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Article

The Apparent Tidal Decay of WASP-4 b Can Be Explained by the Rømer Effect

by
Jan-Vincent Harre
* and
Alexis M. S. Smith
Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstraße 2, 12489 Berlin, Germany
*
Author to whom correspondence should be addressed.
Universe 2023, 9(12), 506; https://doi.org/10.3390/universe9120506
Submission received: 6 November 2023 / Revised: 1 December 2023 / Accepted: 3 December 2023 / Published: 5 December 2023
(This article belongs to the Special Issue The Royal Road: Eclipsing Binaries and Transiting Exoplanets)
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Abstract

:
Tidal orbital decay plays a vital role in the evolution of hot Jupiter systems. As of now, this has only been observationally confirmed for the WASP-12 system. There are a few other candidates, including WASP-4 b, but no conclusive result could be obtained for these systems as of yet. In this study, we present an analysis of new TESS data of WASP-4 b together with archival data, taking the light–time effect (LTE) induced by the second planetary companion into account as well. We make use of three different Markov chain Monte Carlo models: a circular orbit with a constant orbital period, a circular orbit with a decaying orbit, and an elliptical orbit with apsidal precession. This analysis is repeated for four cases. The first case features no LTE correction, with the remaining three cases featuring three different timing correction approaches because of the large uncertainties of the ephemeris of planet c. Comparison of these models yields no conclusive answer to the cause of WASP-4 b’s apparent transit timing variations. A broad range of values of the orbital decay and apsidal precession parameters are possible, depending on the LTE correction. However, the LTE caused by planet c can explain on its own—in full—the observed transit timing variations of planet b, with no orbital decay or apsidal precession being required at all. This work highlights the importance of continued photometric and spectroscopic monitoring of hot Jupiters.

1. Introduction

The study of exoplanets has unveiled a great diversity in terms of their physical properties, orbital characteristics, and formation mechanisms. Among the most intriguing exoplanet classes are the hot Jupiters. These relatively rare companions (e.g., [1]) stand out as a group of gas giants that orbit their host stars at exceptionally close distances and have challenged our understanding of planetary formation and evolution. One of the key phenomena that has piqued the interest of astronomers is the tidal orbital decay of hot Jupiters, and as of now only WASP-12 b could be observationally confirmed to experience this effect [2,3,4]. However, there seems to be evidence for the occurrence rates of hot Jupiters in orbit around Sun-like stars to decrease with stellar age, as opposed to the occurrence rates of cold Jupiters, as found by Miyazaki and Masuda [5]. This is supported by the likely observation of planetary engulfment by a Sun-like star made by the Zwicky Transient Facility [6].
Due to the close proximity of hot Jupiters to their stellar hosts, there are strong tidal interactions between the two bodies. These gravitational interactions can manifest themselves in the planet, raising a bulge on the surface of the star. Depending on the rotational period of the star and the planetary orbital period, the viscosity of the stellar plasma can lead to a lag between the position of the tidal bulge and the virtual line connecting the stellar and planetary centers. If the star is rotating slower than the planet is orbiting around it, the orbital angular momentum of the planet will be transferred to the star (equilibrium tide). The dynamical tide, arising from stellar oscillations, also contributes to this [7]. This means that the star will spin up, and the planetary orbit will shrink gradually [8,9]. This provides us with insights into the long-term stability of these systems.
WASP-4 b is a hot Jupiter discovered by Wilson et al. [10]. It shows TTVs that have been examined before, and other effects, mimicking the orbital decay signature, were ruled out previously, like the Applegate mechanism [11,12]. However, apsidal precession due to a small eccentricity of the planetary orbit, or TTVs arising from the Rømer effect (or light–time effect, LTE hereafter; Irwin [13]), due to a companion in the system, have not yet been ruled out as the cause of the TTVs, even after the discovery of the companion candidate. Previous studies examining the decay rate in the case of tidal orbital decay found values in the range from − 4.8 ms year − 1 to − 12.6 ms year − 1  [12,14,15,16,17,18], with the most recent estimate being P ˙ = ( − 6.2 ± 1.2 ) ms year − 1 [19]. The differences between the measurements arise from the influence of additional timing data in each subsequent study.
In this work, we re-examine the TTVs in this system by making use of recently acquired TESS photometry and combine them with archival data from previous works. In particular, we account for the time-shift due to the LTE induced by the additional planet candidate, as discovered by Turner et al. [18]. The observations are described in Section 2, with our modeling and the results thereof being described in Section 3 and Section 4, respectively. The latter are discussed in Section 5, and our final conclusions can be found in Section 6.

2. Observations

For our analysis, we made use of the previously described data set in Harre et al. [19] for WASP-4 b. In short, this data set is mainly based on the homogeneous re-analysis of previously published light curves from Baluev et al. [16], with the addition of TESS Sectors 28 and 29, as well as a re-analysis of the TESS Sector 2 data. Furthermore, in [19], we re-fitted the publicly available light curves from the ExoClock project [20] and from WASP [10] and added eight transit observations taken with the CHEOPS space telescope [21]. Included in this data set are also four occultation timings from the literature [22,23,24]. All these timings, including three different corrections, can be found in Appendix B, Table A1, for the transit timings and Table A2 for the occultation timings.
To this data set, we add new TESS observations from Sector 69 at a cadence of 120 s. We make use of the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) flux for the analysis of the light curves, which consists of data produced by the TESS Science Processing Operations Center (SPOC) at the NASA Ames Research Center [25]. These data are publicly available at MAST1.

3. Modeling

3.1. Light Curve Modeling

For the analysis of the TESS transits, we made use of the Transit and Light Curve Modeler (TLCM, Csizmadia [26]), as described in Harre et al. [19]. In a first run, we fitted all transits together to obtain the combined shape of all transits to reduce the impact of stellar activity on the transit timings. The respective priors are shown in Table 1. During a second run, we fixed the shapes of the transits to those of the combined model and fitted all transits individually, with only the transit ephemerides being free. In both cases, we used the median solution for the final results.

