\end{displaymath}$

$\begin{displaymath} w_{i}(s+b) = \frac{\gamma} {\lbrack \sigma_{i}^{2}(s+b)_{1} + \sigma_{i}^{2}(s+b)_{2}\rbrack}\end{displaymath}$

$\begin{displaymath} w_{i}(b) = \frac{\gamma} {\lbrack\sigma_{i}^{2}(b1) + \sigma_{i}^{2}(b2)\rbrack}\end{displaymath}$

$\begin{displaymath} w_{i}(s) = \frac{\gamma}{\lbrack\sigma_{i}^{2}(s+b)_{1} + \... ...}(s+b)_{2} + \sigma_{i}^{2}(b1) + \sigma_{i}^{2}(b2)\rbrack}.\end{displaymath}$

$\gamma$ has the same meaning as in the previous subsection.

8.4.3 Determine the in-band powers averaged over a measurement

Operation: determine the average source and background powers of all chopper cycles in a measurement.

For all chopper cycles in a measurement, the weighted average is computed from the parameters per chopper cycle. For a given set of powers ${\rm P_i(X)}$ with weights $\rm w_{i}$ obtained over a measurement, the weighted mean is derived from:

$\begin{displaymath} {\rm P(X) = \frac{\sum_{i=1}^{N} w_{i}(X)P_{i}(X)}{\sum_{i=1}^{N} w_{i}(X)}.}\end{displaymath}$ (8.3)

P(X) can be either the power of the source or background. The corresponding uncertainty is calculated from the weights:

$\begin{displaymath} {\rm \sigma(P(X)) = \sqrt{\frac{1}{\sum_{i=1}^{N} w_{i}(X)}}}.\end{displaymath}$ (8.4)

8.4.3.1 Rectangular mode

In rectangular mode the following mean powers are derived for each pixel:

${\rm P(s)\pm\sigma(P(s)) }$
${\rm P(b)\pm\sigma(P(b)) }$
${\rm P(s+b)\pm\sigma(P(s+b)) }$

8.4.3.2 Sawtooth and Triangular mode

In sawtooth and triangular mode the following mean powers are derived for each pixel:

${\rm P(s)\pm\sigma(P(s)) }$
${\rm P(b)\pm\sigma(P(b)) }$
${\rm P(s+b)\pm\sigma(P(s+b)) }$
${\rm P(b1)\pm\sigma(P(b1)) }$
${\rm P(b2)\pm\sigma(P(b2)) }$

8.4.4 Staring mode

Operation: Derive the average target powers in staring mode.

This is a null operation as there is only one record per measurement.

8.4.5 Determine the mean off-source background

Operation: Compute the background averaged over all pixels in the array for a chopper reference position.

A weighted mean of all the pixels is found - if there are n of these where

$\begin{displaymath} w_{j} = \frac{1}{\sigma^2(P_{j}(b))}\end{displaymath}$

then

$\begin{displaymath} P_{back} = \frac{\sum_{j=1}^{n} w_{j}P_{j}(b)}{\sum_{j=1}^{n} w_{j}}\end{displaymath}$

(8.5)

$\begin{displaymath} \sigma(P_{back}) = \sqrt{\frac{1}{\sum_{j=1}^{n} w_{j}}},\end{displaymath}$ (8.6)

where j refers to the pixels in an array. For sawtooth and triangular chopped modes, this is done at both background positions.

8.4.6 Empirical correction for signal losses in chopped measurements

Operation: Correct the background subtracted source in-band power for possible signal losses due to the application of chopped mode.

In Derive_SPD, see section 7.3.9, the empirical correction factors per detector pixel were determined for signal losses in chopped mode. The correction factors are transferred to Derive_AAR processing via the SPD product header (section 8.3.1). The correction can only be applied as soon as the difference signal between [source+background] and background is known which is only computed in Derive_AAR. Since the in-band power is proportional to signal the correction can be carried out at the present processing stage.

In case of the C100 array, it has been observed that the chopped signal of pixel 6 responds completely differently compared to the other pixels in the array. It is therefore decided for OLP Version 7 to remove the observed background subtracted in-band power and to replace this by the average of the in-band powers of the equivalent pixels - in terms of illumination - on the array (cf Table 3.1):

P(s, pixel 6) = (P(s, pixel 2)+P(s, pixel 4)+P(s, pixel 8))/3,
(8.7)

where s refer to the source flux after background subtraction. Subsequently, to correct for signal losses, for a given pixel j: >

$\displaystyle P^c(s,~j) =$	$\textstyle P(s, pixel~j)/C_{chop}(j)$		(8.8)
$\displaystyle \sigma(P^c(s,~j)) =$	$\textstyle \sigma(P(s, pixel~j))/C_{chop}(j)$		(8.9)

where P^c refer to the corrected in-band power and C_chop(j) is the signal loss correction for pixel j.

