Planet-star interactions with precise transit timing. II. The radial-velocity tides and a tighter constraint on the orbital decay rate in the WASP-18 system
G. Maciejewski, H. A. Knutson, A. W. Howard, H. Isaacson, E. Fernandez-Lajus, R. P. Di Sisto, C. Migaszewski
aa r X i v : . [ a s t r o - ph . E P ] A p r Planet-star interactions with precise transit timing. II. Theradial-velocity tides and a tighter constraint on the orbital decay ratein the WASP-18 system
G. M a c i e j e w s k i , H. A. K n u t s o n , A. W. H o w a r d ,H. I s a a c s o n , E. F e r n á n d e z - L a j ú s , , R. P. D i S i s t o , ,C. M i g a s z e w s k i Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, NicolausCopernicus University, Grudziadzka 5, 87-100 Toru´n, Poland, e-mail: [email protected] Division of Geological and Planetary Sciences, California Institute of Technology, 1200E. California Blvd., Pasadena, CA 91125, USA Institute for Astronomy, University of Hawaii, Honolulu, HI 96822, USA Department of Astronomy, University of California Berkeley, Berkeley CA 94720, USA Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata,1900 La Plata, Buenos Aires, Argentina Instituto de Astrofísica de La Plata (CCT La Plata C CONICET/UNLP), Paseo delBosque s/n, La Plata, BA, B1900FWA, Argentina
April, 2020
ABSTRACTFrom its discovery, the WASP-18 system with its massive transiting planet on a tight orbit wasidentified as a unique laboratory for studies on tidal planet-star interactions. In an analysis of Dopplerdata, which include five new measurements obtained with the HIRES/Keck-I instrument between2012 and 2018, we show that the radial velocity signal of the photosphere following the planetarytidal potential can be distilled for the host star. Its amplitude is in agreement with both theoreticalpredictions of the equilibrium tide approximation and an ellipsoidal modulation observed in an orbitalphase curve. Assuming a circular orbit, we refine system parameters using photometric time seriesfrom TESS. With a new ground-based photometric observation, we extend the span of transit timingobservations to 28 years in order to probe the rate of the orbital period shortening. Since we found nodeparture from a constant-period model, we conclude that the modified tidal quality parameter of thehost star must be greater than 3 . × with 95% confidence. This result is in line with conclusionsdrawn from studies of the population of hot Jupiters, predicting that the efficiency of tidal dissipationis 1 or 2 orders of magnitude weaker. As the WASP-18 system is one of the prime candidates fordetection of orbital decay, further timing observations are expected to push the boundaries of ourknowledge on stellar interiors. Key words: planet-star interactions – stars: individual: WASP-18 – planets and satellites: individ-ual: WASP-18 b
1. Introduction
Hot Jupiters, i.e. massive exoplanets on extremely tight orbits, are recognisedas unique laboratories for studying planet–star interactions. Their orbital distancesof 0.02–0.03 AU are small enough to produce detectable tidal deformations of theirhost stars. Departure from a spherical symmetry of the gravitational potential givesa rise to apsidal precession that could be detected via transit and occultation timing( e.g. , Ragozzine & Wolf 2009) and radial velocity (RV) variations ( e.g. , Csizmadia et al. et al. parts per milion (ppm) of the observed flux (Pfahl etal. − in the Doppler domain (Arras et al. V = . M Jup planet on a 0.94-day orbit (Hellier et al. ∼ M ⊙ companion at a projected separationof ∼ et al. et al. Q ′ ⋆ = (Meibom & Mathieu 2005, Ogilvie &Lin 2007, Milliman et al. × yr and acumulative departure from a linear transit ephemeris would reach about 30 s after10 years (Hellier et al. et al. et al.