3.2. Transit Timing Variation Analysis

For the analysis of the mid-transit times, obtained from our light curve modeling, we employ the same three models as Harre et al. [19]. The first is a model assuming a circular Keplerian orbit, describing a linear ephemeris with a constant orbital period:
t tra ( N ) = T 0 + N P ,
t occ ( N ) = T 0 + P 2 + N P ,
where t tra ( N ) and t occ ( N ) are the calculated mid-transit and mid-occultation timings at the epoch N, T 0 denotes the reference mid-transit time, and P denotes the planetary orbital period. This gives us two free parameters in the fit.
The second model describes the case of a decaying orbit due to the transfer of angular momentum (see, e.g., Counselman [8], Rasio et al. [9]). These are quadratic functions with a constant change in the orbital period of the planet:
t tra ( N ) = T 0 + N P + 1 2 d P d N N 2 ,
t occ ( N ) = T 0 + P 2 + N P + 1 2 d P d N N 2 .
This constant change in the orbital period is denoted by the decay rate d P d N , which can be converted to the period derivative P ˙ = d P d t = 1 P d P d N . There are three free parameters ( T 0 , P, and d P d N ) when fitting this model. The period derivative is linked to the stellar tidal modified quality factor Q ★ ′ via the constant-phase lag model, as defined in Goldreich and Soter [27]:
P ˙ = f π Q ★ ′ M p M ★ R ★ a 5 ,
where f denotes the tidal factor, M p and M ★ the planetary and stellar masses, R ★ the stellar radius, and a the semi-major axis of the planetary orbit. Depending on the ratio of the planetary orbital period to the stellar rotational period and the orbital (mis-)alignment, f takes different values. If the planetary orbital period is shorter than the rotation period of the star, as is the case for WASP-4 b [28], we obtain f = − 27 2 . Refer to Section 4 of Harre et al. [19] for a more detailed description, which also includes the case of inclined orbits and different planet-to-star period ratios.
The third model we are using is an apsidal precession orbit, where a small eccentricity e leads the planetary orbit to precess around the star. This can, on relatively short timescales, induce the same TTV signature as orbital decay, which makes it hard to differentiate the two models if the orbital eccentricity is only loosely constrained. This sinusoidal model follows the descriptions of Giménez and Bastero [29]:
t tra ( N ) = t 0 + N P s − e P a π cos ω ( N ) ,
t occ ( N ) = t 0 + P a 2 + N P s + e P a π cos ω ( N ) ,
where P s is the sidereal period, P a the anomalistic period, and ω the argument of pericenter. The sidereal and anomalistic periods are related via the following:
P s = P a 1 − 1 2 π d ω d N ,
with a constant change in the argument of pericenter d ω d N . The relationship between ω and N can be described as follows:
ω ( N ) = ω 0 + d ω d N N ,
with ω 0 being the argument of pericenter at the reference time T 0 . In total, this model gives us five free parameters in the fitting process.
These three models are then fitted to the data via MCMC optimization using the emcee Python package [30]. Per model, we use 100 walkers and use a burn-in period of 10 , 000 steps with a total chain length of 75 , 000 steps. Convergence is ensured by checking that the chains are longer than 50 times the autocorrelation time of the parameters. The priors for our models can be found in Table 2.

3.3. Light–Time Effect

The presence of a candidate planetary companion in a 7000 d orbit, as established by Turner et al. [18], would induce an orbital motion onto WASP-4. Due to the high mass of this candidate (“planet c” hereafter), the system’s center of mass is shifted by about 8.9 times the radius of WASP-4 (see Figure 1). This will have a significant impact on the observed mid-transit times because of the time difference that it takes the light to travel from the far side to the near side of the orbit from our point of view. In some cases, this could induce TTVs akin to the imprint of tidal decay on short timescales, depending on the orbital period of planet c. To include a correction of the LTE, we apply the formula of Schneider [31] to find the maximum time-shift from this effect for circular orbits:
Δ T max = 2 M p M ★ a sin i c ,
where Δ T max is the maximum resulting time-shift due to the LTE, i is the inclination of the planetary orbit, and c is the speed of light. For WASP-4, using the planetary parameters of planet c, and assuming a circular orbit with i = 90 ∘ , we obtain a maximum time-shift of Δ T max = 37.7  s. To validate this result, we simulate the system using the N-body code REBOUND [32] and measure the orbit of the star around the system’s center of mass, as shown in Figure 1. From this, we obtain Δ T max = 37.6  s. Using the non-circular solution from Turner et al. [18] for planet c, we obtain Δ T max , ell = 41.1  s. However, since the non-circular solution is not preferred in their paper, we adopt the circular solution and correct our transit times according to the following:
Δ T ( t ) = Δ T max 1 2 cos 2 π t − T 0 , c P c + 1 2 ,
where Δ T ( t ) is the time-shift due to the LTE at the time t, T 0 , c is the time of inferior conjunction of planet c, and P c = 7001.0 d is the orbital period of planet c. This formula arises because the star is at its furthest point from us at the time of inferior conjunction of planet c. Due to the uncertainty of the time of conjunction of planet c ( T 0 , c = 2,455,059 − 2100 + 2300 d (BJD TDB )), we examine three cases of the LTE correction: firstly, assuming the best-fit value of the time of conjunction; secondly, the minimum value of the 1 σ interval; and lastly, the maximum value of the 1 σ interval.

4. Results

4.1. Transit Fitting with TLCM

The priors for and results from the combined TLCM fit of all TESS transits from Sector 69 to constrain the transit shape are listed in Table 1, with the fit of the median model to the data shown in Figure 2. We find a root-mean-square scatter of 2.56 ppt between the data and our model. The resulting mid-transit times can be found in Table A1 in Appendix B.

4.2. TTV Fits without LTE Correction

The priors for the MCMC modeling of the circular orbit, orbital decay, and apsidal precession models and their resulting parameters are listed in Table 2.
We find an orbital decay rate of P ˙ = ( − 5.75 ± 0.52 ) ms year − 1 , leading to a stellar modified quality factor for WASP-4 of Q ★ ′ = ( 6.10 ± 0.55 ) × 10 4 and a decay timescale of τ = ( 20.2 ± 1.8 ) Myear from the median solution. The BIC values of our solutions are obtained via the following calculation:
B I C = χ 2 + k · ln ( N ) ,
with k being the number of free parameters and N the number of data points. For the orbital decay fit, we find Δ BIC decay = 119.4 between the linear and tidal decay models, in favor of the latter. In the case of apsidal precession, we find a precession rate of ω ˙ = d w d t = 1 P d w d N = ( 0.033 ± 0.007 ) ∘ d − 1 and a small eccentricity of e = 0.0013 ± 0.0005 . The fit yields Δ BIC precession = 110.7 between the linear and apsidal precession models. This leads to a difference of Δ BIC dec . , prec . = 8.7 in favor of the orbital decay model.
The result of the MCMC fit to the transit and occultation timing data including the final median models is shown in Figure 3. The latest data points highlight the deviation from a linear ephemeris in this system.

