Issue 
A&A
Volume 598, February 2017



Article Number  A90  
Number of page(s)  9  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201629716  
Published online  07 February 2017 
Aeronomical constraints to the minimum mass and maximum radius of hot lowmass planets
^{1} Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria
email: luca.fossati@oeaw.ac.at
^{2} Federal Research Center “Krasnoyarsk Science Center” SB RAS, “Institute of Computational Modelling”, Krasnoyarsk 36, Russian Federation
^{3} Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
^{4} Institut für Geophysik, Astrophysik und Meteorologie, KarlFranzensUniversität, Universitätsplatz 5, 8010 Graz, Austria
Received: 14 September 2016
Accepted: 15 December 2016
Stimulated by the discovery of a number of closein lowdensity planets, we generalise the Jeans escape parameter taking hydrodynamic and Roche lobe effects into account. We furthermore define Λ as the value of the Jeans escape parameter calculated at the observed planetary radius and mass for the planet’s equilibrium temperature and considering atomic hydrogen, independently of the atmospheric temperature profile. We consider 5 and 10 M_{⊕} planets with an equilibrium temperature of 500 and 1000 K, orbiting early G, K, and Mtype stars. Assuming a clear atmosphere and by comparing escape rates obtained from the energylimited formula, which only accounts for the heating induced by the absorption of the highenergy stellar radiation, and from a hydrodynamic atmosphere code, which also accounts for the bolometric heating, we find that planets whose Λ is smaller than 15–35 lie in the “boiloff” regime, where the escape is driven by the atmospheric thermal energy and low planetary gravity. We find that the atmosphere of hot (i.e. T_{eq} ⪆ 1000 K) lowmass (M_{pl} ⪅ 5 M_{⊕}) planets with Λ< 15–35 shrinks to smaller radii so that their Λ evolves to values higher than 15–35, hence out of the boiloff regime, in less than ≈500 Myr. Because of their small Roche lobe radius, we find the same result also for hot (i.e. T_{eq}⪆ 1000 K) higher mass (M_{pl} ⪅ 10 M_{⊕}) planets with Λ< 15–35, when they orbit Mdwarfs. For old, hydrogendominated planets in this range of parameters, Λ should therefore be ≥15–35, which provides a strong constraint on the planetary minimum mass and maximum radius and can be used to predict the presence of aerosols and/or constrain planetary masses, for example.
Key words: planets and satellites: atmospheres / planets and satellites: fundamental parameters / planets and satellites: gaseous planets
© ESO, 2017
1. Introduction
Thanks to the large number of extrasolar planets (exoplanets) discovered to date by ground and spacebased facilities, such as SuperWASP (Pollacco et al. 2006), HATNet (Bakos et al. 2004), CoRoT (Auvergne et al. 2009), Kepler (Borucki et al. 2010), and K2 (Howell et al. 2014), we are beginning to classify the large variety of detected exoplanets on the basis of their properties. One of the greatest recent surprises in planetary sciences was the discovery of a large population of planets with mass and radius in between that of terrestrial and giant planets of the solar system (Mullally et al. 2015). These planets, hereafter subNeptunes, typically have masses and radii in the 1.5–17 M_{⊕} and 1.5–5 R_{⊕} range. SubNeptunes fill a gap of physical parameters that are absent from the solar system. Accurately deriving their masses and radii is therefore crucial to our overall understanding of planets.
The high quality of the Kepler light curves allowed us to obtain precise transit radii, even for small planets, but for most of them, the low mass and faint apparent magnitude of their host stars hampers a precise enough determination of the planetary mass through radial velocity. For several multiplanet systems, planetary masses have been inferred from transittiming variations (TTVs), but some of the resulting values are at odds with those derived from radial velocity (e.g., Weiss & Marcy 2014). SubNeptunes for which both mass and radius have been measured present a large spread in bulk density (≈0.03–80 g cm^{3}; low average densities imply the presence of hydrogendominated atmospheres), which finding is currently greatly debated (e.g., Lopez et al. 2012; Howe et al. 2014; Howe & Burrows 2015; Lee & Chiang 2015, 2016; Owen & Morton 2015; Ginzburg et al. 2016).
It is therefore important to find external independent constraints to planetary masses and radii that could be applied to a large number of planets, for example to independently test the masses derived from TTVs, identify the possible presence of highaltitude aerosols, and estimate a realistic range of planetary radii/masses given a certain mass/radius. We show here how basic aeronomical considerations, supported by hydrodynamic modelling and previous results (Owen & Wu 2016), can constrain the mass/radius of old subNeptunes given their radius/mass and equilibrium temperature (T_{eq}).
