ABSTRACT
Recent observations revealed a bimodal radius distribution of small, short-period exoplanets with a paucity in their occurrence, a radius ‘valley’, around 1.5–2.0 R⊕. In this work, we ...investigate the effect of a planet’s own cooling luminosity on its thermal evolution and atmospheric mass loss (core-powered mass-loss) and determine its observational consequences for the radius distribution of small, close-in exoplanets. Using simple analytical descriptions and numerical simulations, we demonstrate that planetary evolution based on the core-powered mass-loss mechanism alone (i.e. without any photoevaporation) can produce the observed valley in the radius distribution. Our results match the valley’s location, shape and slope in planet radius–orbital period parameter space, and the relative magnitudes of the planet occurrence rate above and below the valley. We find that the slope of the valley is, to first order, dictated by the atmospheric mass-loss time-scale at the Bondi radius and given by d logRp/d logP ≃ 1/(3(1 − β)) that evaluates to −0.11 for β ≃ 4, where Mc/M⊕ = (Rc/R⊕)β(ρc∗/ρ⊕)β/3 is the mass–radius relation of the core. This choice for β yields good agreement with observations and attests to the significance of internal compression for massive planetary cores. We further find that the location of the valley scales as $\rho _{\rm c*}^{-4/9}$ and that the observed planet population must have predominantly rocky cores with typical water–ice fractions of less than ${\sim } 20{{\, \rm per\, cent}}$. Furthermore, we show that the relative magnitude of the planet occurrence rate above and below the valley is sensitive to the details of the planet-mass distribution but that the location of the valley is not.
ABSTRACT
Recent studies have shown that atmospheric mass-loss powered by the cooling luminosity of a planet’s core can explain the observed radius valley separating super-Earths and sub-Neptunes, ...even without photoevaporation. In this work, we investigate the dependence of this core-powered mass-loss mechanism on stellar mass (M*), metallicity (Z*), and age (τ*). Without making any changes to the underlying planet population, we find that the core-powered mass-loss model yields a shift in the radius valley to larger planet sizes around more massive stars with a slope given by dlog Rp/dlog M* ≃ 0.35, in agreement with observations. To first order, this slope is driven by the dependence of core-powered mass-loss on the bolometric luminosity of the host star and is given by dlog Rp/dlog M* ≃ (3α − 2)/36 ≃ 0.33, where (L*/L⊙) = (M*/M⊙)α is the stellar mass–luminosity relation and α ≃ 4.6 for the CKS data set. We therefore find, in contrast to photoevaporation models, no evidence for a linear correlation between planet and stellar mass, but cannot rule it out either. In addition, we show that the location of the radius valley is, to first order, independent of stellar age and metallicity. Since core-powered mass-loss proceeds over Gyr time-scales, the abundance of super-Earths relative to sub-Neptunes increases with age but decreases with stellar metallicity. Finally, due to the dependence of the envelope’s cooling time-scale on metallicity, we find that the radii of sub-Neptunes increase with metallicity and decrease with age with slopes given by dlog Rp/dlog Z* ≃ 0.1 and dlog Rp/dlog τ* ≃ −0.1, respectively. We conclude with a series of observational tests that can differentiate between core-powered mass-loss and photoevaporation models.
Abstract
Recent observations identify a valley in the radius distribution of small exoplanets, with planets in the range 1.5–2.0 R⊕ significantly less common than somewhat smaller or larger planets. ...This valley may suggest a bimodal population of rocky planets that are either engulfed by massive gas envelopes that significantly enlarge their radius, or do not have detectable atmospheres at all. One explanation of such a bimodal distribution is atmospheric erosion by high-energy stellar photons. We investigate an alternative mechanism: the luminosity of the cooling rocky core, which can completely erode light envelopes while preserving heavy ones, produces a deficit of intermediate sized planets. We evolve planetary populations that are derived from observations using a simple analytical prescription, accounting self-consistently for envelope accretion, cooling and mass-loss, and demonstrate that core-powered mass-loss naturally reproduces the observed radius distribution, regardless of the high-energy incident flux. Observations of planets around different stellar types may distinguish between photoevaporation, which is powered by the high-energy tail of the stellar radiation, and core-powered mass-loss, which depends on the bolometric flux through the planet's equilibrium temperature that sets both its cooling and mass-loss rates.