2. RV reanalysis
We acquired 5 RV measurements with the High Resolution Echelle Spectrom-eter (HIRES, Vogt et al. et al. (2014). The originaldataset comprises 6 RV observations secured between 2010 February and October2012. Their errors were in a range 3.7–5.6 m s − with a median value of 4.0 m s − .Our new observations were performed between October 2012 and August 2019.The instrumental setup and data-reduction pipeline was adopted from the Califor-nia Planet Search consortium (Wright et al. et al. etal. et al. ( BJD
TDB ) . Thecomplete dataset is given in Table 1.Additional RV measurements were taken from Triaud et al. (2010) and Albrecht et al. (2012). Thirty seven of them were acquired between 2007 September and2009 January using the CORALIE high resolution échelle spectrograph paired withthe 1.2 m Euler Swiss Telescope (La Silla, Chile). The errors were in a range 8.2–14.2 m s − with a median value of 9.7 m s − . Forty eight measurements comefrom the Planet Finder Spectrograph (PFS) and the Magellan II 6.5 m telescope(Las Campanas, Chile). They were gathered on a single night in October 2011 inorder to study the Rossiter–McLaughlin (RM) effect. They have errors between 4.9and 9.2 m s − with a median value of 5.9 m s − . Twenty three measurements weresecured in 2008 August with the high resolution échelle spectrograph HARPS at the3.6 m ESO telescope at La Silla. They were also used to investigate the RM effect.The errors were between 4.4 and 10.7 m s − with a median value of 6.1 m s − .The procedure, we followed, was adopted from Maciejewski et al. (2020).Thirty seven RV data points fall in transit phase and were affected by the RM effect.Since our analysis procedure does not take this effect into account, the appropriatecorrections were calculated using the RM model obtained by Albrecht et al. (2012)and then subtracted from the original RV measurements. The final RV sample com-prised 119 RV data points. In order to place a constraint on a transit ephemeris, aset of mid-transit times (Sect. 4) was added together with mid-occultation timestaken from Nymeyer et al. (2011), Maxted et al. (2013), and Wilkins et al. (2017),corrected for the light-travel time across the WASP-18A b’s orbit. The Keplerianmodel of the orbit was characterised by 9 free parameters: the orbital period P orb ,RV amplitude K orb , mean anomaly for a given epoch, apparent orbital eccentricity e orb , longitude of periastron ω , and four RV offsets for the individual RV datasets.The best-fitting Keplerian solution was found with the Levenberg-Marquardt al-gorithm. The parameters’ uncertainties were estimated with the bootstrap methodusing the median absolute deviations for 10 resampled datasets.As compared to a circular-orbit model, the eccentric model is favoured by the T a b l e 1
New and reprocessed RV measurements acquired with HIRES/Keck I.Mid-exposure ( BJD
TDB ) Relative RV ∗ m s − RV error ∗ (m s − ) Remarks2455231.725706 − .
832 5.828 (1)2455427.048972 909 .
848 3.519 (1)2456167.071476 1371 .
783 3.951 (1)2456193.094044 − .
056 4.255 (1)2456197.032589 − .
433 4.344 (1)2456207.967066 129 .
533 4.526 (2)2456209.021271 − .
126 4.685 (1)2456913.036796 1092 .
564 4.290 (2)2457241.125035 − .
939 4.793 (2)2458393.977843 797 .
761 2.886 (2)2458720.096038 − .
098 5.313 (2)Remarks: (1) reprocessed from Knutson et al. (2014), (2) new measurement ∗ higher numerical precision left intentionally. Bayesian information criterion (BIC),BIC = χ + k ln N , (1)where k is the number of fit parameters and N is the number of data points, with ∆ BIC ≈ − . This RV jitter was added in quadratureto the RV errors in the final iteration. We found e orb = . ± .