4.3. TTV Fits with LTE Correction

In case the candidate planet c does exist, we corrected for the induced LTE with a maximum time-shift of about 38 s in three different cases. These cases are (1) planet c has the best-fit time of inferior conjunction from model #3 of Turner et al. [18], (2) the time of inferior conjunction is at the lower boundary of the respective 1 σ confidence interval, and (3) the time of inferior conjunction is at the upper boundary of the 1 σ confidence interval. The corrections, according to Equation (11), are subtracted for each transit and occultation timing.
The results in Table 3 show that the orbital decay model provides the best fit to the data in case (1), with the nominal LTE correction, although this model is only slightly preferred over the apsidal precession model. In this case, the decay rate would be P ˙ = ( − 7.04 ± 0.52 )  ms year − 1 , leading to Q ★ ′ = ( 4.98 ± 0.37 ) × 10 4 and a decay time-scale of τ = ( 16.5 ± 1.2 ) Myear. The apsidal precession model yields ω ˙ = ( 0.024 ± 0.007 ) ∘ d − 1 with an eccentricity of e = 0.0029 ± 0.0019 . This case is displayed in Figure 4.
In case (2), where the lower boundary of the 1 σ interval of T 0 , c is used for the LTE correction, the apsidal precession model is preferred with a Δ BIC of 20 in comparison to the orbital decay model. The linear model is heavily disfavored. Assuming the orbital decay model to be true, we find an enhanced decay rate of P ˙ = ( − 9.35 ± 0.52 ) ms year − 1 , yielding Q ★ ′ = ( 3.75 ± 0.21 ) × 10 4 and τ = ( 12.4 ± 0.7 ) Myear. From the apsidal precession model, we obtain ω ˙ = ( 0.041 ± 0.005 ) ∘ d − 1 with e = 0.0016 ± 0.0003 . This case is shown in Appendix A, Figure A1.
Case (3), using the upper boundary of the time of inferior conjunction of planet c, shows that the orbital decay model provides the best fit to the data, with the circular orbit model fitting only slightly worse ( Δ BIC < 3 ), whereas the apsidal precession model fits worse with Δ BIC ≈ 25 − 30 . In this case, the TTV signature is flattened due to the LTE correction. This results in a small decay rate of P ˙ = ( − 1.45 ± 0.50 ) ms year − 1 , leading to Q ★ ′ = ( 2.42 ± 0.83 ) × 10 5 and τ = 80.2 ± 27.7 Myear. In the apsidal precession case, we obtain ω ˙ = ( 0.027 ± 0.012 ) ∘ d − 1 with a small eccentricity of e = 0.0010 ± 0.0007 . The TTV models for this case are shown in Appendix A, Figure A2.
According to Turner et al. [18], the TTVs induced by planet c onto planet b should not exceed 2 s, which is why they are neglected here.

5. Discussion

5.1. TTVs of WASP-4 b

As is evident from the results of our TTV modeling in Table 2 and Table 3, the consideration of the LTE is very important in this system. The results of the orbital decay and apsidal precession modeling are only comparable to each other to a certain degree for the cases that were examined here.
If the detection of WASP-4 c should turn out spurious, then the measured decay rate value is comparable, albeit slightly smaller than those from previous studies [12,14,15,16,18,19,33]. In this case, the apsidal precession model would be disfavored by Δ BIC = 8.5 , which is a bit higher than in the latest study [19]. The decay rates are consistent within 1 σ , with the eccentricity in the precession model being doubled, compared to the previous result, which is still in agreement with our value and uncertainty. The linear model is disfavored by Δ BIC ≈ 110 − 120 .
The three different LTE corrections lead to three vastly different results. The nominal LTE correction agrees within 3 σ with the results without the LTE correction for most of the parameters. The other solutions do not. Depending on the correction, we obtain decay rates in the range from ( − 9.35 ± 0.52 ) × 10 4 ms year − 1 to ( − 1.45 ± 0.50 ) × 10 4 ms year − 1 . This means that the LTE induced by a planetary companion can solely explain the observed TTVs. The quite broad range of results can only be further constrained with more radial velocity measurements to reduce the size of the error bars on the parameters of WASP-4 c. However, a value near the middle of this range is more probable. From these results, we obtain modified stellar quality factors in the range from 3.8 × 10 4 to 2.4 × 10 5 , which includes the value of Q ★ ′ = 1.8 × 10 5 obtained for WASP-12 b [2]. Moreover, this leads to decay timescales ranging from 12 Myear to 80 Myear. The apsidal precession models show a similar range of possible values, like the orbital decay models. However, with more observations, either transits or especially occultations, the models should be relatively easy to distinguish. Occultation observations with JWST, for example, could perform a double duty in this case. They can help to refine atmospheric properties and provide very accurate occultation timing measurements, which might be able to rule out either of the TTV models.
These results highlight the importance of continued radial velocity observations, even, or especially, in hot Jupiter systems. One theory of how hot Jupiters get into their tight orbits is high-eccentricity migration, which necessitates the presence of a massive second companion for the excitation of the high eccentricities that are required for this migration pathway [34,35]. Finding such a body requires long observational baselines due to the distance of the perturber to the host star.

