Atmospheric Escape From TOI-700 d: Venus versus Earth Analogs

We simulate the stellar wind of TOI-700 using the Alfvén Wave Solar Model (AWSoM) within the Space Weather Modeling Framework (Tóth et al. 2012), a data-driven global magnetohydrodynamic (MHD) model initially developed for simulating the solar atmosphere and solar wind (van der Holst et al. 2014). AWSoM has been subjected to extensive validation and has accurately reproduced both the solar corona environment (e.g., van der Holst et al. 2014) and dynamics during coronal mass ejection (CME) eruptions (e.g., Jin et al. 2016, 2017). The model was eventually adapted to simulate stellar winds of several stars (e.g., Cohen et al. 2014; Dong et al. 2018b).

To adapt the AWSoM for modeling the TOI-700 stellar wind, we utilize the rotational period (54 days), radius (0.420 R_⊙), and mass (0.416 M_⊙) of the star from the estimates determined via TESS (Gilbert et al. 2020). Due to the lack of direct surface magnetic field observations for TOI-700, we employ a solar magnetogram under the solar minimum condition, viz., Global Oscillation Network Group magnetogram of Carrington Rotation 2077 (Jin et al. 2012); the choice of magnetogram may be justified by the relatively low levels of stellar activity associated with TOI-700 (Gilbert et al. 2020). We scale the mean magnetic flux density to 36 G, which is estimated based on the X-ray luminosity of the star (∼2.4 × 10²⁷ erg, as per Gilbert et al. 2020) as well as the solar X-ray luminosity (Judge et al. 2003).

The steady-state stellar wind solution is shown in Figure 1. The stellar wind speeds at the orbits of TOI-700 planets are comparable to the solar wind speed at 1 au (∼200–600 km s⁻¹). However, because of the much closer distances to the star, the stellar wind density and dynamic pressure are higher, as seen from panels (b) and (c) of Figure 1. The critical surface, defined as the region where stellar wind speed is equal to the fast magnetosonic speed, is also computed and shown in Figure 1. As with our solar system, all the planets in the TOI-700 system are outside this critical surface, i.e., the stellar wind environment of the planets is always "superfast."

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To simulate the stellar wind interaction with TOI-700 d, and study resultant atmospheric ion losses, we employ the BATS-R-US multifluid MHD model that was originally developed for solar system planets such as Earth, Venus, and Mars (Tóth et al. 2012; Dong et al. 2014, 2015, 2018a). The MHD model solves separate continuity, momentum, and pressure equations for each ion species. In addition, the electron pressure equation is also evolved separately in our model calculations. The whole upper atmosphere is included in the simulation domain, implying therefore that all types of elastic and inelastic collisions and the associated heating and cooling processes are explicitly present in the MHD equations as source and sink terms.

For the Venus-like case, we suppose that the 1 bar atmosphere of TOI-700 d is dominated by CO₂ analogous to modern Venus. For the Earth-like case, the 1 bar atmosphere of TOI-700 d is akin to that of the modern Earth, i.e., primarily N₂ and O₂. In this study, we utilize and rescale the 1D averaged upper atmospheric profiles from the thermosphere general circulation models of Venus and Earth (Bougher et al. 2008). In the case of Venus/Venus-like planets, the dominant neutral species transitions from molecular CO₂ to atomic O with increasing altitude in the thermosphere. On the other hand, the entire thermosphere is dominated by atomic oxygen for Earth/Earth-like planets (Mendillo et al. 2018).

It is noteworthy that in addition to photoelectron heating, Joule heating (or Ohmic heating) is also incorporated in our calculations due to the presence of electron–ion (ν_ei) and electron–neutral (ν_en) elastic collisions within the electron pressure equation; see Tóth et al. (2012) for more details. Photochemical reactions such as charge exchange, electron impact ionization, and electron recombination inherently comprise a part of the inelastic collision terms, based on the rate coefficients delineated in Schunk & Nagy (2009). In the case of photoionization, we calculate the rescaled ionization frequencies at TOI-700 d, based on the X-ray luminosity from Gilbert et al. (2020) and the empirical extreme ultraviolet (EUV) scaling from Sanz-Forcada et al. (2011). Note that the optical depth of the upper atmosphere is determined by employing the numerical formula in Smith & Smith (1972).

By utilizing a five-species multifluid MHD model, we solve e⁻, H⁺, O⁺, ${{\rm{O}}}_{2}^{+}$, and ${\mathrm{CO}}_{2}^{+}$ for the Venus-like case, and the dominant three fluid species (e⁻, H⁺, and O⁺) for the Earth-like case, with these species specified by the dominant ionospheric components (Bougher et al. 2008); we deliberately adopt a Venus-like atmospheric composition given its importance in exoplanetary science (Kane et al. 2019). At the model inner boundary, the species densities satisfy the photochemical equilibrium conditions (Schunk & Nagy 2009). Absorbing boundary conditions are applied to both the velocity and magnetic fields. Due to collisions, both ions and electrons have roughly the same temperature as the neutrals at the inner boundary.