001 and ω = . ◦ ± . ◦ .Arras et al. (2012) demonstrated that an orbital configuration with a nonzeroeccentricity and longitude of periastron close to 270 ◦ might be de facto a sig-nal comprising a circular orbit component V orb ( φ ) and a tidal component V tide ( φ ) ,where φ is an orbital phase. The observed RV signal V obs ( φ ) can be written as V obs ( φ ) = γ + V orb ( φ ) + V tide ( φ ) , (2)where γ is the barycentre velocity and V orb ( φ ) = − K orb sin ( π ( φ − φ )) , (3)and V tide ( φ ) = K tide sin ( π ( φ − φ )) . (4)In those formulae, φ is the phase offset for a reference epoch, and K orb and K tide are the amplitudes of the orbital motion and the tidal component, respectively. Thebest-fitting solution was found with the Markov chain Monte Carlo (MCMC) em-ploying the emcee sampler (Foreman-Mackey et al. steps long each, were used to produce marginalised posteriori probability dis-tributions for the free parameters. The first 10% of steps were discarded in a burn-inphase. Median values of the cumulative distributions were taken as the best fittingparameters, and 15.9 and 84.1 percentile values of those distributions were used asthe 1 σ uncertainties.The best fitting model is presented in panel (a) in Fig. 1 together with the resid-uals in panel (b). The orbital RV component is plotted in panel (c), and the tidalRV signal is shown in panel (d). We obtained K orb = . ± . − and K tide = . ± . − . The phase offsets φ = ( ± ) × − was found tobe consistent with zero well within 1 σ . Although the barycentre velocity was sub-tracted from the RV measurements prior to the fitting procedure, its uncertainty wastaken into account in the error budget by allowing an RV shift to float. This shiftwas found to be − . ± . − , i.e. , consistent with zero well within 1 σ . -2000-1000010002000 a) V ob s ( m s - ) CORALIEPFSHARPSHIRES -2000-1000010002000-60-3003060 -0.50 -0.25 0.00 0.25 0.50b) O - C ( m s - ) Phase-60-3003060 -0.50 -0.25 0.00 0.25 0.50 -2000-1000010002000 c) V o r b ( m s - ) -2000-1000010002000-60-3003060 -0.50 -0.25 0.00 0.25 0.50d) V t i de ( m s - ) Phase-60-3003060 -0.50 -0.25 0.00 0.25 0.50
Fig. 1.
Panel (a) : relative RV measurements observed for WASP-18A, phase-folded with the orbitalperiod of WASP-18A b. Open symbols mark measurements taken from the literature. Dots show ournew and reanalysed observations acquired with HIRES. The error bars of individual measurementsare increased by the value of jitter of 10.7 m s − , added in quadrature. The red line shows the best-fitting model comprises two components: the orbital motion of the planet on a circular orbit and thetidal RV signal. Panel (b) : the residuals from the best-fitting model.
Panel (c) : orbital RV component.
Panel (d) : tidal RV component.
3. TESS photometry
The space-borne photometry from the Transiting Exoplanet Survey Satellite(TESS, Ricker et al. et al. (2019) to refine system or-bital parameters under the assumption that the orbit of WASP-18A b is noncircular.As the observed eccentricity is likely a manifestation of the tidal RV signal, wereanalysed the photometric time series from TESS to refine system parameters fora circular orbit scenario.TESS observed the WASP-18 system in sectors 2 (from August 22 to Septem-ber 20, 2018) and 3 (from September 20 to October 18, 2018) with Camera 2. Thephotometric data of a 2 minute cadence were downloaded through the exo.MASTportal . The Presearch Data Conditioning (PDC) light curve was extracted for fur-ther analysis. A median value of recorded counts was calculated for each sectorseparately and then used for light-curve normalisation. To remove variability otherthan transits, a 12 hour boxcar was applied with in-transit and in-occultation datapoints masked. A transit ephemeris from Shporer et al. (2019) was used to extractdata collected in transit windows and extended by 90 minutes of out-of-transit ob-servations before and after each event. The set of 47 complete transit light curveswas prepared for modelling with the Transit Analysis Package (TAP, Gazak et al. T mid , were modelled separately for each transit light curve. Transitparameters, such as the orbital inclination i orb , the semi-major axis scaled in starradii a / R ⋆ , and the ratio of planet to star radii R p / R ⋆ , were linked together for alllight curves. The value of P orb was taken from the transit-timing analysis (Sect. 4).The coefficients of the quadratic limb-darkening (LD) law – the linear u lin andthe quadratic u quad – were allowed to float. Their initial values were bi-linearlyinterpolated from tables of Claret & Bloemen (2011).The best-fitting parameters and their uncertainties were determined from themarginalised posteriori probability distributions produced from 10 MCMC chains( i.e. , the median value, and 15.9 and 84.1 percentiles). The random walk pro-cess was driven by the Metropolis-Hastings algorithm and a Gibbs sampler. Thewavelet-based technique (Carter & Winn 2009) was employed to account for thecorrelated noise. Each chain was 10 steps long. The first 10% of trials wererejected to minimise the influence of the initial values. The best-fitting model isplotted in Fig. 2, and its parameters are listed in Table 2. We also give the resultsreported by Shporer et al. (2019) for comparison purposes.In order to verify our procedure and its reliability of error estimates, the mod-elling was repeated for a scenario with a noncircular orbit with the initial conditions https://exo.mast.stsci.edu N o r m a li s ed f l u x Time from mid-transit time (d)0.980.991.00 -0.10 -0.05 0.00 0.05 0.10
Fig. 2. Phase folded transit light curve from TESS with the best-fitting model. The residuals areplotted below.
