Discussion

We have provided significant evidence that the pile up in hot Jupiters is real and not an observational artefact. The central value of the peak has been constrained to 3.7 days and the behaviour outside the peak is a rising power law. This implies that an excess of giant planets exist at a characteristic period, which must be a marker for their formation.

Disk migration is driven by torques exerted by material at resonant locations in the protoplanetary disk. The inward migration torque is predominantly provided by the most distant outer Lindblad resonant point where m = 1 in Eq. 1.24. The material orbiting interior to the corotation radius of the disk (the radius at which the material in the disk orbits with the same period as the stellar rotation period) is evacuated due to the experienced drag, so opens up a gap in the centre of the disk. When the outer Lindblad resonance point is interior to the inner edge of this gap, the planet no longer experiences inward torque from the disk and the migration halts. This point is when the inner disk edge is in 2:1 resonance with the planetary orbit [Lin et al., 1996; Kuchner & Lecar, 2002]. We therefore expect to see an over abundance of planets at a period half that of the rotation period at the time of migration, which is likely much faster than that seen today due to magnetic braking. The subsequent evolution of the host star has decreased the spin period so we must consider T Tauri stars. To explain the 3.7 day pile up observed in this study the rotation periods must be near 8 days, which has been found when studying the rotation rates of T Tauri stars in the Orion Nebula Cluster (ONC) [Herbst & Mundt, 2005]. The distribution of rotation rates in the ONC is shown in Fig. 3.25, though other clusters studied do not show such a clear feature indicating that this may be an interesting coincidence.

Another proposed argument for smooth disk migration leading to a pile up is where the inward migration is halted by mass loss due to Roche lobe overflow, widening the orbit [Trilling et al., 1998]. This transfer of mass provides a balancing torque to the inward torque from the disk. Once the disk dissipates the planet remains at the distance required to be Roche lobe filling, with depleted mass. This would lead to a mass deficit for planets which orbit within the pile up. Figure 3.26 shows the planetary mass against orbital period. A mass deficit does seem to be apparent around the peak pile up value calculated, and is visible especially when taking the running median of the dataset. This perhaps could be explained instead by low mass giants being evaporated close to the star, and low mass giants being undetectable to radial velocity measurements further from the star.

Figure 3.25: Rotation rates for T Tauri stars in the Orion Nebula Cluster. The vertical line represents twice the central peak value found when fitting the WASP and Kepler data. Adapted from Herbst & Mundt [2005].

1

2

5

10

Orbital period / days

0.1

0.2

0.5

1

2

5

10

20

50

Pla

ne

t m

as

s /

Mean

Median

Figure 3.26: Planetary mass against orbital period for planets Rp ≥ 8 R⊕. The red line represents the average of the dataset when grouped into period bins. The blue line represents the median of the dataset when grouped into period bins. The vertical dashed line marks the best fit pile up centre.

1 2 5 10 Roche fraction (x =a/aRL)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Norm ali se d un its

Figure 3.27: Roche limit distribution for the WASP planets before (grey) and after (red) correcting for selection effects. Distributions are normalised to unit area con- tained in the histogram. The vertical dashed line represents the inner edge value proposed by Ford & Rasio [2006].

Ford & Rasio [2006] predict an inner edge of pile up to occur at twice the Roche limit, by equating the initial orbital angular momentum with high eccentricity

e≈1 and final orbital angular momentum. We correct the WASP sample for selec- tion effects and plot the distribution ofx=a/aRL the ratio of observed separation to Roche separation in Fig. 3.27. By correcting for selection effects the central peak of the distribution moves further out to aroundx= 3.3 and arguably an inner edge does occur atx= 2. We observe a pile up rather than an abrupt cutoff, as perhaps tidal circularisation has not been strong enough to bring planets with larger Roche fractions into their expected value ofx= 2. If the dominant explanation for the pile up is from tidal recircularisation due to conversion of orbital energy to heat in the planet, we might expect to see inflated planets in the pile up region, depending on how long it is since migration. Each successive pass through the periastron causes the planetary orbit to circularise, and causes heating of the planet, which increases the radius, and as the cooling timescale the planet is longer than the orbital period, the planet remains inflated during the remainder of its orbit. Figure 3.28 shows the radius of hot Jupiters as distributed by period. It is uncertain at this point if this is the case due to the low number of inflated hot Jupiters, but a small peak in the radius is apparent at close to the pile up periods. If this peak becomes more significant it might be evidence to point to this migration method.

1 2 5 10 Orbital period / days

10 20 P la ne ta ry ra di us / R⊕

Figure 3.28: Radius for hot Jupiter planets withRp ≥8R⊕. Horizontal dashed lines mark the edges of the radius bins from Howard et al. [2012]. The vertical dashed line marks the peak of the pile up.

Our model of the period distribution includes a power-law termkPPβ which defines the longer period planets. The calculated power law slope is consistent with the power-law slope found by Howard et al. [2012] for the middle planet class where 4 ≤ Rp < 8 R⊕. This slope was found to be (Table 3.3) β = 0.79±0.50 and we find β = 1.01±0.25, and may suggest that the period distribution of both Jupiter class and Neptune class planets is similar, apart from clear excess at 3.7 days for the Jupiters.

The excess modelled for the radius distribution is likely due to the degen- erate mass radius relation for giant planets. Mass radius relations for giant plan- ets [e.g. Fortney et al., 2007] result in a 1 RJplanet for a wide range of masses and

so an excess of planets of this size is expected.

We note that the consistency between the WASP and Kepler occurrence rates is very high, and is likely a coincidence. WASP is constantly observing and detecting new planets, but the number of stars observed is not increasing at the same rate, which would have the effect of increasing the measured occurrence rate from WASP. Conversely repeat observations of the same stars provide a longer timebase of observations which allows longer periods to be investigated. Under a white noise dominated system adding more observations decreases the noise, which increases the significance of shallower transits and the detectability of smaller planets. This would alter the WASP sensitivity which would not increase the occurrence rates. In reality the true effect is likely to be between the two. The Kepler false positive rate was found to be 17.7±2.9% [Fressin et al., 2013], or 18.2±6.7% [Santerne et al., 2012] for giant planets with orbital periods < 10 days, which would decrease the

Howard et al. [2012] normalisation factor and cause a further discrepancy. Howard et al. [2012] note that the occurrence of hot Jupiters in the Kepler field is only 40% of that in the solar neighbourhood, in comparison with Marcy et al. [2005]. WASP typically targets stars in the solar neighbourhood which typically have solar metallicity. Conversely Kepler observe typically low metallicity stars [Howard et al., 2012]. Correlations between giant planet occurrence and host star metallicity have been found [e.g. Santos et al., 2003; Fischer & Valenti, 2005] so discrepancy in the absolute occurrence rate is expected.