Planet-induced Stellar Pulsations in HAT-P-2's Eccentric System

We model transits and occultations following Mandel & Agol (2002) and allow the eclipse times to vary to search for possible orbital evolution. We model HAT-P-2's transient heating using the functional forms introduced in Lewis et al. (2013). We find that the asymmetric Lorentzian function is favored over the function based on harmonics in orbital phase (ΔBIC = −7). This difference is due to 2 new partial phase curves obtained after periastron passage together with 16 new occultations that provide a stronger leverage on HAT-P-2 b's flux modulation around occultation allowing a better disentanglement from the instrumental systematics.

The change in position of the target's point-spread function (PSF) over a detector with non-uniform intrapixel sensitivity leads to apparent flux variations that are strongly correlated with the PSF position. We account for this effect using the same implementation of the pixel-mapping method as in Lewis et al. (2013)—see, e.g., Wong et al. (2015) and Ingalls et al. (2016) for detailed method comparisons.

IRAC observations often display an exponential increase in flux at the beginning of a new observation. The standard procedure for 4.5 μm observations is to trim the first hour at the start of each observation and subsequent downlinks, which reduces the complexity of fits with minimal loss of information on the eclipse shape and time (Lewis et al. 2013). However, we find here that the photometry around the primary and secondary eclipses shows a positive trend. Individual primary and secondary eclipse fits allowing for the correction of a linear trend in the photometry show that the trend is consistent across all epochs and has a slope of 65 ± 6 ppm hr⁻¹ (see Table 1). Similarly to Stevenson et al. (2012), we find that this trend is best accounted for with a simple exponential function.

While fitting HAT-P-2's photometry with the standard models introduced above, we observed additional signals in the photometry in the forms of oscillations (Figure 1). We account for the oscillations by including sine functions in our global fit—details in Sections 3.4 and 4.1.

We find that the standard deviation of our best-fit residuals is a factor of 1.11 higher than the predicted photon noise limit at 4.5 μm. We expect that this noise is most likely instrumental in nature and account for it in our error estimates using a scaling factor β_red for the measurement errobars (Gillon et al. 2010). We find that the average β_red is 1.3, the maximum values corresponding to a timescale of ∼50 minutes.

We find no sign of variability across the 18 occultations obtained with Spitzer at 4.5 μm. The occultation depths estimated via individual fits are shown in Figure 2(A) and are all consistent with the global-fit estimate within ∼1σ. This implies that the thermal structure of HAT-P-2 b's 4.5 μm photosphere around occultation shows no sign of orbit-to-orbit variability. This lack of significant orbit-to-orbit variability is consistent with the predictions presented in Lewis et al. (2014).

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The reproducibility of the parameter estimates across 18 occultations obtained over 4 years also further emphasizes the reliability of current techniques used for the acquisition and analysis of Warm Spitzer measurements (see also, e.g., Wong et al. 2015, 2016; Ingalls et al. 2016).

We find no sign of orbital evolution from the estimated times of primary and secondary eclipses (Table 1). We derive a new ephemeris from a simultaneous fit of our measured primary and secondary eclipse times, those from the literature (all but the 4.5 μm eclipse times gathered in Lewis et al. 2013) and HAT-P-2's radial velocity (RV) measurements¹⁹ (Table 1). Our new observations extend the previous baseline by a factor of two and the number of occultations by a factor of five yielding ultra-precise estimates of HAT-P-2 b's orbital period, midtransit time, orbital eccentricity, and argument of periastron:

$$\begin{eqnarray}P & = & 5.6334675\pm 1.3\times {10}^{-6}{\rm{}}\,{\rm{days}}\\ {T}_{c,0} & = & 2455288.84969\pm 0.00039{\rm{}}\,{\rm{BJD}}\\ e & = & 0.51023\pm 0.00042\\ \omega & = & 188\buildrel{\circ}\over{.} 44\pm 0\buildrel{\circ}\over{.} 43.\end{eqnarray} \tag{ 1 }$$

Our global fit of HAT-P-2's photometry also yields improved constraints on the transit depth (4941 ± 39 ppm), the occultation depth (971 ± 21 ppm, which translates into a hemisphere-averaged brightness temperature of 2182 ± 27 K), the orbital inclination (8616 ± 026), and stellar density (0.434 ± 0.020 g cm⁻³). We also derive from the ultra-precise eclipse times the 3σ upper limit on the variation of HAT-P-2 b's orbital eccentricity and argument of periastron, respectively, 00012 and 090 per year.

