Constraining the entropy of formation from young transiting planets
MMNRAS , 1–13 (2015) Preprint 10 September 2020 Compiled using MNRAS L A TEX style file v3.0
Constraining the entropy of formation from youngtransiting planets
James E. Owen (cid:63)
Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Recently K2 and TESS have discovered transiting planets with radii between ∼ ⊕ around stars with ages < Myr. These young planets are likely to be the progenitorsof the ubiquitous super-earths/sub-neptunes, that are well studied around stars withages (cid:38) Gyr. The formation and early evolution of super-earths/sub-neptunes arepoorly understood. Various planetary origin scenarios predict a wide range of possibleformation entropies. We show how the formation entropies of young ( ∼ (cid:46) %can be achieved for the discovered young planets around DS Tuc A and V1298 Tau,then insights into their origins can be obtained. For these planets, higher measuredplanet masses would be consistent with standard core-accretion theory. In contrast,lower planet masses ( (cid:46) − M ⊕ ) would require a “boil-off” phase during protoplanetarydisc dispersal to explain. Key words: planets and satellites: interiors — planets and satellites: formation —planets and satellites: individual: DS Tuc Ab — stars: individual: V1298 Tau
Exoplanet surveys and demographic studies have revealed apreviously unknown class of planets. These close-in, super-earth/sub-neptunes have radii in the range 1-4 R ⊕ (e.g.Borucki et al. 2011; Thompson et al. 2018) and masses (cid:46) M ⊕ (e.g. Mayor et al. 2011; Wu & Lithwick 2013a;Marcy et al. 2014; Weiss & Marcy 2014; Hadden & Lithwick2014; Jontof-Hutter et al. 2016). With orbital periods of lessthan a few hundred days, these planets have been shownto be incredibly common, with most sun-like and later-typestars hosting at least one, if not many (e.g. Fressin et al.2013; Silburt et al. 2015; Mulders et al. 2018; Zink et al.2019; Hsu et al. 2019).Combined mass and radius measurements for individ-ual planets revealed a vast spread in densities; from as highas ∼ g cm − to as low as ∼ . g cm − . The former isconsistent with rocky bodies with Earth-like (iron/silicate)compositions (e.g. Dressing et al. 2015; Dorn et al. 2019) andthe latter with solid cores surrounded by larger H/He at-mospheres (e.g. Jontof-Hutter et al. 2016). At intermediatedensities a plethora of compositions are possible, including (cid:63) E-mail: [email protected] some combination of iron, silicates, water and H/He (e.g.Rogers & Seager 2010; Zeng et al. 2019).H/He envelopes on low-mass planets close to their hoststars are vulnerable to mass-loss (e.g. Lammer et al. 2003;Baraffe et al. 2005; Murray-Clay et al. 2009; Owen et al.2012). Evolutionary models of solid-cores surrounded byH/He envelopes suggested that mass-loss would sculpt theplanet population into two classes: those which completelylose their H/He envelope leaving behind a “stripped core”,and those planets that retain about 1 % by mass in theirH/He envelope (e.g. Owen & Wu 2013; Lopez & Fortney2013; Jin et al. 2014). Early indications of such a dichotomywere found in planetary density measurements (e.g. Weiss &Marcy 2014; Rogers 2015). However, it was not until accu-rately measured stellar properties allowed precise planetaryradii to be determined that a bi-modal radius distributionwas revealed (e.g. Fulton et al. 2017; Fulton & Petigura 2018;Van Eylen et al. 2018).Incorporating these evolutionary clues into the com-positional determination suggests most planets larger thanabout 1.8 R ⊕ consist of an Earth-like core surrounded by aH/He envelope, where this envelope contains a few percent ofthe planet’s mass (Wolfgang & Lopez 2015). Further, mass-loss models suggest that the vast majority of those plan- © a r X i v : . [ a s t r o - ph . E P ] S e p Owen, J. E ets that do not possess H/He envelopes today, formed withone which they then lost (e.g. Owen & Wu 2017; Wu 2019;Rogers & Owen 2020).The majority of exoplanet host stars are billions of yearsold. Recent work by Berger et al. (2020) has shown thatthe median age of the
Kepler planet host stars is ∼ Gyrand only ∼ % of the Kepler host stars are < Gyr old.Therefore, most of our knowledge about the demographics ofclose-in exoplanets is restricted to old stars. The fact manyof these planets possess H/He envelopes necessitates theyformed before gas-disc dispersal. These gas-discs have life-times of − Myr (e.g. Haisch et al. 2001; Mamajek 2009).Thus, most exoplanet host-stars are significantly older thanthe planetary formation timescale ( (cid:46) Myr).Compositional uncertainty and lack of knowledge of theexoplanet demographics has led to considerable discussionand debate about how these planets form. In many casesthe size of the H/He envelopes implies they must have ac-creted from a protoplanetary disc. The idea of accretion of aH/He envelope by a solid core fits within the general frame-work of core-accretion (e.g. Rafikov 2006; Ikoma & Genda2006). In the core-accretion model, once the planet is mas-sive enough that its Bondi-radius resides outside its physicalradius (crudely at a few tenths of a Lunar mass, e.g. Mas-sol et al. 2016) it can gravitationally bind nebula gas. Asthe planet’s solid mass continues to grow it can bind everlarger quantities of gas; eventually at around core masses of ∼ ⊕ the H/He envelope’s cooling time becomes compa-rable to the disc’s lifetime (e.g. Lee & Chiang 2015). Thus,beyond solid core masses of ∼ ⊕ accretion of a H/Heenvelope is dependant on how fast current gas in the planet’senvelope evolves thermally; heat the envelope gas via solidaccretion, and gas flows from the envelope to the disc, letit cool and contract, and high-entropy nebula gas flows intothe envelope.The expected thermal envelope structure typically con-sists of a convective zone surrounding the core, where heatgenerated by solid accretion or gravitational contraction ofthe atmosphere is rapidly convected outwards. Eventually,as the temperature and density drops a radiative zone forms,which connects the envelope to the disc, where the lower en-tropy interior is connected to the higher entropy disc (e.g.Rafikov 2006). Since opacity typically increases with pres-sure and temperature it’s the radiative-convective bound-ary that controls the thermal evolution of the envelope, andhence the rate of accretion. This makes gas accretion fairlyinsensitive to the nebula’s properties (e.g. density and tem-perature).Despite the basic success of the core-accretion modelin explaining that low-mass planets can acquire a H/He en-velope, it has failed to quantitatively explain the proper-ties of observed sub-Neptunes (e.g. Lee et al. 2014; Ogihara& Hori 2020; Alessi et al. 2020). In fact the issue is notthat they have accreted H/He atmospheres, it is that theyhave only accreted a few percent by mass, even after mass-loss processes like photoevaporation (Jankovic et al. 2019;Ogihara & Hori 2020) and/or core-powered mass-loss (e.g.Ginzburg et al. 2018) have been taken into account. Specif- Obviously this precise value is uncertain and depends on detailsof the opacity, e.g. Venturini et al. 2016; Lee & Connors 2020. ically, Rogers & Owen (2020) has recently shown standardcore-accretion models over-predict the H/He envelope massby a factor of ∼ during proto-planetary disc dispersal, decreasing the entropy of the inte-rior (e.g. Owen & Wu 2016; Ginzburg et al. 2016; Fossatiet al. 2017; Kubyshkina et al. 2018) (see the discussion for adetailed description of these models). More fundamentally,these proposed solutions all modify the thermodynamic evo-lution of the H/He envelope. Without any intervention onewould expect the planet to have an initial cooling timescale(or Kelvin-Helmholtz contraction timescale) comparable tothe time over which it has been forming, i.e. a few millionyears. Thus, in the first solution where accretion is slowedby increasing the entropy of the interior, the envelope’s cool-ing time becomes shorter (probably less than a few millionyears). In the second solution, where rapid mass-loss dur-ing disc dispersal decreases the planet’s radius and entropy,planets end up with much longer cooling times, closer to ∼ Myr (Owen & Wu 2016). This mechanism is knownas “boil-off”.As planets age they cool; by the time they reach a Gyrold they gone through many initial cooling times and havecompletely forgotten their initial thermodynamic properties.Therefore, our population of old planets is generally unableto tell us about the thermodynamic properties of forming orrecently formed planets.With the advent of wider searches for transiting ex-oplanets (e.g. K2 Howell et al. 2014, TESS Ricker et al.2014 & NGTS Wheatley et al. 2018) discovering young( (cid:46)
Myr) close-in planets has become possible. Notablerecent examples are the four planets in the V1298 Tau sys-tem at an age of ∼
24 Myr (David et al. 2019), a ∼ Myr oldplanet around DS Tuc A (Newton et al. 2019), a ∼ Myrold planet around AU Mic (Plavchan et al. 2020) as well asother recent results from the thyme project (e.g. Rizzutoet al. 2020).In this work, we show how the combination of a massand radius measurement for young planets can be used toplace a constraint on their initial entropy. In Section 2 wedemonstrate how young planets with measured mass andradii can be used to constrain their initial entropies, usingsimple models. In Section 3 we use numerical planetary evo-lution models to explore this further, investigating real sys-tems in Section 4. In Section 5 we discuss the implicationsof our work and summarise in Section 6.
