Characterization of Terrestrial Exoplanet Atmospheres through Lyman-alpha Transit Observations

Introduction

     Lyman-alpha transmission spectroscopy has been a powerful tool for observing hydrogen escape from close-in exoplanets. For instance, Ly-α observations of GJ 436b showed a maximum transit depth of 56% – compared to a 0.69% transit depth in optical wavelengths – due to the hydrodynamic escape of hydrogen from the planet (1). Ly-α observations assist in understanding the evolution of such exoplanets, including characterization of the atmosphere.

     To date, the Ly-α transit of terrestrial-sized exoplanets has yielded only non-detections for exoplanets such as Trappist 1b/c (2) and 55 Cn e (3) using the Hubble Space Telescope. These non-detections possibly indicate that terrestrial exoplanets do not have enough hydrogen escape to be observed in Ly-α. Despite these non-detections, Earth’s hydrogen exosphere has been shown to extend out past 38 Earth-radii, and modelling has suggested that an exoplanet with an Earth-like hydrogen exosphere orbiting an M-dwarf star would be observable with future space telescopes (4).

     This work models the Ly-α transit of varying terrestrial exoplanets to diagnose atmospheric composition. For example, how does the Ly-α transit of a desiccated planet like Venus compare to a water-rich planet like Earth? Can we discern these differences from future space telescope observations? To test this, we model the thermal escape of hydrogen from terrestrial exoplanets and compute the associated Ly-α transit depth. Atmospheric parameters in the upper atmosphere, such as the mixing ratio of hydrogen, are varied to analyze the resulting Ly-α transit depth. From this, we examine trends in the transit depth to characterize the atmospheres of terrestrial exoplanets. In addition, we derive key attributes of exoplanets that would be detectable with future space telescopes.

Methods

     In this work, the hydrogen exosphere is modelled using the Chamberlain approach (5). Key input parameters needed are the exobase height, the number density of hydrogen, and the exobase temperature. The exobase height and hydrogen number density are calculated by setting up a 2-component atmosphere at the homopause (the altitude where different species can diffusively separate, situated at 100 km from the surface with a total species number density of 10¹⁹  molecules cm^-3), with a given hydrogen mixing ratio. The species diffuse upward from the homopause until they reach the exobase where the mean free path of the atmosphere is equal to the scale height. We also account for diffusion-limited escape. In a scenario where Jean’s escape is larger than the diffusion-limited escape, the number density of hydrogen is scaled to the diffusion-limited value. The exobase temperature is a free parameter in our model, though observations from the solar system indicate that a CO₂ dominated exobase is cooler than an atomic oxygen-dominated exobase such as Earth’s, due to infrared cooling.

     Once the hydrogen exosphere is modelled, the radiative transfer code from the open-source model Sunbather (6) is used to model the transit. By default, we consider the planet to be mid-transit with an impact factor of 0. Factors affecting the transit depth, such as thermal line broadening are included.

     We consider a wide parameter space of planets between 0.5 to 2 Earth radii, with atmospheric temperatures between 100 to 700 K, and hydrogen mixing ratios between 10^-10 and 1.  For each run of the model, the hydrogen exosphere is computed at several exospheric temperatures between 100 and 1000 K to span the range of exobase temperatures observed in the solar system.

Results

     An example of the number density of hydrogen in the exosphere and the resulting Ly-α transit in shown in Figure 1. In this case, we have modelled an Earth-sized planet with a hydrogen mixing ratio of 10^-6 at the homopause that diffuses through an upper atmosphere of atomic oxygen. The exobase is 228 km from the surface with a hydrogen number density of 3.4 x 10⁴, limited by the diffusion. The exobase is dominated by atomic oxygen, and modelled to be at 1000 K. The Ly-α transit depth around a sun-sized star is 275 ppm, much smaller than observed for close-in Neptunes, but possibly observable with future space telescope technology at a distance where geocoronal contamination is minimal.

   
Figure 1: (a) Hydrogen number density in the exosphere as a function of distance from the planet (b) The Ly-α transit in ppm.

     Figure 2 shows the transit depth of exoplanets with varying masses and hydrogen mixing ratios at the homopause. Here a fixed exobase temperature of 1000 K  was used though the simulations were also run with other exobase temperatures (not shown here, though the Ly-α transit typically increases with increasing exobase temperature due to the hydrogen atoms escaping the exobase more easily). The transit depth shows a strong dependence on hydrogen mixing ratio, with higher hydrogen mixing ratios resulting in a larger transit depth. The mass of the planet also affects the transit depth, with lower mass planets allowing for hydrogen escape more easily than more massive planets.

Figure 2: Ly-α transit depths for exoplanets with varying masses and hydrogen mixing ratios.

We will also present the effects of photoionization, making the hydrogen no longer observable in Ly-α. Lastly, we will test the ability to detect these Ly-α transits with space telescopes.

 

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