Could the Migration of Jupiter Have Accelerated the Atmospheric Evolution of Venus?

As we have learned more about the solar system, it has become clear that the giant planets must have migrated during their formation and evolution, before settling at their current locations. Evidence for that migration abounds in the system's various small body populations. These include the sculpting of the asteroid belt and orbital distribution of the Jovian Trojans (e.g., Morbidelli et al. 2005; Minton & Malhotra 2009, and references therein), and the excitation of the Plutinos and distribution of objects beyond Neptune (e.g., Malhotra 1993, 1995; Hahn & Malhotra 2005; Levison et al. 2008; Lykawka et al. 2009, 2010). Although significant giant planet migration is now firmly established, there remains debate over the extent of that migration, its smoothness or chaoticity, and its timing. Migration models have ranged from the late (∼700 Myr after planet formation) chaotic interactions of the Nice Model (e.g., Gomes et al. 2005; Morbidelli et al. 2005) to the early but dramatic "Grand Tack" migration of Jupiter and Saturn, in which it is suggested that Jupiter might have migrated inward to approach the current orbit of Mars, before tacking outward to reach its current location (e.g., Walsh et al. 2011; Raymond et al. 2014; Nesvorný 2018)—a process that would have had a significant impact on the hydration of the inner solar system (e.g., O'Brien et al. 2014; Raymond & Izidoro 2017). In particular, the predominant version of the Grand Tack model assumes a disk-dominated migration of Jupiter that was completed before the terrestrial planets formed (e.g., Chambers 2014). Alternatively, models of planetesimal-driven migration of the giant planets can occur over much longer timescales, far beyond the dissipation of the disk required by the Grand Tack model (Hahn & Malhotra 1999; Malhotra 2019). Further migration models consider the possibility of non-uniform, stochastic migration (e.g., Morbidelli et al. 2010; Nesvorný et al. 2018), whereas other models consider more sedate, smoother migration (e.g., Lykawka et al. 2009, 2010; Pirani et al. 2019). Additional complications include evidence via isotopic measurements from meteorites that suggest terrestrial protoplanets completed accretion quickly and while the gas was still present (Schiller et al. 2018), and the prevalence of proto-atmospheres for terrestrial planets while still present in the gas phase (Mai et al. 2020). A summary of migration and formation models is provided by Raymond & Morbidelli (2020).

Regardless of the true nature of the giant planet migration, the past few decades have seen extensive investigation of the dynamical evolution of the solar system (e.g., Laskar 1988; Duncan & Quinn 1993; Levison & Agnor 2003; Batygin & Laughlin 2008; Brasser et al. 2009; Zeebe 2015). In addition to investigations of an early solar system, dynamical simulations by Laskar (1988) considered the current stability of the solar system, finding that the orbital eccentricities of both Venus and Earth remain largely circular over several-million-year timescales, based on their present orbital parameters. As described previously, several theories regarding the formation and migration of the giant planets, Jupiter and Saturn, suggest that there may have been periods during which they were significantly closer to the Sun, including the Nice model (Gomes et al. 2005; Tsiganis et al. 2005) and the complementary Grand Tack model (Walsh et al. 2011, 2012). Given the aforementioned variety in migration theories discussed in the literature, the effects of that migration on the inner solar system are now the subject of investigation. It has been suggested that the dynamical stability of the terrestrial planets may serve as a useful discriminant between the different Jupiter migration models (Nesvorný 2018), with such models considered likely to help explain the relatively high eccentricity of Mercury's orbit (Roig et al. 2016).

The purpose of this work is not to evaluate the relative merits of these and other similar models, but rather to assess the impact of Jupiter's location on the orbital dynamics of the inner planets. However, it is worth noting that our model largely depends on Jupiter's migration occurring after Venus's formation is complete or a proto-Venus that retains eccentricity from formation processes. Numerous models described herein allow for these possibilities, and so the subsequent discussion describes the results of a detailed dynamical simulation and the implications for the orbital eccentricity of Venus.

The dynamical simulations required for this study were conducted as part of a broader investigation of the degree to which Jupiter's semimajor axis and eccentricity drive the time-variability of dynamical behavior in the solar system. A detailed description of the dynamical simulations can be found in Horner et al. (2020), which concentrated on the effect of Jupiter on the Milankovitch cycles of the Earth. Here we provide a brief description of the simulations and the components of the simulation that are utilized in this study.