2. Generalisation of the Jeans escape parameter
The Jeans escape parameter λ is classically defined at the exobase and for a hydrostatic atmosphere. It is the ratio between the escape velocity υ_{∞} and the most probable velocity υ_{0} of a Maxwellian distribution at temperature T, squared (Jeans 1925; Chamberlain 1963; Öpik 1963; Bauer & Lammer 2004). We generalise the Jeans escape parameter at each atmospheric layer r and corresponding temperature T for a hydrodynamic atmosphere composed of atomic and molecular hydrogen as (1)where G is Newton’s gravitational constant, k_{B} is Boltzmann’s constant, M_{pl} is the planetary mass, υ_{th} is the thermal velocity , and υ_{hy} is the bulk velocity of the particles at each atmospheric layer. In Eq. (1), m is the mean molecular weight (2)where n_{X} and m_{X} are the density and mass of each atom/molecule (X) in the atmosphere. In this work, we consider atomic and molecular hydrogen.
The value of υ_{0} in the hydrodynamic case is that of a shifted Maxwellian distribution, where υ_{hy} is the shift. The Maxwellian velocity distribution gives the number of particles between υ and υ + dυ and can be written as (3)where n is the number density and m the particle mass. The most probable velocity υ_{0} is found where Eq. (3) has its maximum and can therefore be derived by setting dF/ dυ = 0. This condition results in a quadratic equation for υ, (4)The solution of this equation is (5) where only this positive solution is physical (the negative solution yields a negative υ_{0}). Note that a direct derivation of υ_{0} by setting υ_{hy} = 0 in Eq. (3) or in Eq. (5) yields υ_{0} = υ_{th}. From Eq. (5) it also follows that if υ_{th} → 0, then υ_{0} → υ_{hy}, as expected.
The formulation of the Jeans escape parameter given in Eq. (1) is reminiscent of the “solar breeze” used before Parker’s solar wind model was accepted (e.g., Chamberlain 1960, 1961). If υ_{hy} is negligible compared to υ_{th} (i.e. hydrostatic atmosphere), the Jeans escape parameter returns to the classical form of (6)We recall that for the classical Jeans escape parameter (hydrostatic atmosphere), a layer is completely bound to a planet for λ≳ 30 and escape is important for λ< 15, while for λ≲ 1.5 the atmosphere is in hydrodynamic “blowoff” (Jeans 1925; Chamberlain 1963; Öpik 1963; Bauer & Lammer 2004). This last condition occurs when the thermal energy of the gas is very close to, or even exceeds, the gravitational energy.
The vast majority of the exoplanets known to date orbits at close distance to their host stars. We therefore consider Rochelobe effects. Following the procedure described in Sect. 2 of Erkaev et al. (2007), in Eq. (1) we substitute the gravitational potential difference between the planetocentric distance r and infinity (GM_{pl}/r) by the gravitational potential difference between r and the Rochelobe radius (Δφ). We therefore obtain (7)where (8)(see Eq. (7) of Erkaev et al. 2007) and (9)In Eq. (9), M_{⋆} is the stellar mass, d is the semimajor axis, and R_{RL} is the Roche lobe radius. Therefore, Eq. (7) gives the generalised form of the Jeans escape parameter.
2.1. Planet atmosphere modelling
Fig. 1
Left: synthetic transmission spectra calculated for a planet with M_{pl} = 5 M_{⊕}, R_{pl} = 4 R_{⊕}, and a 1000 K isothermal atmosphere with 0.01 (blue), 1.0 (orange), and 100 (green) times solar metallicity. The circles of corresponding colour denote the transmission curves integrated over the CoRoT spectral response curve (red dashed curve). Right: contribution functions for the vertical optical depth integrated over the CoRoT spectral response curve. 
To draw profiles of λ^{∗} and we derive the temperature, pressure, velocity, and density structure of planetary atmospheres employing a stellar highenergy (XUV; 1–920 Å) absorption and 1D hydrodynamic upperatmosphere model that solves the system of hydrodynamic equations for mass, momentum, and energy conservation, and also accounts for ionisation, dissociation, recombination, and Lyα cooling. The full description of the hydrodynamic code adopted for the simulations is presented in Erkaev et al. (2016).
Hydrodynamic modelling is valid in presence of enough collisions, which occurs for Knudsen number Kn = l/H< 0.1 (Volkov et al. 2011), where l is the mean free path and H is the local scale height; in the domain of our models, from R_{pl} to R_{RL}, this criterion is always fulfilled. Throughout our calculations, we adopt a net heating efficiency (η) of 15% (Shematovich et al. 2014) and use stellar XUV fluxes (I_{XUV}) estimated from the average solar XUV flux (Ribas et al. 2005), scaled to the appropriate distance and stellar radius. We note that Xray heating is not relevant in our case, because we do not consider active young stars (Owen & Jackson 2012). We also assumed that at R_{pl} hydrogen is completely in molecular form (i.e. H_{2}), which is true for planets with T_{eq}< 2000 K (Koskinen et al. 2010).