T a b l e 2
Transit parameters for WASP-18A b re-determined for the circular orbit. The results from Shporer et al. (2019) and from the trial noncircular scenario are given for comparison purposes.Parameter Circular Shporer et al. (2019) Noncircular R p / R ⋆ . + . − . . + . − . . + . − . a / R ⋆ . + . − . . + . − . . + . − . i orb . + . − . . ± .
33 84 . + . − . u lin . ± .
034 0 . ∗ . ∗ u quad . + . − . . ∗ . ∗∗ value taken from Claret (2017). set as in Shporer et al. (2019). We entered e orb = . ω = ◦ and the LDcoefficients were fixed at the theoretical values u lin = . u quad = . et al. (2019) very well. The parameters of our trial noncircularmodel agree with those of Shporer et al. (2019) within 0.2–0.4 σ . The errors aresimilar for i orb and a / R ⋆ , and our estimate of uncertainty for R p / R ⋆ appears to begreater by 30%. As an additional test showed, this is a consequence of the inclusionof the quadratic term in the de-trending procedure.Differences between parameters of the circular-orbit model and those of Sh-porer et al. (2019) are noticeable at a 1.7–2.6 σ level. The values of both i orb and a / R ⋆ were found to be slightly smaller which is a direct consequence of the sys-tem’s geometry. The transits were found to be deeper. The source of this effectis seen in the LD coefficients, which we set as the free parameters of the model.Because of inconsistencies of theoretical stellar limb darkening tables, it is advo-cated to keep the LD coefficients free in modelling of transit light curves if thephotometry is of sufficient quality and these coefficients can be determined reliably(Csizmadia et al. u lin greater by 2.2 σ and u quad smaller by 2.6 σ if compared to the theoretical expectations from Claret (2017). We note that thesimilar though less significant trend can be found for V -band data in Southworth et al. (2009).
4. Transit timing
The transit model obtained in Sect. 3 was used as a template for the ground-based light curves in order to determine their mid-transit times. We used twofollow-up light curves from Heller et al. (2009), five from Southworth et al. (2009),and two from Kedziora-Chudczer et al. (2019). Timestamps were converted toBJD
TDB and if needed magnitudes were rescaled into fluxes normalised to unityoutside the transits. The photometric time series from Maxted et al. (2013), Wilkins et al. (2017), Cortés-Zuleta et al. (2020), and Patra et al. (2020) were not available.In addition, we acquired a new transit light curve on September 26, 2019 usingthe 0.6 m Helen Sawyer Hogg (HSH) telescope located at Complejo AstronomicoEl Leoncito (CASLEO, San Juan, Argentina). An SBIG STL-1001E CCD camerawith 1024 × × µ m pixels was used as a detector. The instrument offereda field of view of 9 . × . I -band filter with exposure times of 20–25 s (depending on seeing conditions),giving an average cadence of 32 s. The observations were reduced with a standardprocedure carried out with AstroImageJ software (Collins et al. steps. The transit parameters – R p / R ⋆ , a / R ⋆ , and i orb – were allowed to vary under Gaussian penalties of the templatemodel. The LD coefficients were bi-linearly interpolated from tables of Claret &Bloemen (2011) for stellar parameters determined by Heller et al. (2009), and alsoallowed to vary under a Gaussian prior of a width of 0.1. The coefficients of asecond-order polynomial, which accounts for a possible trend in the time domain,and T mid were kept free.A signature of transits of WASP-18A b was detected in broadband Hipparcos photometry by McDonald & Kerins (2018). The star was sparsely sampled be-tween December 1989 and March 1993 with 130 measurements in total. Suchearly epochs are especially important for timing studies. The data were extractedfrom
Hipparcos Epoch Photometry , a complement to
The Hipparcos and TychoCatalogues (ESA 1997), and then phase folded following a preliminary ephemeris. N o r m a li s ed f l u x Time from mid-transit time (d)0.940.960.981.001.02 -0.10 -0.05 0.00 0.05 0.10
Fig. 3. Our new transit light curve observed on September 26, 2019 with the HSH telescope. Individ-ual measurements are marked with dots, the best-fitting model is marked with a line. The quality ofthe light curve is degraded mainly by faintness of comparison stars available in the field of view. Thephotometric scatter is 3.0 parts per thousand of the normalised flux per minute of observation. Theresiduals are shown below.