The full suite of simulations was constructed using N-body integrations calculated using the Hybrid integrator within the Mercury N-body dynamics package (Chambers 1999). The simulations included the eight major planets of the solar system and incorporated the current orbital elements extracted from the Horizons DE431 ephemerides (Folkner et al. 2014). The full suite of simulations covered a large grid of Jupiter semimajor axes (3.2–7.2 au) and eccentricities (0.0–0.4), totaling 159,201 simulations. Each simulation was run for 10⁷ simulation years, with a time step of 1 day to ensure perturbation accuracy, and first-order post-Newtonian relativistic corrections were accounted for (Gilmore & Ross 2008). The output of each simulation was recorded with a time step of 1000 yr. Simulations were halted early if any of the planets became "lost," including collisions with the Sun or each other, or moving beyond a heliocentric distance of 40 au.

Our study of Venus uses 399 of the total simulation set, and includes starting orbital parameters of Jupiter with zero eccentricity in the semimajor axis range a_J = 3.2–7.2 au. We restricted this study to those simulations for which the initial eccentricity of Jupiter's orbit was zero, in order to minimize assumptions regarding the potential migration scenarios for Jupiter and to focus our work on the influence of the giant planet's location alone.

As noted in Section 1, the present orbital eccentricity of Venus is the lowest of all solar system major planets, with a value of 0.006. The results from our simulation for the case of a Jupiter semimajor axis of 5.20 au and eccentricity of 0.048 (present configuration) are shown in Figure 1. These results demonstrate that the present orbital eccentricity of Venus is at a periodic low point, with the maximum value of the periodic eccentricity oscillations of ∼0.07 in agreement with the results from Laskar (1988) and Bills (2005). A Fourier analysis of the data represented in Figure 1 revealed that the eccentricity variations exhibit high-frequency periodicity of 4.1 × 10⁵ yr and a low-frequency periodicity of 2.2 × 10⁶ yr, primarily a result of perturbations from the Earth and Jupiter. For comparison, the eccentricity of the Earth varies in the range 0.0034–0.058 and is currently 0.0167 (Laskar 1988), the variations of which were also validated by our model (Horner et al. 2020).

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The relatively mild variations in the orbital eccentricity of Venus exhibited in Figure 1 are sensitive to the location of Jupiter. Using the results of our extensive suite of dynamical simulations, we extracted the minimum and maximum orbital eccentricities attained by Venus for the full range of Jupiter semimajor axis values. These results are shown in Figure 2, where the dotted (blue) and solid (red) lines indicate the minimum and maximum eccentricity, respectively, for the given Jupiter semimajor axis. The present location of Jupiter is shown as a vertical dashed line. The top axis provides the equivalent period ratio between Jupiter (P_J) and Venus (P_V) as a guide for possible resonance sources of instability.

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The eccentricity data shown in Figure 2 demonstrate the non-trivial relationship between the location of Jupiter and the orbital variability of Venus. It also shows that there are numerous locations at which the influence of Jupiter would cause the range of eccentricity values for Venus to be significantly larger than the present range shown in Figure 1. For example, at a_J = 5.18 au, only 0.02 au away from its present location, the maximum eccentricity of Venus rises to 0.10. There are also Jupiter locations farther from its present position that result in large increases in the maximum eccentricity, most particularly at a_J = 5.71 au, where the maximum Venusian eccentricity rises to 0.18.

However, it is evident from the eccentricity data that the most powerful perturbations to the Venusian orbit occur when Jupiter is located in the vicinity of a_J ∼ 4.3 au. The maximum Venusian eccentricity of 0.31 occurs at a Jupiter semimajor axis of a_J = 4.31 au. It should be noted that not all of the simulations represented in Figure 2 are dynamically viable for the full 10⁷ yr duration. Moreover, as noted by Horner et al. (2020), placing Jupiter at ∼4.3 au produced instability within the system at all Jovian eccentricities, even zero. Even so, those models of planetary migration that invoke extreme excursions in Jupiter's orbital elements (such as the Grand Tack model) require the planet to have moved through this location, albeit probably on timescales that allow unstable regions to be effectively circumnavigated, and so such locations and resulting eccentricities are potentially valid regions of exploration.