8.4.7 Summation of the in-band powers of all detector pixels

Operation: Derive the sum of the in-band powers of all pixels for a given PHT-C array.

In case the observer requested the measurement of a point source flux with the PHT-C detector arrays, the total [source+background] power as well as the background power on the array is determined by summing the respective powers over all pixels.

$\begin{displaymath} {\rm P(s) = \sum_{all~pixels} {P_{pixel}(s)}},\end{displaymath}$ (8.10)

and for the [source+background] and background power:

$\displaystyle P(s+b) =$	$\textstyle \sum_{all~pixels} {P_{pixel}(s+b)},$		(8.11)
$\displaystyle P(b) =$	$\textstyle \sum_{all~pixels} {P_{pixel}(b)}.$		(8.12)

The uncertainties are computed according to:

$\begin{displaymath} {\rm {\sigma}(P(s)) = \sqrt{ \sum_{all~pixels} \sigma^2(P_{pixel}(s))}} .\end{displaymath}$ (8.13)

The relations for ${\rm {\sigma}(P(s+b))}$ and ${\rm {\sigma}(P(b))}$ are similar.

8.4.8 Fit two-dimensional Gaussian to point sources in the C-arrays

Operation: In case the observer has requested a point source measurement, a 2 dimensional Gaussian function is fitted to the intensity pattern on the array. This processing is done in addition to the sum of all pixel in-band powers (section 8.4.7).

Caveat: This method of providing point source photometry is not scientifically validated. In particular in the case of faint sources and noisy data, the derived fluxes and uncertainties are not reliable.

To secure a converging fit, an interpolation is performed whenever there are undefined in-band powers for some pixels. The fitting of the 2 dimensional Gaussian itself is performed using standard iterative fitting routines provided by the NAG mathematical routines library (routines E04FDF and E04YCF).

The following parameters are obtained:

${\rm P_{gauss}(s), \sigma(P_{gauss}(s))}$ : peak power value plus associated error (indicated with d in section 8.4.8.2);
${\rm P_{gauss}(b), \sigma(P_{gauss}(b))}$ : background power value plus error (indicated with c in section 8.4.8.2);
${\rm \alpha, \sigma(\alpha)}$ : x offset of peak in arcsec plus error;
${\rm \beta, \sigma(\beta)}$ : y offset of peak in arcsec plus error;
${\rm g_{fit} }$ : goodness of fit
fit status, if status=0, fit is successful.

Details of the procedure are given in the next subsections (sections 8.4.8.1 and 8.4.8.2).

8.4.8.1 Interpolate missing pixels

The fitting routine described in section 8.4.8.2 requires only valid pixel intensities on the detector array. Interpolation is necessary in case there are `bad' data pixels. The observed in-band power on a pixel is flagged `bad' if there is no valid chopper plateau cycle during a given measurement.

A check is performed to determine whether there is a sufficient number of good pixels for interpolation. For C200 one pixel is allowed to be missing. For C100 the criteria are (1) the presence of the centre pixel (pixel 5) where the source is expected to be and (2) there must be at least 2 good pixels on any side of the array. Criterion (2) is imposed to avoid interpolation using an already interpolated value.

The rules for interpolation are

C200:  a b    :   a = b + c - d
       c d

C100:  a b c  :   b = (a + c)/2
       d e f
       g h i      a = b + d - (c + g)/2

Note that there is a rotational symmetry about each side; only one orientation is given. The accuracy of the method depends on

the location of the source in the array,
the presence of any extended features, and
the uniformity of the pixel response.

8.4.8.2 Determination of Gaussian parameters

The height of the source peak, its position, and the background level is obtained by fitting a Gaussion function to the data. The accuracy of this method depends on the correctness of the assumption of a Gaussian on top of a constant background. The in-band power dstribution is considered as a 2 dimensional array:

$\begin{displaymath} g(x,y) = c + de^{\frac{-z^{2}}{2}}\end{displaymath}$

where

$\begin{displaymath} z^{2} = (x-\alpha)^{2} +(y-\beta)^{2}\end{displaymath}$

and

${\alpha}$ is the x-axis offset of the peak from the centre of the array.
$\beta$ is the y-axis offset of the peak from the centre of the array.
c is a constant background.
d is the (background subtracted) height of the source peak.