For all calculations, and throughout the paper, we consider that R_{pl} lies at a fiducial atmospheric pressure (p_{0}) of 100 mbar. To justify this assumption, we calculated the photospheric deposition level using an updated version of the radiative transfer code described in Cubillos (2016) and Blecic (2016). The model considers opacities from linebyline transitions from HITEMP for H_{2}O, CO, and CO_{2} (Rothman et al. 2010) and HITRAN for CH_{4} (Rothman et al. 2013). In addition, it includes opacities for H_{2}–H_{2} and H_{2}–He collisioninduced absorption from Borysow (2002), Borysow et al. (2001), and Jørgensen et al. (2000), H_{2} Rayleigh scattering from Lecavelier Des Etangs et al. (2008), and sodium and potassium doublets from Burrows et al. (2000).
In Fig. 1 we present transmission spectra for a fiducial subNeptune with M_{pl} = 5 M_{⊕}, R_{pl} = 4 R_{⊕}, and an isothermal atmosphere at 1000 K, in hydrostatic and thermochemical equilibrium. We explored three different cases varying the atmospheric elemental metallicities, considering 0.01, 1.0, and 100 times solar abundances (Figs. 1 and 2). We adjusted the pressureradius reference level such that the resulting transmission radius (integrated over the optical band) matches the fiducial planetary radius, adopting the CoRoT spectral response curve, as an example. We find that the planetary transmission radii correspond to pressure levels of 130, 50, and 10 mbar for the 0.01, 1.0, and 100.0× solarmetallicity models, respectively (Fig. 1, left panel).
After we obtained the pressureradius relationship, we computed the contribution functions (in the optical band) for the vertical optical depth. The barycenter (i.e., average) of the contribution functions indicate where the atmosphere becomes optically thick. This is the position of the planetary photosphere, where the lower boundary for the hydrodynamic calculation would need to be set. We find that for the planet considered here the photospheric deposition level is approximately located at 551, 159, and 33 mbar for the 0.01, 1.0, and 100.0× solarmetallicity models, respectively (Fig. 1, right panel).
We performed the same procedure for all planets analysed in this work and list the pressure corresponding to the barycenter of the contribution function in the fifth column of Table 1. The pressure values range between about 100 and 700 mbar, where the lower pressure values are obtained for the cooler, lower density planets. Figure 1 shows that a higher metallicity, as expected for lowmass planets, would lead to a slight decrease in pressure values, hence justifying our assumption of placing R_{pl} at an average 100 mbar pressure level.
Fig. 2
Molemixing fractions of the atmospheric species (see legend in the bottom panel) in thermochemical equilibrium for isothermal (1000 K) models calculated for 0.01 (top), 1.0 (middle), and 100 (bottom) times solar metallicity. 
Our hydrodynamic model implicitly considers the stellar continuum absorption by setting the temperature at the lower boundary, hence at R_{pl} (i.e. where most of the stellar radiation is absorbed), equal to T_{eq}. We return to the validity of this approximation in Sect. 3. The planets considered here are old, hence heating from the planet interior can be neglected.
Input parameters and results of the simulations performed with η = 15%.
2.2. λ^{∗} and profiles
As an example to show the differences between λ^{∗} and , we modelled a closein lowdensity 5 M_{⊕} and 4 R_{⊕} (average density ρ of 0.4 g cm^{3}) planet with T_{eq} of 1000 K, orbiting an early Ktype star (see Table 1). The parameters adopted for this idealised planet are similar to those of Kepler87c (Ofir et al. 2014). We derived the mean molecular mass at each atmospheric layer from the modelled H and H_{2} mixing ratios. Figure 3 shows the obtained profiles.
In the 1–2 R_{pl} range, λ^{∗} decreases with increasing r because the gravitational potential decreases and the H_{2} molecules dissociate under the action of the stellar XUV flux. All H_{2} molecules are dissociated at ~2 R_{pl}. Then, at larger radii, as the temperature continues to decrease due to adiabatic cooling, λ^{∗} increases and remains above 30 for radii grater than 6.5 R_{pl}. This implies that no particles could escape, regardless of their proximity to R_{RL}, which is nonphysical. Instead, monotonically decreases with increasing r. The bottom panel of Fig. 3 shows that for such a closein planet, despite the hydrodynamic nature of the atmosphere, in most layers υ_{hy} is negligible compared to υ_{th}, therefore λ^{∗}≈λ and ≈.
Fig. 3
Top: temperature (black solid line) and pressure (red dashed line) profiles as a function of radius r in units of R_{pl} for a 5 M_{⊕} and 4 R_{⊕} planet with T_{eq}=1000 K, orbiting an early Ktype star (see Table 1). The right axis indicates the pressure scale. Middle: λ^{∗} (black solid line) and (red dashed line) profiles as a function of radius r in units of R_{pl}. The horizontal lines mark the critical values of the Jeans escape parameter in the hydrostatic case: 1.5, 15, and 30. The blue dotted lines show the λ^{∗} and profiles calculated assuming that the whole atmosphere is made of atomic hydrogen. The filled circle indicates the Λ value (see Sect. 3). Bottom: υ_{th} (black solid line) and υ_{hy} (red dashed line) profiles in km s^{1}. 