The case of a_J = 4.31 au is shown in Figure 3, where the simulation halted after 1.6 × 10⁶ yr. Placing Jupiter at this particular location has little effect on the orbits of Mars and Saturn, but Figure 3 shows that angular momentum is transferred from Jupiter to Earth and then subsequently passed to Venus, as the orbital eccentricities oscillations of Venus and Earth are directly out of phase with one another. This further demonstrates that although a_J = 4.31 au does not appear to correspond to a significant mean motion resonance location (see Figure 2), the chain of perturbations between Jupiter, Earth, and Venus produces exceptionally nonlinear eccentricity distributions.

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As a final example, among those simulations for which the solar system remained stable for the full 10⁷ yr duration of our simulations, the highest values for the Venusian eccentricity occur when Jupiter is located at a_J = 4.05 au. The results for the orbital evolution of Venus in this case are shown in Figure 4. The planetary orbit experiences high-frequency oscillations and achieves an eccentricity as high as 0.24, but remains stable nonetheless. Such cases can therefore accommodate a broader range of Jupiter migration models.

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If Venus once had an orbital eccentricity as high as 0.31, then the question remains as to how the orbit circularized to its current state. One of the most efficient mechanisms to circularize a planetary orbit is through tidal interactions between the planet and its host star. Using a simple single-planet circularization model based on the methodology described by Goldreich & Soter (1966), the tidal circularization timescale is far older than the age of the solar system. However, numerous other factors, including mutual interactions between planets, contribute to tidal dissipation that reduces the tidal circularization timescale (Laskar et al. 2012). Hence, to include both planet–planet interaction and tides in our study, we made use of the Posidonius N-body code (Bolmont et al. 2015; Blanco-Cuaresma & Bolmont 2017). Posidonius implements the constant time-lag model (CTL) to account for tidal interactions, where a planet is modeled as a weakly viscous fluid that is deformed due to the gradient of the gravitational potential of the central body (i.e., the host star; see, e.g., Mignard 1979; Hut 1981; Eggleton et al. 1998; Leconte et al. 2010) The implementation of this model in Posidonius has been successfully used in a number of recent studies (see, e.g., Bolmont et al. 2020; Nielsen et al. 2020; Pozuelos et al. 2020). It is interesting to note that an alternative tides model exists, namely the constant phase model (CPL), which may yield faster tidal dissipation timescales relative to those found via CTL. We refer the reader to Barnes (2017) and references therein for a detailed review of existing models used to study the evolution of tides in a planetary context.

The main free parameters in the CTL model are the degree-2 potential Love number, k₂, and the constant time lag, Δτ, of each considered planet. While the k₂ parameter accounts for the self-gravity and elastic properties of the planet, which can have values between 0 and 1.5, Δτ represents the lag between the line connecting the two centers of mass (star–planet) and the direction of the tidal bulges, which can span orders of magnitudes (Barnes 2017). In the case of the present-day Earth, the product of these two parameters (i.e., ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$) has been estimated to be ∼213 s (Neron de Surgy & Laskar 1997).

It has been demonstrated that tidal dissipation on present-day Earth is mostly dominated by friction induced by its seafloor topography on tidal gravito-inertial waves that propagate in the oceans (see, e.g., Egbert & Ray 2000, 2003). However, on icy-moons such as Enceladus, tidal dissipation occurs in the sub-surface ocean(s) (e.g., Hay & Matsuyama 2017, and references therein). Thus the tidal braking of an Earth-like planet is mostly dependent on the unknown properties of any putative oceans, whose properties may evolve over time as a consequence of losing/gaining oceanic mass, changes in continental distribution, seafloor topography, and so forth (Neron de Surgy & Laskar 1997; Barnes 2017). Hence, when exploring the tidal evolution of a terrestrial exoplanet, the most commonly used approach is to adopt Earth's current value as a reference for a planet with surface oceans, where values in the range of (0.1–10) × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$ are used to account for different planetary conditions, such as rocky planets without oceans and those that are volatile-rich, respectively. This range allows us to explore a plausible spectrum of values and behaviors (see e.g., Bolmont et al. 2014, 2020).