5.2. Optimal Observing Strategy

For hot Jupiter systems in general, the most favorable tidal orbital decay systems have, according to Equation (5) and Table 2 in Harre et al. [19], first and foremost with the highest impact, large stellar radii and short orbital separations. Small stellar tidal quality factors, high planetary masses, and low stellar masses are also beneficial. However, stellar tidal quality factors can only be constrained after the observations. Furthermore, orbital periods of the companion bodies smaller than the stellar rotation periods are essential, but this only applies to equatorial orbits. For polar orbits, orbital decay should always be present, even for fast-rotating stars. Still, slow stellar rotation rates are beneficial here as well. In summary, as is known, hot Jupiters are prime targets to examine the effect of tidal orbital decay. Moreover, brown dwarfs on close-in orbits should even experience a more pronounced effect due to their higher masses. In addition, companions orbiting evolved or (sub-)giant stars should present the best laboratories to explore orbital decay. Yet, the expected lifetimes of these close-by companions will be relatively short, and observing them is only possible in just the right time window, providing us with only a very small sample of targets (see, e.g., Grunblatt et al. [36]).
In terms of radial velocity monitoring, after sufficient data have been accumulated for the characterization of hot Jupiters, it would be optimal for the target to be re-observed every few months so that even farther out companions could be detected in principle. This would shed light not only onto the architecture of these systems but also on their formation and migration pathways. This case study is an excellent example of this, where an outer companion candidate has been detected [18], but the whole detection hinges on the latest set of observations not being spurious or having an incorrect offset in relation to the previously accumulated RV data. Due to the possibility for the outer planet to induce a small eccentricity onto the orbit of the inner planet, a non-detection constraining possible planet masses and distances from the host star could rule out the apsidal precession models in some cases. This could be achieved by comparing the eccentricity damping timescale to the orbital decay timescale.
For photometric observations, it would be ideal to obtain high-precision transit observations every few months. Long-term monitoring is essential to detect orbital decay signatures with a baseline of more than 15 years having been necessary for WASP-12 b [2]. Occultation observations would be even more helpful to differentiate the orbital decay and apsidal precession models, not only from the point-of-view of TTV fitting but also because they can indicate eccentricities directly. In addition to the timing, secondary eclipse observations can give clues about the atmopsheric composition or the thermal structure of the atmosphere and the presence of clouds (see, e.g., Jackson et al. [37], Scandariato et al. [38], van Sluijs et al. [39], Shi et al. [40], Hoyer et al. [41]).
The PLATO mission [42] has the potential to revolutionize the field with its long, uninterrupted observations. Due to the expected excellent precision of the observations, occultations of hot Jupiters are likely to be detected as well. This should allow discrimination between the orbital decay and apsidal precession models. In addition, even relatively far-out companions could be detected with radial velocity follow-up from PLATO’s ground segment, should they not transit the host star. As well as PLATO, ESA’s Gaia mission [43], starting with its fourth data release, has the potential to improve our understanding of already known hot Jupiter systems. Its precise astrometric measurements will allow for the detection of giant companions in these systems and provide detailed characterization of their orbits. This will provide information on the formation and migration history of each system.

6. Conclusions

By analyzing new TESS observations of WASP-4 in combination with archival timing data and taking into account the additional planet candidate in the system, we obtain a broad range of results for the cause of the TTVs of WASP-4 b. We examined a total of four cases. The first case does not take into account the presence of planet c and yields comparable results to the hitherto published results for this system in terms of the orbital decay parameters. In addition, the remaining cases consider, for the first time, the light–time effect from the host star’s orbital motion around the system’s center of mass, induced by planet c. Depending on the application of the timing correction, we find results that range from a slight preference ( Δ BIC ≈ 4 ) of the orbital decay model in the nominal case (using the nominal value for the time of inferior conjuction of planet c from Turner et al. [18]), over a strong preference ( Δ BIC ≈ 20 ) of the apsidal precession model using the lower boundary of the respective 1 σ confidence interval, to a nearly indistinguishable result between the linear ephemeris and orbital decay models ( Δ BIC ≈ 2 ) using the upper boundary of the respective interval for the time of inferior conjunction of planet c. These results leave us with no conclusive answer to the question of what the true origin of the TTVs of WASP-4 b is. We need more radial velocity observations to better constrain the phase of planet c. Only then will we be able to determine whether the LTE solely explains the observed TTVs, or whether other mechanisms, like tidal decay or apsidal precession, are present. It is, however, possible for it to be a mix of all of the effects mentioned here.
This case study highlights the importance of continued monitoring of hot Jupiter systems, in terms of photometric and radial velocity measurements. Only in this way can small effects, like orbital decay or apsidal precession, be measured and differentiated. An additional benefit is given by the chance of discovering and characterizing companions to hot Jupiters, providing hints of the formation and migration scenarios that lead to these special systems. This is a necessary step to inform our theoretical models of planet formation and migration and towards the final answer to the question of the origin of these planets.

Author Contributions

Formal analysis, J.-V.H.; writing—original draft preparation, J.-V.H.; writing—review and editing, A.M.S.S.; supervision, A.M.S.S.; funding acquisition, A.M.S.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets (SM 486/2-1)”.

Data Availability Statement

The data underlying this article are available in Appendix B.

Acknowledgments

We thank the referees for their helpful comments and suggestions. This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by the NASA Explorer Program. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. This work made use of Astropy:2 a community-developed core Python package and an ecosystem of tools and resources for astronomy [44,45,46].

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. TTV Fits with LTE Correction

The TTV fits with the LTE correction at the lower and upper boundaries of the 1 σ interval of the ephemeris of planet c from Turner et al. [18] are shown in Figure A1 and Figure A2.
Universe 09 00506 g0a1
Figure A1. Same as Figure 3, but with the LTE correction applied to the timing data, using the lower limit of the 1 σ interval of the time of inferior conjunction of planet c.
Figure A1. Same as Figure 3, but with the LTE correction applied to the timing data, using the lower limit of the 1 σ interval of the time of inferior conjunction of planet c.
Universe 09 00506 g0a1
Universe 09 00506 g0a2
Figure A2. Same as Figure 3, but with the LTE correction, using the upper limit of the 1 σ interval around the nominal time of inferior conjunction of planet c, applied to the timing data.
Figure A2. Same as Figure 3, but with the LTE correction, using the upper limit of the 1 σ interval around the nominal time of inferior conjunction of planet c, applied to the timing data.
Universe 09 00506 g0a2

