4 Data simulation

4.1 Models of DISPIs

The model of the spectrophotometric output from DIVA used in this work was developed by Scholz (1998). This software requires a spectral energy distribution as input and creates a two-dimensional signal output image on the detector, containing the dispersed intensity. These images have 114 $\times$ 150 pixels where the latter number, refering to the dispersion direction, is in effective pixels and the former number, refering to the scanning direction is in physical pixels. Figure 5 shows such an image. Ultimately, only a narrow window around the DISPI will be read from the focal plane data stream and trasmitted to ground, the so-called Spectroscopic (SC) window.

As input spectra we used synthetic spectra from Lejeune et al. (1997, 1998). In total there were about 5600 spectra covering a parameter grid with 68 values for $T_{\rm eff}$ between 2000 K and 50 000 K (in steps of 200 K for the low temperature star, and 2500 K for the high temperature stars), 19 possible values for $\log g$ ranging from $-1.02 \leq\$$\log g$ $\ \leq 5.5$ in steps of approx. 0.1 to 0.3 dex and 13 values for [M/H] with $-5 \leq\$[M/H]$\ \leq 1$ in steps of 0.5 and 0.1 dex. Note that in our tests there were no input data for metallicities in the range from -2.5 to -0.3 dex.

The obvious advantages of using synthetic spectra are the complete wavelength range from 200 to 1200 nm and the large number of spectra over a large parameter space. We are currently constructing a library of (previously published) real stellar spectra. However, since it combines spectra from many different available catalogues, there is a considerable heterogeneity among these data. Moreover, few stars have been observed with the desired wavelength range from the UV to the IR.

Interstellar extinction was modelled by using a synthetic extinction curve for R=3.1 given as $\frac{A(\lambda)}{E(B-V)}$ versus $\lambda$. We used the extinction curve from Fitzpatrick (1999), simulating 7 different extinction values in steps of 0.15 and 0.2 in the range $0.0 \leq\$E(B-V) $\ \leq 1.0$ mag. Note that the zeroth order was omitted in this and all other simulations, so that we only worked with dispersed images made up of the first to third spectral order. Since the data were area-normalized before passing them through the neural network, the magnitude information of the zeroth order is lost anyway. For the simulation of the UV-telescope (see below), the same extinction curve was applied. This procedure was done for five different visual magnitudes in the range from $8 \leq V \leq 12$ mag.

Noise was added to these two-dimensional intensity distributions by passing them through another software tool developed by Ralf Scholz. Here, a mean sky backround of sky = 0.04 e^-/(pix s) and a dark current of dark = 2 e^-/(pix s) were added with additional source and sky Poisson noise. The CCD's read out noise was 2 e^-/eff.pix.

Figure 5: A dispersed image for an M type star, V = 10 mag (with noise) as generated by the simulation software from Scholz (1998). Note the contribution of the zeroth order seen as a single intensity "blob" in the upper part of the image. The first, second and third spectral order are all overlapping (lower intensity "stretch") due to the grating. The intensity stripes to the side of the dispersed image are due to diffraction at the telescope's aperture.

The size of the SC window to be cut from the on-board data stream around the DISPI is crucial as it determines the data rate which is in turn related to the satellite's overall performance: a smaller window permits a larger number of SC windows (objects) to be transmitted. This would, for example, permit a fainter magnitude limit. The optimum window size, i.e. the window around a dispersed image with the highest amount of important and lowest amount of redundant information, was investigated in earlier studies. Concerning the window size in the cross dispersion direction it was found that the innermost 7 pixel are sufficient (Hilker et al. 2001) in terms of highest S/N. However, due to the satellite's intrinsic attitude uncertainty it is required that the smallest acceptable window size in the scanning direction be 12 pixel (see Bastian & Schilbach 2001). For our studies we therefore summed up the TDI-rows over the innermost 13 rows (6 pixels in each direction about the central row). Future work will use a profile fit to obtain the stellar intensity.

