On the ground

As we saw both principal difficulties of photometry on the ground are the presence of a nonconstant bottom under the supernova (the galaxy host), the variation of the transmission of the systems of observation (telescope, instrument, filter and atmosphere) and the variations of PSF.

The first stages of reduction are in any point identical to those of research (chapter 7 ). They will enable us starting from the rough images to build the images which will be then used for photometry.

We enumerate them the principal stages of the reduction for recall:

1. correction for the quantum effectiveness of CCDs (``flatfielding ' ') and indexing of the images.

2. Construction of the catalogues of photometry and subtraction of the bottom by using the software SExtractor (bertin1996).

3. Construction of the charts of weight.

4. Alignment and rééchantillonnage of all the images on a common geometrical reference on the grid of pixel of the geometrical reference. This geometrical reference is selected like the image of better seeing for the batch of image of smaller pixels.

5. Coaddition of the images of the same night and the same filter.

We consequently have the images summoned for each night and in each filter, aligned between them with identical pixélisations.

We want to estimate the flux of the supernova on each image. The summoned image is the sum of the galactic bottom $I^l_{i, j}$ is thus written as the sum of these various components:

$$I^l_{i,j} = Gal^l_{i,j} + f_l PSF^l_{i,j} + \epsilon^l_{i,j}$$ (8.10)

and its variance (under the assumption of an image in photoelectrons):

$$Var(I^l_{i,j}) = b_l + Gal^l_{i,j} + f_l~ PSF^l_{i,j}$$ (8.11)

In general, PSFs and the model of galaxy will be different from one night to another because of the effects of seeing. To be able to estimate the flux of the supernova, it is initially necessary to bring all the images to the same resolution.

This harmonization of PSFs is made in a way strictly identical to the method employed at the time of research. One seeks a core of convolution $K$ which makes it possible to bring the image of better on the image of good resolution. If the image of better seeing is considered $I^{best}_{i, j}$, one can thus express an image like:

$$I_{i,j}^l = I_{i,j}^{best}\otimes K_l$$ (8.12)

One can thus reexpress, the one night image current like:

$$I^l_{i,j} = Gal^{best}_{i,j}\otimes K_l + f_l PSF^{best}_{i,j}\otimes K_l$$ (8.13)

The knowledge of the PSF of the image of the night of better seeing and the cores of convolution thus makes it possible in theory to estimate the flux of the supernova provided that one has an image of reference of good quality, i.e. , where the noise can be neglected, this image being used as model of galaxy. In practice, even if the images of reference gain from a consequent duration, making it possible to have images of good quality, quality is however insufficient for this kind of estimate. The estimate is thus made by least squares.

The structure of the galaxy is difficult to model, moreover, for the remote supernovæ, it is very often unsolved on the image of reference. The method employed is thus an adjustment of all the pixels of the galaxy on all the images, including those of follow-up.

Minimization is made compared to the parameters according to:

1. The flux $f_l$ of the supernova for each night.
2. The position of the supernova which is supposed to be constant from one night to another.
3. Pixels of the galaxy (presumedly constant).

This is done in each filter, the contribution of the galaxy being significantly different according to the color considered. To force the adjustment, the flux of the supernova supposed no one during the adjustment for the images of reference from where the supernova disappeared (references is in general taken a year or more after the follow-up of the supernova).

The result of this minimization is a file by filter which contains fluxes of the supernova and their errors for the various nights of follow-up, the dates juliennes and the name of the image of the corresponding night. It provides also a file containing the matrix of covariance between various measurements of flux.

Figure 8.6 represents the lightcurve of a supernova which gained from a broad follow-up on the ground. The figure represents the evolution of flux according to the date Julienne . The unit for fluxes is the photometric unit of the image of the night of better seeing.

The framework of the bottom of the figure presents an estimate of the systematic errors made on fiducial objects. For each fiducial object, one adjusts in a simultaneous way the constant bottom and a flux of PSF, a way strictly identical to the curve fitting of light of the supernova. The single difference is that the position of the PSF is fixed a priori and will not be adjusted.

Except for the variable objects of the field, the flux estimated by this method must be on average null. The estimates on the fiducial objects make it possible to test the method and to control biass which could come to deteriorate measurement. One does not see systematic bias according to the bottom of sky, seeing of the images of the various nights. The method thus seems robust and optimal. It was thus selected to build the lightcurves for the points measured on the ground of the supernovæ of our batch of study.

With this stage of the analysis, the lightcurves are still in arbitrary units.

We will see in the continuation how this lightcurve is gauged so as to pass in the standard system magnitudes.

Figure 8.6: This figure represents the lightcurve in flux of a supernova discovered at the time of a search campaign for supernovæ with shift towards the intermediate red, its redshift is 0.18. The lightcurve was obtained starting from observations made during the summer 2002 with the INT for the follow-up. The lightcurve (tallies the top) was obtained by simultaneous adjustment. In bottom, an average of the residuals estimated on the fiducial objects in units is presented of $\sigma$. For more details, to refer to the text.