Eclipse depth uncertainty estimates

Uncertainties for the eclipse depths in each channel, for the models usingNf = 5−7,

were assessed in two ways: using a Markov Chain Monte Carlo (MCMC) routine and using a ‘prayer-bead’ method.

The MCMC routine I used here was different to that described and used in Chapters 2–4. I used the Python program emcee [Foreman-Mackey et al., 2013], which utilises a slightly different algorithm to the Metropolis-Hastings algorithm described in Section 2.4.2. Briefly, the algorithm simultaneously evolves an ensemble of points in the model parameter space (known as ‘walkers’). At each step in the chain the proposal step for a given walker is found by drawing randomly from the current state of all the other walkers in the ensemble and then perturbing from this, based on the parameter differences of the two walkers. Comparison of the posterior probabilities of the current and proposed model fits (given by equation 2.26) are then used to determine if the proposal is accepted, in a similar way to the Metropolis- Hastings algorithm. If the proposal is rejected, the state of the walker remains the same for the next step in the chain (note the slight difference here compared to the mcmctransit algorithm). All of the walkers in the ensemble are updated in this way before moving onto the next step.

The light curve model used to fit the WASP-33 data in the MCMC routine was as described in Sections 5.3.5 and 5.3.6, with separate instances of equation 5.6 being fitted to each channel, phase values being fixed across the three channels

−4.0 −3.5 −3.0 −2.5 −4.0 −3.5 −3.0 −2.5 log10 (norm ali se d re sid ua l RM S) −5.0 −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 −1.5

log

Figure 5.13: Binned normalised residual RMS values as a function of bin width (dt). From top to bottom, plots are for the red, green and blue channels of ULTRACAM. The different line colours represent fits using different numbers of frequencies,Nf.

Black is forNf = 5, red is for Nf = 6 and blue is forNf = 7. The dotted lines show

the 1/√nexpectation for white noise, where n is the number of data points per bin. This expectation is fixed to the unbinned RMS value for the fit using Nf = 5. In

each channel the binned RMS values diverge from the the white noise expectation, motivating the use of the prayer bead error analysis in Section 5.3.7.

and frequencies fixed to the values given in Table 5.1. The MCMC jump parameters were therefore the same as the free parameters used in the Levenberg-Marquardt fit in Section 5.3.6. In order to obtain appropriate eclipse depth uncertainties, the differential flux errors were rescaled such that the best fit (found in Section 5.3.6) gave a reduced χ2 of 1 in each channel. I also note here that uniform priors were placed on all of the jump parameters.

I used an ensemble of 500 walkers for which the parameter values were ini- tialised using the Levenberg-Marquardt minimisation solution (from Section 5.3.6). Each walker was perturbed from this, in each dimension of the model parameter space, by a Gaussian random number with standard deviation of 5σ (where σ val- ues were found for each jump parameter from an initial MCMC run). A burn-in

Table 5.2: Eclipse depth and uncertainty estimates derived from light curve models usingNf = 5–7 (see equation 5.6 and Table 5.1). The eclipse depths are taken from

the Levenberg-Marquardt fit (Section 5.3.6), while uncertainty estimates derived from the MCMC and prayer bead analyses (Section 5.3.7) are given in the fourth and fifth columns, respectively. The adopted uncertainty value was taken as the largest value from these two analyses.

Channel Nf Eclipse depth (%) MCMC (%) Prayer (%) Adopted values (%)

Red 5 0.052 _±0.006 −+00..016017 0.052+0−0..016017 Red 6 0.039 _±0.007 −+00..020018 0.039+0−0..020018 Red 7 0.057 _±0.007 −+00..008017 0.057+0−0..008017 Green 5 -0.012 _±0.008 +0−₀._.017₀₁₅ −0.012 +0.017 −_0.015 Green 6 -0.033 _±0.010 −+00..016039 −0.033+0−0..016039 Green 7 0.037 _±0.017 +0−₀._.066₀₀₃ 0.037 +0.066 −_0.017 Blue 5 -0.065 _±0.021 +0−₀._.071₀₅₈ −0.065 +0.071 −_0.058 Blue 6 -0.107 _±0.023 −+00..072060 −0.107+0−0..072060 Blue 7 -0.001 _±0.028 +0−₀._.024₀₂₀ −0.001±0.028

phase of 2_×103 steps was carried out before a production run of 2_×104 steps. From the resulting posterior distribution, the uncertainties on the eclipse depths in each channel were found from the 68 % confidence intervals of the marginalised dis- tributions for these parameters (i.e. their histograms). Separate MCMC runs were carried out for models using Nf = 5, 6 and 7. The resulting uncertainty values can

be found in Table 5.2.