The optimum size in the dispersion direction was evaluated by S/N studies and the (spectral) information content. This amount of information was measured by the ability of Neural Networks to determine the stellar parameters $T_{\rm eff}$, $\log g$ and [M/H] for different ranges of DISPIs. It was found (Willemsen et al. 2001) that these parameters can be adequately retrieved from approximately 45 effective pixels around the maximum intensity in the DISPIs (which is at about effective pixel 60). However, since these earlier studies included only DISPIs with $T_{\rm eff}$ $\geq$ 4000 K and since the overall intensity distribution moves to smaller effective pixel values for lower temperatures, we chose the range from 30 to 80 effective pixels in this work. This should also be appropriate for very red objects like L and M dwarfs with $T_{\rm eff}$ $\simeq$ 1200-4000 K.

For further processing, the simulated sky was subtracted from the dispersed image by evaluating the background level from a single column in scanning direction next to the dispersed image.

The UV imaging telescope will make use of the same type of CCD's as the main instrument. The UV magnitudes in the two different passbands next to the Balmer jump were calculated from the same synthetic spectra as described above, simply by integrating the flux in the ranges from 310 to 360 nm and 380 to 410 nm. Of course, the true filters will not have exactly square transmission curves, but this approximation is sufficient for a first analysis of the influence of the UV channel. The two UV flux values were fed into the network in three different ways. First, we calculated the $a\sin h$ of the flux ratio, i.e. $a\sin h({UV_{\rm short}}/{UV_{\rm long}}$) (note that the $a\sin h$ - function is not undefined for negative values, in contrast to the log-function. Negative values might occur due to noise for very low temperature stars with almost no flux in the UV). This ratio is designed to be sensitive to the Balmer jump thus yielding additional information about gravity and temperature. Second, we summed up the intensity in a DISPI in the range 70 to 80 effective pixel ( $\sum_{i=70}^{80}I_i$) and calculated the ratios ( ${UV}_{\rm short}/\sum_{i=70}^{80}I_i$) and ( ${UV}_{\rm long}/\sum_{i=70}^{80}I_i$). Since the first order's contribution in the selected effective pixel range corresponds to a wavelength range from about 550 to 600 nm (see Fig. 1), these ratios should be a good measure of extinction due to the long "lever" ranging from the UV to the visual/red part of the spectrum.

4.2 Noise in single DISPIs versus end of mission stacked DISPIs

The results reported in this paper (Sect. 6) have been obtained using single DISPIs. However, by the end of the mission, DIVA will have imaged each star about 120 times. Thus the final signal-to-noise ratio for any given magnitude will be much better than from a single measurement. Therefore, the parametrization performance will also be improved or, equivalently, will be achieved at a fainter magnitude. We calculate the final S/N from a sum of 100 two-dimensional intensity distributions. From the ratio of this final S/N to the single DISPI S/N, we can find the equivalent magnitude difference which gives the magnitude to which our parametrization results for a single DISPI can be applied to, without having to do a set of separate simulations on summed DISPIs. The resulting $\Delta V$ is given in Fig. 6 (see further Sect. 6). We see, for example, that a DISPI made up of one-hundred frames each with V=14 mag has the same S/N as a single DISPI of a star with magnitude V=10.8 mag. Unless stated otherwise, all results below will refer to end-of-mission data quality.

Figure 6: One-hundred added frames each of the "single'' visual magnitude given on the x-axis yield the same S/N as a star which is $\Delta V$ magnitudes brighter. For example, adding 100 frames of V=14 mag stars (two-dimensional intensity distributions) and calculating a DISPI from these yields a DISPI which has the same S/N as a single DISPI of a star with V=10.8 mag. Clearly, for fainter stars, the noise is dominated by the read-out noise, while for brighter stars only the Poisson noise of the signal is relevant, thus yielding a full magnitude shift of 5 mag. This curve was calculated by making use of the specific DIVA's CCDs noise characteristics.