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arxiv: 1906.10330 · v1 · pith:7JN2TJOFnew · submitted 2019-06-25 · 🌌 astro-ph.EP

Thermal Tides in Rotating Hot Jupiters

Umin Lee , Daiki Murakami This is my paper

Pith reviewed 2026-05-25 16:40 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords hot Jupitersthermal tidestidal torqueoscillation modesresonanceradiative coolingplanetary rotation
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The pith

Resonances with g-, r-, and inertial modes enhance the tidal torque from thermal tides in rotating hot Jupiters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates the tidal torque from semi-diurnal thermal tides on rotating hot Jupiters while including radiative cooling in the envelope and the effects of planetary rotation. It uses a model with a nearly isentropic convective core and a thin radiative envelope, representing the responses through spherical-harmonic series with multiple l values for each m. At low forcing frequencies, the calculation finds resonances with g- and r-modes in the envelope and inertial modes in the core that increase the torque. Resonances with g- and r-modes produce broad peaks in torque versus frequency, while inertial-mode resonances produce very sharp peaks, and the torque curve differs between prograde and retrograde forcing at long periods because r-modes exist only on the retrograde side.

Core claim

Resonance with the g- and r-modes produces broad peaks and that with the inertial modes very sharp peaks in the tidal torque as a function of forcing frequency, with different behavior between prograde and retrograde forcing for long periods due to r-modes existing only on the retrograde side.

What carries the argument

Spherical-harmonic series expansions (multiple l for fixed m) of the non-adiabatic tidal responses in a two-layer model consisting of a convective core and radiative envelope.

If this is right

  • The tidal torque increases at the resonant forcing frequencies set by the planet's oscillation modes.
  • Resonance with g- and r-modes in the envelope produces broad peaks in the torque-frequency relation.
  • Resonance with inertial modes in the core produces very sharp peaks in the torque-frequency relation.
  • The torque as a function of forcing period differs between prograde and retrograde cases at long periods.
  • Non-adiabatic effects associated with the modes control the width of the resonance peaks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The direction-dependent torque could produce different spin-orbit evolution depending on whether the planet is being forced prograde or retrograde.
  • The narrow inertial-mode peaks might create isolated frequency windows where thermal tides strongly affect orbital migration or spin.
  • Incorporating these resonances could change estimates of how thermal tides contribute to the long-term evolution of hot-Jupiter orbits and atmospheres.

Load-bearing premise

A simple model of a nearly isentropic convective core plus thin radiative envelope is adequate to represent the tidal responses of rotating hot Jupiters when rotation and radiative cooling are included.

What would settle it

Tidal torque or orbital-decay measurements on hot Jupiters that show no peaks at the resonant frequencies of g-, r-, or inertial modes and no difference between prograde and retrograde forcing at long periods.

Figures

Figures reproduced from arXiv: 1906.10330 by Daiki Murakami, Umin Lee.

Figure 1
Figure 1. Figure 1: Square of the Brunt-V¨ais¨al¨a frequency N (solid line) and the Lamb frequency Ll for l = 2 (dotted line) plotted as a function of log10 p for the Jovian model, where N2 and L 2 l are normalized by GM/R3 with M and R being the mass and radius of the model. In this paper, we use pb = 100bar = 108dyn/cm2 and set the outer boundary Re at p = 0.01dyn/cm2 and define the planet’s radius R = Re/1.01. We use M = 0… view at source ↗
Figure 2
Figure 2. Figure 2: Tidal torque, given in erg, due to thermal tides (left panel) and gravitational tides (right panel) as a function of the forcing frequency ¯ω = ω/σ0 for Ω = 0, where the red (black) lines are for positive (negative) ¯ N , and the dash-dotted lines, solid lines, and dotted lines are for τ∗ = 0.1, 1, and 10 days, respectively [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Tidal torque, given in erg, due to thermal tides (j∗ 6= 0 and ψ = 0) as a function of the forcing frequency ¯ω for Ω = 0 ¯ .05, where positive (negative) ¯ω indicates prograde (retrograde) forcing, and the red (black) lines are for positive (negative) N . 25 26 27 28 29 30 31 32 33 -1 -0.5 0 0.5 1 log10|N| ω/σ 0 25 26 27 28 29 30 31 32 33 -0.2 -0.1 0 0.1 0.2 log10|N| ω/σ 0 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Color maps of −Im(ρ ′ ) for φ = 0 produced by semi-diurnal thermal tides, from left to right panels, at the forcing frequency tabulated in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Tidal torque, given in erg, due to thermal tides for τ∗ = 1 day versus the tidal forcing period τtide = 2π/ω in days, where the rotation speed Ω of the planet is given by Ω = Ωorb − π/τtide as a function of τtide for a given Ωorb, and we use Ω¯ orb = 0.0537 for the left panel and Ω¯ orb = −0.0537 for the right panel. Here, the red lines and black lines respectively indicate positive and negative torque N .… view at source ↗
Figure 10
Figure 10. Figure 10: Comparing to Fig. 7, for which we assumed [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Normalized energy dissipation rate D due to thermal tides versus forcing frequency ¯ω for τ∗ = 10day (dash-dotted line), 1day (solid line), and 0.1day (dotted line) for Ω = 0 ¯ .1 where D is defined by equation (51). 27 28 29 30 31 32 33 34 35 -1 -0.5 0 0.5 1 log10|N| ω/σ 0 27 28 29 30 31 32 33 34 35 -0.2 -0.1 0 0.1 0.2 log10|N| ω/σ 0 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same as [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: max |ρ ′ 2 /ρ|  as a function of the forcing period τtide (day) for j∗ 6= 0 and ψ = 0 and for τ∗ = 1day, where the rotation speed Ω is given by Ω = Ωorb − π/τtide. This figure corresponds to [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
read the original abstract

