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8.5 Mapping

Two dimensional raster and spatially oversampled mapping observations are processed to images in the equatorial (RA and Dec) coordinate system. For the construction of the images it is assumed that a detector footprint (or beamprofile) can be represented by a simple ``top-hat'' function: the emission is 1 inside the detector area and zero outside. In addition, the circular apertures have perfectly circular footprints and the square PHT-P apertures and PHT-C detector pixels are considered to be perfectly square. The positions of each detector pixel per raster point have already been determined in the processing steps described in section 8.3.3. This section describes the co-addition steps to obtain an image.

Note that in the remainder of this section a clear distinction should be made between detector pixel and image pixel, the former being one of the 9 (for C100) or 4 (for C200) elements in the detector array and the latter being a picture element of an image.

The mapping software yields three products:

surface brightness map in units of MJy/sr, this is the average surface brightness of all detector pixel exposures for each image pixel;
uncertainty map in units of MJy/sr, mean brightness uncertainty per image pixel;
exposure map in s, giving for each image pixel the total time that it has been covered by any detector pixel.

8.5.1 Establish image pixel size

Operation: Define the size of the image pixel

The angular dimensions of an image pixel depend only on the detector subsystem used and is independent on the filter/aperture combination. The values used in the current version of the OLP are given in Table 8.1.

**Table 8.1:** image pixel sizes
Detector	Shortest	Airy disc	Pixel
used	wavelength	diameter	size
	( ${\mu}$ m)	(arcsec)	(arcsec)
P1	3.2	2.8	8.0
P2	20	17.7	8.0
P3	60	50.3	8.0
C100	70 (C_50)	41.9	15.0
C200	120	101.0	15.0

Based on the image pixel size, the fine-grid is defined. The fine-grid is used to compute the fractional pixel coverage. The optimum fine-grid size in terms of realistic processing efforts has been decided to be:

$\begin{displaymath} {\rm S_{fine~grid} = S_{pixel}/5~~~~~~~~~~~~~~~~~~(arcsec)}\end{displaymath}$ (8.22)

Ancillary data needed

None, parameters are hardcoded

8.5.2 Establish maximum image size

Operation: Calculate the angular dimensions of the image required to contain the full raster scan in equatorial coordinates.

The maximum size of the image is calculated using the following parameters:

M: the number of raster steps along a raster leg;
N: the number of raster legs;
dM: step size in arcsec along raster leg;
dN: step size in arcsec in between raster legs;
${\alpha}_c,~{\delta}_c$ : the J2000 RA and Dec position in decimal degrees of the raster centre;
${\alpha}_{m,n},~{\delta}_{m,n}$ : the J2000 RA and Dec position in decimal degrees of raster point (m,n), with m= 1...M, and n= 1...N;
D_A: the diameter in arcsec of the aperture used, in case of a C-array: the sidelength of the array in arcsec;
$\theta_{chop}$ : the maximum chopper deflection while mapping (only for PHT32 maps).

The centre of the raster scan is defined as the image centre. The dimensions of the frame in equatorial coordinates are derived as follows. For all raster points (m,n) determine:

$\begin{eqnarray*} {\alpha}_{max} = & max( \Delta\alpha(m,n) )\\ {\alpha}_{min} ... ...elta\delta(m,n) )\\ {\delta}_{min} = & min( \Delta\delta(m,n) ) \end{eqnarray*}$

where:

$\Delta\alpha(m,n)$ : RA component of the difference vector between ( ${\alpha}_{m,n},{\delta}_{m,n}$ ) and ( ${\alpha}_c,{\delta}_c$ )
$\Delta\delta(m,n)$ : Dec component of the difference vector between ( ${\alpha}_{m,n},{\delta}_{m,n}$ ) and ( ${\alpha}_c,{\delta}_c$ )

The components $\Delta\alpha(m,n)$ and $\Delta\delta(m,n)$ are computed using spherical trigonometry. The margins around the raster grid is determined by D_A and the maximum chopper deflection $\theta_{chop, max}$ , in raster orientation:

$\begin{eqnarray*} x_{m} = & D_{A}/2+\theta_{chop, max}\\ x_{n} = & D_{A}/2 \\ \end{eqnarray*}$

Then the map dimensions $L_{\alpha}$ and $L_{\delta}$ are computed from:

$\begin{eqnarray*} L_{\alpha} = & {\alpha}^*_{max}-{\alpha}^*_{min}\\ L_{\delta} = & {\delta}^*_{max}-{\delta}^*_{min} \end{eqnarray*}$

with

$\begin{eqnarray*} {\alpha}^*_{max} = &{\alpha}_{max}+(\vert x_{m}cos(\phi)\vert+... ...lta}_{min}-(\vert x_{m}sin(\phi)\vert+\vert x_{n}cos(\phi)\vert) \end{eqnarray*}$

where $\phi$ is the spacecraft roll angle.

The rectangular image frame is therefore chosen so large that the entire area covered by the detectors in the raster fits inside. Depending on the rotation angle of the raster as well as the sampling step sizes with respect to the size and rotation of the detector, undefined frame areas may be present.