Appendix B. Transit and Occultation Timings

The transit and occultation timings of WASP-4 b, including those with the different LTE corrections, are given in Table A1 and Table A2.
Table A1. Transit timings of WASP-4 b including three different corrections for the LTE. Time is given in BJDTDB − 2,450,000. “Time” denotes the mid-transit times without any light–time correction. “Time LTE” denotes LTE-corrected timings using the nominal value for the time of inferior conjunction of planet c from Turner et al. [18], “Time LTE lower ” denotes the timings corrected using the lower boundary of the respective 1 σ interval. “LTE upper ” denotes the timings corrected using the upper boundary of the interval. The “Source” column denotes the source of the timings, with 0–4 defined as the homogeneous reanalysis from Baluev et al. [16] (0), TESS timings (1) from [19] and this work (last sector), and the ExoClock project (2) [20] as fitted by Harre et al. [19], as well as their CHEOPS timings (3), and WASP timings (4), respectively.
Table A1. Transit timings of WASP-4 b including three different corrections for the LTE. Time is given in BJDTDB − 2,450,000. “Time” denotes the mid-transit times without any light–time correction. “Time LTE” denotes LTE-corrected timings using the nominal value for the time of inferior conjunction of planet c from Turner et al. [18], “Time LTE lower ” denotes the timings corrected using the lower boundary of the respective 1 σ interval. “LTE upper ” denotes the timings corrected using the upper boundary of the interval. The “Source” column denotes the source of the timings, with 0–4 defined as the homogeneous reanalysis from Baluev et al. [16] (0), TESS timings (1) from [19] and this work (last sector), and the ExoClock project (2) [20] as fitted by Harre et al. [19], as well as their CHEOPS timings (3), and WASP timings (4), respectively.
TimeTime LTETime LTElowerTime LTEupperError (d)EpochSource
3960.4314803960.4311433960.4314793960.4311270.001827−3054
4361.9004844361.9000904361.9004634361.9002000.001889−54
4396.6961634396.6957654396.6961384396.6958850.000085210
4697.7981544697.7977304697.7980954697.7979340.0000632460
4697.7983064697.7978824697.7982474697.7980860.0001332460
4701.8130104701.8125864701.8129514701.8127910.0002172490
4701.8129804701.8125564701.8129214701.8127610.0002062490
4701.8127954701.8123714701.8127364701.8125760.0001482490
4705.8272944705.8268694705.8272344705.8270750.0002102520
4728.5779324728.5775064728.5778694728.5777180.0002742690
4732.5923084732.5918834732.5922454732.5920950.0005362720
4732.5922914732.5918654732.5922274732.5920780.0001452720
4740.6217274740.6213014740.6216634740.6215160.0002642780
4740.6214724740.6210464740.6214074740.6212600.0000932780
4748.6511314748.6507044748.6510654748.6509210.0000562840
5041.7237625041.7233265041.7236505041.7236080.0001215030
5045.7386365045.7382015045.7385235045.7384830.0000415060
5049.7532255049.7527905049.7531125049.7530730.0000425090
5053.7677595053.7673245053.7676465053.7676080.0000785120
5069.8269195069.8264845069.8268035069.8267710.0002265240
5069.8263675069.8259325069.8262515069.8262190.0002495240
5069.8265915069.8261565069.8264755069.8264430.0002275240
5069.8267405069.8263055069.8266245069.8265920.0002215240
5069.8264915069.8260565069.8263745069.8263420.0001285240
5073.8409285073.8404935073.8408115073.8407810.0002515270
5073.8411505073.8407155073.8410335073.8410030.0002325270
5073.8411105073.8406755073.8409935073.8409630.0001685270
5073.8411235073.8406885073.8410065073.8409760.0002095270
5096.5913585096.5909235096.5912375096.5912150.0001485440
5100.6059325100.6054975100.6058105100.6057890.0001015470
5112.6503715112.6499365112.6502475112.6502300.0002365560
5112.6502965112.6498615112.6501725112.6501560.0001775560
5112.6501625112.6497275112.6500385112.6500210.0002075560
5112.6500865112.6496515112.6499625112.6499450.0002615560
5132.7233255132.7228915132.7231985132.7231880.0002575710
5385.6494825385.6490565385.6493075385.6493880.0003427600
5424.4569505424.4565265424.4567685424.4568620.0011277894
5425.7961865425.7957625425.7960045425.7960990.0001887900
5468.6194205468.6189995468.6192295468.6193390.0003208220
5473.9723345473.9719135473.9721425473.9722540.0005248260
5502.0760075502.0755895502.0758105502.0759320.0004718470
5506.0904895506.0900715506.0902915506.0904140.0006168500
5551.5902055551.5897915551.5899995551.5901370.0003788840
5777.7509555777.7505645777.7507055777.7509160.00022010530
5785.7805515785.7801605785.7802995785.7805120.00019610590
5811.2095605811.2091725811.2093035811.2095240.00091810784
5820.5740645820.5736785820.5738055820.5740290.00028610850
5828.6045405828.6041555828.6042805828.6045060.00049310910
5832.6184925832.6181075832.6182315832.6184580.00016310940
5832.6190445832.6186605832.6187835832.6190110.00042610940
5844.6629305844.6625475844.6626675844.6628980.00013811030
