Hot Super-Earths with Hydrogen Atmospheres: A Model Explaining Their Paradoxical Existence

Using Figure 1 as reference, the total centrifugal force, F_Total,1, at an arbitrary latitude is

$$\begin{eqnarray}&&{F}_{\mathrm{Total},1}={\omega }^{2}{R}_{P}\cos \left(\phi \right)\times \delta m,\end{eqnarray} \tag{ 1 }$$

where δm is an arbitrary unit of mass and ω is the angular velocity of the spin and orbit of 55 Cancri e, which is calculated using the period (P_P) shown in Table 1. However, if we want to find the transverse component, we need to multiply this by $\sin (\phi )$ to get

$$\begin{eqnarray}&&{F}_{\mathrm{spin}}={F}_{t,1}=\displaystyle \frac{1}{2}{\omega }^{2}{R}_{P}\sin \left(2\phi \right)\times \delta m.\end{eqnarray} \tag{ 2 }$$

This equation proposes that the maximum transverse component is at a latitude of 45°. At the limits, the centrifugal force from the planet's own spin is zero. This is because as $\phi \to 0^\circ$, the transverse component is nonexistent, and when $\phi \to 90^\circ$, the total centrifugal force (F_Total,1) vanishes.

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In order to calculate this force, we can also use Figure 1 as a reference with the only difference being that the semimajor axis (a_P) also needs to be considered:

$$\begin{eqnarray}&&{F}_{\mathrm{Total},2}={\omega }^{2}\left({a}_{P}+{R}_{P}\cos \left(\phi \right)\right)\times \delta m.\end{eqnarray} \tag{ 3 }$$

Similarly, if the transverse component is to be found, Equation (3) needs to be multiplied by $\sin (\phi )$, which gives

$$\begin{eqnarray}&&{F}_{\mathrm{orb}}={F}_{t,2}={\omega }^{2}\left({a}_{P}+{R}_{P}\cos \left(\phi \right)\right)\sin \left(\phi \right)\times \delta m.\end{eqnarray} \tag{ 4 }$$

Assuming a_P ≫ R_P, the maximum force from the transverse component of the orbital centrifugal force occurs at ϕ = 90° and the minimum is at ϕ = 0°.

One important force that is often ignored is the star's gravity on the atmosphere. This force acts against the centrifugal forces and will pull the atmospheric gases toward the dayside. Figure 2 shows the different components of the force from which Equation (5) can be intuitively derived:

$$\begin{eqnarray}&&{F}_{\mathrm{Total},3}=\displaystyle \frac{{{GM}}_{* }\delta m}{{\left({a}_{P}+{R}_{P}\cos \left(\phi \right)\right)}^{2}}.\end{eqnarray} \tag{ 5 }$$

From this, the transverse component can be found:

$$\begin{eqnarray}&&{F}_{{\rm{g}}}={F}_{t,3}=\displaystyle \frac{{{GM}}_{* }\delta m\sin \left(\phi \right)}{{\left({a}_{P}+{R}_{P}\cos \left(\phi \right)\right)}^{2}}.\end{eqnarray} \tag{ 6 }$$

According to Equation (6), the maximum transverse force occurs at ϕ = 90° and the minimum at ϕ = 0°. As for the planet's own gravity, because this force would only act radially, we can ignore it.

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The final force considered here is the thermal gas force. Unsurprisingly, this is the hardest force to model, as many effects such as those related to thermodynamics, viscous forces, or turbulence should be considered. In order to avoid this complexity and analyze the situation analytically, we can focus on the maximum possible gas force, which occurs when there is a free expansion. Regarding temperature gradients, being on the nightside, the temperatures should be relatively constant, which is consistent with the phase curve data of 55 Cancri e (Demory et al. 2016). In our simple model, we will therefore assume a constant nightside temperature. Furthermore, the high nightside temperatures inferred by Demory et al. (2016) mean that intermolecular forces become exceedingly small so it is acceptable to treat the gas ideally. With regard to the incoming thermal radiation, being tidally locked, these rays should be partially blocked out. This can be shown by using an altered version of Equation (1) of Caldas et al. (2019), which gives the maximum extent that radiation from the host star can interact with the atmosphere on the nightside:

$$\begin{eqnarray}&&\phi =\arcsin \left(\displaystyle \frac{{R}_{P}}{{R}_{P}+z}\right).\end{eqnarray} \tag{ 7 }$$