Present day sub-neptunes with voluminous H/He atmo-spheres cool over time, contracting under the release of heatleft over from formation (both in their atmospheres and solidcores). Evolutionary tracks (e.g. Baraffe et al. 2005; Lopez &Fortney 2014) predict these planets were substantially largerwhen younger, even when mass-loss is not factored in. InFigure 1 we show tracks from planetary evolution modelsfor planets that evolve into typical sub-neptunes after bil-
MNRAS , 1–13 (2015) oung planets P l a n e t R a d i u s [ R ⊕ ] KH timescale ∼
500 Myr, M env /M c ∼ ∼ M env /M c ∼ ∼
500 Myr, M env /M c ∼ ∼ M env /M c ∼ Time since disc dispersal [yr]5 . . . . . P l a n e t M a ss [ M ⊕ ] Figure 1.
The planet’s radius (top) and total planet mass (bot-tom) evolution of low-mass, close-in exoplanets with H/He atmo-spheres. The solid and dashed lines show planets whose envelopeshave initial Kelvin-Helmholtz contraction timescales of 500 and 5Myr respectively. The thick and thin lines show planets that haveinitial envelope fractions ( M env / M c ) of 0.1 and 0.04 respectively.The evolution calculations are computed using mesa as describedin Section 3 and include thermal evolution and photoevaporation.The planet contains a core with a mass of ∼ M ⊕ and is locatedat 0.1 AU around a sun-like star. lions of years of evolution. These planets have radii in therange 4-15 R ⊕ at ages (cid:46) Myr.In fact the upper envelope of the radius evolution forpresent-day sub-neptunes is expected to become (cid:38) R Jup atyoung ages. This is demonstrated in Figure 1 indicating thatat the youngest ages these planets could be conflated witha giant planet, as they have radii similar to present day hotjupiters. In fact, young hot jupiters are expected to haveradii in excess of ∼
13 R ⊕ (Fortney & Nettelmann 2010).This indicates that even those young planets that have mea-sured radii ∼ R ⊕ (e.g. V1298 Tau b, David et al. 2019 andHIP 67522b, Rizzuto et al. 2020) are likely to be “proto-sub-neputunes” rather than young, jupiter mass planets.Unlike stars, planet’s have no method of preventinggravitational collapse (other than Coulomb or degeneracypressure), therefore, the observed size of a planet dependson the thermodynamic state of it’s interior, or how muchentropy it currently possesses. As the planet’s envelope con-tracts the interior entropy falls. Eventually, the interior willeither be supported by degeneracy pressure at high massesor, at low-masses by Coulomb pressure (Seager et al. 2007). It is well known that with measurements of the massand radius of a single composition self-gravitating sphere(e.g. a pure H/He mixture), then the internal thermody-namic state and interior can be determined. However, bothobservational (Wu & Lithwick 2013b; Jontof-Hutter et al.2016; Fulton et al. 2017; Van Eylen et al. 2018; Bennekeet al. 2019b,a), and theoretical (Owen & Wu 2017; Wu2019; Rogers & Owen 2020) evidence suggests present-daysub-neptunes are not of a single composition, but mostlylikely composed of a solid core surrounded by a H/He enve-lope . As we show below, this makes the envelope mass andits internal thermodynamic state degenerate. Specifically, aplanet with a known mass and radius can have more massin its core and less in its envelope if the envelope is hot-ter (and therefore has a larger scale height) or vice-versa, asdemonstrated by works where extra heating mechanisms areincluded (e.g. Pu & Valencia 2017; Millholland et al. 2020).This is even before the degeneracy between the compositionof the core and envelope mass is taken into account. The degeneracy between entropy and envelope mass fractionis trivial to identify. Given a planet composed of a core ofknown density surrounded by a H/He envelope, there arethree parameters which specify its structure: core mass, en-velope mass and internal entropy. Thus, with only measure-ments of a planet’s mass and radius it is not possible to con-strain these three structure parameters, and thus we cannotdetermine its internal entropy.This can be seen explicilty if we build a simple model forthe planet’s internal structure. This simple model is basedon the assumption that self-gravity can be neglected in theenvelope (e.g. Rafikov 2006; Piso & Youdin 2014; Lee & Chi-ang 2015; Ginzburg et al. 2016). As derived by Owen &Wu (2017) & Owen & Campos Estrada (2020) the envelopemass, M env , surrounding a core of mass M c and radius R c with equation-of-state relating pressure, P to density, ρ via P ∝ ρ γ is given, approximately by: M env ≈ π R ρ rcb (cid:18) ∇ ab GM c c s R rcb (cid:19) /( γ − ) I ( R c / R rcb , γ ) (1)where R rcb is the radius of the radiative-convective boundary, ∇ ab is the adiabatic gradient, c s the isothermal sound-speedat the radiative-convective boundary and I n is a dimension-less integral of the form: I n ( R c / R rcb , γ ) = ∫ R c / R rcb x n (cid:16) x − − (cid:17) /( γ − ) d x (2)Due to the extreme irradiation level most close-in planetsreceive, the radiative-convective boundary occurs at opticaldepths much higher than the photosphere. The radiative-convective boundary sets the point at which the energy Although alternative ideas do exists, e.g. Zeng et al. (2019). It can also be seen by considering “loaded” polytropes (Hunt-ley & Saslaw 1975) which include atmosphere self-gravity andsmoothly tend to the mono-composition models as the envelopemass exceeds the core mass. However, such analysis is not neces-sary to elucidate the basic point.MNRAS000
13 R ⊕ (Fortney & Nettelmann 2010).This indicates that even those young planets that have mea-sured radii ∼ R ⊕ (e.g. V1298 Tau b, David et al. 2019 andHIP 67522b, Rizzuto et al. 2020) are likely to be “proto-sub-neputunes” rather than young, jupiter mass planets.Unlike stars, planet’s have no method of preventinggravitational collapse (other than Coulomb or degeneracypressure), therefore, the observed size of a planet dependson the thermodynamic state of it’s interior, or how muchentropy it currently possesses. As the planet’s envelope con-tracts the interior entropy falls. Eventually, the interior willeither be supported by degeneracy pressure at high massesor, at low-masses by Coulomb pressure (Seager et al. 2007). It is well known that with measurements of the massand radius of a single composition self-gravitating sphere(e.g. a pure H/He mixture), then the internal thermody-namic state and interior can be determined. However, bothobservational (Wu & Lithwick 2013b; Jontof-Hutter et al.2016; Fulton et al. 2017; Van Eylen et al. 2018; Bennekeet al. 2019b,a), and theoretical (Owen & Wu 2017; Wu2019; Rogers & Owen 2020) evidence suggests present-daysub-neptunes are not of a single composition, but mostlylikely composed of a solid core surrounded by a H/He enve-lope . As we show below, this makes the envelope mass andits internal thermodynamic state degenerate. Specifically, aplanet with a known mass and radius can have more massin its core and less in its envelope if the envelope is hot-ter (and therefore has a larger scale height) or vice-versa, asdemonstrated by works where extra heating mechanisms areincluded (e.g. Pu & Valencia 2017; Millholland et al. 2020).This is even before the degeneracy between the compositionof the core and envelope mass is taken into account. The