On the concavity of the arithmetic volumes

In this note, we study the differentiability of the arithmetic volumes along arithmetic R-divisors, and give some equality conditions for the Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big arithmetic R-divisors.Comment: 35 page

On subfiniteness of graded linear series

Hilbert's 14th problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the field of fractions of the algebra. It has a negative answer due to the counterexample of Nagata. We show that a subfinite version of Hilbert's 14th problem has a confirmative answer. We then establish a graded analogue of this result, which permits to show that the subfiniteness of graded linear series does not depend on the function field in which we consider it. Finally, we apply the subfiniteness result to the study of geometric and arithmetic graded linear series

Impact of photoevaporative mass loss on masses and radii of water-rich sub/super-Earths

Recent progress in transit photometry opened a new window to the interior of super-Earths. From measured radii and masses, we can infer planetary internal compositions. It has been recently revealed that super-Earths are diverse in composition. Such a diversity is thought to arise from diversity in volatile content. The stability of the volatile components is to be examined, because hot super-Earths undergo photo-evaporative mass loss. While several studies investigated the impact of photo-evaporative mass loss on hydrogen-helium envelopes, there are few studies as to the impact on water-vapor envelopes. To obtain theoretical prediction to future observations, we also investigate the relationships among masses, radii, and semimajor axes of water-rich sub/super-Earths that have undergone photo-evaporative mass loss. We simulate the interior structure and evolution of sub/super-Earths that consist of a rocky core surrounded by a water envelope, including mass loss due to the stellar XUV-driven energy-limited hydrodynamic escape. We find that the photo-evaporative mass loss has a significant impact on the evolution of hot sub/super-Earths. We then derive the threshold planetary mass and radius below which the planet loses its water envelope completely as a function of the initial water content, and find that there are minimums of the threshold mass and radius. We constrain the domain in the parameter space of planetary mass, radius, and semimajor axis in which sub/super-Earths never retain water envelopes in 1-10 Gyr. This would provide an essential piece of information for understanding the origin of close-in low-mass planets. The current uncertainties in stellar XUV flux and its heating efficiency, however, prevent us from deriving robust conclusions. Nevertheless, it seems to be a robust conclusion that Kepler planet candidates contain a significant number of rocky sub/super-Earths.Comment: 13 pages, 14 figures, accepted for publication in Astronomy & Astrophysic

Critical Core Masses for Gas Giant Formation with Grain-Free Envelopes

We investigate the critical core mass and the envelope growth timescale, assuming grain-free envelopes, to examine how small cores are allowed to form gas giants in the framework of the core accretion model. This is motivated by a theoretical dilemma concerning Jupiter formation: Modelings of Jupiter's interior suggest that it contains a small core of < 10 Earth mass, while many core accretion models of Jupiter formation require a large core of > 10 Earth mass to finish its formation by the time of disk dissipation. Reduction of opacity in the accreting envelope is known to hasten gas giant formation. Almost all the previous studies assumed grain-dominated opacity in the envelope. Instead, we examine cases of grain-free envelopes in this study. Our numerical simulations show that an isolated core of as small as 1.7 Earth mass is able to capture disk gas to form a gas giant on a timescale of million years, if the accreting envelope is grain-free; that value decreases to 0.75 Earth mass, if the envelope is metal-free, namely, composed purely of hydrogen and helium. It is also shown that alkali atoms, which are known to be one of the dominant opacity sources near 1500 K in the atmospheres of hot Jupiters, have little contribution to determine the critical core mass. Our results confirm that sedimentation and coagulation of grains in the accreting envelope is a key to resolve the dilemma about Jupiter formation.Comment: 17 pages, 2 figures, 1 table, Accepted for publication in Ap