The Intrinsic Temperature and Radiative-Convective Boundary Depth in the Atmospheres of Hot Jupiters
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The Intrinsic Temperature and Radiative-Convective Boundary Depth in the Atmospheres of Hot Jupiters
Daniel Thorngren,
1, 2
Peter Gao, and Jonathan J. Fortney Department of Physics, University of California, Santa Cruz Institut de Recherche sur les Exoplan`etes, Universit´e de Montr´eal, Canada
51 Pegasi b Fellow, Department of Astronomy, University of California, Berkeley Department of Astronomy and Astrophysics, University of California, Santa Cruz
ABSTRACTIn giant planet atmosphere modelling, the intrinsic temperature T int and radiative-convective bound-ary (RCB) are important lower boundary conditions. Often in one-dimensional radiative-convectivemodels and in three-dimensional general circulation models it is assumed that T int is similar to thatof Jupiter itself, around 100 K, which yields a RCB around 1 kbar for hot Jupiters. In this work,we show that the inflated radii, and hence high specific entropy interiors (8-11 k b / baryon), of hotJupiters suggest much higher T int . Assuming the effect is primarily due to current heating (ratherthan delayed cooling), we derive an equilibrium relation between T eq and T int , showing that the lattercan take values as high as 700 K. In response, the RCB moves upward in the atmosphere. Using one-dimensional radiative-convective atmosphere models, we find RCBs of only a few bars, rather than thekilobar typically supposed. This much shallower RCB has important implications for the atmosphericstructure, vertical and horizontal circulation, interpretation of atmospheric spectra, and the effect ofdeep cold traps on cloud formation. Keywords: planets and satellites: atmospheres – planets and satellites: gaseous planets – planets andsatellites: interiors – planets and satellites: physical evolution INTRODUCTIONSoon after the discovery of strongly irradiated giantplanets (Mayor & Queloz 1995), it was realized thatthey would have strikingly different atmospheres fromthe giant planets in our own solar system (Guillot et al.1996; Seager & Sasselov 1998; Marley et al. 1999). ForJupiter at optical wavelengths, one can see down to theammonia cloud tops ( ∼ T eq > K , seeMiller & Fortney 2011), it was appreciated that theiratmospheres could remain radiative to a considerablygreater depth due to incident fluxes that are often thou-sands of times that of Earth’s insolation, which force theupper atmosphere to a much higher temperature (typ-ically 1000-2500 K) than for an isolated object. (Guil-lot & Showman 2002; Showman & Guillot 2002; Su-darsky et al. 2003). This leads to a significant depar-ture of the atmospheric temperature structure from anadiabat, and has major consequences for atmosphericcirculation (Showman & Guillot 2002; Showman et al.2008; Rauscher & Menou 2013; Heng & Showman 2015).As such, there has long been significant interest in un- derstanding what controls the pressure level of the hotJupiter radiative-convective boundary (RCB).Radiative-convective atmosphere models for hotJupiters found that, for Jupiter-like intrinsic fluxes (pa-rameterized by T int , of 100 K) but incident stellar fluxes10 times larger, one typically found RCB pressuresnear 1 kbar (e.g., Guillot & Showman 2002; Sudarskyet al. 2003; Fortney et al. 2005). While it was under-stood early on that the RCB depth strongly depends onthe value of T int (e.g. Sudarsky et al. 2003, their Figure16), cooling models suggested that T int values would fallwith time to Jupiter-like values (Guillot & Showman2002; Burrows et al. 2004; Fortney et al. 2007), and the ∼ a r X i v : . [ a s t r o - ph . E P ] J a n Flux (erg s − cm − ) T i n t ( K ) Gaussian ProcessParametric1000 1500 2000 2500 T eq (K) Figure 1.
The intrinsic temperatures of hot Jupiters in equi-librium as a function of incident flux (bottom) or equilibriumtemperature (top). These were derived from the two favoredheating models (Gaussian process and Gaussian parametric)of Thorngren & Fortney (2019), using Eq. 2, with corre-sponding uncertainties. The two models yield nearly identi-cal results. Importantly, the intrinsic temperatures must bequite high – up to 700K – to match the hot interiors requiredto explain the radii of hot Jupiters. T eq to better inform the thermalstructure of 1D and 3D atmosphere models. MODELLINGWe will parameterize the rate at which heat escapesfrom a planet’s deep interior using the intrinsic temper-ature T int . Its value is primarily driven by the entropyof the underlying adiabat. Thus, high T int is typical ofyoung exoplanets and inflated hot Jupiters. If the mech-anism inflating hot Jupiters involves the deposition ofheat into the interior, then they will eventually reach athermal equilibrium where E in = E out . The hotter theplanet, the faster this equilibrium will be reached, in aslittle as tens of megayears (Thorngren & Fortney 2018).Once equilibrium is reached, the intrinsic temperature Mass ( M J ) E n tr o p y ( k b / b a r y o n ) Figure 2.