ABSTRACT Some recently discovered short-period Earth- to Neptune-sized exoplanets (super-Earths) have low observed mean densities that can only be explained by voluminous gaseous atmospheres. Here, ...we study the conditions allowing the accretion and retention of such atmospheres. We self-consistently couple the nebular gas accretion onto rocky cores and the subsequent evolution of gas envelopes following the dispersal of the protoplanetary disk. Specifically, we address mass-loss due to both photo-evaporation and cooling of the planet. We find that planets shed their outer layers (dozens of percent in mass) following the disk's dispersal (even without photo-evaporation), and their atmospheres shrink in a few Myr to a thickness comparable to the radius of the underlying rocky core. At this stage, atmospheres containing less particles than the core (equivalently, lighter than a few percent of the planet's mass) can be blown away by heat coming from the cooling core, while heavier atmospheres cool and contract on a timescale of Gyr at most. By relating the mass-loss timescale to the accretion time, we analytically identify a Goldilocks region in the mass-temperature plane in which low-density super-Earths can be found: planets have to be massive and cold enough to accrete and retain their atmospheres, but not too massive or cold, such that they do not enter runaway accretion and become gas giants (Jupiters). We compare our results to the observed super-Earth population and find that low-density planets are indeed concentrated in the theoretically allowed region. Our analytical and intuitive model can be used to investigate possible super-Earth formation scenarios.
Recent observations by the Kepler space telescope have led to the discovery of more than 4000 exoplanet candidates consisting of many systems with Earth- to Neptune-sized objects that reside well ...inside the orbit of Mercury around their respective host stars. How and where these close-in planets formed is one of the major unanswered questions in planet formation. Here, we calculate the required disk masses for in situ formation of the Kepler planets. We find that if close-in planets formed as isolation masses, then standard gas-to-dust ratios yield corresponding gas disks that are gravitationally unstable for a significant fraction of systems, ruling out such a scenario. We show that the maximum width of a planet's accretion region in the absence of any migration is 2upsilon sub(esc)/Omega, where upsilon sub(esc) is the escape velocity of the planet and Omega is the Keplerian frequency, and we use it to calculate the required disk masses for in situ formation with giant impacts. Even with giant impacts, formation without migration requires disk surface densities in solids at semi-major axes of less than 0.1 AU of 10 super(3)-10 super(5) g cm super(-2), implying typical enhancements above the minimum-mass solar nebular (MMSN) by at least a factor of 20. Corresponding gas disks are below but not far from the gravitational stability limit. In contrast, formation beyond a few AU is consistent with MMSN disk masses. This suggests that the migration of either solids or fully assembled planets is likely to have played a major role in the formation of close-in super-Earths and mini-Neptunes.
We assess the multi-planet systems discovered by the Kepler satellite in terms of current ideas about orbital migration and eccentricity damping due to planet-disk interactions. Our primary focus is ...on first order mean motion resonances, which we investigate analytically to lowest order in eccentricity. Eccentricity damping (parameterized by tau sub(e) = e/|e|) offers a plausible resolution. Estimates suggest tau sub(e)/tau sub(n) ~ (h/a) super(2) ~ 10 super(-2), where h/a is the ratio of disk thickness to radius. Together, eccentricity damping and orbital migration give rise to an equilibrium eccentricity, e sub(eq) ~ (tau sub(e)/tau sub(n)) super(1/2). Most Kepler planet pairs have e sub(eq) > mu super(1/3). Since tau sub(n) >> tau sub(e), is the timescale for migration between neighboring resonances, only a modest percentage of pairs end up trapped in resonances after the disk disappears. Planet pairs close to a mean motion resonance typically exhibit period ratios 1 %- 2 % larger than those for exact resonance.