The x and y axes are the first (along spacecraft Z-axis) and second (along spacecraft Y-axis) dimension of the pixel array, respectively, with origin at the centre of the array. A chi-squared ``goodness of fit'' function is defined as

$\begin{displaymath} \chi^{2} = \sum_{x,y} (P_{x,y}(s+b) - g(x,y))^{2}\end{displaymath}$

The best fit can be found by determining the minimum of $\chi^{2}$ .

For C100, estimates of the uncertainties on the parameters can be derived from the Jacobian of the function at the solution. Detailed discussion of the method is beyond the scope of this document; the NAG algorithm E04YCF is used. The nominal uncertainty of the fit is

$\begin{displaymath} s = \sqrt{\frac{\chi^{2}}{\delta}}, \end{displaymath}$

where $\delta$ is the number of degrees of freedom which is determined by the number of pixels n, the number of parameters (4), and the number of interpolations $\iota$ performed on C100 (section 8.4.8.1):

$\begin{displaymath} \delta = n - 4 - \iota \end{displaymath}$

Since the position of the peak $(\alpha,\beta)$ is irrelevant of the size of it and the background it is located, $(\alpha,\beta)$ and (c,d) are largely independent of each other. Thus adding 2 to the degrees of freedom is justified. This argument implies that the uncertainties for C100 may be overestimated. For C200 it is assumed that:

$\begin{displaymath} \delta = 2 \end{displaymath}$

The variances are calculated from:

$\begin{displaymath} \sigma_{\alpha}^{2} = \frac {\chi^{2}} {(\delta+2) \cdot m_{pix}^{2}} \end{displaymath}$

$\begin{displaymath} \sigma_{\beta}^{2} = \frac {\chi^{2}} {(\delta+2) \cdot m_{pix}^{2}} \end{displaymath}$

$\begin{displaymath} \sigma^{2}_{c} = \frac {\chi^{2}} {(\delta+2)} \end{displaymath}$

$\begin{displaymath} \sigma^{2}_{d} = \frac {\chi^{2}} {(\delta+2)} \end{displaymath}$

where m_pix is the mean pixel value used to scale into the correct units:

$\begin{displaymath} m_{pix}= \frac{\sum_{pixels} P_{pix}}{N_{pix} \cdot (P^{max}-P^{min})}\end{displaymath}$

The uncertainty of the fit is estimated as

$\begin{displaymath} s' = \sqrt{\frac{\chi^{2}}{\delta+2}} \end{displaymath}$

Ancillary data needed

Estimates of reasonable boundariess for $\alpha, \beta, c, d$ , these are obtained by considering the residuals of the fit during the iteration.

8.4.9 Convert in-band power to point source flux density

Operation: Convert the mean in-band power on a detector to a monochromatic flux density (Jy) assuming a ${\nu}^{-1}$ or - equivalently - constant ${\rm {\nu}F_{\nu}}$ spectral energy distribution.

Caveat: The uncertainty in the PHT-S flux densities contains a systematic term coming from the uncertainty in the spectral response function (see eqn. 8.17 and 8.21), this term should have been removed.

The PHT-S subsystem is not calibrated by performing an FCS measurement within the AOT. Therefore the PHT-S signals are directly converted to flux density assuming a fixed responsivity for each PHT-S pixel and a fixed spectral response function. This requires a flux density computation with is different from all other AOTs. The distinction is made in the following subsections.

Ancillary data needed

power calibration factors C1 for all filters read from Cal G file PFLUXCONV, see sections 5.2.3 and 13.17.1
central wavelengths $\lambda_{pixel}$ for all PHT-P and C bands also stored in PFLUXCONV, see section 13.17.1
point spread function factors which are stored in Cal G file PPSF, see section 13.15, note: for the C arrays the fraction of the psf on the entire array;
PHT-S spectral response function i.e. signal calibration factors C_phts,pixel plu uncertainties for all PHT-S pixels. The values are read from the SPD product header, for description of contents see section 13.16.1;
PHT-S central wavelengths $\lambda_{pixel}$ for all PHT-S pixels, these are also read from the SPD product header, see also section 13.16.1.