5852.6928715852.6924895852.6926065852.6928400.00055711090
5856.7067245856.7063425856.7064585856.7066930.00021011120
5856.7066665856.7062845856.7064005856.7066350.00009811120
5915.5885335915.5881595915.5882575915.5885080.00029711560
6086.8824276086.8820786086.8821186086.8824140.00016212840
6149.7787216149.7783826149.7784026149.7787130.00129413310
6181.8973906181.8970566181.8970656181.8973830.00068013550
6212.6764546212.6761266212.6761246212.6764490.00042813780
6216.6911806216.6908526216.6908496216.6911750.00006713810
6450.8804466450.8801606450.8800796450.8804460.00089515560
6556.6022686556.6020016556.6018876556.6022670.00031516350
6572.6598876572.6596246572.6595046572.6598860.00031116470
6576.6755686576.6753056576.6751846576.6755670.00005516500
6639.5725616639.5723106639.5721706639.5725580.00038816970
6873.7631386873.7629336873.7627246873.7631230.00060218720
6885.8071126885.8069106885.8066966885.8070960.00038618810
6889.8212956889.8210936889.8208796889.8212790.00019818840
6889.8211046889.8209026889.8206886889.8210880.00053518840
6924.6155006924.6153056924.6150816924.6154810.00008619100
6956.7340516956.7338626956.7336306956.7340300.00156819340
7249.8053687249.8052347249.8049337249.8053150.00039421530
7257.8350297257.8348977257.8345957257.8349760.00008221590
7261.8496577261.8495257261.8492227261.8496030.00012921620
7265.8647817265.8646507265.8643477265.8647260.00014721650
7613.8046497613.8045757613.8042197613.8045420.00008424250
7621.8338157621.8337427621.8333857621.8337060.00019124310
7625.8486817625.8486097625.8482527625.8485720.00022624340
7675.3630837675.3630187675.3626567675.3629660.00019624710
7679.3781567679.3780917679.3777307679.3780380.00022824740
7961.7453977961.7453667961.7449937961.7452270.00070726850
7973.7888247973.7887957973.7884217973.7886520.00016226940
7993.8623047993.8622767993.8619037993.8621280.00017227090
8004.5676278004.5676008004.5672278004.5674490.00048127170
8020.6260708020.6260458020.6256728020.6258900.00070227290
8020.6266088020.6265838020.6262108020.6264270.00034527290
8262.8477148262.8477068262.8473478262.8474860.00038729100
8290.9486078290.9486008290.9482438290.9483730.00082429310
8325.7449608325.7449558325.7446018325.7447200.00061729570
8341.8024138341.8024098341.8020578341.8021700.00036129690
8343.1401768343.1401718343.1398208343.1399320.00059129700
8345.8163408345.8163368345.8159858345.8160960.00065729720
8345.8176988345.8176948345.8173438345.8174540.00179729720
8345.8169878345.8169838345.8166328345.8167430.00010329720
8349.8317508349.8317478349.8313968349.8315060.00021229750
8353.8467178353.8467138353.8463638353.8464710.00065029780
8357.8614978357.8614938357.8611448357.8612510.00078329810
8357.8610778357.8610738357.8607238357.8608300.00010529810
8406.0371058406.0371038406.0367598406.0368490.00065430170
8355.1849678355.1849638355.1846138355.1847210.00033029791
8356.5223918356.5223888356.5220388356.5221450.00036729801
8357.8609508357.8609478357.8605978357.8607040.00031629811
8359.1995798359.1995768359.1992268359.1993330.00031129821
8360.5369478360.5369448360.5365948360.5367000.00034329831
8361.8754288361.8754248361.8750758361.8751800.00030729841
8363.2142098363.2142058363.2138568363.2139610.00036129851
8364.5518758364.5518718364.5515228364.5516270.00034829861
8365.8907288365.8907258365.8903768365.8904800.00039329871
8369.9051468369.9051438369.9047958369.9048980.00036129901
8371.2429348371.2429318371.2425838371.2426850.00031029911
8372.5812018372.5811988372.5808508372.5809520.00038429921
8373.9197488373.9197458373.9193978373.9194990.00034829931
8375.2579608375.2579578375.2576098375.2577100.00031929941
8376.5963248376.5963218376.5959748376.5960740.00033629951
8377.9342418377.9342388377.9338918377.9339910.00035229961
8379.2730098379.2730068379.2726598379.2727590.00034429971
8380.6110588380.6110558380.6107088380.6108070.00037429981
8653.6102378653.6102378653.6099338653.6099350.00026532020
8692.4183518692.4183508692.4180548692.4180420.00083332310
8705.8010028705.8010008705.8007088705.8006910.00049232410
8712.4917078712.4917058712.4914138712.4913940.00078232460
8764.6832928764.6832888764.6830088764.6829710.00036632850
8827.5814668827.5814608827.5811948827.5811340.00031433322
8831.5946918831.5946848831.5944208831.5943580.00052133350
8835.6103268835.6103208835.6100568835.6099930.00058133380
9006.9034169006.9033999006.9031799006.9030560.00037934662
9063.1088129063.1087909063.1085859063.1084440.00039735081
9064.4475619064.4475399064.4473359064.4471920.00036235091
9065.7850809065.7850589065.7848549065.7847110.00043935101
9067.1243069067.1242849067.1240809067.1239370.00039035111
9068.4615709068.4615489068.4613459068.4612010.00041735121
9069.7999769069.7999549069.7997519069.7996070.00046335131
9071.1384559071.1384329071.1382309071.1380850.00053035141
9076.4911139076.4910909076.4908899076.4907430.00043135181
9077.8290809077.8290579077.8288569077.8287090.00045335192