degeneracy between entropy and envelope mass fractionis trivial to identify. Given a planet composed of a core ofknown density surrounded by a H/He envelope, there arethree parameters which specify its structure: core mass, en-velope mass and internal entropy. Thus, with only measure-ments of a planet’s mass and radius it is not possible to con-strain these three structure parameters, and thus we cannotdetermine its internal entropy.This can be seen explicilty if we build a simple model forthe planet’s internal structure. This simple model is basedon the assumption that self-gravity can be neglected in theenvelope (e.g. Rafikov 2006; Piso & Youdin 2014; Lee & Chi-ang 2015; Ginzburg et al. 2016). As derived by Owen &Wu (2017) & Owen & Campos Estrada (2020) the envelopemass, M env , surrounding a core of mass M c and radius R c with equation-of-state relating pressure, P to density, ρ via P ∝ ρ γ is given, approximately by: M env ≈ π R ρ rcb (cid:18) ∇ ab GM c c s R rcb (cid:19) /( γ − ) I ( R c / R rcb , γ ) (1)where R rcb is the radius of the radiative-convective boundary, ∇ ab is the adiabatic gradient, c s the isothermal sound-speedat the radiative-convective boundary and I n is a dimension-less integral of the form: I n ( R c / R rcb , γ ) = ∫ R c / R rcb x n (cid:16) x − − (cid:17) /( γ − ) d x (2)Due to the extreme irradiation level most close-in planetsreceive, the radiative-convective boundary occurs at opticaldepths much higher than the photosphere. The radiative-convective boundary sets the point at which the energy Although alternative ideas do exists, e.g. Zeng et al. (2019). It can also be seen by considering “loaded” polytropes (Hunt-ley & Saslaw 1975) which include atmosphere self-gravity andsmoothly tend to the mono-composition models as the envelopemass exceeds the core mass. However, such analysis is not neces-sary to elucidate the basic point.MNRAS000 , 1–13 (2015)
Owen, J. E released by gravitational contraction in the interior is no-longer transported by the convection, but rather by ra-diation. Hence the luminosity of the planet (and there-fore internal entropy) is related directly to the density atthe radiative-convective boundary. Since luminosity and en-tropy are rather non-intuitive quantities and do not facil-itate easy comparison across planet-mass, age or forma-tion models we follow Owen & Wu (2017) and choose theatmosphere’s Kelvin-Helmholtz contraction timescale τ KH (or cooling timescale) as our quantity to describe the en-tropy and thermodynamic state of the planetary interior.We choose this parameterisation of entropy as this coolingtimescale can be directly compared to quantities like theprotoplanetary disc lifetime. This allows us to write the lu-minosity as: L ≈ τ KH GM c M env R rcb I I (3)For clarity, we restrict ourselves to a constant opacityenvelope, with opacity κ (while this is clearly incorrect, noth-ing we demonstrate below is invalidated by this). Thus, solv-ing for the density at the radiative-convective boundary (e.g.Owen & Wu 2017) we find: ρ rcb ≈ (cid:18) µ k b (cid:19) (cid:20)(cid:18) I I (cid:19) πσ T R rcb τ KH κ M env (cid:21) (4)with µ the mean molecular weight, k b Boltzmann’s constant, σ Stefan-Boltzmann’s constant and T is the equilibrium tem-perature of the planet. Finally, noting that the radiativelayer is approximately isothermal (and therefore, well de-scribed by an exoponential density profile) and thus thin,we simply take R rcb ≈ R p . Combining Equations 1 and 4 wecan derive a mass-radius relationship for our planets of theform: R ( γ − )/( γ − ) p M /( γ − ) p ∝ M τ − I I (5)where we have dropped unimportant variables. Equation 5clearly demonstrates that even with the measurement ofa planet’s mass and radius, the Kelvin-Helmholtz contrac-tion timescale cannot be determined. Now, in the limit M env → M p this degeneracy disappears . For old planetsthis degeneracy is bypassed by the reasonable assumptionthat the planet has cooled sufficiently that its initial ther-modynamic state has been forgotten and τ KH has simplytended towards the age of the planet ( T age ). However, foryoung planets such statements cannot be made, all we cansafely say is τ KH (cid:38) T age . Thus, at young ages, measurementsof a planet’s mass and radius are not sufficient to determineeither its internal composition (fraction of mass in the atmo-sphere compared to the core) or its internal thermodynamicstate.The dependence in the core composition is encapsu-lated in the dimensional integrals I and I . In the limit R p / R c (cid:29) , relevant for young planets, both integrals tendto a constant for γ > / and I tends to a constant for > / . Inspection of our numerical models (Section 3) indi-cates that planetary interiors span the full range of possible However, one needs to include self-gravity in the analysis. limits, with γ < / close to the planetary cores when the in-teriors are high-entropy and γ > / closer to the planetarysurface for low-entropy interiors. In order to assess whethercore-composition will effect our analysis we calculate howthe ratio I / I varies with core-composition at different val-ues of γ , we do this for a 7 R ⊕ , 5 M ⊕ planet. For an extremevariation in core composition of 1/3 ice, 2/3 rock to 1/3iron, 2/3 rock we find a variation of 4% for γ = / anda factor of two for a of γ = . (the lowest found at anypoint in the numerical models). These variations are muchsmaller than the order of magnitude changes in the Kelvin-Helmholtz timescale we are investigating, especially whenconsidering detailed fits to the exoplanet data suggest thespread in core-composition is narrow (e.g. Dorn et al. 2019;Rogers & Owen 2020). Specifically, for the spread in corecomposition inferred by Rogers & Owen (2020) the ratio I / I varies by a maximum of 15% for γ = . for a 7 R ⊕ ,5 M ⊕ planet. Thus, we consider our results to be robust touncertainties in the core-composition, and certainly smallerthan variations arising from the observational uncertaintieson age, mass and radius . Given we have already adopteda constant opacity, we chose to adopt a constant value of γ = / and ignore the variation of I and I for simplicityin the rest of the section, while noting no choice of a sin-gle value of γ is justifiable. We emphasise that this sectionis purely an illustrative demonstration of the method andthese choices do not affect the general idea. In our numer-ical models in Section 3 the appropriate equation-of-stateand opacities are used. Fortunately, for close-in planets we do have a way ofconstraining the mass in a planet’s envelope. The high-irradiation levels experienced by young planets cause themto lose envelope mass over time (e.g. Baraffe et al. 2005;Lopez & Fortney 2013; Owen & Wu 2013). Specifically, givena planet with a known mass and radius there is a minimumenvelope mass it could have retained given its age. Make theenvelope less massive and it could not have survived mass-loss until its current age. Therefore, the envelope mass-losstimescale t (cid:219) m , must satisfy: t (cid:219) m ≡ M env (cid:219) M (cid:38) T age (6)Since the mass-loss rate (cid:219) M depends only on planet mass andradius (for externally driven loss processes), or planet mass,radius and Kelvin-Helmholtz contraction timescale (for in-ternally driven