8.4.9.1 All AOTs except PHT40

The monochromatic flux density ${\rm F_{\nu}}$ in Jy is derived as follows (cf eqn. 5.6):

$\begin{displaymath} {\rm F_{\nu}(\lambda_{c}) = 10^{26} \frac{P}{C1{\cdot}f_{psf}(\lambda_{c},aperture)}~~~~~~~~~~~~(Jy)}\end{displaymath}$ (8.14)

with uncertainty

$\begin{displaymath} {\rm {\Delta}F_{\nu}(\lambda_{c}) = 10^{26} \frac{{\sigma}(P)}{C1{\cdot}f_{psf}(\lambda_{c},aperture)}~~~~~~~~~~~~(Jy)}\end{displaymath}$ (8.15)

where,

${\rm F_{\nu}(\lambda_{c})}$ in Jy is the flux density at the centre wavelength of the filter, and
${\rm {\Delta}F_{\nu}(\lambda_{c})}$ the corresponding uncertainty;
${\rm P}$ in W is the in-band power on the detector in the selected filter band, and
${\rm {\sigma}(P)}$ the corresponding uncertainty;
${\rm C1}$ in m²Hz is a constant related to each filter band and describes the total transmission of the bandpass along the optical path onto the detector; this includes the size of the mirror, reflections, filter transmission, spectral response of the detector, etc.;
${\rm f_{psf}(\lambda_{c}, aperture)}$ (unity) is the fraction of the telescope point-spread function falling into given aperture for PHT-P or onto the full array for PHT-C. at a given wavelength.

8.4.9.2 PHT40

In the case of PHT-S (AOT P40) the signal s_pixel with uncertainty ${\rm {\sigma}(s_{pixel})}$ (in V/s) for each pixel is converted to a flux density through:

$\begin{displaymath} {\rm F_{\lambda}(\lambda_{pixel}) = 10^{-20} \frac {s_{pixel... ... {c} {{\lambda}_{pixel}^2}~~~~~~~~~~~~~~(W\,m^{-2}{\mu}m^{-1})}\end{displaymath}$ (8.16)

with uncertainty

$\begin{displaymath} {\rm {\Delta}F_{\lambda}(\lambda_{pixel}) = \vert F_{\lambd... ...gma}{s_{pixel}}}{s_{pixel}})^2 } ~~~~~~~(W\,m^{-2}{\mu}m^{-1})}\end{displaymath}$ (8.17)

where,

${\rm F_{\lambda}(\lambda_{pixel})}$ in ${\rm W\,m^{-2}{\mu}m^{-1}}$ is the flux density in wavelength units, and
${\rm {\Delta}F_{\lambda}(\lambda_{pixel})}$ the corresponding uncertainty;
${\rm C_{phts,pixel}}$ in (V/s)/Jy is the spectral response function which gives the conversion from signal in V/s to Jy for a given pixel, and
${\rm {\Delta}C_{phts,pixel}}$ the corresponding uncertainty;
${\rm \lambda_{pixel}}$ in ${\mu}$ m is the centre wavelength of a pixel;
${\rm c}$ = 299792500 m/s is the speed of light used in OLP.

8.4.10 Convert in-band power to surface brightness

Operation: Convert flux density ${\rm F_{\nu}}$ in Jy to surface brightness ${\rm I_{\nu}}$ in MJy/ster.

The surface brightness calculation assumes that the point source flux density has been derived. Based on the point source flux density the surface brightness is determined.

Ancillary data needed

Effective solid angles $\Omega_{\lambda}$ for each filter/aperture and filter/pixel combination. The values are read from Cal G files PPOMEGA in case of PHT-P, PC1OMEGA in case of PHT-C100 and PC2OMEGA in case of PHT-C200. See also appendix A.1.2, and section 13.18.
PHT-S spectral response function for extended emission i.e. signal calibration factors C^s_phts,pixel plus uncertainties for all PHT-S pixels. The values are read from the SPD product header, for a description of contents see section 13.16.1.
PHT-S central wavelengths $\lambda_{pixel}$ for all PHT-S pixels, these are also read from the SPD product header, see also section 13.16.1.

