Tidal Quality of the Hot Jupiter WASP-12b
Round 1
Reviewer 1 Report
This paper discusses the tidal evolution of the very short period hot Jupiter WASP-12 b. This planet has become a hot topic over the past few years because it is the only hot Jupiter for which there are observed shifts in transit times that provide reasonably strong evidence in favour of tidally-driven orbital decay on a timescale of ~3 Myr. Previous studies have generally focused on tidal dissipation in the host star WASP 12, since this is thought to be able to drive planetary inspiral because the star rotates much slower than the planet orbits the star (i.e. the opposite tidal evolution of the Earth-Moon systems due to tides in the Earth). This paper discusses this possibility and argues against it, and proposes the new idea that the observed orbital migration could be caused by tidal dissipation inside the planet WASP-12b rather than the star if the planet can continue to maintain a sufficiently eccentric orbit against tidal damping.
The scenario for WASP-12 b’s migration that has been discussed the most in the literature, and which has been proposed to explain the observed orbital decay of WASP-12 b is tidal dissipation of internal gravity waves in the radiation zone of the star WASP-12. This is predicted to be the most efficient tidal mechanism in hot Jupiter host stars that rotate slowly, and the resulting rate of tidal migration due to the dissipation of these waves can explain the observed value if the star has a radiative core (in which the waves can break where they reach large amplitudes), but it is difficult to explain this value if the star is on the main sequence and has a convective core. This is for two reasons: 1) the dissipation of these waves is likely to be inhibited if the star has a convective core that can reflect gravity waves before they reach the stellar centre where they can break and be efficiently dissipated, and 2) the excitation is weaker in more massive stars with thinner outer convection zones, in which the density at the radiative/convective interface for these waves is smaller. As a result, Weinberg et al. (2017) proposed that WASP-12 could be a subgiant star, which has a radiative core. If correct, this would neatly explain the observations, but it has turned out to be difficult to get stellar models to agree with the observed properties of the star in such subgiant models. The author have revisited this and use Gaia data to argue against this hypothesis.
The present paper discusses the alternative idea that if the planet's orbit is eccentric, excited by chance events like e.g. dynamical interactions with a recently destroyed planet, then dissipation of eccentricity tides in the planet could cause the planetary orbit to change in a similar way to that observed if the planet is similarly as tidally dissipative as Jupiter in our solar system. In my opinion, while I’m not convinced that the idea the authors are proposing is going to be the correct one for WASP-12 b, I think this paper is sufficiently interesting and topical to be worth publishing. It tackles a hot topic in astrophysics, is timely, and provides a new perspective on this problem. Before I would be happy to recommend this paper for publication, I would suggest the authors consider addressing further the following issues:
- Personally, I don’t find the proposed scenario very plausible because we would have to explain the maintenance of the eccentricity of WASP-12 b. Dynamical interactions with an external companion with a misaligned orbit have been proposed, but rely on the existence of a non-detected planet. A recently destroyed inner planet (i.e. an inner planet in 1:2 MMR that excited WASP-12 b’s eccentricity in the past) may well be possible but sounds implausible. This is surely a low probability event. It is true however that WASP-12 b is just one system, so a low probability event for explaining it is not completely ruled out. I think it would be worth commenting a bit further on the plausibility (or otherwise) of such an explanation.