Figure 3 shows that the value of approaches unity at atmospheric layers where the pressure, hence density, is high enough to power high escape rates (see Table 1). These upper layers are in a blowoff regime where the escaping gas is continuously replenished by the hydrodynamically expanding atmosphere, with the expansion being driven by the high thermal energy and low planet gravity. This escape regime, here presented from an aeronomical point of view, has been discovered and thoroughly described by Owen & Wu (2016), who called it “boiloff”, in relation to the study of the evolution of young planets that are just released from the protoplanetary nebula (see also Ginzburg et al. 2016).
3. Using escape rates to identify planets in the boiloff regime
We define Λ as the Jeans escape parameter λ (without accounting for Rochelobe effects and hydrodynamic velocities) at R_{pl}, evaluated at the T_{eq} of the planet and for an atomichydrogen gas (see the full dot and the blue dotted lines in Fig. 3) (10)This quantity, which we call the restricted Jeans escape parameter, is useful because it can be derived for any planet for which mass, transit radius, and T_{eq} are measured, and without the need of any atmospheric modelling or calculation of R_{RL}. We aim here at roughly finding the threshold Λ values (Λ_{T}), as a function of M_{pl}, R_{pl}, and T_{eq}, below which the atmosphere transitions towards the boiloff regime. For this we use escape rates, as described below.
In addition to the escape rates derived from the hydrodynamic model (L_{hy}), we consider the maximum possible XUVdriven escape rates, which can be analytically estimated using the energylimited formula (e.g., Watson et al. 1981; Erkaev et al. 2007), (11)where R_{XUVeff} is the effective radius at which the XUV energy is absorbed in the upper atmosphere (see Table 1; Erkaev et al. 2007, 2015) and η is the heating efficiency (see Sect. 2.1). The factor accounts for Rochelobe effects (Erkaev et al. 2007). We note that Rochelobe effects are also considered in the hydrodynamic model.
By construction, XUV heating and the intrinsic thermal energy of the atmosphere are considered in the computation of L_{hy}, while only XUV heating is taken into account when deriving L_{en}. It follows that the boiloff regime, that is, when the intrinsic thermal energy of the atmosphere becomes the efficient main driver of the escape, occurs for L_{hy} greater than L_{en}. For this situation, L_{hy}/L_{en}>1 cannot be achieved purely from XUV heating, implying that the outflow must be driven by the heat present at the lower boundary of the atmosphere. We can therefore use the L_{hy}/L_{en}≈1 as an empirical condition to estimate Λ_{T}.
To identify the Λ_{T} value, which is the Λ value satisfying the L_{hy}/L_{en}≈1 condition, we ran a set of hydrodynamic simulations for two idealised old planets of 5 and 10 M_{⊕} orbiting an early G, K, and Mtype star at distances such that T_{eq} is equal to 500 and 1000 K, assuming a Bond albedo of 0.3. Table 1 lists the complete set of input parameters and results, which are visually displayed in Fig. 4.
Fig. 4
Ratio between the hydrodynamic (L_{hy}) and energylimited (L_{en}) escape rates as a function of Λ for the modelled planets orbiting the G2 (top), K2 (middle), and M2 (bottom) star. Within each panel, the legend indicates the mass (in M_{⊕}) and temperature (in K) of the modelled planets. The dashed line indicates the equality between L_{hy} and L_{en}, while the dotted line indicates where the L_{hy}/L_{en} ratio is equal to 2.0. The value of Λ_{T} lies between 15 and 35. 
Figure 4 shows that the L_{hy}/L_{en}≈1 condition is reached for Λ values between 15 and 35, with a slight dependence on stellar type and T_{eq}. In particular, for the planets orbiting the G and Ktype stars, the Λ_{T} values appear to be lower at higher temperature, hence Λ_{T} decreases with increasing tidal gravity. This does not seem to be the case for the planets orbiting the Mtype star, particularly for the 10 M_{⊕} planet.
We discuss here the uncertainties related to the computation of the L_{hy}/L_{en} ratio. Since we do not consider real planets, there are no observational uncertainties connected to the system parameters. The R_{XUVeff} value present in Eq. (11) is an output of the hydrodynamic code, and it is used to calculate L_{hy} as well. For these reasons, there are no uncertainties on the R_{XUVeff} value. The heating efficiency η is therefore the only input parameter for which its uncertainties may affect the L_{hy}/L_{en} ratio.
Generally, the heating efficiency varies with altitude, and Shematovich et al. (2014) concluded that for hot Jupiters the value of η in the thermosphere varies between ≈10% and 20%. Because our model does not selfconsistently calculate η with height, we assume an average value of 15% (Sect. 2.1). This agrees well with calculations by Owen & Jackson (2012), who also estimated that η values higher than 40% are unrealistically high. More recently, Salz et al. (2016) calculated the average heating efficiency for a set of planets with different masses and radii. They concluded that for planets with log (GM_{pl}/R_{pl}) smaller than 13.11 (the case of the planets considered here), η is about 23%, independent of the planet parameters.
