Suppressed Far-UV Stellar Activity and Low Planetary Mass Loss in the WASP-18 System^*

Ions abundant in the interstellar medium (ISM) absorb strongly at the wavelengths of their resonance lines, including some that are measurable in the COS spectrum. Consequently, we examine the ISM absorption along the line of sight of WASP-18.

We started by fitting synthetic fluxes, calculated with marcs models (Gustafsson et al. 2008) and rescaled for the measured distance of 126.4 ± 4.6 pc (Gaia Collaboration et al. 2016), to the observed Johnson (Kharchenko et al. 2009), TYCHO (Høg et al. 2000), 2MASS (Cutri et al. 2003), and WISE (Cutri et al. 2012) photometry, converted to physical units. We converted the photometry adopting the calibrations provided respectively by Bessell et al. (1998), Grossmann et al. (1995), van der Bliek et al. (1996), and Wright et al. (2010). From the fit, we derived a stellar radius of 1.30 ± 0.05 ${R}_{\odot }$ and a low color excess $E(B-V)$ of 0.023 ± 0.009 mag, where the error bars include the uncertainties in the stellar distance and photometry. The stellar radius is in agreement with the value of 1.255 ± 0.028 ${R}_{\odot }$ given by Maxted et al. (2013).

The slightly larger stellar radius, which is most likely caused by the adoption of a larger distance, has an impact on both planetary radius and brightness temperature. We therefore re-derived both of them, obtaining a significant increase in the planetary radius of 0.1 R_J and a corresponding decrease in the brightness temperature of 130 K, compared to the values given respectively by Southworth (2012) and Nymeyer et al. (2011).

An increase in the stellar radius leads to an increase in the inferred stellar luminosity, which may therefore have an effect on the estimated age of the star. We made use of the Bayesian PARAM¹¹ tool (da Silva et al. 2006), employing the stellar evolutionary tracks by Bressan et al. (2012) and as priors the stellar initial mass function by Chabrier (2001) and a constant star formation rate. Taking into account the HIPPARCOS distance, PARAM returned a stellar mass of 1.23 ± 0.04 ${M}_{\odot }$, radius of 1.20 ± 0.06 ${R}_{\odot }$, and age of 0.61 ± 0.58 Gyr, while by employing the GAIA distance, we obtained a mass of 1.27 ± 0.04 ${M}_{\odot }$, radius of 1.26 ± 0.05 ${R}_{\odot }$, and age of 0.84 ± 0.76 Gyr. The stellar radius derived on the basis of the GAIA distance is in agreement with those given by Maxted et al. (2013) and derived here from the fitting of the spectral energy distribution. More importantly, these results indicate that the larger distance has no significant effect on the previously inferred stellar age estimate, further confirming that WASP-18 is a rather young star, in particular when considering that the main-sequence lifetime of a 1.2 ${M}_{\odot }$ star is of the order of 4 Gyr (Bressan et al. 2012).

We further derived the $E(B-V)$ value from the Galactic extinction maps of Amôres & Lépine (2005) obtaining $E(B-V)$ = 0.008 mag and from the Infrared Science Archive¹² dust maps obtaining a value of $E(B-V)$ = 0.0115 ± 0.0012. This last value lies in between that derived from the fitting of the spectral energy distribution and from the maps of Amôres & Lépine (2005). Although the star lies more than 100 pc away from us, the low ISM absorption in the direction of WASP-18 is confirmed by the results of the HIPPARCOS mission, which indicate that along the WASP-18 line of sight, particularly within the first 50 pc, the ISM is sparse (see e.g., Figure 1 of Frisch et al. 2011).

We therefore used the maximum and minimum $E(B-V)$ values (i.e., 0.023 ± 0.009 and 0.008 mag) to estimate the range of ISM column densities for H i, C i, C ii, C iii, C iv, Si iv, Mg ii, Ca ii, and Na i. Following the results and considerations of Fossati et al. (2015a, 2017), we derived the column densities employing the N_{H i}–$E(B-V)$ conversion¹³ provided by Savage & Mathis (1979) and the ISM abundances and ionization mix given by Frisch & Slavin (2003, their model 2—Table 5). The results are listed in Table 1. We remark that the use of other N_{H i}–$E(B-V)$ conversions, such as those of Diplas & Savage (1994) or Güver & Özel (2009), does not significantly affect the results (e.g., Fossati et al. 2017, 2015a).