In the solar system, k₂ has been measured for several objects, while Δτ is, in general, poorly known. In the case of Venus, from the Magellan mission, the value of k₂ was found to be in the range of 0.23–0.36 (Konopliv & Yoder 1996). Moreover, Dumoulin et al. (2017) computed tidal viscoelastic deformation using different models of internal structure, and established k₂ in the range of 0.265–0.270 when a solid pure-iron core was considered and 0.27–0.29 when the core was partially or entirely liquid. These values of k₂ are similar to that found for Earth, which was estimated to be ∼0.298 (Jagoda et al. 2018). Hence, as a plausible value for present-day Venus (i.e., a rocky terrestrial planet without oceans), we assumed 0.1 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$.

With all this information, we explored the circularization of the Venus orbit by performing a suite of simulations considering three different scenarios within the solar system: (1) initial Venusian orbital eccentricity excited to 0.31 but with no tidal forces; (2) initial Venusian orbital eccentricity excited to 0.31 and 0.1 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$ (i.e., the plausible current value representing Venus's tidal dissipation); and (3) initial Venusian orbital eccentricity excited to 0.31 and 200 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$. The scenario (3) tidal dissipation of 200 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$ is unrealistic but accelerates the circularization process and allows us to explore the possible timescale of circularization due to the effects of tides. Indeed, as the timescales of the tidal effects scale linearly with Δτ, we are able to estimate the circularization time for a range of realistic dissipation factors (1–10 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$) in a straightforward manner by using this result (Neron de Surgy & Laskar 1997; Barnes 2017). For self-consistency, we also included the effects of tides for Mercury, which has a k₂ value of 0.45 (Noyelles et al. 2014). However, for simplicity, we did not account for the tidal effects for other planets beyond Venus. This choice was motivated by the very small tidal effects experience by terrestrial planets in the Habitable Zone (and beyond) around solar-like stars (Heller et al. 2011). We integrated the scenarios described above for 10⁷ yr with a time step of 1 day, and we accounted for relativistic corrections.

We found that scenario (1) yielded a system which became unstable within a very short timescale of 2 Myr. This hints that if Venus reached an eccentricity of 0.31 during its early history, some mechanism was needed not only to circularize the orbit to reach its current value but also to stabilize it. In scenario (2), we found that the system was more stable with respect to scenario (1); however, it became unstable after 6 Myr. This suggests that the current tidal dissipation of Venus is not enough to stabilize and circularize its orbit. In scenario (3), the system was stable for the full integration time and the orbit started to circularize by decreasing its initial eccentricity by ∼35%. To estimate the circularization time needed to reach its current value of 0.006, we performed a linear extrapolation of the eccentricity evolution over 10⁷ yr, finding a circularization time of ∼30 Myr to reach the current value. Making use of the aforementioned linear-scale property of tides, we estimated the circularization times for realistic tidal dissipation as 600 Myr for 10 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus };$ 1200 Myr for 5 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus };$ 3000 Myr for 2 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus };$ and 6000 Myr for 1 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$. These results hint that the current Earth dissipation is not enough to circularize the Venus orbit on a timescale that is less than the age of the solar system. However, these results also clearly show that slightly larger dissipation factors may be valid. The results corresponding to scenarios (2) and (3) are shown in Figure 5.

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Taken together, our results suggest that if Venus's eccentricity was excited up to 0.31 due to Jupiter's migration in the early solar system, the effects of tides may have played a key role in circularizing and stabilizing Venus's orbit. We found that the current value of Venus's tidal dissipation is not enough to achieve this, suggesting that Venus was not as dry in the past as it is today. To circularize its orbit over the timescale of the age of the solar system (∼4000 Myr, post gas phase and migrations), the dissipation factor needed is ∼1.5 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$. This suggests that Venus might have had a water-rich past, possibly in the form of surface or sub-surface oceans. Due to the similarities between Venus and Earth, it seems more realistic to speak about surface oceans, such as those found on Earth, but we cannot favor any specific location for the water bodies on ancient Venus due to the limitations of the models used in this study. This result agrees with the current formation theories of the solar system, where it has been surmised that Venus and Earth likely received similar water inventories (see, e.g., Raymond et al. 2006). Moreover, it has recently been suggested by Way & Del Genio (2020) that Venus may have possessed early oceans that could have created significant tidal dissipation over ∼3000 Myr. While it is not currently possible definitive to ascertain the true tidal dissipation scenario experienced by the early Venus, a plausible range of values seems to be 1.5–5 × ${k}_{2,\oplus }{\rm{\Delta }}{\tau }_{\oplus }$, which implies circularization times of 4000–1200 Myr.

