In order to convince myself that I am measuring the halo of the LAEs, I spent this week looking into measuring and understanding the point spread function (PSF). The PSF is a consequence of observation and is due to the atmospheric seeing on the night of observations. Different filters are affected differently by the conditions, higher redshift sources will have a lower PSF as a result of observing in redder wavelengths. It is a quantitative measure of how extended a point source will be in an image and since a star is a point source, it should appear so in an image under perfect conditions. However, it will never appear entirely point like in observational images, any extension seen in a star’s appearance is not real and is entirely due to these conditions of observation and can be quantified by the PSF of the image. Thus, when we are wanting to measure the extension of the Lyman alpha emission from our LAEs, the halo, we can only measure this beyond the PSF. The LAH is already extended by this amount due to observation. We can measure the effects of the PSF and subtract this from our surface brightness (SB) measurements of the LAH. It is also important to know the extent of the PSF as we can only begin to fit our exponential profile to the LAH at radii exceeding this.
We measured the PSF per redshift using the stars in the corresponding filter images of the COSMOS field. The PSF was measured by performing annuli photometry on the stars and constructing a SB profile from these measurements, over the same range of radii and by measuring at the same intervals as the SB profiles for the LAEs. The SB profile of the stars were then normalised to start from 1 and all the stars in the catalogue median combined. This gave us a PSF for this redshift, and the same was repeated for each filter. I then plotted the PSF onto the SB profiles of the LAEs by normalising this to the maximum of the SB profile, in order to see visually the similarities and differences between the PSF and the SB profile of the LAEs.
By finding the point where the PSF diminishes, I determined a radius to start my exponential fit, for determining the scale length of the LAEs. We only want to be measuring beyond the PSF. I started by looking at the radius that corresponds to 50% of the flux, then 68% and then finally 96%, which corresponds to plus or minus 3 sigma. Since the SB profiles are normalised, the radius that corresponds to say 68% of the flux, is just the radius where the SB drops to a value of 0.32. These radii were determined for each filter from the corresponding PSF and for the three different flux cuts, see Table 1 and Figure 1.
What we find is useful as it allows us to be able to make our exponential fits to the SB profile starting from a radius of less than 2 arc-seconds. Other studies, e.g. Momose et al 2014, start there fit from 2” but when we do this, at most of the redshifts, we are unable to measure a scale length as there are not enough detections to be able to make the fit. Our data is not deep enough. Therefore, being able to fit from lower radii is very useful and using the PSF we are quantitatively able to demonstrate why we are able to do this.
We then re-plot our SB profile with the PSF subtracted where the errors on this are now the errors on the PSF plus the SB profile errors originally.
Since all the stars SB measurements are median combined per radii, we calculated the errors on the PSF using the 16thand 68thpercentiles of the variation of the measurement from the median, per radii, as the down and up error respectively. We noticed that these errors were growing large towards the tail of the PSF, see Figure 2, and for some bands the PSF lies above the SB profile, see Figure 3. So, we decided to visually check the stars in the catalogue to make sure we were measuring the PSF correctly. The catalogue contains over 300 stars which we investigate in each of the 14 filters. Using DS9 imaging software we are able to view the images of these stars and easily inspect them. We remove double stars, stars that are too faint and also stars that are saturated, see Figure 4 for examples of these. Once we removed the flagged stars that may be affecting our PSF measurements, we end up having a catalogue of 196 stars to use. As well as measuring the SB for each individual star and then median combining these into a PSF, we decided to also stack the 196 stars images in 2D. Then from this measure the SB profile as the PSF. This way is more stable and the errors on the PSF are quantified by measuring the noise in the background of the stacked image, as opposed to using the percentiles of the spread of the individual stacks. It also means we are being consistent in the way we measure our PSF from the images of stars and the SB profile from the stacked images of LAEs. We find that the two methods are consistent in the PSF they produce, although we minimise the errors significantly with the 2D stack. This is what we use going forward with our measurements.
A check we can do is to fit the PSF profiles with the exponential profiles like we do with the LAEs. Stars have no Lyman alpha halo and so this fit is not physical but it is useful to see that it produces the same results as a fit to the LAEs over the central regions, r=0-2”. The two are plotted together in Figure 5 such that the similarities can be highlighted. It is reassuring to see that the PSF includes the central regions of the galaxy and thus by investigating at our SB profiles past the PSF we are indeed measuring the halo, and not the light from stars in the galaxy.
My next task was to then fit the exponential profile to the PSF subtracted SB profile detections, like has been done with the SB profile previously over a set radius range. The difference now was that I could fit across the whole range of radii in which there were detections as the central regions have been removed by subtracting the PSF. We thus are able to fit from lower radii now and get a better fit to the halo and a better estimate for our scale lengths.
My results seem to suggest no redshift evolution of LAE scale-lengths over redshift. The PSF subtracted measurements appear to show a small decrease with redshift but it is not significant. We notice that the different approaches to fitting the SB profile and determining the scale-length produce different results.
The median scale length for each fit is shown by the dashed lines in the corresponding colours in Figure 7. The fit from 2” gives a value of 6.1645, the fit from 96% of the PSF gives a much lower value of 3.8067 and the fit to the PSF subtracted SB profile gives a value of 6.3981, which is in agreement with the first. The fit from 2” can only be applied to the data for two of the redshift slices and so the median that results from this data is not necessarily the most reliable. The scale-length derived from the fit to the PSF subtracted SB profile, is trusted the most as this method is more robust and we can be sure we are measuring the halo only and no components of the bright central regions.
It can be seen in the SB profile for IA527, z =3.33, in Figure 6, why the three fits produce different values and that the PSF subtracted (halo only) and fit from 2” are in more of an agreement with each other than the fit from 96% of the PSF.