Table 1.  ISM Column Densities (in cm⁻²)

$E(B-V)$	0.023 ± 0.009	0.008
Element	$\mathrm{log}{N}_{{\rm{X}}}$
H i	${20.13}_{-0.22}^{+0.14}$	19.67
C i	${13.21}_{-0.22}^{+0.14}$	12.75
C ii	${16.53}_{-0.22}^{+0.14}$	16.07
C iii	${15.05}_{-0.22}^{+0.14}$	14.59
C iv	${0.00}_{-0.00}^{+0.00}$	0.00
Si iv	${10.41}_{-0.22}^{+0.14}$	9.95
Mg ii	${14.71}_{-0.22}^{+0.14}$	14.25
Ca ii	${12.77}_{-0.22}^{+0.14}$	12.53
Na i	${11.59}_{-0.22}^{+0.14}$	11.13

Note. The column densities are given for the derived maximum and minimum $E(B-V)$ values (see the text). The uncertainties account for the error bar on $E(B-V)$, only.

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Unfortunately, none of the stars analyzed by Redfield & Linsky (2002, 2004) are close to the WASP-18 line of sight. However, there are a few stars that lie 20°–50° away and for which Ca ii and Mg ii column densities have been measured. For these stars, Redfield & Linsky (2002, 2004) listed maximum column density values of 11.0 and 14.5 for Ca ii and Mg ii, respectively, values that are consistent with those in Table 1, as these stars lie much closer to us than WASP-18. We also looked for hot, massive stars, lying close to us and to the WASP-18 line of sight. Hot stars facilitate direct measurements of the ISM column density for various ions. Within an angular distance of 10 ^◦, there are two nearby hot, massive stars, κ Eri and ϕ Eri, which lie, respectively, 155.8 ± 3.6 and 47.1 ± 0.3 pc from us (van Leeuwen 2007). Because of the distance (∼156 pc) similar to that of WASP-18 (∼126 pc), κ Eri is the most interesting of the two stars. For this star, Welsh et al. (1994, 1997) reported ISM column densities for H i, Ca ii, and Na i of log(N_X[cm⁻²]) = 19.58, 12.29 ± 0.05, and 11.27, respectively. They also listed a $E(B-V)$ value of 0.02 mag. These ISM column density values agree well with those given in Table 1 and in particular with those based on the lower $E(B-V)$ value of 0.008 mag.

For the elements considered here, C ii, Si ii, Mg ii, Ca ii, and Na i are the main ionization stages in the local ISM. At the low $E(B-V)$ value of WASP-18, almost all of the carbon in the diffuse ISM is in the singly ionized state (Snow & McCall 2006). C iv is not typically observed in the local ISM, which is why Table 1 lists a null column density (Frisch & Slavin 2003; Frisch et al. 2011), though a single detection of C iv in the local ISM has been reported by Welsh et al. (2010).

From the values listed in Table 1, we conclude that only H i and C ii are significantly affected by ISM absorption (i.e., log N_X ≫ 15) and that Mg ii and C iii may be affected too, though high-resolution UV spectra would be necessary to identify the ISM absorption features. We also remark that the core of the Ca ii H&K and Na i D lines of WASP-18, observed with the high-resolution UVES spectrograph of the ESO Very Large Telescope (VLT), does not present any signature of ISM absorption (Fossati et al. 2014). We also used this optical spectrum to look for features of diffuse interstellar bands, finding none. We can therefore safely conclude that the discrepancy between the young stellar age and the low activity is not caused by extrinsic ISM absorption, in agreement with the conclusions of Fossati et al. (2014) and Pillitteri et al. (2014).

We inspected the COS spectrum and found just four clearly detected stellar emission features, namely the C iii multiplet at ∼1176 Å, the C ii doublet at ∼1335 Å, the Si iv line at ∼1394 Å, and the C iv doublet at ∼1548 Å (see Figure 1). To measure the fluxes of these features, we employed the procedure adopted for WASP-13 (Fossati et al. 2015a). The wavelength regions considered for each feature are shown in Figure 2, while the resulting integrated fluxes and line detections are listed in Table 2.