There are numerous other processes that also contribute to the circularization of the Venusian orbit. Tidal dissipation and circularization are tied to the thermal evolution of the planet, decreasing the circularization timescale as the planet's interior cools (Driscoll & Barnes 2015). Furthermore, the interaction between the atmosphere and surface, strengthened by recent observations of standing gravity waves, is calculated to be a source of tidal dissipation for the Venus system (Dobrovolskis & Ingersoll 1980; Fukuhara et al. 2017). It should additionally be noted that since the orbital eccentricity oscillations of Venus presented here depend on Jupiter's location, subsequent Jupiter migration can further impact the Venusian orbital variations. For example, if Jupiter's migration occurred on a timescale longer than that of the periodic eccentricity oscillations of the Venus orbit, then this would act to dampen the amplitude and frequency of the oscillations. The summation of these sources provides ample opportunity for the orbit of Venus to arrive at its present state from the significantly non-circular cases described previously.

As described in Section 2.4, tidal dissipation plays a role in circularizing eccentric orbits with time. The internal heat budget of the Earth is well known to be sourced primarily from radiogenic heating and primordial heat. Radiogenic heating of the Earth's interior, and the resulting heat flux at the surface, can be calculated using the known mass concentrations of potassium, thorium, and uranium and their corresponding half-lives (Turcotte & Schubert 2002). The current total heat flux at the Earth's surface is ∼47 TW (Davies & Davies 2010), but at 4 Ga the radiogenic heating alone produced a heat flux of ∼80 TW (Arevalo et al. 2009).

A third component of internal heat production is gravitational tides. Tidal heating of planetary interiors and tidal dissipation as a result of orbital perturbations can result in planets remaining tidally active for extended periods (Renaud & Henning 2018). Such heat dissipation contributes to the surface heat flow budget and thus the overall energy budget of the atmosphere. Tidal energy potentially extended the duration of the molten surface scenario for a young Venus, increasing the timescale of hydrodynamic escape described by Hamano et al. (2013). It has been further suggested by Barnes et al. (2009) that tidal heating of short-period planets can result in "Super-Io" outcomes with significant resurfacing, as is potentially the case for CoRoT-7b (Barnes et al. 2010). Under extreme circumstances, tidal dissipation can dramatically alter the course of the atmospheric evolution, such as steering that evolution into a runaway greenhouse state (Barnes et al. 2013).

To perform our tidal heating calculations, we adopt the methodology of Jackson et al. (2008a, 2008b). Specifically, we use the equation for the tidal heating rate, H:

$$\begin{eqnarray}&&H=\displaystyle \frac{63}{4}\displaystyle \frac{{({{GM}}_{\star })}^{3/2}{M}_{\star }{R}_{p}^{5}}{(3{Q}_{p}/2{k}_{2})}{a}^{-15/2}{e}^{2},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, M_⋆ is the stellar mass, R_p is the planet radius, Q_p is the tidal dissipation parameter, k₂ is the Love number, a is the semimajor axis, and e is the eccentricity. Note that Equation (1) applies to radial tides, rather than those tides resulting from non-synchronous rotation. Based on the analysis of Magellan and Pioneer Venus data by Konopliv & Yoder (1996) and the reanalysis of those data by Dumoulin et al. (2017), we use values of Q_p = 12 and k₂ = 0.299. Equation (1) shows the dramatic dependency of the tidal heating on a. Thus the distance of Venus from the Sun results in a relatively low rate of heating due to radial tides (see Equation (1)). Specifically, for the maximum eccentricity of 0.31 (a_J = 4.31 au scenario) described in Section 2.3, we calculate a tidal heating flux at the surface of Venus of ∼11 GW. The results of the tidal heating calculations were validated using the tide modules of the Virtual Planet Simulator (Barnes et al. 2020). Although the contribution of the tidal heating to the total atmospheric energy budget is low, it occurred at a time when the contribution from the insolation flux was also significantly lower than that at the present epoch.

The incident (insolation) flux for the solar system planets has evolved with time as the Sun evolves on the main sequence. Such evolution can have a profound effect on the time-dependent insolation flux for terrestrial planets (Kulikov et al. 2007), as is seen in calculations of water loss for planets around M dwarfs (Luger & Barnes 2015). In particular, it is known that the Sun was ∼30% less luminous in the early era of the solar system (Gough 1981), during which the radiation environment of Venus was substantially different.