We calculate tidal torque due to semi-diurnal thermal tides in rotating hot Jupiters, taking account of the effects of radiative cooling in the envelope and of the planets rotation on the tidal responses. We use a simple Jovian model composed of a nearly isentropic convective core and a thin radiative envelope. To represent the tidal responses of rotating planets, we employ series expansions in terms of spherical harmonic functions $Y_l^m$ with different $l$s for a given $m$. For low forcing frequency, there occurs frequency resonance between the forcing and the $g$- and $r$-modes in the envelope and inertial modes in the core. We find that the resonance enhances the tidal torque, and that the resonance with the $g$- and $r$-modes produces broad peaks and that with the inertial modes very sharp peaks, depending on the magnitude of the non-adiabatic effects associated with the oscillation modes. We also find that the behavior of the tidal torque as a function of the forcing frequency (or period) is different between prograde and retrograde forcing, particularly for long forcing periods because the $r$-modes, which have long periods, exist only on the retrograde side.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript calculates the tidal torque due to semi-diurnal thermal tides in rotating hot Jupiters. It employs a two-layer model consisting of a nearly isentropic convective core and a thin radiative envelope, incorporates radiative cooling and rotation via spherical-harmonic expansions that allow multiple l for each m, and reports that resonances with g- and r-modes produce broad torque peaks while inertial modes produce sharp peaks, with prograde/retrograde asymmetry arising because r-modes exist only on the retrograde side.

Significance. If the resonance calculations hold, the work supplies a concrete forward model of how non-adiabatic thermal tides can drive spin-orbit evolution in hot Jupiters, distinguishing the width of g/r-mode resonances from the narrow inertial-mode features and the one-sided r-mode contribution. This supplies a useful benchmark for more elaborate interior models.

minor comments (3)
  1. [Abstract] The abstract states that the model is 'nearly isentropic' in the core; the manuscript should quantify the small entropy gradient adopted and show that the reported resonance locations are insensitive to that choice within the stated range.
  2. Clarify the truncation level in the spherical-harmonic series (i.e., the maximum l retained for each m) and demonstrate convergence of the torque peaks with respect to that truncation.
  3. The distinction between broad g/r-mode peaks and sharp inertial-mode peaks is central; a figure or table that tabulates the quality factor or damping rate for representative modes of each class would make the dependence on non-adiabatic effects explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work on thermal tidal torques in rotating hot Jupiters. The referee correctly identifies the key results regarding resonances with g-, r-, and inertial modes and the resulting torque behavior. We appreciate the recommendation for minor revision and will incorporate any editorial or minor clarifications in the revised version. No major comments requiring substantive changes were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper performs a direct forward calculation of tidal torques from a specified Jovian model (nearly isentropic core + thin radiative envelope) using spherical-harmonic expansions that incorporate rotation and radiative cooling. Resonance peaks with g/r-modes and inertial modes are reported as numerical outcomes of the linear non-adiabatic equations; no parameter is fitted to the target torque curve and then re-labeled as a prediction, and no load-bearing step reduces to a self-citation or self-definition. The derivation chain is therefore self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the adequacy of the simplified two-layer model and the validity of the spherical-harmonic expansion for capturing non-adiabatic tidal responses; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption A simple Jovian model with nearly isentropic convective core and thin radiative envelope suffices to represent hot Jupiters for tidal torque calculations.
    Explicitly stated as the basis for the calculations in the abstract.

pith-pipeline@v0.9.0 · 5734 in / 1225 out tokens · 34808 ms · 2026-05-25T16:40:07.211595+00:00 · methodology

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Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Andr\'e Q., Barker A.J., Mathis S., 2017, A&A, 605, A117

    work page 2017

  2. [2]

    et al, 2017, A&A, 603, A108

    Auclair-Desrotour P. et al, 2017, A&A, 603, A108

    work page 2017

  3. [3]

    Auclair-Desrotour P., Leconte J., 2018, A&A, 613, A45

    work page 2018

  4. [4]

    Auclair-Desrotour P., Mathis S., Laskar J., 2018, A&A, 609, A118

    work page 2018

  5. [5]

    Arras P., Bildsten L., 2006, ApJ, 650, 394

    work page 2006

  6. [6]