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Figure 2 and Table 2 indicate that all four lines are clearly detected. Note that the background level around the C iv doublet at ∼1548 Å is much higher than that of the other three lines because the stellar photospheric flux rises significantly between 1400 and 1500 Å. From the COS spectrum, we also derived an upper limit on the strength of the C i multiplet at ∼1657 Å, obtaining a value of < 5.7 × 10⁻¹⁶ erg s⁻¹ cm⁻².

On the basis of the results presented in Section 3.1, we can safely conclude that the fluxes measured for C iv and Si iv are unaffected by ISM absorption. The same applies to the measured C iii multiplet because this does not arise from ground-state transitions (i.e., no resonant lines). On the other hand, the flux measured for C ii is almost certainly reduced by ISM absorption. However, thanks to the high spectral resolution and good data quality, we can separate the C ii doublet at ∼1335 Å into the two components (i.e., 1334 and 1335 Å), measuring them separately. We did this by fitting the observed profile with a double-peak Gaussian, taking the background into account (Figure 3). This is useful because C ii ISM absorption affects mostly the 1334 Å feature, which rises from a resonance transition, while the 1335 Å feature is only marginally affected as a result of the large ISM C ii column density. For the 1335 Å line, we obtained a continuum-subtracted integrated flux of 3.83(±0.16) × 10⁻¹⁶ erg s⁻¹ cm⁻², which, because of the possible contamination of ISM absorption, should be considered as a lower limit. The presence of C ii emission is already an indication that extrinsic absorption, in particular from circumstellar material, is probably not at the origin of the anomalously low activity level.

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In the COS spectrum of WASP-13, we obtained tentative 2–4σ detections of the Si iv line at 1403 Å and the C i multiplets at 1560 and 1657 Å that are not detected in the FUV spectrum of WASP-18. The Si iv line at 1403 Å is intrinsically weaker than the Si iv 1394 Å feature. We attempted to detect this weaker feature in WASP-18, but obtained just a ≈3σ detection, which a visual inspection of the spectrum suggests being possibly spurious. The non-detection of C i could be attributed to the fact that the C i lines may not be strong enough to emerge over the stellar photospheric flux and/or by the fact that the C i lines are intrinsically weaker for WASP-18 than for WASP-13, hence lost in the noise. This last possibility would in turn imply either the presence of a hotter or thicker (i.e., more emitting power or volume) chromosphere compared to that of WASP-13. For a meaningful comparison of the FUV spectra of WASP-13 and WASP-18, it is important to note that the spectrum of WASP-13 is of much lower quality than that of WASP-18, as indicated, for example, by the fact that for WASP-13, we could not separate the two components of the C ii doublet at ∼1335 Å.

To identify which of these two possibilities is correct, we compare the flux ratios obtained for the lines detected and measured for both stars. Because of ISM absorption, we cannot rely on the fluxes measured for C ii¹⁴ , which is why we compare the flux ratios between the C iv and Si iv features, which form in the transition region. We note that the formation temperature of C iv is ∼107 kK, while that of Si iv is ∼62 kK (see e.g., Table 6 of Pagano et al. 2004). For WASP-18, we obtained a C iv/Si iv flux ratio of 1.87 ± 0.35, while for WASP-13, the values reported by Fossati et al. (2015a) lead to a flux ratio of 3.37 ± 0.63. These line flux ratios give the ratios of the emission measures at the respective line formation temperatures, assuming thermodynamic equilibrium. This implies that the emission measure at higher temperature is relatively lower in WASP-18 than in WASP-13, thus indicating a lower non-radiative heating of the upper atmosphere of the former. This further suggests that the transition region of WASP-13 is probably hotter or thicker than that of WASP-18, allowing us to conclude that the C i multiplets at 1560 and 1657 Å are probably intrinsically weak and therefore not able to emerge significantly above the photospheric flux. This is interesting because in Fossati et al. (2015a) we concluded that WASP-13 has a chromosphere and transition region similar to those of other inactive solar-type stars, very similar to those of the Sun and α Cen A. This indicates that WASP-18 may indeed have an intrinsically low activity level, consistent with the lack of detection in the X-rays (Pillitteri et al. 2014). To confirm this, we compare then the FUV line flux ratios with those of other stars of similar temperature.

