The presence of the in-transit anomaly motivated the use of star spot models. Star spot occultations have been seen in the transit light curves of several planets, such as HD 189733b (Pont et al., 2007; Sing et al., 2011a), TrES-1b (Dittmann et al., 2009; Rabus et al., 2009) and CoRoT-2b (Huber et al., 2009; Wolter et al., 2009; Silva-Valio et al., 2010).
If a planet occults a spot (a region cooler than the surrounding photosphere) it blocks less of the stellar flux than compared with its transit across the hotter pristine stellar disc. This results in a bump during transit and therefore a smaller derived planetary radius. Star spot activity is not unexpected for WASP-52 since H´ebrard et al. (2013) found modulations in its light curve and chromospheric emis-
sion peaks in the CaII H+K lines. Using these modulations they calculated the
rotation period of WASP-52 to be 16.4 d.
Spot crossing events have been modelled with a variety of software, such as
Figure 4.6: These plots show different time-varying extinction coefficients fitted to the blue channel’s data. The left column shows the logarithm of the target’s flux (top panel), comparison’s flux (middle panel) and differential flux (bottom panel) against airmass. The differential flux, in particular, shows the hook-like feature indicating the time-varying nature of the extinction coefficient. If the extinction coefficient was constant in time, the flux would follow airmass along a straight line. The solid, colour lines in all of these plots indicate fits to the out of transit data (after masking the in-transit data) with a non-varying extinction coefficient (red line), linearly varying extinction coefficient (green line) and quadratically varying extinction coefficient (blue line). The right hand column shows the same plots but against time, not airmass.
0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Normalised flux 4 3 2 1 0 1 2
Time from mid-transit (hours) 0.030 0.025 0.020 0.015 0.010 0.005 0.000 Residuals
Figure 4.7: MCMC fits of analytic quadratic limb-darkened transit light curves (Mandel & Agol, 2002) revealing the presence of an in-transit anomaly. The upper
panel shows the fits to each of the three wavelengths; NaI (red), g0 (green) and u0
(blue), which are offset for clarity. The lower panel shows the residuals from these fits, which are again offset for clarity.
(B´eky et al., 2014). For this analysis I used spotrod to construct quadratically
limb-darkened transit light curves that included the effects of spot crossings. I
incorporated spotrod into an MCMC framework and fitted the light curves si-
multaneously and with the same long time-scale trends as before. I chose to use
spotrod due to the speed of its integration, which uses polar coordinates in the
projection plane. The integration with respect to the polar coordinate is done ana- lytically so that only the integration with respect to the radial coordinate needs to be performed numerically. To calculate the projection of the planet on the stellar
surface,spotrodcalculates the arrays of planar orbital elementsξand η, using the
formalism of P´al (2009), and assumes the same limb darkening law for the spot as
for the star.
Since my study of WASP-52b a recent paper introduced pytranspot, a
code to model multi-wavelength light curves that are affected by spots (Juvan et al., 2018). This improves on the pre-existing software for modelling spot-affected light curves, which are normally only set-up to deal with a single light curve.
When using my own fitting routines, the system parameters were again fit across the three light curves simultaneously (although this time fitting for impact
parameter, b, rather than the inclination, as required by spotrod) and with the
addition of the parameters defining the spots. The fitting of one spot was tested but was unable to fit both bumps on either side of the transit mid-point, therefore two spots were used in further analysis. The parameters defining each spot were the longitude, latitude, radius ratio of spot to star, and ratio of the spot flux to stellar flux (with 1 being a spot with the same flux as the pristine photosphere and
0 being a spot with zero flux). I heldu1 fixed as before but now also put Gaussian
priors on u2 with means equal to the values from Claret & Bloemen (2011) and
standard deviations from the propagated errors in the effective temperature and surface gravity of the host star. This prior was necessary as the limb darkening and spot models can play off each other in trying to fit the transit shape.
The MCMC was initiated with the system parameters equal to those in H´ebrard et al. (2013) and was run for 10 000 steps in burn in and another 10 000 steps in the production run. There were 31 fitted parameters with 124 walkers. For the spot starting parameters, I started one spot in the western hemisphere and one in the eastern hemisphere (as this configuration was necessary to explain the features either side of the transit, Fig. 4.7), and both at latitudes near the transit chord. I initiated them with radii 10% of the stellar radii and with contrasts of 1 (i.e. the same flux as the surrounding photosphere). This flux ratio was selected so as not to impose a spot on the star if the model did not need it. After the burn in
phase the error bars in the data points were rescaled to give a reducedχ2 of unity.
After the first MCMC chain, a second MCMC was run but this time with the parameters that were tied across channels fixed to the results from the first run
(a/R∗, b,Tc and the spot sizes and positions). Correlations with these parameters
causeRP/R∗to move up and down together across the three wavelengths, contribut-
ing to the uncertainty in the absolute planetary radius in each of the bands. Since we are concerned with the shape of the transmission spectrum, we are interested in the relative radii between the bands and not the absolute planetary radius, thus motivating the second run of the MCMC with fixed system parameters.
I present the best fitting spot model in Fig. 4.8, after the second MCMC run,
with the results in Table 4.1 and transmission spectrum in Fig. 4.16 (blue squares)2.
With the sizes and contrasts calculated fromspotrod, I was able to create
a schematic of the stellar surface (Fig. 4.9) and consider what filling factor would reproduce the derived contrasts (section 4.4.1 and Fig. 4.10). Fig. 4.9 displays the large regions of stellar activity along the transit chord. The second spot crossing
event composed of a smaller region of higher contrast (0.2 in theg0 band, Table 4.1).