The equilibrium entropy of hot Jupiters as a func-tion of their mass for various equilibrium temperatures. Eachline has models of the same intrinsic and equilibrium tem-perature, but variations in the resulting surface gravity leadto different internal entropies. In particular, heat escapesmore efficiently through the compact atmospheres of mas-sive objects for a given T int . The composition was assumedto be typical (from Thorngren et al. 2016), using the SCvH(Saumon et al. 1995) and ANEOS 50-50 rock-ice (Thompson1990) equations of state; different compositions will shift theentropy somewhat. is a function of the equilibrium temperature:4 πR σT = πR F (cid:15) ( F ) (1) T int = (cid:18) F (cid:15) ( F )4 σ (cid:19) = (cid:15) ( F ) T eq (2) ≈ . T eq exp (cid:18) − (log( F ) − . . (cid:19) (3)Here, F is the incident flux on the planet (so F =4 σT ), σ is the Stefan-Boltzmann constant, and (cid:15) is thefraction of the flux which heats the interior. This varieswith flux, and was inferred by matching model planetswith the observed hot Jupiter population in Thorngren& Fortney (2018). The resulting intrinsic temperaturesare shown in Figure 1, and the associated entropy (whichdepend on mass and composition), are shown in Figure2. The high intrinsic temperatures this relation producesare important due to the effect they have on the atmo-sphere. In particular, the radiative-convective bound-ary moves to lower pressures for higher T int . In con-trast, larger T eq tends to push the RCB to higherpressures. As these temperatures are related, it isnot immediately obvious where the RCB ends up forplanets at high equilibrium temperatures. To evaluatethe RCB depth, we generate model atmospheres us-ing a well-established thermal structure model for ex-oplanets and brown dwarfs (e.g. McKay et al. 1989;Marley et al. 1996, 1999; Fortney et al. 2005, 2008;Saumon & Marley 2008; Morley et al. 2012). The modelcomputes temperature–pressure (TP) and compositionprofiles assuming radiative–convective–thermochemicalequilibrium, taking into account depletion of molecularspecies due to condensation.Model atmospheres are generated for a grid of cloud-free giant exoplanets with 1 bar gravities of 4, 10, 25,and 75 m s − and a range of T eq from ∼
700 to ∼ T eq of nearly all observed hot Jupiters. Values of T eq were computed assuming full heat redistribution, mean-ing that incoming stellar radiation is reradiated from theentire planetary surface. Functionally, we positioned themodel planets at various semi-major axes around a sun-like star. Two grids were computed, one assuming solaratmospheric metallicity and one assuming 10 × solaratmospheric metallicity (similar to Saturn), with anyadditional heavy elements sequestered in a core, suchthat the bulk metallicities matched the median of theobserved mass–metallicity relationship given by Thorn-gren et al. (2016). The RCB depth for each model planetis then defined, when traveling from the deep interiorinto the atmosphere, as the first pressure level wherethe local lapse rate transitions from adiabatic to suba-diabatic.We calculate the masses, radii, and adiabat entropies(Figure 2) of our model planets (from the gravity and T eq ) using the planetary interior model of Thorngren& Fortney (2018), which solves the equations of hydro-static equilibrium, mass and energy conservation, andan appropriately chosen equation of state (EOS). Weuse the SCvH (Saumon et al. 1995) EOS for a solar ra-tio mixture of hydrogen and helium, and the ANEOS50-50 rock-ice EOS (Thompson 1990) for the metals. RESULTS AND DISCUSSIONOur results for the location of the RCB are shown inFigure 4. The RCB moves to lower pressures at higherequilibrium temperatures , roughly in line with how T int varies with T eq . At the inflation cutoff of around 1000K (Miller & Fortney 2011), the RCB is found at around100 bars. At the extremum around T eq = 1800 K, it isfound at roughly 1 bar. Higher gravity moves the RCBto higher pressures, and higher metallicity moves it tolower pressures. At equilibrium, no hot Jupiter withgravity <
25 m s − and solar or supersolar atmosphericmetallicity should have an RCB as deep as 1 kbar.3.1. Relation to the Heating Mechanism
These results have important implications for under-standing the anomalous heating of hot Jupiters. Heating
Temperature (K) P r e ss u r e ( b a r ) int = 100 K T int = 100 KMg SiO FeCaTiO CO-CH Figure 3.