loss processes such as core-powered mass-lossGinzburg et al. 2018; Gupta & Schlichting 2019), then theinequality in Equation 6 can be used to place a lower boundon the Kelvin-Helmholtz timescale of the planet. Workingwithin the framework where the loss is driven by photoe-vaporation, we show that with a planet radius alone one can In fact retaining the dimensional integrals and an arbitrarychoice of γ , we find the dependence on the constrain of τ KH inEquation 10 scales linearly with I / I , compared to much higherpowers of mass, radius and age, indicating that for typical 10-20% errors, the variation in the dimensional integrals with core-compositions will have a small effect on our constraint on theKelvin-Helmholtz timescale. MNRAS , 1–13 (2015) oung planets place a minimum value on the planet mass to be consistentwith the simplest (“vanilla”) picture of planetary formationvia core accretion. Further, with a measured mass and radiuswe demonstrate how a lower bound on the Kelvin-Helmholtztimescale can be found. The most na¨ıve expectation is that planet formation issmooth and continuous and disc dispersal gently releasesthe planet into the circumstellar environment wherin mass-loss can proceed. In this scenario, with no violent processes,the planet’s Kelvin-Helmholtz contraction timescale shouldroughly track age. Therefore, at young-ages we can followwhat is done at old ages, where we accept that a planet isseveral cooling times old and set τ KH ∼ T age . Combining thisanszat with Equation 5 and inequality 6 we find: M / p (cid:38) A (cid:219) MT / R − / p (7)where the term A incorporates all the terms we have dropped(e.g. temperature, opacity, mean-molecular weight and fun-damental constants). Setting the mass-loss rate to: (cid:219) M ∝ F HE π R p GM p ∝ R p / M p (8)as found in the case of energy-limited photoevaporation(with F HE the high energy flux received by the planet), wefind: M p (cid:38) A (cid:48) R p T / (9)with A (cid:48) ≡ A / . Put simply, for a young planet with a mea-sured radius from a transit survey there is a minimum massfor it to be consistent with the na¨ıvest expectation of planetformation. Put another way, if a planet is consistent withthe criterion in Equation 9, then limited constraints can beplaced on its formation entropy and history. Noting in thescenario above where T ∝ a − / (with a the orbital separa-tion) and (cid:219) M ∝ a − we find A (cid:48) ∝ a − / , validating the ex-pectation that the higher irradiation levels closer to the starlead to higher mass-loss and therefore higher required planetmasses. Finally (in the case of photoevaporation), the rapiddrop in XUV flux when the star spins down (e.g. Tu et al.2015) means T age cannot be set arbitrarily long, but is ratherconstrained to be the saturation time of the XUV output ofthe star. Hence, the typically quoted values of ∼ Myr forsun-like stars (e.g. Jackson et al. 2012).While the above style of calculation is unlikely to pro-vide interesting analysis for real planets, it could be usefulfor selecting which planets to target with radial velocity,transit-timing variation (TTV) or spectroscopic follow-up.
Doing away with the anszat that τ KH ∼ T age , we can nowgeneralise to the possibility that at young ages τ KH (cid:38) T age .Now we can follow a similar argument to that in the preceed-ing section, and show that with measurement of a planet’smass and radius, one can place a lower bound on the planet’sKelvin-Helmholtz contraction timescale. Again combining Equation 5 for the mass-radius rela-tionship with the mass-loss criterion in Equation 6 we find: τ KH (cid:38) B R / p M − / p T (10)where like the A factor above, B encapsulates all the termsand fundamental constants we have dropped from our anal-ysis. The dependence of the inequality in Equation 10 iseasy to understand. Larger and less massive planets whichare older have experienced more mass-loss. The higher totalmass-loss requires a higher atmosphere mass to resist, ne-cessitating a lower entropy interior to give a planet with thesame total mass and radius (Equation 5). Again for the casewhere T ∝ a − / and (cid:219) M ∝ a − we find B ∝ a − / indicatingthat it is those planets that are closest to their host stars(and experience more vigorous mass-loss) that are the mostconstraining. Now clearly, if one finds a constraint on theKelvin-Helmholtz timescale that is shorter than its age, onehas not learnt anything other than it is consistent with the“vanilla” scenario for core-accretion, and satisfies the con-straint in Equation 9.With a sample of young planets with ages less than a few100 Myr with measured masses and radii it is possible to con-strain their current Kelvin-Helmholtz contraction timescalesand hence gain insights into their formation entropies andthe physical processes that lead to their formation and earlyevolution. On the flip-side, if all young planets appear to beconsistent with τ KH ∼ T age at young ages we can also makeinferences about their formation pathways.We caution that in the previous sections we deliberatelychose an incorrect opacity-law (a constant opacity) and asimple mass-loss model, in order that the powers in the pre-vious expressions did not become large integer ratios, and assuch they should not be used for any quantitative analysis.Switching to more realistic opacity and mass-loss laws doesnot change the facts identified in Section 2.3 and 2.4. Before we switch to using full numerical solutions of plane-tary evolution we can get a sense of the range of interestingplanet properties by using the semi-analytic planet structuremodel developed by Owen & Wu (2017), where all choices(opacity-law, mass-loss model etc) follow those in Owen &Campos Estrada (2020). In all cases we assume an Earth-like core composition with a 1/3 iron to 2/3 rock mass-ratio,which is consistent with the current exoplanet demograph-ics (but as mentioned above, such a choice does not stronglyaffect our results).
In Figure 2 we show the minimum mass required for the“vanilla” scenario where τ KH ∼ T age at all ages. The dottedlines on these figures show the radius evolution of planets(not undergoing mass-loss), which begin at 10 R ⊕ at 10 Myr.These evolutionary curves do not cross many minimum masscontours indicating that there is little strong age preferencein the range of 10 to 100 Myr for selecting planets for thiskind of analysis (although we will investigate this more pre-cisely in Section 3). This is fairly easy to understand; as the MNRAS000
In Figure 2 we show the minimum mass required for the“vanilla” scenario where τ KH ∼ T age at all ages. The dottedlines on these figures show the radius evolution of planets(not undergoing mass-loss), which begin at 10 R ⊕ at 10 Myr.These evolutionary curves do not cross many minimum masscontours indicating that there is little strong age preferencein the range of 10 to 100 Myr for selecting planets for thiskind of analysis (although we will investigate this more pre-cisely in Section 3). This is fairly easy to understand; as the MNRAS000 , 1–13 (2015)
Owen, J. E ( e v a p o r a t i o n v a ll e y ) Separation=0.05 AU1 10Planet Radius [R ⊕ ]10100 P l a n e t A g e [ M y r ] N o s o l u t i o n Separation=0.10 AU 2468102040 M i n i m u m P l a n e t M a ss [ M ⊕ ] Figure 2.
