The String Method

Our first attempt at calibrating the PASS0 lightcurves consists of trying to determine what cor- rections need to be applied to the reference fluxes in order to remove their LST dependence. The method we use is applied on a star by star basis since each star experiences unique changes in air- mass, extinction and position, especially across such a wide field, and each star enters and leaves the CCD at a different LST.

Consider a single star with observations atN LSTs on each night over a period ofM nights. Then we haveM N data points, and we requireN ₋1factors to correct theN reference fluxes. So long asM is greater than 1 then we are guaranteed to be able to determine the factors.

We apply a minimization technique based on the idea behind the string length technique used for determining periods in periodic variables with unknown forms of variations [43]. Stars are most likely constant but they may also be variables of any type and hopefully transit candidates. Hence we cannot make any prior assumption about the lightcurve shape. Instead we calculate a string length between consecutive data points and sum the lengths to get a total lengthdgiven by:

d=

N M−₁ X

i=1

mi+1−mi (3.10)

where mi are the lightcurve magnitudes. Then we use the AMOEBA algorithm to adjust the reference flux factorsKjin order to minimize this total string length by recalculating the lightcurve magnitudes as follows: mi =M0−2.5 log KjFref,j+ ∆fi p (3.11)

whereFref,j is the relevant reference flux for the LSTjthat applies to the data pointi.

To help illustrate why we do this, imagine the data points as beads threaded by a string. The beads are allowed to move vertically in small groups (grouped by LST). By pulling the string taught you try to minimize the string length between the beads and the system will settle into its natural shape. This shape depends on the underlying variations in the light from the star. Since we are only interested in adjusting the reference fluxes, we do not consider the dependence of the string length on epoch of observation.

78 CHAPTER 3. PASS0 DATA REDUCTION 0.001 0.01 0.1 1 4 5 6 7 8 9 10 RMS Scatter (mag) PASS0 Magnitude (a) 0.001 0.01 0.1 1 4 5 6 7 8 9 10 RMS Scatter (mag) PASS0 Magnitude (b)

Figure 3.9: (a): Plot of RMS scatter against mean magnitude for the PASS0 lightcurves before (green points) and after (red points) calibration using the string method. The red line shows the aperture photometry theoretical limit using the special extended aperture suitable for the trailed star images, and the blue dashed line shows the theoretical PSF photometry limit calculated using the known PASS0 PSF. Scintillation noise at 5.2 mmag has been included. (b): The same as (a) except that the red points now represent the PASS0 lightcurves after calibration withPASSCAL.

In Figure 3.8 we also show the reference fluxes (top panels) and lightcurves (bottom panels) for the two example stars after calibration with the string method (solid black points). In correcting the reference fluxes using the string method, it is now obvious which star is the constant star and which star is the variable star.

In Figure 3.9(a) we plot the RMS scatter in the lightcurves versus PASS0 mean magnitude (calibrated to Tycho-2 catalogue magnitudes). The green points show the RMS scatter for the raw lightcurves from the pipeline, and the red points show the improvement after applying the string method that we have developed. In calculating the theoretical aperture photometry limit we have assumed that a special aperture is necessary. This aperture consists of a rectangle with a semicircle at each end (on the ends in the direction of the star trail; see Figure 2.6). The aperture is defined by the diameter of the semicircles and the length of the rectangle. For observations near the celestial equator we calculate that the trailing in 20 seconds requires a rectangle of length

∼5.2 pix (or∼5.0 arcmin; see Equation 2.45) and the PASS0 PSF requires a semicircle diameter of 7.0 pix (exactly the same size as for the SuperWASP pipeline in Chapter 4). The theoretical aperture photometry limit is plotted in Figure 3.9(a) as the solid red line. Finally the theoretical PSF photometry limit is calculated using Equations 2.17, 2.18 and 2.23 with the known PASS0 PSF and it is plotted as the dashed blue line. A scintillation noise of 5.2 mmag (as calculated in Section 2.3.10) has been added in quadrature with the noise models to account for this effect.

It is clear from Figure 3.9(a) that the string method greatly improves the raw PASS0 lightcurves but it does not reach to the theoretical PSF photometry limit and it does only slightly better than the aperture photometry limit towards the faint end (from 9th mag and fainter). The spread in RMS at each magnitude is due to the fact that most PASS0 lightcurves have fewer epochs than the maximum number since stars drift on, over and off the CCD. Also, the wide field-of-view of the PASS0 camera means that each star has a slightly different airmass at each epoch and the varying amounts of scintillation noise present for each star contribute to this spread in RMS. The theoretical limits presented have been calculated assuming that all epochs are present in a lightcurve and that the scintillation noise is the same as that in the field centre for the whole field- of-view.