Determining the origin of volatiles on terrestrial planets and quantifying atmospheric loss during planet formation is crucial for understanding the history and evolution of planetary atmospheres. ...Using geochemical observations of noble gases and major volatiles we determine what the present day inventory of volatiles tells us about the sources, the accretion process and the early differentiation of the Earth. We further quantify the key volatile loss mechanisms and the atmospheric loss history during Earth’s formation. Volatiles were accreted throughout the Earth’s formation, but Earth’s early accretion history was volatile poor. Although nebular Ne and possible H in the deep mantle might be a fingerprint of this early accretion, most of the mantle does not remember this signature implying that volatile loss occurred during accretion. Present day geochemistry of volatiles shows no evidence of hydrodynamic escape as the isotopic compositions of most volatiles are chondritic. This suggests that atmospheric loss generated by impacts played a major role during Earth’s formation. While many of the volatiles have chondritic isotopic ratios, their relative abundances are certainly not chondritic again suggesting volatile loss tied to impacts. Geochemical evidence of atmospheric loss comes from the
He
3
/
22
Ne
, halogen ratios (e.g., F/Cl) and low H/N ratios. In addition, the geochemical ratios indicate that most of the water could have been delivered prior to the Moon forming impact and that the Moon forming impact did not drive off the ocean. Given the importance of impacts in determining the volatile budget of the Earth we examine the contributions to atmospheric loss from both small and large impacts. We find that atmospheric mass loss due to impacts can be characterized into three different regimes: 1) Giant Impacts, that create a strong shock transversing the whole planet and that can lead to atmospheric loss globally. 2) Large enough impactors (
m
cap
≳
2
ρ
0
(
π
h
R
)
3
/
2
,
r
cap
∼
25
km
for the current Earth), that are able to eject all the atmosphere above the tangent plane of the impact site, where
h
,
R
and
ρ
0
are the atmospheric scale height, radius of the target, and its atmospheric density at the ground. 3) Small impactors (
m
min
>
4
π
ρ
0
h
3
,
r
min
∼
1
km
for the current Earth), that are only able to eject a fraction of the atmospheric mass above the tangent plane. We demonstrate that per unit impactor mass, small impactors with
r
min
<
r
<
r
cap
are the most efficient impactors in eroding the atmosphere. In fact for the current atmospheric mass of the Earth, they are more than five orders of magnitude more efficient (per unit impactor mass) than giant impacts, implying that atmospheric mass loss must have been common. The enormous atmospheric mass loss efficiency of small impactors is due to the fact that most of their impact energy and momentum is directly available for local mass loss, where as in the giant impact regime a lot of energy and momentum is ’wasted’ by having to create a strong shock that can transverse the entirety of the planet such that global atmospheric loss can be achieved. In the absence of any volatile delivery and outgassing, we show that the population of late impactors inferred from the lunar cratering record containing 0.1%
M
⊕
is able to erode the entire current Earth’s atmosphere implying that an interplay of erosion, outgassing and volatile delivery is likely responsible for determining the atmospheric mass and composition of the early Earth. Combining geochemical observations with impact models suggest an interesting synergy between small and big impacts, where giant impacts create large magma oceans and small and larger impacts drive the atmospheric loss.
•Planetesimal impacts likely played a mayor role in atmospheric mass loss of the terrestrial planets.•Current difference in the atmospheres of Earth and Venus can be explained by modest difference in ...their initial atmospheric masses.•The current atmosphere of the Earth could have resulted from an equilibrium between atmospheric erosion and volatile delivery by planetesimals.•Global atmospheric mass loss due to giant impacts is given by Xloss=0.4x+1.4x2−0.8x3 where x=(vimpm/vescM).
