Planetary evolution with atmospheric photoevaporation I. Analytical derivation and numerical study of the evaporation valley and transition from super-Earths to sub-Neptunes. (arXiv:2002.02455v1 [astro-ph.EP])

Planetary evolution with atmospheric photoevaporation I. Analytical derivation and numerical study of the evaporation valley and transition from super-Earths to sub-Neptunes. (arXiv:2002.02455v1 [astro-ph.EP])
<a href="http://arxiv.org/find/astro-ph/1/au:+Mordasini_C/0/1/0/all/0/1">Christoph Mordasini</a>

Observations have revealed in the Kepler data a depleted region separating
smaller super-Earths from larger sub-Neptunes. This can be explained as an
evaporation valley between planets with and without H/He that is caused by
atmospheric escape. First, we conduct numerical simulations of the evolution of
close-in low-mass planets with H/He undergoing escape. Second, we develop an
analytical model for the valley locus. We find that the bottom of the valley
quantified by the radius of the largest stripped core $R_{rm b}$ at a given
orbital distance depends only weakly on post-formation H/He mass. The reason is
that a high initial H/He mass means that there is more gas to evaporate, but
also that the planet density is lower, increasing loss. Regarding stellar
$L_{rm XUV}$, $R_{rm b}$ scales as $L_{rm XUV}^{0.135}$. The same weak
dependency applies to the efficiency factor $varepsilon$ of energy-limited
evaporation. As found numerically and analytically, $R_{rm b}$ varies as
function of orbital period $P$ for a constant $varepsilon$ as $P^{-2 p_{rm
c}/3}approx P^{-0.18}$ where $M propto R^{p_{rm c}}$ is the mass-radius
relation of solid cores. $R_{rm b}$ is about 1.7 $R_{oplus}$ at a 10-day
orbit for an Earth-like composition, increasing linearly with ice mass
fraction. The numerical results are explained very well with the analytical
model where complete evaporation occurs if the temporal integral over the
stellar XUV irradiation absorbed by the planet is larger than binding energy of
the envelope in the gravitational potential of the core. The weak dependency on
primordial H/He mass, $L_{rm XUV}$ and $varepsilon$ explains why
observationally the valley is visible, and why theoretically models find
similar results. At the same time, given the large observed spread of $L_{rm
XUV}$, the dependency on it is still strong enough to explain why the valley is
not completely empty.

Observations have revealed in the Kepler data a depleted region separating
smaller super-Earths from larger sub-Neptunes. This can be explained as an
evaporation valley between planets with and without H/He that is caused by
atmospheric escape. First, we conduct numerical simulations of the evolution of
close-in low-mass planets with H/He undergoing escape. Second, we develop an
analytical model for the valley locus. We find that the bottom of the valley
quantified by the radius of the largest stripped core $R_{rm b}$ at a given
orbital distance depends only weakly on post-formation H/He mass. The reason is
that a high initial H/He mass means that there is more gas to evaporate, but
also that the planet density is lower, increasing loss. Regarding stellar
$L_{rm XUV}$, $R_{rm b}$ scales as $L_{rm XUV}^{0.135}$. The same weak
dependency applies to the efficiency factor $varepsilon$ of energy-limited
evaporation. As found numerically and analytically, $R_{rm b}$ varies as
function of orbital period $P$ for a constant $varepsilon$ as $P^{-2 p_{rm
c}/3}approx P^{-0.18}$ where $M propto R^{p_{rm c}}$ is the mass-radius
relation of solid cores. $R_{rm b}$ is about 1.7 $R_{oplus}$ at a 10-day
orbit for an Earth-like composition, increasing linearly with ice mass
fraction. The numerical results are explained very well with the analytical
model where complete evaporation occurs if the temporal integral over the
stellar XUV irradiation absorbed by the planet is larger than binding energy of
the envelope in the gravitational potential of the core. The weak dependency on
primordial H/He mass, $L_{rm XUV}$ and $varepsilon$ explains why
observationally the valley is visible, and why theoretically models find
similar results. At the same time, given the large observed spread of $L_{rm
XUV}$, the dependency on it is still strong enough to explain why the valley is
not completely empty.

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