The minimum mass required to be consistent with ascenario where τ KH ∼ T age at all ages.This figure uses the more so-phisticated calculation described in Section 2.5, rather than thesimple inequality given in Equation 9. The top panel shows aplanet with a separation from a sun-like star of 0.05 AU and thebottom panel 0.1 AU. The sun-like star is assumed to have a sat-urated XUV flux of − . L (cid:12) at all ages. The white regions in theleft of the plot, labelled as “no solution (evaporation valley)” areregions of parameter space planet where a H/He envelope wouldhave undergone run-away loss and is a manifestation of the evap-oration valley. The dotted lines show planetary radius evolutioncurves (with no mass-loss) that begin at 10 R ⊕ at 10 Myr andtrack τ KH = T age . planet cools and contracts the absorbed XUV falls, reducingthe mass-loss rate. However, the total time to resist mass-loss increases. These two competing effects approximatelybalance, resulting in a minimum mass that does not changestrongly with age. Once the XUV flux is no-longer saturatedand rapidly falls with time, the mass-loss rate drops precip-itously and the minimum mass will also drop rapidly withage. The difference between the two panels in Figure 2 indi-cates, as expected from the previous analysis, that close-inplanets require higher masses. For those young planets dis-covered to date with radii in the range 5-10 R ⊕ , minimummasses in the range of 5-15 M ⊕ are required. While the minimum masses provide a useful guide they donot provide much insight into planetary formation. Here, weelaborate on the much more interesting case of young planetswith well measured masses and radii. ⊕ ]110 P l a n e t M a ss [ M ⊕ ] Consistent with τ KH ∼ T age Requires τ KH > T age Solidcores
Figure 3.
The planet mass and radius plane separated into thoseplanets consistent with τ KH ∼ T age and those which require longerinitial Kelvin-Helmholtz timescales. As in Figure 2 this figure usesthe more sophisticated calculation described in Section 2.5, ratherthan the simple inequality given in Equation 10 . The diagram isshown for a planet located at 0.1 AU around an XUV saturated(10 − . L (cid:12) ), 50 Myr old sun-like star. Planets that sit the white re-gion would require a long Kelvin-Helmholtz contraction timescaleat formation. The point shows a representative young planet witha measured radius of 7 R ⊕ and mass of 5 M ⊕ shown with indica-tive 10% error-bars. The solid-line and dotted line show curveswith a constant τ KH , with values of 382 Myr and 50 Myr respec-tively . Thus, this representative planet would require a current(and hence formation) contraction timescale of (cid:38) T age . In Figure 3 we show how the mass-radius plane is par-titioned into regions of parameter space that are consistentwith τ KH ∼ T age and those requiring τ KH (cid:38) T age . This anal-ysis is performed for a planet located at 0.1 AU around anXUV saturated, 50 Myr old Sun-like star. Placing a repre-sentative young planet with a measured radius of 7 R ⊕ andmass of 5 M ⊕ on this diagram indicates it would require alonger Kelvin-Helmholtz contraction timescale (and hencelower entropy) than predicted by simple formation scenar-ios. Given theoretical ideas, such as “boil-off”, predict ini-tial Kelvin-Helmholtz contraction timescales of ∼
100 Myr(Owen & Wu 2016), this figure indicates they could be de-tected with radii and mass measured to the ∼ % precisionlevel.Its important to emphasise (as demonstrated in Equa-tion 10) that this type of analysis only provides a boundon the Kelvin-Helmholtz contraction timescale, where theequality holds when a planet is on the limit of stability dueto mass-loss. Therefore, finding a planet in the region con-sistent with τ KH ∼ T age does not imply that it doesn’t havea longer contraction timescale, rather it could just be verystable to envelope mass-loss. In the previous section we have used analytic tools to illus-trate the basic physics; however, these models can only bepushed so far. For robust and quantitative results full nu-merical models are a must. This is for several reasons, most
MNRAS , 1–13 (2015) oung planets importantly, many of the transiting planets discovered todate are large and thus may contain quite significant enve-lope mass-fractions ( (cid:38) ). While self-gravity of such anenvelope is small it is not negligible, and not included inthe previous analytic model. Additionally, in the previoussection we assumed an ideal equation-of-state with constantratio of specific heats and power-law opacity model. Whilethese choices are acceptable for understanding demographicproperties, these assumptions induce unnecessary errors inthe analysis of individual systems. Finally, by characterisingthe full evolutionary history (rather than the instantaneousstate) we are able to leverage even more power. This is be-cause not all planetary structures that are consistent witha planet’s current state are consistent with its evolution-ary history once mass-loss is taken into account. The lastpoint is demonstrated by the fact Owen & Morton (2016)were able to provide a (albeit weak) constraint on the en-tropy of formation for the old planet Kepler- τ KH (cid:38) T age . In this section, by including the full evolutionaryhistory, we are able to compare to the planet’s initial KelvinHelmholtz timescale, which we define to be the envelope’sKelvin Helmholtz timescale at the end of disc dispersal. Thiscomparison is more powerful, as it allows to to explore ini-tial Kelvin-Helmholtz timescales which are shorter than theplanet’s current age.Therefore, to overcome the above shortcomings we solvefor the full planetary evolution using mesa (Paxton et al.2011, 2013, 2015). The mesa models are identical to thoseused in Owen & Morton (2016) and Owen & Lai (2018),and include the impact of stellar irradiation (which tracksthe Baraffe et al. 1998 stellar evolution models) and pho-toevaporation using the Owen & Jackson (2012) mass-lossrates.
Here we return to our example planet from Figure 3, a planetlocated at 0.1 AU around a 50 Myr sun-like star. Nominally,we consider this planet to have a measured radius of 7 R ⊕ and measured mass of 5 M ⊕ , but we will investigate howchanges to the mass, as well as measurement precision, willaffect constraints on the planet’s initial Kelvin-Helmholtztimescale. One of the big uncertainties at young ages is how long theplanet has been exposed to XUV irradiation, and hence pho-toevaporating. When embedded in the protoplanetary discit is protected from XUV photons. Thus, the age of the staronly provides an upper bound on the time the planet hasspent photoevaporating. Since disc lifetimes vary between ∼ and ∼ Myr, at young ages this is not a trivial un-certainty. We include this uncertainty in our analysis byderiving the probability distribution for the time a planethas spent photoevaporating after disc dispersal T p . We thenmarginalise over this probability when determining our lowerbound on the planet’s initial Kelvin-Helmholtz timescale.We take the star to have a Gaussian age uncertaintywith mean t ∗ and error σ ∗ . We further assume after a time t d , the disc fraction decays exponentially with the form ∝ exp (cid:18) − T d − t d σ d (cid:19) (11)where T d is the disc’s lifetime and σ d is the decay time for thedisc fraction. Such a phenomenological form describes theevolution of the protoplanetary disc fraction (e.g. Mamajek2009). Now given a star’s actual age ( T ∗ ) is a sum of theunknown disc’s lifetime and the unknown time the planethas been undergoing photoevaporation ( T p ), then we know T ∗ = T p + T d . Therefore the probability distribution for T p can be written as: P ( T p ) = σ d exp (cid:34) σ ∗ + σ d (cid:0) T p + t d − t ∗ (cid:1) σ d (cid:35) × (cid:40) − erf (cid:34) σ ∗ + σ d (cid:0) T p + t d − t ∗ (cid:1) √ σ ∗ σ d (cid:35)(cid:41) (12)In this work we set t d = Myr and σ d = Myr asthis reproduces the fact that all (single) stars host discs atan age of 1 Myr, but by 10 Myr the vast majority of starshave dispersed their discs. Thus, we adopt an initial Kelvin-Helmholtz timescale of 10 Myr as the upper limit that canbe reached in standard core-accretion theory.