As discussed by Lammer et al. (2016), the heating efficiency enters in the calculation of both L_{hy} and L_{en}, although with a slightly different dependence. To quantitatively estimate the effects of the uncertainty on the heating efficiency on the L_{hy}/L_{en} ratio, we ran a set of simulations for two planets orbiting the K2 star with two different Λ values (Λ = 21, M_{pl} = 5 M_{⊕}, R_{pl} = 1.8 R_{⊕} and Λ = 8, M_{pl} = 5 M_{⊕}, R_{pl} = 4.5 R_{⊕}) and T_{eq} = 1000 K, varying η between 10 and 40%, leaving all other parameters fixed. The results, displayed in Fig. 5, indicate that variations of η by a factor of two from the adopted value of 15% (e.g. between 10 and 30%) modify the L_{hy}/L_{en} ratio by a factor of about 1.5 in the case of low Λ and of about 1.05 in the case of high Λ. The sensitivity of the L_{hy}/L_{en} ratio on variations of η therefore decreases with increasing Λ. On the basis of these results, to be conservative, we consider the L_{hy}/L_{en}≈1 condition to be fulfilled when L_{hy}/L_{en}≤2.0.
Fig. 5
Variation of the L_{hy}/L_{en} ratio, normalised to the value of the L_{hy}/L_{en} ratio obtained with η = 15% (adopted for our calculations), as a function of heating efficiency η for two planets orbiting the K2 star with two different Λ values (dashed line: Λ = 21, M_{pl} = 5 M_{⊕}, R_{pl} = 1.8 R_{⊕}; solid line: Λ = 8, M_{pl} = 5 M_{⊕}, R_{pl} = 4.5 R_{⊕}) and T_{eq} = 1000 K. 
Figure 6 shows the atmospheric structure of the 5 M_{⊕} planet considered in Sect. 2.2, but with a radius of 1.8 R_{⊕} (i.e. out of the boiloff regime). Close to R_{pl} the atmosphere is hydrostatic, as indicated by the temperature increase (i.e. no adiabatic cooling), with the highenergy stellar flux providing a considerable amount of heating. The rise in temperature close to the lower boundary in Fig. 6 is caused by XUV heating, which is the driver of the outflow. In contrast, the monotonic temperature decrease (caused by adiabatic cooling) shown in Fig. 3 indicates that XUV heating is not important, implying that the outflow is driven by the high thermal energy of the planet. In our modelling we do not consider cooling from H. However, H cooling is not relevant in our case, because it does not affect the thermally driven escape rates in the boiloff regime (H is produced much above the lower boundary of the atmosphere; Chadney et al. 2016).
On the basis of detailed evolution modelling of young planets immediately after the disk dispersal, Owen & Wu (2016) concluded that planets exit the boiloff regime when their radius becomes smaller than 0.1 Bondi radii (R_{B}). The Bondi radius is defined as , where c_{s} is the isothermal sound speed. The R_{pl}/R_{B} = 0.1 condition for the occurrence of boiloff given by Owen & Wu (2016) is therefore mathematically identical to the Λ_{T}=20 condition, when an adiabatic gas index γ equal to 1 is considered, or in other words, isothermal gas.
We arrived at a result similar to that of Owen & Wu (2016), who properly took into account the various heating and cooling sources, which indicates that the assumptions and simplifications we made for our modelling are robust. In particular, it shows the validity of (i) simplifying the processes leading to the planet’s thermal balance by setting the temperature of the atmosphere equal to T_{eq} at the lower boundary; and (ii) setting the lower boundary at the pressure level where the optical depth is roughly unity, which is where most of the stellar radiation is absorbed^{1}. We note that modifications to these two assumptions affect the shape of the atmospheric profiles, but not the escape rates, if L_{hy} is equal to or smaller than L_{en}. For example, Fig. 2 of Lammer et al. (2016) shows that by varying the pressure at the lower boundary from 100 mbar to 1 bar only affects the L_{hy}/L_{en} ratio in the boiloff regime (when L_{hy}>L_{en}), while the radius at which the L_{hy}/L_{en}≈1 condition is reached (namely the value of Λ_{T}) is not affected. This implies that, within our scheme, the Λ_{T} values are independent of the two assumptions described above. It should also be noted that our results apply to any planet, independent of the internal structure, for which the 100 mbar pressure level lies above the solid core, if any is present.
4. Constraints on M_{pl} and R_{pl}
To explore whether the knowledge of the value of Λ_{T}, or equivalently of the R_{pl}/R_{B} = 0.1 condition, can help to constrain the parameters of old planets, it is necessary to consider the atmospheric evolution of planets in the boiloff regime. To roughly estimate how much time the modelled planets need to evolve out of the boiloff regime, we follow the same procedure as adopted by Lammer et al. (2016) to study the case of CoRoT24b.