9077.8299289077.8299059077.8297049077.8295570.00051135192
9077.8292789077.8292559077.8290549077.8289080.00041835191
9079.1676819079.1676589079.1674589079.1673100.00040335201
9080.5061539080.5061309080.5059309080.5057820.00046735211
9081.8444219081.8443979081.8441989081.8440500.00044135221
9083.1818579083.1818349083.1816359083.1814860.00039835231
9084.5205359084.5205129084.5203139084.5201640.00059935241
9088.5347319088.5347079088.5345099088.5343590.00034635271
9089.8736739089.8736499089.8734529089.8733010.00030935281
9091.2117339091.2117099091.2115129091.2113610.00034735291
9092.5497889092.5497649092.5495689092.5494160.00037935301
9093.8876199093.8875959093.8873999093.8872470.00034135311
9095.2260179095.2259929095.2257969095.2256440.00036435321
9096.5648539096.5648299096.5646339096.5644810.00045335331
9097.9028149097.9027899097.9025949097.9024410.00036835341
9103.2554389103.2554139103.2552199103.2550640.00036635381
9104.5941959104.5941699104.5939769104.5938210.00033435391
9104.5959449104.5959199104.5957269104.5955700.00047135392
9105.9325019105.9324759105.9322839105.9321270.00044035401
9107.2707859107.2707599107.2705679107.2704100.00051535411
9108.6089059108.6088799108.6086879108.6085300.00049935421
9108.6086959108.6086699108.6084779108.6083210.00073135422
9109.9468429109.9468169109.9466259109.9464670.00036135431
9111.2849699111.2849439111.2847529111.2845940.00035035441
9112.6222799112.6222539112.6220629112.6219040.00121935451
9124.6676729124.6676449124.6674579124.6672950.00039535542
9191.5780539191.5780199191.5778529191.5776680.00039536042
9203.6230269203.6229919203.6228279203.6226390.00056336132
9411.0487239411.0486639411.0485649411.0483140.00015837683
9436.4752459436.4751819436.4750909436.4748340.00015037873
9444.5046799444.5046149444.5045269444.5042670.00028237933
9457.8867529457.8866859457.8866029457.8863390.00029138033
9465.9163429465.9162749465.9161939465.9159280.00029038093
9480.6367249480.6366539480.6365779480.6363080.00021938203
9502.0482499502.0481759502.0481069502.0478320.00025738363
9800.4742009800.4740789800.4741099800.4737670.00027040593
10,183.20810510,183.20791210,183.20806610,183.2076740.00034143451
10,184.54676110,184.54656810,184.54672210,184.5463300.00030743461
10,185.88443510,185.88424210,185.88439610,185.8840040.00032543471
10,187.22371810,187.22352410,187.22368010,187.2232870.00039143481
10,188.56088510,188.56069110,188.56084710,188.5604540.00034443491
10,189.89991810,189.89972410,189.89988010,189.8994880.00038643501
10,191.23656810,191.23637410,191.23653010,191.2361380.00031843511
10,192.57493910,192.57474410,192.57490110,192.5745090.00041043521
10,196.59068710,196.59049210,196.59065010,196.5902570.00029543551
10,197.92913110,197.92893510,197.92909410,197.9287010.00037643561
10,199.26689210,199.26669610,199.26685510,199.2664620.00034843571
10,200.60560110,200.60540510,200.60556410,200.6051710.00035243581
10,201.94359610,201.94339910,201.94355910,201.9431660.00031443591
10,203.28211810,203.28192110,203.28208110,203.2816880.00047943601
10,204.61955710,204.61936010,204.61952110,204.6191270.00039843611
10,205.95831110,205.95811410,205.95827510,205.9578810.00037043621
Table A2. Occultation timings of WASP-4 b including three different corrections for the LTE. Time is given in BJD TDB − 2,450,000 . “Time” denotes the mid-occultation times without the light–time correction due to planet c, “Time LTE” denotes LTE-corrected timings using the nominal value for the time of inferior conjunction of planet c from Turner et al. [18], “Time LTE lower ” denotes the timings corrected using the lower boundary of the respective 1 σ interval, and “LTE upper ” denotes the timings corrected using the upper boundary of the interval. The “Source” column denotes the source of the timings, with (5) the timing from Cáceres et al. [22], (6) the two from Beerer et al. [23], and (7) the one from Zhou et al. [24].
Table A2. Occultation timings of WASP-4 b including three different corrections for the LTE. Time is given in BJD TDB − 2,450,000 . “Time” denotes the mid-occultation times without the light–time correction due to planet c, “Time LTE” denotes LTE-corrected timings using the nominal value for the time of inferior conjunction of planet c from Turner et al. [18], “Time LTE lower ” denotes the timings corrected using the lower boundary of the respective 1 σ interval, and “LTE upper ” denotes the timings corrected using the upper boundary of the interval. The “Source” column denotes the source of the timings, with (5) the timing from Cáceres et al. [22], (6) the two from Beerer et al. [23], and (7) the one from Zhou et al. [24].
TimeTime LTETime LTElowerTime LTEupperError (d)EpochSource
5102.6116205102.6111855102.6114985102.6114780.000740548.55
5172.2015905172.2011565172.2014565172.2014600.001300600.56
5174.8778005174.8773665174.8776655174.8776710.000870602.56
6907.8871406907.8869426907.8867236907.8871230.0029001897.57