In Figure 4 we show joint probability distributions for theinitial Kelvin-Helmholtz timescale, initial envelope massfraction and core mass, as well as the marginalised proba-bility distribution for the initial Kelvin-Helmholtz timescale.This analysis has been performed assuming
Gaussianerrors on stellar age, radius and mass. Similar to our anal-ysis in the earlier section for our 7 R ⊕ and 5 M ⊕
50 Myrold planet we find that it would require an initial Kelvin-Helmholtz timescale significantly longer than would be pre-dicted by standard core-accretion theory. In this example,we would place a 99% lower limit on the initial Kelvin-Helmholtz timescale of ∼ Myr. The joint probabilitydistributions are also correlated as expected with our earlieranalysis. Lower mass planets require longer initial Kelvin-Helmholtz timescales and higher initial envelope mass frac-tions.We explore the role of planet mass in Figure 5 wherewe consider measured planet masses between 4 and 8 Earthmasses (again for our 7 R ⊕ , 50 Myr old planet with 10%measurement uncertainties). We note very few 4 M ⊕ modelsare consistent with the measured radius, as most have initialenvelope mass fractions ∼ , making them extremely vul-nerable to photoevaporation (Owen 2019). As expected, asthe planet mass increases, the bound on the initial Kelvin-Helmholtz timescale decreases (as the higher-mass core isable to hold onto a less massive, and thus higher entropyenvelope).While (cid:46) measurement uncertainties on planet ra-dius and stellar age have been achieved for known youngplanets, stellar activity may mean obtaining radial-velocity Much of the uncertainty in the time a planet has spent photo-evaporating is dominated by the uncertainty in the disc dispersaltimescale at ages (cid:46)
Myr.MNRAS000
Myr.MNRAS000 , 1–13 (2015)
Owen, J. E I n i t i a l K e l v i n - H e l m h o l t z T i m e s c a l e [ y r s ] − Initial Envelope Mass Fraction10 ⊕ ]10 − I n i t i a l E n v e l o p e M a ss F r a c t i o n Initial Kelvin-Helmholtz Timescale [yrs]0 . . . . . . M a r g i n a li s e d R e l a t i v e P r o b a b ili t y Consistentwith vanillascenario Requiresboil-off . . Figure 4.
The top left, top right and bottom left panels show joint probability distributions for the initial Kelvin-Helmholtz timescale,core mass and initial envelope mass fractions for a 50 ± Myr old planet with a radius of ± . R ⊕ and mass of ± . M ⊕ . The bottomright panel shows the marginalised probability distribution for the initial Kelvin-Helzholtz timescale, with the point indicating the 99%lower-limit at a value of 168 Myr. mass measurements at a ∼ % precision is difficult. There-fore, in Figure 6 we show how sensitive our constraints on theinitial Kelvin-Helmholtz timescale are to mass uncertaintiesin the range of 5-25%. As you would naturally expect, in-creasing the uncertainty means higher mass planets becomeconsistent with the measured mass, allowing shorter initialKelvin-Helmholtz timescales. However, even with a tentative ∼ mass detection, for this example we would still be ableto place a useful constraint on the initial Kelvin-Helmholtztimescale.This gives us confidence that measured masses can placeuseful constraints on the entropy of formation of young tran-siting planets, even if those mass measurements are tenta-tive. One question that remains is what is the best system ageto do this experiment for? Obviously young planets allowyou to constrain shorter and shorter initial Kelvin-Helmholtztimescales as they’ve had less chance too cool. Yet, at youngages there are two confounding effects. First photoevapora-tion may not have had enough time to significantly controlthe planet’s evolution. Second, at very young ages, the timethe planet has spent photoevaporating after disc dispersalis not dominated by the uncertainty in the age of the sys-tem, but rather by the unknown disc lifetime. For examplea 10 Myr old planet could have spent anywhere between 0and ∼ Myr photoevaporating. However, wait too long andthe planet will have cooled sufficiently that knowledge of itsinitial thermodynamic state will have been lost, especiallyat ages (cid:38)
Myr when photoevaporation no-longer domi-nates.
MNRAS , 1–13 (2015) oung planets Initial Kelvin-Helmholtz Timescale [yrs]0 . . . . . . M a r g i n a li s e d R e l a t i v e P r o b a b ili t y ⊕ ⊕ ⊕ ⊕ ⊕ Figure 5.
Marginalised probability distributions for the initialKelvin-Helmholtz timescale for a 50 ± Myr old planet with aradius of ± . R ⊕ . Different lines show different planet masses(with 10% uncertainty). For this particular planet a measuredmass of (cid:46) M ⊕ would be required to claim evidence of boil-off. Initial Kelvin-Helmholtz Timescale [yrs]0 . . . . . . M a r g i n a li s e d R e l a t i v e P r o b a b ili t y Figure 6.
Marginalised probability distributions for the initialKelvin-Helmholtz timescale for a 50 ± Myr old planet with aradius of ± . R ⊕ and mass of M ⊕ . Different lines show differentuncertainties on the measured planet mass. Thus, we expect there to be an optimum range of stel-lar ages at which this experiment is most stringent. In or-der to assess this we take the evolution of a planet with a4.375 M ⊕ core, with an initial envelope mass fraction andKelvin-Helmholtz timescale of . and Myr respectively.This model roughly corresponds to our 5 M ⊕ , 7 R ⊕ , 50 Myrold planet studied earlier. We then use our method to con-strain its initial Kelvin-Helmholtz timescale as a function ofage assuming 10% errors on planet mass, radius and stellarage. The minimum initial Kelvin-Helmholtz timescale (atthe 99% confidence level) is shown as a function of age forthis exercise in Figure 7. The system age that is most con-straining lies around ∼ Myr, where even with errors anda fairly robust confidence level we recover the actual initialKelvin-Helmholtz timescale within a factor of 3. Ages in therange ∼ − Myr have a constraint that varies by lessthan of the absolute maximum. We can clearly see that
10 100System Age [Myr]10 M i n i m u m I n i t i a l K H T i m e s c a l e [ y r ] Figure 7.
The constraint on the minimum inferred initial Kelvin-Helmholtz timescale as a function of stellar age. after 100 Myr the constraint on the initial Kelvin-Helmholtztimescale becomes uninformative. Therefore, for real sys-tems our method should provide meaningful constraints onthe initial Kelvin-Helmholtz timescale and hence formationentropy for planets around stars with ages in the range 20-60 Myr.