Phase of each data point was transformed into BJD
TDB of an artificial transit lo-cated near a middle of the time span of the observations. We note that the orbitalperiod could be in principle a subject of variation but the scale of this variation isexpected to be relatively small in a time scale of a few months or years, and anycumulative shift in transit times would be preserved. A mid-transit time, which isdetermined in this manner, is de facto an average representative for a time coveredby the phase-folded observations. The magnitudes were transformed into fluxes andnormalised using a median value of magnitude. The final light curve was trimmedto ± et al. σ clip-ping of outlying data points, 230 measurements were qualified for further analysis.The SuperWASP database provides 3360 and 4330 observations done in 2006 and2007, respectively. Because of the large number of data points and a high cadence,data in both observing seasons were analysed separately. Trimming and 5 σ clip-ping left the final light curves with 1430 and 2060 data points for 2006 and 2007,respectively. A subsequent procedure of the analysis was similar to that one whichwas applied to the single follow-up light curves. The only differences were that notime-domain trends were considered and the parameter R p / R ⋆ was allowed to floatin order to prevent underestimation of errors.The compilation of mid-transit times is listed in Table 3. As the photometrictime series from Maxted et al. (2013), Wilkins et al. (2017), Cortés-Zuleta et al. (2020), and Patra et al. (2020) were unavailable, the original mid-transit times weretaken then. Southworth et al. (2010) note that the time stamps in photometric timeseries of Southworth et al. (2009) might be offset by an unknown amount becausea clock of a computer which was used to record observations was not synchronisedproperly. We used the photometric data which are available via CDS and convertedtheir midpoints of observations given in BJD
UTC into BJD
TDB . However, we foundno systematic shift of this subset of mid-transit times. This finding indicates that theobservations of Southworth et al. (2009) were practically unaffected by the faultytime service.The transit timing analysis was performed following a procedure adopted fromMaciejewski et al. (2018). The MCMC algorithm running 100 chains, 10 stepslong each with the first 1000 trials discarded, was employed to refine the lineartransit ephemerides T mid ( E ) [ BJD
TDB ] = . ( ) + . ( ) × E , (5)where E is a transit number counted from a reference epoch given in Hellier et al. (2009). The posterior probability distributions of the fitted parameters were used todetermine their best-fitting values (medians) and their uncertainties (15.9 and 84.1percentile values of the cumulative distributions). The best-fitting solution yields χ = . lin = . T mid = T + P orb × E + dP orb dE × E , (6)where T is the reference mid-transit time for the transit number 0 and dP orb dE isthe change in the orbital period between succeeding transits, yields dP orb dE = ( . ± . ) × − days per epoch, χ = .
9, and BIC quad = .
0. The quadratic termis indistinguishable from zero well within 1 σ and the quadratic ephemeris is un-ambiguously disfavoured by the statistics.The timing residuals against the linear ephemeris are plotted in Fig. 4 togetherwith uncertainties of the quadratic term.The parameter dP orb dE is related to Q ′ ⋆ with the formula (see e.g. , Maciejewski etal. Q ′ ⋆ = − π (cid:18) M p M ⋆ (cid:19) (cid:18) aR ⋆ (cid:19) − (cid:18) dP orb dE (cid:19) − P orb , (7)where M p is a planet mass and M ⋆ is a mass of a host star mass. Since no or-bital decay was detected for WASP-18A b, the lower constraint on Q ′ ⋆ at the 95%confidence level can be placed from the 5th percentile of the posterior probabilitydistribution of dP orb dN tr . Adopting the stellar mass M ⋆ = . ± .