Along with the assessment of uncertainties from the MCMC routine, I was motivated to perform a ‘prayer bead’ analysis [Gillon et al., 2007] after assessing the trend in the binned residual RMS values in each channel, as shown in Figure 5.13. The RMS values tend to diverge from the white noise expectation in each of the channels - a sign that the data is affected by red noise. Note that the red noise signal is suppressed for fits using more frequency modes, suggesting that residual pulsations are driving the red noise signal.

The prayer bead analysis I performed here was similar to that described for WASP-3 in Section 3.3.10, where the residuals to the light curve fits were sub- tracted from the raw differential flux data and then added back in with a cyclic offset in phase. I used 5000 residual shifts for this analysis and for each shift the model described in Section 5.3.6 (and equation 5.6) was fitted to the data using a Levenberg-Marquardt χ2 minimisation algorithm [Press et al., 1992]. From the

Table 5.3: Final eclipse depth results for the secondary eclipses in each of the UL- TRACAM channels.

Channel Filter Eclipse depth Red z’ 0.057+0−₀._.008₀₁₇% Green Bcont <0.155 % (3σ)

Blue u’ <0.069 % (3σ)

resulting distribution of eclipse depths, the 1σ eclipse depth uncertainty was found as the 68 % confidence interval centred on the value from the original fit (that had no residual shift applied). This analysis was applied to each of the ULTRACAM channels separately, e.g. to assess the prayer bead errors in the red channel eclipse depth, the blue and green channel data were not modified. As with the MCMC analysis, prayer bead errors were derived for models using Nf = 5, 6 and 7. The

resulting eclipse depth errors from this prayer bead analysis can be found in Table 5.2.

The final column of Table 5.2 gives the adopted error values in each channel, for each Nf. The adopted error was chosen as the largest error resulting from the

MCMC and prayer bead analyses. The chosen errors are plotted on the eclipse depth values for theNf = 5−7 fits in Figure 5.12.

The main conclusion from these error analyses is that the eclipse in the red channel (the z’ band) is detected significantly and consistently across the model fits usingNf = 5−7. The z’ band eclipse depths for other Nf values are also consistent

within the errors derived here. Because theNf = 7 solution gives the best fit to the

data in the red channel, I use the eclipse depth and uncertainty values from this fit as the final value.

For the green and blue channels, the error analysis shows that eclipse depths are not reliably found across the Nf = 5−7 fits, and are not clearly distinguished

from 0. As a result, I only place upper limits on the eclipse depths for the green and blue channels.

5.4

Results

The final results from the analysis described above are given in Table 5.3. The main result I report in this chapter is the detection of a secondary eclipse signal from WASP-33b in the z’ band. The eclipse depth is ∆Fz= 0.057+0−0..008017% and this

method described in Section 2.6.1).

The results for the other two ULTRACAM bands do not show clear detections of an eclipse signal and so I have placed 3σ upper limits on the eclipse depth values here. The upper limits were taken as the 3σ limit of the distribution resulting from either the MCMC or the prayer bead analysis, whichever was largest. The model fits usingNf = 7 were used here. For the Blue continuum band (green channel) the

limit is ∆FBcont <0.155 % (resulting from the prayer bead analysis), while for the

u’ band (blue channel) it is ∆Fu <0.069 % (resulting from the MCMC analysis).

The top row of Figure 5.14 shows the normalised differential fluxes in each channel along with theNf = 7 model, which was used to derive the eclipse depth in

the z’ band. Note that the models shown for the Blue continuum and u’ bands do not reflect the final results in these bands - upper limits were given on the eclipse depths here instead. The middle row of Figure 5.14 shows the normalised differential fluxes that have had the pulsation and airmass model components removed, leaving the planetary flux signal. In the z’ band the eclipse is clearly seen. A non-zero eclipse depth is also seen in the Blue continuum band, but this is not adopted due to the variations seen across the fits usingNf = 5−7 (but see Section 5.5.3). Finally, in

the u’ band no eclipse is seen.

5.5

Discussion