I assessed the relative quality of the fits given by the detrending functions de- scribed in Sections 3.3.3.1 and 3.3.3.2 using the Bayesian Information Criterion [BIC, Schwarz, 1978], as described in Section 2.5.
Initially I performed MCMC analyses on individualSpitzer datasets to find the best 2 or 3 detrending functions in each case (i.e. with the lowest BIC value). For these tests I used datasets where the source flux had been extracted using aperture sizes of 2.5, 3.0 and 2.5 pixels in channels 1, 2 and 4 respectively. The results of these tests are given in Tables 3.3, 3.4 and 3.5. Note that in all of these comparisons of detrending functions, secondary eclipse errors were not rescaled to ensure BIC values were not affected by differences in rescaling between different models.
The best detrending functions, highlighted in these tables, were then tested in various combinations, using combined MCMC fits to all the IRAC datasets. Com- bined fits were used because I wanted to ensure parameters that were simultaneously constrained by all the datasets (e.g. ∆φ) were not affecting my conclusions about the detrending functions. In addition, motivated by systematic features in the channel 1 residuals (see Section 3.3.9) that could have been biasing parameters like ∆φ, I tested combinations of detrending functions using combined MCMC fits to the chan- nel 2 and 4 data only. My final choice for the channel 1 detrending function came from the combined fits to all the IRAC data, while the final choices for channels 2 and 4 came from the combined fits to the channel 2 and 4 data.
For channel 1 I found the best detrending function from the simultaneous fits to all IRAC data was:
Table 3.3: Eclipse depths and BIC values for fit to the channel 1 data only, using a variety of detrending models. Fits including quadratic terms are clearly favoured by the BIC and the eclipse depths derived from these fits are consistent with the final result of 0.209+0−0..040028%. The best two models from these individual fits were used in
the subsequent tests of simultaneous fits to data in all channels. The worst three detrending models given here show markedly poorer BIC values. This is a result of the lack of spatial detrending terms for these models.
Channel 1 detrending model (−Fˆ⋆) Eclipse depth (%) BIC =χ2+klnN a0+axdx+aydy+axxdx2+ayydy2 0.210±0.009 14969.7 a0+axdx+aydy+axxdx2+ayydy2+atdt 0.212±0.009 14977.1 a0+axdx+aydy+axydxdy+axxdx2+ ayydy2 0.212±0.009 14978.5 a0+axdx+aydy+axxdx2+ayydy2+ atln(dt+toff) 0.212±0.009 14984.9 a0+axdx+aydy 0.244±0.008 16012.6 a0+axdx+aydy+atdt 0.246±0.008 16028.2 a0+axdx+aydy+atln(dt+toff) 0.242±0.008 16040.9 a0+atln(dt+toff) 0.179±0.008 24218.0 a0+atdt 0.131±0.008 24331.1 a0 0.072±0.008 25899.3
Table 3.4: As Table 3.3, but for channel 2. Again detrending models with quadratic terms are favoured. The best three models here have eclipse depths consistent with the final result of 0.282±0.012%. As in channel 1, the best two models from these individual fits were used in the subsequent tests of simultaneous fits to data in channels 2 and 4. The models with no spatial detrending terms are not as markedly poor here as they were for channel 1 because the intra-pixel sensitivity effect is not as strong in channel 2.
Channel 2 detrending model (−Fˆ⋆) Eclipse depth (%) BIC =χ2+klnN a0+axdx+aydy+axxdx2+ayydy2+atdt 0.282±0.012 1637.7 a0+axdx+aydy+axxdx2+ayydy2 0.279±0.012 1639.1 a0+axdx+aydy+axydxdy+axxdx2+ ayydy2+atdt 0.281±0.012 1645.0 a0+axdx+aydy+axxdx2+ayydy2+ atln(dt+toff) 0.271±0.012 1647.4 a0+axdx+aydy 0.286±0.012 1661.5 a0+axdx+aydy+atdt 0.289±0.012 1667.5 a0+axdx+aydy+atln(dt+toff) 0.279±0.012 1677.2 a0+atdt 0.277±0.011 1815.9 a0+atln(dt+toff) 0.254±0.011 1827.8 a0 0.266±0.012 1860.9
Table 3.5: As Tables 3.3 and 3.4, but for channel 4 data. The best three detrending functions were chosen for the subsequent tests of simultaneous fits to data in all channels. The difference in eclipse depth for these three detrending functions is discussed in the text. The very poor performance of the worst model in this table highlights the necessity of including temporal terms in the detrending model.
Channel 4 detrending model (−Fˆ⋆) Eclipse depth (%) BIC =χ2+klnN a0+a1exp(a2dt) 0.371±0.046 1794.2 a0+atln(dt+toff) 0.332±0.034 1795.4 a0+a1exp(a2dt) +a3exp(a4dt) 0.339±0.050 1798.5 a0+atdt+attdt2 0.328±0.048 1801.4 a0+axdx+aydy+a1exp(a2dt) 0.366±0.047 1803.2 a0+atln(dt+toff) +attln(dt+toff)2 0.316±0.035 1807.9 a0+atdt 0.107±0.030 1827.3 a0 0.827±0.028 2119.2
Table 3.6: Eclipse depths and BIC values for channel 4, from the simultaneous fits to the channel 2 and 4 data. The first 10 minutes of channel 4 data were removed for these fits as these were the source of the inconsistencies in the channel 4 eclipse depths seen in Table 3.5. The channel 2 data was detrended using equation 3.3, while the detrending for channel 4 is shown in this table. The eclipse depths found were all consistent with each other, with the single exponential function being favoured by the BIC, as in the individual fits.
Detrending model (−Fˆ⋆) Eclipse depth (%) BIC =χ2+klnN a0+a1exp(a2dt) 0.329±0.052 3329.2
a0+atln(dt+toff) 0.333±0.032 3338.5
a0+a1exp(a2dt) +a3exp(a4dt) 0.326±0.053 3343.6
This is the same as was found for the individual fits. Table 3.3 shows that for the individual fits, there was little change in the eclipse depth for the best fitting detrending functions. This was also the case for the simultaneous fits. Equation 3.10 was therefore chosen as the detrending function for the final analysis of the channel 1 data.
In channel 2 the results followed a similar pattern. The best detrending func- tion from the simultaneous fits (both including and excluding the channel 1 data) was:
F⋆,model = ˆF⋆+a0+axdx+aydy+axxdx2+ayydy2+atdt. (3.11) as was found for the individual fits. Again there was little variation in the eclipse depths found from both the individual (see Table 3.4) and simultaneous fits. Equa- tion 3.11 was therefore chosen as the detrending function for the final analysis of the channel 2 data.
In channel 4, as in channel 2, there was consistency between the best fit to both the individual and combined datasets. The best detrending function was found to be the single exponential in time:
F⋆,model= ˆF⋆+a0+a1exp(a2dt). (3.12)
As shown in Table 3.5, this function was only marginally favoured over the linear logarithmic time (equation 3.5 with att = 0) and double exponential (equation 3.6) functions. In addition, the eclipse depth from the single exponential function only agrees with the other two at 1σ. I found this difference stemmed from data at the beginning of the time-series, where the single exponential function did not
give a good fit. With the first 10 minutes of data removed, these three models gave consistent channel 4 eclipse depths, with the single exponential again being favoured by the BIC (see Table 3.6). As a result, for the final analysis in Section 3.3.10 the channel 4 data was detrended using the single exponential function with the first 10 minutes of data removed.