Selected pressure-temperature profiles of our 10 × solar atmospheric metallicity models for various semi-majoraxes around a sun-like star and the resulting T eq (in brack-ets), from which we derive the intrinsic temperature. The1-bar gravity is set to 10 m s − . The thick lines indicateconvective regions, whereas thin lines correspond to radiativeregions. Alternative pressure-temperature profiles for a T int = 100 K model for the 0.05 AU and 0.01 AU cases are plot-ted as dotted curves. The condensation curves for Mg SiO ,CaTiO , and iron are shown as dashed curves; T int can beseen to strongly affect their condensation pressures. TheCO-CH coexistence curve (Visscher 2012) is also shown; ingeneral, hot Jupiters should be well on the CO side. In thehottest cases, a second convective region forms; however, wewill use the term RCB to refer exclusively to the outer edgeof the interior adiabatic envelope. This boundary is visible inthe plot for the profiles given, and moves to lower pressuresat higher equilibrium temperatures. Flux (erg s − cm − ) R C B P r e ss u r e ( b a r s )
10 m/s
25 m/s
75 m/s T eq (K) Figure 4.
The RCB pressure as a function of incident flux(or T eq ), shown for different surface gravities (colors, see leg-end) and 1 × (pale dotted) and 10 × (dashed) solar metallicityatmospheres. Due to binning effects, there are small uncer-tainties around the modeled points, so we drew smooth linesthrough the data using Gaussian process interpolation with asquared exponential kernel whose parameters were optimizedvia the maximum likelihood. deposited below the RCB is much more effective for in-flating planets than heat deposited above (Komacek &Youdin 2017; Batygin & Stevenson 2010). This is par-ticularly important for Ohmic dissipation. For the lowerRCB pressures that we predict, models like Ohmic dis-sipation will be more efficient than previously consid-ered. Since giant planets are born quite hot, they willhave RCBs at low pressures at young ages that could bemaintained there by this heating. However, this wouldnot necessarily allow for reinflation of a planet whoseinterior has already cooled (see Lopez & Fortney 2016),as the RCB might already be at tens of bars or deeperwhen the heating started. We refer the reader to Ko-macek & Youdin (2017) for a more detailed and broaderdiscussion of these heating deposition depth effects.It is also worthwhile to consider the effect that ourassumptions about the hot Jupiter heating have on themodel. In Thorngren & Fortney (2018), it was assumedthat the heating was proportional to and a function ofthe incident flux, based on the results of Weiss et al.(2013). There may be additional factors that affect theheating, such as planet and stellar mass, but since (cid:15) ( F )seems to predict planetary radii quite well, these likelyadd at most modest uncertainty to our estimates of T int .An additional consideration is whether the anomalousradii are caused entirely by heating, or whether thereis a delayed cooling effect as well; for example, Ohmicdissipation (Batygin et al. 2011) may be a combinationof these (Wu & Lithwick 2013; Ginzburg & Sari 2016).Delayed cooling effects would alter the apparent T int fora given internal adiabat entropy, and delay arrival atthermal equilibrium. However, many anomalous heat-ing models do not rely on delayed cooling (e.g. Arras &Socrates 2009; Youdin & Mitchell 2010; Tremblin et al.2017), and signs of possible reinflation (Hartman et al.2016; Grunblatt et al. 2016, 2017) seem to favor these. Ifreinflation is conclusively shown to occur, then anoma-lous heating must be the dominant cause of radius in-flation (Lopez & Fortney 2016), and our T int estimateswill be particularly good. Finally, the usual uncertain-ties in the equation of state (see e.g. Militzer & Hubbard2013; Chabrier et al. 2019) and planet interior structure(Baraffe et al. 2008; Leconte & Chabrier 2012) discussedin Thorngren et al. (2016) also apply to this work.3.2. Effect on Atmospheric Models