Quantifying the atmospheric mass loss during planet formation is crucial for understanding the origin and evolution of planetary atmospheres. We examine the contributions to atmospheric loss from both giant impacts and planetesimal accretion. Giant impacts cause global motion of the ground. Using analytic self-similar solutions and full numerical integrations we find (for isothermal atmospheres with adiabatic index γ=5/3) that the local atmospheric mass loss fraction for ground velocities vg≲0.25vesc is given by χloss=(1.71vg/vesc)4.9, where vesc is the escape velocity from the target. Yet, the global atmospheric mass loss is a weaker function of the impactor velocity vImp and mass mImp and given by Xloss≃0.4x+1.4x2-0.8x3 (isothermal atmosphere) and Xloss≃0.4x+1.8x2-1.2x3 (adiabatic atmosphere), where x=(vImpm/vescM). Atmospheric mass loss due to planetesimal impacts proceeds in two different regimes: (1) large enough impactors m≳2ρ0(πhR)3/2 (25km for the current Earth), are able to eject all the atmosphere above the tangent plane of the impact site, which ish/2R of the whole atmosphere, where h,R and ρ0 are the atmospheric scale height, radius of the target, and its atmospheric density at the ground. (2) Smaller impactors, but above m>4πρ0h3 (1km for the current Earth) are only able to eject a fraction of the atmospheric mass above the tangent plane. We find that the most efficient impactors (per unit impactor mass) for atmospheric loss are planetesimals just above that lower limit (2km for the current Earth). For impactor flux size distributions parametrized by a single power law, N(>r)∝r-q+1, with differential power law index q, we find that for 1<q<3 the atmospheric mass loss proceeds in regime (1) whereas for q>3 the mass loss is dominated by regime (2). Impactors with m≲4πρ0h3 are not able to eject any atmosphere. Despite being bombarded by the same planetesimal population, we find that the current differences in Earth’s and Venus’ atmospheric masses can be explained by modest differences in their initial atmospheric masses and that the current atmosphere of the Earth could have resulted from an equilibrium between atmospheric erosion and volatile delivery to the atmosphere from planetesimal impacts. We conclude that planetesimal impacts are likely to have played a major role in atmospheric mass loss over the formation history of the terrestrial planets.
ABSTRACT
Super-Earths and sub-Neptunes are commonly thought to have accreted hydrogen/helium envelopes, consisting of a few to ten percent of their total mass, from the primordial gas disc. ...Subsequently, hydrodynamic escape driven by core-powered mass-loss and/or photoevaporation likely stripped much of these primordial envelopes from the lower mass and closer-in planets to form the super-Earth population. In this work, we show that after undergoing core-powered mass-loss, some super-Earths can retain small residual H/He envelopes. This retention is possible because, for significantly depleted atmospheres, the density at the radiative–convective boundary drops sufficiently such that the cooling time-scale becomes shorter than the mass-loss time-scale. The residual envelope is therefore able to contract, terminating further mass-loss. Using analytic calculations and numerical simulations, we show that the mass of primordial H/He envelope retained as a fraction of the planet’s total mass, fret, increases with increasing planet mass, Mc, and decreases with increasing equilibrium temperature, Teq, scaling as $f_\mathrm{ret} \propto M_\mathrm{c}^{3/2} T_\mathrm{eq}^{-1/2} \exp {M_\mathrm{c}^{3/4} T_\mathrm{eq}^{-1}}$. fret varies from <10−8 to about 10−3 for typical super-Earth parameters. To first order, the exact amount of left-over H/He depends on the initial envelope mass, the planet mass, its equilibrium temperature, and the envelope’s opacity. These residual hydrogen envelopes reduce the atmosphere’s mean molecular weight compared to a purely secondary atmosphere, a signature observable by current and future facilities. These remnant atmospheres may, however, in many cases be vulnerable to long-term erosion by photoevaporation. Any residual hydrogen envelope likely plays an important role in the long-term physical evolution of super-Earths, including their geology and geochemistry.
Abstract
The population of small, close-in exoplanets is bifurcated into super-Earths and sub-Neptunes. We calculate physically motivated mass–radius relations for sub-Neptunes, with rocky cores and ...H/He-dominated atmospheres, accounting for their thermal evolution, irradiation, and mass loss. For planets ≲10
M
⊕
, we find that sub-Neptunes retain atmospheric mass fractions that scale with planet mass and show that the resulting mass–radius relations are degenerate with results for “water worlds” consisting of a 1:1 silicate-to-ice composition ratio. We further demonstrate that our derived mass–radius relation is in excellent agreement with the observed exoplanet population orbiting M dwarfs and that planet mass and radii alone are insufficient to determine the composition of some sub-Neptunes. Finally, we highlight that current exoplanet demographics show an increase in the ratio of super-Earths to sub-Neptunes with both stellar mass (and therefore luminosity) and age, which are both indicative of thermally driven atmospheric escape processes. Therefore, such processes should not be ignored when making compositional inferences in the mass–radius diagram.
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