A visual representation of this can be seen in Figure 1 of Caldas et al. (2019). In this equation, z is the height of the atmosphere, which is of the order $\sim {k}_{{\rm{B}}}{T}_{{\rm{night}}}/\bar{\mu }{g}_{P},$ where k_B is Boltzmann's constant, $\bar{\mu }$ is the mean molecular weight of the atmosphere, and g_P is the gravitational acceleration of the planet. To understand the amount of XUV irradiation that comes in contact with the nightside, we first need to build our atmospheric model and then apply Equation (7). It is important to note that Figure 3 does not represent the configuration of 55 Cancri e—it just depicts the case where the gas force would be maximized. Instead, Equation (7) should be used in conjunction with Figure 10 in order to visualize its applicability. We will therefore come back to this equation later and instead focus on modeling the gas force. We begin with the ideal gas equation,

$$\begin{eqnarray}&&P=\displaystyle \frac{\rho }{\bar{\mu }}{k}_{{\rm{B}}}{T}_{\mathrm{night}},\end{eqnarray} \tag{ 8a }$$

where ρ is the density of the atmosphere. To find the acceleration (a), we multiply by the cross-sectional area (A) and divide by the encased atmospheric mass (M). However, mass is volume multiplied by density, so the formula can be simplified further:

$$\begin{eqnarray}&&a=\displaystyle \frac{{k}_{{\rm{B}}}{T}_{\mathrm{night}}}{\bar{\mu }}\displaystyle \frac{A}{V}.\end{eqnarray} \tag{ 8b }$$

As shown in Figure 3, the area A is the green section, which is approximately $A\approx 2\pi {R}_{P}z\sin \phi$, and the volume is $V\approx 2\pi {R}_{P}^{2}z\left(1-\cos \phi \right)$,

$$\begin{eqnarray}&&a=\displaystyle \frac{{k}_{{\rm{B}}}{T}_{\mathrm{night}}}{\bar{\mu }{R}_{P}}\cot \left(\displaystyle \frac{\phi }{2}\right).\end{eqnarray} \tag{ 8c }$$

Therefore, the gas force is

$$\begin{eqnarray}&&{F}_{\mathrm{gas}}={F}_{t,4}=\displaystyle \frac{{k}_{{\rm{B}}}{T}_{\mathrm{night}}}{\bar{\mu }{R}_{P}}\cot \left(\displaystyle \frac{\phi }{2}\right)\times \delta m.\end{eqnarray} \tag{ 8d }$$

Equation 8(d) assumes that the atmosphere is expanding into a vacuum, which is a strong simplification. Because of this, it overestimates the gas force as a realistic atmosphere is continuous and does not have a discrete end. However, if we show that the centrifugal and gravitational forces can balance out the maximum possible gas force, then it follows that for more realistic situations, the condition would also hold.

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We can first begin with a primordial atmosphere that has a mean molecular mass of $\bar{\mu }\approx 2.3$ amu if there is >99% H₂ (${\mu }_{{{\rm{H}}}_{2}}\approx 2$ amu) and <1% volcanic gases (μ_other ≈ 50 amu). Because the atmosphere is dominated by hydrogen, we can ignore the chemical reactions with the volcanic gases and the subsequent products. If we combine Equations (2), (4), (6), and 8(d) and divide by the arbitrary mass unit δm, we get the total acceleration as a function of the latitude, which is shown in Figure 4. To calculate the kinetic energy and momentum of the gases, we follow a simple algorithm that considers the acceleration at each latitude for each chemical species. The results are shown in Figures 5 and 6.

