Searching for Rapid Orbital Decay of WASP-18b

The TRAnsiting Planets and PlanetesImals Small Telescope—South (TRAPPIST-S; Gillon et al. 2011; Jehin et al. 2011) is a ground-based, 60 cm robotic telescope based at the La Silla Observatory used to study both exoplanets and small bodies in the solar system. TRAPPIST observed two WASP-18b photometric transits in the fall of 2015 in the broadband Sloan-z filter, centered at 0.9134 μm.

The HST observed WASP-18b in 2014 in spatial scan mode (Deming et al. 2013) over its full phase (PID 13467, PI: Bean), including one full transit, one full secondary eclipse, and one extra eclipse ingress. All observations were made with the Wide Field Camera 3 (WFC3) G141 infrared grism, covering 1.1–1.7 μm. While the primary deliverable from such observations is the spectrum, we sum over wavelength to extract a photometric light curve. To maximize observing efficiency, the scan reverses direction, rather than taking the time to reset to the starting point, at the end of each scan. This introduces a non-constant offset requiring separate analysis of the forward and reverse scans.

We reduce our TRAPPIST data in the methods described by Gillon et al. (2012). We calculated the best-fit transit curve for each observation using the TRAPPIST MCMC procedure (Gillon et al. 2009 and references therein), executing the Mandel & Agol (2002) algorithm to find the new best-fit light curve parameters. We generated the curve plotted in Figure 1 with the BATMAN procedure (Kreidberg 2015), given those orbital parameters.

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As has been studied extensively (e.g., Sing et al. 2016), the HST WFC3 camera, while improved over its predecessor NICMOS, has persistent systematic errors that seem to be a function of incident flux (Wilkins et al. 2014), with three distinctive effects: a visit-long ramp, an orbit-long ramp, and a "hook" within orbits (Berta et al. 2012; Wilkins et al. 2014). We reduce the WFC3 data and mitigate systematics in a modified divide-oot method—a method of averaging out all three systematic effects (Deming et al. 2013; Wilkins et al. 2014), including the correction to the STScI wavelength calibrations found in Wilkins et al. (2014). To fit the transit, we use the nonlinear, fourth-order limb darkening coefficients from Claret (2000) in the Mandel & Agol (2002) light curve models, and derive "prayer-bead" error bars as in Gillon et al. (2009). To fit the the secondary eclipse, we use the same procedure in the limit of no limb darkening, such that the shape is that of a trapezoid. We analyze the forward and reverse scans independently, as mentioned in Section 2, due to a nonlinear offset between the two; the final timing results agree and are thus shown as an average in the table.

Tides raised within a central star by a planetary companion are dissipated within the star, and angular momentum is transferred between the stellar spin and planetary orbit in the process (e.g., Ogilvie 2014). For short-period planets (orbiting sub-synchronously rotating stars), like WASP-18b, that also have approximately circular orbits, tidal dissipation in the star causes the planet to lose angular momentum and spiral inward, because the tidal bulge raised in the star lags the planet when the planet's orbital period is less than the star's rotational period (i.e., P_orb < P_rot). This is the opposite of the Earth–Moon system, in which the Moon recedes from the Earth because the bulge leads the Moon (since P_orb > P_rot). The rate of change of the orbit depends on the efficiency of tidal dissipation within the host star; this is where stellar structure becomes important.

The tide in the star is often decomposed into two contributions: an equilibrium tide and a dynamical tide (e.g., Zahn 1977). Dissipation of both components is expected to become less efficient in stars slightly more massive than the Sun (i.e., F stars). While we often generalize Sun-like stars (typically defined as 0.5${M}_{\odot }$ ≲ M_* ≲ 1.3 ${M}_{\odot }$) to have radiative cores and convective envelopes and more massive stars to have the opposite, development of convective cores and radiative envelopes is actually a continuum. WASP-18, for example, is a 1.2 ${M}_{\odot }$ F6 star and, according to MESA stellar structure models (Paxton et al. 2011), should have a convective core within the innermost 6% of the stellar radius and a convective envelope in the outer 15%; it is therefore intermediate between a solar-mass and high-mass star. For tidal dissipation, therefore, an F star like WASP-18 is not "Sun-like."