Having shown that by using the photoevaporation model itis possible to constrain a young planet’s entropy of forma-tion we turn our attention to detected young planets, andconsider how their inferred entropy of formation varies as afunction of possible measured mass. We choose to focus hereon DS Tuc Ab and V1298 Tau c, out of the handful of knownyoung planets, as these are the most strongly irradiated, andtherefore most likely to result in strong constraints on theirinitial Kelvin-Helmholtz contraction timescale.
DS Tuc Ab (Benatti et al. 2019; Newton et al. 2019) isa . ± . R ⊕ planet discovered around a ± Myr,1.01 M (cid:12) star, orbiting with a period of 8.1 days. Using ex-actly the same formalism as applied in Section 3 we considerthe constraints on entropy of formation and initial Kelvin-Helmholtz timescale as a function of planet mass. We findthat a measured mass (cid:46) . M ⊕ would be inconsistent withthe current properties of DS Tuc Ab. In Figure 8 we showhow the inferred lower-limit on the initial Kelvin-Helholtztimescale varies with both the measured planet mass andthe measurement uncertainty. Our results indicate that ameasured mass (cid:46) . M ⊕ with a uncertainty of 10% (or (cid:46) ⊕ with a 20% uncertainty) would require a longer thanna¨ıvely expected initial Kelvin-Helmholtz timescale and re-quire with a “boil-off” phase. A mass of 7.5 M ⊕ correspondsto a radial velocity semi-amplitude of ∼ . m s − , eminently We choose to use the stellar and planetary parameters fromNewton et al. (2019).MNRAS000
DS Tuc Ab (Benatti et al. 2019; Newton et al. 2019) isa . ± . R ⊕ planet discovered around a ± Myr,1.01 M (cid:12) star, orbiting with a period of 8.1 days. Using ex-actly the same formalism as applied in Section 3 we considerthe constraints on entropy of formation and initial Kelvin-Helmholtz timescale as a function of planet mass. We findthat a measured mass (cid:46) . M ⊕ would be inconsistent withthe current properties of DS Tuc Ab. In Figure 8 we showhow the inferred lower-limit on the initial Kelvin-Helholtztimescale varies with both the measured planet mass andthe measurement uncertainty. Our results indicate that ameasured mass (cid:46) . M ⊕ with a uncertainty of 10% (or (cid:46) ⊕ with a 20% uncertainty) would require a longer thanna¨ıvely expected initial Kelvin-Helmholtz timescale and re-quire with a “boil-off” phase. A mass of 7.5 M ⊕ correspondsto a radial velocity semi-amplitude of ∼ . m s − , eminently We choose to use the stellar and planetary parameters fromNewton et al. (2019).MNRAS000 , 1–13 (2015) Owen, J. E ⊕ ]10 M i n i m u m I n i t i a l K H T i m e s c a l e [ Y e a r s ] Requires boil-offConsistent withvanilla case 10% error20% error
Figure 8.
The minimum initial Kelvin-Helmholtz timescale (atthe 99% confidence limit) for DS Tuc Ab shown as a function ofmeasured planet mass for a 10% and 20% measured mass uncer-tainty. A measured mass of (cid:46) − M ⊕ would require a “boil-off”phase to explain. detectable with current instrumentation, stellar noise notwithstanding. The 23 ± Myr old, 1.1 M (cid:12) , V1298 Tau system containsfour large transiting young planets (David et al. 2019). Allplanets are between 5-11 R ⊕ in radii and orbit close to thestar with periods (cid:46) days indicating it is likely to bea precursor to the the archetypal Kepler multi-planet sys-tems. Given our analysis in Section 2.4 indicated that plan-ets much closer to their star will provide the most stringentlimits (due to more vigorous photoevaporation) we selectplanet c to investigate here. V1298 Tau c is a . ± . R ⊕ planet with an orbital period of 8.2 days. Since the V1298Tau system is a multi-planet system, dynamical argumentshave already put constraints on the sum of planet c’s andd’s mass to be + − M ⊕ . Like DS Tuc Ab above we calcu-late the minimum initial Kelvin-Helmholtz timescale, takento be the 99% lower limit, as a function of measured planetmass (with both 10% and 20% measurement uncertainties)which is shown in Figure 9We note measured planet masses (cid:46) M ⊕ are inconsis-tent with V1298 Tau c’s current properties. A mass measure-ment of (cid:46) . M ⊕ with a 10% uncertainty (or (cid:46) ⊕ witha 20% uncertainty) would require a boil-off phase to explain.This corresponds to a RV semi-amplitude of ∼ m s − , againwithin the realm of possibility for radial velocity characteri-sation (stellar noise not withstanding). Further, since V1298Tau is a multi-planet system, this permits the possibility ofmass constraints through Transit Timing Variations. Typical sub-neptune and super-earth planets are expected tobe much larger at early ages. In this work we have shown that ⊕ ]10 M i n i m u m I n i t i a l K H T i m e s c a l e [ Y e a r s ] Requires boil-offConsistent withvanilla case 10% error20% error
Figure 9.
The minimum initial Kelvin-Helmholtz timescale (atthe 99% confidence limit) for V1298 Tau c shown as a function ofmeasured planet mass for a 10% and 20% measured mass uncer-tainty. A measured mass of (cid:46) − M ⊕ would require a “boil-off”phase to explain. with a combined mass and radius measurement of a proto-sub-neptune/super-earth ( (cid:46) Myr old), a lower boundcan be placed on its initial Kelvin-Helmholtz timescale. Thislower bound provides valuable insight into the accretion andearly evolution of its H/He envelope. This lower bound is es-sentially found by answering how low-mass a H/He envelopecan exist on the planet given it’s been undergoing photoe-vaporation. For a fixed planet radius, a higher entropy en-velope contains less mass and is therefore more vulnerableto mass-loss. Whereas, lower entropy envelopes need to bemore massive and thus are able to resist mass-loss for longer.One might expect that the younger the planet this ex-periment is done for the tighter a constraint can be ob-tained. While this is generally true, at the youngest ages thefact protoplanetary disc lifetimes vary means one cannot becertain how long a planet has been undergoing mass-loss.Thus, we find that the optimum ages for this experimentare around 20-60 Myr.Further, more accurate measurements obviously resultin tighter constraints. In this work, we showed that mea-surement precision in the range of 10-20% on radius, massand age are required to perform this analysis. Current (andprevious) transit surveys have already reached and exceededthis requirement on known young planets (e.g. the planetsin the V1298 Tau system have radius uncertainties in therange 6-7%). Further, the already published young planetshave age uncertainties at the 10% level.What is difficult to ascertain is whether the mass mea-surements at the (cid:46) precision are achievable. As dis-cussed in Section 4, the problem is not RV precision. Ratherit is intrinsic stellar variability, particularly due to spots (e.g.Huerta et al. 2008), which are more prevalent on youngerstars. Recent work using Gaussian Processes have shownthat it is possible to model intrinsic stellar variability andobtain mass measurements for planets (Haywood et al. 2014;Grunblatt et al. 2015). Using this technique Barrag´an et al.(2019) recently obtained an RV mass measurement for K2-100b, which is a moderately young star showing significant