As an example, we take the simulations we carried out for the M_{pl} = 5 M_{⊕} planet with T_{eq} = 1000 K orbiting the Ktype star. We assumed a core mass of 5 M_{⊕} and used formation and structure models by Rogers et al. (2011, see their Fig. 4) to estimate for each modelled radius the atmospheric mass fraction f. We then used the L_{hy} values to roughly estimate the evolution of the atmospheric mass over time. Figure 7 shows that the atmospheric mass for a radius above 1.8 R_{⊕} (where L_{hy}/L_{en}≈ 1) would be lost within ≈500 Myr. This is therefore the timescale needed for this planet to evolve out of the boiloff regime.
Fig. 7
Atmospheric mass M_{AT} evolution normalised to the atmospheric mass corresponding to R_{pl} =1.8 R_{⊕} (where L_{hy}/L_{en}≈1) estimated from the L_{hy} escape rates obtained for the M_{pl} = 5 M_{⊕} planet with T_{eq} = 1000 K orbiting the Ktype star. The dashed line indicates M_{AT} = M_{AT}(1.8 R_{⊕}). The initial time is arbitrarily set at 0.1 Myr. The legend lists the atmospheric mass fraction corresponding to each radius. 
Table 2 lists the timescales for each modelled planet from Table 1. We find the shorter timescales for the less massive and hotter planets. In particular, for planets with M_{pl} = 5 M_{⊕} and T_{eq} = 1000 K, the timescale to evolve out of boiloff is shorter than 500 Myr. The same also occurs for the hot (i.e. T_{eq} = 1000 K) 10 M_{⊕} planet orbiting the Mtype star, likely because of the effect of the smaller Rochelobe radius compared to the case of the same planet orbiting the G and Ktype stars. In general, we therefore find that hot (i.e. T_{eq} 1000 K) lowmass (M_{pl} 5 M_{⊕}) planets with hydrogendominated atmospheres, unless very young, should not have Λ<Λ_{T}. Because of their small Roche lobe, this conclusion also extends to hot (i.e. T_{eq} 1000 K) higher mass (M_{pl} 10 M_{⊕}) planets if they are orbiting Mdwarfs.
From the above considerations, it follows that for hot lowmass planets with hydrogendominated atmospheres with observed values leading to Λ< 15–35 there must be problems with the estimation/interpretation of the measured mass (i.e. too low), or radius (i.e. too large), or both. Large transit radii may be caused by the presence of aerosols lying far above R_{pl} or by an incorrect estimation of the stellar radius. We note, however, that the atmosphere of planets with a large enough atmospheric mass may stably lie in the boiloff regime, as described above.
Fig. 8
Colourscaled value of Λ as a function of planetary mass and radius for T_{eq} = 1000 K. The white straight lines indicate equal Λ values given in the plot. The red solid lines indicate lines of equal average densities of 0.6, 1.6, 3.2, and 5.5 g cm^{3}. The symbols correspond to the observed (blue bar and arrow) and possible massradius combination (black points) for CoRoT24b. 
The presence of aerosols may indeed lead to a misinterpretation of the observed transit radius. Lee et al. (2015), for example, calculated from first principles the formation of aerosols in the atmosphere of the hot Jupiter HD 189733 b (T_{eq}≈1000 K, similar to that of the hottest planets considered in this work), obtaining that clouds start forming in the 10–100 μbar pressure range. For the planet considered in Fig. 6 (M_{pl} = 5 M_{⊕}; R_{pl} = 1.8 R_{⊕}; T_{eq} = 1000 K), this pressure level corresponds to about 1.2–1.4 R_{pl}, that is, a radius of 2.2–2.5 R_{⊕} or 5.3–9.3 pressure scale heights above R_{pl}. The presence of highaltitude clouds/hazes in the atmosphere of such a planet would therefore lead to an overestimation of R_{pl} measured through broadband optical transit observations of about 20–40%. Lee et al. (2015) investigated a hot Jupiter, which has physical characteristics different from those of the planets considered here, but this is what is currently available, showing that similar cloud formation calculations, tuned for lowermass planets, are clearly needed for a more appropriate interpretation of the results.
For hot lowmass planets it is therefore possible to use the Λ≥Λ_{T} condition to constrain the minimum mass, given a certain radius, or maximum radius, given a certain mass. The only assumptions are the presence of a hydrogendominated atmosphere, which is likely for lowdensity planets, and an old age (i.e. >1 Gyr). Most of the extremely lowdensity planets discovered by Kepler fall into this regime.