The extensive coverage of HAT-P-2 b's occultation and post-periastron passage allows us to disentangle further systematics from the planetary flux modulation. The result is a phase curve (Figure 1) whose main difference with Lewis et al. (2013) is a lower minimum—322 ± 69 ppm, which translates into a hemisphere-averaged temperature of 1327 ± 106 K. We find an excellent agreement on the timing and amplitude of the planetary flux peak, 5.40 ± 0.57 hr after periastron passage and 1178 ± 34 ppm—which translates into a hemisphere-averaged temperature of 2425 ± 40 K. The timescales for the planetary flux increase and decrease are respectively 5.5 ± 1.1 hr and 10.3 ± 1.5 hr. Because of the orbital geometry of HAT-P-2 b with periastron passage occurring midway between transit and occultation, the measured flux increase and decrease timescales are likely an overestimate of the true radiative timescale of HAT-P-2 b's atmosphere near the 4.5 μm photosphere, but are consistent with a short atmospheric radiative timescale of a few hours. The results presented here are in excellent agreement with the predictions from three-dimensional general circulation models presented in Lewis et al. (2014).

Our analysis also reveals distinct pulsations with a period of approximately 87 minutes in HAT-P-2's photometry (Figure 1). The oscillations are observed during the occultations, which implies that if they are astrophysical in nature they must originate from the host star and not the planet. We verify that the pulsations are not due to a strong instrumental signal in a subset of the AORs that would then be diluted by the inclusion of additional observations without this signal. We first explore this possibility by showing that the oscillations are consistently visible across subsets of AORs. To do so, we randomly picked off the 18 occultation AORs, 25 random sets of 9 AORs, 25 random sets of 5 AORs, and 25 random sets of 3 AORs and included in our model a sine function. We found that all but two of the sets of three AORs and one of five AORs yield a consistent detection of the pulsation. The consistency of the retrieved pulsation implies that, although the pulsation is not detectable from an individual AOR, the SNR of five AORs combined is sufficient to yield a >2σ detection. Furthermore, its consistency over these different AOR subsets indicates its coherence. We explore this further by including a sine curve function in the model and allowing its amplitude to vary individually for each occultation AOR. We find that the individual amplitudes scatter around 40 ppm, while the significance of the pulsation detection in each individual visit is less than 1σ (Figure 2(B)). The pulsations are hence present in each individual occultation AOR, but its study requires fitting multiple AORs.

The transit photometry, however, shows no sign of such short-period oscillations (Figure 1(B) and Table 1). Similar individual fits lead to an oscillation amplitude of 3 ± 15 ppm in transit. The pulsations build coherently when we combine 18 distinct epochs of data, which implies that the pulsation is a harmonic of the planet orbital frequency, as shown by the periodogram of the residuals (Figure 3). The two most significant peaks correspond to HAT-P-2 b's 79th and 91st orbital harmonics, which are found to interfere constructively around occultations but destructively around transits as a result of the relative phase of the two harmonics. The pulsations are therefore astrophysical in nature unless the instrumental systematics appear to be harmonics of HAT-P-2 b's orbital frequency phased in such a way that they build coherently over all the different epochs. This scenario would require that the modulation of the detector gain/noise or the PSF position/shape²⁰ follows a repeated pattern of oscillatory changes that are not only harmonics of the planet's orbital frequency, but also constructively phased. We use the noise-pixel parameter (Mighell 2005; Lewis et al. 2013) as a proxy for any changes in the PSF position and shape. We use the "sky" background as a proxy for gain variations and electronic noise. We find that the periodograms of the PSF position (Figures 4(B) and (C)) and of the noise-pixel parameter (Figure 4(D)) show no significant peak in the period range related to the pulsations while showing a significant signal at Spitzer's pointing oscillation period. Similarly, the periodogram of the "sky" background shows no specific peak around the 79th and 91st planetary harmonics as revealed in the photometry, ruling out the scenario of constructively phased and adequately modulated instrumental systematics.

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We therefore conclude that the pulsations are stellar and not instrumental in nature, and we account for them by including two sine functions in the final version of our global fits. We find that the pulsations have respective amplitudes and frequencies of 35 ± 7 and 28 ± 6 ppm and 162.335 ± 0.015 and 186.976 ± 0.013 μHz—respectively, 79.006 ± 0.007 and 0.006 ± 0.007 times the orbital frequency.

HAT-P-2's RV measurements provide a complementary perspective to Spitzer's. In addition to showing a high level of stellar jitter (∼36 m s⁻¹; discussed in Section 4.2), the RV measurements suggest that HAT-P-2 b's orbit is evolving in a way that is inconsistent with our previous fits to the photometry alone. Figure 5 shows that an independent analysis of the first half of HAT-P-2's RV data (obtained prior to BJD 2454604) yields values for e and ω that are inconsistent at the 4.5σ level with those from a fit of the second half of the data (obtained after BJD 2455466). This translates into respective variations of e and ω of 8.9 ± 2.8E⁻⁴ and 091 ± 031 per year, which is inconsistent with the limits derived from the primary and secondary eclipse times alone.