We inspected the HST Spectroscopic Legacy Archive (HSLA)¹⁵ and the StarCAT catalog (Ayres 2010) looking for FUV spectra of main-sequence stars of various ages and activity levels, and with a spectral type ranging between G0 and F3 (WASP-18's spectral type is F6V) to measure their C iii/C iv and C ii¹⁶ /C iv flux ratios to be then compared to those of WASP-18. The comparison stars are listed in Table 3, where we also give their age and $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ values. All stars but HD 220657 were already considered as comparison stars for WASP-13 by Fossati et al. (2015a), and we adopt the same age and $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ values employed there.

Table 3.  Comparison Stars

Star	Spectral	Distance	$\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$	Age
	Type	(pc)		(Gyr)
HD 39587	G0V	8.7 ± 0.1	−4.43	0.29
HD 97334	G0V	21.9 ± 0.2	−4.42	0.55
HD 209458	G0V	49.6 ± 1.9	−4.97a	4.00b
HD 142373	G0V	15.9 ± 0.1	−5.11c	7.40d
HD 33262	F9V	11.6 ± 0.1	−4.65e	0.25d
HD 220657	F8IV	52.2 ± 0.5	−4.54e	0.85d

Notes.  The spectral types (column (2)) are taken from Simbad. Columns (3), (4), and (5) list the stars' distance, $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ and age (in Gyr). The stellar distances are taken from TGAS (Gaia Collaboration et al. 2016), while, unless otherwise stated, the stars' $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ values and ages are taken from Barnes (2007). If multiple $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ were available, we opted for the lowest one. The typical uncertainty on the $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ values is of 0.05–0.1 dex, while for the age the typical uncertainty is of 500 Myr (e.g., Casagrande et al. 2011).

^aFrom Figueira et al. (2014). ^bFrom Melo et al. (2006). ^cFrom Mamajek & Hillenbrand (2008). ^dFrom Casagrande et al. (2011). ^eFrom Pace (2013).

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Unfortunately, all these comparison stars are cooler than WASP-18, with F8 being the earliest spectral type. This is not an ideal situation, in particular because the surface convective layers of stars corresponding to mid-F spectral types become progressively thinner, hence affecting the chromospheric and coronal structures. We searched the IUE archive for FUV high-resolution spectra of stars with spectral type earlier than F8, but we were able to detect measurable features just for HD 11443, which is an evolved star (F5III) and hence unsuitable. The only way to fully solve this problem would be to obtain HST FUV spectra of nearby main-sequence stars of known age in the F8–F3 spectral range. Such observations would be crucial to improve our understanding of stellar structure and activity, and not just to provide appropriate comparisons for WASP-18.

The spectra of HD 142373, HD 220657, HD 33262, HD 39587, and HD 97334 were obtained with STIS (E140M grating), and we downloaded them from the StarCAT catalog (Ayres 2010), while that of HD 209458 was obtained with COS and downloaded from HSLA (France et al. 2010). The COS spectra obtained with the G130M and G160M gratings have been co-added with the same tool employed to co-add the WASP-18 COS spectra. Before measuring the line fluxes, all spectra were convolved with a Gaussian to match the spectral resolution of the COS spectrum of WASP-18.

Figure 4 shows the C iii/C iv (top) and C ii/C iv (bottom) flux ratios as a function of stellar age (left) and $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ value (right). We do not consider here the C iv/Si iv flux ratio because it does not vary significantly among the considered comparison stars and hence it is not constraining of the characteristics of WASP-18. We also looked at whether the upper limit on the C i 1657 Å line flux provides meaningful constraints on the properties of WASP-18, but the lower limit on the C iv/C i ratio of 0.65 allows one to conclude just that WASP-18 is younger than 6 Gyr (see Equation (2) of Fossati et al. 2015a).