Figure 8 shows the Λ value as a function of planetary mass and radius (at the 100 mbar level) for T_{eq} = 1000 K. We use the subNeptune CoRoT24b as an example of the constraining power of this plot. CoRoT24b has a mass lower than 5.7 M_{⊕}, a transit radius of 3.7 ± 0.4 R_{⊕}, and an equilibrium temperature of 1070 K (blue bar and arrow in Fig. 8; Alonso et al. 2014). CoRoT24b therefore has a Λ value lower than 10.9, well below Λ_{T}. For a value of Λ_{T} of 25 and when we assume that M_{pl} is equal to 5.7 M_{⊕} (Lammer et al. 2016, excluded masses smaller than ≈5 M_{⊕}), Fig. 8 (bottom black point) indicates that the 100 mbar pressure level, and hence where the transit radius would be if the planet were possessed of a clear atmosphere, lies around 2 R_{⊕} (≈1.7 R_{⊕} less than the transit radius), in agreement with the detailed analysis of Lammer et al. (2016). When we instead assume a clear atmosphere, hence R_{T} = R_{100 mbar}, M_{pl} should be M_{⊕} (right black cross in Fig. 8), although this is unlikely given the nondetection of the planet in the radialvelocity measurements.
The atmospheric pressure profile of CoRoT24b shown by Lammer et al. (2016) indicates that if we assume that the 100 mbar level lies at 2 R_{⊕}, then the transit radius is at a pressure of 1–10 μbar, which is about 10 times smaller than the lowest pressure at which Lee et al. (2015) predicts cloud formation. For this particular planet, the most likely scenario is therefore a combined effect of the presence of aerosols and of a slight mass underestimation.
Table 2 shows that for most of the more massive planets (M_{pl} 10 M_{⊕}) and all the cooler (T_{eq} 500 K) ones, the timescale for the atmosphere to evolve out of the boiloff regime is longer than 10 Gyr and in some cases even longer than the mainsequence life time of the host stars. This clearly shows that although the atmosphere of these planets may be in boiloff, the escape rates are not high enough to significantly affect the atmosphere in a short time, in agreement with the results of Ginzburg et al. (2016).
From the results of Table 2, it follows that in the 5–10 M_{⊕} planetary mass and 500–1000 K equilibrium temperature range with increasing temperature and/or decreasing mass the escape rates start affecting the longterm evolution of the atmosphere. This transition region depends not only on the planetary parameters, but also on the stellar properties and orbital separation, which affect the escape rates through the XUV flux and size of the Roche lobe. We will explore this transition region in detail in a forthcoming work.
5. Conclusions
We generalised the expression of the Jeans escape parameter to account for hydrodynamic and Rochelobe effects, which is important for closein exoplanets. We use a planetary upper atmosphere hydrodynamic code to derive the atmospheric temperature, pressure, and velocity structure of subNeptunes with various masses and radii and draw the profiles of the Jeans escape parameter as a function of height. We used our simulations and the generalised Jeans escape parameter to describe the boiloff regime (Owen & Wu 2016), which is characterised by very high escape rates driven by the planet’s high thermal energy and low gravity.
We introduce the restricted Jeans escape parameter (Λ) as the value of the Jeans escape parameter calculated at the observed planetary radius and mass for the planet’s equilibrium temperature, and considering atomic hydrogen. We used the L_{hy}/L_{en}≤1 empirical condition, where L_{en} is derived analytically from the energylimited formula, to estimate Λ_{T}, the critical value of Λ below which efficient boiloff occurs. We ran simulations with varying planetary mass, stellar mass, and equilibrium temperature, concluding that Λ_{T} lies between 15 and 35, depending on the system parameters. This result, mostly based on aeronomical considerations, is in agreement with that obtained by Owen & Wu (2016), namely R_{pl}/R_{B}> 0.1.
From the analysis of our simulations, we find that the atmosphere of hot (i.e. T_{eq} 1000 K) lowmass (M_{pl} 5 M_{⊕}) planets with Λ<Λ_{T} would be unstable against evaporation because they lie in an efficient boiloff regime that would shrink their radius within a few hundreds of Myr. We find the same result also for hot (i.e. T_{eq} 1000 K) higher mass (M_{pl} 10 M_{⊕}) planets with Λ<Λ_{T}, when they orbit Mdwarfs. We conclude that for old hydrogendominated planets in this range of parameters, Λ should be ≥Λ_{T}, which therefore provides a strong constraint on the planetary minimum mass/maximum radius.
This information can be used to predict the presence of highaltitude aerosols on a certain planet without the need to obtain transmission spectra, or inform on the reliability of planetary masses. Our results could also be used to indicate the possible presence of contaminants in the images used to derive the transit light curves, which would lead to the measurement of a planetary radius larger than what is in reality (Dalba et al. 2017). Our results are relevant because of the various present and future ground and spacebased planetfinding facilities (e.g. K2, NGTS, CHEOPS, TESS, PLATO), which will detect subNeptunes orbiting bright stars, hence amenable to atmospheric characterisation. Our results will help prioritisation processes: for instance, hot lowdensity, lowmass planets, with masses measured through radial velocity, are good targets for transmission spectroscopy, but their large radii may be caused by highaltitude clouds, which would therefore obscure the atmospheric atomic and molecular features. An application of our results to the transiting subNeptune planets known to date is presented by Cubillos et al. (2017).