MNRAS , 1–13 (2015) oung planets intrinsic RV variability. Alternatively, if the planets happento reside in multi-transiting systems, then TTVs could beused. While we acknowledge the difficulty in obtaining themass measurements we require, we advocate that the scien-tific value of constraints on planets’ initial entropy is impor-tant enough to motivate the effort.Since we are using photoevaporation to constrain theentropy of formation, our results are sensitive to the ac-curacy of theoretical photoevaporation calculations. In thiswork we use the mass-loss rates of Owen & Jackson (2012)which are consistent with the location and slope of the“evap-oration valley” (Van Eylen et al. 2018) , and are generallyin good agreement with observed outflows (Owen & Jack-son 2012). Only more theoretical and observational workcalibrating photoevaporation models can assess the impactchanging the mass-loss rates may have on entropy con-straints. Since the discovery of sub-neptunes and super-earths therehas been much work on their origin (e.g. Ikoma & Genda2006; Ikoma & Hori 2012; Lee et al. 2014; Venturini et al.2015, 2016; Ginzburg et al. 2018). It is clear that the onlyway to explain (at least some of) their current densities isto have a have a planetary core (made of some mixture ofrock, iron and ices) surrounded by a H/He envelope whichcontains ∼ − % of the planet’s total mass (e.g. Jontof-Hutter et al. 2016).Such a planetary composition would naturally arisethrough the core-accretion mechanism, whereby the grow-ing solid core accretes a H/He envelope over the disc’s life-time. In this standard picture, the accreting planetary en-velope smoothly connects to the disc, but remains in quasi-hydrostatic and thermal equilibrium. As the envelope cools,it contracts and slowly accretes. This process happens onthe envelope’s Kelvin-Helmholtz timescale, which withoutany strong internal heating sources, quickly equilibrates toroughly the envelope’s age. If disc dispersal allows the enve-lope to remain in quasi-hydrostatic and thermal equilibrium,then a planet’s “initial Kelvin-Helmholtz timescale” (whichwe define as the Kelvin-Helmholtz timescale after disc dis-persal) is essentially the time it has spent forming, which isbounded by the protoplanetary disc lifetime (e.g. (cid:46) Myr).While the basic picture appears to fit, there is growingevidence that the standard core accretion model significantlyover-predicts the amount of H/He a core of a given massshould accrete (Jankovic et al. 2019; Ogihara & Hori 2020;Alessi et al. 2020; Rogers & Owen 2020). In some cases theproblem is so acute that it’s not clear why certain planetsdid not become giant planets (e.g. Lee et al. 2014; Lee 2019).Several solutions have been proposed to solve this problem.Chen et al. (2020) suggested enhanced opacity from dustcould slow the atmosphere’s accretion. Using numerical sim-ulations, Ormel et al. (2015) and Fung et al. (2015) suggestedthat the envelope was not in quasi-hydrostatic equilibriumwith the disc, but rather high-entropy disc material contin-ually flowed into the envelope, preventing it from cooling. As does the core-powered mass-loss model (Gupta & Schlichting2019, 2020).
Lee & Chiang (2016) hypothesised instead these plan-ets do not spend the entire disc lifetime accreting fromthe nebula, but rather formed rapidly (over a timescale of − years) in the final “transition” disc stage of the pro-toplanetary disc. The much lower gas surface densities andthe shorter lifetime of the transition disc phase gave rise tosmaller accreted atmospheres. The above modifications tothe “vanilla” core accretion theory model will typically re-sult in higher entropy envelopes and therefore initial Kelvin-Helmholtz contraction timescales, significantly shorter thanthe standard value of a few Myr.An alternative solution to the over accretion problem is the introduction of additional mass-loss. While it doesnot seem energetically feasible to increase the rates of ei-ther photoevaporation or core-powered mass-loss (as theyare already fairly efficient), the assumption that the enve-lope maintains some sort of dynamical equilibrium as thedisc disperses seems unlikely. Protoplanetary discs are ob-served to live and evolve slowly over their 1-10 Myr lifetimes.However, the dispersal process is rapid with a timescale of ∼ years (e.g. Kenyon & Hartmann 1995; Ercolano et al.2011; Koepferl et al. 2013; Owen 2016).As argued by Owen & Wu (2016) and Ginzburg et al.(2016) this means accreted H/He envelopes cannot main-tain dynamical and thermal balance with the gas in the dis-persing disc. As such the envelopes become over-pressurised,and expand hydrodynamically into the forming cicumstellarvacuum. This “boil-off” process results in mass-loss (in ex-treme cases up to 90% of the initial envelope is lost), butalso importantly cooling of the interior. This is because thebottleneck for cooling (the radiative-convective boundary)is replaced by an advective-convective boundary and ther-mal energy is removed from the interior quickly by advectionand mass-loss. Using simulations, Owen & Wu (2016) foundthat after this boil-off process, the remaining envelopes hadtheir entropies reduced. Their Kelvin-Helmholtz contrac-tion timescales at the end of disc dispersal were around ∼ Myr.Thus, any constraints of the initial Kelvin-Helmholtzcontraction timescale of proto-sub-neptunes/super-earthswill be invaluable for constraining and testing our modelsfor their origins.
The formation of sub-Neptunes and super-Earths is uncer-tain and many formation models have been proposed to ex-plain their origin. These formation models are essentiallyunconstrained by the old, evolved exoplanet population thathas a typical age of 3 Gyr. However, various planet formationmodels predict vastly different entropies at the end of pro-toplanetary disc dispersal. Characterising the entropies atthe end of disc dispersal in terms of initial Kelvin-Helmholtzcontraction timescales, these predictions range from (cid:46) Myrto (cid:38)
Myr.A young proto-sub-neptune/super-earth with a mea-sured mass, radius and age can be used to place a lower Although it does not prohibit the modifications to core-accretion theory described above.MNRAS000
Myr.A young proto-sub-neptune/super-earth with a mea-sured mass, radius and age can be used to place a lower Although it does not prohibit the modifications to core-accretion theory described above.MNRAS000 , 1–13 (2015) Owen, J. E bound on its initial Kelvin-Helmholtz contraction timescale.This requires the planet to be close enough to its host starthat photoevaporation has had an impact on its evolution.This constraint is obtained by answering how low-mass aH/He envelope can exist on the planet given the mass-lossit experienced. For a fixed planet radius, a higher entropyenvelope contains less mass and is therefore more vulnerableto mass-loss. Whereas, lower entropy envelopes need to bemore massive and thus are able to resit mass-loss for longer.We have shown that planets around host stars with ages20-60 Myr are the optimum targets for this kind of analysis.Applying our hypothesised method to detected young plan-ets DS Tuc Ab and V1298 Tau c we show planet mass con-straints (with (cid:46) precision) in the range 7-10 M ⊕ wouldbe consistent with our standard picture of core-accretion.Mass measurements (cid:46) M ⊕ would favour a “boil-off” pro-cess, where a planet loses mass and its interior cools signifi-cantly during dispersal.While precise mass measurements of low-mass planetsorbiting young stars are likely to be challenging, the insightsinto planet formation that could be obtained warrant theeffort. ACKNOWLEDGEMENTS
JEO is supported by a Royal Society University ResearchFellowship and a 2019 ERC starting grant (PEVAP).
DATA AVAILABILITY
The code used to create the planet structure models in Sec-tion 2.5 is freely available at: https://github.com/jo276/EvapMass . The custom mesa code used to calculate theplanet evolution models in Section 3 and 4 is freely avail-able at: https://github.com/jo276/MESAplanet . The re-maining data underlying this article will be shared on rea-sonable request to the corresponding author.
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