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The graphs show that when the atmosphere is dominated by hydrogen, there is a positive force which causes the gases to advect toward the dayside where they would be dissociated by the incoming XUV irradiation. An interesting outcome is that because all the gases are accelerated equally, the heavier volcanic and mineral species have greater absolute energies because of their larger masses. Our calculations show that for a primordial atmosphere, the gas pressure would overcome the tidal and centrifugal forces so gas would be able to flow approximately freely into the dayside. In other words, atmospheric destruction would be balanced out by the mass flux from the nightside. The balanced phase would take place until the loss of hydrogen, and active volcanism would raise the mean molecular weight of the atmosphere to a level where tidal and centrifugal forces became important. Our model predicts that for a planet like 55 Cancri e, the critical mean molecular weight is at $\bar{\mu }\approx 35$ amu (∼70% volcanic gases and ∼30% H₂), beyond which tidal forces dominate over the maximum gas force.

Once the planet reaches the diffusive phase, the atmospheric dynamics change. To demonstrate this phenomenon, we assume that the nightside atmosphere is ∼10% hydrogen and ∼90% volcanic and mineral species, which gives $\bar{\mu }\approx 45$ amu. Clearly, a full chemical analysis would be required to account for the interactions between carbon and hydrogen, which have a tendency to react. However, as a proof of concept, we will work with these values. Our results for the acceleration, energy, and momentum are shown in Figures 7, 8, and 9 respectively.

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For an atmosphere that extends to ϕ ≈ 80°, the volcanic species would need ∼4k_BT_night to escape while the H₂ molecules would need ∼ 0.1k_BT_night. It is important to note that if the atmospheric viscosity, friction, turbulence, and a less extreme gas force were used, then the required escape energies would be greater. Notwithstanding these, due to the Maxwell–Boltzmann distribution of velocities, it would be easier for the H₂ molecules than the heavier species to escape to the dayside. However, because the volcanic and mineral gases are more hydrodynamically stable, due to requiring more energy to escape, the H₂ molecules would not be able to flow unobstructed into the dayside. Instead, the hydrogen gas would need to diffuse through the atmosphere in a similar fashion to how hydrogen molecules on Earth can only escape when they reach the exosphere. This diffusion will slow down the H₂ loss considerably which can be shown by applying Ficks first law of diffusion:

$$\begin{eqnarray}&&\displaystyle \frac{{dM}}{{dt}}=\displaystyle \frac{{\rm{\Delta }}{\rho }_{{{\rm{H}}}_{2}}\cdot A\cdot D}{L},\end{eqnarray} \tag{ 9 }$$

where ${\rm{\Delta }}{\rho }_{{{\rm{H}}}_{2}}$ is the hydrogen density contrast between both hemispheres, A is the cross-sectional area of the atmosphere, D is the diffusion constant, and L is the transverse length scale of the diffusive motions. Calculating the diffusive flux is a complex dynamic problem but using Equation (9) with a few "guesstimates" for the parameters will show how in principle the mass flux would be greatly suppressed. Using ${\rm{\Delta }}{\rho }_{{{\rm{H}}}_{2}}\sim 1$ kg m⁻³ (unimportant because $D\propto {\rho }_{{H}_{2}}^{-1}$), A ∼ 2πR_Pz, L ∼ R_P, and D ∼ 10⁻⁵ m² s⁻¹ (calculated using the Chapman–Enskog theory) gives ${dM}/{dt}\sim 1$ kg s⁻¹. The uncertainty in this value could be of several orders of magnitude especially as the diffusion constant can vary greatly depending on the turbulence present, but the important point to take from this calculation is that diffusive transport is slower than free expansion. Notwithstanding this, due to there being more hydrogen molecules traversing into the dayside than volcanic and mineral species, the terminator region would become hydrogen rich as illustrated in Figure 10. The hydrogen gas would then be removed from the planet due to XUV irradiation (e.g., Erkaev et al. 2007; Ehrenreich & Désert 2011; Lammer et al. 2013; Owen & Wu 2013, 2017; Jin et al. 2014; Kubyshkina et al. 2018a, 2018b; Locci et al. 2019).