The simulations presented in this work, only sparsely cover the typical parameter space of the discovered systems hosting subNeptunes, also in terms of highenergy stellar flux. In the future, we will extend our work to a larger parameter space and aiming at its more homogeneous coverage. In particular, we will better identify the dependence of the Λ_{T} value on the planetary (e.g. mass, radius, and temperature/pressure at the lower boundary) and stellar (e.g. mass and highenergy flux) parameters.
This is also how Owen & Wu (2016) set their upper boundary.
Acknowledgments
We acknowledge the Austrian Forschungsförderungsgesellschaft FFG projects “RASEN” P847963 and “TAPAS4CHEOPS” P853993, the Austrian Science Fund (FWF) NFN project S11607N16, and the FWF project P27256N27. N.V.E. acknowledges support by the RFBR grant Nos. 150500879a and 165214006 ANF_a. We thank the anonymous referee for the comments that led to a considerable improvement of the manuscript.
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All Tables
All Figures
Fig. 1
Left: synthetic transmission spectra calculated for a planet with M_{pl} = 5 M_{⊕}, R_{pl} = 4 R_{⊕}, and a 1000 K isothermal atmosphere with 0.01 (blue), 1.0 (orange), and 100 (green) times solar metallicity. The circles of corresponding colour denote the transmission curves integrated over the CoRoT spectral response curve (red dashed curve). Right: contribution functions for the vertical optical depth integrated over the CoRoT spectral response curve. 

In the text 
Fig. 2
Molemixing fractions of the atmospheric species (see legend in the bottom panel) in thermochemical equilibrium for isothermal (1000 K) models calculated for 0.01 (top), 1.0 (middle), and 100 (bottom) times solar metallicity. 

In the text 
Fig. 3
Top: temperature (black solid line) and pressure (red dashed line) profiles as a function of radius r in units of R_{pl} for a 5 M_{⊕} and 4 R_{⊕} planet with T_{eq}=1000 K, orbiting an early Ktype star (see Table 1). The right axis indicates the pressure scale. Middle: λ^{∗} (black solid line) and (red dashed line) profiles as a function of radius r in units of R_{pl}. The horizontal lines mark the critical values of the Jeans escape parameter in the hydrostatic case: 1.5, 15, and 30. The blue dotted lines show the λ^{∗} and profiles calculated assuming that the whole atmosphere is made of atomic hydrogen. The filled circle indicates the Λ value (see Sect. 3). Bottom: υ_{th} (black solid line) and υ_{hy} (red dashed line) profiles in km s^{1}. 

In the text 
Fig. 4
Ratio between the hydrodynamic (L_{hy}) and energylimited (L_{en}) escape rates as a function of Λ for the modelled planets orbiting the G2 (top), K2 (middle), and M2 (bottom) star. Within each panel, the legend indicates the mass (in M_{⊕}) and temperature (in K) of the modelled planets. The dashed line indicates the equality between L_{hy} and L_{en}, while the dotted line indicates where the L_{hy}/L_{en} ratio is equal to 2.0. The value of Λ_{T} lies between 15 and 35. 

In the text 
Fig. 5
Variation of the L_{hy}/L_{en} ratio, normalised to the value of the L_{hy}/L_{en} ratio obtained with η = 15% (adopted for our calculations), as a function of heating efficiency η for two planets orbiting the K2 star with two different Λ values (dashed line: Λ = 21, M_{pl} = 5 M_{⊕}, R_{pl} = 1.8 R_{⊕}; solid line: Λ = 8, M_{pl} = 5 M_{⊕}, R_{pl} = 4.5 R_{⊕}) and T_{eq} = 1000 K. 

In the text 
Fig. 6
Same as Fig. 3, but for a 5 M_{⊕} planet with a radius of 1.8 R_{⊕}. 

In the text 
Fig. 7
Atmospheric mass M_{AT} evolution normalised to the atmospheric mass corresponding to R_{pl} =1.8 R_{⊕} (where L_{hy}/L_{en}≈1) estimated from the L_{hy} escape rates obtained for the M_{pl} = 5 M_{⊕} planet with T_{eq} = 1000 K orbiting the Ktype star. The dashed line indicates M_{AT} = M_{AT}(1.8 R_{⊕}). The initial time is arbitrarily set at 0.1 Myr. The legend lists the atmospheric mass fraction corresponding to each radius. 

In the text 
Fig. 8
Colourscaled value of Λ as a function of planetary mass and radius for T_{eq} = 1000 K. The white straight lines indicate equal Λ values given in the plot. The red solid lines indicate lines of equal average densities of 0.6, 1.6, 3.2, and 5.5 g cm^{3}. The symbols correspond to the observed (blue bar and arrow) and possible massradius combination (black points) for CoRoT24b. 

In the text 
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