We quantify the efficiency of tidal dissipation using the tidal quality factor, Q, defined as (Goldreich 1963)

$$\begin{eqnarray}&&Q\equiv \displaystyle \frac{\mathrm{energy}\ \mathrm{stored}\ \mathrm{in}\ \mathrm{tidal}\ \mathrm{distortion}}{\mathrm{energy}\ \mathrm{dissipated}\ \mathrm{in}\ \mathrm{one}\ \mathrm{cycle}}=2\pi {E}_{0}{\left(\oint -\dot{E}{dt}\right)}^{-1},\end{eqnarray} \tag{ 1 }$$

where E₀ is the maximum energy stored in the tidal bulge and $\dot{E}$, intrinsically negative, is the energy dissipated in one tidal period. We use the modified Q (i.e., Q') convention throughout this Letter:

$$\begin{eqnarray}&&{Q}_{* ,0}^{\prime }\equiv \displaystyle \frac{3Q}{2{k}_{2}},\end{eqnarray} \tag{ 2 }$$

where k₂ is the tidal Love number. Q' is almost certainly not a single constant number for all stars (even of the same spectral type), but is instead a complicated function of the stellar mass, structure, rotation, and tidal periods, as well as the planetary properties (e.g., Ogilvie 2014). Q' is the Q of an equivalent homogeneous body (k₂ = 3/2). A large Q' corresponds to weak or inefficient tidal dissipation, and a smaller Q' corresponds to strong or efficient dissipation. We investigate here the Q' of the star (WASP-18), not the planet (WASP-18b); the planet's tidal Q' is relevant for its own tidal evolution, and leads to synchronization of its rotation and circularization of its orbit.

The equilibrium tide is dissipated within the convective envelope of the star by the effective viscosity of convective turbulence (Zahn 1966; Goldreich & Nicholson 1977); however, the effective viscosity may be significantly reduced in the case of a short-period planet (Penev & Sasselov 2011; Ogilvie & Lesur 2012). In addition, in F stars, the outer convection zone is thin and of very low mass, so it is expected to be much less dissipative than in G stars; the effective tidal Q' could be as high as 10¹¹ (Barker & Ogilvie 2009) for a star like WASP-18 at the tidal frequencies of interest. Dissipation in the convective core of an F star is also likely weak (e.g., Zahn 1977).

The dynamical tide primarily consists of internal gravity (g-mode) waves that are tidally excited at the convective-envelope–radiative-core boundary and propagate inward to the center of the star. These waves are thought to be damped by radiative diffusion or nonlinear effects. If they can reach the center, they become geometrically focused, and if the planet exciting them is sufficiently massive—like WASP-18b—they may reach sufficiently large amplitudes such that they break, leading to significantly enhanced tidal dissipation (Goodman & Dickson 1998; Ogilvie & Lin 2007; Barker & Ogilvie 2010; Barker 2011). This process deposits angular momentum into the star, thereby removing angular momentum from the planet's orbit; the star's rotation gets faster ("spin-up"), while the planet's orbit shrinks. If WASP-18 were Sun-like, WASP-18b would be sufficiently massive to cause wave breaking, and we would expect the planet to rapidly spiral into its star. However, in the case of an F star like WASP-18, the convective core prevents the tidally excited gravity waves from reaching the center where they would be focused, so that they may never reach such large amplitudes to break, though they may be subject to weaker nonlinear effects (e.g., Barker & Ogilvie 2011; Weinberg et al. 2012; Essick & Weinberg 2016). The dissipation would be significantly reduced, save for select resonant tidal frequencies, so that we would expect the planet to remain in the orbit in which it was discovered (Barker & Ogilvie 2009). Furthermore, the lingering thin outer convective envelope in an F star of WASP-18's mass would inhibit radiative damping of the waves near the top of the radiative zone (relative to more massive A stars). The dissipation that Valsecchi & Rasio (2014) find for WASP-71 may be moderately higher than we would expect in WASP-18 precisely because it is a more massive (1.5 versus 1.2 M_⊙) star, and therefore has a thinner outer convective envelope than WASP-18, but what they obtain is still very weak.

