IRAC systematic features

The secondary eclipse light curves presented in Chapters 3 and 4 of this thesis were taken using a staring mode for IRAC, where the target star is centred on the same pixel throughout the observation. Observations typically last for around 8 h, to allow for out-of-eclipse baseline measurements (an eclipse typically lasts 2₋ 3 h). Staring mitigates systematics caused by array location dependent effects which affect observations using the more standard dithering technique [Harrington et al., 2007]. However, significant systematic features still remain, namely the intra-pixel

sensitivity variation effect for channels 1 and 2 (InSb detectors) and the detector ramp effect for channels 3 and 4 (Si:As detectors).

Intra-pixel sensitivity variations

The intra-pixel sensitivity effect is a quasi-periodic variation in flux seen in the channel 1 and 2 detectors for staring data of bright sources. The variations have a typical amplitude of 1 % on a time scale of 1 h. Figure 2.5 shows observations of the star-planet system TrES-4 by Knutson et al. [2009]. The effect is clearly seen and is stronger in channel 1 than in channel 2 - a feature that has been found generally. Charbonneau et al. [2005] found that these variations correlated with the centroid position of the point-spread function (PSF), which typically varies by 0.2 pixels across a secondary eclipse observation.

The physical origin of this effect is believed to be a variation of the sensitivity within each pixel, where the centre of a pixel is more sensitive than its edges. The observed PSF is therefore a combination of the true PSF and the intra-pixel sensi- tivity functions of the illuminated pixels. Since the point source full-width at half maximum (FWHM) for these detectors is 1.5 pixels, the PSFs are under-sampled and the central brightest pixel dominates the flux and the observed flux variation. The contribution to the flux variation from other pixels is also suppressed by the symmetry of the PSF. As the centre of the PSF moves from the centre of the bright- est pixel to the edge, the total measured flux of the star decreases (and vice versa). The _∼1 h variations in flux seen in light curves result from the telescope pointing variations described in Section 2.1.1.

In this thesis I corrected for this effect using a polynomial fit to the measured xandypositions of the PSF centre. This is a standard technique that has been used in many exoplanet eclipse studies [e.g. Charbonneau et al., 2005; Knutson et al., 2008; Todorov et al., 2012; Anderson et al., 2013]. In general the variations were modelled using subsets of the equation:

F lux=a0+axdx+aydy+axxdx2+ayydy2+atdt, (2.1) where dx = x₋xˆ and dy = y₋yˆ are the positions of the PSF centre relative to their weighted means,dt is the time since the first observation anda0,ax,ay,axx, ayy and at are coefficients (determined as described in Section 2.4.2.3).

In some cases I tested additional terms, such as a cross term in position (axydxdy) or a log temporal term (atln (dt)), although these were never adopted for the final analyses. The linear temporal term (atdt) has been found to be necessary

Figure 2.5: Raw Spitzer IRAC light curves of the secondary eclipse of TrES-4b by Knutson et al. [2009]. The quasi-periodic flux variations caused by the intra-pixel sensitivity effect is clearly seen in channels 1 and 2, while the detector ramp effect can be seen in channel 4. The slight decrease in flux in channel 3 is unexplained, but may be due a linear temporal trend in the weakly illuminated background pixels that continues after the bright source pixels have levelled off. A linear model was used for the intra-pixel sensitivity variations, while a quadratic log-time model was used for the ramps.

in some studies [e.g. Knutson et al., 2009], perhaps due to a weak detector ramp in these detectors (see below). Figure 2.5 shows a linear model inx and y applied to secondary eclipse data for the TrES-4, for channels 1 and 2. It can be seen that the intra-pixel sensitivity variations are well accounted for by such a model. The disagreements between the models and the data in the middle of the observations are the eclipse signals.

Alternative techniques to remove this systematic are used also in the liter- ature. For example, Ballard et al. [2010] introduced a technique to create a point- by-point sensitivity map based on the flux measurements weighted by Gaussian functions in both spatial directions. Similarly, Stevenson et al. [2012] create a sen- sitivity map based on bi-linear interpolation (in space) of a grid of modelled flux ‘knots’, that span the range of x and y. The idea of these techniques is to account

for both large and small scale sensitivity variations. These techniques can improve the signal-to-noise on the residuals, but give consistent results when compared with polynomial fits [Stevenson et al., 2012; Blecic et al., 2013].

Detector ramp

The detector ramp effect is a systematic effect seen in Spitzer staring mode obser- vations for the channel 3 and 4 detectors. It is seen as a temporal rise in measured flux from individual pixels that is dependent on its illumination history [Knutson et al., 2007]. The effect for highly illuminated pixels tends to level off after a few hours, while less bright pixels have a linear trend over the time-scale of a typical exoplanet eclipse observation. In many cases the target object is too faint for the ramps in individual pixels to be seen (changes in pixel fluxes due to telescope point- ing variations dominate). However, the total flux from a star does show a clear ramp which is a sum of the individual pixel ramps (with the periodic flux variations being suppressed).

The physical explanation for this effect is believed to be that impurities in the Si:As detector material create charge traps which mask some fraction of the photoelectrons from the readout. In time, more and more charge traps are filled, meaning that fewer photo-electrons are lost from the readout i.e. the gain for the pixel increases. These charge traps also decay, releasing electrons which are then detected in the readout, though on a longer time-scale than the traps are filled. Eventually, the total decay rate of the traps will equal the rate at which the traps are filled and the pixel gain will remain constant [Agol et al., 2010].

In this thesis, I corrected for this effect by testing models that reproduce the qualitative behaviour of the ramp effect seen in the total source flux. These models take the form:

F lux=a0+atdt+attdt2, (2.2) F lux=a0+atln(dt+toff) +attln(dt+toff)2, (2.3)

F lux=a0+a1exp(a2dt) +a3exp(a4dt), (2.4)

with subsets of these equations being tested.dtis the time since the first observation anda0,a1,a2,a3,a4,at, andatt are coefficients (determined as described in Section 2.4.2.3). The parameter toff was used to ensure the argument of the logarithm did

not go to 0.

The latter model is physically motivated, as the behaviour of individual pixels is expected to be exponential in nature, as described in the toy model of Agol

et al. [2010]. They found correcting for individual pixels was not possible due to the pointing variations, but that a double exponential modelled the total flux sufficiently. Presumably, one of the exponential terms accounts for the ramp effect from the highly illuminated pixels, while the other models the fainter pixels.