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The exact correspondence of the stellar pulsation frequencies with harmonics of the planet's orbital frequency implies a connection between the two. Two scenarios are possible; either the pulsations are altering the planet's orbit or the planet is causing the pulsation. Mode-planet gravitational interactions could synchronize the planet's orbital frequency to an integer harmonic of a mode oscillation period, analogous to two weakly coupled oscillators that synchronize in the presence of weak damping. However, the detection of multiple harmonics of the planet's orbital frequency rules out this scenario.

We next consider the hypothesis that the observed stellar pulsations are induced by the orbiting planet. Tidally excited pulsations have been previously observed in several eccentric stellar binaries (Welsh et al. 2011; Thompson et al. 2012; Beck et al. 2014) known as "heartbeat" stars. The characteristic feature of a tidally excited pulsation is that its frequency is an exact integer harmonic of the orbital frequency because it arises from a resonantly forced stellar pulsation mode (Derekas et al. 2011; Burkart et al. 2012; Fuller & Lai 2012; O'Leary & Burkart 2014; Smullen & Kobulnicky 2015).

In order to understand the nature of the detected pulsations, we calculate the stellar response to tidal forcing. We first calculate the non-adiabatic stellar pulsation mode spectrum of HAT-P-2 using the MESA evolution code (Paxton et al. 2011, 2013) to describe the stellar evolution and the GYRE pulsation code (Townsend & Teitler 2013) to compute the pulsation modes. We then calculate the response of each pulsation mode to the tidal forcing and compute the amplitude and frequencies of each tidally excited pulsation. Our models indicate that although HAT-P-2 b can in principle create luminosity fluctuations of this amplitude in its host, we expect these pulsations to have much lower frequencies, in the vicinity of twice the periastron orbital frequency (Hut 1981)—approximately 10 μHz (period of approximately 28 hr). Our models are unable to reproduce a tidally excited pulsation at the 79th or 91th orbital harmonics at the observed amplitude; the strength of the forcing at these harmonics is many orders of magnitude weaker than at the tidal forcing peak near the 6th orbital harmonic.

Although HAT-P-2 lies near the extreme cool ends of the γ-Doradus and δ-Scuti instability strips (T_eff ∼ 6300 K, log₁₀g ∼ 4.16, R ∼ 1.54 R_Sun; Uytterhoeven et al. 2011; Balona et al. 2015), it shows no evidence of the large amplitude (∼mmag) pulsations typically observed in these pulsators. Nonetheless, it is possible that the observed pulsations correspond to low-amplitude δ-Scuti pulsation modes, as our non-adiabatic pulsation calculations indicate that l = 1 and l = 2 pulsation modes are unstable near the observed frequencies of f ∼ 175 μHz, although we caution that mode growth rates—the parameter indicating stability of a pulsation mode—may be altered by a more realistic treatment of convective flux perturbations. In our models, the unstable modes correspond to low-order (n < 3) mixed pressure-gravity modes. In principle, these modes can couple strongly with HAT-P-2 b because they produce relatively large gravitational perturbations (i.e., they are similar to fundamental modes). However, our models suggest that they have frequencies that are too large to resonantly interact with the planet.

The failure of our models to explain the observed high-frequency pulsations suggests that current tidal theories may be incomplete. One tentative possibility is that nonlinear effects are important. The difference between the observed pulsations is ≃12 times the orbital frequency where resonant interactions with the planet can occur. This sort of nonlinear mode splitting at orbital harmonics has been observed in stellar systems (see, e.g., Hambleton et al. 2013) and could occur in the HAT-P-2 system as well.

Stellar pulsations cause RV variations of the stellar photosphere (e.g., Willems & Aerts 2002; Welsh et al. 2011). The RV amplitude of the photospheric motion due to a pressure mode is Δv ∼ 2πfΔR, where ΔR is the mode surface displacement. Our non-adiabatic pulsation calculations indicate that the stellar surface displacements and luminosity variations are approximately related by ΔR/R ∼ 2ΔL/L for modes near the observed frequencies. The observed photometric amplitudes imply Δv ∼ 60 m s⁻¹, consistent with HAT-P-2's large RV jitter. Mode identification would be required for a more precise calculation, but we posit that HAT-P-2's RV jitter and HAT-P-2 b's apparent orbital evolution in the RV data is mainly produced by the pulsations.