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We find no significant trend in the C iii/C iv flux ratio as a function of age and $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ value, while the C ii/C iv flux ratio increases significantly with increasing age and $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ value. The two fits are

$$\begin{eqnarray}\displaystyle \frac{{\rm{C}}\,{\rm{II}}\,1335}{{\rm{C}}\,{\rm{IV}}\,1548} & = & 0.207(\pm 0.071)\\ & & +\,0.182(\pm 0.026)\times \mathrm{log}(\mathrm{Age}\,[\mathrm{Myr}])\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}\displaystyle \frac{{\rm{C}}\,{\rm{II}}\,1335}{{\rm{C}}\,{\rm{IV}}\,1548} & = & -1.588(\pm 0.278)\\ & & -\,0.496(\pm 0.060)\times \mathrm{log}\ {R}_{\mathrm{HK}}^{{\prime} }.\end{eqnarray} \tag{ 2 }$$

For the fits given in Equations (1) and (2), we excluded WASP-18 and did not consider the uncertainties on age and $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ values.

Interestingly, the lower limit on the C ii/C iv flux ratio measured for WASP-18 is at the same level as that of the old, inactive comparison stars, in agreement with the low activity level inferred from the Ca ii H&K lines and the X-ray fluxes.

To further confirm this result, we rescaled the fluxes measured for HD 39587 and HD 33262, the youngest stars in our sample, to the distance and radius of WASP-18. For the two comparison stars, we considered the distance listed in Table 3 and a radius of 1.047 and 1.100 solar radii, respectively (van Belle & von Braun 2009; Chandler et al. 2016). We found that the FUV line fluxes measured for WASP-18 are, in general, about a factor of 10 smaller than expected if WASP-18 had the activity level of a young, active star like HD 39587 and HD 33262.

Our results strongly support the postulated intrinsically low stellar activity, which Pillitteri et al. (2014) suggested is due to the tidal influence of the massive planet (tidal star–planet interaction) affecting the outer stellar structure and activity of WASP-18. The presence of an intrinsically low stellar activity is further strengthened by comparing the measured upper limit on the X-ray fluxes with what expected on the basis of the stellar age. Following the relations of Sanz-Forcada et al. (2011), we obtain an upper limit on the EUV fluxes of $\mathrm{log}{L}_{\mathrm{EUV}}([\mathrm{erg}\,{{\rm{s}}}^{-1}])$ < 27.9 ± 0.2 dex, where the uncertainty is caused by the scatter in the data-points, particularly at low activity levels, considered by Sanz-Forcada et al. (2011). We further converted this upper limit into a stellar age, obtaining a minimum age of about 9.6 Gyr, which for a star like WASP-18 is longer than the main-sequence lifetime, thus in stark contrast with the star's young age. When using the age estimated from the X-ray luminosity vs age relationship of Sanz-Forcada et al. (2011), we would predict an X-ray luminosity of 1.93 × 10²⁸ erg s⁻¹, which is about 27 times larger than the upper limit derived from the Chandra observations.

Bonomo et al. (2017) reconsidered all of the available observations of the WASP-18 system obtaining lower limits to the modified tidal quality factors of the star ${Q}_{{\rm{s}}}^{{\prime} }$ and of the planet Q^'_p. We recall that the larger Q^', the lower the dissipation of the tidal kinetic energy into the corresponding body (see Ogilvie 2014, for details). They found ${Q}_{{\rm{s}}}^{{\prime} }\gt {10}^{8}$ and ${Q}_{{\rm{p}}}^{{\prime} }\gt {10}^{7}$ in WASP-18 that implies that dynamical tides are likely not excited into the star, as suggested by Ogilvie & Lin (2007). The absence of a detectable tidal decay of the orbit of WASP-18b supports this conclusion (Wilkins et al. 2017). The consequence is that stellar rotation is probably not significantly affected by tides in this system and the remaining lifetime of the planet before it is engulfed by the star is at least of ≈100 Myr. Even if dynamical tides are likely not excited, the relatively large amplitude of the equilibrium tide in WASP-18 (see Pillitteri et al. 2014) may imply a perturbation of the dynamo by the associated tidal flow. For example, it could increase the turbulence in the convection zone thus accelerating the decay of the magnetic fields in a stellar dynamo. A sufficiently large turbulent dissipation may quench a large-scale solar-like dynamo leaving only a small-scale turbulent dynamo that can sustain stellar activity only at the basal level. Nevertheless, this is just a qualitative conjecture that requires detailed physical modeling to be proven.