Were a resonance present, Q' could indeed be very low, and therefore the star could be quite dissipative. However, the above arguments and those of, e.g., Lanza et al. (2011) and Barker & Ogilvie (2009), support a high-Q', generally minimally dissipative scenario for a star like WASP-18.

When a planet transits its host star, we have a convenient time point from which to measure any changes in the orbit, which we infer through a shift in the transit (or secondary eclipse) arrival time. Birkby et al. (2014) show that the expected shift can be reduced to

$$\begin{eqnarray}&&{T}_{\mathrm{shift}}=-\left(\displaystyle \frac{27}{8}\right)\left(\displaystyle \frac{{M}_{p}}{{M}_{* }}\right){\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{5}\left(\displaystyle \frac{2\pi }{P}\right)\left(\displaystyle \frac{1}{{Q}_{* }^{\prime} }\right){T}^{2},\end{eqnarray} \tag{ 3 }$$

where ${M}_{p}/{M}_{* }$ is the planet-to-star mass ratio (for WASP-18, ${M}_{p}/{M}_{* }$ = 0.007843), ${R}_{* }/a$ is the stellar-radius-to-semi-major-axis ratio (for WASP-18, ${R}_{* }/a$ = 0.2789), T is the elapsed time, and P is the orbital period of the planet. Therefore, in a quadratic fit of the form

$$\begin{eqnarray}&&t={{qT}}^{2}+{pT}+c,\end{eqnarray} \tag{ 4 }$$

where the linear coefficient p corresponds to the period of the planet's orbit, the quadratic term is defined by Equation (3). Rearranging, we find that Q' depends on the quadratic coefficient q as

$$\begin{eqnarray}&&{\text{}}Q^{\prime} =-\left(\displaystyle \frac{27}{8}\right)\left(\displaystyle \frac{{M}_{p}}{{M}_{* }}\right){\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{5}\left(\displaystyle \frac{2\pi }{P}\right)\left(\displaystyle \frac{1}{q}\right).\end{eqnarray} \tag{ 5 }$$

We fit for the coefficients in Equation (4), and thus the period and Q', as discussed in Section 4 and shown in Figure 2.

Equation (3) makes it clear that a planet must be close-in and massive (relative to the radius and mass of the host star), its orbital period must be short, and it must orbit a star with a favorable $Q^{\prime}$, in order to produce any discernible shift in time. Currently, in the NASA Exoplanet Archive, only eight confirmed planets have both masses larger than 1.0 M_J and orbital periods of roughly one day or less. The addition of recently announced KELT-16b (Oberst et al. 2016) makes nine. Of those, one is around a pulsar and four (WASP-18b, KELT-16b, WASP-12b, and WASP-103b) are around stars more massive than 1.2 M_⊙ and therefore likely possessing convective cores that preclude tidal wave breaking at the center. Of the remaining four, WASP-43b has already demonstrated no rapid tidal decay (Hoyer et al. 2016), but WASP-19, WTS-2, and K2-22 all orbit stars less massive than the Sun, and may be reasonable testbeds for dissipation within a star with a larger convective envelope and a smaller radiative core; Birkby et al. (2014) have already suggested that WTS-2 should have a barely discernible shift for Q' = 10⁶ (17 s over 16 years).

Equation (3) assumes a stellar obliquity of zero and neglects tidal dissipation in the planet, assuming its orbit to be circularized and its spin to be synchronized and aligned with the orbit. The canonical value of Q' is 10⁶, as derived for stars from measurements of the orbits of binary star systems (e.g., Meibom & Mathieu 2005) and for solar system giant planets from the orbits of their satellites (Zhang & Hamilton 2008).

We return to Figure 2, as we can now interpret the findings for q (and therefore Q') physically. The 95th percentile posterior probability distribution for q is effectively zero; given that Q' ∝ $\tfrac{1}{q}$, this means we only can definitively extract a lower limit, Q' ≥ 1 × 10⁶, taken at the 5th percentile of the q distribution. Continued monitoring of this system should further constrain WASP-18's Q', and it follows from the discussion above that we will continue to find an increasing lower limit, i.e., no evidence of rapid tidal decay.