The minimum number of image pixels required to map the whole of the raster area is calculated from the dimensions of the raster and the image pixel size s_pixel, as follows :

$\begin{eqnarray*} n_{\alpha}& = & 2(\Vert\frac{L_{\alpha}}{2s_{pixel}}\Vert + 1)... ..._{\delta}& = & 2(\Vert\frac{L_{\delta}}{2s_{pixel}}\Vert + 1)+1, \end{eqnarray*}$

in which $\Vert$ denotes ``the integer value of'' - i.e. the initial value is rounded down to the nearest integer.

Also calculated at this stage are the angular coordinates of the image pixel origin which is defined as the bottom left hand corner (blhc) of image pixel (1,1).

$\begin{eqnarray*} x_{0} & =& -\frac{n_{\alpha}s_{pixel}}{2}\\ y_{0} & =& -\frac{n_{\delta}s_{pixel}}{2}\\ \end{eqnarray*}$

These are used subsequently to translate between the coordinate origin at the centre of the image and the origin of the pixel coordinates.

The blhc of an image pixel (i,j) is thus given by :

$\begin{eqnarray*} x_o(i,j) & = & (i-1-\frac{n_{\alpha}}{2})s_{pixel}\\ & = & (... ...frac{n_{\delta}}{2})s_{pixel}\\ & = &(j-1)s_{pixel} + y_{0}\\ \end{eqnarray*}$

8.5.3 Determination of the coverage by a sky sample in the image

Operation: Map data from a given detector pixel, raster position, and chopper offset combination onto the image.

We define a sky sample to be a sky observation for a given detector pixel, at raster position (m,n), and chopper deflection $\theta_{chop}$ .A sky sample has a raster dwell time or chopper dwell time (in case of PHT32 maps).

The fine grid (section 8.5.1) is used to determine the fractional coverage of the image pixel by the detector footprint. The fine-grid size is chosen such that it is a simple factor of the image pixel size: there is an integral number of grid points per image pixel.

Therefore for each sky sample the following steps are performed:

determine the image pixel range ([x_min,x_max],[y_min,y_max] in which the detector pixel footprint can completely be contained;
for each image pixel contained in this range determine the number of fine-grid positions N_fg which fall within the detector footprint.
for each image pixel in the range determine the coverage factor F = N_fg/N_total, where N_total is the total number of fine-grid positions on a pixel.

8.5.4 Binning sky samples onto image pixels

Operation: Accumulate the sky samples and derive the average surface brightness, uncertainty and exposure.

The contribution from the n^th sky sample to the mean brightness, brightness uncertainty, and exposure time at image pixel i,j are denoted b_n(i,j), u_n(i,j) and e_n(i,j) respectively. The accumulated sums of the contributions from the first n pointings are denoted by B_n(i,j), U_n(i,j) and E_n(i,j).

The effective exposure time of sky sample n at image pixel i,j is the fraction of the image pixel covered by the aperture for a given sample multiplied by the dwell time t_n:

$\begin{displaymath} e_{n}(i,j) = F(i,j){\cdot}t_{n}\end{displaymath}$ (8.23)

To calculate the total exposure on the image pixel, the contributions from all sky samples are summed. The surface brightness values and the corresponding uncertainties are summed thereby using the exposure times as weights.

$\begin{eqnarray*} E_{n+1}(i,j) = & E_{n}(i,j) + e_{n+1}(i,j)\\ B_{n+1}(i,j) = &... ...)\\ U_{n+1}(i,j) = & U_{n}(i,j) + e_{n+1}(i,j)u_{n+1}^{2}(i,j). \end{eqnarray*}$

The accumulated sums for all sky samples are used to derive images of total exposure time E_total, mean brightness $\overline {B}$ and brightness uncertainty U_rms.

$\begin{eqnarray*} E_{total}(i,j) & = & E_{N}(i,j)\\ & = & \sum_{n=1}^{N} e_{n}(i,j) \end{eqnarray*}$

$\begin{eqnarray*} {\overline B}(i,j) & = & B_{N}(i,j)/E_{N}(i,j)\\ & = & \frac{1}{E_{N}} \sum_{n=1}^{N} e_{n}(i,j)b_{n} \end{eqnarray*}$

$\begin{eqnarray*} U_{rms}(i,j) & = & \left(\frac{1}{E_{N}} {U_{N}(i,j)} \right)... ...\sum_{n=1}^{N} e_{n}(i,j)u_{n}^{2}(i,j)} \right)^{\frac{1}{2}} \end{eqnarray*}$

8.5.5 Write image products

Operation: Write a set of complete photometric image products.

First write the product FITS header followed by the processed data in a primary array. The derived surface brightness (in MJy/sr), brightness uncertainty (in MJy/sr) and exposure time (in s) images are stored in separate products.

Detailed product descriptions can be found in section 12.4, subsections 12.4.6 (product PGAI), 12.4.7 (product PGAU), and 12.4.8 (product PGAT) for the brightness, uncertainty and exposure time images.

Next: 8.6 Spectroscopy Up: 8 Data Processing Level: Previous: 8.4 Photometry

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