To simulate the expected insolation flux of Venus during a possible early era with the high eccentricity described in Section 2.3, we adopt a solar luminosity that is 75% of the current value. At the semimajor axis of Venus, this results in an insolation flux of S/S₀ = 1.43, where S₀ is the present-day solar flux received at Earth. One possible scenario is thus the a_J = 4.31 au scenario (see Section 2.3), where the evolution of the maximum flux (perihelion) and minimum flux (aphelion) received by Venus is represented in the top panel of Figure 6. As for Figure 3, Venus starts in a circular orbit, then the rise in eccentricity results in a maximum insolation flux that rapidly starts to oscillate high above its present value, indicated by the horizontal dashed line.

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To demonstrate the effect of the eccentricity on the flux received at the top of the atmosphere, we simulated the latitudinal flux map of the planet for the case where the eccentricity reaches 0.31 using the methodology of Kane & Torres (2017). The flux map is shown in the bottom panel of Figure 6 as a function of orbital phase, where zero phase corresponds to perihelion passage. We assumed a zero obliquity for the rotational axis and also that the early Venus was a rapid rotator prior to its present near spin–orbit synchronization. The consequence of the assumed rotation means that the incident flux is calculated as an average at each latitude, accounting for the reduced flux as a particular longitude rotates away from maximum solar elevation as well as the zero flux received at the night side of the planet. As such, the average flux values in the contour map are significantly lower than the flux received at the top of the Venusian atmosphere at the sub-solar point. The maximum average flux received (1297 W m⁻²) therefore occurs at the equator during the perihelion passage. For comparison, the maximum average flux received by the Earth is 563 W m⁻². Similarly, the minimum average flux of 360 W m⁻² occurs at the Venusian equator during aphelion, which is smaller than the minimum average flux received at the equator of the present Earth during its aphelion (386 W m⁻²).

As described earlier, the loss of water can occur rapidly for planets orbiting M dwarfs (Luger & Barnes 2015), and eccentricity-induced tidal heating can force a runaway greenhouse scenario (Barnes et al. 2013). Significant water loss can also occur during the early period of coronal mass ejection for young solar-type stars (Dong et al. 2017) and the incident XUV flux plays an important role in water loss from CO₂ dominated atmospheres (Wordsworth & Pierrehumbert 2013). Here, we combine several of these aspects to explore the impact of early Venusian eccentricity on the rate at which the planet would have lost its water.

A recent study of the water-loss implications for eccentric orbits conducted by Palubski et al. (2020) used 1D climate models to explore water loss for planets orbiting stars of a range of spectral types. We utilize the data generated for the G2 main-sequence star case, while extending the water-loss calculations to high insolation flux regimes. Shown in Figure 7 are the data for the faint young Sun scenario described in Section 3.2, where the insolation flux at the semimajor axis of Venus is S/S₀ = 1.43. The horizontal axis is expressed in terms of the percentage effect of eccentricity on the timescale for losing one Earth ocean worth of water (Palubski et al. 2020). For zero eccentricity, this timescale is ∼3 Gyr. The dashed lines in Figure 7 highlight the water-loss effect for the eccentricity of 0.31 described in Section 2.3. In this case, the time taken to remove all of the water is reduced by 4.7% due to the high eccentricity.

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An aspect that is not completely accounted for in the provided water-loss model is the changing luminosity of the Sun. The evolution of the solar luminosity affects the solar wind received by the planets (Pognan et al. 2018), which in turn can have significant effects on atmospheric retention (Ribas et al. 2005; Howe et al. 2020). Furthermore, the pre-main-sequence period of the solar evolution can be characterized by a relatively high XUV flux that also results in substantial atmospheric and water loss (Lammer et al. 2008; Ramirez & Kaltenegger 2014; Gronoff et al. 2020). As a result, the 4.7% reduction in water-loss timescale should be considered a lower limit, and the true water-loss rate would probably be substantially higher as a result of the evolution of the young Sun.

Note that the migration of Jupiter and Saturn likely occurred before 4 Ga, prior to when water would have had an opportunity to be both substantially delivered and condensed on the surface of Venus and Earth. However, the tidal circularization timescale estimated in Section 2.4 demonstrates that the perturbing effect of Jupiter may have had long-lasting consequences. Thus an increased eccentricity for Venus could have been sustained long after Jupiter and Saturn settled to their final (present) locations.