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This is consistent with spectroscopic studies of the terminator, which have found evidence of a hydrogen-rich envelope on 55 Cancri e (e.g., Tsiaras et al. 2016; Esteves et al. 2017) despite it being old and hot. Furthermore, this is also compatible with observations that were unable to detect significant hydrogen destruction in the exosphere of 55 Cancri e (Ehrenreich et al. 2012; Bourrier et al. 2018b).

Now we can come back to Equation (7), which describes the maximum latitude within which stellar light can interact with the nightside's atmosphere. In the diffusive phase, the mean molecular mass will be large (≳35 amu) so the atmospheric scale height will be small. Plugging reasonable values into Equation (7) shows that latitudes ≲85° would be safe from XUV irradiation. This is consistent with our previous assumption that for an atmosphere that extends to ϕ ≈ 80°, thermal fluctuations can be ignored.

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One important aspect that has been readily ignored in previous studies of exoplanet evaporation is the migration induced due to the conservation of angular momentum; as an exoplanet loses mass, it must migrate outwards. However, due to the Roche limit (Aggarwal & Oberbeck 1974), there is a bound to how close an exoplanet could be to its host star in the past:

$$\begin{eqnarray}&&{a}_{\mathrm{RL}}\approx \mu {R}_{* }{\left(\displaystyle \frac{{\rho }_{* }}{{\rho }_{0}}\right)}^{1/3}.\end{eqnarray} \tag{ 16 }$$

Equation (16) is the Roche limit where μ is a constant of value 1.957, R_* is the radius of the host star (m), ρ_* is the density of the host star (kg m⁻³), and ρ₀ is the initial density of the planet (kg m⁻³; Aggarwal & Oberbeck 1974). Therefore, combining the conservation of angular momentum (Equation (13)) with the Roche limit (Equation (16)), one can arrive at the theoretically maximum initial mass that a super-Earth could have had:

$$\begin{eqnarray}&&{a}_{0}\gt {a}_{\mathrm{RL}}.\end{eqnarray} \tag{ 17a }$$

Substituting in Equations (13) and (16),

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{M}_{P}}{{M}_{0}}\right)}^{7/4}{a}_{P}\gt \mu {R}_{* }{\left(\displaystyle \frac{{\rho }_{* }}{{\rho }_{0}}\right)}^{1/3}.\end{eqnarray} \tag{ 17b }$$

For hot exoplanets with large hydrogen envelopes ρ₀ ∼ ρ_*,

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{M}_{P}}{{M}_{0}}\right)}^{7/4}{a}_{P}\gtrsim \mu {R}_{* }.\end{eqnarray} \tag{ 17c }$$

Using trivial algebra,

$$\begin{eqnarray}&&{M}_{0}\lesssim \beta \cdot {M}_{P}\cdot {a}_{P}^{4/7},\end{eqnarray} \tag{ 17d }$$

where $\beta ={\left(\mu {R}_{* }\right)}^{-4/7}$. By using R_* ≈ 6.8 × 10⁸ m (Crida et al. 2018) and μ ≈ 1.957 (Aggarwal & Oberbeck 1974) we calculate the constant to be β ≈ 6 × 10⁻⁶. Substituting this into Equation 17(d) gives the formula for the maximum initial mass physically possible:

$$\begin{eqnarray}&&{M}_{0}\lesssim 6\times {10}^{-6}\cdot {M}_{P}\cdot {a}_{P}^{4/7}.\end{eqnarray} \tag{ 18 }$$

To reach this conclusion, we made several assumptions. In the following, we discuss their implications, and if and how they limit the use of our model to super-Earths with high temperatures orbiting G-, K-, M-type stars:

• 1.  
  In this paper, we assume that 55 Cancri e has an atmosphere with a considerable hydrogen component.  The geology of 55 Cancri e is an active subject of dispute although more precise mass and radius measurements from Crida et al. (2018) have lowered the degeneracy considerably. Using the geological models from Zeng & Sasselov (2013) and Zeng et al. (2016) and considering the limits permitted by the uncertainty in Crida et al. (2018), it can be shown that the mass and radius of 55 Cancri e are not consistent with a denuded terrestrial planet even if it is coreless. However, other studies that measured the mass and radius of 55 Cancri e (e.g., Bourrier et al. 2018a) claim values that could be consistent with a coreless terrestrial planet, although they state it is unlikely. In addition, having no core introduces new problems, such as the planet lacking a magnetic field which negates the possibility of magnetic planet–star interactions (e.g., Winn et al. 2011) that could explain the observed ∼100 ppm modulation in the optical (Winn et al. 2011; Dragomir et al. 2014) and infrared (Tamburo et al. 2018) phase curves. Other proposed models for 55 Cancri e include the possibility of it being an icy planet (Zeng & Sasselov 2013; Zeng et al. 2016) or having a carbon-rich composition (Madhusudhan et al. 2012; Miozzi et al. 2018). We understand that the situation concerning 55 Cancri e is complex as there is limited observational data, which may appear contradictory in some occasions. Nonetheless, here we propose a model that could potentially explain how some hot super-Earths can host atmospheres with large hydrogen components for timescales comparable to the life span of their host stars.
• 2.  
  We assumed hydrogen does not chemically interact with the volcanic molecules in the diffusive phase.  We would expect some hydrogen to react with other species and become locked up, therefore resulting in a diminished XUV-induced atmospheric erosion. Consequently, our model overestimates the amount of available hydrogen, which implies that the presence of an atmosphere on 55 Cancri e is even more problematic in the absence of a countermechanism.
• 3.  
  For Equations (12) and (14), we assumed that the host star's X-ray and UV flux remain constant with time.  This approximation is appropriate for exoplanets orbiting G-type stars that become tidally locked within their first ∼50 Myr because X-ray and UV radiation is somewhat uniform. However, beyond this time, our model gives an overestimation for the minimum mass required for a super-Earth to have a molecular hydrogen atmosphere. This, however, can easily be resolved by implementing the time dependence accounted for in Penz et al. (2008) and Sanz-Forcada et al. (2011) and integrating with respect to time.
• 4.  
  The dayside has no atmosphere so it is modeled as a vacuum (Equations 8(d)).  The dayside is certainly not a vacuum; it instead might host a low-altitude atmosphere composed of volcanic and vaporized surface materials (e.g., Schaefer & Fegley 2009; Schaefer et al. 2012). We assumed a vacuum in order to consider the maximum possible gas force and whether it could be balanced by the tidal and centrifugal forces. Because we were able to balance out the forces even in the extreme case, we argue that the tidal and centrifugal forces can also counteract the gas force for more realistic atmospheres.
• 5.  
  In Equation (13), we assume that ∼50% of the angular momentum from the mass lost is conserved.  The true amount of angular momentum conserved is not easily calculated and would most probably require a fully hydrodynamic treatment. For instance, if 0% of all angular momentum were conserved, then according to Verbunt (1993), the final orbital distance would be almost identical to the initial one. Therefore, the initial mass of 55 Cancri e would be set by the amount of XUV it received during its lifetime. Locci et al. (2019) performed a backwards reconstruction of 55 Cancri e using this assumption and arrived at an initial mass of ∼110M_⊕. Conversely, if 100% of all angular momentum were conserved, then the exponent in Equation (13) would change from $7/4\Rightarrow 2$. This change would decrease the maximum initial mass (Equation (18)) and increase the minimum initial mass required for 55 Cancri e to have a hydrogen atmosphere (Equation (15)). In other words, when more momentum is conserved, the required initial conditions have to be more fine-tuned in order for our proposed mechanism to take effect. We assumed that ∼50% of the angular momentum was conserved as it is the middle point.
• 6.  