    Arras P., Socrates A., 2010, ApJ, 714, 1

    work page 2010

  7. [7]

    Baraffe I., Chabrier G., Barman T.S., Allard F., Hauschildt P.H., 2003, A&A, 402, 701

    work page 2003

  8. [8]

    Bodenheimer P., Lin D.N.C., Mardling R.A., 2001, ApJ, 548, 466

    work page 2001

  9. [9]

    Deming D., Mumma M.J., Espenak F., Jennings D.E., Kostiuk T., Wiedemann G., Loewenstein R., Piscitelli J, 1989, ApJ, 343, 456

    work page 1989

  10. [10]

    Clayton D.D., 1983, Principles of Stellar Evolution and Nucleosynthesis, The University of Chicago Press, Chicago

    work page 1983

  11. [11]

    Fuller J., Lai D., 2013, MNRAS, 430, 274

    work page 2013

  12. [12]

    Goldreich P., Nicholson P.D., 1977, Icarus, 30, 301

    work page 1977

  13. [13]

    Goldreich P., Soter S., 1966, Icarus, 5, 375

    work page 1966

  14. [14]

    Goodman J., Dickson E.S., 1998, ApJ, 507, 938

    work page 1998

  15. [15]

    Greenspan H.P., 1969, The Theory of Rotating Fluids, Cambridge University Press, Cambridge

    work page 1969

  16. [16]

    Gerkema T., Shrira V.I., 2005, Journal of Fluid Mechanics, 529, 195

    work page 2005

  17. [17]

    Iro N., B\'ezard B., Guillot T., 2005, A&A, 436, 719

    work page 2005

  18. [18]

    Ivanov P.B., Papaloizou J.C.B., 2007, MNRAS, 376, 682

    work page 2007

  19. [19]

    Jermyn A.D., Tout C.A., Ogilvie G.I., 2017, MNRAS, 469, 1768

    work page 2017

  20. [20]

    Kumar P., Goodman J., 1996, ApJ, 466, 946

    work page 1996

  21. [21]

    Lai D., 1997, ApJ, 490, 847

    work page 1997

  22. [22]

    Lee U., 2019, MNRAS, 484, 5845

    work page 2019

  23. [23]

    Lee U., Saio H., 1986, MNRAS, 221, 365

    work page 1986

  24. [24]

    Lee U., Saio H., 1987, MNRAS, 224, 513

    work page 1987

  25. [25]

    Lee U., Saio H., 1987b, MNRAS, 225, 643

    work page

  26. [26]

    Lee U., Saio H., 1989, MNRAS, 237, 875

    work page 1989

  27. [27]

    Lee U., Saio H., 1990a, ApJ, 359, 29

    work page

  28. [28]

    Lee U., Saio H., 1990b, ApJ, 360, 590

    work page

  29. [29]

    Lee U., Saio H., 1997, ApJ, 491, 839

    work page 1997

  30. [30]

    Magalh\ aes J.A., Weir A.L., Conrath B.J., Gierasch P.J., Leroy S.S., 1989, Nature, 337, 444

    work page 1989

  31. [31]

    Mihalas D., Weibel-Mihalas B., 1999, Foundations of Radiation Hydrodynamics, Dover Publishing, New York

    work page 1999

  32. [32]

    Ogilvie G.I., 2014, Annu. Rev. Astron. Astrophys., 52, 171

    work page 2014

  33. [33]

    Ogilvie G.I., Lin N.D.C., 2004, ApJ, 610, 477

    work page 2004

  34. [34]

    Papaloizou J., Pringle J.E., 1978, MNRAS, 182, 423

    work page 1978

  35. [35]

    Press W.H., Teukolsky S.A., 1977, ApJ, 213, 183

    work page 1977

  36. [36]

    Savonije G.J., Papaloizou J.C.B., 1984, MNRAS, 207, 685

    work page 1984

  37. [37]

    Savonije G.J., Papaloizou J.C.B., 1997, MNRAS, 291, 633

    work page 1997

  38. [38]

    Astrophy

    Stevenson D.J., Geophys. Astrophy. Fluid. Dynamics, 1979, 12, 139

    work page 1979

  39. [39]

    Stevenson D.J., Salpeter E.E., 1977a, ApJS, 35, 221

    work page

  40. [40]

    Stevenson D.J., Salpeter E.E., 1977b, ApJS, 35, 239

    work page

  41. [41]

    Turner J.S., 1979, Buoyancy Effects in Fluids, Cambridge University Press, Cambridge

    work page 1979

  42. [42]

    Unno W., Osaki Y., Ando H., Saio H., Shibahashi H., 1989, Nonradial Oscillations of Stars, 2nd ed., University of Tokyo Press, Tokyo

    work page 1989

  43. [43]

    Witte M.G., Savonije G.J., 2002, A&A, 386, 222

    work page 2002

  44. [44]

    Yoshida S., Lee U., 2000, ApJ, 529, 997

    work page 2000

  45. [45]

    Zahn J.P., 1977, A&A, 57, 383

    work page 1977