Analytical modeling of helium absorption signals of isothermal atmospheric escape

Abstract

Atmospheric escape driven by extreme ultraviolet (EUV) radiation is a critical process shaping the evolution of close-in exoplanets. Recent observations have detected helium triplet absorption in numerous (>20) close-in exoplanets, highlighting the importance of understanding upper atmospheric thermo-chemical structure. While super-solar metallicity has been observed in the atmospheres of some close-in exoplanets, the impact of metal species on both atmospheric escape dynamics and observed absorption features remains poorly understood. In this study, we derive a simplified yet accurate formula for the equivalent width of helium absorption in the limit of an isothermal temperature for the upper atmosphere. Our results demonstrate that planets with lower temperature (metal-rich atmosphere) exhibit lower mass-loss rate although the equivalent width of helium triplet absorption remains largely independent of atmospheric temperature (metallicity) because the low temperatures in these atmospheres enhance the fraction of helium in its triplet state. Additionally, we present a hydrodynamic model based on radiation-hydrodynamics simulations that incorporates the effects of metal cooling. Our analytical model can predict the helium triplet equivalent width of the atmosphere of simulations. The analytical model provides a comprehensive framework for understanding how metal cooling in the upper atmosphere influences the thermo-chemical structure and observable helium features of close-in exoplanetary atmospheres, offering valuable insights for interpreting current and future observational data.

Figures (11)

• Figure 1: Upper panel: the real part of the incomplete gamma function $\Gamma(-1/2,-a/b)i$ (black points) with fitting with $f_1(x)$(blue line), $f_2(x)$(orange line), and $f_3(x)$ (green line). Lower panel: deviations from the incomplete gamma function.
• Figure 2: The radial profiles of temperature (top), metastable helium density(upper-middle), and heating/cooling rate(lower-middle), neutral fraction ($n_{\rm neutral}/n_{\rm total}$ bottom) with high-mass planets different metallicity ($M_{\rm p}=0.7{\,{\rm M_{\rm J}}},R_{\rm p}=1.4{\,{\rm R_{\rm J}}}, l=0.05\,\mathrm{au}, F_{\rm EUV} = 5600\mathrm{\, erg/s/cm^2}, Z=0,10~Z_{\odot},30~Z_{\odot}$). The photoionization heating (solid), Mg cooling (dashed), and Ly$\alpha$ cooling (dotted) are shown in the bottom panel.
• Figure 3: Same as Fig. \ref{['fig:profile_HJ']}, but for low-mass planets ($M_{\rm p}=10.2~M_{\oplus},R_{\rm p}=0.25~{\,{\rm R_{\rm J}}}, l=0.05\,\mathrm{au}, F_{\rm EUV} = 5600\mathrm{\, erg/s/cm^2}$).
• Figure 4: Thermospheric balance gas temperatures as a function of metallicity. The temperature from Eq. \ref{['eq:T_th']} (solid curve, orange; Mg cooling, green; OI cooling) and gas temperature at the sonic point from simulations (dots) are shown. The power-law fit of \ref{['eq:Teqfit']} is also shown as a dotted curve.
• Figure 5: Relationship between equivalent width and mass-loss rate. The solid blue curve represents the order-of-magnitude estimate by Zhang_2023c. The solid orange curve represents our 1D analytical model. The results of our 1D simulations with different metallicity ($Z=0,1,10,30\, Z_{\odot}$) are also shown as black dots.