We employ a nonuniform spherical grid for modeling all cases. The angular (i.e., horizontal) resolution is chosen to be 3° × 3°. However, as the upper atmosphere of a Venus-like planet is cooler than that of a Earth-like planet due to the efficient CO₂ cooling (Bougher et al. 2008), the scale height associated with the former's atmosphere is correspondingly smaller than the latter; therefore, we adopt the smallest radial resolution of 5 km and 10 km for Venus and Earth analogs, respectively, to span the whole upper atmospheric region. The code is run in the planet–star–orbital coordinate system, where the x-axis is directed from the planet toward the star, the z-axis is perpendicular to the planet's orbital plane, and the y-axis completes the right-hand system. The simulation domain is set by −90 ≤ x/R_p ≤ 45, −90 ≤ y/R_p ≤ 90, and −90 ≤ z/R_p ≤ 90, where R_p denotes the planetary radius.

We also analyze the role of the planetary magnetic field for the Earth-like case by turning the global dipole magnetic field on and off; to facilitate direct comparison, the dipole moment is taken to be the same as modern Earth. Table 1 summarizes the six cases along with the associated atmospheric ion escape rates. We select two locations in the orbit, namely, P_min and P_max, because they ought to yield the minimum and maximum ion escape rates (e.g., Lammer et al. 2008; Dong et al. 2017b; Persson et al. 2020), thus providing the range of values associated with this system. Furthermore, we hold the orientation of the planetary dipole fixed with respect to the interplanetary magnetic field (IMF), as varying the orientation causes the escape rates to change only by a modest factor of ≲2 (Dong et al. 2019). As with previous publications, a fiducial 1 bar atmosphere is chosen for both Venus- and Earth-like cases (e.g., Dong et al. 2017a, 2018b).

The modeled characteristics of the stellar wind were already presented in Section 2.1. In addition, there is another key stellar characteristic that we numerically compute. Based on our MHD numerical simulations, we estimate a stellar mass-loss rate of ${\dot{M}}_{\star }$ ≈ $1.3\times {10}^{-14}$ ${M}_{\odot }\,{\mathrm{yr}}^{-1}$ for TOI-700. It is instructive to compare this result with the stellar mass-loss rate calculated using the semi-analytical approach prescribed in Johnstone et al. (2015):

$$\begin{eqnarray}\begin{array}{rcl}{\dot{M}}_{\star } & \approx & 1.4\times {10}^{-14}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\ {\left(\displaystyle \frac{{{\rm{\Omega }}}_{\star }}{{{\rm{\Omega }}}_{\odot }}\right)}^{1.33}\\ & & \times \ {\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{-3.36}{\left(\displaystyle \frac{{R}_{\star }}{{R}_{\odot }}\right)}^{2},\end{array}\end{eqnarray} \tag{ 1 }$$

where R_⋆, Ω_⋆, and M_⋆ are the radius, rotation rate, and mass of the star, respectively. For TOI-700, we adopt the stellar parameters from Gilbert et al. (2020, Table 1) and substitute them into (1), there obtaining ${\dot{M}}_{\star }$ ≈ $1.9\times {10}^{-14}$ ${M}_{\odot }\,{\mathrm{yr}}^{-1}$. This estimate exhibits good agreement with the prior numerical prediction for ${\dot{M}}_{\star }$ because the two results differ by a factor of <1.5.

The calculation of the stellar wind parameters for TOI-700 enables us to address another crucial issue: the detection of the putative magnetic field of TOI-700 d via radio emission generated by the cyclotron maser instability (Zarka 2007). The frequency ν at which maximal emission occurs is

$$\begin{eqnarray}&&\nu \approx \displaystyle \frac{{{eB}}_{p}}{2\pi {m}_{e}c}\approx 2.8\,\mathrm{MHz}\ \left(\displaystyle \frac{{B}_{p}}{1\,{\rm{G}}}\right),\end{eqnarray} \tag{ 2 }$$

where B_p is the planetary magnetic field, while e and m_e denote the electron charge and mass, respectively. As the expected frequency is <10 MHz, an ultra-low-frequency radio telescope in space or the lunar surface is necessary to detect such emission.