  We did not account for the effect of mass lost on the planetary rotation.  This is also a complicated hydrodynamical problem as it depends on how much of the planet's angular momentum is dissipated into the evaporating gases. In addition, other countermechanisms need to be considered, such as how friction caused by a large primordial atmosphere could slow down a planet's rotation. For instance, as an atmosphere is evaporated, it becomes thinner. Thus, while on the one hand angular momentum could be extracted from the interior, on the other hand the loss of gas might result in less atmospheric friction. The balance of these mechanisms requires a full analysis in order to accurately determine the effects of the atmosphere on the planet's rotation. In any case, because we did not account for these effects, we overestimated how long it would take for the planet to become tidally locked, meaning that in reality less strict initial conditions would have been required for the planet to become tidally locked before losing the entirety of its primordial atmosphere.
• 7.  
  When calculating the maximum initial mass (Equation 17), we assumed that the initial density of the exoplanet was equal to its host star's one.  Our assumption is based on the observation that planets with large hydrogen envelopes such as the outer planets in our solar system (except Saturn) have densities very similar to our Sun.
• 8.  
  In the case of 55 Cancri e, we did not account for planet–planet interactions and how they could induce large-scale migration.  Hansen & Zink (2015) point out that in multiplanetary systems where the stellar-induced tidal evolution is coupled to the long-term nonperiodic perturbations caused by the other planets, the inner planet can experience a strong semimajor axial evolution. They argue that the eccentricities and inclinations of 55 Cancri b and c changed in order to compensate for the significant migration experienced by 55 Cancri e. If this is the case, the maximum theoretical mass predicted by Equation (18) may be an underestimate and the true value may be significantly greater, meaning that the initial mass can be constrained less thoroughly. Nevertheless, as shown in Figure 13, a higher maximum initial mass increases the range of permissible initial rotational periods for the tidal-locking timescale to be shorter than the mass-loss timescale. This raises the probability of an exoplanet having the right initial conditions for it to retain hydrogen in its atmosphere.
• 9.  
  In our simple analytic model, we did not account for magnetic fields.  The effects of magnetism on atmospheric erosion is an active area of research where it is generally accepted that the presence of a B-field can slow down atmospheric destruction. For instance, planets are mostly well shielded from coronal mass ejections, even for relatively weak intrinsic fields (e.g., Cohen et al. 2011). While the magnetic field of 55 Cancri e is unknown, using the dynamo scaling law from Christensen & Aubert (2006) with the core radius estimate from Crida et al. (2018) and the radiogenic models from Frank et al. (2014), it can be shown that a B-field of ∼B_Earth or potentially higher is reasonable. At this strength, the ionized hydrogen atoms in the outer layers of the atmosphere would be strongly influenced and mass loss would be mitigated. However, even in the absence of a geomagnetic field, XUV irradiation should result in an ionosphere that would interact with escaping ions. A similar mechanism has been observed on Venus (Zhang et al. 2012). In conjunction, this implies that XUV atmospheric erosion would be more inefficient when a magnetic field is present. Therefore, a larger range of initial rotational periods would be permitted in order for condition 14(a) to hold. Consequently, including a magnetic field would increase the probability of super-Earths and sub-Neptunes having the right conditions for our mechanism to take place.

Fulton et al. (2017) showed that there is a bimodal distribution in exoplanet radii with one peak at ∼1.3R_⊕, the other at ∼2.4R_⊕, and the minimum at ∼1.8R_⊕. They argue that instead of using arbitrary definitions for super-Earths and sub-Neptunes, these should be defined more objectively. Specifically, super-Earths should be defined as having radii between 1 and 1.75R_⊕, while sub-Neptunes should have radii ranging from 1.75 to 3.5R_⊕. One of the most thorough analyses of the bimodal distribution in exoplanet sizes comes from Owen & Wu (2013, 2017). In their model, they argued that XUV-induced mass loss results in the observed ∼1.8R_⊕ minima as Neptune-mass objects closer than ∼0.1 au lose their hydrogen envelopes. As a general rule of thumb, we agree with the conclusions derived in the aforementioned study. Our model just states that in some cases, small and hot exoplanets may be able to retain atmospheres on their nightsides if they have the right configuration. It may be the case that the presence or lack of a hydrogen component in an atmosphere on a hot super-Earth could give information on its initial mass and day length. Constraining the birth conditions of an exoplanet should provide more information on its planetary system's history and potentially improve our overall understanding of exoplanetary statistics.