Notes

1
https://mast.stsci.edu/, (accessed on 13 October 2023)
2
http://www.astropy.org, (accessed on 15 October 2023)

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Universe 09 00506 g001
Figure 1. The orbits of WASP-4 (star symbol, dashed line) and WASP-4 b (black dot, solid line) with reference to the system’s center of mass (black plus symbol) from REBOUND. The x- and y-axes lie within the orbital plane of the planets, which are assumed to be coplanar. The orbit of planet c is not visible in this view, since its semi-major axis is assumed to be 6.82 AU.
Figure 1. The orbits of WASP-4 (star symbol, dashed line) and WASP-4 b (black dot, solid line) with reference to the system’s center of mass (black plus symbol) from REBOUND. The x- and y-axes lie within the orbital plane of the planets, which are assumed to be coplanar. The orbit of planet c is not visible in this view, since its semi-major axis is assumed to be 6.82 AU.
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Figure 2. Phase-folded TESS light curve of WASP-4 b from Sector 69, including 16 transits. The data are shown as black dots after modeling and correction with TLCM, with the median solution model shown as the red line.
Figure 2. Phase-folded TESS light curve of WASP-4 b from Sector 69, including 16 transits. The data are shown as black dots after modeling and correction with TLCM, with the median solution model shown as the red line.
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Figure 3. O-C plot showing all transit timing (top) and occultation data (bottom) together with the orbital decay and apsidal precession models. The transit number from the reference epoch is shown on the x-axis, while the deviation from the median linear ephemeris is shown on the y-axis. Colors are according to the legend, with the pink shaded area showing the 1 σ interval around the median solution orbital decay fit. For the available TESS data, the weighted mean timings with their respective error bars for each sector have been added. No LTE correction was applied in this case.
Figure 3. O-C plot showing all transit timing (top) and occultation data (bottom) together with the orbital decay and apsidal precession models. The transit number from the reference epoch is shown on the x-axis, while the deviation from the median linear ephemeris is shown on the y-axis. Colors are according to the legend, with the pink shaded area showing the 1 σ interval around the median solution orbital decay fit. For the available TESS data, the weighted mean timings with their respective error bars for each sector have been added. No LTE correction was applied in this case.
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Figure 4. Same as Figure 3, but with the nominal LTE correction applied to the timing data to account for the influence of planet c.
Figure 4. Same as Figure 3, but with the nominal LTE correction applied to the timing data to account for the influence of planet c.
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Table 1. The resulting parameters from the TLCM fit to the TESS S69 data. The impact parameters are given by b, and u a and u b describe the quadratic limb-darkening parameters. T 0 is given in [BJD TDB − 2,450,000 ], and U denotes a uniform prior.
Table 1. The resulting parameters from the TLCM fit to the TESS S69 data. The impact parameters are given by b, and u a and u b describe the quadratic limb-darkening parameters. T 0 is given in [BJD TDB − 2,450,000 ], and U denotes a uniform prior.
Parameter (Unit)PriorResult
a R ★ − 1 U ( 5.4773 ± 0.1000 ) 5.39795 ± 0.01864
R P R ★ − 1 U ( 0.1540 ± 0.0100 ) 0.1524 ± 0.0009
b U ( 0.5 ± 1.0 ) 0.219 ± 0.015
P (d) U ( 1.338231 ± 0.000100 ) 1.338239 ± 0.000014
T 0 U (10,192.575 ±  0.050 ) 10,192.57567 ± 0.00008
u a U ( 0.5 ± 1.0 ) 0.35 ± 0.10
u b U ( 0.5 ± 1.0 ) 0.22 ± 0.14
Table 2. Priors and results from our MCMC modeling of the transit timings without any LTE correction. T 0 is given in BJD TDB − 2,450,000 . Units are given in parentheses if applicable. Priors were chosen according to the results of Harre et al. [19] with enough flexibility for the walkers to sufficiently explore the parameter space.
Table 2. Priors and results from our MCMC modeling of the transit timings without any LTE correction. T 0 is given in BJD TDB − 2,450,000 . Units are given in parentheses if applicable. Priors were chosen according to the results of Harre et al. [19] with enough flexibility for the walkers to sufficiently explore the parameter space.
ModelParameterPriorResult
Circular orbit T 0 U ( 7000 , 8000 ) 7490.68717 ± 0.00002
P (d) U ( 1.3 , 1.4 ) 1.33823133 ± 0.00000001
  BIC-382.45
Orbital decay T 0 U ( 7000 , 8000 ) 7490.68735 ± 0.00002
P (d) U ( 1.3 , 1.4 ) 1.33823122 ± 0.00000001
d P / d N (d/orbit) U ( − 10 − 8 , − 10 − 11 ) ( − 2.43 ± 0.22 ) × 10 − 10
  BIC-263.01
Apsidal precession T 0 U ( 7000 , 8000 ) 7490.68690 ± 0.00019
P (d) U ( 1.3 , 1.4 ) 1.33823140 ± 0.00000009
d ω / d N (rad/orbit) U ( 0.0001 , 0.002 ) ( 7.68 ± 1.60 ) × 10 − 4
e U ( 10 − 6 , 10 − 2 ) 0.0013 ± 0.0005
ω 0 (rad) U ( 2 , 2 π ) 3.770 ± 0.242
BIC-271.62
Table 3. Results from our MCMC modeling for the three cases of the LTE correction. The priors are the same as those given in Table 2. T 0 is given in BJD TDB − 2,450,000 . Units are given in parentheses if applicable. LTE refers to case (1), and LTE low and LTE up refer to cases (2) and (3), respectively. Δ BIC is defined relative to the circular Keplerian orbit case for each LTE correction. Note that the Δ BIC values should only be compared within a column, not between columns.
Table 3. Results from our MCMC modeling for the three cases of the LTE correction. The priors are the same as those given in Table 2. T 0 is given in BJD TDB − 2,450,000 . Units are given in parentheses if applicable. LTE refers to case (1), and LTE low and LTE up refer to cases (2) and (3), respectively. Δ BIC is defined relative to the circular Keplerian orbit case for each LTE correction. Note that the Δ BIC values should only be compared within a column, not between columns.
ModelParameterResult LTEResult LTE lower Result LTE upper
Circ. orbit T 0 7490.68699 ± 0.00002 7490.68697 ± 0.00002 7490.68690 ± 0.00002
P (d) 1.33823145 ± 0.00000001 1.33823129 ± 0.00000001 1.33823126 ± 0.00000001
   Δ BIC000
Orb. decay T 0 7490.68721 ± 0.00002 7490.68727 ± 0.00002 7490.68695 ± 0.00002
P (d) 1.33823132 ± 0.00000001 1.33823112 ± 0.00000001 1.33823124 ± 0.00000001
d P / d N (d/orbit) ( − 2.97 ± 0.22 ) × 10 − 10 ( − 3.94 ± 0.22 ) × 10 − 10 ( − 0.61 ± 0.21 ) × 10 − 10
   Δ BIC−181.47−323.87−2.65
Aps. prec. T 0 7490.68608 ± 0.00055 7490.68677 ± 0.000096 7490.68684 ± 0.00018
P (d) 1.33823113 ± 0.00000026 1.33823141 ± 0.00000008 1.33823142 ± 0.00000013
d ω / d N (rad/orbit) ( 5.51 ± 1.69 ) × 10 − 4 ( 9.69 ± 1.16 ) × 10 − 4 ( 6.31 ± 2.76 ) × 10 − 4
e 0.0029 ± 0.0019 0.0016 ± 0.0003 0.0010 ± 0.0007
ω 0 (rad) 2.795 ± 0.274 3.922 ± 0.105 4.455 ± 0.515
Δ BIC−177.46−343.83+25.66
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Harre, J.-V.; Smith, A.M.S. The Apparent Tidal Decay of WASP-4 b Can Be Explained by the Rømer Effect. Universe 2023, 9, 506. https://doi.org/10.3390/universe9120506

AMA Style

Harre J-V, Smith AMS. The Apparent Tidal Decay of WASP-4 b Can Be Explained by the Rømer Effect. Universe. 2023; 9(12):506. https://doi.org/10.3390/universe9120506

Chicago/Turabian Style

Harre, Jan-Vincent, and Alexis M. S. Smith. 2023. "The Apparent Tidal Decay of WASP-4 b Can Be Explained by the Rømer Effect" Universe 9, no. 12: 506. https://doi.org/10.3390/universe9120506

APA Style

Harre, J. -V., & Smith, A. M. S. (2023). The Apparent Tidal Decay of WASP-4 b Can Be Explained by the Rømer Effect. Universe, 9(12), 506. https://doi.org/10.3390/universe9120506

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