The evolution of the atmosphere of a planet is mostly driven by atmospheric escape (e.g., Jin et al. 2014), which, in turn, is controlled by the high-energy stellar irradiation (XUV: X-ray [1–100 Å] + EUV [100–912 Å]). This is therefore one of the most critical parameters for exoplanet studies (e.g., Ribas et al. 2005; Sanz-Forcada et al. 2011; Linsky et al. 2014; France et al. 2016). For all stars other than the Sun, direct measurements of the EUV flux are not possible because of absorption by neutral hydrogen in the ISM. To circumvent this problem, various indirect methods and scaling relations have been developed to estimate the XUV flux on the basis of accessible observables, such as X-ray fluxes (Sanz-Forcada et al. 2011; Chadney et al. 2015), Lyα fluxes (Linsky et al. 2014), and stellar rotational velocities (Wood et al. 1994; Johnstone et al. 2015; Tu et al. 2015).

As detailed above, no adequate proxy for the high-energy spectrum of WASP-18 is available. X-ray and Lyα fluxes are not available, and the stellar age does not correlate with the fluxes measured in the X-rays (corona) and in the FUV emission lines (chromosphere). To estimate the stellar XUV fluxes, we can only follow the procedure we previously adopted for WASP-13 (Fossati et al. 2015a). We rescaled the solar irradiance reference spectrum (Woods et al. 2009) to match the observed WASP-18 flux in the Si iv λ1394 feature, accounting for WASP-18's distance and radius. This Si iv feature is detected at a significance higher than that of the C iv λ1548 feature. By integrating the rescaled solar spectrum, we obtained an XUV (10–1180 Å) flux at the planetary distance of 10.2 mW m⁻² (or erg s⁻¹ cm⁻²), which is about twice that of the average Sun (Ribas et al. 2005). This result is supported by a comparison of the observed EUV (100–600 Å) fluxes of Procyon, which has a spectral type F5V (hence close to that of WASP-18), and the Sun (Figure 5) from which we find that the integrated EUV flux of Procyon is about three times larger than solar. We further inferred the X-ray luminosity from the scaled solar fluxes obtaining a value of 1.2 × 10²⁶ erg s⁻¹, which is about a factor of six smaller than the upper limit derived with Chandra, thus in agreement with the observations.

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We use the model of Koskinen et al. (2013a, 2013b) to obtain the mass-loss rate from WASP-18b. We calculate the basic atmospheric structure to set the lower boundary conditions of our escape model by assuming an isothermal lower atmosphere with a dayside temperature of 3150 K¹⁷ (Nymeyer et al. 2011; Sheppard et al. 2017). This is a rather extreme choice that leads to significant dissociation of H₂ above the 0.1 bar level. In reality, somewhat more moderate temperature profiles fit the secondary eclipse data too, leading to less efficient dissociation and hence a less extended upper atmosphere (Iro & Maxted 2013). A more detailed exploration of the thermal structure is required to properly resolve the conditions in the middle atmosphere between ∼10 mbar and 1 μbar. A model of thermal structure applicable to WASP-18b in this region of the atmosphere, however, does not currently exist and the exploration is well beyond the scope of the present work. Fortunately, accurate modeling of the middle atmosphere is not required for our purposes. As we demonstrate below, the escape rate from WASP-18b is very low and this conclusion will not be changed by detailed radiative transfer models that instead are likely to produce even lower escape rates.