13 (Hellier et al. Q ′ ⋆ > . × .0 -40-30-20-10010-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 Hipparcos ASAS SuperWASP TESS O − C ( m i n ) Time from cycle zero (yr)-40-30-20-10010-18 -15 -12 -9 -6 -3 0 3 6 9 12 15-4-2024 -2 0 2 4 6 8 10 12 14
TESS O − C ( m i n ) Time from cycle zero (yr)-4-2024 -2 0 2 4 6 8 10 12 14
Fig. 4. Transit-timing residuals against the refined linear ephemeris. The new mid-transit time ac-quired with the HSH telescope is marked with a diamond. The re-determined mid-transit times fromTESS are marked with dots. Data points from the literature are marked with open circles. The greylines illustrate uncertainty of the trial quadratic ephemeris and are drawn for 100 sets of parameters,randomly chosen from the Markov chains. Lower panel shows the transit-timing residuals zoomedin on observations acquired in the last 2 decades. The greyed vertical strap marks additional TESSobservations which are scheduled in August–November 2020.
5. Discussion
In all studies addressing the issue of the orbital eccentricity of WASP-18A b,its value was found to be non-zero within 2.5-8.9 σ , depending on the data used andthe methodology of their analysis (Hellier et al. et al. et al. et al. e b which differs from 0 ata 10 σ level. In Maciejewski et al. (2020), we show that our procedure of the RVanalysis provides reliable, not underestimated uncertainties. Following Eq. (3) ofAdams & Laughlin (2006), a circularisation timescale for WASP-18A b is ∼ . This is morethan one order of magnitude shorter than the age of the system, which falls in arange 0.5–1.5 Gyr (Hellier et al. et al. (2012) predict thetidal RV amplitude to be ∼
32 m s − . Using their Eq. (20), which arises from asimplification of general considerations for a circular orbit, and adopting the stellarmass M ⋆ = . ± .
13 (Hellier et al. ∼
20 m s − . The main source of uncertainties in the theoretical predictions isa parameter f , which contains information on LD, and is proportionally relatedto K tide . In our calculations, we followed Arras et al. (2012) who used the LDunder the Eddington approximation, for which f ≈ .
1. Since the equilibrium tideapproximation is supposed to be accurate to a factor of about 2 (Arras et al. K tide = . ± . − can be considered asbeing consistent with the theoretical prediction. After transformation of Eq. (20) ofArras et al. (2012), we obtained f = π K tide R ⋆ M ⋆ M b (cid:18) aR ⋆ (cid:19) P orb ( sin i orb ) − = . ± .
12 (8)for WASP-18A. This result agrees with the Eddington approximation at the 1 σ level.A rough estimate of a hight of the tides relative to the unperturbed stellar radiuscomes from the ratio of the tidal acceleration to the star’s surface gravity (Pfahl etal. H exp = M b M ⋆ (cid:18) aR ⋆ (cid:19) − R ⋆ . (9)For the WASP-18 system, we obtained H exp = ±
10 km. This quantity can beempirically determined by calculating a distance traveled by a stellar photospherein a time t equal to a quarter of a tidal cycle H tide = pK tide Z P tide sin (cid:18) π P tide t (cid:19) d t , (10)where p = .
36 is a projection factor, which scales disk integrated RVs into actualphotospheric velocities (Getting 1934, Burki et al. P tide = P orb /
2. Weobtained H tide = ±
16 km which is consistent with H exp well within 1 σ .Using Eq. (8) of Shporer et al. (2019), we redetermined the expected amplitudeof the photometric modulation A ellip = ±
12 ppm. On the other hand, simplegeometrical considerations lead to the relation A ellip = H tide R ⋆ − H tide , (11)2which yields A ellip = ±
21 ppm. This value, which was de facto derived fromthe RV tides, is in excellent agreement with both the theoretical predictions and theobserved amplitude of 190 . + . − . ppm (Shporer et al. et al. (2019) has recently postulated that the eccentric orbit of WASP-18A b is undergoing apsidal precession. The rate of this precession was found to be˙ ω = . + . − . degrees per day using the literature RV together with the transitand occultation timing datasets. While trying to reproduce this result, we noticedthat the best-fitting solution is found for ˙ ω = . ± . χ distribution was found around the value of ˙ ω postulatedby Csizmadia et al. (2019). Compared to the model with no precession (Sect. 2),the precession is disfavoured by the statistics. The non-precession model gives χ = . = . χ = .