These results have important implications for globalcirculation models (GCMs) of hot Jupiters. It has longbeen a convention in this field to use intrinsic temper-atures similar to Jupiter’s (e.g. Showman et al. 2015;Amundsen et al. 2016; Komacek et al. 2017; Lothringeret al. 2018; Flowers et al. 2019, and many others),around 100 K (Li et al. 2012). Our work shows thatmore realistic values for T int should depend strongly on the incident flux and will typically be several hundredsof Kelvin, as shown in Figure 1. This difference is impor-tant for vertical mixing and circumplanetary circulation,as it shifts the RCB to considerably lower pressures. Itwas recently demonstrated in the hot Jupiter contextthat changing the lower boundary conditions can yieldsignificantly different atmospheric flows in these simula-tions (see Carone et al. 2019).The higher implied intrinsic fluxes could also impactinterpretations of the observed flux from hot Jupiters.For phase curves, night-side fluxes will be a mix of intrin-sic flux, which in many cases can no longer be thought ofas negligible, in addition to energy transported from theday side. Even on the day side, in near-infrared opacitywindows that probe deeply, one might be able to ob-serve this intrinsic flux as a small perturbation on theday-side emission spectrum (e.g., Fortney et al. 2017)The value of T int is also important for the location andabundance of condensates in hot Jupiter atmospheres.Figure 3 compares the condensation curves of severalspecies, including forsterite, iron, and perovskite, to ourmodel TP profiles; the intersection between the conden-sation curve and the TP profile delineates the cloudbases. Previous works that considered low T int atmo-spheres have hypothesized the existence of deep “coldtraps” for hot Jupiter clouds, where a cloud base athigh pressures ( >
100 bars) removes condensates andcondensate vapor from the visible layers of the atmo-sphere (e.g. Spiegel et al. 2009; Parmentier et al. 2016).However, higher T int values increase deep atmospherictemperatures, such that the cloud base is much shal-lower in the atmosphere. For example, at T eq = 1244 K,whether T int = 100 K or the nominal value computedin this work can results in differences in the forsteritecloud base pressure of ∼ abundances aretypically out of equilibrium in cool gas giants and browndwarfs due to the mixing times being faster than thetimescale for CO to convert to CH (Cooper & Showman2006; Moses et al. 2011; Zahnle & Marley 2014; Drum-mond et al. 2018). For the cooler planets modeled inFigure 3, if the quench pressure is ∼
10 to 1000 bars (forinstance), then the disequilibrium chemical abundanceswill differ when the RCB is moved, (see also Drummondet al. 2018), as the local TP profile in the deep atmo-sphere will move in reference to the local CO-CH equalabundance curve. This effect could be seen in the poten-tial detectability of CH in only the very coldest planetsmodeled here, where the upper atmosphere and deepatmospheres are both relatively cool.3.3. Observational Tests and Future Work
We suggest several approaches to further verify ourfindings observationally. The previously supposed T int of 100K could be ruled out if clouds are detected whenthey would otherwise be cold trapped, particularly forplanets with T eq ∼ dominated) atmospheres across a wide T eq range(at least for solar-like C/O ratios), including nearly allmodels shown in Figure 3, would suggest lower pres-sures RCBs. Transmission and emission spectroscopyare well-suited to these characterization tasks and thehigher precision that will be attained with JWST willbe important in this area. More directly, as suggestedby Fortney et al. (2017), high intrinsic fluxes may bemeasured directly by higher fluxes in the near-IR, par-ticularly in windows in water opacity. Finally, recentdetections of strong magnetic fields suggest that highintrinsic temperatures are the reality (Yadav & Thorn-gren 2017; Cauley et al. 2019), since intrinsic tempera-ture is tied to magnetic field strength (Christensen et al.2009). This work now needs to be tied back into revisedestimates for the Ohmic dissipation that occurs in theseatmospheres.Future work should focus on the effects that these al-tered boundary conditions have on the cloud structure,chemical abundances, spectra, and day-night contrastsof hot Jupiters. As we learn more about hot Jupiterinteriors through theoretical developments, populationstudies (especially from new TESS discoveries) and po-tentially reinflated giants (Grunblatt et al. 2017), we canbetter characterize these important atmospheric bound-ary conditions.REFERENCES