In addition to the temperature, the atmospheric structure depends on the composition through the mean molecular weight. We assumed a solar He abundance (Lodders 2003) and that the relative abundances of H₂ and H are in thermochemical equilibrium. The presence of trace species such as H₂O and CO in solar abundance does not significantly affect the mean molecular weight. Once the mean molecular weight as a function of pressure is known, the altitudes are obtained from the hypsometric equation

$$\begin{eqnarray}&&{\int }_{{h}_{0}}^{h}\displaystyle \frac{m(h^{\prime} )g(h^{\prime} ){\rm{d}}h^{\prime} }{{kT}(h^{\prime} )}=-\mathrm{ln}\left[\displaystyle \frac{p(h)}{p({h}_{0})}\right],\end{eqnarray} \tag{ 3 }$$

where m is the mean molecular weight, k is the Boltzmann constant, g is gravity, T is temperature, p is pressure, and the altitude coordinate h = h(r, θ, ϕ) is measured perpendicular to surfaces of constant gravitational potential at a given location (e.g., Koskinen et al. 2015b).

The distortion of the planet by stellar gravity (i.e., the "Roche lobe" effect) and rotation are negligible (the L1 point is at 5.1 R_p) and thus we simply treat the planet and its gravitational potential as spherical, i.e., h = r. We set the lower boundary of the escape model at 1 μbar where the altitude based on our structure model is 1452 km and the mixing ratios of H₂, He, and H are 1.5 × 10⁻³, 7.35 × 10⁻², and 0.925, respectively. We need to specify species densities, temperature, and the bulk velocity at the lower boundary. We do this by fixing the pressure and use it with the lower boundary temperature to calculate the total density and obtain species densities by fixing the lower boundary mixing ratios. We use mass flux conservation to set the lower boundary velocity during the simulation.

Given a composition dominated by atomic hydrogen, the temperature begins to increase with altitude immediately above the lower boundary. Figure 6 shows the temperature profile calculated by the model. We note here that our calculation also includes cooling by Lyα emissions introduced by Murray-Clay et al. (2009) and later studied in detail by Menager et al. (2013). The temperature increases steeply in the lower atmosphere, reaching an isothermal value of 13,660 K below the exobase, which is located at a relatively low radial distance of 1.22 R_p with a pressure of 1.4 × 10⁻¹³ bar. Even at such a high temperature, the atmosphere does not expand to high altitudes and, because of the large planetary mass, the Jeans escape parameter for H remains large ${X}_{{\rm{H}}}({r}_{\mathrm{exo}})$ = 70.9. This shows that the upper atmosphere of WASP-18b is not in a hydrodynamic regime, which would require values of the Jeans parameter for H around 2–3 (Volkov et al. 2011). The escape lies therefore in a Jeans escape regime, where the atmospheric particles move with a speed controlled by a Maxwellian distribution, while in the hydrodynamic regime the velocity distribution function evolves to a drifting Maxwellian that incorporates the bulk flow and the exobase extends to very high altitudes.

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Figure 7 shows the composition of the thermosphere-ionosphere on WASP-18b. Not surprisingly, the upper atmosphere is strongly ionized and H⁺ overtakes as the dominant species above 1.05 R_p (p = 2.4 nbar). Exceptionally for hot Jupiters, escape is negligible and the abundance of He and He⁺ decreases with altitude more rapidly than that of H and H⁺ due to diffusive separation. This is one of the few well known hot Jupiters to date where diffusive separation is relevant—in most other cases, hydrodynamic escape leads to near uniform mixing of the species of interest. It means that the escape of heavy atoms and ions from the atmosphere of WASP-18b is even less likely than that of hydrogen.

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As described above, the atmosphere of WASP-18b does not escape hydrodynamically due to the high mass of the planet, while Jeans escape holds. We treat this by imposing kinetic Jeans escape upper boundary conditions at the exobase (where the Knudsen number based on all collisions, evaluated and adjusted during the simulation, is Kn = 1). The bulk outflow velocity at the upper boundary is based on the sum of the Jeans outflow fluxes for each species. In addition, we impose upper boundary temperature and species density gradients consistent with Jeans escape. In effect, we adapt the Type-I upper boundary conditions from Zhu et al. (2014, Equations (14)–(19)) that were generalized for multiple species by Koskinen et al. (2015a).