1, engaging ˙ ω as the additional free parameterresults in no significant improvement in BIC = .
8. Adding jitter results in a de-crease of ˙ ω down to ˙ ω = − . ± . − . A negativevalue of ˙ ω cannot be induced by the tidal deformations. It could be produced in aresult of the rotational deformation of the rapidly rotating host star if the planetaryorbit were significantly misaligned (Migaszewski 2012). However, the rotation pe-riod of WASP-18A of ≈ . et al. ω ≈ − . et al. et al. ω is consistent with zero regardless the amount of jitter added, the detection ofapsidal precession of WASP-18A b appears to be premature.As noted by McDonald & Kerins (2018), early Hipparcos observations of WASP-18A b provide rather weak constraints in transit timing studies. In a test run withthose data skipped, the constraint on Q ′ ⋆ was found to differ by a marginal valueof 3%. While our value of the timing residual of about − .
018 d is consistentwith − .
021 d derived from the mid-transit time reported by McDonald & Kerins(2018), our timing errors were found to be 2.5–2.6 times greater. To check if ourprocedure overestimates timing uncertainties, we calculated a ratio of our timingerrors for TESS data to errors derived by Shporer et al. (2019). We found that thisratio is between 0.93 and 1.36 with a median value of 1.19. Thus, we conclude thatthe timing errors reported by McDonald & Kerins (2018) might be underestimated.We also note that the timing errors from the
Hipparcos photometry are significantlyasymmetric with σ + / σ − = .
6. This effect is also visible in the original results ofMcDonald & Kerins (2018) with σ + / σ − = .
5. This asymmetry is a consequenceof a non-uniform data point distribution in the
Hipparcos light curve.Our homogenous transit-timing analysis has provided the tightest constraint on Q ′ ⋆ for WASP-18A. Wilkins et al. (2017) used all timing data available then andgot Q ′ ⋆ > at 95% confidence. Although McDonald & Kerins (2018) obtained Q ′ ⋆ ≈ × , Shporer et al. (2019) could only place a constraint on Q ′ ⋆ > . × . The same constraint has been obtained recently by Patra et al. (2020). Ourdetermination of Q ′ ⋆ excludes values smaller than 3 . × with 95% confidence.However, it is still not enough to verify theoretical findings of Collier Cameron &Jardine (2018). They predict that the stars hosting hot Jupiters could have Q ′ ⋆ ofthe order of 2 × if the equilibrium-tide regime is considered. Under favourablecircumstances, the dynamical-tide mechanism could operate in stellar interiors andthe efficiency of tidal dissipation would be boosted by one order of magnitude thattranslates into Q ′ ⋆ ≈ × .WASP-18A will be observed again with TESS between August and November2020. As it is shown in Fig. 4 (lower panel), new mid-transit times will definitelyplace tighter constraint on Q ′ ⋆ . Adopting Q ′ ⋆ = × , a departure of 1 minutefrom a linear ephemeris could be detected after 2 decades of precise observations.However, it is more likely that the system is far from the dynamical-tide regimeand the host star dissipates the tidal energy less efficiently. In such case, the cumu-lative time shift of 1 minute would be noticed after about 60 years. Nevertheless,the WASP-18 system still remains one of the best candidates for an infalling hotJupiter orbiting a main sequence star. The rapid decay rate of the WASP-12 b,so far the only planet for which the orbital evolution due to tidal interactions hasbeen observed (Maciejewski et al. et al. et al.
6. Conclusions
As with the WASP-12 system (Maciejewski et al. Q ′ ⋆ of WASP-18A must be greaterthan the canonical value of 10 reported in studies of binary stars in stellar clusters(Meibom & Mathieu 2005, Ogilvie & Lin 2007, Milliman et al. Q ′ ⋆ from the dynamical-tide regime.4 Acknowledgements.