- In my opinion, the presence of planets with P_orb <10 days around 0.49% of low-luminosity red giants (Grunblatt et al., 2019) does not rule out stellar tidal dissipation becoming efficient due to structural changes that lead to gravity wave breaking, for two reasons: 1) this mechanism would only disrupt the closest hot Jupiters inside 2-3 days before the end of the MS, as the inspiral time would otherwise be too long. 2) tidal dissipation of equilibrium tides is likely to be important in the giant phase of stellar evolution, and some of these planets could have migrated rapidly inside 10 days due to this mechanism after the star became a red giant (see e.g. Mustill & Villaver 2012, ApJ, 761, Issue 2, article id. 121)
- The modelling of Remus et al. 2012 etc. is based on old pictures for the interior structures of Jupiter/Saturn. Recent Juno observations (e.g. Wahl et al. 2017, GRL, 44, 10, 4649) indicate that the heavy elements are likely to be distributed throughout a large portion of the interior (e.g. out to 0.5 to 0.6 of the planetary radius), and a core like that considered in the old interior models might not exist. Another mechanism that has been proposed is gravity waves in stably-stratified layers of the planet (e.g. Fuller at al. 2016 https://arxiv.org/abs/1601.05804 since published in MNRAS, André et al. 2019 https://arxiv.org/abs/1902.04848 since published in A&A), and inertial waves in the convective envelope (e.g. Ogilvie & Lin 2004). The idea of resonance locking of these two types of modes has also been proposed (e.g. Fuller at al. 2016). These are possible alternatives to viscoelastic dissipation in a core, which could be briefly commented on. However this will not change your conclusions as Lainey et al. observations have constrained such values of K_2 for Jupiter & Saturn due to migration of the moons.
- In my opinion, the inferred eccentricity being instead due to semidiurnal variation of the radial velocity due to the equilibrium tide, has not been ruled out. Please see some updated work on this by e.g. Bunting & Terquem (2021, MNRAS: https://ui.adsabs.harvard.edu/abs/2021MNRAS.500.2711B/abstract, particularly their section 5.3) who studied this using a compressible and non-adiabiatic linearized numerical approach. They found significant differences with Arras et al. 2012. You could perhaps comment on possibility this further.
More minor issues in the order in which they appear in the manuscript:
Title: Should the title be “Tidal quality factor of” rather than “Tidal quality of”? The latter looks strange to me.
Abstract line 12: the measured rate of -> the measured rate of change of ?
Since the modified tidal quality factor in yellow dwarfs is insufficient to warrant such a decay rate -> I’m not sure that this is true, since it strongly depends on the stellar model parameters. See e.g. Barker (2020) that you cite.
In what way does the present paper “employ a more advanced theory of frequency-dependent tidal dissipation”? Your arguments only imply or rely on a constant value of K_2 i.e. Q’_2, rather than result from a first principles study of K_2 or Q_2’ that would probably imply frequency-dependent values.
Line 45: “surprising Q’~ 1.8 × 10^5, which is at least two orders of magnitude lower (i.e., the tidal dissipation rate is two orders of magnitude higher) than what is expected from a solar-type MS dwarf” -> Note that this statement is not correct. The result for a solar-mass MS dwarf is in fact given by e.g. Eq.54 and Fig. 8 of Barker (2020) and is compatible with this number for Q’ if gravity wave breaking has occurred (or if gravity waves are otherwise fully damped).
Line 69 -> dynamics -> dynamical ?
Figure 1: If straightforward to do so, I would suggest making the blue point for the observed properties of WASP-12 a different colour or symbol so that it is easier to spot in the figure.
I would expect that with further tweaking of stellar model parameters it would be possible to get an even better match with the observed properties for WASP-12 (not that this would necessarily be any more meaningful though).
Line 119: Please define l, m, p, q when they are first used.
Line 126: I would not say that WASP-12 is similar to the Sun in mass and radius, particularly because it is likely to have a very different interior structure (being an F star with a convective core on the MS). Note also that internal gravity waves are believed to be the dominant tidal mechanism in “slowly-rotating stars” (for which inertial waves cannot be excited, as opposed to in more rapidly rotating stars).
Line 149: What does “incremental potential” mean? Do you mean linearized perturbation of the Eulerian gravitational potential perturbation?
Line 179: Note that Bunting & Terquem recent re-analyzed this problem using a compressible and non-adiabiatic numerical approach. They found significant differences with Arras et al. 2012. This is referred to above.
Line 186: Note that the toroidal terms in the velocity field are reasonably justifiably ignored by Arras et al. because they are small for slowly rotating stars like WASP-12, though please also see Bunting & Terquem’s paper listed above. This star is believed to rotate as slow as 23-38 days, which is much longer than the planetary orbit, and so Coriolis modifications (and modifications to the tidal frequency) to the tidal flow are likely to be extremely small.
Line 188: anomalistic -> anomalous ?