As the upper atmosphere of WASP-18b is strongly ionized, we follow Koskinen et al. (2013a) in writing the thermal escape (Jeans) parameter for ions and electrons (species s) as

$$\begin{eqnarray}&&{X}_{{\rm{s}}}({r}_{\mathrm{exo}})={X}_{\mathrm{sg}}({r}_{\mathrm{exo}})+{X}_{\mathrm{se}}({r}_{\mathrm{exo}})=\displaystyle \frac{{\rm{\Delta }}{\phi }_{\mathrm{exo}}{m}_{{\rm{s}}}}{{{kT}}_{\mathrm{exo}}}-\displaystyle \frac{{q}_{{\rm{s}}}{\phi }_{{\rm{e}}}({r}_{\mathrm{exo}})}{{{kT}}_{\mathrm{exo}}},\end{eqnarray} \tag{ 4 }$$

where ϕ_e is the ambipolar electric potential, q_s is the charge of the ion/electron, Δϕ_exo is the gravitational potential difference from the exobase to the Roche lobe, T_exo is the temperature at the exobase, r_exo is the exobase radius, and X_sg and X_se are the gravitational and electrostatic escape parameters, respectively. The value for ${\phi }_{{\rm{e}}}({r}_{\mathrm{exo}})$ is obtained iteratively by imposing the condition of zero electric current through the upper boundary. As electrons are much lighter than ions, they escape more readily and drag ions along. The ambipolar electric field therefore has the effect of accelerating ion escape (decreasing X_s for positive q_s) while slowing down electron escape (increasing X_s for electrons i.e., q_s < 0) until the zero electric current condition is satisfied. Note that our upper boundary conditions, as indicated, also account for stellar gravity (i.e., "Roche lobe effect") although this is negligible for WASP-18b.

It is interesting to explore the effect of ambipolar electric fields on X_s, even though in this case the effect does not lead to significantly enhanced mass loss. The direct effect of electric fields is of course negligible for neutrals (although collisions with ions can impart some acceleration on them) and the abundance of He⁺ near the exobase is negligible. Thus, we focus on H and H⁺. As stated above, ${X}_{{\rm{H}}}({r}_{\mathrm{exo}})$ = 70.9. Our calculation shows that ${X}_{{{\rm{H}}}^{+}{\rm{e}}}$(r_exo) = 37.4 and thus ${X}_{{{\rm{H}}}^{+}}({r}_{\mathrm{exo}})$ = 33.5 for H⁺. A Jeans parameter X_s of 33.5, however, is still high and the bulk outflow velocity at the exobase is only 4 × 10⁻¹⁰ m s⁻¹, giving a negligible mass loss rate of 2.8 × 10⁻⁹ kg s⁻¹ (8.8 × 10⁷ kg Gyr⁻¹ or 4.6 × 10⁻²⁰ M_J Gyr⁻¹). Thus mass loss from WASP-18b is not energy limited. Energy-limited mass loss would be inversely proportional to planet mass, whereas Jeans escape is instead proportional to $(1+{X}_{{\rm{s}}}){e}^{-{X}_{{\rm{s}}}}$ and thus the rate decreases very rapidly with increasing X_s, or planet mass.

To strengthen this result, we calculate the exobase temperature required for the atmosphere to be in the hydrodynamic escape regime and thus dramatically increase the escape rates. The approximate temperature can be directly inferred from the Jeans escape parameter¹⁸ considering that hydrodynamic escape requires the Jeans escape parameter to be below two. Figure 6 shows that the upper atmosphere has a temperature of about 1.3 × 10⁴ K. To reach a Jeans escape parameter of about two, the temperature should therefore be of the order of 2.2 × 10⁵ K, which is impossible to reach through heating from the stellar EUV flux. This further indicates that even much larger EUV fluxes would not significantly affect the atmospheric escape of WASP-18b.

The presence of an enhanced atmospheric metallicity for WASP-18b (Sheppard et al. 2017) would not significantly affect our results. A higher metallicity would produce an even more compact atmosphere, which is less likely to escape. Given that our pure H/He atmosphere presents a very low escape rate, the high metallicity derived by Sheppard et al. (2017) is most likely to be primordial and not caused by selective erosion (i.e., lighter elements are more likely to escape compared to heavier ones), even considering that the star may have been more active in the past.