We are grateful to Dr. Coel Hellier and Dr. Lucyna Kedziora-Chudczer for sharing the light curves with us. GM and CM acknowledge the finan-cial support from the National Science Centre, Poland through grant no. 2016/23/B/ST9/00579.This work was based on observations at the W. M. Keck Observatory granted bythe University of Hawaii, the University of California, and the California Instituteof Technology. We thank the observers who contributed to the measurements re-ported here and acknowledge the efforts of the Keck Observatory staff. We extendspecial thanks to those of Hawaiian ancestry on whose sacred mountain of MaunaKea we are privileged to be guests. This paper includes data collected with theTESS mission, obtained from the MAST data archive at the Space Telescope Sci-ence Institute (STScI). Funding for the TESS mission is provided by the NASAExplorer Program. STScI is operated by the Association of Universities for Re-search in Astronomy, Inc., under NASA contract NAS 5-26555. This paper makesuse of data from the DR1 of the WASP data (Butters et al.
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Mid-transit times for WASP-18 b.Epoch T mid (BJD TDB ) + σ (d) − σ (d) Light curve source-6283 48466.366 0.033 0.021 Hipparcos , ESA (1997)-747 53518.2126 0.0084 0.0067 ASAS, Pojma´nski (1997)-267 53970.11352 0.00092 0.00100 SuperWASP, Butters et al. (2010)156 54368.3462 0.0010 0.0011 SuperWASP, Butters et al. (2010)471 54664.90568 0.00058 0.00058 Heller et al. (2009)472 54665.84803 0.00063 0.00064 Heller et al. (2009)915 55082.910597 0.00060 0.00063 Southworth et al. (2009)916 55083.852439 0.00032 0.00030 Southworth et al. (2009)917 55084.793919 0.00024 0.00024 Southworth et al. (2009)918 55085.734997 0.00027 0.00026 Southworth et al. (2009)919 55086.677153 0.00034 0.00035 Southworth et al. (2009)969 55133.7472 0.0012 ∗ ∗ Cortés-Zuleta et al. (2020)970 55134.6914 0.0012 ∗ ∗ Cortés-Zuleta et al. (2020)971 55135.6331 0.0012 ∗ ∗ Cortés-Zuleta et al. (2020)1062 55221.30420 ∗ ∗ ∗ Maxted et al. (2013)1286 55432.18970 ∗ ∗ ∗ Maxted et al. (2013)1327 55470.78850 ∗ ∗ ∗ Maxted et al. (2013)1330 55473.61440 ∗ ∗ ∗ Maxted et al. (2013)1416 55554.57860 ∗ ∗ ∗ Maxted et al. (2013)1433 55570.58400 ∗ ∗ ∗ Maxted et al. (2013)1689 55811.5970 0.0041 ∗ ∗ Cortés-Zuleta et al. (2020)1758 55876.5559 ∗ ∗ ∗ Maxted et al. (2013)2841 56896.14780 ∗ ∗ ∗ Wilkins et al. (2017)3223 57255.78320 ∗ ∗ ∗ Wilkins et al. (2017)3291 57319.80100 ∗ ∗ ∗ Wilkins et al. (2017)3311 57338.6296 0.0011 0.0011 Kedziora-Chudczer et al. (2019)3312 57339.57210 0.00052 0.00051 Kedziora-Chudczer et al. (2019)3649 57656.84078 0.00097 ∗ ∗ Cortés-Zuleta et al. (2020)3650 57657.78359 0.00097 ∗ ∗ Cortés-Zuleta et al. (2020)3651 57658.72404 0.00097 ∗ ∗ Cortés-Zuleta et al. (2020)3684 57689.79147 0.00075 ∗ ∗ Patra et al. (2020)4042 58026.8319 0.0011 ∗ ∗ Cortés-Zuleta et al. (2020)4390 58354.45788 0.00019 0.00019 TESS4391 58355.39931 0.00018 0.00018 TESS4392 58356.34077 0.00022 0.00021 TESS4393 58357.28206 0.00022 0.00024 TESS4394 58358.22352 0.00021 0.00021 TESS4395 58359.16514 0.00018 0.00017 TESS4396 58360.10664 0.00019 0.00019 TESS4397 58361.04799 0.00021 0.00022 TESS4398 58361.98976 0.00021 0.00022 TESS4399 58362.93133 0.00019 0.00019 TESS4400 58363.87260 0.00019 0.00018 TESS4401 58364.81379 0.00021 0.00020 TESS4402 58365.75525 0.00019 0.00019 TESS ∗ Value taken from a source paper. T a b l e 3
Concluded.Epoch T mid (BJD TDB ) + σ (d) − σσ