Line 194: What are the observational error bars on the planetary eccentricity? e=0.04, but what is the lowest acceptable value? Is it known to be necessarily incompatible with a circular orbit?
Line 209: What evidence is there that the outer layer of WASP-12b is locked in pseudosynchronous rotation? I do not think any evidence has been obtained for this so far.
Line 213: demonstrated -> hypothesized ?
Linear 215: I would add that your results show this “in the case that this mechanism is the only contributor to orbital evolution” i.e. neglecting stellar tides etc.
Is Table 1 needed? I would suggest this could be omitted without making the paper any more difficult to read, but I’ll leave this for the authors to decide.
References: Perhaps reduce the number of authors listed for the Gaia papers, since they take up a lot of space? Ultimately this is a decision for the journal though.
Note that there is a repeated reference to Weinberg et al. 2017, so one of these should be removed.
Author Response
Please see the attached document.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
This paper addresses an interesting and important exoplanetary system, with implications for the important topic of the internal dynamics and structure of exoplanets. Unfortunately, I do not deem the paper to be acceptable in its current form. I am prepared to re-consider, but I raise a major issue that must first be addressed:
I think that the authors are cherry-picking an orbital eccentricity that supports their hypothesis of tidal dissipation in the planet. The timing of the secondary eclipse is *by far* more sensitive to eccentricity than are radial velocity measurements, and it's incorrect to adopt an eccentricity from RV measurements more than a decade ago (Husnoo et al. 2011), when those results are contradicted by multiple secondary eclipse timings of high precision. Recent secondary eclipse times are quoted in Yee et al. (2020, ApJ 188, L5), Garhart et al. (2020, AJ, 159, 137) and Wong et al. (2021, AJ, 162, 127). Those eclipse times are all consistent with a circular orbit. The orbit can only be eccentric by 0.04 if our line of sight is closely parallel to the major axis of the orbit. But this paper quotes omega = 270 degrees (on line 171), so that solution does not appear to be promising. The conclusion of significant tidal dissipation in the planet is extraordinary, and extraordinary conclusions require extraordinary evidence. The planet's tidal dissipation can only be high if the orbital eccentricity is significantly non-zero. The authors must present a joint orbital analysis of the RV and eclipse timing data to support their extraordinary claim that the eccentricity is 0.04.
Minor comments:
Figure 1 needs observational error bars on both the X- and Y-positions of the star, and those uncertainties should be factored into the discussion in Section 2.
Section 3.1, WASP-12a >> WASP-12A (the convention is to capitalize the star's letter)
Author Response
Please see the document attached.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
In my opinion, the current version of the paper depends on an eccentricity value that can probably be proven to be wrong, with a deeper look at all of the data, not just relying on RV data. RV data are preeminent for measuring planet masses, but not very sensitive to orbital eccentricity. Even a small eccentricity can cause the secondary eclipse time to shift by hours, an obvious and easily measured shift (whereas the RV effects are subtle). Even in the case where our line of sight is aligned with the semi-major axis, the duration of the secondary eclipse (compared to transit) will be significantly affected, and thereby the eccentricity can be measured. I suggest that the authors mention that their adopted value of the eccentricity will have significant implications for the timing and/or duration of the secondary eclipse, pending an in-depth analysis of all of the data (see below).
The authors reply: "We do agree that this planet’s eccentricity remains a matter of discussion and, largely, of interpretation of the available data. We do not agree that the existing RV data must be ignored in the face of the data on timing of the secondary eclipses."
I didn't advocate ignoring the RV data. The best response to my critique would be a joint analysis of the RV data in combination with the timing and shape (i.e., duration) of the secondary eclipse (and also the transit), to constrain the orbital parameters more realistically. The authors will probably claim that to be beyond the scope of their paper. If so, the next best revision would be a more balanced discussion of the possible eccentricity values, and mentioning the implications for the timing and shape of the eclipse. I also think it would be valuable to expand on the remark that "our conclusions remain correct also for lower values of e" and add more discussion of that issue.
Author Response
Please see the document attached.
Author Response File: Author Response.pdf