8.4.10.1 All AOTs except PHT40

The surface brightness is obtained by applying the relationship:

$\begin{displaymath} {\rm I_{\nu}(\lambda_{c}) = \frac {F_{\nu}(\lambda_{c}){\cdo... ...- \epsilon^{2}){\cdot}{\Omega}_{\lambda}} ~~~~~~~~~~(MJy/ster)}\end{displaymath}$ (8.18)

with the uncertainty computed according to

$\begin{displaymath} {\rm {\Delta}I_{\nu}(\lambda_{c}) = \frac {{\Delta}F_{\nu}(\... ...- \epsilon^{2}){\cdot}{\Omega}_{\lambda}} ~~~~~~~~~~(MJy/ster)}\end{displaymath}$ (8.19)

where with the same definitions as in the previous sections,

${\rm I_{\nu}(\lambda_{c})}$ in MJy/ster is the surface brightness at the centre wavelength of the filter, and
${\rm {\Delta}I_{\nu}(\lambda_{c})}$ the corresponding uncertainty;
$\Omega$ in ster is the effective solid angle on the sky of the detector pixel or selected aperture;
(1- $\epsilon^2$ )=0.91 ( $\epsilon$ =0.30) is the correction for the obscuration by the secondary mirror.

The values of $\Omega_{\lambda}$ were computed by using a model which takes into account the ISO telescope mirrors as well as the physical sizes of the apertures in case of PHT-P or detectors in case of PHT-C. The model provided the 2-dimensional beam profile (or ``footprint'') of each possible aperture/filter (PHT-P) or pixel/filter (PHT-C) combination. The value of $\Omega_{\lambda}$ was eventually obtained from the integral of the footprint.

8.4.10.2 PHT40

In the case of PHT-S (AOT P40) the signal s_pixel (in V/s) with uncertainty ${\sigma}{s_{pixel}}$ is directly converted to surface brightness:

$\begin{displaymath} {\rm I_{\lambda}(\lambda_{pixel}) = 10^{-26} \frac {s_{pixel... ...\lambda}_{pixel}^2}~~~~~~~~~~~~~~(W\,m^{-2}{\mu}m^{-1}sr^{-1})}\end{displaymath}$ (8.20)

with uncertainty

$\begin{displaymath} {\rm {\Delta}I_{\lambda}(\lambda_{pixel}) = \vert I_{\lambd... ...pixel}}}{s_{pixel}})^2 } ~~~~~~~(W\,m^{-2}{\mu}m^{-1}sr^{-1})},\end{displaymath}$ (8.21)

where

${\rm I_{\lambda}(\lambda_{pixel})}$ in ${\rm W\,m^{-2}{\mu}m^{-1}sr^{-1}}$ is the surface brightness for a given pixel, and
${\rm {\Delta}I_{\lambda}(\lambda_{pixel})}$ is the corresponding uncertainty;
$\lambda_{pixel}$ in m is the central wavelength of a PHT-S detector pixel;
${\rm C^E_{phts,pixel}}$ in (V/s)/(MJy/sr) is the spectral response function for the conversion from signal (in V/s) to surface brightness (in MJy/sr);
${\rm {\Delta}C^E_{phts,pixel}}$ is the corresponding uncertainty;
${\rm c}$ = 299792500 m/s is the speed of light.

8.4.11 Write PHT-P point source photometry product

Operation: Write a complete PHT-P point source photometry product.

Write the product FITS header followed by the processed data in a binary table with each record containing the data for a single filter or aperture.

Detailed product descriptions can be found in section 12.4, subsection 12.4.2 (product PPAP).

8.4.12 Write PHT-P extended source or raster photometry product

Operation: Write a complete PHT-P extended source or raster photometry product.

Write the product FITS header followed by the processed data in a binary table with each record containing the data for a single filter, aperture or raster point.

Detailed product descriptions can be found in section 12.4, subsections 12.4.3 (PPAE) and 12.4.9 (PPAS) for single pointing and raster products, respectively.

8.4.13 Write PHT-C point source photometry product

Operation: Write a complete PHT-C point source photometry product.

Write the product FITS header followed by the processed data in a binary table with each record containing the data for a single filter.

Detailed product descriptions can be found in section 12.4, subsection 12.4.4 (PCAP).

8.4.14 Write PHT-C extended source or raster photometry product

Operation: Write a complete PHT-C extended source or raster photometry product.

Write the product FITS header followed by the processed data in a binary table with each record containing the data for a single filter or raster point.

Detailed product descriptions can be found in section 12.4, subsections 12.4.5 (PCAE) and 12.4.10 (PCAS) for single pointing and raster products, respectively.

Next: 8.5 Mapping Up: 8 Data Processing Level: Previous: 8.3 Common processing steps

ISOPHOT Data Users Manual, Version 4.1, SAI/95-220/Dc