To determine whether the radio emission is detectable, the radio flux density (S) is estimated as

$$\begin{eqnarray}&&S=\displaystyle \frac{{P}_{R}}{4\pi {\rm{\Delta }}\nu {d}^{2}},\end{eqnarray} \tag{ 3 }$$

where d is the Earth-to-planet distance, P_R signifies the planetary radio power, and Δν ≈ ν/2 (Zarka 2007). As the radio emission is potentially dictated by the stellar wind's magnetic power incident on the planet, we employ Vidotto & Donati (2017) to determine P_R:

$$\begin{eqnarray}&&{P}_{R}\sim 2\times {10}^{-3}\ \left(\displaystyle \frac{\pi {B}_{\mathrm{sw}}^{2}{R}_{M}^{2}{v}_{\mathrm{sw}}}{4\pi }\right),\end{eqnarray} \tag{ 4 }$$

where B_sw denotes the IMF, v_sw is the velocity of the stellar wind, and R_M represents the magnetospheric radius given by

$$\begin{eqnarray}&&{R}_{M}={R}_{p}{\left(\displaystyle \frac{{B}_{p}^{2}}{4\pi {P}_{\mathrm{sw}}}\right)}^{1/6},\end{eqnarray} \tag{ 5 }$$

where P_sw ≈ m_pn_swv_sw² is the dynamical pressure, with n_sw representing the stellar wind density and m_p is the proton mass. After making use of (3)–(5) along with Table 1 and R_p ≈ 1.2 R_⊕, we are in a position to calculate S. For the maximum pressure (P_max) scenario in Table 1, after simplification we have

$$\begin{eqnarray}&&S\approx 1.4\times {10}^{-7}\,\mathrm{Jy}{\left(\displaystyle \frac{{B}_{p}}{1{\rm{G}}}\right)}^{-1/3}.\end{eqnarray} \tag{ 6 }$$

While the estimate is smaller than the predicted radio emission of some nearby exoplanets by orders of magnitude, two essential factors must be noted. As seen from (6), the radio emission could increase by an order of magnitude if B_p is a few mG. More importantly, during large CMEs (e.g., Jakosky et al. 2015; Luhmann et al. 2017), S would increase by a couple of orders of magnitude (Dong et al. 2018b), thereby potentially attaining values as high as ∼0.1 mJy for weak magnetic fields.

Next, we direct our attention to the central issue of atmospheric ion escape facilitated by the stellar wind in conjunction with the stellar radiation.

Figure 2 depicts the pressure balance arising from the stellar wind–planet interactions. The first and second columns present the unmagnetized Venus-like and magnetized Earth-like cases, respectively. As a result of the supermagnetosonic nature of the stellar wind at TOI-700 d, bow shocks are formed ahead of this planet, regardless of the inclusion of the planetary magnetic field. Compared to the unmagnetized Venus analog, the presence of the global magnetic field results in a larger interaction cross-section with the stellar wind for the magnetized Earth-like case. An inspection of the bottom panels of Figure 2 reveals that the planetary magnetosphere (or induced magnetosphere for unmagnetized cases) is compressed, and the total pressure becomes intensified due to the stronger stellar wind pressure.

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The salient results concerning atmospheric escape are depicted in Figure 3, which shows the calculated oxygen ion density with the associated magnetic field lines in the meridional plane for all six cases. This figure yields some general conclusions. To begin with, as seen from Table 1, we note that O⁺ constitutes the dominant ion species undergoing escape for all configurations considered here. The atmospheric oxygen escape rates vary from ${ \mathcal O }({10}^{26})$ s⁻¹ to ${ \mathcal O }({10}^{28})$ s⁻¹, indicating that they are higher by a few orders of magnitude than the typical escape rates of ${ \mathcal O }({10}^{25}$) s⁻¹ for the terrestrial planets in our solar system (e.g., Dong et al. 2017b, 2018c).

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Let us first compare the unmagnetized cases, namely, the first and second columns. We find that the Earth-like case exhibits a stronger and broader flux of escaping O⁺ compared to the Venus-like case; this result is also consistent with the atmospheric ion escape rates shown in Table 1. The reason chiefly stems from the fact that the upper atmosphere of a Venus analog is cooler than its Earth-like counterpart due to the efficient CO₂ cooling caused by 15 μm emission (Bougher et al. 2008). Consequently, the exobase of a Venus-like planet is situated lower than that of an Earth-analog, thereby making the former more tightly confined. In other words, the extent of the atmospheric reservoir that is susceptible to erosion by stellar wind is smaller for the Venus analog as compared to an Earth-like atmosphere.

Now, let us hold the atmospheric composition fixed and vary the magnetic field, i.e., we compare the second column (unmagnetized Earth-analog) with the third column (magnetized Earth-analog). Despite the fact that the magnetized cases exhibit a larger interaction cross-section with the stellar wind (also refer to Figure 2), the planetary magnetic field exerts a net shielding effect for the configurations studied herein for TOI-700 d. The presence of the global magnetic field reduces the atmospheric loss rate by roughly one order of magnitude relative to the unmagnetized case (see Table 1). Hence, of the three different scenarios considered in this work, unmagnetized Earth-like worlds are characterized by the highest atmospheric ion escape rates.