Distant Supernova Photometry

For high red-shift supernovae we are interested in, separation from the host galaxies is very small, typically far less then .5". The usual ground-base images have seeing (=FWHM of point spread function) of about 1" or worse. This means that aperture photometries of these supernovae will inevitably include significant amount of host galaxy light. Supernova Cosmology Project has been using host galaxy subtraction technique where host galaxy contamination is subtracted out from a supernova plus host galaxy image (NEW), by a reference image (REF) of the same host galaxy without the supernova.

For each NEW image and REF image, we calculate their seeing by fitting a gaussian to the stellar objects in the image. We then convolve the better seeing image to match the worse seeing image. For the supernova and its nearby "fiducial" objects, one FWHM radius aperture photometries are performed on the two images separately to avoid unnecessary resampling of pixels to superimpose the two images. Fiducial objects are any objects (mostly galaxies in our case) close enough to the SN so that they are under the similar seeing when the seeing varies spatially over the image. In this process, we fix the position of the supernova, i.e., position of the supernova is predetermined using all the available images and then translated back to each image. (This positioning usually requires some iterations). Multicolor (R & I) imaging allows us to do rough color cut on fiducial objects to select object of similar color to the supernova host galaxy. From this subset of fiducial objects we can find "fidratio", the ratio between (convolved) NEW and REF. (The idea of subset of fiducial objects are not yet implemented and currently we are using all the fiducial objects for the fidratio.) Then one FWHM radius aperture of SN only at each new image is given by

Subtraction above takes care of underlying host galaxy but the problem of calibrating each data point still remains, especially when many of the SN images are taken under non-photometric condition. Calibration of SNe magnitude is also more complicated by the rapid evolution of SNe spectra. Color correction and instrumental correction require color of the object, which depends on the date of maximum light. This means that iterations are inevitable in SN photometry. As a starting point, we can ignore some of the corrections needed for the final photometry and define an crude magnitude system to save time and effort.

Thus we choose one particular image called "Primary Reference" for of each SN for a practical purpose, requiring it to have good seeing, good signal-to- noise ratio, and SN preferably in the center of the image, not to close to the edges of the image. (The last requirement is to be able to match it with the most other images of the same SN.) Then we transform each subtraction data point to our primary reference system by multiplying by the stellar object ratio of primary reference and each new image in 1 FWHM radius aperture.

We do take some calibration fields, mostly Landolt stars, while we take SN fields. Using 4 FWHM radius aperture for these standard stars, we fit the zeropoints, A, B, C, and D of the following formula on each night.

Once magnitudes of our secondary standard stars are determined, we may define R zeropoint for 4 FWHM radius aperture on our primary reference image as follows;

The magnitude in section 3 is in Bessel system for stars. In other words, if there were a "star" which happened to have the same counts and the same R-I color as our SN in our image, Bessel magnitude for that star is given by the procedures described in section 3. Note that having the same counts means having the same instrumental magnitude. Therefore,

1) Introduction

In reality, measurements of flux is done only in a small window of wavelength, called filter pass-band. The luminosity distance will remain the same for these filtered measurements of flux as long as luminosity L is also "filtered correspondingly". Therefore, in the absence of red-shift and reddening, the differences between the apparent magnitudes and the absolute magnitudes of any arbitrary object will be the same across any filter band of non-zero flux.

(2) Traditional K-correction

Let F(y) be the spectral distribution of flux of the nearby object. Then that of the redshifted object can be written as f(y) = A*(1/(1+z))*F(y/(1+z)), where A is a scalar representing all the dimming due to luminosity distance effect and (1/(1+z)) shows the fact that the energy in a unit y bin is move to a (1+z)-strethed y bin. Flux measurements are then integration of these with the filter function S(y).

(3) Generalized K-correction

For that purpose alone, the ideal filter for high z observation is the filter used in nearby observation red-shifted by the same z. But continuously red-shiftable filters are not realistic and also very difficult calibrate. Still, we are better off using different filters that will match our target redshift range than insisting one identical filter for both nearby and distant observation. For instance, when z is around 0.5, R filter matches very well to redshifted B filter and it is better to compare R filter measurement of SNe at z~5 with not R but B magnitude of nearby ones.

Let G(y) be the filter function used for the distant observation. Then the flux measurement is :

First we undo the extinction in our galaxy by using E(B-V) value of Burstein and Heiles. A(V) = 3.1 * E(B-V) is the normalization specifying the amount of dust. Integration of A(x) * (red-shifted SN spectra) * (filter function) gives the extinction due to dust in our Milky Way for that filter.

For the extinction in the host galaxy, we have to come up with color excess at the host galaxy. We use our light-curve fit in R and I to determine stretch and (observed) R & I magnitudes. Since we do generalized K-correction for B to R and V to I, R-I we got here is B-V at the host galaxy. Unextincted B-V is given by integration of theoretical SN spectra. Once we get E(B-V) = (B-V)_observed - (B-V)_theory at the host galaxy, we do the similar procedure as above, using un-redshifted spectra of SN there.